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Impedance model of a solid oxide fuel cell for degradation diagnosis Gazzarri, Javier Ignacio 2007

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i   IMPEDANCE MODEL OF A SOLID OXIDE FUEL CELL FOR DEGRADATION DIAGNOSIS  by JAVIER IGNACIO GAZZARRI Ingeniero Mecánico, Universidad de Buenos Aires, 1998 M.A.Sc., The University of British Columbia, 2003  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)    THE UNIVERSITY OF BRITISH COLUMBIA December 2007 © Javier Ignacio Gazzarri, 2007 ii Abstract A numerical model of the steady state and alternating current behaviour of a solid-oxide fuel cell is presented to explore the possibilities to diagnose and identify degradation mechanisms in a minimally invasive way using impedance spectroscopy.  This is the first report of an SOFC impedance model to incorporate degradation, as well as the first one to include the ribbed interconnect geometry, using a 2-D approximation.  Simulated degradation modes include: electrode/electrolyte delamination, interconnect oxidation, interconnect/electrode interface detachment, and anode sulfur poisoning.  Detailed electrode-level simulation replaces the traditional equivalent circuit approach, allowing the simulation of degradation mechanisms that alter the shape of the current path.  The SOFC impedance results from calculating the cell response to a small oscillatory perturbation in potential.  Starting from the general equations for mass and charge transport, and assuming isothermal and isobaric conditions, the system variables are decomposed into a steady-state component and a small perturbation around the operating point.  On account of the small size of the imposed perturbation, the time dependence is eliminated, and the original equations are converted to a new linear, time independent, complex-valued system, which is very convenient from a numerical viewpoint. Geometrical and physical modifications of the model simulate the aforementioned degradation modes, causing variations in the impedance.  The possibility to detect unique impedance signatures is discussed, along with a study of the impact of input parameter inaccuracies and parameter interaction on the presented results.  Finally, a study of pairs of concurrent degradation modes reveals the method’s strengths and limitations in terms of its diagnosis capabilities. iii Table of Contents  Abstract……………………………………………………………………………………ii Table of Contents…………………………………………………………………………iii List of Tables……………………………………………………………………………... v List of Figures……………………………………………………………………………. vi Nomenclature……………………………………………………………………..…….. xii Acknowledgements…………………………………………………………………….. xiv Dedication………...…………………………………………………………………….. xv Co-Authorship Statement………………………………………………………………. xvi 1. Chapter One.  Introduction and Background. ............................................................... 1 1.1 Introduction......................................................................................................... 1 1.2 The Solid Oxide Fuel Cell (SOFC)..................................................................... 1 1.2.1 Principle of Operation..................................................................................... 2 1.3 Degradation Phenomena in Solid Oxide Fuel Cells ........................................... 5 1.4 SOFC Modeling .................................................................................................. 8 1.5 Scope of this thesis............................................................................................ 13 1.6 References......................................................................................................... 15 2. Chapter Two. Mathematical Model ............................................................................ 18 2.1 Introduction....................................................................................................... 18 2.2 General Description and Assumptions.............................................................. 18 2.3 Mathematical Formulation................................................................................ 23 2.3.1 Steady State Equations.................................................................................. 24 2.3.1.1 Steady-State Charge Balance................................................................ 24 2.3.1.2 Steady-State Mass Balance ................................................................... 28 2.3.2 AC Equations ................................................................................................ 30 2.3.2.1 AC Charge Balance............................................................................... 30 2.3.2.2 AC Mass Balance.................................................................................. 33 2.4 Calculation of the Impedance ........................................................................... 36 2.5 Model Parameters ............................................................................................. 37 2.5.1 Electrical conductivity .................................................................................. 39 2.5.2 Gas species diffusion .................................................................................... 40 2.5.3 Electrochemistry ........................................................................................... 41 2.6 On the Validity of the Model Equations and Results ....................................... 44 2.7 Conclusions....................................................................................................... 45 2.8 References......................................................................................................... 47 3. Chapter Three.  Electrode Delamination .................................................................... 51 3.1 Introduction  [      ] ............................................................................................ 51 3.2 Results and Discussion ..................................................................................... 53 3.2.1 Electrolyte-supported button cell .................................................................. 53 3.2.1.1 Intact cell............................................................................................... 53 3.2.1.2 Cathode Delamination .......................................................................... 56 3.2.1.3 The capacitance of the delamination gap.............................................. 59 3.2.2 Electrolyte supported rectangular cell .......................................................... 60 3.2.2.1 Intact cell............................................................................................... 60 3.2.2.2 Cathode Delamination .......................................................................... 63 iv 3.3 The Normalized Series and Polarization Resistances....................................... 66 3.4 The Influence of the Supporting Configuration................................................ 69 3.5 Anode Delamination ......................................................................................... 71 3.6 On the Aspect Ratio .......................................................................................... 73 3.7 Comparison with Previous Experimental Observations ................................... 74 3.8 Conclusions....................................................................................................... 75 3.9 References......................................................................................................... 77 4. Chapter Four.  Degradation of the Interconnect - Electrode Interface ....................... 80 4.1 Introduction        ............................................................................................... 80 4.2 Results and Discussion ..................................................................................... 81 4.2.1 Oxide Layer .................................................................................................. 81 4.2.1.1 The effect of interconnect geometry ..................................................... 84 4.2.2 Rib Detachment between the Interconnect and the Cathode ........................ 86 4.2.2.1 The effect of interconnect geometry ..................................................... 90 4.3 Conclusions....................................................................................................... 91 4.4 References......................................................................................................... 93 5. Chapter Five. Sulfur Poisoning................................................................................... 94 5.1 Introduction....................................................................................................... 94 5.2 Results and Discussion ................................................................................... 100 5.3 Conclusions..................................................................................................... 106 5.4 References....................................................................................................... 107 6. Chapter Six.  Sensitivity Analysis and Combined Degradation Mode Scenario...... 108 6.1 Introduction..................................................................................................... 108 6.2 Sensitivity of the Method to Inaccuracies in the Input Data and Interaction among Parameters....................................................................................................... 109 6.2.1 Delamination............................................................................................... 110 6.2.2 Oxide Layer Growth ................................................................................... 114 6.2.3 Sulfur poisoning.......................................................................................... 115 6.2.4 Interconnect detachment ............................................................................. 120 6.2.5 Summary..................................................................................................... 123 6.3 On the Simultaneous Occurrence of Multiple Degradation Modes................ 124 6.3.1 Tracking an individual electrode process ................................................... 126 6.3.2 Tracking the Degradation Path ................................................................... 128 6.4 Conclusions..................................................................................................... 134 6.5 References....................................................................................................... 136 7. Chapter Seven.  Conclusions and Outlook. .............................................................. 137 7.1 Conclusions..................................................................................................... 137 7.2 Outlook ........................................................................................................... 143 7.3 Concluding remarks ........................................................................................ 144 7.4 References....................................................................................................... 145 I. Appendix One.  Geometry and Mesh Generation..................................................... 146 II. Appendix Two.  On the linearization of nonlinear PDEs around the steady state point   ................................................................................................................................. 162 III. Appendix Three.  Fitting to an experimental polarization curve. ............................. 164 IV. Appendix Four.  Uniform design parameter grids .................................................... 166  v   List of Tables  Table 1-1. Traditional materials used in planar SOFCs ...................................................4 Table 1-2. Study of SOFC degradation ............................................................................8 Table 1-3. Recent research work on impedance modeling of SOFC-relevant electrochemical systems................................................................................11 Table 2-1. Modeled variables.........................................................................................20 Table 2-2. Standard operating conditions and geometry for typical simulations...........38 Table 2-3. Electrical conductivity and related properties...............................................39 Table 2-4. Literature review of correction factors used for the calculation of the effective conductivity of phase k ..................................................................40 Table 2-5. Gas diffusion properties................................................................................41 Table 2-6. Anode and cathode experimental data used to calculate the exchange current and the double layer capacitance by fitting ......................................42 Table 2-7. Electrochemical data adopted from the literature and results of the fitting.............................................................................................................43 Table 2-8. Reported values for the polarization resistance of LSM-YSZ cathodes reported by researchers at Risø, Denmark .....................................44 Table 3-1. Series, polarization, and normalized resistances calculated based on reported degradation attributed to electrode delamination. ..........................75 Table 5-1. Resistance change (ohm cm2) reported by Xia and Birss (Table 1 in [2]).  Conditions: 800ºC, OCV, 18 hs exposure to 10 ppm H2S, 3% H2O, balance H2.  Three-electrode measurement. ........................................98 Table 5-2  Series and anodic polarization resistance (Ohm), and normalized resistances, before and after exposure to different concentrations of hydrogen sulfide, at two different temperatures, at OCV, Pt counter- electrode [3].  Balance stream composition: 79% H2, 21% H2O..................98 Table 5-3. Kinetics of sulfur poisoning according to [3] ...............................................99 Table 5-4. Series and polarization resistance (ohm cm2) according to [1] before and after exposure to 1ppm H2S in H2. .........................................................99 Table 5-5. Estimated anodic Rp increase as a function of H2S concentration [1]. ......100 Table 6-1. Extent of degradation used in the example of combined degradation. .......126 Table 6-2. Range of variability of input parameters for the sensitivity analysis of the results ................................................................................................112 Table 6-3. Average and percent deviation in normalized series and polarization resistance for the 30 Uniform Design runs, for each degradation mode, varying 21 factors according to Table 6-2.  Cases 1 and 2 indicate before and after locking the parameter identified as primary source of deviation.  Gray cells indicate relevant dispersion values affected by this change................................................................................124 Table 7-1. Summary of the qualitative changes expected in the series and polarization resistance, and in the peak frequency, for the four degradation modes under study. .................................................................141  vi  List of Figures  Figure 1-1. Principle of operation of an SOFC unit cell.................................................3 Figure 1-2. Cross-flow configuration for a planar-stack unit cell ..................................4 Figure 2-1.  Modeled geometry for the button-cell configuration.  Variables anot, elyt, and catt are the anode, electrolyte, and cathode thicknesses, and they vary according to the supporting configuration.  Values for typical simulations are L = 10mm, anot (ESC) = 40 microns, anot (ASC) = 1mm,  catt = 40 microns, elyt (ESC) = 150 microns, elyt (ASC) = 10 microns, where ASC and ESC stand for anode-supported cells and electrolyte-supported cells, respectively.....................................................21 Figure 2-2.  The rectangular cell geometry includes the ribbed interconnect plates.  This two-dimensional approximation is valid for co- and counter-flow configurations.......................................................................................22 Figure 3-1.  Solid line: Polarization curve for the electrolyte-supported cell configuration.  Model parameters detailed in Tables 2-2 to 2-7.  Dashed line:  Ohmic component of the polarization curve.  Kinetic polarization accounts for approximately 75% of the total polarization losses, as also indicated in Figure 3-2. ..............................................................................................54 Figure 3-2.  Nyquist (bottom) and imaginary impedance vs. logarithmic frequency plots of the impedance corresponding to the ESC configuration.  The curved arrow indicates the direction of increasing frequency.  VCELL = 0.7V...........................................................................................55 Figure 3-3.  A gap with the dielectric characteristics of air simulates cathode delamination...............................................................................................................57 Figure 3-4.  Effect of cathode delamination on the impedance spectrum. Both series and polarization resistances increase, without change in relaxation frequencies.  VCELL = 0.7V. ......................................................................57 Figure 3-5.  Ionic current density (A/cm2) within the electrodes and the electrolyte, after a concentric, circular cathode delamination of radius a in a button cell of radius L.  The shadowing effect caused by the delamination makes the current density within the affected zone close to zero.  VCELL = 0.7V................................................................................................58 Figure 3-6.  Electronic current density profile from the centre to the edge along the cathode current collecting line. ..................................................................59 Figure 3-7.  Reactant partial pressure contours inside the porous cathode (left) and anode (right), in atm.  The limited access of reactant to regions underneath the interconnect rib causes local depletion.  The lines of equal reactant concentration appear to be straight because of the large aspect ratio of the cell.  Note the difference in colour scale range.  The picture shows half the rib and half the channel for symmetry reasons.  Axes indicate length in metres. ..................................................61 Figure 3-8.  Polarization curve corresponding to the ribbed interconnect geometry (thick line), calculated for an equivalent set of parameters as vii Figure 3-1 (thin line).  The maximum current density is not high enough for the polarization curve to show mass transport limitation........................62 Figure 3-9.  A new time constant, whose frequency depends on rib and channel size, appears at 10Hz as an anodic diffusional effect (A) in the intact case, for a ribbed interconnect geometry.  Processes B and C correspond to anodic and cathodic electrochemical reactions, respectively.  Labels indicate rib and channel half-width, both equal in the three cases shown.  Numbers below the real axis indicate the high and low frequency intercept impedance values.  VCELL = 0.7V.................................63 Figure 3-10.  Electronic current density magnitude (colour scale, A cm-2) and streamlines (red lines) for a cathode-electrolyte delamination the length of 2 rib widths and 2 channel widths (extending 0.008 m from the left).  Axes indicate spatial dimensions in meters.  ch indicate channels, r indicate ribs.  The zone affected by delamination is completely deactivated.  Mid-way across the intact ribs (r4 and r5) the current density value is small (~0.25 A/cm2) because of the reactant depletion underneath the rib, which makes the current density bend towards the channels, where reactant concentration is higher.  At the corners of ribs r3, r4, r5, and r6 with their neighbour channels, the current density is higher than 0.5 A/cm2 (top of scale), thus the region appears white.  This scaling strategy was needed for better visualization of the overall figure....................................................................................................64 Figure 3-11.  Vertical component of the ionic current density at the mid- plane in the electrolyte for the delamination case shown in the previous figure.  The first eight millimetres show almost no ionic species flow. ....................65 Figure 3-12.  Impedance change caused by a cathode delamination the length of two rib widths and two channel widths.  VCELL = 0.7V. ............................66 Figure 3-13.  Normalized series and polarization resistances as a function of normalized degradation for a circular concentric delamination of radius a, on a button cell of total radius L.  Both resistances scale with delamination area. ......................................................................................................68 Figure 3-14.  Normalized resistances corresponding to the case of delamination of the cathode-electrolyte interface for ESC rectangular cells including the interconnect.  The series resistance is affected uniformly with increasing loss of contact.  The polarization resistance shows an edge effect, by which the shadowing is lesser for delamination of the cathode between the first ribs, where the electrochemical activity is lower in the intact case due to limited access of reactant.  Labels refer to rib and channel numbers (see Figure 3-10) under which the electrode delamination from the electrolyte occurs. .......................69 Figure 3-15.  The influence of cathode delamination on the impedance spectrum for an anode-supported cell.   VCELL = 0.7V. .............................................70 Figure 3-16.  Normalized resistances corresponding to the anode-supported configuration.  The series resistance shows an increase that is larger than proportional to the delaminated area.  The origin of this “overshadowing” is not clear at present.....................................................................71 viii Figure 3-17.  A delamination of the electrolyte from the anode causes essentially the same effect as a delamination of the cathode from the electrolyte in this simulation for an anode-supported button cell.  VCELL = 0.7V. .......................................................................................................................72 Figure 3-18.  Vertical component of the electronic current density for cathode (top) and anode (bottom) delamination in an ASC button cell. Note that the position of the electrodes is reversed for computational convenience, hence the inversion in the direction of the current.  Axes indicate distance in metres.  Scale bar shows electronic current density in A/cm2. ....................................................................................................................73 Figure 3-19.  Normalized series and polarization resistance as a function of the ratio of button cell radius to electrolyte thickness, for 30% cathode delamination.  Aspect ratios ≥ 20 approach R0/R = 0.7.  0.7 is the normalized resistance expected for 30% cathode delamination for the case of complete shadowing.  Labels indicate radii in mm of button cells. ...........................................................................................................................74 Figure 4-1.  Change in current magnitude and streamline distribution produced by a 20 micron thick oxide layer between the cathode and the interconnect................................................................................................................82 Figure 4-2.  Impedance spectrum change produced by the growth of a 20 micron thick layer of chromia on the interconnect surface in contact with the cathode.  VCELL = 0.7V. ...............................................................................83 Figure 4-3.  Normalized resistance behaviour for oxide layer growth between the interconnect and the cathode.  The total degradation state definition is arbitrary, and corresponds to a thickness of 20 microns, the expected growth for 40,000 hrs at 850°C in an oxidant atmosphere. ........................84 Figure 4-4.  The effect of interconnect rib and channel width on the normalized resistances for oxide layer growth, for the case of equal widths.  Performance deterioration is almost insensitive to variations in rib and channel width.................................................................................................85 Figure 4-5.  The effect of interconnect rib and channel width on the normalized resistances for oxide layer growth, for the case of total width equal to 4 mm.  Fractional numbers in the label indicate rib/channel width in mm.  The series resistance increases to a larger extent, the narrower the rib. .......................................................................................86 Figure 4-6.  Redistribution of the electronic current density upon the detachment of the first two cathode interconnect ribs from the left.  Red lines: current density path.  Colour scale indicates A/cm2.  The current density increases at the first intact rib as a result of this rib taking up the current produced at the electrodes within zones above and below the first two ribs. ........................................................................................................87 Figure 4-7.  Impact on the impedance of the rib detachment shown in Figure 4-6.  The reduction of the conductive path increases the series resistance.  The high electronic conductivity of the electrodes redistributes the current density, thereby limiting the shadowing effect. ix Almost no change in characteristic frequency is observed. VCELL = 0.7V............................................................................................................................88 Figure 4-8.  Normalized series and polarization resistances as a function of the number of interconnect ribs detached from the cathode. .....................................89 Figure 4-9.  Ionic current density at the mid-plane in the electrolyte. Compared with the effect caused by electrode delamination (Figure 3-11, Chapter Three), the detached zone deterioration is less severe. .......................90 Figure 4-10.  Influence of rib and channel width in the relative performance loss.  Left: Ribs and channels of equal width, 1 mm, 2 mm, and 5 mm. Relative deterioration increases with increasing width.  Right: Constant rib+channel width equal to 4 mm.  Relative deterioration increases with narrower ribs and wider channels. .............................................................................91 Figure 5-1.  The influence of anode surface area loss on the impedance spectrum of an electrolyte supported SOFC.  The anodic process arc increases in diameter with decrease in anode active surface area.  The anode time constant changes due to an overall change in polarization. This change would have not been observed if the calculation had been done at OCV.  Calculated using a 1-D ESC geometry, i.e., with no interconnect. VCELL = 0.7V. .....................................................................................101 Figure 5-2.  Normalized series and polarization resistances as a function of loss of active area in the anode, for the ESC configuration.  The deterioration is less severe than for cathode delamination, for equivalent fractional degradation.............................................................................102 Figure 5-3.  Thick electrodes are relatively less fully utilized than thin electrodes (calculation performed using a 1-D version of the model, ESC configuration).  Top: normalized local Faradaic current density as a function of the normalized distance from the electrolyte.  Bottom: local Faradaic current density as a function of the distance from the electrolyte.  Legends show electrode thickness in microns. ....................................104 Figure 5-4.  Surface area loss has a stronger impact on thin electrodes than on thick electrodes, because thin electrodes are more fully utilized in the intact state, whereas thick electrodes experience a redistribution of the electrochemical activity away from the electrolyte, thereby tolerating larger uniform surface area loss.  ESC configuration..............................105 Figure 6-1.  Normalized resistance results for 30% cathode area delamination (button cell) and for a delamination of 36% of the cathode length (two ribs and two channel widths in a rectangular cell) are almost insensitive to input parameter variation, and no unexpected interaction is apparent. Complete list of values corresponding to each run # are in Appendix IV, Table IV-1, page 162. ....................................................113 Figure 6-2.  Some variability is observed in the normalized resistance results for 5 micron interconnect oxide layer growth.  The variability in polarization resistance is associated with the non-symmetric polarization behaviour of the cathode.  Series resistance variability occurs because electrolyte thickness is among the varied input parameters.  If electrolyte thickness is fixed at the default value (thin x line), the variability decreases. Complete list of values corresponding to each run # is in Appendix IV, Table IV-2, page 163...............................................115 Figure 6-3.  While the normalized series resistance shows, as expected, no variability, a high level of interaction among parameters yields a large variability in the results for polarization resistance.  The kinetic parameters, inlet partial pressures, and the operating point are responsible for much of this large variability.  This variability decreases when these parameters are fixed.  The simulation shows 90% loss in the anode active area.  Complete list of values corresponding to each run # is Appendix IV, Table IV-2, page 162. ..................................................116 Figure 6-4.  Uniform design experiment corresponding to 90% surface area loss due to sulphur poisoning, for the anode-supported configuration. After fixing the parameters responsible for the scatter, the magnitude of this scatter is smaller than it was in the ESC case.  Complete list of values corresponding to each run # is in Appendix IV, Table IV-3, page 163............................................................................................................................117 Figure 6-5.  Cathode polarization resistance dependence on overall (external) cathodic overpotential vs. 1.12V.  Bold line: standard case of non-equal cathodic charge transfer coefficients (1.5 and 0.5).  Thin line: Equal cathodic charge transfer coefficients (0.5 and 0.5)................................119 Figure 6-6.  Both series and polarization resistance show a variability that is comparable to that of oxide layer growth.  Cathode porosity and thickness, the factors that affect cathode conductance the most, are responsible for the series resistance variability (solid circles).  When these parameters are locked at the default values, the series resistance variability decreases (thin line).  Complete list of values corresponding to each run # are in Appendix III, Table III-2, page 162. ........................................121 Figure 6-7. Both Rp magnitude and Rp scatter increase when the number of detached ribs increase from 2 to 3.  Equivalently to the 2-rib case, the scatter in Rs decreases upon locking the parameters relevant to cathode electronic conductance: porosity and thickness Complete list of values corresponding to each run # are in Appendix III, Table III-2, page 162. ................122 Figure 6-8. Interconnect detachment of two ribs on the anode side presents a lower scatter than it does on the cathode side, because of the very high electronic conductance of the anode.  Complete list of values for each run # in Appendix IV, Table IV-3, page 163...........................................................122 Figure 6-9.  Impedance spectra corresponding to the first three stages of combined sulfur poisoning + interconnect oxidation degradation shown in Table 6-3.  Above: Complete spectrum.  Below: Detail of the anodic contribution.  VCELL = 0.7V. ....................................................................................127 Figure 6-10.  Approximate overall degradation path followed by the mechanisms under study, when acting individually. ...............................................128 Figure 6-11.  Degradation path plot showing the normalized resistance behaviour of:  Interconnect oxidation plus sulfur poisoning, and  cathode delamination.  Although the final degradation state is approximately the same for both cases, the difference in degradation xi history presents a possible way of distinguishing between the two cases.  The error bars correspond to the scatter calculated in section 6.2. ESC. .........................................................................................................................130 Figure 6-12.  Degradation path plots showing two different degradation scenarios: combination of delamination with interconnect oxidation (filled circles), and sulfur poisoning (filled triangles).  The delamination, oxide layer growth, and sulfur poisoning characteristic paths are shown for comparison.  Electrolyte-supported configuration. The error bars correspond to the scatter calculated in section 6.2. ..........................131 Figure 6-13.  Degradation plot for the same combinations shown in Figure 6-12, for the anode-supported configuration.  The error bars were recalculated by re-running the 30 uniform design runs for the anode- supported configuration. ..........................................................................................132 Figure 6-14.  Normalized resistance of the two combined degradation scenarios of the example above, as a function of the result of their direct addition, according to equation (3), with the assumption of no interaction.  The error bars correspond to the scatter calculated in section 6.2. ...............................................................................................................134 Figure I-1.  Geometry details.  Upon a change in physical properties, a rectangle between the rib and the cathode becomes an oxide layer or a detachment, and a rectangle between the cathode and the electrolyte becomes a delamination...........................................................................................147 Figure I-2.  Global meshing strategy and detail showing densification around the rib/cathode/channel corner.....................................................................148 Figure III-1.  Polarization curve fitting of the present model (line) to experimental data (dots) presented by Kim et al. Adapted from Fig 3a, J. Electrochem Soc. 146 (1) 69-78 (1999). ..............................................................165   xii Nomenclature Note: The mathematical model uses SI units throughout.  However, it is customary in the SOFC community to normalize currents using centimeters, because the relevant electrode areas of both test cells and commercial prototypes are of the order of a few square centimeters.  Therefore, cm2 is the preferred unit to express output values for current densities and area-specific resistances.  Likewise, millimeters occasionally describe geometrical quantities such as lengths and widths.  Symbol  units anot anode thickness  m ASC anode supported cell APU auxiliary power unit elyt electrolyte thickness  m catt anode thickness  m dlC  electrochemical double layer capacitance F m -2  cy concentration of species y mol m-3 c0y reference concentration of species y mol m-3 pc  steady state concentration of species p mol m -3  pc ~  oscillatory concentration of species p mol m-3 Deffp-q binary effective diffusivity  m2 s-1 ESC electrolyte supported cell f frequency  Hz F Faraday’s constant  Coul mol-1 pqf  RT Fpqα    V-1 i current density  A m-2  or  A cm-2 iF Faradaic current density  A m-2 iION ionic current density  A m-2  or  A cm-2 iELE electronic current density A m-2  or  A cm-2 i0,q exchange current density electrode q A m-2 j imaginary unit  - k generic conductivity  S m-1 kp,q effective conductivity of species p, electrode q S m-1 L cell length or radius  m M number of concurrent degradation modes - n number of electrons transferred in a reaction - n   normal unit vector  - Np flux of species p  [species] m-2 s-1 NTOT total (bulk) species flux  [species] m-2 s-1 xiii OCV open circuit voltage  V PDE partial differential equation p system gas pressure  Pa or atm pq partial pressure of gas q  Pa or atm Q generic species source term  [species] m-3 s-1 r radial coordinate Rg universal gas constant  J mol-1 K-1 R* universal gas constant  cal mol-1 K-1 Rs series resistance  ohm cm2 Rp polarization resistance  ohm cm2 sR  normalized series resistance - pR   normalized polarization resistance - S electrochemically active surface area m-1 tp thickness of layer p  microns T absolute temperature  K CELLV  operating cell potential  V xy molar fraction of species y - xp,q volume fraction of p phase in electrode q - z axial coordinate Z area specific impedance  ohm cm2   αpq charge transfer coefficient, electrode p, direction q - pε  porosity fraction of electrode p  - pη  local overpotential of electrode p  V η  REFIONELE Φ−Φ−Φ   V η~  IONELE Φ−Φ ~~   V pτ  gas phase tortuosity of electrode p  - Φ  generic electric potential  V ELEΦ  electronic potential  V IONΦ  ionic potential  V REFΦ  reference potential  V ELEΦ  steady-state electronic potential V ELEΦ ~  AC electronic potential  V IONΦ  steady-state ionic potential V ON ~ IΦ   AC ionic potential  V 0 ~Φ∆  perturbation amplitude  V ω   angular frequency  s-1   xiv Acknowledgements I am most grateful to my research supervisor, Dr. Olivera Kesler, for her trust and support throughout the duration of my PhD project.  Thank you, Olivera, for so much time, patience and intellectual generosity dedicated to my research work. The knowledge, experience, and patience of Professor Colin Oloman were essential in shaping my research work and thesis. Thank you very much to all my other Committee members: Drs. Walter Mérida, Brian Wetton, and Gary Schajer for their input throughout the years.  This work would have never been possible without the unconditional love and understanding of my wife Verónica and my son Matias.  xv            a Verónica y Matias xvi Co-Authorship Statement The research work presented in this thesis (and the publications stemming from it) corresponds to the work of the author, under the supervision of Dr. Olivera Kesler, co- author of the publications.  The author of the thesis was responsible for the development of the equations, and code programming, and Dr. Kesler provided continuous academic guidance, critical review of the results, and supervision.  1 1. Chapter One.  Introduction and Background. 1.1 Introduction Modern lifestyles have led to a relentless increase in energy consumption. Traditional ways to generate power include combustion of fossil fuels and coal, and hydroelectric and nuclear energy conversion.  An undesirable consequence of fossil fuel and coal combustion, the most widespread power generation methods, is environmental contamination, as a result of the products of combustion released to the atmosphere.  In addition, the uneven distribution of the finite fossil fuel sources worldwide causes geopolitical unrest.  Therefore, there is a need for better ways to satisfy the energy demands of society. Fuel cells appear as an attractive alternative to traditional power generation methods.  A fuel cell is a reactor that generates electricity by combining a fuel and an oxidant electrochemically, rather than thermochemically, as is the case with fossil fuel combustion.  Electrochemical power generation has many advantages over fossil fuel combustion, including higher efficiency, low pollution, low equipment maintenance, and modularity.  The fuel cell receives the reactants that take place in the energy conversion process in a continuous manner, unlike batteries, which use chemical energy that is stored within the electrodes.  A fuel cell based stationary generation plant could also provide independence of a power distribution grid, an attractive choice for remote settlements and mining operations. Since the first demonstration of the fuel cell principle described by Sir William Grove in 1839, many types of fuel cells have been developed.  Among them, Solid Oxide Fuel Cells (SOFC) appear especially suitable for stationary electricity generation, both for power plant scales, and for small domiciliary scale. 1.2 The Solid Oxide Fuel Cell (SOFC) The SOFC is a completely solid-state type of fuel cell that presents many advantages as an energy conversion device, especially for stationary applications.  The main components of a fuel cell are the electrolyte and the electrodes.  The electrolyte is 2 an ionically conducting ceramic membrane that prevents direct combination of fuel and oxidant.  This membrane is sandwiched between two porous, electronically (or mixed) conducting electrodes: the anode, or fuel electrode, and the cathode, or air electrode.  The electrolyte is a dense, gas-tight ceramic layer, traditionally made of yttria-stabilized zirconia (YSZ), whose high mobile vacancy content makes it a good conductor of oxide ions, O2-.  An electronically conductive, gas tight interconnect plate provides the series connection of unit cells, constituting an SOFC stack. Because ionic conductivity in ceramics is a thermally activated process, high operating temperatures are required to obtain sufficient overall conductivity in the fuel cell.  YSZ exhibits acceptable conductivity in the 700º - 1100ºC temperature range.  The oxide ion as the main charge carrier allows the use of a variety of fuels other than hydrogen, such as carbon monoxide or even hydrocarbons, either by direct oxidation or via steam reforming.  Another important feature of the high operating temperature is the possibility to use the exhaust heat in a power generating bottoming cycle, using either a gas or steam turbine. These features give the SOFC great versatility compared to other fuel cell types, as well as creating the possibility of generating electrical power at very high efficiencies. On the other hand, the elevated temperature of operation requires long start-up periods, which are inconvenient for mobile applications, and results in large thermal stresses on the delicate ceramic components during thermal cycling to and from the operating temperature, which jeopardizes the device performance.  In addition, the uneven distribution of electrochemical reaction sites also generates large thermal gradients that are a source of stress.  Finally, fabrication of the cell components also requires high temperatures that can cause detrimental residual stresses within the cell components. 1.2.1 Principle of Operation  Oxygen molecules in the cathode pores dissociate and combine with free electrons coming from the external circuit, and the resulting oxygen ions migrate through the electrolyte toward the anode.  In the anode, the oxygen ions combine with hydrogen, carbon monoxide, or hydrocarbon molecules to form water (the only exhaust product if 3 H2 is used) and CO2.  The electrons released in the oxidation reaction have to escape through an external circuit, creating usable electricity and closing the loop.  Figure 1-1 depicts the case of hydrogen being used as a fuel. H2 H2O O2 O2- e- cathode electrolyte anode 1/2 O2 + 2e-   O2- H2+ 1/2 O2-   H2O + 2e- e-  Figure 1-1. Principle of operation of an SOFC unit cell A unit cell generates approximately 1V at open circuit conditions, with exact voltage depending on temperature, oxygen partial pressure, and reactant and product concentration.  This value is commonly known as open-circuit voltage, or OCV.  Thus, most practical applications require a series connection between adjacent cells to increment this voltage.  The series connection is provided by the interconnect layer, which also acts as a gas separator in the planar configuration. Under operating conditions, the delivered voltage is smaller than the OCV due to irreversible losses or cell polarization.  In ceramic fuel cells, the most important contribution to these losses is typically the ohmic polarization, i.e. the voltage drop due to internal resistance. Table 1-1 shows the most commonly used materials for each cell unit of planar SOFC stacks.  Each stack component is the subject of a great deal of research, aimed at optimizing its chemical, mechanical and micro-structural characteristics and compatibility with other cell components, as well as improving the fabrication methods and long-term stability of the cells.  Ivers-Tiffée et al. published a very good summary of SOFC material properties in [1]. 4 Table 1-1.  Traditional materials used in planar SOFCs Component Material Denomination Acronym Anode Ni - Yx Zr1-x O2-x/2 nickel-yttria stabilised zirconia Ni-YSZ Electrolyte Yx Zr1-x O2-x/2 yttria stabilised zirconia YSZ Cathode SrxLa1-xMnO3-δ +  Yx Zr1-x O2-x/2 doped lanthanum manganite LSM-YSZ Interconnect  stainless steel -  A great variety of techniques has been developed to fabricate each component. Traditionally, each layer was deposited on top of the substrate (typically the electrolyte or the anode) by a wet method, which consists in painting the surface with a well-mixed slurry and subsequently drying it in a furnace and consolidating it by sintering, or creating pores by adding a pore-former to the slurry [2] in the case of the electrodes. cell interconnect air fuel  Figure 1-2. Cross-flow configuration for a planar-stack unit cell Internal or external manifolds can be used to transport reactants to the reaction sites.  The reactants spread over the plane of the cell by means of channels of varying thickness and pattern.  These channels are usually grooved on the interconnect surface. In planar SOFC stacks, the reactants can be manifolded in different ways: co-flow, counter-flow, cross-flow (Figure 1-2), or radial flow. 5 1.3 Degradation Phenomena in Solid Oxide Fuel Cells  The widespread commercialization of SOFCs is subject to the solution of problems related to their durability, reliability, and cost.  Associated with durability and reliability is the important problem of cell degradation.  The severe environmental conditions within an SOFC limit the choice of suitable materials and poses important challenges to the stability of the different cell components.  The mismatch in thermal expansion characteristics of the cell’s different layers is a major cause of stresses during thermal cycling, during both fabrication and operation. In addition, the uneven thermodynamics of chemical reactions throughout the cell also generates large thermal gradients that are also a source of stress.  A consequence of the aforementioned thermo-mechanical phenomena is the degradation of the contact between adjacent cell layers, namely the electrolyte-electrode interface, or the interconnector- electrode interface.  This degradation can lead to detachment of two adjacent layers, with the consequent loss of conductive area. Another example of a degradation mechanism affecting SOFCs is the growth of an electrically less conductive oxide layer between the interconnect plate and the electrodes, especially the cathode.  A consequence of this phenomenon is also an increase in the cell’s internal resistance, due to the increased Ohmic resistance along the conductive path.  The elevated temperature of operation favours the kinetics of chromium oxide growth on the surface of the stainless steel interconnect plate. Extended periods of operation at elevated temperature and exposure to contaminants contained in the reactants are responsible for another important phenomenon: microstructural degradation.  The electrochemical combination of reactants in a fuel cell occurs inside the porous electrodes.  In state-of-the-art SOFCs, these reactions are not restricted to the interface with the electrolyte, but take place within a few microns to tens of microns into the volume of the electrode.  The (3D) electrode microstructure is of great importance in providing adequate active surface area for electrochemical reaction and sufficient porosity for gas species transport.  Consequently, degradation mechanisms that affect electrode microstructure can compromise the cell performance, stability, and durability.  Examples of microstructure degradation mechanisms include: 6 1. Sintering of the electrode microstructure, typically in the nickel-YSZ cermet anode [3,4].  Solid mobility phenomena are thermally favoured.  The fine-grained nickel in the porous anode shows a tendency to agglomerate after long exposure to high temperatures.  This evolution results in loss of active surface area and changes in the porosity distribution, thereby lowering overall performance. 2. Poisoning of the cathode microstructure with chromium products outgassed from the interconnect plates [5,6].  This electrochemically driven degradation mode has been extensively reported, and constitutes a major disadvantage of stainless steel interconnectors.  The chromium rich compound deposits preferentially on the most electrochemically active sites, resulting in a loss of active area. 3. Carbon deposition in the anode [7].  An undesirable reaction favoured by nickel, which catalizes coking in addition to hydrogen oxidation.  Solid carbon deposits on nickel, affecting electrochemistry, diffusion, and structural integrity. 4. Sulfur poisoning of the anode, a consequence of the traces of sulphur present in hydrocarbon fuels or hydrogen produced by reforming [8,9].  Sulfur present in small quantities as H2S or thiols adsorbs on nickel as a molecular monolayer, blocking electroactive sites.  Nonetheless, unlike the two mechanisms previously mentioned, sulfur desorbs upon re-establishment of a sulfur-free fuel stream, making it reversible at low concentrations.  Furthermore, it does not affect the anode gas diffusion properties, at least at early stages of poisoning. Despite their difference in nature, the effect of these degradation modes on cell performance measured in long-term degradation testing is common to all of them: a loss in available potential at constant current load, or a drop in delivered current density at constant voltage.  In other words, it is not possible to identify a specific degradation mechanism or combination of mechanisms by observing only changes to the DC behaviour of the cell.  Thus, there is a need to develop a diagnostic technique that allows the identification of specific degradation mechanisms of SOFCs in a minimally invasive way.  Such a technique would be a useful tool for diagnosis of both a cell in service or under research, where identifying the nature of a degradation mechanism could save the disassembly time required for a direct observation.  In the case of reversible degradation, the method would indicate the possibility to correct the failure while the fuel cell is in 7 operation.  In the case of irreversible degradation, the method would aid in identifying the specific cause of failure of a component that needs replacement, so that operating conditions or cell or stack materials or designs could potentially be adjusted in subsequent tests to minimize further degradation.  At a research stage, a diagnostic tool of this kind would be a useful complement to post-mortem microstructural observation.  A minimally invasive tool that provides a wealth of information about electrochemical systems, and that has become a standard in fuel cell research, is electrochemical impedance spectroscopy [10]. Impedance spectroscopy is a well-known technique used to study electrochemical characteristics of systems such as batteries, capacitors, and fuel cells, and which is widely used in disciplines such as corrosion and materials science.  It consists of measuring the impedance of a system at different frequencies by superimposing a small voltage or current perturbation onto the voltage or current operating point.  The underlying idea is that individual processes will appear in the impedance spectrum at different frequencies, according to their inherent rate.  For example, a charge transfer process is likely to appear at higher frequencies than a slower diffusional process.  This technique is convenient because it is minimally invasive, and it can be used on systems in operation.  Its resolution in frequency allows the separate study of simultaneous cell processes, potentially enabling their individual optimization. As mentioned above, a common observation during SOFC single-cell testing at R&D stages is a change in performance over time.  Regardless of its origin, this change is usually manifested as a decrease in output power, and, if this measurement is available, as an evolution of the impedance spectrum, which varies in shape and size over time. Although the state-of-the-art knowledge provides general ideas about the nature of the degradation mechanism that is taking place within the cell or stack, there is still no systematic procedure to link a certain degradation mode to its effect on the impedance spectrum.  A technique that is capable of providing information about the nature of the degradation modes affecting the cell or stack would be most useful to a researcher who is testing new materials, geometries, or fabrication procedures. In summary, SOFC degradation is an important research area, and successful commercialization of SOFCs depends to a large extent on the understanding of the 8 degradation phenomena.  Table 1-2 gives a summary of recent research done in the area of SOFC degradation. This thesis focuses on degradation modes associated with contact impedance, namely electrode delamination, interconnect oxidation, and interconnect detachment, together with the simplest case of microstructural degradation, sulfur poisoning.  Details of relevant previous work done in this area will appear in the appropriate chapters. Table 1-2.  Study of SOFC degradation Research Group Main Researchers Field of Study Ref Illinois I. of T. Y. Hsiao, R. Selman Degradation of SOFC electrodes [11] Risø Nat. Lab. A. Hagen, R. Barfod, P. Hendriksen, Y. Liu, S. Ramousse, S. Primdahl, M. Mogensen Delamination Chromium poisoning Anode stability [12,13] FZ Jülich D. Simwonis, F. Tietz, D. Stöver Nickel coarsening [3] Fuji Electric T. Iwata Nickel coarsening [4] Univ. Calgary S. Paulson, V. Birss Chromium poisoning [5] FZ Jülich E. Konysheva et al. Chromium poisoning [6] Univ. Calgary S. Xia, V. Birss Sulfur poisoning [8] Georgia Inst. Tech. S. Zha, Z. Cheng, M. Liu Sulfur poisoning [9] EPFL D. Larrain, J. van Herle, D. Favrat Interconnect oxidation [14] Tokyo Gas K. Fujita et al. Interconnect oxidation [15] Georgia Inst. Tech. G. Nelson, C. Haynes Delamination [16] Univ. Karlsruhe M. Heneka, N. Kikillus, E. Ivers-Tiffeé Delamination [17] PNNL S. Simner et al. Delamination [18] Tohoku Univ. K. Sato, J. Mizusaki, et al. Delamination and cracking detection using acoustic emmission [19] FZ Jülich J. Malzbender, T. Wakui, R. Steinbrech, L. Singheiser Interconnect detachment [20] Univ. Florida J. Smith, E. Wachsman Interfacial reactions [21]  1.4 SOFC Modeling  Increasing computational power at affordable costs has permitted the widespread use of modeling as an invaluable tool for design, optimization and analysis of fuel cells at many levels, from detailed electrochemistry electrode level to system, cell and balance- of-plant level.  Models can be used to predict performance variations upon the change of 9 a certain design parameter(s), as well as to identify the relative importance of operational variables.  Advantages of modeling analysis include cost reduction, both from time and materials saving, and flexibility of parameter variations.  Disadvantages include difficulty in the interpretation of results and in conducting unambiguous experimental validation, and the challenges involved in the choice of reliable input parameters.  A fuel cell is a complex nonlinear system where several interacting processes occur simultaneously in a highly convoluted manner, and care must be taken in the choice of input parameters since the same overall results may be obtained using different combinations of these parameters. A core contribution of this thesis is the combination of an SOFC impedance model with degradation.  Modeling the impact of degradation on performance and on the impedance of a working SOFC requires, in the first place, modeling the intact cell. Several numerical simulation studies currently published assume lateral invariance of the modeled quantities (e.g. [24,27,29,31,33]), leading to the important simplification of the governing equations to one dimension, i.e. the main direction of current flow.  Although this assumption is useful to study the cell performance under certain conditions, it cannot account for effects that are inherently two- or three-dimensional, such as the uneven reactant concentration distribution within the porous electrode resulting from the ribbed interconnect geometry.  Furthermore, many degradation modes do not have a uniform impact on the cell components such that they can be represented using a one-dimensional model.  A clear example of this statement is electrode delamination, which leads to an in- plane redistribution of the current within the cell, destroying any initial in-plane invariance of the system. A requirement for modeling cell degradation is to have detailed information about the cell processes at the electrode level.  The need to know the current redistribution after degradation is an example that supports this need.  Modeling at the electrode level provides information about space distribution of operating variables such as currents, temperature, and concentrations.  The charge, energy, and mass conservation equations describe the behaviour of these quantities within the different cell components.  In solving these equations, the macrohomogeneous modeling approach is especially suitable if no a priori assumption is made on the relative importance of the transport processes. 10 Newman and Tobias first described this approach in 1962 [22].  In that work, the authors treat the porous electrodes as continuum materials characterized by effective properties, corresponding to their averaged porous composite properties.  Every phase, ionic conductor, electronic conductor, and pore, coexists at every point in the modeled electrode in proportion to its presence in the real electrode.  A balance equation is solved for each species throughout the electrode, with appropriate boundary conditions. The microscopic details of each of the reactions that take place within the SOFC electrodes remain largely unknown despite decades of research in the field.  Although there is general consensus in that the electrochemical combination of fuel and oxidant takes place at or near the triple-phase-boundary in a porous solid electrode, there is disagreement on the details of the elementary processes that constitute the overall reactions.  The characteristics of these processes seem to depend on the type of reactants, the electrode and electrolyte materials and microstructure, the operating temperature, and the power demand.  From a modeling perspective, these uncertainties pose the need for a number of assumptions about the nature and characteristics of the modeled processes. Furthermore, the difficulty (or impossibility) in measuring certain quantities results, unavoidably, in the need to treat some input variables as free parameters, whose values would be determined by fitting to experimental data. Several research groups have developed or are currently developing SOFC impedance models.  Table 1-3 summarizes this information, indicating the main topic of their modeling work.  In [24], Adler et al. postulate mechanisms other than charge transfer as responsible for the Gerischer-type impedance observed in solid mixed conducting electrodes.  Their simulation is exemplified with a model of an LSFC cathode on SDC.  In [25], Fleig and Maier study the influence on the impedance of imperfect contact between porous electrodes on solid electrolytes, using finite elements.  The anode reaction is studied in [26], where Bieberle and Gauckler propose a series of steps as responsible for the overall electrochemical oxidation of hydrogen.  Also interested in the anode reaction, Bessler [27] modeled the impedance of patterned and porous anodes, in an attempt to assess the validity of the mechanism postulated in [26].  In [28], Bessler applies a simpler approach to reproduce the conversion impedance arc reported by Primdahl and Mogensen [23] in their anode experiments.  Using a two-dimensional 11 geometry that describes the fuel impingement on the anode, the author investigates the influence of gas flow rates and concentration variations on the impedance. Table 1-3.  Recent research work on impedance modeling of SOFC-relevant electrochemical systems Research Group Main Researchers Field of Study Ref Ceramatec, Imperial College S. Adler, J. Lane, B. Steele Electrode kinetics of porous mixed ionic- electronically conductive electrodes [24] Max-Planck Int. Festkörperforsch. J. Fleig, J. Maier Influence of current constriction on impedance [25] ETH Zürich A. Bieberle, L. Gauckler Anode processes, state-space modeling [26] Univ. Heidelberg W. Bessler Anode processes, concentration effects [27, 28] SINTEF S. Sunde Monte Carlo simulations, porous electrodes [29] Univ. Karlsruhe H. Schichlein, E. Ivers-Tiffeé Deconvolution of impedance spectrum [30] LEPMI J. Deseure, Y. Bultel, L. Dessemond Cathode processes, influence of electrode parameters [31] Univ. Twente B. Boukamp Equivalent circuit approach to study impedance behaviour of alternative anodes [32] Colo. Sch. Mines H. Zhu, R. Kee 1-D Time-domain analysis of button cells using H2 or CH4 [33]  Sunde [29] utilized an alternative to the equivalent circuit approach, using Monte Carlo simulations to calculate the impedance of porous anodes.  His original approach, albeit computationally intensive, allowed him to account for non-percolating clusters in the porous microstructure.  Schichlein et al. [30] developed a promising technique to increase the resolution of impedance deconvolution, proposing an infinite string of RC parallel circuits as an alternative to the traditional equivalent circuit approach that lumped each electrode’s impedance properties into a more or less complicated set of circuit elements.  The group at LEPMI simulated the impedance behaviour of one-phase mixed conducting cathodes [31].  In their 1-D model, the authors solved the O2- transport equations assuming Langmuir adsorption of oxygen and solid-state diffusion as the governing processes, with charge transfer in equilibrium.  The authors predict a non- monotonic decrease-increase in polarization resistance with increasing cathodic bias (Figs 2b, 3b in [31]), and a monotonic increase in polarization resistance with increasing oxygen partial pressure (Fig 5a, [31]).  This prediction is in contradiction with respect to 12 the commonly encountered behaviour of Butler-Volmer kinetic electrodes (e.g. Fig 3 in [34], and Fig 4 in [35]), probably because of the fact that the assumed limiting process in [31] is solid-state diffusion.  The milestone work described in [32] describes the analysis of electrochemical systems by linking a circuit element to electrochemical processes such as charge transfer, double-layer charging, and diffusion.  Values for the circuit elements are assigned using complex non-linear least-squares fitting.  The work of Zhu and Kee [33] describes a 1-D model of the impedance of a button cell where the authors consider methane reforming among the modeled reactions.  They use finite volumes to solve the problem in the time domain, letting the response stabilize after a certain number of periods.  The system is, however, assumed to respond linearly to the small amplitude of the perturbation.  The exchange current is a fitting parameter to reproduce typical performance, and the charge transfer coefficients are taken as 1.5 and 0.5 for the anodic and cathodic directions, respectively, for both electrodes. A common feature of all these modeling approaches, except for the recent anode model published by Bessler [28], is the assumption of in-plane uniformity, leading to the simplification of the system to one dimension: the main direction of current.  This assumption is not applicable in the present case, since degradation modes such as electrode delamination and interconnect detachment dramatically modify the direction of the current path, making the problem inherently two- or three-dimensional.  Moreover, the presence of the ribbed interconnect in contact with the electrode also produces local reactant depletion underneath the ribs, thereby invalidating the assumption of in-plane invariance, as will be shown in later chapters of this work.  Modeling in two or three dimensions is clearly essential in view of the objective sought by this thesis. A second important aspect is that the electrode-level impedance simulations have mostly focused on one electrode at a time.  This is true for all the cited previous work except for [33].  Using the method proposed in this thesis in practical-size, operating SOFCs requires modeling the full cell, and also not counting on a reference electrode to separate the individual contributions to the impedance of each electrode, since reference electrodes are not feasible in practical SOFC systems, especially those with thin electrolytes. A third aspect to take into account is computational cost.  In the first place, 13 gradients close to the electrochemical interfaces in SOFCs require local densification of the numerical grid.  Secondly, two-dimensional modeling demands slender elements due to the large aspect ratio of the planar SOFC.  In the third place, having flexibility in the creation of the modeled geometry was an attractive option to consider, so as to test the model results on diverse supporting configurations, as well as to perform important parametric studies.  These requirements do not pose major difficulties to one-dimensional models, since the number of total degrees of freedom in such models is limited.  In the present case, the equations, the geometry, and the meshing were developed taking into account the computational requirements that were specific to the problem being solved. Isothermal and isobaric conditions are an assumption common to all the aforementioned research work, as well as to the work presented in this thesis. 1.5 Scope of this thesis This thesis explores, from a modeling viewpoint, the possibility of using impedance spectroscopy as a minimally invasive tool to diagnose degradation in a working SOFC.  The work focuses on contact resistance degradation modes, namely electrode delamination, interconnect detachment, and oxide layer growth between the interconnect and the electrode, and one microstructural degradation mode, sulfur poisoning.  Chapter Two presents a novel SOFC impedance model based on the finite element method.  In subsequent chapters, geometrical, physical, and electrochemical parameter modifications simulate different degradation conditions.  This is the first time that an SOFC model combines impedance behaviour and degradation.  Chapter Three focuses on electrode delamination, describing its effects, electrochemical implications, and impedance signatures.  A comparison with published experimental observations supports the modeling results.  Chapter Four describes two further contact degradation problems, this time related to the interconnect-electrode interface: interconnect oxidation and interconnect detachment.  An analysis of their impact on performance and on the impedance spectrum reveals interesting differences with delamination, as well as possible ways to diagnose them.  Reducing the anode electro-active surface area simulates sulfur poisoning in Chapter Five, and an analogous analysis is presented in terms of diagnosis. Chapter Six studies a scenario of two simultaneous degradation modes and assesses the 14 method’s capability to diagnose them.  Secondly, it presents a sensitivity analysis of the presented results to variations in the model’s input parameters.  Chapter Seven presents the overall conclusions, and recommendations for future work stemming from this thesis.  Original research contributions of this thesis include: 1. The development of the first two-dimensional impedance model of a working full-cell SOFC, and the first one to include the ribbed interconnect geometry. 2. The incorporation of degradation features into an impedance model of an SOFC. 3. The linearization of the time-dependent AC equations around the operating point to yield a linear, time-independent, complex-valued system of equations for the charge and material balance for an SOFC. 4. The definition of novel and convenient non-dimensional variables that represent extent of degradation, and that are indicative of the degradation mechanism present in the SOFC. 5. A study of the impact of inaccurate estimation of, and of the interaction among input parameters on the model predictions. 6. A study of possible interactions among different degradation modes and input parameters. 7. The coding of an automated generic geometry and mesh generator for use in any finite element two-dimensional analysis of a fuel cell comprising a ribbed interconnect and a cell, using high aspect ratio rectangular elements.  Not restricted to just SOFCs, this code is expected to be useful in performing parametric studies involving geometrical and physical variables on a variety of fuel cell types.  15 1.6 References   [1] E. Ivers-Tiffée, Q. Weber, D. Herbstritt, “Materials and technologies for SOFC- components”, J. Eur. Ceram. Soc., 21, 1805-1811 (2001). [2] M. Boaro, J. Vohs, R. Gorte, “Synthesis of highly porous YSZ by tape-casting methods”, J. Am. Ceram. Soc, 86 [3], 395-400 (2003). [3] D. Simwonis, F. Tietz, and D. Stöver, “Nickel coarsening in annealed Ni/8YSZ anode substrates for solid oxide fuel cells”, Solid State Ionics 132, 3-4, 2 (2000) 241-251. [4] T. Iwata. “Characterization of Ni-YSZ Anode Degradation for Substrate-Type Solid Oxide Fuel Cells”, J. Electrochem. Soc. 143 (1996), 1521. [5] S. Paulson and V. Birss, “Chromium Poisoning of LSM-YSZ SOFC Cathodes I. Detailed Study of the Distribution of Chromium Species at a Porous, Single-Phase Cathode”, J. Electrochem. Soc., 151, 11, (2004) A1961-A1968 [6] E. Konysheva et al, “Chromium Poisoning of Perovskite Cathodes by the ODS Alloy Cr5Fe1Y2O3 and the High Chromium Ferritic Steel Crofer22APU”, Journal of The Electrochemical Society, 153 (4) A765-A773 (2006) [7] H. He and J. Hill, “Carbon deposition on Ni/YSZ composites exposed to humidified methane”, Applied Catalysis A: General, 317, 2, (2007), 284-292. [8] S. Xia and V. Birss, “Deactivation and recovery of Ni-YSZ anode in H2 fuel containing H2S”, Proc. SOFC IX, Vol. 2, 2005, 1275-1283. [9] S. Zha, Z. Cheng, and M. Liu, “Sulfur Poisoning and Regeneration of Ni-Based Anodes in Solid Oxide Fuel Cells”, Journal of The Electrochemical Society, 154 (2) B201-B206 (2007) [10] E. Barsoukov and J. MacDonald (editors), “Impedance Spectroscopy”, 2005. [11] Y. Hsiao and R. Selman, “The degradation of SOFC electrodes”, Solid State Ionics 98, (1998) 33-38 [12] R. Barfod, M. Mogensen, T. Klemensoe, A. Hagen, Y. Liu, “Detailed characterisation of anode supported SOFCs by impedance spectroscopy”, Proc. SOFC IX, Vol. 1, 2005, 524-533. 16  [13] A. Hagen, R. Barfod, P. Hendriksen, and Y. Liu, “Effect of operational conditions on long term stability of SOFCs”, Proc. SOFC IX, Vol. 1, 2005, 503-513. [14] D. Larrain, J. van Herle, D. Favrat, “Simulation of SOFC stack and repeat elements including interconnect degradation and anode reoxidation risk”,  J. Power Sources 161, (2006) 392-403 [15] K. Fujita et. al., “Relationship between electrochemical properties of SOFC cathode and composition of oxide layer formed on metallic interconnects”, J. Power Sources 131, (2004) 270-277 [16] G. Nelson, “Solid oxide cell constriction resistance effects”, MSc Thesis, GIT, 2006. [17] M. Heneka, N. Kikillus, and E. Ivers-Tiffée, “Design of experiments for lifetime modelling of SOFC” , Proc. SOFC IX, Vol. 1, 2005, 908-919. [18] S. Simner, M. Anderson, M. Engelhard, and J. Stevenson, “Degradation Mechanisms of La–Sr–Co–Fe–O3 SOFC Cathodes”, Electrochemical and Solid- State Letters, 9 (10)  (2006) A478-A481 [19] K. Sato et al. “Mechanical damage evaluation of SOFC under simulated operating conditions”, J. Ceram. Soc. Japan, 113 (8) 2005 562-564 [20] Malzbender, J., Wakui, T. and Steinbrech, R. W., “Deflection of planar solid oxide fuel cells during sealing and cooling of stacks”. Proceedings of the sixth Euro. SOFC Forum, vol 1 (2004) 329–338. [21] J. Smith and E. Wachsman, “Effect of harsh anneals on the LSM/YSZ interfacial impedance profile”, Electrochimica Acta 51 (2006) 1585–1591 [22] J. Newman and C. Tobias, “Theoretical analysis of current distribution in porous electrodes”, J. Electrochemical Soc., Vol. 109, No. 12, 1962, 1183-1191. [23] S. Primdahl and M. Mogensen, “Gas Conversion Impedance: A Test Geometry Effect in Characterization of Solid Oxide Fuel Cell Anodes”, J. Electrochem. Soc., Volume 145, Issue 7, pp. 2431-2438 (1998) [24] S. Adler, J. Lane, and B. Steele, "Electrode Kinetics of Porous Mixed-Conducting Oxygen Electrodes", J. Electrochem. Soc., 144, (5), (1997) 1884-1890 17  [25] J. Fleig and J. Maier, “The influence of current constriction on the impedance of polarizable electrodes”,  J. Electrochem. Soc., 144, (11), (1997) L302-L305 [26] A. Bieberle and L. Gauckler, “State-space modeling of the anodic SOFC system Ni, H2–H2OYSZ”, Solid State Ionics,146, 1-2, (2002), 23-41 [27] W. Bessler, “A new computational approach for SOFC impedance based on detailed electrochemical reaction-diffusion models,” Solid State Ionics 176, (2005) 997-1011. [28] W. Bessler, "Gas concentration impedance of solid oxide fuel cell anodes. I. Stagnation point flow geometry," J. Electrochem. Soc. 153, (2006) A1492-A1504. [29] S. Sunde, "Calculations of Impedance of Composite Anodes for Solid Oxide Fuel Cells", Electrochimica Acta, 42, (1997) 2637 [30] H. Schichlein, A. Müller, M. Voigts, A. Krügel and E. Ivers-Tiffée, “Deconvolution of electrochemical impedance spectra for the identification of reaction mechanisms in solid oxide fuel cells”, Journal of Applied Electrochemistry 32 (2002) 875–882 [31] J. Desseure, Y. Bultel, L. Dessemond, and E. Siebert, “Modelling of dc and ac responses of a planar mixed conducting oxygen electrode”, Solid State Ionics, 176, 3-4 (2005) 235-244. [32] B. Boukamp, “A package for impedance/admittance analysis”, Solid State Ionics 18-19 (1986) 136-140. [33] H. Zhu and R. Kee, “Modeling Electrochemical Impedance Spectra in SOFC Button Cells with Internal Methane Reforming”, J. Electrochem. Soc., 153 (9) (2006) A1765-A1772 [34] C. Xia, W. Rauch, F. Chen and M. Liu, “Sm0.5Sr0.5CoO3 cathodes for low- temperature SOFCs” Solid State Ionics, 149, 1-2, (2002), 11-19 [35] A. Co, S. Xia, and V. Birss, “A kinetic study of the oxygen reeduction reaction at LaSrMnO3-YSZ composite electrodes”, J. Electrochemical Soc., Vol. 152, No. 3, A570-A576 (2005) 18 2. Chapter Two. Mathematical Model1 2.1 Introduction Several physical phenomena take place simultaneously within a working SOFC. The reactants flow along the distribution channels and diffuse into the porous electrodes toward the electrolyte, driven by the concentration gradient established due to electrochemical consumption.  A result of this reaction is the production or consumption of ionic (O2-) or electronic (e-) species, which move through the ionically- or electronically-conductive phases.  Ions travel across the electrolyte, and electrons flow through an external circuit, producing usable electricity.  In addition, there is production or consumption of heat, depending on the type of chemical or electrochemical reaction taking place within the porous electrode, as well as heat transfer among the different cell components.  These many concurrent physical processes pose important difficulties in modeling fuel cell systems, since the fundamental partial differential equations that describe the involved variables form a coupled, nonlinear system, with many parameters not independently measurable.  A further complication appears when a perturbation is imposed that causes the system to evolve dynamically.  Impedance studies are an example of this situation, where the system oscillates around the operating point in response to a small sinusoidal perturbation. 2.2 General Description and Assumptions The simulation of localized degradation mechanisms such as partial electrode delamination or interconnect detachment requires modeling at the electrode level, to provide information about changes in the current flow patterns.  This requirement rules  1  This chapter constitutes the mathematical formulation used in the models published as: a. J. I. Gazzarri and O. Kesler, “Non-destructive delamination detection in solid oxide fuel cells”, Journal of Power Sources, 167 (2) 430-441 (2007). b. J. I. Gazzarri and O. Kesler, “Electrochemical AC impedance model of a solid oxide fuel cell and its application to diagnosis of multiple degradation modes”, Journal of Power Sources, 167 (1) 100-110 (2007). c. J. I. Gazzarri and O. Kesler, “Short stack modeling of degradation in solid oxide fuel cells - Part I: Contact degradation”, Journal of Power Sources, doi: 10.1016 / j.jpowsour.2007.10.047. d. J. I. Gazzarri and O. Kesler, “Short Stack Modeling of Degradation in Solid Oxide Fuel Cells - Part II: Sensitivity and Interaction Analysis”, Journal of Power Sources, doi : 10.1016 / j . jpowsour . 2007.10.046. 19 out one-dimensional approaches, such as equivalent circuit (e.g. [1]) or 1-D transmission- line [2] modeling, which inherently assume no variation in the plane perpendicular to the current flow. The present model describes the cell behaviour by solving the material and charge balance within the porous electrodes.  Hydrogen and water are the only species considered at the anode, while oxygen diffuses through nitrogen at constant partial pressure at the cathode.  The electrochemical reactions modeled at each electrode are, then: ( ) ( ) ( ) −− −− →+ +→+ 2 2 2 2 2 2 2 1 2 OecathodeO eanodeOHOanodeH  ( 1 ) Hydrogen flowing through anode pores combines with an oxide ion coming from the ionically conductive phase, forming water and releasing two electrons.  Oxygen gas at the cathode is reduced to oxide ions when the molecule accepts the electrons coming from the external circuit, and the oxide ion fills a vacancy within the ionic conductor, and then travels towards the anode through the electrolyte.  Within the electrodes, two electrical processes take place in parallel, and with no mutual interaction (this statement implies that the double-layer capacitance is assumed independent of the local overpotential.  Chapter Six of this thesis explores the impact of this and other assumptions on the reported results): 1. A charge transfer current is established between the ionic and electronic phases.  This current is known as Faradaic current, and its magnitude depends on the local overpotential.  Ions and electrons migrate according to the generalized Ohm’s Law, i.e. at a rate proportional to the gradient in ionic and electronic potential, respectively. 2. A non-Faradaic current flows between the two phases as a result of the charge and discharge of the interfacial double layer.  The magnitude of this current depends on frequency. The electrochemical and diffusional processes are linked via the concentration dependence of the reaction rates.  The nature of the rate-limiting step(s) in electrode reactions has been the subject of numerous research studies, and consensus in this regard has yet to be achieved.  In the context of this model, it is sufficient to describe this 20 mechanism as a polarization- and concentration-dependent process whose result is the electrochemical (as opposed to direct) combination of hydrogen and oxygen.  At least one step of the overall process at each electrode involves a charge transfer between phases. Later chapters in the thesis will present a discussion of the implications of these assumptions.   Table 2-1 summarizes the fundamental and derived variables solved throughout the SOFC, the corresponding domain, and the coupling mechanisms among them.  Ionic and electronic potentials are the variables solved for in the charge balance equations.  These potentials determine the local overpotential within each electrode, which determines the local current density via the Butler-Volmer relationship. Table 2-1.  Fundamental and derived variables solved over the model domains Fundamental Variable link Derived Variable Domain ionic potential Ohm's law ionic current density anode, cathode, electrolyte electronic potential Ohm's law electronic current density anode, cathode, interconnect hydrogen concentration mass conservation water concentration anode oxygen concentration cathode charge conservation concentration dependence of electrochemical reaction rates Coupling  Internal reforming is not considered, as well as the presence of any non-reacting species other than nitrogen at the cathode.  This assumption justifies the use of binary diffusion as the mass transfer process that describes the concentration profiles within the electrodes. The model presented in this thesis is restricted to two dimensions, with two different modeled configurations: a. A small circular button cell, representative of typical laboratory-scale cell geometry (Figure 2-1), resulting in an axisymmetric model. b. A small rectangular cell, including a ribbed-interconnect / current collector, which constitutes a representative unit cell in a small planar SOFC stack in co- or counter-flow configurations (Figure 2-2). The two dimensions represent the through-thickness direction (the main current path), and a transversal direction.  The latter represents the radial dimension in the axisymmetric case, or an in-plane direction perpendicular to the interconnect reactant flow channels in the rectangular case. 21 L cathode electrolyte anode an o t ca tt el yt r z (axisymmetry)  Figure 2-1.  Modeled geometry for the button-cell configuration.  Variables anot, elyt, and catt are the anode, electrolyte, and cathode thicknesses, and they vary according to the supporting configuration.  Values for typical simulations are L = 10mm, anot (ESC) = 40 microns, anot (ASC) = 1mm, catt = 40 microns, elyt (ESC) = 150 microns, elyt (ASC) = 10 microns, where ASC and ESC stand for anode-supported cells and electrolyte- supported cells, respectively. The intact cell behaviour is adequately described using a single half-rib / half- channel repeating unit (inset in Figure 2-2), at substantially less computational cost compared to the geometry in Figure 2-2.  Using symmetry boundary conditions along the lateral edges, this geometry is representative of a cell consisting of the side-by-side concatenation of the repeating unit.  However, this configuration cannot be used to describe localized, non-uniform degradation, as will become clear in the next chapter.  22 cathode electrolyte rib cathode current collector anode current collector channel anode rib channel distance (mm) di st an ce   (m m ) 0 5 10 15 20 2 2 2 2 r1 r2 r3 r4 r5 r6ch1 ch2 ch3 ch4 ch5 rj : rib j chj : channel j (interconnect) (interconnect) half-rib / half-channel geometry  Figure 2-2.  The rectangular cell geometry includes the ribbed interconnect plates.  This two-dimensional approximation is valid for co- and counter-flow configurations.  The system is assumed to be isothermal.  Previous modeling work has revealed that the main temperature variation in a planar SOFC occurs in the in-plane direction [3,4].  Temperature gradients are the result of the progressive variation in reaction rate, and consequently heat generation, caused by reactant consumption.  The much smaller through-thickness dimension is expected to present far less temperature variation, especially if internal reforming is not considered.  Ackmann et al calculated a maximum difference of 1.5ºC for a ribbed geometry, considering methane reforming [4].  The main temperature variation is expected to occur along the channel direction.  The impact of this three-dimensional effect on the results presented in this work will be addressed in Chapter Six.  For a two-dimensional model, the addition of energy balance is not 23 expected to improve the predictive capabilities due to the relatively uniform in-plane temperature.  Energy balance will be needed if the present model is expanded to three dimensions.  The simulation of the impedance of an SOFC requires calculating the response of the system to a small sinusoidal perturbation.  If this perturbation consists of a small sinusoidal voltage, then the response will be a small sinusoidal current of the same frequency, generally shifted in phase due to the capacitive behaviour of the electrochemical interfaces within the cell.  The small amplitude of the perturbation is a requirement for the linearity in the response, i.e. the absence of induced harmonics.  All the variables detailed in Table 2-1 will have a steady state and an oscillatory component, as will be described later in this chapter. As far as the author knows, this is the first report of a two-dimensional model of the impedance behaviour of a working SOFC, and also the first one to include the interconnect plates in the geometry.  Up to the present, SOFC impedance modeling work has aimed at elucidating the influence on the impedance spectrum of the many factors that determine the cell behaviour, or at finding optimal parameters characterizing the system, but no attempt has been made to develop a numerical model that uses the impedance behaviour as a diagnostic tool. 2.3 Mathematical Formulation Each cell component is treated as a continuum with quantities that obey material and charge transport laws.  The general equation for the transport of species x is [5] xx QNt x +⋅∇−= ∂ ∂  ( 2 ) This fundamental equation states that the time variation in the amount of species x equals the negative divergence of its flux, Nx, plus the rate of production or consumption of x, Qx, within an elemental volume.  This general equation takes the form of mass and charge conservation equations to solve for electric potential and concentrations throughout the cell components. For example, if x represents ionic or electronic charge per unit volume (Coul/m3), Nx will represent a current density (A/m2), and Qx will take the form of a Faradaic current 24 source or sink (A/m3).  In this case, the fundamental variable to be solved is the electric potential, and the accumulation term on the left hand side will describe the charge of an interfacial double layer, given by a time change in potential.  The current density is proportional to the gradient in electric potential using Ohm’s law, using an effective conductivity as the proportionality constant. If x represents concentration of gaseous species within the electrode pores (mol/m3), Nx will become a diffusive mass flux (convection within small pores can be neglected), and Qx will take the form of a Faradaic production or consumption of gas species.  Species concentration is the fundamental variable to solve; the accumulation term indicates the time variation of concentration within an elemental volume.  The diffusive mass flux is, in this case, proportional to the concentration gradient with the effective diffusivity as the proportionality constant. A core feature of the present model is the partial decoupling of the steady state and oscillatory component of all the variables solved for.  This mathematical approach results in a convenient and efficient computational formulation, useful for systems of the size presented here, on the order of 30,000 degrees of freedom.  As will be demonstrated in Section 2.3.2, the oscillatory variables can be calculated separately from the steady state variables, once the steady state variables are known.  It is a requirement to start by solving for the steady state equations of the system, and solve the oscillatory equations secondly.  For simplicity, the next section first presents the steady state formulation, and the oscillatory equations immediately after. 2.3.1 Steady State Equations 2.3.1.1 Steady-State Charge Balance The general form of the ionic or electronic charge balance for an elemental volume within a porous electrode is [6, 7] )(    ηη Fdl iSitCS +⋅−∇=∂ ∂ −  ( 3 ) which states that the time variation of charge density equals the negative divergence of the current density, and the production or consumption of current density via electrochemical reactions.  The time change in charge density results from the charging 25 and discharging of the interfacial double layer.  This phenomenon can be visualized considering the porous electrode as a two-path transmission line, in which the paths are connected via parallel RC elements.  These elements are commonly known as leaky capacitors.  The capacitive element is the only source of time dependence, and is responsible for the phase shift between an input signal and the system’s response.  In this expression, i (A m-2) is the ionic or electronic current density, S (m-1) is the electroactive surface area per unit volume of the porous electrode, iF (A m-2) is the Faradaic current density, established between electronically- and ionically-conductive phases, Cdl (F m-2) is the double layer capacitance of the (distributed) interface between electronic and ionic conductors, and REFIONELE Φ−Φ−Φ=η  is the local overpotential (V), with respect to a reference electrode of the same kind as the one under study.  This formulation for porous electrodes was first described by Newman and Tobias [8], and states that the local overpotential equals the potential difference between the electronic and ionic phases.  In a full-cell model, one of the reference potentials can be arbitrarily set to zero (the anode reference potential, in this work), and the other reference potential must be equal to the open-circuit potential of the system.  The term on the l.h.s. of equation ( 3 ) is null for DC operation because the time derivative vanishes at steady state.  Combined with Ohm’s law, which relates current density and potential gradient as Φ∇−=  ki , we obtain an expression as a function of the potential only: )(  2 ηFiSk =Φ∇−  ( 4 ) where it is assumed that the effective conductivity k (S m-1) is independent of position. Here Φ (V) denotes ionic or electronic potential, the fundamental unknowns for which the equations are solved.  Applying Ohm’s law to the solid phases in the electrodes assumes that the transport of ionic and electronic species occurs mainly by migration. This is a good assumption for electronic transport and solid-state ionic transport [9,10]. Another fundamental principle involved in the calculation is the conservation of charge, which states that the divergence of the electronic ( ELEi ) and ionic ( IONi ) current densities must add to zero, since there is a continuous conversion of current from electronic to ionic, from the current collector toward the electrolyte.  Equation ( 5 ) 26 indicates that the source terms for the ionic and electronic species are equal in magnitude and opposite in sign.  Equivalently, a source of ions is a sink of electrons. 0=⋅∇+⋅∇ ELEION ii  ( 5 ) The Faradaic current density constitutes the current source term of equation ( 3 ). A function of the local overpotential, its magnitude reflects the electrochemical activity at every point in the electrode.  In this work, we assume a Butler-Volmer relationship between overpotential and Faradaic current density, modified to include mass transport effects.  The source term is positive or negative, indicating production or consumption of the corresponding current.  For example, the cathode is a source of oxide ions, and consequently a sink of ionic current (according to the convention of positive current being the flow of positively charged species).  The source term for ionic current will be negative for cathodic polarization.  Therefore, the Faradaic ionic current generated in the DC cathodic reaction is given by [11, 12]                 −−         = CAT g CATCAT O O CAT g ANOCAT CATCATIONF TR F c c TR F ii ηαηαη ,0 , ,0,, expexp)( 2 2  ( 6 ) where i0,CAT is the cathodic exchange current density (A m-2), αij is the charge transfer coefficient for the reaction with the first sub-index indicating the electrode and the second sub-index indicating the anodic or cathodic direction, F is the Faraday's constant (A s mol-1), Rg is the universal gas constant (J mol-1 K-1), T is the absolute temperature (K), cO2 (mol m-3) is the local oxygen concentration, and c0O2 (mol m-3) is the reference oxygen concentration, at which i0,CAT is measured, and it is equal to the channel or bulk concentration in this model.  The second term on the r.h.s. of ( 6 ) is larger than the first term for cathodic polarization, resulting in a negative source term.  The local reactant concentrations are unknown, and they must be solved for simultaneously using the mass transport equations explained in the next section.  The temperature dependence of the exchange current density is not stated explicitly because the model is isothermal; i.e. i0,CAT is a constant for the single operating temperature considered in this model. Expression ( 6 ) incorporates both activation and concentration overpotentials [13].  The cathodic (second on the r.h.s.) term becomes very small at sites of reactant depletion. The concentration ratio of products (oxide ions) is approximated as one, since the oxide 27 ion concentration is largely independent of the atmosphere for YSZ, because its ionic conductivity is an extrinsic property.  Combining expression ( 6 ) with expression ( 4 ), we obtain the DC ionic potential equation for the cathode.                 −−         =Φ∇− CAT g CATCAT O O CAT g ANOCAT CATCATIONCATION TR F c c TR F iSk ηαηα ,0 , ,0 2 , expexp 2 2  ( 7 ) The product CATCAT iS ,0 constitutes a volumetric exchange current.  The two parameters CATS  and CATi ,0  are, in the context of this model, inseparable, and their product reflects the microstructure dependence of the electrode kinetics.  This point is revisited in the Model Parameters section of this chapter.  An analogous expression applies to the electronic current, using the appropriate electronic conductivity parameters, and equation ( 5 ) to calculate the source term:                 −−         −=Φ∇− CAT g CATCAT O O CAT g ANOCAT CATCATELECATELE TR F c c TR F iSk ηαηα ,0 , ,0 2 , expexp 2 2  ( 8 ) The anode is a sink of O2- ions, thus, a source of ionic current.  The DC ionic expression results:                 −−         =Φ∇− ANO g CATANO OH OH ANO g ANOANO H H ANOANOIONANOION TR F c c TR F c c iSk ηαηα ,0 , 0,0 2 , expexp 2 2 2 2  ( 9 ) with a positive source term for anodic polarization.  Finally, the electronic current equation is:                 −−         −=Φ∇− ANO g CATANO OH OH ANO g ANOANO H H ANOANOELEANOELE TR F c c TR F c c iSk ηαηα ,0 , 0,0 2 , expexp 2 2 2 2  ( 10 ) In the purely ionically conductive electrolyte, no reaction takes place, and therefore the source term is absent: 02 =Φ∇ ION  ( 11 ) Equivalently, the expression for the interconnect domain becomes 02 =Φ∇ ELE  ( 12 )  Prescribed potential boundary conditions for the steady state equations are only required for the electronic potential, along the current collecting lines at the top of the cathode interconnect and at the bottom of the anode interconnect, as seen in Figure 2-1: 28 0 collectorcurrent  anode collectorcurrent  cathode =Φ =Φ ELE CELLELE V  ( 13 ), where VCELL is the operating voltage of the cell.  2.3.1.2 Steady-State Mass Balance The driving force for reactant transport within a porous electrode is the concentration gradient between the electrode/channel and the electrode/electrolyte boundaries.  The change in reactant or product concentration at a point within an electrode is proportional to the divergence of the negative flux of that species, and to its electrochemical production or consumption. nF iSN t c F k k )( η+⋅−∇= ∂ ∂  ( 14 ) Here, ck is the concentration of species k (mol m-3), and Nk is the species’ molar flux (mol s-1m-2), n is the number of electrons participating in the relevant electrochemical production or consumption reaction per molecule of reactant, and the other terms are as defined above.  Equation ( 14 ) indicates that accumulation of species within a volume element can be the result of species flux and of electrochemical production or consumption.  For binary gas mixtures, Fick’s law of diffusion gives an adequate representation of the diffusive flux, under the assumption that Knudsen diffusion effects are small for pore sizes of about 1 micron: TOTkkkk NxcDN +∇−=  ( 15 ) where Dk is the effective diffusivity of the gas in the binary mixture (m2 s-1), xk is the molar fraction of species k, and NTOT is the total flux for all species. At the anode, hydrogen diffuses toward the electrolyte, while gaseous water diffuses away from it.  This combination obeys the laws of binary equimolar diffusion, since the sum of molar hydrogen and water fluxes is null at every point because one hydrogen molecule is transformed into one water molecule: 0 22 =+= OHHTOT NNN  ( 16 ) 2222 HOHHH cDN ∇−= −  ( 17 ) 29 The following steady state mass balance equation for the hydrogen at the anode results, assuming that the effective diffusion coefficient is spatially invariant, i.e., that the porosity and microstructure are invariant throughout the electrode, and independent of gas composition within the range of interest (Table II in [14]):                 −−         −=∇− − ANO g CATANO OH OH ANO g ANOANO H HANOANO HOHH TR F c c TR F c c F iS cD ηαηα ,0 , 0 ,02 expexp 2 2 2 2 2 222  ( 18 ) The water concentration is calculated using the relation: 1 22 =+ OHH xx  ( 19 ) with xk indicating molar fraction of species. At the cathode, with dry air as the oxidant, the situation is slightly different, since only oxygen takes part in the reaction, while the nitrogen remains unreacted.  The net total flux at a reaction site is equal to the oxygen flux, since the net flux of nitrogen at any specific reaction site is null: 222222 OOONOO NxcDN +∇−= −  ( 20 ) Simplifying and substituting for the mole fraction xO2 from the ideal gas law, 2 222 2 222 2 11 O g ONO O ONO O c p TR cD x cD N − ∇− = − ∇− = −−  ( 21 ) where p is the total pressure of the system, often very close to atmospheric pressure.  The resulting mass transport equation, again assuming that the diffusivity is spatially invariant, is:                 −−         =             − ∇ ⋅∇− − CAT g CATCAT O O CAT g ANOCATCATCAT O g O NO TR F c c TR F F iS c p TR c D η α η α , 0 ,,0 expexp 4 1 2 2 2 2 22  ( 22 ) The boundary conditions for hydrogen and oxygen concentrations are prescribed concentrations at the boundaries between the electrodes and channels, and no-flux everywhere else: 22 0 HH cc =   at the channel / anode boundary 0 2 =⋅∇ ncH   everywhere else on the anode side 30 22 0 OO cc =   at the channel / cathode boundary 0 2 =⋅∇ ncO   everywhere else on the cathode side Here n is the unit vector normal to each boundary.  The DC mass transport equations are solved within the porous electrodes, simultaneously with the DC charge transport equations.  The equations are nonlinear and coupled.  The nonlinearity arises from the exponential dependence of current density on overpotential.  The coupling is a result of the concentration dependence of the Butler-Volmer source terms.  Comsol Multiphysics (Comsol, Inc.) general purpose PDE solver was used to solve the equations using finite elements.  The choice of initial guess is of importance to achieve convergence.  The default value for the initial guess is zero for the Comsol package, but a better value is the Dirichlet boundary condition value.  The Analytical Jacobian method used by Comsol makes it possible to input the term inside the divergence in equation ( 22 ) directly. 2.3.2 AC Equations 2.3.2.1 AC Charge Balance One premise in impedance spectroscopy analysis is that the system responds linearly to a small perturbation.  If the imposed perturbation is small, all system variables can be decomposed into a steady state, or DC, component, and an oscillatory component. tj OHOHOH tj OOO tj HHH tj IIONION tj ELEELEELE eccc eccc eccc e e ω ω ω ω ω 222 222 222 ~ ~ ~  ~  ~ ON += += += Φ+Φ=Φ Φ+Φ=Φ  ( 23 ) Here e is the base of the natural logarithm, ω is the angular frequency of the applied perturbation, and j is the imaginary unit.  Appendix II provides a demonstration of this statement for a general, nonlinear PDE.  In ( 23 ), the bar indicates the steady state part, and the tilde indicates the oscillatory part.  The oscillatory component is a complex variable of small magnitude that represents the phase shift and the excursion from the 31 equilibrium point of the corresponding variable.  In practical terms, “small” means “lower than the system thermal voltage”, a temperature dependent quantity given by: KmV F TR V gTH 1123@97≈=  ( 24 ) In practice, 20mV is an adequate compromise between linearity and acceptable signal-to-noise ratio.  The AC charge balance at the anode results from placing the anode- relevant expressions from ( 23 ) in ( 2 ).  The equation for ionic charge is: ( ) ( ) ( )( ) ( )( )         +− + −+ + =Φ+Φ∇−+ ∂ ∂ − tj ac OH tj OHOHtj aa H tj HH ANOANO tj IIONANOION tj ANOdlANO ef c ecc ef c ecc iS eke t CS ω ω ω ω ωω ηηηη ηη ~exp ~ ~exp ~  ~ ~ 2 22 2 22 00,0 ON 2 ,,  ( 25 ) Here, for notation simplicity, the following replacements were made: TR Ff RT Ff g CATANO ac ANOANO aa IONELE REFIONELE , , ~~ ~ α α η η = = Φ−Φ= Φ−Φ−Φ=  ( 26 ) Equation ( 25 ) contains terms that can be expressed in simpler form, on account of the small amplitude of the perturbation.  Linear order Taylor expansion of the exponential functions yields: ( )( ) ( )( ) ( )( ) ( )( )tjacactjac tj aaaa tj aa effef effef ωω ωω ηηηη ηηηη ~1exp~exp ~1exp~exp −−≈+− +≈+  ( 27 ) Substituting simplified expressions ( 27 ) in ( 25 ) and expanding the time derivative, 32 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                           −+ +−− +−+ +−− =Φ∇−Φ∇−− ηηηη ηη ηηηη ηη ηω ωωωω ωω ωω ωω ~exp ~ ~exp ~ exp ~ exp ~ ~exp~exp expexp ~ ~ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 00 00 00 00 ,0 ON 2 , 2 ,, ac tj ac tj OH OH aa tj aa tj H H tj ac OH OHtj aa H H tj acac OH OHtj aaaa H H ac OH OH aa H H ANOANO I tj ANOIONIONANOION tj ANOdlANO fefe c cfefe c c ef c c ef c c eff c c eff c c f c cf c c iS ekkejCS  ( 28 ) All terms in equation ( 28 ) that do not have the pulsation tje ω as a factor add up to zero because they are identical to those of the steady state equation, whose solution we already have (equation ( 9 )).  Furthermore, the last two terms in brackets are of second order, since they contain the product of two oscillatory terms, η~~ 2H c  and η~~ 2OHc , each of small magnitude, and can therefore be neglected with respect to the other terms.  These two simplifications allow the removal of the tje ω term from all the remaining terms, yielding a linear, time independent PDE with respect to the oscillatory potential and concentration amplitudes [15, 16]: ( ) ( ) ( ) ( )            −− +−+ =Φ∇−− ηη ηηηη ηω ac OH OH aa H H acac OH OH aaaa H H ANOANO IANOIONANOdlANO f c cf c c ff c cff c c iS kjCS exp ~ exp ~ ~exp~exp ~ ~ 2 2 2 2 2 2 2 2 00 00 ,0 ON 2 ,,  ( 29 ) The first two terms in the brackets correspond to the Faradaic local current that appears as a consequence of the imposed AC overpotential η~ .  The last two terms in the brackets indicate the AC concentration contribution to the source term.  The first term on the l.h.s. constitutes the non-Faradaic contribution to the local current density, assumed to be additive with the Faradaic component, and without mutual interaction [16]. This formulation is numerically very convenient, since the DC component of the solution is decoupled from the AC component of the solution.  On account of the small size of the AC perturbation, it does not have any influence on the DC solution.  This fact allows the solution of the AC equations based on the previously and independently 33 obtained DC solution.  The steady state equations are then solved using a nonlinear iterative routine yielding values for η , and the steady state solution is then used as a base on which the AC solution is superimposed.  No iterative method is required for the latter, since the equations are linear in the oscillatory unknowns.  The electronic potential equation is obtained similarly, as explained in the previous section: ( ) ( ) ( ) ( )            −− +−+ − =Φ∇−− ηη ηηηη ηω ac OH OH aa H H acac OH OH aaaa H H ANOANO ELEANOELEANOdlANO f c cf c c ff c cff c c iS kjCS exp ~ exp ~ ~exp~exp ~ ~ 2 2 2 2 2 2 2 2 00 00 ,0 2 ,,  ( 30 ) The derivation of the cathode ionic charge balance equation is omitted, but it is obtained in the same way, resulting in: ( ) ( ) ( )      −−−+ =Φ∇−− ηηηηη ηω cc O O cccc O O cacaCATCAT ICATIONCATdlCAT f c cff c cffiS kjCS exp ~ ~exp~exp ~ ~ 2 2 2 2 00,0 ON 2 ,,  ( 31 ) Finally, a similar expression corresponds to the electronic charge balance at the cathode: ( ) ( ) ( )      −−−+− =Φ∇−− ηηηηη ηω cc O O cccc O O cacaCATCAT ELECATELECATdlCAT f c cff c cffiS kjCS exp ~ ~exp~exp ~ ~ 2 2 2 2 00,0 2 ,,  ( 32 ) Prescribed potential boundary conditions for the AC equations are only required for the electronic potential: 0collectorcurrent  anode 0collectorcurrent  cathode ~~ ~~ Φ∆+=Φ Φ∆−=Φ ELE ELE  ( 33 ) where 0 ~Φ∆  is the perturbation amplitude.  Neumann boundary conditions apply everywhere else for the electronic potential, and everywhere for the ionic potentials. 2.3.2.2 AC Mass Balance In order to solve for the oscillatory concentrations we need another set of equations to be solved simultaneously with equations ( 30 ) to ( 33 ).   The decomposition 34 shown in eq. ( 23 ) can be applied to the mass balance equations, in a similar way as shown for the charge balance.  The equation describing the distribution of hydrogen concentration at the anode can be calculated starting with the general form of the mass balance within the pores at the anode: ( ) ( ) ( ) ( )( ) ( ) ( )( )      −− −∇= ∂ ∂ − txf c txc txf c txc F iS txcD t txc ac OH OH aa H HANOANO HOHH H ,exp , ,exp , 2 , , 2 2 2 2 222 2 00 ,0 2 ηη  ( 34 ) Here, the concentration and the overpotential depend on time t and on the spatial coordinates ( )yxx ,= .  Replacing them using the expressions ( 23 ): ( ) ( ) ( )( ) ( )( )         +− + −+ + −+∇= ∂ +∂ − tj ac OH tj OHOHtj aa H tj HHANOANO tj HHOHH tj HH ef c ecc ef c ecc F iS eccD t ecc ω ω ω ω ω ω ηηηη ~exp ~ ~exp ~ 2 ~ ~ 2 22 2 22 2222 22 00 ,0 2  ( 35 ) After a simplification equivalent to that shown for the charge balance, equation ( 35 ) results: ( ) ( ) ( ) ( )            −− +−+ −∇= − ηη ηηηη ω ac OH OH aa H H acac OH OH aaaa H H ANOANO HOHHH f c cf c c ff c cff c c F iS cDcj exp ~ exp ~ ~exp~exp 2 ~~ 2 2 2 2 2 2 2 2 2222 00 00 ,0 2  ( 36 ) The derivation of the AC equation for oxygen concentration on the cathode side is more complicated, because of the presence of the extra term in the expression for the oxygen flux (eq. ( 20 ), second term on the r.h.s.).  Revisiting eq. ( 21 ), 2 222 2 1 O g ONO O c p TR cD N − ∇− = −  ( 21 ) it is apparent that we cannot directly incorporate the AC terms as before, because of the presence of the concentration in the denominator.  However, it is possible to expand the 35 2 1 1 O g c p TR −  term around some convenient value for 2O g c p TR .  If the expansion is performed around zero, the following expression holds if we neglect second order terms and higher:       +∇−≈ − ∇− − − 2222 2 222 1 1 O g ONO O g ONO c p TR cD c p TR cD  ( 37 ) as long as 1 2 <O g c p TR .  Usually, the cathode gas is air, and for air this inequality always holds, since the maximum value it can take is 21.0 2 =O g c p TR .  The point around which the expansion is centred does not have to be zero, but it can also be, for example, some average oxygen concentration along the cathode channel.  If this is the case, the linearization of equation ( 21 ) is, for a given intermediate concentration 2Oc∗ :               −      −+      −∇−≈ − ∇− ∗ − ∗ − ∗ − − 2222222 2 222 21 11 1 O g O g O g O g ONO O g ONO c p TR c p TR c p TR c p TR cD c p TR cD  ( 38 ) A sensitivity analysis of the results revealed that the choice of intermediate concentration has a negligible effect on the solution in the partial pressure range from zero to 0.21 atm.  Our choice was to expand the term around atmospheric oxygen concentration 21.02 =∗O g c p TR , resulting in:       +∇−≈ − ∇− − − 2222 2 222 21 1 O g ONO O g ONO c p TR AAcD c p TR cD  ( 39 ) where 602.1,929.0 21 == AA .  Now, substituting tj OOO eccc ω 222 ~+= as before: ( ) ( )      +++∇−=      +∇− −− tj OO gtj OONOO g ONO eccp TR AAeccDc p TR AAcD ωω 2222222222 ~~ 2121  ( 40 ) 36 Expanding this expression, reorganizing the time-dependent and the time-independent terms, and neglecting second order terms, the term on the l.h.s. in equation ( 37 ) becomes:       ∇+∇+∇−∇      +− −− 22222222222 ~~~ 22121 OO g OO g ONO tj OO g NO ccp TR Acc p TR AcADecc p TR AAD ω  ( 41 ) The first term in ( 41 ) is the first-order approximation of ( 21 ). When we combine equation ( 41 ) with the other terms that constitute the mass balance for oxygen, the first term in equation ( 41 ) vanishes because it is part of the steady state solution, similarly to the case of equation ( 35 ) above.  The remaining AC terms form the AC mass balance for oxygen in the cathode. ( ) ( ) ( ) ( )      −−−+ +      ∇+∇+∇⋅∇= − ηηηηη ω cc O O cccc O O caca CATCAT OOOO g ONOO f c cff c cff F iS cccc p TR AcADcj exp ~ ~exp~exp 4 ~~~~ 2 2 2 2 22222222 00 ,0 21  ( 42 ) Comparing equations ( 42 ) and ( 36 ), it can be concluded that the second term inside the first set of brackets of equation ( 42 ) represents the influence of the unreacting nitrogen in the oxygen flux.  Although oxygen diffusion through stagnant nitrogen on porous SOFC cathodes has been described for steady state conditions [17], no previous reference was found about its influence on the AC equations.  The boundary conditions used for the AC concentrations are: 0~ 2 =Hc  at the channel / anode boundary 0~ 2 =⋅∇ ncH  everywhere else 0~ 2 =Oc  at the channel / cathode boundary 0~ 2 =⋅∇ ncO  everywhere else 2.4 Calculation of the Impedance Upon calculation of the four unknown potentials, the current densities (in A m-2) are calculated as the negative gradient of the potential times the corresponding conductivity: 37 ELEELEELE IONIONION ELEELEELE IONIONION ki ki ki ki Φ∇−= Φ∇−= Φ∇−= Φ∇−= ~~ ~~  ( 43 ) The impedance is the ratio between the applied voltage perturbation magnitude and the AC component of the current density computed as the average AC current density component at either of the current collectors. ( ) ( ) collectorcurrentELE i Z  0  ~ ~2 ω ω Φ∆ =  ( 44 ) The AC current density is a complex number, and in general it has a phase shift with respect to the applied potential.  As a consequence, the impedance is also complex- valued.  In practice, the perturbation size 0 ~Φ∆ is the result of a compromise between response linearity and adequate signal to noise ratio.  In the context of this model, this value is transparent to the results, since the impedance is the result of the solution of the already linearized equations, and the perturbation is only a calculation variable. Traditionally, the impedance is presented in Cartesian plots of its negative imaginary part as a function of its real part (Nyquist plot), or of its magnitude and phase shift as a function of the logarithm of the frequency (Bode plots).  The Nyquist plot of the impedance spectrum of SOFCs usually exhibits one or more flattened semicircles and/or straight lines, due to the capacitive behaviour of the electrochemical processes involved. The relative position of these semicircles, their size, the apex frequencies, and the values of their intercepts with the real axis yield important information on the different processes within the cell. Appendix I presents the Matlab/Comsol code written and used to solve the above described equations. 2.5 Model Parameters An adequate choice of input parameters is, undoubtedly, of tremendous importance in the development of numerical models.  A nonlinear, convoluted electrochemical system such as a full SOFC is a good example of this statement.  Many important system parameters are very difficult, or impossible, to measure independently. 38 Exchange current densities, transfer coefficients, symmetry factors, tortuosity, and interfacial double-layer capacitance, are examples of these quantities.  In addition, the porous nature of the electrodes in a fuel cell further complicates the analysis, since many of their properties are not defined by a value, but by a distribution.  Finally, the experimental difficulties inherent to solid-state electrochemical systems make validation of modeling results extremely difficult.  Among these difficulties, contact resistance, system inductance, reference electrode placement, and precursor powder contamination, are examples of external sources of error that hinder accuracy and repeatability.  The influence of inaccuracies in the input parameters on the present model results is discussed in Chapter Six.  The present chapter details the model parameters used in typical simulations, and presents a brief discussion on those of controversial nature.  Table 2-2 shows the general operating conditions and geometries for typical simulations. Table 2-2.  Standard operating conditions and geometry for typical simulations. Property Symbol Value Units Temperature T 1123 K Operating point VCELL 0.7 V Anode channel hydrogen partial pressure pH20 0.97 atm Anode channel water partial pressure pH2O0 0.03 atm Cathode channel oxygen partial pressure pO20 0.21 atm Open circuit potential  OCV 1.13 V Electrolyte thickness (ESC) tELY 150 micron Electrolyte thickness (ASC) tELY 10 micron Anode thickness (ASC) tANO 1000 micron Anode thickness (ESC) tANO 40 micron Cathode thickness (all cases) tCAT 40 micron Cell radius (button cell) L 10 mm Rib width (rectangular cell)  2 mm Channel width (rectangular cell)  2 mm  Model parameters can be classified into three groups, according to the physical process that they characterize: 39 1. Electrical conductivity 2. Gas species diffusion 3. Electrochemistry 2.5.1 Electrical conductivity Power generation in fuel cells involves the conversion of electronic current (going through the external circuit) to ionic current (going through the electrolyte).  Bulk ionic and electronic conductivities are well known for typical materials, in particular those used in this model.  Table 2-3 shows bulk conductivities and volume fractions of the materials used for the electrodes, electrolyte, and interconnect in this model. Table 2-3.  Electrical conductivity and related properties Property Symbol Value Units Ref Electrolyte ionic conductivity @ 1123 K kION,ELY 4 S/m [18, 19] Ni bulk electronic conductivity @ 1123 K kbELE,ANO 1.90·106 S/m [20, 21] LSM bulk electronic conductivity @ 1123 K kbELE,CAT 2.66·104 S/m [19] Anode porosity volume fraction εANO 0.4  this work Cathode porosity volume fraction εCAT 0.4  this work Anode electronically conductive phase volume fraction† xELE,ANO 0.4  this work Cathode electronically conductive phase volume fraction† xELE,CAT 0.5  this work Anode ionically conductive phase volume fraction† xION,ANO 0.6  this work Cathode ionically conductive phase volume fraction† xION,CAT 0.5  this work Interconnect conductivity @ 1123 K kELE,IC 8.0·105 S/m [22] †  Based on total solid phase The calculation of conductivity in porous electrodes requires a correction for the presence of non-conducting phases.  Table 2-4 shows examples of such correction, taken from the SOFC literature, with the exception of the classical Bruggeman correction.  In all other cases, the author(s) make reference to a previous work.  The preferred method used in this thesis is [26], approximating the probability of percolation with the volume fraction of the corresponding phase [23]. The large electronic conductivity of the electrodes and the interconnect makes them practically equipotential.  The ionic conductivity of the electrodes is one of the 40 factors that influence the extent of penetration of the electrochemical reaction from the electrolyte interface into the electrode volume. Table 2-4.  Literature review of correction factors used for the calculation of the effective conductivity of phase k Correction factor value Ref ANO CAT  Source Correction factor expression ELE ION ELE ION Bruggeman  † ( ) bulkkNCFeffk x σεσ 5.11 −−=     0.12 0.22 0.16 0.16 [24] Deseure et al  ‡ ( ) bulk kk eff k x σ ε σ 6.1 1− =  0.15 0.23 0.19 0.19 [7] Costamagna et al  †† ( ) ( ) bulk k c ck eff k n nn σ γ σ − −= 1 2  0.01 0.06 0.03 0.03 [25] Kenney et al  ‡‡ ( ) ( ) bulkkkkeffk xPx σεσ ⋅⋅−= 1  0.10 0.22 0.15 0.15 [26] D-Y Jeon et al ( ) ( )( ) bulkkkkeffk xPx σεσ 5.11 ⋅⋅−=  0.04 0.21 0.13 0.13 [27] †   xNCF indicates fraction of non-conductive solid phase ‡   xk indicates fraction of the phase of interest ††  nk and nc are the number fraction of particles of the phase of interest and at the percolation threshold, and γ is an adjustable parameter = 0.5, aimed at accounting for the presence of inter-particle necks ‡‡   P(xk) indicates the percolation probability of the phase of interest  2.5.2 Gas species diffusion  Reactants and products diffuse throughout the pores in the electrodes, with a flux proportional to the concentration gradient.  Effective diffusivity in porous media is smaller than bulk diffusivity, and it can be calculated using a correction analogous to that of conductivity [5].  Unlike the latter, there is reasonable consensus on the type of correction to use: bulkeff DD τ ε =  ( 45 ) Where ε and τ equal to the porosity and tortuosity of the pore phase, respectively. Despite the relative agreement on the functional form of the diffusivity correction, the tortuosity value is still a source of discrepancy.  This quantity depends on the level of porosity, as well as on microstructural morphology.  Typical values for tortuosity 41 reported in the literature range from 2 to 7, sometimes reaching 10 [28].  A dedicated study [29] was recently published stating τ = 3 as a reliable estimate for SOFC electrodes based on tortuosity measurements, and that is the value adopted in this work.  As far as the bulk diffusivities are concerned, they can be calculated using the Lennard-Jones formulation [30].  Their values, and the corresponding effective values, are summarized in Table 2-5. Table 2-5.  Gas diffusion properties Property Symbol Value Units Ref Hydrogen bulk diffusivity in H2 – H2O @ 1123 K OHHbD 22 −  8.9·10-4 m2/sec [30] Oxygen bulk diffusivity in O2 – N2  @ 1123 K 22 NO bD −  2.3·10-4 m2/sec [30] Anode and cathode porosity εANO  , εCAT 0.4 - this work Anode and cathode tortuosity τANO  , τCAT 3 - [29] Hydrogen effective diffusivity in H2 – H2O @ 1123 K OHHD 22 −  1.2·10 -4  m2/sec this work Oxygen effective diffusivity in O2 – N2  @ 1123 K 22 NOD −  0.3·10 -4  m2/sec this work 2.5.3 Electrochemistry  Factors affecting the rate of the electrochemical reactions at the SOFC electrodes include: electroactive surface area, exchange current density, charge transfer coefficients, reactant concentration, overpotential, and temperature.  The AC behaviour is also characterized by the capacitance of the electrochemical double-layer existing at the interfaces.  The nature and details of the reactions taking place at a microscopic level in SOFCs remain largely unknown, and one reason for this limitation is the difficulty to measure the electrochemical parameters mentioned above.  From a modeling viewpoint, it is necessary to make estimates based on fitting model results to experimental data.  A problem inherent to electrode-level modeling of 3D electrodes is the indirect relationship between what can be measured and what has to be modeled.  When measuring a polarization curve or an impedance spectrum, the quantities that are accessible to the measurement are the current and potentials at the current collectors.  It is not possible to measure current density within the electrode volume.  Therefore, the relationship between, e.g., a measured exchange current and i0 in equations ( 6 ) - ( 10 ), is not 42 straightforward.  A similar situation occurs with the observed charge transfer coefficients, recently shown in [31] and [32].  Patterned electrode experiments are useful in trying to measure electrochemical parameters on a simple geometry [33], and provide evidence of the inverse relationship between polarization resistance and triple-phase-boundary length, reflected in this model in the electroactive surface area.  It is clear, then, that the products S i0 and S Cdl are to be considered together as volumetric exchange current and volumetric double layer capacitances, and whose values need to be adjusted to fit relevant experimental data. The procedure adopted in the present work is as follows.  Charge transfer coefficients were adopted from published results on single electrode studies at the modeled temperature.  Surface areas were estimated based on a published model that uses particle size and packing theory [25].  The exchange current density and the double layer capacitance are the only free parameters, tuned to match published apparent polarization resistance and peak frequency in impedance spectroscopy performed on single electrode experiments.  Co, Xia, and Birss [34] showed the calculation of exchange current density using impedance data in a similar way, for an LSM-YSZ composite cathode, demonstrating the validity of their approach independently of the assumed reaction mechanism.  Table 2-6 and 2-7 summarize these data. Table 2-6.  Anode and cathode experimental data used to calculate the exchange current and the double layer capacitance by fitting Property and conditions Symbol Value Units Ref Cathode polarization resistance @ OCV, 3-electrode measurement, 1123 K Rp ~0.8 ohm cm 2  Fig 8a  in [35] Cathode summit frequency @ OCV, 2-electrode measurement, 1123 K fp ~0.5 Hz Fig 8a in [35] Anode polarization resistance @ OCV, 2-electrode measurement, 1123 K, 97%H2, 3% H2O Rp 0.4 ohm cm2 Fig 5.4 in [21] Anode summit frequency, 3-electrode measurement, 1123 K fp 3600 ‡  Hz Fig 4 in [36] ‡   data not available at OCV, but taken at 0.3 A/cm2 43  Table 2-7.  Electrochemical data adopted from the literature and results of the fitting Property Symbol Value Units Ref Anode charge transfer coefficient, anodic direction αAA 1.2 - [37] Anode charge transfer coefficient, cathodic direction αAC 0.8 - [37] Cathode charge transfer coefficient, anodic direction αCA 1.5 - [19, 38] Cathode charge transfer coefficient, cathodic direction αCC 0.5 - [19, 38] Anode and cathode active surface area Sano ,  Scat 106 m-1 [25] Anode volumetric exchange current density Sano i0,ano 2.5·107 † A m-3 [21] Cathode volumetric exchange current density Scat i0,cat 5·107 † A m-3  [35] Anode double layer capacitance 1123 K Cdl,ano 0.4 † F m-2 Fig 4 in [39] Cathode double layer capacitance 1123 K Cdl,cat 90 † F m-2 Fig 8a in [35] †  fitted to reproduce empirical impedance results It is important to mention that the literature shows large discrepancies in the reported values of polarization resistance and relaxation frequencies for both anode and cathode.  This statement holds even within a single piece of work for nominally identical electrodes (e.g., Fig 4 in [35], details in Table 2-8).  The multiplicity of factors that affect impedance measurements in cells with solid electrolytes, the strong dependence of electrode performance on microstructure, misalignment of electrodes, inadequate placement of the reference electrodes, and the presence of contaminants in powder precursors, are just examples of the many variables that affect the repeatability in experiments of this kind.  Under certain circumstances also explored in this thesis, however, an accurate knowledge of polarization resistance and double layer capacitance is not essential for the implementation of this model, since its results are based on the study of changes, rather than relying on absolute values.  Chapter Six of this thesis presents a more detailed assessment of the impact of these uncertainties on the results. 44  Table 2-8.  Reported values for the polarization resistance of LSM-YSZ cathodes reported by researchers at Risø National Laboratory, Roskilde, Denmark Property and conditions Value Units Ref Cathode polarization resistance @ OCV in air, 2- electrode measurement, 1123 K § ~0.8 ohm cm 2  Fig 8a in [35] Cathode polarization resistance @ OCV in air, 3- electrode measurement, 1123 K § ~0.5 ohm cm 2  Fig 10a in [35] Cathode polarization resistance @ OCV in air, 3- electrode measurement, 1123 K ~1.5 ohm cm 2  Fig 4 in [40] § nominally identical cathodes, prepared simultaneously 2.6 On the Validity of the Model Equations and Results  An essential aspect of modeling physical phenomena is experimental validation. Fully accomplishing this requirement is a tremendously difficult task for complex systems such as a working fuel cell.  A valid model should be able to reproduce experimental results for a range of different operating conditions without ambiguity. Strictly speaking, fitting model results to experimental data does not constitute validation of the model unless it can be performed for a wide range of operating conditions, and even in that case, it is not clear whether some quantities would be properly measurable for the conditions and microstructures that are characteristic of SOFC electrodes.  An example of such a quantity is the exchange current density for each electrode reaction. Very important from the performance point of view, the exchange current density is inseparably tied to an active surface area, over which the reaction is assumed to take place.  This active area is also difficult to estimate for composite porous electrode microstructural morphologies.  One way to overcome this limitation is to use patterned electrodes of very well specified geometry, as described in [41].  That work shows how electrode performance increases with triple-phase boundary length.  However, it is not obvious whether the results of the patterned electrode experiments are quantitatively transferable to the much more complicated random microstructure of SOFC electrodes. Moreover, the pattern electrode manufacturing methods (e.g. sputtering) make them prone to change their morphology during testing at high temperature, undermining their 45 fundamental advantage of having an accurately determined geometry.  The extent to which the electrochemical reaction proceeds into the catalyst phase is not known with certainty, since the bulk path contribution seems to play a non-negligible role in the overall reaction, as shown in [42] using microelectrodes. The aim of the model presented in this thesis is to find distinct changes in the impedance spectrum induced by diverse degradation mechanisms.  The validity of the results relies on the assumption that the simulated degradation modes do not alter the kinetic input parameters used to model the intact case.  This assumption is reasonable for all the degradation modes considered in this thesis.  Loss of contact between the cathode and the electrolyte, for example, is not expected to change the exchange current density of the cathodic process within the electrode volume. Appendix III presents a comparison of the DC behaviour of this model with experimentally measured polarization curves. Chapter Six explores the robustness of the model results for each degradation mode to variations in cell characteristics and operating conditions. 2.7 Conclusions  A 2-D impedance model of a working SOFC is presented, based on the simultaneous solution of the steady state and oscillatory mass and charge conservation equations at the porous electrodes, the electrolyte, and the interconnector.  The electrode- level model, a necessary alternative to equivalent-circuit models, provides detailed two- dimensional information about steady-state and oscillatory potentials, current densities, and concentrations. On account of the small size of the imposed perturbation, the time dependence of the oscillatory variables is eliminated, transforming the problem into a complex valued system of linear equations, valid in the vicinity of the steady state operating point.  The method results in a system of equations for the electronic and ionic potentials, coupled with a system of equations for the hydrogen and oxygen concentrations, all of them having a steady state and an oscillatory component.  An overall charge transfer process, whose rate depends on reactant concentration, microstructure, and overpotential, rules the 46 electrochemical conversion of reactants.  Mass transport is modeled using a Fickean formulation for binary gas mixtures. This formulation is convenient for detailed electrode-level models using finite elements.  The method is applied for the first time to an SOFC.  It is also the first two- dimensional impedance model, and the first to include the interconnect plates.  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Kendall, Eds., High-temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications, Elsevier, Oxford, 2003, 73. [19] B. Kenney and K. Karan, “Mathematical micromodel of an SOFC composite cathode”, Proceedings of Hydrogen and Fuel Cells 2004, 2004, 1-11. [20] D. Pollock, Physical Properties of Materials for Engineers, 2nd ed., CRC Press, p210 (1993) [21] R. Stübner, “Untersuchungen zu den Eigenschäften der Anode der Festoxid- Brennstoffzelle (SOFC)”, PhD thesis, p24, Technische Universitaet Dresden (2002) [22] S. Loo, “High Temperature Power Factor Measurement System for Thermoelectric Materials”, Abs. 1150, 204th Meeting, The Electrochemical Society (2003). [23] A. Ioselevich, A. Kornyshev, and W. Lehnert, “Statistical geometry of reaction space in porous cermet anodes based on ion-conducting electrolytes”, Solid State Ionics 124 221-237 (1999) [24] D. Bruggeman, Dielectric constant and conductivity of mixtures of isotropic materials, Ann. Phys. (Leipzig) 24, 636–679 (1935). [25] P. Costamagna, P. Costa, and V. Antonucci, “Micro-modelling of SOFC electrodes”, Electrochimica Acta,  Vol. 43, No. 3-4,  375-394 (1998) [26] B. Kenney and K. Karan, “Impact of nonuniform potentials in SOFC composite cathodes on the determination of electrochemical kinetic parameters”, J. Electrochemical Soc., Vol. 153, No. 6,  A1172-A1180 (2006) 49  [27] D-H Jeon,  J-H Nam, and C-J Kim, “Microstructural optimization of anode- supported SOFC by a comprehensive microscale model”, J. Electrochemical Soc., Vol. 153, No. 2,  A406-A417 (2006) [28] P. Costamagna and K. Honegger, “Modeling of Solid Oxide Heat Exchanger Integrated Stacks and Simulation at High Fuel Utilization”, J. Electrochem. Soc. 145, 3995 (1998) [29] R. Williford, L. Chick, G. Maupin, S. Simner, and J. Stevenson, “Diffusion limitations in the porous anodes of SOFCs”, J. Electrochemical Soc., 150, Vol. 8, 2003, A1067-A1072. [30] J. Welty, C. Wicks, R. Wilson, G. Rorrer, “Fundamentals of Momentum, Heat, and Mass Transfer”, 4th ed, John Wiley & Sons, (2001) pp432. [31] J. Soderberg, A. Co, A. Sirk, and V. Birss, “Impact of porous electrode properties on the electrochemical transfer coefficient”, J. Phys. Chem. B, 110, 10401-10410 (2006) [32] B. Kenney and K. Karan, “Impact of nonuniform potential in SOFC composite cathodes on the determination of electrochemical kinetic parameters”, J. Electrochemical Soc., Vol. 153, No. 6,  A1172-A1180 (2006) [33] R. Radhakrishnan, A. Virkar, and S. Singhal, “Estimation of charge-transfer resistivity of LSM Cathode on YSZ Electrolyte Using Patterned Electrodes, Journal of The Electrochemical Society, 152 (1) (2005) A210-A218 [34] A. Co, S. Xia, and V. Birss, “A kinetic study of the oxygen reeduction reaction at LaSrMnO3-YSZ composite electrodes”, J. Electrochemical Soc., Vol. 152, No. 3, A570-A576 (2005) [35] M. Jørgensen and M. Mogensen, “Impedance of Solid Oxide Fuel Cell LSM/YSZ Composite Cathodes”, J. Electrochem. Soc. Vol. 148, No. 5 (2001) A433-A442. [36] S. Primdahl, M. Mogensen, “Durability and thermal cycling of Ni/YSZ cermet anodes for solid oxide fuel cells”, J. App. Electrochem. 30 (2000) 247-257. [37] M. Gonzalez-Cuenca, “Novel anode materials for solid oxide fuel cells”, PhD thesis, Univ. Twente (2002) 50  [38] F. van Heuveln and H. Bouwmeester, “Electrode properties of LSM on YSZ. II Electrode kinetics”,  J. Electrochem. Soc. Vol. 144, (1997) 134-139. [39] S. Primdahl, M. Mogensen, “Durability and thermal cycling of Ni/YSZ cermet anodes for solid oxide fuel cells”, J. App. Electrochem. 30 (2000) 247-257. [40] M. Jørgensen and M. Mogensen, “Characterisation of composite SOFC cathodes using electrochemical impedance spectroscopy”, Electrochimica Acta 44, (1999) 4195-4201. [41] R. Radhakrishnan, a. Virkar, and S. Singhal, “Estimation of charge-transfer resistivity of La0.8Sr0.2MnO3 cathode on Y0.16Zr0.84O2 electrolyte using patterned electrodes”, J. Electrochem. Soc. Vol. 152, No. 1 (2005) A210-A218 [42] V. Brichzin, J. Fleig, H. Habermeier, and J. Maier, “Geometry Dependence of Cathode Polarization in Solid Oxide Fuel Cells Investigated by Defined Sr-Doped LaMnO3 Microelectrodes”, Electrochemical and Solid-State Letters, 3 (9) 403-406 (2000) 51 3. Chapter Three.  Electrode Delamination2 3.1 Introduction  [1  2  3  4] Thermal cycling may lead to degradation of the interfaces between the electrolyte and one of the electrodes, usually the cathode, with eventual detachment occurring.  This phenomenon is known as delamination, and it has been documented as an important mechanical degradation mechanism in SOFCs [1-4].  Recently, Virkar presented a model in which he showed how single cell deterioration in a stack could lead to cathode delamination because of local oxygen partial pressure build-up resulting from voltage reversal [5]. Upon the occurrence of delamination, the cell performance deteriorates because the open gap, perpendicular to the main current path, constitutes an insulating barrier to charge conduction, and destroys electrochemical reaction sites.  These processes result in the under-utilization of the fuel cell’s effective area.  Under standard degradation test conditions, the cell delivers a constant current (or voltage) as its voltage (or current) is monitored over time.  Upon the occurrence of delamination, the power output decreases because of both the Ohmic and polarization resistance increase, manifested as a drop in potential or in current.  As mentioned in the introductory chapter, other degradation mechanisms also cause a similar effect in the aforementioned degradation test conditions. From the maintenance standpoint, however, it is important to recognize and distinguish between different degradation mechanisms because, in general, each of them requires a different corrective action.  Disassembling the fuel cell stack to diagnose it is an option, but it is undesirable because it requires taking the stack out of service, and introduces additional degradation caused by the extra thermal cycle.  For the reasons expressed in Chapter One, a diagnostic technique that can distinguish between different modes of degradation in a minimally-invasive manner while the stack is still in operation is preferable to disassembly and inspection of the stack for diagnosis.  Furthermore, even at  2  The results described in this chapter have been published as: a. J. I. Gazzarri and O. Kesler, “Non-destructive delamination detection in solid oxide fuel cells”, Journal of Power Sources, 167 (2) 430-441 (2007). b. J. I. Gazzarri and O. Kesler, “Short stack modeling of degradation in solid oxide fuel cells - Part I: Contact degradation”, Journal of Power Sources, doi: 10.1016 / j.jpowsour.2007.10.047. 52 early R&D stages, a sample cell can experience delamination due to thermal cycling or inadequate firing.  Cell testing often includes monitoring the impedance after some time at operating temperatures and polarization.  Very often this impedance evolves, usually increasing in magnitude.  However, no systematic technique exists that links this evolution to a certain degradation mode, and in particular, to delamination. There are several SOFC applications or operating conditions in which delamination is particularly likely to occur, and where a minimally-invasive diagnostic tool to assess mechanical damage could be very useful.  Auxiliary power units, also known as APUs, are power generators used for peripheral devices on vehicles such as trucks and motor-homes, in order to avoid extended periods of combustion engine idling. In this application, repeated thermal cycling is unavoidable, and mechanical damage is prone to occur due to thermal expansion coefficient mismatch of SOFC components. A second example is redox instability of Ni-YSZ SOFC anodes.  Nickel is the most popular catalyst used in SOFC anodes.  However, it easily oxidises to NiO if the oxygen partial pressure inside the anode chamber is not kept sufficiently low.  This is a common scenario during emergency power shutdowns in which fuel is no longer supplied.  Further examples of causes of high PO2 in the anode are air leaks due to seal failure, periodic air treatment of the anode to prevent incipient carbon deposition, or very high fuel utilization conditions, resulting in high concentrations of steam at the anode exhaust [6].  This oxidation of Ni produces a volume expansion of the anode functional layer that can lead to mechanical damage, including delamination of the anode from the electrolyte.  Although oxidation of nickel in the anode is a reversible phenomenon (the NiO can be re-reduced back to nickel on re-introduction of H2 to the anode), it can lead to irreversible cell damage. Both thermal cycling of APUs and anode redox cycling are examples of the potential benefits of having a minimally-invasive diagnostic technique that can be used to detect delamination in-situ, thus justifying the need to disassemble the stack to replace or bypass the affected cell or cells.  [7 8]  [9 10 11] The idea of identifying mechanical damage in materials using impedance spectroscopy has been reported in the literature since the early 1990's.  Working with structural ceramics, Kleitz and co-workers [7,8] showed a change in the overall resistivity 53 and permittivity of an object as a function of internal crack growth.  They attributed this behaviour to the creation of blocking interfaces that acted as a hindrance to ion movement.  Tiefenbach et al. [9-11] found a new relaxation time in the impedance spectrum of an 8 mol% yttria-stabilized zirconia (8YSZ) sample caused by the growth of a crack.  They attributed this result to the different permittivity of the air contained in the gap compared to the permittivity of the surrounding material.  The aim of their work was to detect the presence of phase transformation induced microcracks in the bulk of a ceramic.  Fleig et al. [12] found that imperfect contact of silver electrodes on CaF2 gives rise to a new semicircle in the impedance spectrum.  They proposed an equivalent circuit to describe the influence of the gap created by the imperfect contact on the impedance behaviour of the ceramic. These studies showed the potential of impedance spectroscopy for the detection of mechanical degradation in ceramics.  However, the concept has not previously been applied to ceramic fuel cells.  In SOFCs, the fuel cell internal processes have their own impedance signatures superimposed on the characteristic impedance behaviour of the cell materials, whereas in the inert ceramics previously studied, the impedance spectrum corresponds to the ceramic material electrical behaviour only.  The superposition of the impedance features associated with mechanical degradation on the cell's own impedance features poses additional challenges in terms of the proposed method resolution.  In addition, the operating temperatures of the SOFCs are much higher than the temperatures studied in the previous contributions (200-400ºC), thus complicating the identification of some defect-related signatures at frequencies that are experimentally accessible. 3.2 Results and Discussion 3.2.1 Electrolyte-supported button cell 3.2.1.1 Intact cell Figure 3-1 shows the polarization curve of the simulated cell in solid line, and the Ohmic contribution in dashed line.  Tables in Section 2.5, Chapter Two, show the cell parameters.  The slope of the solid line at the 0.7V operating point is –1.52 ohm cm2. The current density corresponding to this point is approximately 0.25 A / cm2.  The poor kinetics of this cell makes the activation region extend beyond 0.3 A/cm2, hence the 54 absence of a steep and well-defined activation region at very low current sometimes seen in polarization curves.  The comparison of total losses (solid line) with Ohmic losses (dashed line) shows the large relative contribution (about 75%) of the kinetic polarization to the total polarization.  At 0.25 A / cm2, the Ohmic loss is approximately 0.1 V, and the kinetic loss is approximately 0.3 V. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 baseline case ohmic polarization component VC EL L (V ) i  (A/cm2)  Figure 3-1.  Solid line: Polarization curve for the electrolyte- supported cell configuration.  Model parameters detailed in Tables 2-2 to 2-7.  Dashed line:  Ohmic component of the polarization curve.  Kinetic polarization accounts for approximately 75% of the total polarization losses, as also indicated in Figure 3-2.  A typical model result for the Nyquist impedance plot of an intact cell can be seen in Figure 3-2 (bottom), as well as a plot of the imaginary impedance as a function of logarithmic frequency (top).  The Nyquist plot represents the complex impedance in a Cartesian plot with the negative imaginary part on the ordinate and the real part on the abscissa, with frequency as the parameter.  Each arc corresponds to an electrode process: the high frequency arc corresponds, in this case, to the anode, and the low frequency arc 55 corresponds to the cathode.  The diameter of the arcs depends on the reaction rate of the corresponding processes: the faster the processes, the smaller the diameter.  In this model, the exchange current density and the charge transfer coefficient determine the rate of the electrochemical reaction.  The difference in relaxation constants between anode and cathode determines the extent to which individual arcs are distinguishable.  The absence of lateral concentration gradients restricts the number of impedance arcs to just two.  The small feature at 1Hz reflects the transition from the high frequency range (linear in the Nyquist plot) to the cathode arc.  Plotting the imaginary impedance as a function of log(f) is useful to determine relaxation frequencies.  Later in this chapter, additional arcs will appear as a consequence of the lateral concentration gradient that results from the presence of the interconnect ribs. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 re(Z / ohm cm2) -0.4 -0.2 0 0.2 0.4 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) - im (Z  / o hm  cm 2 ) - im (Z  / o hm  cm 2 ) ω  Figure 3-2.  Nyquist (bottom) and imaginary impedance vs. logarithmic frequency plots of the impedance corresponding to the ESC configuration. The curved arrow indicates the direction of increasing frequency.  VCELL = 0.7V. The frequencies studied range from 1mHz to 1MHz, increasing counterclockwise around the Nyquist plot arcs, logarithmically spaced.  The high and low frequency arcs correspond to the anodic and cathodic reactions, respectively.  The semicircular shape of the spectrum reflects the capacitive behaviour of the double layer at the electrode / 56 electrolyte interfaces.  The linear portion at high frequencies is typical of porous electrodes, where the reaction sites extend beyond the electrode / electrolyte boundary interface into the electrode volume.  The magnitude of the Faradaic reaction tapers down away from the interface with the electrolyte, and becomes small near the current collector for sufficiently thick electrodes, in agreement with the transmission line model [13, 14]. That model describes the electrode as a network of resistors and capacitors that represent the ionic and electronic resistance, the double layer capacitance, and the Faradaic resistance.  The high frequency intercept, known as the series resistance, corresponds to the Ohmic resistance of the electrolyte. The impedance plot in Figure 3-2 forms the baseline for subsequent studies using the button cell geometry.  The low-frequency intercept corresponds to the total resistance of the cell, and is the sum of the series resistance (high frequency intercept) and the electrode polarization resistances (arc diameters).  These two characteristic points are of great importance because they provide important information about the cell.  The series resistance is associated with the ohmic contributions to the overall impedance.  In SOFCs, this contribution comes mainly from the electrolyte, plus any added resistance resulting from interlayers or detachments.  Its contribution can be isolated at high frequencies, at which the electrode processes are bypassed due to their capacitive component.  The polarization resistance represents the electrodes electrochemical and diffusional contributions to the impedance.  The spectrum in Figure 3-2 intercepts the real axis at 1.52 ohm cm2 at the low frequency end, a point that corresponds to the total resistance of the cell at the operating point, i.e. the slope of the polarization curve (Figure 3-1). 3.2.1.2 Cathode Delamination Let L be the radius of a circular cell, a the radius of a delamination in the centre of the cell, t its thickness, and kDELAM its (complex) conductivity.  An array of “air” elements inserted between the cathode and the electrolyte with a/L = 0.55 (which implies that 30% of the original electrode area is delaminated, as shown in Figure 3-3), t = 0.2µ m, 0ωεjkDELAM = , and insulation (Neumann) boundary conditions prescribed on all surfaces of the delamination, result in the impedance spectrum shown in Figure 3-4. 57 cathode electrolyte anode L a gap t  Figure 3-3.  A gap with the dielectric characteristics of air simulates cathode delamination 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 re(Z / ohm cm2) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) intact 30% cathode delaminated - im (Z  / o hm  cm 2 ) - im (Z / o hm  cm 2 )  Figure 3-4.  Effect of cathode delamination on the impedance spectrum.  Both series and polarization resistances increase, without change in relaxation frequencies.  VCELL = 0.7V. Two changes are apparent when comparing the impedance spectra of the delaminated and intact cells: 1. The high frequency intercept shifts to the right by approximately 0.14 ohm cm2, and 2. Both electrode arcs increase in diameter, by an amount that is approximately proportional to the delaminated area. 58 The first phenomenon indicates an increase in series resistance due to the reduction of ionic conduction area caused by the delamination.  In fact, the new high- frequency intercept is 0.51 ohm cm2, 39% higher (In general, an increase in resistance due to a reduction in area is: R/R0  = A0/(A0 - A) = 1 / (1-(a/L)2) = 1/0.7 = 1.43 = 43%) than the corresponding intact cell intercept.  The second phenomenon is related to the reduction of available reaction sites caused by the shadowing effect of the delamination on the electrodes.  The high aspect ratio of the cell makes in-plane ionic conduction negligible, resulting in the effective non-utilization of the electrode areas above and below the delamination.  These changes in the impedance spectrum have been experimentally observed and qualitatively reported, and tentatively attributed to anode delamination by Hsiao et al. [15], and to cathode delamination by Barfod et al. [2], Hagen et al. [3] and Heneka et al. [4].  However, these reports make no reference to the equivalence in the resistance increase, nor do they attempt to link an impedance change to delamination.  Figure 3-5 shows the ionic current density within the electrolyte and the electrodes.  The current density is nearly zero in the areas affected by delamination. Figure 3-6 shows the electronic current density profile along the cathode current collector.  Similarly to what occurred with the ionic current, the region affected by delamination exhibits almost no activity.  The current density in the intact zone is equal to that of the intact case, about 0.25 A / cm2. A/cm2delaminated zone intact zone a L ax isy m m et ry  lin e delamination  Figure 3-5.  Ionic current density (A/cm2) within the electrodes and the electrolyte, after a concentric, circular cathode delamination of radius a in a button cell of radius L.  The shadowing effect caused by the delamination makes the current density within the affected zone close to zero.  VCELL = 0.7V. 59  0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 radial distance (mm) i (A /cm 2 )  Figure 3-6.  Electronic current density profile from the centre to the edge along the cathode current collecting line. A further observation in Figure 3-4 is that the relaxation frequency for both electrode processes remains unchanged after the occurrence of delamination.  This behaviour is expected, since delamination does not change the electrochemistry of the system, but causes degradation by blocking the conductive path between the electrode and the electrolyte.  This observation provides further evidence of the fact that delamination cannot be studied using the equivalent circuit approach, since a change in polarization resistance (i.e. impedance arc diameter) could have not been explained using the equivalent circuit simplification, without a change in double layer capacitance for a constant relaxation frequency. The effect caused by delamination is insensitive to whether the delamination occurs at the centre of the cell, or as a concentric ring with radii a1 < a2 ≤ L [16].  This result suggests that an irregular shape delamination over an arbitrary portion of the cell area will cause the same result as that predicted with the present model. 3.2.1.3 The capacitance of the delamination gap Located perpendicular to the main current path of an electrical device such as the SOFC, the delamination gap is electrically analogous to a capacitor:  filled with air or other gases, it is blocking to direct current, but becomes a conductor of displacement 60 current, in particular alternating current, especially at high frequencies.  The capacitance per unit area of this newly-formed capacitor is determined by the permittivity of air and the thickness of the discontinuity.  Previous research work on crack detection in structural ceramics showed that the air gaps created within the bulk of the material had a detectable capacitive effect reflected as a new time constant added to the total impedance [8-11]. This time constant depends on the crack dimensions, as well as on the bulk conductivity of the ceramic (i.e. on temperature), since the “air capacitor” formed by the crack is in parallel with the “resistor” constituted by the bulk material.  The relaxation frequency of the crack-induced process is proportional to the conductivity, and inversely proportional to the permittivity.  The low temperature (200-400ºC) at which the mentioned experimental work was performed made this relaxation frequency observable within the capabilities of the instrumentation.  In the case of the SOFC, the extremely low capacitance of this gap, along with the relatively high conductivity of the ceramic, would make this capacitive behaviour visible as an extra arc only at extremely high frequencies. Running the calculation for an extended frequency range (of the order of tens or hundreds of MHz) resulted in a small semicircle at the high frequency end of the bold curve in Figure 3-4, joining the value of Rs with the original Rs0 as a result of the high frequency current “bridging” the delamination gap. 3.2.2 Electrolyte supported rectangular cell 3.2.2.1 Intact cell The relatively low voltage given by a single SOFC requires the series stacking of about 20 or 40 cells to provide a potential that suits practical demands.  Stacking individual cells is equivalent to interconnecting them by electrically joining the anode of one cell with the cathode of its adjacent neighbour.  A flat plate with channels on both sides has traditionally accomplished this task for planar SOFCs.  The interconnecting piece is a purely electronic conductor that must also be gas-tight, since its other important purpose is to provide a means of conveying reactant along the electrode surface.  In this way, the channels convey the reactants and the rib landings provide electrical contact. The relative size of landings and channels results from a compromise between good electrical conductivity and adequate supply of reactant to the electrodes. 61  The partial contact between the interconnect and the electrode has performance implications even in the absence of degradation.  Depending on rib/channel width, supporting geometry, and diffusional characteristics of the electrode microstructure, there will be zones underneath the rib landing where the limited access of reactant will cause localized depletion, and consequently poor electrochemical activity [17, 18].  Figure 3-7 a and b illustrate this idea for cathode- and anode-interconnect interfaces, respectively.  Figure 3-7.  Reactant partial pressure contours inside the porous cathode (left) and anode (right), in atm.  The limited access of reactant to regions underneath the interconnect rib causes local depletion.  The lines of equal reactant concentration appear to be straight because of the large aspect ratio of the cell.  Note the difference in colour scale range. The picture shows half the rib and half the channel for symmetry reasons.  Axes indicate length in metres.  Figure 3-8 shows the polarization curve corresponding to this geometry (thick line), using the same electrochemical parameters as in the axisymmetric example.  The polarization curve of the axisymmetric cell is shown as a thin line for comparison. Reactant depletion underneath the interconnect ribs reduces the overall performance with respect to the axisymmetric case.  The loss in performance increases as the current density increases, because of the higher demand for reactant. 0 0.04 0.08 0.1 0.16 0.21 electrolyte cathode interconnect rib pO2 (atm) cathode channel distance (m) di st an ce  (m ) 0.8 0.83 0.85 0.87 0. 94 0.96 electrolyte anode interconnect rib pH2 (atm) anode channel 0.97 di st an ce  (m ) distance (m) 62 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 i ( A / cm2 ) VC EL L (V ) Rectangular Axisymmetric  Figure 3-8.  Polarization curve corresponding to the ribbed interconnect geometry (thick line), calculated for an equivalent set of parameters as Figure 3-1 (thin line).  The maximum current density is not high enough for the polarization curve to show mass transport limitation.  Local reactant concentration gradients also influence the impedance spectrum shape, in some cases introducing a new time constant, beyond those associated with the interfacial double layer capacitance.  This phenomenon has been experimentally observed [19, 20, 21] and numerically reproduced [22], and it seems to be the result of laterally- varying gas composition.  Primdahl and Mogensen studied this phenomenon on the SOFC anode, describing it as a gas diffusion impedance [20, p2433].  Stübner [21, p97] observed its dependence on interconnect geometry and independence on temperature, thus attributing it to diffusional processes.  He found this diffusional time constant to relax at about 10 Hz. Figure 3-9 shows the spectrum of an intact cell for different rib/channel widths, where the new time constant appears at a frequency of about 10Hz.  The wider the rib, the larger the diffusional effect becomes. 63 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 re(Z / ohm cm2) -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) 0.5mm 1 mm 2.5mm - im (Z / o hm  cm 2 ) - im (Z / o hm  cm 2 ) A AB B C C 0.38 0.37 0.43 2.01.6 2.8  Figure 3-9.  A new time constant, whose frequency depends on rib and channel size, appears at 10Hz as an anodic diffusional effect (A) in the intact case, for a ribbed interconnect geometry.  Processes B and C correspond to anodic and cathodic electrochemical reactions, respectively. Labels indicate rib and channel half-width, both equal in the three cases shown.  Numbers below the real axis indicate the high and low frequency intercept impedance values.  VCELL = 0.7V.  An increase in the rib width changes the overall electrochemistry of the cell.  The reactant depletion underneath the rib accentuates the difference in electrochemical behaviour in the cell between ribs and channels.  The resulting behaviour is an average of two different “sub-cells” in parallel (centred at abscissa = 0mm and 2mm in either graph of Figure 3-7).  This combination is also responsible for the change in relaxation frequency of the cathodic process seen in the top graph of Figure 3-9. 3.2.2.2 Cathode Delamination A delamination at the electrode-electrolyte interface for a rectangular cell with interconnect has essentially the same effect as that described for the circular cell, namely 64 a simultaneous and equivalent increase in series and polarization resistance.  The regions affected by delamination become electrochemically inactive throughout the electrode volume.  Figure 3-10 illustrates this statement, showing that the electronic current density is negligible in the regions affected by delamination. cathode electrolyte anode delaminated zone ch1 ch2 ch3 ch4 ch5 ch1 ch2 ch3 ch4 ch5 r1 r2 r3 r4 r5 r6 electronic current density (A/cm2) length (m)  Figure 3-10.  Electronic current density magnitude (colour scale, A cm-2) and streamlines (red lines) for a cathode-electrolyte delamination the length of 2 rib widths and 2 channel widths (extending 0.008 m from the left).  Axes indicate spatial dimensions in meters.  ch indicate channels, r indicate ribs.  The zone affected by delamination is completely deactivated.  Mid-way across the intact ribs (r4 and r5) the current density value is small (~0.25 A/cm2) because of the reactant depletion underneath the rib, which makes the current density bend towards the channels, where reactant concentration is higher.  At the corners of ribs r3, r4, r5, and r6 with their neighbour channels, the current density is higher than 0.5 A/cm2 (top of scale), thus the region appears white.  This scaling strategy was needed for better visualization of the overall figure.  An observation of Figure 3-10 reveals: 1. The almost total deactivation of the affected zone, both on the cathode (top) side, and on the anode (bottom) side, with delamination occurring at the cathode- 65 electrolyte interface.  This is the result of the cell shadowing due to its high aspect ratio. 2. The third rib from the left (intact) carries a lower current density than it does in the intact case, because the cathode segment under channel #2 is inactive and does not generate any demand for charge carriers. 3. The intact ribs adjacent to the delamination do not take up any current from their delaminated neighbours.  This observation contrasts with the case presented in the next chapter, constituting a distinct difference between delamination and interconnect detachment.  However, this difference is not observable experimentally, except indirectly through the consequences of local overheating. Similarly, the ionic current at the mid-plane in the electrolyte is shadowed along the width of the first two ribs and two channels, as can be observed in Figure 3-11.  This phenomenon affects the impedance similarly as it did for the button cell geometry, with a simultaneous increase in both series and polarization resistance and no change in peak frequencies, as shown in Figure 3-12. 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 lateral distance (mm) io n ic  cu rr e n t (A /cm 2 ) ionic current at the electrolyte shadowed zone  Figure 3-11.  Vertical component of the ionic current density at the mid- plane in the electrolyte for the delamination case shown in the previous figure.  The first eight millimetres show almost no ionic species flow.  66 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 re(Z / ohm cm2) -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) 36% cathode delamination intact - im (Z  / o hm  cm 2 ) - im (Z / o hm  cm 2 )  Figure 3-12.  Impedance change caused by a cathode delamination the length of two rib widths and two channel widths.  VCELL = 0.7V. In summary, the ohmic resistance increase and the electrochemical deactivation produced by delamination results in the series and polarization resistance increasing simultaneously and equivalently as delamination progresses.  No change in characteristic frequencies is observed.  This event is likely to occur during thermal cycling or redox cycling, rather than continuously in time. 3.3 The Normalized Series and Polarization Resistances Cathode delamination affects the performance of both electrodes simultaneously. In contrast, microstructural degradation mechanisms affect one electrode at a time.  For example, sulfur poisoning affects the anode, and chromium poisoning affects the cathode. In the absence of a reference electrode, the individual identification of electrode processes in the impedance spectrum is subject to the difference in relaxation frequencies being sufficiently large, e.g. a decade or more.  These frequencies depend on a number of parameters including: nature of the reaction, reactant concentration, electrode microstructure, and temperature.  In general, the identification of individual electrode contributions to cell impedance is not always straightforward.  However, the series resistance and the polarization resistance provide a wealth of information about the cell behaviour, performance, and degradation, and their measurement is not subject to the 67 feasibility of separating individual electrode processes.  A change in series resistance reveals changes in the conductive path of charge carriers, usually interlayer growth, electrode delamination, or interconnect detachment.  A change in polarization resistance indicates changes in the electrode processes, and may correspond to changes in operating conditions and structure degradation.  The study of the change in the series and polarization resistance may provide useful information on the type and extent of degradation occurring in the fuel cell.  This section presents convenient nondimensional variables that allow visualization of the progressive degradation of the SOFC: the normalized resistances. Normalizing the resistance changes and the extent of degradation allows the direct comparison between series and polarization resistance, and among different degradation modes.  The resistance changes are normalized as the ratio of intact to degraded resistance, such that the ratio for the intact cell case corresponds to unity, and that for a completely destroyed cell corresponds to zero.  In Figure 3-13, the fractional changes in polarization and series resistance are shown as a function of (a/L)2 for the case of delamination in the button cell.  The abscissa indicates the extent of delaminated area, expressed as a fraction of the original electrode area.  Figure 3-13 indicates that delamination affects both polarization and series resistance in approximately the same proportion as the delaminated area.  The 1-(a/L)2 line represents the limiting case of complete shadowing, i.e. total destruction of conductive area or of electrochemically active sites within the electrode.  Values larger than those along this line indicate that the shadowing is not total.  Values below the line indicate a second degradation mode superimposed onto delamination (Chapter Six of this thesis). Plotting the normalized resistance as a function of the extent of degradation is a useful way to understand the physical effect of degradation.  However, this technique is not directly applicable as a diagnostic tool in operating fuel cells, since the extent of degradation is not known.  However, the normalized series and polarization resistances can be tracked as a function of time, and their evolution compared with each other.  If series and polarization resistances are found to increase equivalently or proportionally, this is a sign of a delamination process. 68 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ( a / L )2 R0 /R Rs0/Rs Rp0/Rp  Figure 3-13.  Normalized series and polarization resistances as a function of normalized degradation for a circular concentric delamination of radius a, on a button cell of total radius L. Both resistances scale with delamination area. Figure 3-14 shows the normalized resistance plot for the rectangular cell case, where normalized polarization resistance values corresponding to delamination of cathode segments underneath ribs appear slightly less shadowed (i.e. away from the 45º line) than those corresponding to segments underneath channels.  The difference in behaviour is the result of the non-uniform electrochemical activity across the ribbed interconnect cell.  Regions located vertically between ribs on the anode and cathode side interconnect present less reactant availability, and therefore lower electrochemical activity, than regions underneath channels.  Consequently, delamination of electrode segments underneath ribs causes slightly less shadowing than their underneath-channels counterparts, since their relative contribution to cell performance is different.  69 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 a / L R0 /R Rs0/Rs Rp0/Rp r1 ch1 r2 ch2 r3  Figure 3-14.  Normalized resistances corresponding to the case of delamination of the cathode-electrolyte interface for ESC rectangular cells including the interconnect.  The series resistance is affected uniformly with increasing loss of contact.  The polarization resistance shows an edge effect, by which the shadowing is lesser for delamination of the cathode between the first ribs, where the electrochemical activity is lower in the intact case due to limited access of reactant.  Labels refer to rib and channel numbers (see Figure 3-10) under which the electrode delamination from the electrolyte occurs. 3.4 The Influence of the Supporting Configuration The need for increased power density in planar SOFCs has driven the development of thin electrolyte, anode-supported configurations for the past ten years. Although not devoid of problems (e.g. redox stability), this configuration offers current densities several times higher than their electrolyte-supported counterparts.  Their technological importance justifies the study of the impact of degradation on their performance.  Figure 3-15 shows how delamination between the cathode and the electrolyte affects the impedance spectrum, for an extent of electrode delamination of two rib and two channel widths. 70 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 re(Z / ohm cm2) -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) 36% cathode delamination ASC intact ASC - im (Z  / o hm  cm 2 ) - im (Z / o hm  cm 2 )  Figure 3-15.  The influence of cathode delamination on the impedance spectrum for an anode-supported cell.   VCELL = 0.7V.  Delamination affects the ESC and ASC configurations equivalently, but the much smaller series resistance, and thereby its change, may not be within the range of detectability of the instrumentation.  A recent report suggests, however, that very small changes of the order of 0.002 ohm cm2 in series resistance may be accurately detected ([23], Figure 2).  From an experimental standpoint, it is therefore very important to measure the series resistance as accurately as possible, avoiding the inductance effects so commonly observed in test stands where the leads to each electrode come from different sides of the furnace [24,25].  This requirement poses important challenges for the experimental validation of these modeling results, since measuring series resistance with enough repeatability is extremely difficult. The shadowing effect caused by cathode delamination is a result of the large aspect ratio of the electrolyte.  The much larger electrolyte aspect ratio of anode- supported cells is expected to reflect on the extent of shadowing.  Figure 3-16 supports this idea, where the normalized resistances plot is shown for the ASC configuration (parameters in Table 2, Chapter Two). 71 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 a / L R0 /R Rs0/Rs Rp0/Rp  Figure 3-16.  Normalized resistances corresponding to the anode- supported configuration.  The series resistance shows an increase that is larger than proportional to the delaminated area.  The origin of this “overshadowing” is not clear at present. 3.5 Anode Delamination During fabrication, the anode can be fired at higher temperatures than the cathode, and thus it is customary to co-fire the anode and the electrolyte, with no cathode to avoid inter-reactions.  This procedure results in a better bond of the anode-electrolyte interface, compared to the cathode-electrolyte interface.  Consequently, delamination of the anode- electrolyte interface is less likely than cathode delamination during subsequent thermal cycling.  Redox cycling, on the other hand, can lead to anode-electrolyte delamination due to anode volume expansion and contraction. 72 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 re(Z / ohm cm2) -0.4 -0.2 0 0.2 0.4 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) intact 30% cathode delaminated - im (Z / o hm  cm 2 ) - im (Z / o hm  cm 2 )  0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 re(Z / ohm cm2) -0.4 -0.2 0 0.2 0.4 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) intact 30% anode delaminated - im (Z / o hm  cm 2 ) - im (Z / o hm  cm 2 )  Figure 3-17.  A delamination of the electrolyte from the anode causes essentially the same effect as a delamination of the cathode from the electrolyte in this simulation for an anode-supported button cell.  VCELL = 0.7V. Redox cycling occurs when the oxygen partial pressure at the anode increases because of fuel supply interruption, seal leakage allowing air into the fuel chamber, or high steam concentration under high fuel utilization conditions.  The consequence is the re-oxidation of nickel to NiO, a reaction that also causes volume expansion [26].  The electrochemical deactivation caused by delamination of an electrode results in negligible ionic exchange at the intact interface of the electrolyte with the other electrode. Consequently, the result of delamination is the same, regardless of which interface delaminates.  Figure 3-17 shows no difference in the impedance change between delamination of the cathode (left) and of the anode (right).  Figure 3-18 shows the electronic current density distribution for these two cases. 73   Figure 3-18.  Vertical component of the electronic current density for cathode (top) and anode (bottom) delamination in an ASC button cell.  Note that the position of the electrodes is reversed for computational convenience, hence the inversion in the direction of the current.  Axes indicate distance in metres.  Scale bar shows electronic current density in A/cm2. 3.6 On the Aspect Ratio The high aspect ratio of the electrolyte is responsible for the shadowing effect of delamination due to limited ionic conductivity.  “High”, however, is a relative term that needs clarification, to ensure that it applies to a reasonable range of sizes in use, especially for research button cells.  The diameter of button cells varies greatly depending 74 on the test station in use, ranging from a few millimetres to a few centimetres.  Figure 3-19 shows the normalized resistance values for different aspect ratios in an ESC button cell with 30% cathode area delamination, with a 150 micron thick electrolyte.  The range in button cell radius (labels, in mm) covers commonly used button cell geometries.  The figure shows that the normalized resistances approach the theoretical value at radii greater than or equal to 3 mm.  Aspect ratios lower than 20 cannot be considered “high” in terms of the shadowing effect.  This result is important for interpretation of results in cell geometries such as those with microelectrodes or sputtered electrode arrays. 0.60 0.65 0.70 0.75 0.80 1 10 100 1000  radius / electrolyte thickness Rs0/Rs Rp0/Rp R0 /R 2 5 10 203 4  Figure 3-19.  Normalized series and polarization resistance as a function of the ratio of button cell radius to electrolyte thickness, for 30% cathode delamination.  Aspect ratios ≥ 20 approach R0/R = 0.7.  0.7 is the normalized resistance expected for 30% cathode delamination for the case of complete shadowing.  Labels indicate radii in mm of button cells. 3.7 Comparison with Previous Experimental Observations Table 3-1 shows the results for normalized resistance values calculated based on previous experimental work where delamination is reported as responsible for SOFC degradation reflected on the impedance spectrum.  In agreement with the predictions of this model, the series and polarization resistances increase simultaneously in equal proportions. 75 Table 3-1.  Series, polarization, and normalized resistances calculated based on reported degradation attributed to electrode delamination.  Units for R Rs0 Rs Rp0 Rp Rs0/Rs Rp0/Rp Ref Heneka et al mohm 7.20 13 53 90 0.55 0.59 Fig 10 in [4] Hsiao et al ohm 0.40 2.00 2.60 10.20 0.20 0.25 Fig 6 in [15] Hagen et al ohm cm2 0.10 0.19 0.41 0.89 0.53 0.46 Table I in [3] Simner et al§ ohm cm2 0.12 0.15 0.28 0.33 0.8 0.85 Fig 5b in [27] §  Delamination + non-conductive Sr-Zr-O insulating interlayer All studies presented photomicrographs with evident electrode delamination.  In addition to delamination, [27] also reports the growth of a non-conductive Sr-Zr-O insulating interlayer, probably with the same effect as delamination, within the frequency range of the equipment.  It is not possible, however, to know the total extent of delaminated area with reasonable accuracy.  Only in [15] was it possible to check that the peak frequency for the electrode processes remained unchanged.  Performing a similar study on the other results would require the raw experimental data, not available at present. 3.8 Conclusions Electrode delamination degrades both conductivity and electrochemistry in an SOFC.  It constitutes a direct hindrance to the flow of charge between the electrodes, and also deactivates the cell everywhere in the electrodes above and below the delaminated zone as a result of the large aspect ratio of the cell.  This deactivation effectively shadows the cell over the affected area.  The simultaneous and equivalent increase in series and polarization resistances, proportional to the delaminated area, is a distinct change of the impedance spectrum that may be used for identification purposes.  Not causing a change in the electrochemical processes, delamination is expected to leave the relaxation frequencies of the electrochemical processes unchanged. Both button cell and reactangular cell with interconnect yielded equivalent results. In the rectangular cell with ribbed interconnect, incipient delamination starting at the edge did not cause immediate shadowing due to the limited electrochemical activity of 76 the cell near the edges.  Delamination in this edge zone is therefore not as serious as it is everywhere else at inner ribs.  Shadowing in anode-supported cells is more pronounced since the aspect ratio of their electrolytes is larger.  Delamination of the anode has the same effect as delamination of the cathode.  Model predictions are in good agreement with published experimental observations in terms of impedance change, where delamination was responsible for the degradation. 77  3.9 References  [1] E. Ivers-Tiffée, Q. Weber, and D. Herbstritt, “Materials and technologies for SOFC- components”, J. Eur. Ceram. Soc., Vol. 21, 2001, 1805-1811. [2] R. Barfod, M. Mogensen, T. Klemensoe, A. Hagen, Y. Liu, “Detailed characterisation of anode supported SOFCs by impedance spectroscopy”, Proc. SOFC IX, Vol. 1, 2005, 524-533. [3] A. Hagen, R. Barfod, P. Hendriksen, and Y. Liu, “Effect of operational conditions on long term stability of SOFCs”, Proc. SOFC IX, Vol. 1, 2005, 503-513. [4] M. Heneka, N. Kikillus, and E. Ivers-Tiffée, “Design of experiments for lifetime modelling of SOFC” , Proc. SOFC IX, Vol. 1, 2005, 908-919. [5] A. Virkar, “A model for SOFC stack degradation”, ECS Transactions, 7 (1) 443- 454 (2007) [6] D. Waldbillig, A. Wood and D. Ivey, “Electrochemical and microstructural characterization of the redox tolerance of solid oxide fuel cell anodes” Journal of Power Sources, 145, 2, (2005), 206-215 [7] L. Dessemond and M. Kleitz, “Effect of Mechanical Damage on the Electrical Properties of Zirconia Ceramics”, J. Eur. Ceram. Soc., 9, 1992, 35-39. [8] M. Kleitz, C. Pescher, and L. Dessemond, “Impedance Spectroscopy of Microstructure Defects and Crack Characterisation”, Science and Technology of Zirconia, Vol. 5, 1993, 593-608. [9] A. Tiefenbach, “Electrische Charakterisierung mechanischer Schaedigungen in ZrO2-Keramik”, VDI:-Fortschrittsberichte 5/555, VDI-Verlag, Duesseldorf, 1999. [10] A. Tienfenbach and B. Hoffmann, “Influence of a crack on the electrical impedance of polycrystalline ceramics”,  J. Eur. Ceram. Soc. Vol. 20, 2000, 2079-2094. [11] A. Tienfenbach, S. Wagner, R. Oberacker, and B. Hoffmann, “The use of impedance spectroscopy in damage detection in tetragonal zirconia polycrystals”, Ceramics International Vol. 26, 2000, 745-751. [12] J. Fleig and J. Maier, “The impedance of imperfect electrode contacts on solid electrolytes”, Solid State Ionics Vol. 85, 1996, 17-24. 78  [13] R. De Levie, “Electrochemical response of porous and rough electrodes”, Adv. Electrochem. Electrochem. Eng., Vol. 6, 1967, 329. [14] E. Barsoukov and J. MacDonald (editors), “Impedance Spectroscopy”, 2005, 531 [15] Y. Hsiao and R. Selman, “The degradation of SOFC electrodes”, Solid State Ionics 98, (1998) 33-38 [16] J. I. Gazzarri and O. Kesler, “Non-destructive delamination detection in solid oxide fuel cells”, Journal of Power Sources, 167 (2) 430-441 (2007). [17] T. Ackmann, L. de Haart, W. Lehnert, and D. Stolten, “Modeling of mass and heat transport in planar substrate type SOFC”, J. Electrochemical Society, 150 (6) A783- A789 (2003) [18] J. Pharoah, K. Karan, W. Sun, “On effective transport coefficients in PEM fuel cell electrodes: Anisotropy of the porous transport layers”, Journal of Power Sources 161 (2006) 214–224 [19] S. Primdahl and M. Mogensen, “Oxidation of Hydrogen on Ni/Yttria-Stabilized Zirconia Cermet Anodes”, J. Electrochem. Soc., Volume 144, Issue 10, pp. 3409- 3419 (O1997) [20] S. Primdahl and M. Mogensen, “Gas Conversion Impedance: A Test Geometry Effect in Characterization of Solid Oxide Fuel Cell Anodes”, J. Electrochem. Soc., Volume 145, Issue 7, pp. 2431-2438 (1998) [21] R. Stübner, “Untersuchungen zu den Eigenschäften der Anode der Festoxid- Brennstoffzelle (SOFC)”, PhD thesis, p24, (2002) [22] W. Bessler, "Gas Concentration Impedance of SOFC Anodes”, 7th European SOFC Forum, Lucerne, Switzerland, (2006) [23] A. Mai et al, “Time-dependent performance of mixed-conducting SOFC cathodes”, Solid State Ionics, 177, 19-25, (2006), 1965-1968 [24] E. Barsoukov and J. MacDonald (editors), “Impedance Spectroscopy”, 2005. [25] S. Singhal and K. Kendall, Eds., High-temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications, Elsevier, Oxford, 2003, 73. 79  [26] D. Larrain, J. van Herle, D. Favrat, “Simulation of SOFC stack and repeat elements including interconnect degradation and anode reoxidation risk”,  J. Power Sources 161, (2006) 392-403 [27] S. Simner M. Anderson, M. Engelhard, and J. Stevenson, “Degradation Mechanisms of La–Sr–Co–Fe–O3 SOFC Cathodes”, Electrochemical and Solid- State Letters, 9 (10) A478-A481 (2006)    80 4. Chapter Four.  Degradation of the Interconnect - Electrode Interface3 4.1 Introduction   1 2  3  4 The harsh atmospheric conditions on each side of the SOFC interconnect narrow the spectrum of materials from which to fabricate the cell components to only a few candidates.  Expensive lanthanum chromite ceramics are typically used as interconnects in stacks of tubular SOFCs, but cheaper stainless steel interconnects are more commonly used in planar SOFC stacks, thanks to the lower temperature of operation allowed by the planar configuration.  Despite their advantages, chromium-based steels oxidize over time under the physical conditions to which they are exposed in an SOFC, and ceramic coatings are often used to lower the rate of oxidation and reduce the rate of chromia evaporation from the interconnects.  Scales of chromia grow on the interconnect surface, degrading the electrical conductivity of the interconnect-electrode interface.  This oxide layer is dense and adherent, providing the substrate protection against further oxidation. However, it is important to ensure that the conductivity loss will remain within acceptable limits over the expected lifetime of the fuel cell.  Extensive research work on chromia layer growth has focused on the study of oxidation kinetics and on the increase in ohmic resistance over time [1-4].  Park et al [1] studied the kinetics of chromia growth at different temperatures and in different atmospheres.  According to their data, it is expected that a 20 micron layer of chromium oxide grows at 850ºC in cathode-like atmospheres over 40,000 hours, the current target lifetime for prospective commercial systems.  Larrain et al [2] modeled stack degradation derived from interconnect oxidation and anode re-oxidation, analyzing the main factors influencing both processes. According to their model, cells in the inner part of the stack degrade to a greater extent than cells in the outer part degrade, mainly due to temperature differences.  Brylewski  3  Results from this chapter were published in the Journal of Power Sources as:  J. I. Gazzarri and O. Kesler, “Short-Stack Modeling of Degradation in Solid Oxide Fuel Cells - Part I: Contact Degradation”, Journal of Power Sources, doi : 10.1016 / j . jpowsour . 2007.10.047 81 [3], and Huczkowski [4] conducted similar studies, focusing on SOFC conditions, leading to similar results. Typical Solid Oxide Fuel Cell processing routes involve several thermal cycles with temperature amplitudes of the order of 900ºC-1500ºC.  Co-firing and in-situ firing are examples of techniques aiming at minimizing the number of thermal cycles during manufacture.  These thermal excursions, plus those resulting from normal operation, may lead to deformation of the planar cell components, with detrimental consequences associated with contact degradation between the interconnect and the electrodes.  If one or more interconnect ribs lose contact with the adjacent electrode, electron conduction will be interrupted through the affected ribs.  This phenomenon is referred to as rib detachment [5]. In common with electrode delamination, these two degradation modes hinder the flow of current through the cell.  However, their physical nature, impact on performance, and impedance signatures are different from that of electrode delamination and from each other, as explained below. 4.2 Results and Discussion 4.2.1 Oxide Layer Figure 4-1 shows the current density magnitude and distribution before and after the growth of a 20 micron thick oxide layer between the interconnect and the cathode. This thickness corresponds to the extrapolation of the oxidation kinetics data presented in [1] and [3] to 40,000 hours, the target lifetime of a stationary SOFC.  The current density is lower than it was in the intact case, and there is not significant curvature of the current lines toward the channel.  Figure 4-2 shows the change in impedance for this degradation mode.  The thin line curve shows the impedance for the intact case, for comparison.  The increase in series resistance is approximately 0.4 ohm cm2.  For an oxide layer thickness of 20 microns, and an electronic conductivity of 1 S/m, this increase represents twice the value:  because the interconnect rib surface on which the layer is grown is half as wide as the tyconductivi thicknessR ==Ω σ δ  82 total unit width.  As in the case of delamination, the peak frequencies of the electrode processes remain intact after the growth of the oxide layer.  The addition of a highly resistive oxide layer within the electronic current path has an effect on the impedance equivalent to adding a series resistor to the overall electrical path.  cathode interconnect electronic current density (A / cm2) oxide layer channel channel intact 20 micron oxide layer distance (m) di st an ce  (m ) distance (m) rib rib  Figure 4-1.  Change in current magnitude and streamline distribution produced by a 20 micron thick oxide layer between the cathode and the interconnect. 83 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 re(Z / ohm cm2) -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) 20micron oxide layer intact rectangular - im (Z / o hm  cm 2 ) -im (Z / o hm  cm 2 )  Figure 4-2.  Impedance spectrum change produced by the growth of a 20 micron thick layer of chromia on the interconnect surface in contact with the cathode. VCELL = 0.7V. Applying the normalized resistance concept to oxide layer growth is not straightforward, since “total” degradation cannot be defined for a progressively increasing oxide layer thickness.  To circumvent this problem, we defined the “total” degradation oxide layer thickness as the thickness corresponding to 40,000 hours of operation, based on a growth kinetic study reported in the literature at the temperature of operation simulated in the present work: approximately 20 microns, at 0.1 micron/hr0.5 [1].  Using a conductivity of 1 S/m for Cr2O3 at 1123K, Figure 4-3 shows the normalized resistances as a function of extent of degradation.  The graph shows that the performance loss is primarily due to the increase in series resistance.  The polarization resistance, on the other hand, remains almost unchanged.  This trend is different from the trend found for delamination, where both series and polarization resistances change in a simultaneous and equivalent way, proportionally to the amount of delaminated area. It is important to note that diagnosis of this degradation mode may become difficult in cases where the oxide layer thickness is not uniform across the rib.  This non- 84 uniformity could be the result of a gradient in physical quantities such as temperature from the rib core to the rib edge. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 oxide layer thickness / 20 microns R0 /R Rs0/Rs Rp0/Rp  Figure 4-3.  Normalized resistance behaviour for oxide layer growth between the interconnect and the cathode.  The total degradation state definition is arbitrary, and corresponds to a thickness of 20 microns, the expected growth for 40,000 hrs at 850°C in an oxidant atmosphere. 4.2.1.1 The effect of interconnect geometry  The Ohmic nature of interlayer growth suggests that the rib size may have an effect on the impact of this degradation mode on cell performance.  The choice of rib and channel size in planar SOFCs corresponds to a compromise between adequate electronic conductivity from cell to cell, and good reactant diffusion into the porous electrode. Figure 4-4 compares the results obtained above for the default interconnect geometry (rib width=channel width=2mm) with a larger (5mm), and a smaller (1mm) geometry. 85 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 oxide layer thickness / 20 microns R0 /R Rs0/Rs 1mm Rp0/Rp 1mm Rs0/Rs 2mm Rp0/Rp 2mm Rs0/Rs 5mm Rp0/Rp 5mm  Figure 4-4.  The effect of interconnect rib and channel width on the normalized resistances for oxide layer growth, for the case of equal widths.  Performance deterioration is almost insensitive to variations in rib and channel width.  Figure 4-4 indicates that the results are nearly independent of interconnect geometry for the case where rib and channel are of equal size.  In contrast, Figure 4-5 shows that the series resistance is more severely affected for narrow rib sizes, for the case of constant rib+channel total width of four millimetres.  This result is of importance in view of a recent trend in interconnect geometry design, involving the use of narrow compliant contacts instead of the traditional ribs. 86 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 oxide layer thickness / 20 microns R0 /R Rs0/Rs 1/3 Rp0/Rp 1/3 Rs0/Rs 2/2 Rp0/Rp 2/2 Rs0/Rs 3/1 Rp0/Rp 3/1  Figure 4-5.  The effect of interconnect rib and channel width on the normalized resistances for oxide layer growth, for the case of total width equal to 4 mm.  Fractional numbers in the label indicate rib/channel width in mm.  The series resistance increases to a larger extent, the narrower the rib. The performance loss associated with oxide layer growth in the interconnect increases with oxide layer thickness, as expected.  The increase in series resistance depends only on the layer thickness, and its effect will be much more severe, in relative terms, in high performance cells, compared to low performance cells.  In contrast, electrode delamination causes the same relative loss of performance for a given amount of delaminated area.  Small rib width / channel width ratios show good performance in the intact state, but they are more severely influenced by oxide layer growth than cells with large relative rib width. 4.2.2 Rib Detachment between the Interconnect and the Cathode A second degradation mechanism involving the interconnect plates consists of the loss of contact, or detachment, of one or more ribs from the electrode [5].  In common with oxide layer growth, this problem causes an increase in the total resistance of the cell, but it is physically of a different nature.  Figure 4-6 shows the redistribution of the 87 electronic current density after the detachment of the first two interconnect ribs, on the cathode side, from the left.  The electronic current cannot go from the interconnect to the cathode through the affected ribs.  However, it can travel laterally across the cathode because of the relatively high mobility of the electrons.  This results in an increase in current density at the first intact rib adjacent to the group of detached ribs.  This increase in current may cause localized overheating and thereby increased the oxidation rate of the affected zone.  For the case under study (Figure 4-6), the average electronic current at rib #3 on the cathode (top) side is approximately twice as large as the current at ribs #4 and #5.  The Joule heating effect depends on the square of the current; therefore, the overheated rib (#3) will dissipate roughly four times as much heat as it would in the intact state. cathode electrolyte anode detached zone ch1 ch2 ch3 ch4 ch5 ch1 ch2 ch3 ch4 ch5 r1 r2 r3 r4 r5 r6 electronic current density (A/cm2) length (m)  Figure 4-6.  Redistribution of the electronic current density upon the detachment of the first two cathode interconnect ribs from the left.  Red lines: current density path. Colour scale indicates A/cm2.  The current density increases at the first intact rib as a result of this rib taking up the current produced at the electrodes within zones above and below the first two ribs. 88 A comparison of this current density distribution with the one corresponding to delamination in Figure 3-11, Chapter Three, reveals: 1. The anode (bottom) side in this example shows much more activity than it does in the case of delamination.  This observation indicates that the shadowing effect is much less pronounced in the case of detachment than in delamination. 2. Here, the current density in the third rib shows an increase in magnitude, indicating that the current is able to avoid the detachment to some extent.  This behaviour contrasts with that of delamination, where no current density is generated in the affected regions. 3. In common to both cases, the effect of the disturbance practically disappears from the fourth rib to the right.  This observation is important in view of the possible extrapolation of the results to larger cells. Figure 4-7 shows the impact of this phenomenon on the cell impedance, and Figure 4-8 shows the trend in normalized series and polarization resistances for an increasing number of detached ribs. 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 re(Z / ohm cm2) -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) two out of six ribs detached intact rectangular - im (Z / o hm  cm 2 ) - im (Z / o hm  cm 2 )  Figure 4-7.  Impact on the impedance of the rib detachment shown in Figure 4-6.  The reduction of the conductive path increases the series resistance.  The high electronic conductivity of the electrodes redistributes the current density, thereby limiting the shadowing effect.  Almost no change in characteristic frequency is observed. VCELL = 0.7V. 89  0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 # detached ribs / total # of ribs R0 /R Rs0/Rs Rp0/Rp  Figure 4-8.  Normalized series and polarization resistances as a function of the number of interconnect ribs detached from the cathode.  Unlike electrode delamination, interconnect detachment degrades the cell performance by blocking electronic species, instead of blocking ions.  The high mobility of the electrons makes the performance loss caused by this degradation mode less severe than that caused by electrode delamination.  In the case of electrode delamination, the blocking of ionic species deactivates the affected area since ions cannot travel the long path around the delamination due to their limited mobility.  In the case of interconnect detachment, electrons fed into the electrode through the intact interconnect contacts are able to travel laterally, spreading over the electrode and combining with ions coming from the electrolyte.  The shadowing effect caused by electrode delamination, reflecting loss of electrochemical activity within the electrode volume, prevents electrochemical reactions from happening at all.  Figure 4-9 shows the ionic current density at the electrolyte mid-plane.  In contrast with Figure 3-10 of Chapter Three, in this case the cell activity diminishes to a much lesser extent than in the case of delamination, revealing the 90 very different effect of both degradation modes. 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 lateral distance (mm) io n ic  cu rr e n t (A /cm 2 ) ionic current at the electrolyte affected zone  Figure 4-9.  Ionic current density at the mid-plane in the electrolyte. Compared with the effect caused by electrode delamination (Figure 3-11, Chapter Three), the detached zone deterioration is less severe. 4.2.2.1 The effect of interconnect geometry Figure 4-10 shows the effect of interconnect geometry on the results for rib detachment.  For equally wide ribs and channels, the relative deterioration is larger for wider ribs and channels, as seen in the graph on the left.  For rib and channel of constant total rib+channel width (equal to 4 mm in this example), the relative deterioration is larger for narrower ribs and wider channels, as shown in the graph on the right.  It is important to take into account that the normalized resistance value provides information about the relative deterioration, rather than real cell performance. 91 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 # detached ribs / total # of ribs R0 /R Rs0/Rs 1mm Rp0/Rp 1mm Rs0/Rs 2mm Rp0/Rp 2mm Rs0/Rs 5mm Rp0/Rp 5mm 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 # detached ribs / total # of ribs R0 /R Rs0/Rs 1mmR 3mmC Rp0/Rp 1mmR 3mmC Rs0/Rs 2mmR 2mmC Rs0/Rs 2mmR 2mmC Rs0/Rs 3mmR 1mmC Rp0/Rp 3mmR 1mmC  Figure 4-10.  Influence of rib and channel width in the relative performance loss.  Left: Ribs and channels of equal width, 1 mm, 2 mm, and 5 mm.  Relative deterioration increases with increasing width.  Right: Constant rib+channel width equal to 4 mm. Relative deterioration increases with narrower ribs and wider channels. 4.3 Conclusions  Interconnect oxidation is an important degradation mode affecting contact resistance.  Simulated by the insertion of a low-conductivity layer between the interconnect and the cathode, this degradation mode mainly affects performance by increasing the SOFC series resistance.  The extent of degradation increases for small rib/channel width ratios.  A small change in size of the electrode arcs may appear as a consequence of the current density dependence of the polarization resistance, but this contribution is small compared with the change in ohmic resistance.  The characteristic frequency of the electrode arcs remains essentially unchanged.  Contact problems can also appear between the electrode and the interconnect, a type of degradation mode known as interconnect detachment.  The simulation of this type of problem yielded interesting results in terms of the physical difference between interconnect detachment and electrode delamination.  The loss of performance resulting from detachment is relatively less severe than that caused by delamination, since the blocked electrons can move laterally in the electrodes to a much larger extent than ions can move within the electrolyte.  Consequently, this mechanism produces an increase in 92 series resistance, with some increase in polarization resistance, and almost no peak frequency change.  The relative performance loss caused by interconnect detachment is larger: a. the wider the total rib and channel widths for equally wide ribs and channels, and b. as the rib to channel width ratio decreases for a constant total rib + channel width  93 4.4 References  [1] J.-H. Park, K. Natesan, “Electronic transport in thermally grown Cr2O3”, Oxidation of Metals, 33, Nos. 1-2, (1990) 31-53 [2] D. Larrain, J. van Herle, D. Favrat, “Simulation of SOFC stack and repeat elements including interconnect degradation and anode reoxidation risk”,  J. Power Sources 161, (2006) 392-403 [3] T. Brylewski et. al., “Application of Fe-16Cr ferritic alloy to interconnector for a SOFC”,  Solid State Ionics 143, (2001) 131-150 [4] P. Huczkowski et. al. “Growth mechanisms and electrical conductivity of oxide scales on ferritic steels proposed as interconnect materials for SOFCs”, Fuel Cells 06,  No. 2, (2006) 93-99 [5] Malzbender, J., Wakui, T. and Steinbrech, R. W., “Deflection of planar solid oxide fuel cells during sealing and cooling of stacks”. Proceedings of the sixth Euro. SOFC Forum, vol 1 (2004) 329–338.  94 5. Chapter Five. Sulfur Poisoning4 5.1 Introduction Electrode delamination degrades the SOFC performance by making the affected part of the cell area unusable (Chapter Three).  This phenomenon can be interpreted as a direct loss of electrochemically active surface area for reaction.  Although other degradation mechanisms such as anode sintering, incipient coke formation, and chromium or sulfur poisoning also cause a loss in electrode electrochemically active surface area, the impedance behaviour changes observed as a result of these degradation mechanisms are qualitatively and quantitatively distinct from those corresponding to delamination, and proper interpretation of the impedance spectra could yield useful insight into the cell condition.  These latter mechanisms affect the cell performance by decreasing the electroactive surface area available for reaction.  Some of them also affect the diffusional properties of the porous electrode by changing the porosity and the microstructural morphology.  In terms of the present model, the deterioration of the electroactive surface area corresponds to a decrease in the value of S in equation ( 3 ), Chapter Two. [1, 2, 3] Anode degradation mechanisms that deteriorate cell performance by causing a reduction in the available amount of active surface area include sulfur poisoning, nickel sintering, and coke formation.  Sulfur deteriorates the cell performance by adsorbing on the nickel surface, thereby hindering hydrogen adsorption on the triple phase boundary, a necessary step previous to electro-oxidation [1-3].  The loss in surface area is a result of sulfur atoms covering potentially active reaction sites.  Nickel sintering, on the other hand, consists of a thermally-driven coarsening of the anode microstructure [4,5].  The reduction in surface area is a result of a change in microstructural morphology. Therefore, this localized nickel particle sintering also produces a change in porosity and tortuosity, altering the electrode gas transport characteristics, along with reducing the  4  The results described in this chapter were published as: J. I. Gazzarri and O. Kesler, “Electrochemical AC impedance model of a solid oxide fuel cell and its application to diagnosis of multiple degradation modes”, Journal of Power Sources, 167 (1) 100-110 (2007). 95 surface area available for electrochemical reaction.  A similar argument is valid for carbon deposition, or coking, with solid carbon depositing on electro-active sites, but also altering the composite porosity, and consequently, its gas phase transport properties, as well as the structural integrity of the anode materials. The most commonly reported microstructural degradation mode affecting the cathode is chromium poisoning.  Chromium outgassed from the interconnect plates tends to deposit on electrochemically active sites at the cathode, especially those close to the interface with the electrolyte.  This preferred deposition location is a consequence of the electrochemical nature of the process.  Quantitative microscopic studies performed on cathodes that had been exposed to chromium gases revealed the presence of a chromium- containing solid phase at the triple phase boundaries [6, 7].  It is possible, then, that this solid phase also affects local transport properties, apart from reducing the overall electrochemical activity by covering triple-phase boundaries. The phenomena described in the previous two paragraphs suggest that only sulfur poisoning at low concentrations can be reasonably described as purely reducing active surface area, since it is the only mechanism that does not affect gas transport properties. This is the only degradation mode that is realistically simulated by only lowering the anode electroactive surface area.  The model formulation is, however, general enough to incorporate other microstructural changes such as those to porosity and tortuosity (hence, effective diffusivity), but a literature search could not provide any quantitative correlation between microstructure and surface area loss for anode sintering, carbon deposition, or chromium poisoning.  Once reliable experimental results are available, they can be incorporated into the present model.  Sulfur poisoning is a degradation mode that is of great interest within the SOFC community because the anode fuel stream often contains traces of hydrogen sulfide. Only a few ppm of hydrogen sulfide can deteriorate cell performance within minutes in the case of electrolyte-supported cells, and within hours in the case of anode-supported cells.  Sulfur adsorbs on nickel, hindering its catalytic activity by blocking surface area that would otherwise be used by the fuel.  This degradation is purely electrochemical, and produces no change in the cell conductivity, i.e. in the series resistance, according to [1, 3].  Xia and Birss [2] did observe a series resistance increase with sulfur poisoning, but to 96 a much lower extent compared to the increase in polarization resistance.  The authors explain this observation as a contact degradation issue between anode particles, or between the anode and the current collector. In addition, the thin adsorbed sulfur layer that is responsible for the electrochemical deterioration does not change the gas diffusion properties of the electrode, at least at early stages when the degradation is still reversible.  In the context of the present model, this statement suggests that the effect of sulfur poisoning can be reasonably reproduced solely by a change in electroactive surface area.  Xia and Birss measured current density as a function of overpotential using cyclic voltammetry, and found very little change in the Tafel slope [2, Fig 4].  This observation led them to conclude that the anode electrochemical process was essentially unaltered.  This conclusion further supports the assumption of constancy of the charge transfer coefficients.  The microscopic details of the adsorption of sulfur on nickel are beyond the descriptive capabilities of the present method.  Sulfur poisoning must then be modeled based on its macroscopic effect, namely an effective reduction in electrochemically active surface area, reflected as an increase in polarization resistance.  Although it is probably not possible to measure the change in electroactive surface area directly, an estimate of this change can be made indirectly via fitting the model output to match observed changes in polarization resistance.  Furthermore, thermodynamic equilibrium can predict a value for the coverage of nickel with sulfur at a given temperature, if the following assumptions are made: 1. Sulfur follows a Langmuir-type adsorption isotherm on Ni [8,9]: 97       ⋅= + = TR B p p B p p B H SH H SH * 21300 exp55 1 2 2 2 2 θ    0 0.2 0.4 0.6 0.8 1 0 5 10 15 ppm H2S sulfur on Ni @ 850ºC fra ct io n al  co ve ra ge  o f t he  N i s u rfa ce     Here θ is the fractional coverage of the nickel surface, p indicates partial pressure, and R* is the gas constant in cal/mol K.  The constant B in these equations results from an experimental fitting done by Rostrup-Nielsen, cited by [9] as unpublished work.  It was, therefore, not possible to find the total absolute pressure at which this fitting was performed. 2. The presence of YSZ and porosity in the cermet does not invalidate point 1. In the present chapter, the purpose of the coverage calculation is to have a rough estimation of the expected fractional loss of active surface area at the porous anode, as well as to make an approximate assessment of the model results in terms of the change in polarization resistance.  The model in its current form is unable to represent any dependence on current density, although it has been suggested that increasing the cell current is beneficial because of the enhanced oxidation of adsorbed sulfur at the triple phase boundaries [1]. The experimental work of Xia and Birss resulted in the following changes in the impedance after poisoning a Ni-YSZ anode at 800ºC with 10ppm H2S (All reported experiments in this chapter were performed at atmospheric pressure) [2] (Table 5-1).  The authors reported a two-hour incubation period with no detectable degradation from the onset of the exposure to the contaminant, during which the sulfur atoms reach the active triple-phase boundaries.   Most of the degradation occurred within 3hs of exposure, gradually increasing thereafter. 98 Table 5-1.  Resistance change (ohm cm2) reported by Xia and Birss (Table 1 in [2]).  Conditions: 800ºC, OCV, 18 hs exposure to 10 ppm H2S, 3% H2O, balance H2.  Three-electrode measurement.   800ºC ppm H2S Rs0 Rs Rs0/Rs Rp0 Rp Rp0/Rp 10 0.35 0.55 0.64 1.64 6.66 0.25  Matsuzaki and Yasuda [3] poisoned Ni anodes with different amounts of H2S, at different temperatures.  The authors worked at 750ºC, 900ºC, and 1000ºC.  Table 5-2 shows relevant resistance changes, and Table 5-3 indicates the kinetics of the poisoning process. Table 5-2.  Series and anodic polarization resistance (Ohm), and normalized resistances, before and after exposure to different concentrations of hydrogen sulfide, at two different temperatures, at OCV, Pt counter-electrode [3].  Balance stream composition: 79% H2, 21% H2O.   750ºC 900ºC ppm H2S Rs0 Rs Rs0/Rs Rp0 Rp Rp0/Rp Rs0 Rs Rs0/Rs Rp0 Rp Rp0/Rp 0.05 6 6 1 9 12 0.750 2.5 2.5  3.5 0.5 6 6  9   2.5 2.5 1 3.5 5 0.700 0.7 6 6 1 9 18.5 0.486 2.5 2.5  3.5 1 6 6  9   2.5 2.5 1 3.5 6 0.583 2 6 6  9   2.5 2.5 1 3.5 6.8 0.515 4 6 6  9   2.5 2.5 1 3.5 7.2 0.486 8 6 6  9   2.5 2.5 1 3.5 8 0.438  99  Table 5-3.  Kinetics of sulfur poisoning according to [3]   750ºC   0.05ppm t (h) Rs0 Rs Rs0/Rs Rp0 Rp Rp0/Rp 0.0 6 6 1 9 9 1.000 0.2 6 6 1 9 9 1.000 0.5 6 6 1 9 9 1.000 1.0 6 6 1 9 9 1.000 2.5 6 6 1 9 10.5 0.857 3.3 6 6 1 9 11.5 0.783 6.7 6 6 1 9 11.75 0.766   The incubation period is present in this case as well, as shown in Table 5-3.  Zha et al. [1] studied the poisoning effect on Ni-YSZ anodes using full cells, with LSM-YSZ as the cathode material.  In common with [3], they observed no change in series resistance, but a substantial deterioration of performance associated with an increase in polarization resistance.  Table 5-4 and 5-5 summarize their results. Table 5-4.  Series and polarization resistance (ohm cm2) according to [1] before and after exposure to 1ppm H2S in H2.   1 ppm H2S T(ºC) Rs0 Rs Rs0/Rs Rp0 Rp Rp0/Rp 800 0.5 0.5 1 1.10 1.70 0.647 850 0.4 0.4 1 0.64 0.92 0.696 900 0.33 0.33 1 0.45 0.52 0.865  100  Table 5-5.  Estimated anodic Rp increase as a function of H2S concentration [1].  800ºC  OCV 0.7V ppm H2S ∆Rp% Rp0/Rp ∆Rp% Rp0/Rp 2 130 0.435 30 0.769 4 170 0.370 35 0.741 8 180 0.357 40 0.714  According to their experiments, sulfur poisoning occurred very rapidly, within minutes, with no incubation period (This observation is likely due to the supporting configuration, since the authors used thin anodes on electrolyte-supported cells).  The poisoning increased with increasing H2S concentration, with decreasing temperature, and with decreased DC bias. 5.2 Results and Discussion Although not in complete mutual agreement, these previous reports provide useful guidelines for the simulation of sulfur poisoning.  In the first place, it is reasonable to expect surface area reductions of up to 90%.  Both SOFC degradation experiments [1,2,3] and thermodynamic calculations [8] support such high active surface area reductions.  At present, the change in series resistance reported in [2] cannot be reproduced since its physical mechanism has not been completely identified.  Figure 5-1 shows the impedance spectra of an electrolyte-supported cell with 30%, 60%, or 90% active surface area loss at the anode, compared to the intact cell spectrum.  Apparent changes in the impedance spectrum include: 1. An increase in size of the anodic process arc, revealing a decrease in the anode electrochemical activity, and some increase in size of the cathodic arc. 2. A small increase in the anodic relaxation frequency. 3. No change in series resistance, since sulfur poisoning does not alter the charge carriers’ conductive path. 101 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 re(Z / ohm cm2) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) intact 30% surface area loss 60% surface area loss 90% surface area loss - im (Z / o hm  cm 2 ) - im (Z / o hm  cm 2 )  Figure 5-1.  The influence of anode surface area loss on the impedance spectrum of an electrolyte supported SOFC.  The anodic process arc increases in diameter with decrease in anode active surface area.  The anode time constant changes due to an overall change in polarization.  This change would have not been observed if the calculation had been done at OCV.  Calculated using a 1-D ESC geometry, i.e., with no interconnect. VCELL = 0.7V.   An interesting and unexpected observation in Figure 5-1 is that the cathode arc also changes slightly, even though its physical parameters remained unchanged in the model from the intact to the degraded state.  The reason for this behaviour is the coupling between the polarization resistances of both electrodes via the total current density.  The Butler-Volmer dependence of the Faradaic current on local overpotential makes the polarization resistance of both electrodes dependent on the total current density going through the system.  As a consequence, the change in delivered total current density caused by sulfur poisoning also has an impact on the polarization resistance of the cathode in potentiostatic mode, because the total current density decreases with increasing sulfur poisoning.  This interaction between the electrodes will be addressed in Chapter Six. 102  Figure 5-2 shows the progressive deterioration of cell performance for anode electro-active surface area loss, in a normalized resistance graph.  These resistances are defined as in Chapter Three: S S S R R R 0 = P P P R R R 0 = and they are displayed on the ordinate.  The horizontal axis indicates the normalized extent of degradation.  x-axis values of zero and one correspond to the intact case and to total destruction, respectively. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 fractional surface area loss R0 /R Rs0/Rs Rp0/Rp  Figure 5-2.  Normalized series and polarization resistances as a function of loss of active area in the anode, for the ESC configuration.  The deterioration is less severe than for cathode delamination, for equivalent fractional degradation.   The less severe degradation in cell performance due to uniform surface area loss compared to the equivalent amount of surface area loss caused by delamination is related to the uneven utilization of electrode thickness for electrochemical reaction.  In the intact state, there is an uneven utilization of the electrode: the local Faradaic current production or consumption tapers down away from the interface with the electrolyte.  This fact is 103 reflected in the changing magnitude of the local current density over the anode thickness (Figure 5-3).  If the surface area decreases uniformly throughout the electrode volume, this local current profile changes so that the electrode volume is relatively more fully utilized further away from the electrolyte interface.  In this case, the electrochemical reaction takes place more evenly throughout the electrode thickness.  This behaviour implies that the detrimental effect on performance of the mechanisms that degrade the electrode microstructure is less severe for thicker electrodes, under the assumption that all other processes, such as diffusion, remain unchanged by the loss in active surface area (a reasonable assumption for adsorption of a monolayer of sulfur on a portion of the anode reaction sites).   Thin electrodes are more completely utilized in the intact state, and therefore their performance degradation due to surface area loss is more abrupt.  Figure 5-4 illustrates this statement with a comparison of series and polarization resistance changes for anodes of different thicknesses: 10, 40, and 100 microns.  The solid symbols indicate the trend in polarization resistance.  For high surface area loss values, the deterioration is more abrupt for thin anodes (squares) than for thick anodes (triangles). The series resistance (open symbols) remains unchanged in all cases, as expected. 104 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 normalized distance from electrolyte no rm al iz ed  lo ca l F ar ad ai c cu rr en t d en si ty 10micron 40micron 100micron  0 50 100 150 200 250 300 0 20 40 60 80 100 distance from electrolyte (microns) lo ca l F a ra da ic  cu rr e n t d e n si ty  (A /cm 3 ) 10 micron 40 micron 100 micron  Figure 5-3.  Thick electrodes are relatively less fully utilized than thin electrodes (calculation performed using a 1-D version of the model, ESC configuration).  Top: normalized local Faradaic current density as a function of the normalized distance from the electrolyte.  Bottom:  local Faradaic current density as a function of the distance from the electrolyte. Legends show electrode thickness in microns.  105 Unlike the change in impedance behaviour caused by delamination, which is not dependent on electrode thickness, uniform surface area loss caused by poisoning causes slightly less severe changes in polarization resistance for thick electrodes than for a delamination of the equivalent surface area.  It is important to indicate that the observation of this dependence may be obscured by phenomena such as parameter interactions.  The third degradation state (90% loss) shown in Figure 5-4 is an example of this statement.  In this case, the cases for 10 microns and 40 microns are reversed.  This is a consequence of the polarization state, and the polarization resistance dependence on total current density.  This non-monotonic behaviour is not observed at OCV, since the total current density is null.  This and other interaction phenomena will be discussed in detail in Chapter Six. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 fractional surface area loss R0 /R Rs0/Rs 10micron Rp0/Rp 10micron Rs0/Rs 40micron Rp0/Rp 40micron Rs0/Rs 100micron Rp0/Rp 100micron  Figure 5-4.  Surface area loss has a stronger impact on thin electrodes than on thick electrodes, because thin electrodes are more fully utilized in the intact state, whereas thick electrodes experience a redistribution of the electrochemical activity away from the electrolyte, thereby tolerating larger uniform surface area loss.  ESC configuration.  106 5.3 Conclusions  According to previous experimental work, sulfur poisoning affects SOFC performance by adsorption of S on electrochemically active sites within the porous anode, and blocking possible hydrogen adsorption sites.  The thin adsorbed layer of S that is responsible for the anode poisoning does not change the microstructural morphology of the electrode in low concentrations, nor the electrochemical nature of the anodic reaction. Lowering the electroactive surface area simulates sulfur poisoning, under the assumption of constant specific double layer capacitance.  The model predicts an increase in the anodic arc diameter, without a change in series resistance.  This observation is in agreement with experimental observations where the anode arc is distinguishable from the overall impedance spectrum [1,2,3].  In potentiostatic mode, there is a slight change in the polarization resistance of the cathode for increasing amounts of sulfur poisoning, due to the overall reduction in total current density.  This interaction obscures the trend in total polarization resistance, making diagnosis more difficult. For diagnostic purposes, useful signatures of sulfur poisoning are: 1) The change in polarization resistance, mainly of the anode arc, without a change in series resistance. 2) Degradation occurring within a few minutes to hours, reaching a plateau in deterioration, which corresponds to the state of adsorption equilibrium. Sulfur poisoning affects thin anodes faster and to a greater extent than it affects thick anodes, since the coverage of triple phase boundaries takes place more rapidly for a given set of experimental conditions, and with less opportunity to redistribute current. Further work is needed to address issues such as: a. The change in series resistance reported by [2]. b. The potentially beneficial effect of anodic current in oxidizing the adsorbed sulfur at the triple-phase boundaries, thereby alleviating the effect of sulfur poisoning.  107 5.4 References   [1] S. Zha, Z. Cheng, and M. Liu, “Sulfur Poisoning and Regeneration of Ni-Based Anodes in Solid Oxide Fuel Cells”, Journal of The Electrochemical Society, 154 (2) B201-B206 (2007) [2] S. Xia and V. Birss, “Deactivation and recovery of Ni-YSZ anode in H2 fuel containing H2S”, Proc. SOFC IX, Vol. 2, 2005, 1275-1283. [3] Y. Matsuzaki, I. Yasuda, “The poisoning effect of sulfur-containing impurity gas on a SOFC anode: Part I. Dependence on temperature, time, and impurity concentration”, Solid State Ionics 132 (2000) 261–269 [4] D. Simwonis, F. Tietz, and D. Stöver, “Nickel coarsening in annealed Ni/8YSZ anode substrates for solid oxide fuel cells”, Solid State Ionics 132, 3-4, 2 (2000) 241-251. [5] T. Iwata. “Characterization of Ni-YSZ Anode Degradation for Substrate-Type Solid Oxide Fuel Cells”, J. Electrochem. Soc. 143 (1996), 1521. [6] J. Bentzen, J. Bilde-Soerensen, Y. Liu, and M. Mogensen, “Chromium poisoning of composite LSM/YSZ cathodes”, Proc. 26th Risoe International Symposium on Materials Science, Roskilde, Denmark, 2005, 127-132. [7] Y. Matsuzaki, I. Yasuda, “Dependence of SOFC Cathode Degradation by Chromium-Containing Alloy on Compositions of Electrodes and Electrolytes”, J. Electrochem. Soc., Volume 148, Issue 2, (2001) A126-A131 [8] J. Piña, V. Bucalá, and D. Borio, “Influence of the sulphur poisoning on the performance of a primary steam reformer”, International Journal of Chemical Reactor Engineering 1 A11 (2003) 1-20 [9] L. Christiansen, and S. Andersen, “Transient profiles in sulphur poisoning of steam reformers”, Chamical Engineering Science 35, 314-321 (1980)  108 6. Chapter Six.  Sensitivity Analysis and Combined Degradation Mode Scenario5 6.1 Introduction Three important issues associated with the practical applicability of the presented method have not been discussed: 1. The sensitivity of the results to variation in input parameters. 2. The study of possible interactions among input parameters. 3. Multiple degradation mode scenarios. The first two problems listed above are of importance given the high complexity of the system under study.  This chapter assesses the model robustness to variation in input parameters.  The purpose is three-fold: In the first place, to know the method’s sensitivity to variation in parameters that are difficult or impossible to measure independently.  Examples of such parameters were given in Chapter Two.  Secondly, to be aware of possible nonlinear (and non-intuitive) interaction among parameters that may lead to ambiguities.  In the third place, the variation of input parameters such as kinetic constants and inlet partial pressures provides a rough estimation of the limitations of the two-dimensional and isothermal approximations, addressing, at least statistically, the possibility to have in-plane variation of the modeled variables. The third problem is related to the occurrence of multiple simultaneous degradation processes.  Even if the proposed method can provide information about a single degradation process taking place during cell operation, multiple mode scenarios would be much more difficult to diagnose, especially if there is interaction between the degradation processes.  5  Results from this chapter have been published in the Journal of Power Sources as:  J. I. Gazzarri and O. Kesler, “Short Stack Modeling of Degradation in Solid Oxide Fuel Cells - Part II: Sensitivity and Interaction Analysis”, Journal of Power Sources, doi : 10.1016 / j . jpowsour . 2007.10.046. 109 6.2 Sensitivity of the Method to Inaccuracies in the Input Data and Interaction among Parameters The problems of ambiguities in the results and of finding adequate physical parameters that reliably describe the system, inherent to every numerical model, suggest the need for a method to assess the sensitivity of the results to inaccuracies in the input data.  One objective of this work is to find a method to determine the impact of those possible inaccuracies on the model output.  This method is based on the Uniform Design technique [1], which consists of designing a set of experiments using different combinations of input variables, to evaluate their relative influence on the results, and their potential interaction.  The application of this concept to the present model consists of repeating the calculation of a certain study case for a broad, yet reasonable, range of variability of the input parameters whose impact is to be tested.  If the scatter obtained in the results is of the order of the expected experimental scatter, then it can be concluded that the simulation is reasonably insensitive to inaccuracies in the input data.  On the other hand, if the dispersion is significant, the critical input parameters that are most responsible for the scatter should be identified and, if possible, their values should be estimated with higher accuracy.  This study is of interest in a complicated system such as a fuel cell, since its many concurrent physical mechanisms may interact with each other, yielding unexpected results if not properly taken into account. The cross influence between electrodes is an example of this kind of interaction. The polarization resistance of an electrode is inversely proportional to the total current density in the Tafel regime, as long as the system is not under mass transport control6.  If one electrode loses electrochemical activity due to microstructural degradation, the overall cell polarization will change.  This will change, in particular, the polarization state of the other electrode, thereby changing its polarization resistance.  Therefore, both polarization resistances will change due to a performance loss of just one electrode.  In the next sections, the results for each degradation mode are assessed in terms of robustness to input parameter variation.  6  The slope of the polarization curve (i vs η) increases with increasing overpotential.  This slope is the reciprocal of the polarization resistance.  If the system goes into mass transport control, this slope starts to decrease, levelling off at the limiting current value. 110 In addition, a sensitivity analysis of the modeling results with respect to the variation of input parameters can provide a statistical appraisal of some of the major assumptions of the present model.  The two dimensional and isothermal approximations are examples of these assumptions.  A major limitation of the two dimensional approximation is that reactant concentration variation is not accounted for.  Repeating the model calculations for a given degradation mode and for a fixed extent of degradation, using a wide range of input parameter variation, including reactant concentrations, provides an assessment of the impact of this assumption on the results.  A similar statement applies to the effect of temperature variations on kinetics along the plane of the cell. 6.2.1 Delamination In Chapter Three, a distinct trend followed by the normalized series and polarization resistances as a function of progressively increasing electrode delaminated area appeared as a suitable means to identify this type of degradation mode.  For 30% loss in delaminated area, both normalized resistance values became (Figure 3-13, page 68): 7.0 0 ≈= S S S R R R   7.0 0 ≈= P P P R RR In order to assess the sensitivity of these results to input parameter inaccuracies, as well as to possible unexpected parameter interaction, 21 parameters are varied over the range shown in Table 6-1, using a 30-run strategy.  Each parameter will take 30 different values, equispacially chosen from an interval that spans a broad, yet reasonable, range of variability, for a fixed extent of degradation.  Each run consists of the solution of the system for a different combination of parameters, chosen according to the Uniform Design criterion.  Appendix III shows the complete list of the parameter combinations used to perform each numerical experiment.  The range of variability in exchange current densities gives an approximate idea of the range in temperatures covered by this study.  The thermal dependence of the exchange current density for an electrode reaction is given by: 111         −= TR E exp Ai g a 0  ( 1 )  In this expression, the factor A includes any other dependence other than that on temperature.  When all other conditions are unchanged, two values of exchange current density correspond to two values of temperature according to the following derivation. The logarithmic ratio of current densities at two different temperatures is: ( ) ( )      −=         −=      121220 10ln T 1 T 1 R E TR E TR E Ti Ti g a g a g a  ( 2 )  This relationship allows the calculation of a temperature difference corresponding to two exchange current densities, for a given process activation energy, according to: ( ) ( ) 120 10 2 ln 1 T 1 Ti Ti E R T a g +      =  ( 3 )  The extreme values given in Table 6-1 for exchange current densities yield a logarithmic ratio of 100.  Assuming a minimum operating temperature T1 = 700ºC corresponding to the lower bound in exchange current density, the estimated upper bound for temperature is, for an anodic activation energy7 ranging from 0.8 eV [2, Table II, p3412] to 1.1eV [3, p55 and Table 5.4], respectively: ( ) ( ) CeVT CeVT °= °= 12271.1 16088.0 2 2  ( 4 ) These values are beyond the expected maximum temperature of operation of planar SOFCs, implying that the chosen range of exchange current densities covers the expected range of temperature variability within a given system.  Analogously, for the cathode, and for activation energies ranging between 1.5 eV and and 1.9 eV [4, Table I, pA440, process C], the upper values for temperature result: ( ) ( ) CeVT CeVT °= °= 9489.1 10375.1 2 2  ( 5 )  7  This calculation assumes that the activation energy for the exchange current is equivalent to that for the charge transfer resistance, because the reported Rp values were measured at OCV. 112 The higher activation energy of the cathodic process yields lower upper bound temperatures, but these are still reasonably high so as to include very large thermal differences due to reactant utilization.  As far as reactant depletion is concerned, the range considered for channel concentration of reactants covers the reasonable limits of fuel utilization. Table 6-1.  Range of variability of input parameters for the sensitivity analysis of the results Parameter Symbol min max Units Anode (ESC) and cathode thickness (ESC, ASC) tANO,tCAT 10 100 micron Anode (ASC) thickness tANO 500 1000 micron Electrolyte thickness (ESC) tELY 90 200 micron Electrolyte thickness (ASC) tELY 5 20 micron Channel hydrogen partial pressure pH2 0.1 0.97 atm Channel oxygen partial pressure pO2 0.1 0.21 atm Operating point as a fraction of the OCV VxCELL 0.2 0.99 - Anode charge transfer coefficient, anodic direction αAA 0.5 2 - Anode charge transfer coefficient, cathodic direction αAC 0.2 1 - Cathode charge transfer coefficient, anodic direction αCA 0.5 2 - Cathode charge transfer coefficient, cathodic direction αCC 0.2 1 - Anode and cathode active surface area Sano ,  Scat 105 106 m-1 Anode and cathode porosity ε 0.1 0.7 - Anode and cathode gas phase tortuosity τ 2 7 - Log anode double layer capacitance Cdl,ano -1 2 F m-2 Log cathode double layer capacitance Cdl,cat -1 2 F m-2 Log anode exchange current density i0,ANO 1 3 A m-2 Log cathode exchange current density i0,CAT 1 3 A m-2 Delamination thickness t 0.1 5 micron  Figure 6-1 shows the results for both normalized resistances for cathode delamination, where it is apparent that their sensitivity to variations in the input parameters is very small.  It is now possible to complete the values given on the previous 113 page using the standard deviation obtained in this example.  For a delamination of 30% of the cathode area in a button cell: 004.0719.0 0 ±== S S S R R R   006.0716.0 0 ±== P P P R RR For a delamination of 36% of the cathode area in a rectangular cell: 001.0638.0 0 ±== S S S R R R   01.063.0 0 ±== P P P R RR It is important to keep in mind that this scatter in the calculated values is solely due to the variation of input parameters in the model, and does not consider other sources of experimental error in a fuel cell test. Cathode Delamination 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 # run no rm al iz ed  re si st an ce s Rs0/Rs axisymmetric Rp0/Rp axisymmetric Rs0/Rs rectangular Rp0/Rp rectangular  Figure 6-1.  Normalized resistance results for 30% cathode area delamination (button cell) and for a delamination of 36% of the cathode length (two ribs and two channel widths in a rectangular cell) are almost insensitive to input parameter variation, and no unexpected interaction is apparent. Complete list of values corresponding to each run # are in Appendix IV, Table IV-1, page 162.  This result shows that the normalized resistance behaviour caused by delamination is robust to variability in the input parameters.  This invariance is associated with the nature of delamination itself.  As explained in Chapter Three, electrode 114 delamination deteriorates the cell performance by interrupting the flow of charge, and deactivating the affected regions of the cell area because of the large aspect ratio of the cell.  This phenomenon is of a conductance nature, and it will have an equivalent effect for all reasonable cell geometries and electrochemical characteristics.  The normalized series resistance for run k can be approximated as (This approximation implies neglecting any possible in-plane conduction): ( ) ( ) ( ) ( ) 00 0 0 00 1 A A AA AkR kR kR kR D DS S S S −= − = ( 6 ) where A0 and AD are the intact projected area and the delamination-affected area, respectively.  Equation ( 6 ) shows that the normalized series resistance is independent of the intact series resistance.  This property makes the series resistance behaviour different from that corresponding to the following case studied: oxide layer growth. 6.2.2 Oxide Layer Growth If the same concept is applied to interlayer growth, the variability is larger (Figure 6-2, squares and circles).  This graph shows the scatter in normalized series and polarization resistances for a 5 micron-thick oxide layer simulated between the cathode and its corresponding interconnector.  The variability in normalized series resistance is a result of the impact of the oxide layer on the cell impedance.  Unlike the case of delamination, the normalized series resistance does depend on the value of the intact series resistance.  An oxide layer of fixed thickness increases the series resistance by a constant value in all runs: ( ) ( ) ( ) ( ) δ+= kR kR kR kR S S S S 0 00  ( 7 ) where δ is a constant that depends on the oxide layer thickness and conductivity.  The intact series resistance is different in all cases since the electrolyte thickness, by far the most important contributor, is among the varied parameters.  Therefore, the scatter in the normalized series resistance is a result of the scatter in the intact series resistance.  If the numerical experiment is repeated excluding the electrolyte thickness variability, the 115 series resistance scatter disappears (Figure 6-2, thin line).  Electrolyte thickness is one variable that is relatively easy to estimate.  The values for normalized resistances, adjusted by the scatter found using the uniform design experiment for fixed electrolyte thickness for a 5 micron-thick oxide layer on the interconnect on the cathode side is: 005.0778.0 0 ±== S S S R RR   04.098.0 0 ±== P P P R RR Oxide Layer Growth 0 0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 # run no rm al iz ed  re si st an ce s Rs0/Rs Rp0/Rp Rs0/Rs locked electrolyte thickness Rp0/Rp locked electrolyte thickness  Figure 6-2.  Some variability is observed in the normalized resistance results for 5 micron interconnect oxide layer growth.  The variability in polarization resistance is associated with the non-symmetric polarization behaviour of the cathode.  Series resistance variability occurs because electrolyte thickness is among the varied input parameters.  If electrolyte thickness is fixed at the default value (thin line), the variability decreases. Complete list of values corresponding to each run # is in Appendix IV, Table IV-2, page 163.  6.2.3 Sulfur poisoning This degradation mode is the most severely influenced by a change in the input parameters, because it is purely of an electrochemical nature.  The large variability in Rp shown in Figure 6-3 (empty squares) illustrates this statement.  As explained in the 116 introduction to this chapter, the high complexity of the SOFC electrochemistry causes component interdependencies that may mislead an attempt to diagnose a single degradation event.  The reason for this variability is the current density dependence of the polarization resistance, and the fact that the simulation is set as potentiostatic.  The Butler-Volmer kinetics assumed dominant at the electrodes implies a non-monotonic behaviour of the polarization resistance with polarization state for cathodes with different charge transfer coefficients in each reaction direction, thereby causing the change in cathode Rp to vary non-monotonically with a given change in anode Rp, depending on the original operating point.  Sulfur Poisoning 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 # run no rm al iz ed  re si st an ce s Rs0/Rs Rp0/Rp Rs0/Rs locked kinetics, channel concentrations, and Vcell Rp0/Rp locked kinetics, channel concentrations, and Vcell  Figure 6-3.  While the normalized series resistance shows, as expected, no variability, a high level of interaction among parameters yields a large variability in the results for polarization resistance.  The kinetic parameters, inlet partial pressures, and the operating point are responsible for much of this large variability.  This variability decreases when these parameters are fixed.  The simulation shows 90% loss in the anode active area. Complete list of values corresponding to each run # is Appendix IV, Table IV-2, page 162. Figure 6-3 (lines) shows the result of the same numerical experiment with locked operating point, inlet partial pressures, and main charge transfer coefficients.  The 117 variability is clearly smaller, confirming the analysis of the previous paragraph.    The normalized Rp result for this case was: 07.076.0 ±=PR Although some influence of the extent of electrode utilization was expected, repeating the analysis with locked anode thickness (40 microns) did not lead to a different conclusion. The anode-supported case equivalent to the experiment above yielded qualitatively the same results, yet a lower scatter for the case of fixed parameters, and also one case in which the simulation failed to converge (#17), as shown in Figure 6-4. Sulfur Poisoning 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 # run n or m al iz ed  re si st an ce s Rs0/Rs ASC Rp0/Rp ASC Rs0/Rs ASC, locked kinetics Rp0/Rp ASC, locked kinetics  Figure 6-4.  Uniform design experiment corresponding to 90% surface area loss due to sulphur poisoning, for the anode-supported configuration.  After fixing the parameters responsible for the scatter, the magnitude of this scatter is smaller than it was in the ESC case.  Complete list of values corresponding to each run # is in Appendix IV, Table IV-3, page 163. At constant DC potential, the reduction in anode electrochemical activity due to poisoning decreases the current density of the cell, and this change modifies the polarization resistance of both electrodes.  The polarization resistance of the cathode, according to this model, results from the reciprocal of the derivative of equation (6), Chapter Two with respect to the overpotential: 118 ( ) ( ) 1 00,0 1,, exp1exp )( 2 22 2 − −                 −      ∂ ∂ −+== ∂ ∂ η η η η η cc O O cc O O cacaCATCATP CATIONF fc c f c cffiSRi Due to the different charge transfer coefficients in the anodic and cathodic direction, the cathode polarization resistance first increases, and then decreases for larger overpotentials.  Figure 6-5 shows the cathode polarization resistance, calculated parametrically as a function of applied overpotential for the standard case considered in this model of non-equal cathodic charge transfer coefficients (thick line), and the polarization resistance corresponding to the case of both cathodic charge transfer coefficients equal to 0.5, for comparison purposes.  For the standard case, this figure shows three operating points: A, the intact case, B, the state after some mild change in total current density, and C, a severely degraded case.  These two degraded states can occur due to different extents of sulfur poisoning that change the overall delivered current density, and therefore, the polarization state of the cathode, even when degradation occurred at the anode.  B and C indicate an increase and a decrease in polarization resistance, respectively.  The non-monotonic behaviour of Rp with cathodic overpotential is responsible for this behaviour.  In other words, there is interaction between the amount of sulfur poisoning and the operating point.  Consequently, to diagnose sulfur poisoning it is essential to have access to information such as the degradation history, since the initial and final state of resistances may result in ambiguities. 119 0.6 0.7 0.8 0.9 1 1.1 1.2 0 0.1 0.2 0.3 0.4 - cathode overpotential (V) vs. OCV R p CA T (oh m  cm 2 ) [1.5  0.5] [0.5  0.5] C B A  Figure 6-5.  Cathode polarization resistance dependence on overall (external) cathodic overpotential vs. 1.12V.  Bold line: standard case of non-equal cathodic charge transfer coefficients (1.5 and 0.5).  Thin line: Equal cathodic charge transfer coefficients (0.5 and 0.5).  In addition, diagnosis would be greatly improved if the individual anode arc associated with the electrochemical reaction could be identified and tracked upon progressing anode poisoning.  Correcting the normalized resistance values using the improved dispersion (bold lines in Figure 6-3) requires the knowledge of quantities that are usually the most elusive, therefore it is expected that this degradation mode would be difficult to diagnose by tracking the normalized resistances. This is because although none of the other degradation modes studied in this thesis lead to Rp changes with no change in Rs, it is possible that other modes not studied in this thesis, such as anode sintering and carbon deposition, could also cause comparable changes in Rp with potentially small changes to Rs, to different extents that may not be clearly distinguishable from sulfur poisoning. The final values, for 90% loss in the anode electro-active surface area, and fixing the parameters identified as responsible for the scatter, are: ESC:   000.0000.1 0 ±== S S S R R R   07.076.0 0 ±== P P P R R R 120 ASC:   000.0000.1 0 ±== S S S R R R   03.076.0 0 ±== P P P R R R It is useful to remember at this point that the series resistance remains unchanged according to this model because sulfur poisoning causes no alteration to the current path, hence the absence of a finite dispersion value in the two equations above up to the first four significant figures.  However, a very small change may be observed as a result of a change in the peak frequency of the anodic process. 6.2.4 Interconnect detachment The variability in the case of interconnect detachment is comparable to that of oxide layer growth, as shown in Figure 6-6.  The reason for the series resistance scatter, however, is not the same one described in section 6.2.2.  The electronic conductivity of the cathode largely influences the resulting series resistance after cathode-interconnect interface detachment, because of the effective increase in the electronic path length upon detachment.  The increase in series resistance becomes, therefore, dependent on the two factors that determine the electronic conductance for a fixed material set: cathode porosity and cathode thickness.  Upon detachment, the increase in series resistance will be more severe for high cathode porosity and low cathode thickness.  As an example, run #9, the run with minimum normalized series resistance, results from the combination of a medium porosity: 0.41 (porosity range = [0.1 0.7]) and low thickness: 13 microns (thickness range = [10 100] microns).  In contrast, run #11 corresponds to a similar porosity (0.37), but maximum cathode thickness (100 microns).  When cathode thickness and cathode porosity remain fixed at the default values, the dispersion in series resistance diminishes considerably, as shown by the thin line in Figure 6-6. The scatter in polarization resistance is related to the different amount of shadowing produced by the detachment among runs.  Shadowing is the partial deactivation of projected cell area caused by a degradation mechanism affecting contact resistance at an interface, and it manifests itself as an increase in polarization resistance. An interfacial problem at the interconnect-cathode boundary causes an extent of shadowing that depends on the cathode electronic resistance in the in-plane direction, since incoming electrons must circumvent the affected interconnect ribs in order to reach 121 the cathode.  For large amounts of detachment, the relative effect of relevant cathode parameters on in-plane resistance is large; hence the larger scatter in the resulting polarization resistance.  Figure 6-7 clearly shows this trend, comparing the effect of two- and three-rib detachments on normalized resistance. Figure 6-8 shows relatively small scatter in the case of interconnect detachment on the anode side, providing further evidence about the influence of electronic conductance on the variability in the results.  The very high conductance provided by metallic nickel accommodates the electronic current redistribution relatively easily, largely regardless of the input parameter combination.  This statement is especially true for anode-supported cells, because of their very thick anode layer. Interconnect / cathode detachment 0 0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 10 11 12 1314 15 16 1718 19 20 2122 23 24 25 2627 28 29 30 # run n or m al iz ed  re si st an ce s Rs0/Rs 2 ribs detached Rp0/Rp 2 ribs detached Rs0/Rs, cathode porosity = 0.4, cathode thickness = 40 microns Rp0/Rp, cathode porosity = 0.4, cathode thickness = 40 microns  Figure 6-6.  Both series and polarization resistance show a variability that is comparable to that of oxide layer growth.  Cathode porosity and thickness, the factors that affect cathode conductance the most, are responsible for the series resistance variability (solid circles).  When these parameters are locked at the default values, the series resistance variability decreases (thin line).  Complete list of values corresponding to each run # are in Appendix III, Table III-2, page 162. 122 Interconnect - cathode detachment 0 0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 # run no rm al iz ed  re si st an ce s Rs0/Rs 2 ribs detached Rp0/Rp 2 ribs detached Rs0/Rs 3 ribs detached Rp0/Rp 3 ribs detached Rs0/Rs 3 ribs detached - cathode porosity = 0.4, cathode thickness = 40 microns Rp0/Rp 3 ribs detached - cathode porosity = 0.4, cathode thickness = 40 microns  Figure 6-7. Both Rp magnitude and Rp scatter increase when the number of detached ribs increase from 2 to 3.  Equivalently to the 2-rib case, the scatter in Rs decreases upon locking the parameters relevant to cathode electronic conductance: porosity and thickness Complete list of values corresponding to each run # are in Appendix III, Table III-2, page 162. Interconnect - anode detachment 0 0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 10111213 14151617 18192021 222324 25262728 2930 # run no rm al iz ed  re si st an ce s Rs0/Rs anode detachment, ESC Rp0/Rp anode detachment, ESC Rs0/Rs anode detachment, ASC Rp0/Rp anode detachment, ASC  Figure 6-8. Interconnect detachment of two ribs on the anode side presents a lower scatter than it does on the cathode side, because of the very high electronic conductance of the anode.  Complete list of values for each run # in Appendix IV, Table IV-3, page 163.  123 Under the assumption of accurate knowledge of the cathode porosity and thickness, the normalized resistances values, affected with scatter found for the case of interconnect detachment, are: 1. 2-rib detachment, electrolyte-supported cell, cathode side detachment:  01.074.0 0 ±== S S S R RR  04.091.0 0 ±== P P P R RR 2. 3-rib detachment, electrolyte-supported cell, cathode side detachment:  01.057.0 0 ±== S S S R RR  08.075.0 0 ±== P P P R RR 3. 2-rib detachment, electrolyte-supported cell, anode side detachment:  02.097.0 0 ±== S S S R RR   002.0001.1 0 ±== P P P R RR 4. 2-rib detachment, anode-supported cell, anode side detachment:  01.098.0 0 ±== S S S R RR  000.0000.1 0 ±== P P P R RR 6.2.5 Summary Table 6-2 shows the variation in normalized series and polarization resistance for each case detailed above.  The scatter values calculated in this section will be used to affect the results for normalized resistances in the next section.  Lack of convergence for one case of sulfur poisoning in the ASC configuration precluded the calculation of mean and standard errors for this case.  As stated in section 6.2.3, the non-null value in the percentage standard deviation for normalized Rs in the sulfur poisoning case responds to the slight change in the impedance end point because of a change in characteristic frequency of the anode arc, and at the same time a constant maximum frequency of the simulation. 124  Table 6-2.  Average and percent standard deviation in normalized series and polarization resistance for the 30 Uniform Design runs for each degradation mode, varying 21 factors over the ranges given in Table 6-1.  Cases 1 and 2 indicate before and after locking the parameter(s) identified as the primary source(s) of deviation.  Gray cells indicate relevant dispersion values affected by this change. degradation type config interface or electrode affected extent of degradation Rs 0/Rs Rp0/Rp parameter(s) responsible for dispersion     mean % stdev mean % stdev delamination axisym ESC cathode / electrolyte 30% 0.72 0.59 0.72 0.77 - delamination rect ESC cathode / electrolyte 36% 0.64 0.15 0.63 1.92 - IC oxidation 1 ESC cathode / interconnect 5 micron 0.77 5.44 0.98 2.55 IC oxidation 2 ESC cathode / interconnect 5 micron 0.78 0.61 0.98 2.59 tELY IC detachment 1 ESC cathode / interconnect 2 ribs 0.75 4.62 0.91 6.12 IC detachment 2 ESC cathode / interconnect 2 ribs 0.74 1.56 0.91 4.06 εCAT tCAT IC detachment 1 ESC cathode / interconnect 3 ribs 0.58 6.27 0.77 13.58 IC detachment 2 ESC cathode / interconnect 3 ribs 0.56 2.07 0.75 10.53 εCAT tCAT IC detachment ESC anode / interconnect 2 ribs 0.97 2.29 1.00 0.17 - IC detachment ASC anode / interconnect 2 ribs 0.98 1.08 1.00 0.02 - sulfur poisoning 1 ESC anode 90% 1.00 0.02 0.71 32.14 sulfur poisoning 2 ESC anode 90% 1.00 0.02 0.76 9.23 αAA αCC VCELL i0,CAT i0,ANO pHin pOin sulfur poisoning 1 ASC anode 90% NA NA NA NA sulfur poisoning 2 ASC anode 90% 1.00 0.01 0.76 4.43 αAA αCC VCELL i0,CAT i0,ANO pHin pOin  6.3 On the Simultaneous Occurrence of Multiple Degradation Modes Cathode delamination produces a simultaneous increase in series and polarization resistance.  Sulfur poisoning, on the other hand, causes an increase in polarization resistance with little (compared to the change in polarization resistance) [5] or no [6, 7] change in series resistance, since the conductive path for charge carriers remains essentially unaltered.  Interconnect oxidation results almost exclusively in an increase in 125 series resistance.  A conceivable degradation scenario could be the simultaneous occurrence of sulfur poisoning and interconnect oxidation such that both resulting normalized resistance ratios are approximately equal, as in the case of delamination. Based only on the initial and final states of series and polarization resistances, the two situations would be undistinguishable.  However, extra information may help to diagnose such a combination of degradation modes: 1. Tracking the evolution of a single electrode arc or peak, in the case where this feature is reasonably deconvoluted 2. Tracking the cell degradation path to find a distinct pattern, based on previous knowledge about specific degradation mode kinetics Table 6-3 details different stages of degradation in a simulated combined degradation scenario of interconnect oxide layer + sulfur poisoning that produces approximately the same final values of normalized resistances as cathode delamination of one rib and one channel width, starting from an edge.  The characteristics of this degradation mode were estimated as follows:  The target degradation state is fixed by the change in Rs and Rp caused by a cathode delamination of one rib and one channel width. The rectangular cell geometry only allows discrete increments of delamination fractions, namely an integer number of rib and channel widths.  This restriction simplifies the meshing strategy considerably, since the mesh densification is only needed at the ends of the delamination layer, which always coincides with a rib or a channel edge.  The resulting total Rp increase is approximately reproduced by anode active area loss (Chapter Five).  This change causes a polarization resistance increase that is then matched to the series resistance increase caused, independently, by interconnect oxidation.  A 4.3 micron thick oxide layer satisfies this requirement using the oxidation kinetics and chromia resistivity at the operating temperature [8, 9] described in Chapter Four.  Using this information, a combined degradation history is simulated, as shown in Table 6-3. 126  Table 6-3.  Extent of degradation used in the example of combined degradation. time oxide layer growth sulfur poisoning hr oxide layer thickness (microns) fraction of active area loss Rs 0/Rs Rp0/Rp 0 0.00 0 1.00 1.00 50 0.71 0.88§ 0.95 0.81 508 2.25 0.88 0.88 0.80 967 3.11 0.88 0.84 0.80 1425 3.78 0.88 0.82 0.80 1884 4.34 0.88 0.80 0.80 §  The large difference in kinetics of both processes suggests that not considering the two hour incubation period observed by Xia and Birss [2] is expected to be of no appreciable influence in this example.  6.3.1 Tracking an individual electrode process As explained in Chapter Five, the arc corresponding to the main anode process increases in diameter when the anode surface area decreases due to sulfur poisoning. Depending on the difference in relaxation frequencies between electrode processes, individual impedance arcs may be identifiable, in which case individual degradation mode diagnosis becomes easier.  Figure 6-9 illustrates this case for the first three stages in the simultaneous sulfur poisoning and oxide layer growth of Table 6-3.  The increase in Rp is primarily due to the anode polarization resistance, unlike the case of delamination, where both electrode arcs increase by equal proportions, regardless of which one delaminates. 127 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 re(Z / ohm cm2) -1 -0.5 0 0.5 -3 -2 -1 0 1 2 3 4 5 6 log(f / Hz) intact 50h 508h 967h - im (Z  / o hm  cm 2 ) - im (Z  / o hm  cm 2 )  0 0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 re(Z / ohm cm2) -0.3 -0.2 -0.1 0 0.1 0.2 1 2 3 4 5 6 log(f / Hz) intact 50h 508h 967h - im (Z / o hm  cm 2 ) - im (Z / o hm  cm 2 )  Figure 6-9.  Impedance spectra corresponding to the first three stages of combined sulfur poisoning + interconnect oxidation degradation shown in Table 6-3.  Above: Complete spectrum.  Below: Detail of the anodic contribution.  VCELL = 0.7V.  In this case, the anode arc is clearly identifiable among the other features of the spectrum.  In case it is not, Schichlein et al.’s [10] interesting method could be of aid, but 128 care must be taken with the choice of data filtering and windowing strategy. 6.3.2 Tracking the Degradation Path Different combinations of degradation modes may lead to the same final normalized resistances values.  However, the different degradation rates make it unlikely that the degradation path followed by the system is equivalent in different cases.  The concept of degradation space is introduced to clarify this point.  The graph in Figure 6-10 shows normalized polarization resistance vs. normalized series resistance, with the extent of degradation as the implicit parameter.  In this degradation space graph, point (1,1) corresponds to the intact case. 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 Rs0/Rs R p0 /R p interconnect oxidation delamination sulfur poisoning combined degradation combined degradation  Figure 6-10.  Approximate overall degradation path followed by the mechanisms under study, when acting individually. Plotting the evolution of normalized series and polarization resistance starting at (1,1) provides useful visual information about the nature and extent of degradation.  A degradation mode that mainly affects Rs, like oxide layer growth, appears as an almost horizontal line progressing leftward with increasing extent of degradation.  Sulfur poisoning, on the other hand, is reflected as a vertical line down from (1,1), since it 129 mainly affects the anodic Rp.  Finally, a line at 45º from (1,1) towards the origin represents delamination, because it causes a shadowing of the entire cell. In this way, a combination of two or more modes will be, in general, a curve starting at (1,1) and progressing down and left, depending on the relative effect on series and polarization resistance.  Regardless of the end point, the likelihood of two combinations to follow the same entire path is small and can reasonably be neglected, since each degradation mode has its own kinetics.  However, care must be taken with the use of this concept, since the resulting degradation path depends on characteristics of the intact cell, notably supporting configuration.  Figure 6-11 shows the two degradation paths of our working example: a. Cathode delamination b. Sulfur poisoning + oxide layer growth to produce an increase in Rs and Rp equivalent to that in a., according to the degradation history shown in Table 6-3. 130 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1.0 Rs0/Rs R p0 /R p intact cathode delamination Sloss + Oxide  Figure 6-11.  Degradation path plot showing the normalized resistance behaviour of:  Interconnect oxidation plus sulfur poisoning, and  cathode delamination.  Although the final degradation state is approximately the same for both cases, the difference in degradation history presents a possible way of distinguishing between the two cases. The error bars correspond to the scatter calculated in section 6.2.  ESC. In this example, the large difference in degradation kinetics makes the two degradation paths different from each other.  Sulfur adsorption kinetics are much faster than those of chromia growth, as illustrated in Figure 6-11 (circles).  The error bars correspond to the scatter calculated in section 6.2.  Despite the large scatter caused by the uncertainty in the values associated with sulfur poisoning, the difference in kinetics may be useful to distinguish between the two cases, at least for this amount of sulfur poisoning.  This situation would be more difficult to evaluate if the degree of sulfur poisoning were small enough that the deviation in polarization resistance was within the expected scatter.  However, since the extent of coverage of sulfur at equilibrium is comparable to the surface area loss simulated, the extent of sulfur poisoning is likely to be comparable to that simulated here. 131 The combination of delamination with sulfur poisoning and, independently, with interconnect oxidation, are further illustrative application examples of the concepts explained in this section.  In this case, the degradation scenario corresponds to the individual extents of degradation shown in Table 6-3, combined in two different ways, and for both electrolyte- and anode-supported configurations: a. delamination + interconnect oxidation b. delamination + sulfur poisoning Figure 6-12 shows these two combinations, compared with pure delamination, for the ESC configuration.  The offset from the delamination characteristic line reflects the influence of interconnect oxidation and sulfur poisoning. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Rs0/Rs R p0 /R p intact cathode delamination sulfur oxide delam + sulfur delam + oxide  Figure 6-12.  Degradation path plots showing two different degradation scenarios: combination of delamination with interconnect oxidation (filled circles), and sulfur poisoning (filled triangles).  The delamination, oxide layer growth, and sulfur poisoning characteristic paths are shown for comparison.  Electrolyte-supported configuration.  The error bars correspond to the scatter calculated in section 6.2. Figure 6-13 shows the equivalent situation for the ASC configuration.  The performance loss for the delamination + oxide layer is in this case much more severe, 132 because of the much higher relative importance of series resistance deterioration of the anode supported cell. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Rs0/Rs R p0 /R p intact cathode delamination sulfur oxide delam + sulfur delam + oxide  Figure 6-13.  Degradation plot for the same combinations shown in Figure 6-12, for the anode-supported configuration. The error bars were recalculated by re-running the 30 uniform design runs for the anode-supported configuration.  The usefulness of the degradation path plot is subject to a-priori knowledge about the expected kinetics of the degradation modes, exemplified in this case by the much faster sulfur poisoning, compared with interconnect oxidation.  In case of non-interacting multiple degradation modes simultaneously acting on a cell, it is possible to add their contributions to the degradation path plot.  For M concurrent, non-interacting degradation mechanisms, the individual contributions to series and polarization resistance are additive: ( ) ∑∑∑ === −=−=∆=∆=− M k k M k k M k k MRRRRRRRR 1 0 1 0 1 0  ( 8 ) 133 Where R is either a polarization or a series resistance, ∆R indicates its change, and R0 is the intact state series or polarization resistance.  Now, ( ) ( ) ( )∑ ∑ ∑ ∑ = = = = −− =⇒ −− =⇒ −−=⇒ −=− M k k M k k M k k M k k M R RR R RMR R R R RMRR MRRRR 1 0 0 1 0 00 1 0 1 00 1 1 1 1  ( 9 ) But 0R Rk is the reciprocal of the normalized resistance for the kth degradation mode, so ( )∑ = −− = M k k M R R R R 1 0 0 11 1  ( 10 ) This expression relates the normalized resistance for the multiple degradation state, and the individual normalized resistances for each degradation mode.  Plotting the result of (3) in a degradation path plot gives an idea of the relative influence of each degradation mode and its contribution to the overall degraded state.  For known individual degradation mode behaviours, this equation gives the result of their combination, under the assumption of no interaction.  As an application example of weakly interacting degradation modes, Figure 6-14 shows the normalized resistances for the combined degradation modes shown above, calculated using the model for the two concurrent degradation modes shown in Figure 6-12, as a function of the combined normalized resistances calculated using equation ( 3 ), i.e. adding the individual effects assuming no interaction.  The small deviation from the x=y line indicates only weak interaction between each pair of degradation modes. 134 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 no-interaction normalized resistance ca lc ul at ed  n or m al iz ed  re si st an ce Rs del+oxi Rp del+oxi Rs del+sulf Rp del+sulf  Figure 6-14.  Normalized resistance of the two combined degradation scenarios of the example above, as a function of the result of their direct addition, according to equation (3), with the assumption of no interaction.  The error bars correspond to the scatter calculated in section 6.2. A much more complicated situation not considered here would be that of total or partial masking of one mode by another.  An example of such a case would be simultaneous delamination and detachment with overlapping affected areas. 6.4 Conclusions  A parametric study consisting of solving for the polarization and series resistances for a wide but reasonable range of input parameters revealed some of the strengths and limitations of the method.  Repeated for all degradation modes under study, this parameterization revealed that the results for delamination are very robust to inaccuracies in the knowledge of, or interactions among, cell parameters.  An equivalent conclusion applies to oxide layer growth and interconnect detachment, but to a lesser extent.  In contrast, the results for sulfur poisoning are strongly dependent on the operating conditions and cell characteristics, especially the operating point, channel atmosphere, 135 and the charge transfer coefficients.  Further knowledge of the system is required to diagnose sulfur poisoning and distinguish it from other degradation modes not studied here such as anode sintering, namely the change in the anodic arc in the impedance spectrum, and the rapid initial kinetics of adsorption followed by a stabilization as the thermodynamic adsorption equilibrium is reached.  Simulating two concurrent degradation modes challenged the diagnosis capability of the proposed method.  Modeling results indicate that the convolution of the impedance spectrum features may hinder individual diagnosis of each mode.  Tracking the electrode characteristic arc (or peak), and/or the degradation path, provides further insight to the diagnostic technique, but some a-priori knowledge of the expected kinetics of the different mechanisms is essential.  The degradation path plot constitutes a useful tool for the visualization of the degradation history of the cell.  Further work is needed to explore the possibility of diagnosing strongly interacting degradation modes.  136 6.5 References   [1] ttp://www.math.hkbu.edu.hk/UniformDesign/ [2] S. Primdahl and M. Mogensen, “Oxidation of Hydrogen on Ni/YSZ cermet anodes”, Journal of The Electrochemical Society 144 (10) (1997) 3409-3419 [3] R. Stübner, “Untersuchungen zu den Eigenschäften der Anode der Festoxid- Brennstoffzelle (SOFC)”, PhD thesis, Technische Universitaet Dresden (2002) [4] M. Jørgensen and M. Mogensen, “Impedance of Solid Oxide Fuel Cell LSM/YSZ Composite Cathodes”, J. Electrochem. Soc. Vol. 148, No. 5 (2001) A433-A442. [5] S. Xia and V. Birss, “Deactivation and recovery of Ni-YSZ anode in H2 fuel containing H2S”, Proc. SOFC IX, Vol. 2, 2005, 1275-1283 [6] S. Zha, Z. Cheng, and M. Liu, “Sulfur Poisoning and Regeneration of Ni-Based Anodes in Solid Oxide Fuel Cells”, Journal of The Electrochemical Society, 154 (2) B201-B206 (2007) [7] Y. Matsuzaki, I. Yasuda, “The poisoning effect of sulfur-containing impurity gas on a SOFC anode: Part I. Dependence on temperature, time, and impurity concentration”, Solid State Ionics 132 (2000) 261–269 [8] J.-H. Park, K. Natesan, “Electronic transport in thermally grown Cr2O3”, Oxidation of Metals, 33, Nos. 1-2, (1990) 31-53 [9] T. Brylewski et. al., “Application of Fe-16Cr ferritic alloy to interconnector for a SOFC”,  Solid State Ionics 143, (2001) 131-150 [10] H. Schichlein, A. Müller, M. Voigts, A. Krügel, E.Ivers-Tiffée, “Deconvolution of electrochemical impedance spectra for the identification of electrode reaction mechanisms in solid oxide fuel cells”, Journal of Applied Electrochemistry, 32, 8, (2002) 875-882 137 7. Chapter Seven.  Conclusions and Outlook. 7.1 Conclusions A numerical model of the steady state and alternating current behaviour of a solid-oxide fuel cell was presented to explore the possibilities to diagnose and identify degradation mechanisms in a minimally invasive way using impedance spectroscopy. Simulating degradation modes required detailed electrode-level modeling, and ruled out the option of using the equivalent circuit approach, since localized degradation modes change the current flow pattern, destroying in-plane uniformity, and thereby invalidating an essential assumption of the equivalent circuit method.  This is the first attempt to simulate SOFC impedance behaviour using a two-dimensional model that incorporates a realistic ribbed interconnect geometry. The solution of the relevant transport equations was implemented using the finite element method.  The high aspect ratio geometry demanded a slender rectangular element meshing strategy to avoid the ill-posedness associated with long triangular elements. Modeled geometries included a circular button cell and a rectangular cell.  In all cases, the geometry consisted of a set of rectangles whose size could be easily modified, providing flexibility to perform parametric studies.  The rectangular mesh automatically adapted itself to the chosen geometry, based on a desired mesh density specified at critical places.  The MATLAB/COMSOL coding of the geometry is not restricted to the present SOFC model, and is applicable to any similar system in which the dimensions are to be considered among the input parameters.  An example of this kind of study is fuel cell channel design optimization. On account of the small size of the perturbation imposed on the system when performing impedance studies, the system variables were decomposed into a steady state and a small oscillatory component.  The linearization of the time domain equations around the operating point resulted in a system of equations coupled in just one direction, allowing the sequential solution of the nonlinear steady state system, and later the linearized, complex valued, time independent, AC system, superimposed on the steady state condition.  The resulting numerical efficiency of the system is important to allow 138 mesh densification close to the interfaces between domains, where large gradients in the solved variables are expected. The mathematical treatment of the oscillatory concentrations resulted in original expressions for the diffusion of oxygen in unreacting nitrogen, an observation previously unpublished, to the author’s knowledge. Results for the intact cell compared well with published experimental impedance data, using only two fitting parameters per electrode, i0 and Cdl.  The assumptions made regarding the nature and characteristics of the electrochemical reactions proved to be reasonable in view of experimental observations of the impedance from previous work. Nonetheless, the framework of the presented model allows future users to modify the source term of the model equations in accordance with new experimentally observed phenomena. The presence of the ribbed interconnect plates resulted in a decrease in overall performance with respect to the button cell configuration, each in the intact case.  Local reactant depletion underneath the interconnect ribs is responsible for this phenomenon, and it manifests itself even at low current levels. The numerical simulation of diverse degradation mechanisms revealed distinct features in the way they impact the impedance spectrum.  Under certain conditions that are degradation mode specific, it is expected that they would be identifiable.  The thesis focused on electrode delamination, interconnect oxidation, interconnect detachment, and sulfur poisoning.  The path followed by the series and polarization resistances proved to be an important indicator of the kind of degradation mode acting on the cell.  The introduction of two normalized variables, the normalized series resistance and the normalized polarization resistance, provided a consistent way of comparing the effects of different degradation modes.  These normalized resistances change upon increasing the extent of degradation, with the measure of intactness decreasing from unity, in the intact state, toward zero.  The lower the resistance ratios, the larger the relative increase in resistance, and the less intact the cells.  A sensitivity analysis of these normalized resistances assessed the predictive capability of the model as a function of the variation in input parameters.  For a given degradation mode and extent of degradation, the series and polarization resistances were calculated thirty times using thirty different combinations of 139 the relevant input parameters, each spanning a broad dynamic range.  The next few paragraphs describe the robustness of the model in identifying each degradation mode under study. 1,2,3,4,5,6 Electrode delamination affects cell performance by interrupting the ionic current between electrode and electrolyte [1,2].  As a consequence, and because of the very large aspect ratio of the electrolyte and thereby its low in-plane conductance, electrochemical activity is prevented throughout the electrode volumes above and below the delamination area.  Effectively, the fuel cell geometric area becomes smaller by an amount equal to the delamination area.  The combination of flow path blocking and electrochemical shadowing causes a simultaneous and equivalent increase in series and polarization resistance, in proportion to the delamination area.  The present model predictions were in agreement with published experimental impedance data, measured on single-cell experiments in which delamination was observed or suggested as the cause for degradation [3-6].  This is the only degradation mode among those studied in which there is a direct correspondence between series and polarization resistance.  The nature of the electrode reactions remains unaltered, inducing no change in the impedance characteristic frequencies.  This behaviour is a signature of delamination.  Delamination is slightly more severe for thin electrolytes, albeit not to an extent of practical importance, and is equivalent for the delamination of either anode or cathode.  Therefore, the observation of a simultaneous and proportional increase in both series and polarization resistance is an indication of the occurrence of delamination.  A sensitivity analysis revealed that the impedance changes induced by electrode delamination are largely independent of inaccuracies in the model input parameters.  These changes are also quite insensitive to the SOFC geometrical configuration, affecting both button cells and large rectangular cells.  A second degradation mode associated with conductivity hindrance studied in this thesis is interconnect oxidation.  Simulated by the insertion of a low-conductivity layer between the interconnect and the cathode, this degradation mode mainly affects performance by increasing the SOFC series resistance, in proportion to the interlayer thickness [2].  Further evidence of the presence of this degradation mode is the oxide growth rate.  According to experimental studies performed in SOFC relevant conditions, 140 it is expected that the oxide layer thickness increases as t0.5.  If this functional form is observed for the series resistance, interconnect oxidation is a suitable candidate to be responsible for the cell degradation.  A small change in electrode arc size may appear in potentiostatic mode as a consequence of the current density dependence of the polarization resistance, but this contribution is small compared with the change in Ohmic resistance.  The characteristic frequency of the electrodes is expected to stay invariant because of the Ohmic (as opposed to electrochemical) nature of oxide layer growth. Performance deterioration depends on the contact area between the interconnect and the electrode: the lower the rib/(rib+channel) width ratio, the more severe the increase in series resistance.  For equally wide rib and channel, the degradation is essentially insensitive to total width of the rib+channel.  The model predictions for the change in series resistance were largely insensitive to variations in the input parameters, as long as the electrolyte thickness was known with accuracy.  Loss of physical contact may also appear between the electrode and the interconnect, a type of degradation mode known as interconnect detachment [2].   The simulation of this type of problem yielded interesting results in terms of the physical difference between interconnect detachment and electrode delamination.  The loss of performance resulting from detachment is less severe than that caused by delamination for an equivalently affected area, since electrons, the blocked species in this case, move laterally in the electrodes to a much larger extent than ions move laterally in the electrolyte.  Consequently, this mechanism produces an increase in series resistance, with a smaller increase in polarization resistance, whose magnitude increases for larger detached areas.  There is no substantial change in peak frequency.  Unlike oxide layer growth, this phenomenon is prone to occur triggered by thermal cycling, as opposed to continuously over time.  This behaviour is useful to distinguish detachment from interconnect oxidation.  Unlike delamination, the increase in polarization resistance is slower than the increase in series resistance, since they are not in general proportional.  Sulfur poisoning affects SOFC performance by adsorption of sulfur on electrochemically active sites within the porous anode, resulting in blocking of possible hydrogen adsorption sites [7].  The thin adsorbed layer of sulfur or other sulfur containing species that is responsible for the anode poisoning does not change the 141 microstructural morphology of the electrode in small concentrations, and thus it is expected that sulfur poisoning would not affect the diffusional behaviour of the electrode, especially at early stages when deterioration is reversible but important.  By lowering the active surface area to simulate poisoning, the model results show an increase in polarization resistance without change in series resistance, in agreement with experimental observations.  The clear detection of an individual arc increasing in size is subject to its identification within the overall spectrum.  The degradation caused by poisoning is more severe for thin anodes than it is for thick anodes, since thick electrodes offer the possibility for the reaction to redistribute to sites further away from the electrolyte, where the electrochemical activity was lower in the intact case.  Diagnosing this degradation mode using the total polarization resistance may be difficult because of the influence of the change in total current density on the cathodic polarization resistance. Further research work is required for this degradation mode to be diagnosed with confidence.  Table 7-1 summarizes the qualitative and quantitative changes expected in the series resistance, polarization resistance, and peak frequency, for all the degradation modes considered in this thesis. Table 7-1.  Summary of the qualitative and quantitative changes expected in the series and polarization resistance, and in the peak frequency, for the four degradation modes under study. degradation type Rs Rp fp Cathode delamination increases proportionally to delaminated area increases proportionally to delaminated area no change Interconnect oxidation increases proportionally with oxide layer thickness small or no change no change Interconnect detachment increases with extent of detachment small to moderate change no change Sulfur poisoning no change anode arc increases, cathode arc may change due to interactions small increase  142 The strong coupling among concurrent processes within an SOFC, and the large number of input parameters involved in a simulation, suggested the possibility that unexpected interactions may adversely affect the diagnosing capability of the presented method [8].  The model results were assessed in terms of their possible dependence on parameter interaction, as well as in terms of their sensitivity to variation in the input parameters.  Using an experimental design strategy, individual degradation situations were repeated for a set of input parameters statistically varied within a wide and reasonable range.  Normalized resistance results for electrode delamination showed almost no dependence on parameter variation, and were the most insensitive to possible interactions.  This invariance indicates that the predictive capability of this model for delamination is very robust to input parameter inaccuracies.  Sulfur poisoning, on the other hand, presents the largest scatter because of the interaction between the polarization resistances of both electrodes.  The current density dependence of the cathodic polarization resistance caused the cathode arc to change in size upon anode poisoning simulation.  Interconnect oxidation results for the change in series resistance are robust to input parameter variations, as long as the electrolyte thickness is known with accuracy. The same statement applies to cathode interconnect detachment, with the requirement of knowing the thickness and porosity of the cathode in contact with the detached interconnect.  Interconnect detachment on the anode side presented smaller scatter than that on the cathode side. The sensitivity analysis of the results also provided a rough estimation of the validity of the two-dimensional and isothermal approximations.  By including the inlet partial pressures and electrode kinetic parameters among the varied input parameters, the analysis statistically reproduced the situation of in-plane reactant depletion and temperature (hence kinetics) variation. Finally, a study of the effect of two concurrent degradation modes on the impedance revealed the extent to which the proposed method could diagnose degradation in several two-degradation-mode scenarios.  The convoluted nature of a two-degradation- mode situation, in addition to the expected scatter of its experimental measurement, constitutes a major challenge of the proposed method, and substantial further work is needed in this direction.  The method’s ability to separate individual degradation 143 mechanisms improves with knowledge of the degradation path, i.e. the state history. Even when several possible combinations of degradation modes may induce the same final overall impedance change, the path along which that state was reached provides valuable information that may be used for better diagnosis. 7.2 Outlook The two dimensional approximation is a limitation of the presented model. Extending the model to three dimensions would allow accounting for the effect of variation in reactant concentration over the plane of the cell, as well as including the possibility to simulate cross flow manifolding.  Additional requirements of performing this extension include:  a. Incorporating energy balance into the equations to account for heat transfer in the in-plane direction, and the uneven heat generation resulting from the varying reactant concentrations.  b. Using a more efficient meshing strategy with no unnecessary refining, and computational hardware suitable for the much larger resulting system. Secondly, the internal reforming reaction could be an interesting addition to the simulation, since the use of hydrocarbons appears as a very realistic opportunity for SOFCs in the short- to mid-term. Modern SOFC electrodes consist of a support layer, or gas-diffusion layer, and a functional layer in contact with the electrolyte, not modeled in the work presented in this thesis.  The finer structure of the functional layer increases the active surface area, thereby increasing performance.  Although not complicated from the coding viewpoint, incorporating these extra elements also requires reliable data on microstructural properties, especially active surface area and diffusional properties. Degradation mechanisms such as cathode chromium poisoning and anode sintering also affect the cell performance by reducing the electrode surface area.  In addition, the porous structure is also affected, and consequently the diffusional properties. As soon as quantitative experimental data are available on how these changes take place, those changes could be incorporated into the model for the analysis of these degradation modes. 144 The source term in the transport equations describes the nature of the electrochemical processes occurring at the triple-phase boundaries in the SOFC electrodes.  As mentioned in earlier chapters, there is no general consensus on the functional form of this term, or on the microscopic details of the relevant reactions.  The model infrastructure presented in this work could, however, be used along with suitable modifications of the source term, when better understanding of the problem is acquired. One possibility not explored in this thesis is, for example, to express this source term as a generalized series of processes, each having an RC behaviour, weighted according to a previous fitting.  Regardless of the nature of each process, such a combination could be general enough to represent reality to an adequate extent. The impedance signatures of delamination resulted from electrode/electrolyte de- bonding whose area was large compared to the cell layer thicknesses.  Considering a scenario of multiple micro-delaminations would be useful to assess a situation of incipient delamination. Extending the applicability of the presented technique to multi-cell stacks would be a major piece of work stemming from this thesis.  The concepts described in this work in its present form require individual probing of each cell in a stack.  Substantial future work would be necessary to develop a technique that could diagnose cell degradation based on the whole stack impedance. 7.3 Concluding remarks The present work is intended as a first step toward the development of a more comprehensive degradation diagnosis method for in-service SOFCs.  In its present state, it is useful as an aid in the understanding of degradation of SOFCs at the R&D stage, where there is as yet no systematic approach to interpreting impedance evolution over time. It is the hope of the author that the ideas presented in this work are pursued further in our attempt to understand nature and to contribute to its sustainability.  145 7.4 References  [1] J. I. Gazzarri and O. Kesler, “Non-destructive delamination detection in solid oxide fuel cells”, Journal of Power Sources, 167 (2) 430-441 (2007). [2] J. I. Gazzarri and O. Kesler, “Short stack modeling of degradation in solid oxide fuel cells - Part I: Contact degradation”, Journal of Power Sources, doi: 10.1016 / j.jpowsour.2007.10.047. [3] S. Simner M. Anderson, M. Engelhard, and J. Stevenson, “Degradation Mechanisms of La–Sr–Co–Fe–O3 SOFC Cathodes”, Electrochemical and Solid- State Letters, 9 (10) A478-A481 (2006) [4] A. Hagen, R. Barfod, P. Hendriksen, and Y. Liu, “Effect of operational conditions on long term stability of SOFCs”, Proc. SOFC IX, Vol. 1, 2005, 503-513. [5] M. Heneka, N. Kikillus, and E. Ivers-Tiffée, “Design of experiments for lifetime modelling of SOFC” , Proc. SOFC IX, Vol. 1, 2005, 908-919. [6] Y. Hsiao and R. Selman, “The degradation of SOFC electrodes”, Solid State Ionics 98, (1998) 33-38 [7] J. I. Gazzarri and O. Kesler, “Electrochemical AC impedance model of a solid oxide fuel cell and its application to diagnosis of multiple degradation modes”, Journal of Power Sources, 167 (1) 100-110 (2007). [8] J. I. Gazzarri and O. Kesler, “Short Stack Modeling of Degradation in Solid Oxide Fuel Cells - Part II: Sensitivity and Interaction Analysis”, Journal of Power Sources, doi : 10.1016 / j . jpowsour . 2007.10.046. 146 I. Appendix One.  Geometry and Mesh Generation. Unlike the previously reported SOFC impedance models, the present model includes one in-plane dimension in addition to the through-cell direction.  This is an essential requirement for the simulation of electrode delamination and interconnect detachment.  One difficulty associated with this geometry is the large size of the numerical problem to be solved, especially for the case that includes the ribbed interconnect.  In addition to the classical numerical mesh refinement close to the electrochemical interfaces, some element densification is required at the corners formed by the ribs and channels, since there are large variations in the current density flow direction close to these corners.  While the 1-D version of the present model consists of less than 200 degrees of freedom, the 2-D axisymmetric and the 6-rib rectangular geometries present 10,700 and 36,000 degrees of freedom, respectively.  Away from edges and corners, the fuel cell quantities vary to a larger extent in the through-cell direction than they do in the lateral direction.  This suggests the need to use slender elements to avoid unnecessary lateral densification, ruling out the use of equi-axed, automatically generated triangular elements.  The most suitable alternative to these elements was the rectangular element, whose Jacobian matrix tolerates high aspect ratios, at the expense of relinquishing the automated generation.  The geometry of the cell is divided into rectangles of various sizes, and their physical properties set according to the part of the cell that each represents.  The subdivision of the edges of these rectangles defines the elements of the rectangular mesh.  Providing the model with flexibility in the geometry requires the careful coding of both the geometry and the mesh generation, to ensure the matching of adjacent elements throughout.  Figure I-1 shows successive blow- ups of the geometry of the rib-cathode area. 147 oxide layer (5micron) delamination (0.2micron) cathode electrolyte electrolyte cathoderib rib current collector  Figure I-1.  Geometry details.  Upon a change in physical properties, a rectangle between the rib and the cathode becomes an oxide layer or a detachment, and a rectangle between the cathode and the electrolyte becomes a delamination.   Figure I-2 shows the meshing strategy corresponding to the geometry of Figure I-1.  Mesh refinement at interfaces and around corners ensures adequate resolution in the solved variables.  Unnecessary mesh refinement in regions like those indicated by the arrows is an undesired consequence of the rectangular meshing strategy.  A more elaborated geometry/mesh development would improve the overall numerical efficiency and allow the simulation to include a larger number of ribs.  148 cathode electrolyte rib zones of unnecessary densification  Figure I-2.  Global meshing strategy and detail showing densification around the rib/cathode/channel corner Mesh Density and Convergence  An adequate mesh densification at locations where the solved variables present large gradients is a requirement for convergence and precision in the results.  In the simulation of fuel cells, these locations are the interface of the electrodes and the electrolyte, where the local current density changes more abruptly than anywhere else in the electrode domain.  The baseline case presented in this work (Figure 3-2) required 1 micron thick elements (second order Lagrangian) in the main direction of the current.  A simple 1D model is useful for convergence studies, since its mesh can easily be densified until adequate accuracy is achieved.  In this way, the element size is determined using the 1D model, and used thereafter in more elaborate geometries.  Table I-1 details the number of degrees of freedom corresponding to each geometry, along with the CPU time required for a computer using an Intel Pentium 4 CPU with 1GB ram. 149  Geometry # DoF Problem # iter Time (s) Full, ASC 36945 DC (nonlinear) 2 8 -  36945 AC (linear) - 606 61 freq Full, ESC 32361 DC (nonlinear) 2 7 -  32361 AC (linear) - 476 61 freq Half rib, half channel, ASC 9951 DC (nonlinear) 2 2.6 -  9951 AC (linear) - 139 61 freq Half rib, half channel, ESC 7491 DC (nonlinear) 2 2.2 -  7491 AC (linear) - 92.1 61 freq Table I-1.  Computational demand for each intact case considered.  MATLAB/COMSOL code for the programming of the geometry The numerical solution to the problem addressed in this thesis is obtained using the linear and nonlinear solvers of COMSOL 3.2, called from its MATLAB interface, where the geometry and meshing are previously specified.  This strategy is convenient for solving the problem with different physical and geometrical configurations, as well as for the performance of parametric studies. The SOFC is modeled as a set of rectangles.  Each SOFC component constitutes a domain, with one or more differential equations solved for the relevant unknown functions.  The 2-D object rect2 specifies the domain geometry.  Input parameters are specified in SI units (m, kg, sec) throughout. Relevant constants: L total width anot anode thickness catt cathode thickness elyt electrolyte thickness p rib width q channel width r rib and channel height del delamination thickness it interlayer or detachment thickness 150 nlan number of modeled ribs tT anot+elyt+catt cca height current collector anode ccc height current collector cathode  Input file. This function is called at the beginning of the code, and it loads all the input parameters necessary for the run of interest. function [obj R F L a t crack ndel del elyt anot catt Temp T Vcell Vrefc Vrefa Sa Sc kinc kina alphaaa alphaac alphaca alphacc tT ki kLSM kNi pora porc tora torc kia kic kea kec kair kaira kairc Cdla Cdlc epsair epsely expon PhiAC catdel p q r it cca ccc nlan nilr kit kila kilc epsila epsilc DObulk DHbulk DO DH cH0 cO0 varia MUD] = inputparam(k,nrun,varia)  % Universal constants R = 8.314; F = 96485; UD = 0    % Uniform design or regular experiment interl = 0    % interlayer or not crack = 0    % delaminated or intact it = 5E-6;          % interlayer thickness t = 0.2E-6;         % delamination thickness del = 0.2E-6;       % delamination thickness for joint degradation ndel = 1;           % number of rib/channel blocks affected by delamination nlan = 6;           % number of modeled landings (minimum 2) nilr = 2;           % number of detached landings (minimum 1, the intact case is achieved setting interl to 0 as seen below)  % Operating conditions Temp = 850;         % Temperature [C] T = Temp+273; % Temperature [K] catdel = 1;   % = 1 for cathode delamination               % = 0 for anode delamination  % Calculation of OCV based on channel pressures pH0 = 0.97;    % Hydrogen (atm) pH2O0 = 0.03;  % Water (atm) pO0 = 0.21;   % Oxygen (atm) cH0 = pH0*101300/R/T; % Hydrogen (mol/m3) cO0 = pO0*101300/R/T; % Oxygen (mol/m3) cH2O0 = pH2O0*101300/R/T; % Water (mol/m3)  % Reference concentrations that will be compared to local concentrations cHref = 0.97*101300/R/T; cWref = 101300/R/T-cHref; cOref = 0.21*101300/R/T;  Del_H0 = -241.82E3;  % Enthalpy of formation of water Del_S0 = -44.37;  % Entropy of formation of water DG = Del_H0 - T * Del_S0; % Gibbs free energy E0T = -DG/2/F;  % OCV std conditions Vrefc = E0T+R*T/2/F*log(pH0 * pO0^0.5 / (1-pH0)); % OCV Vrefa = 0;   % Anode reference  Vcell = 0.7;   % DC operating point  % Microstructural parameters S = 1e6; % Surface area 151 Sa = S; % Surface area anode Sc = S;  % Surface area cathode  xia = 0.6;   % Volume fraction of ionic phase in the anode xea = (1-xia);  % Volume fraction of electronic phase in the anode xic = 0.5;   % Volume fraction of ionic phase in the anode xec = (1-xic);  % Volume fraction of electronic phase in the anode ria = 1;   % Average radius of ionic particles anode (micron) rea = 1;   % Average radius of ionic particles anode (micron) ric = 1;   % Average radius of ionic particles cathode (micron) rec = 1;   % Average radius of ionic particles cathode (micron)  Zcoord = 6; nia = xia / (xia + xea * (ria/rea)^3); nea = (1-nia); nic = xic / (xic + xec * (ria/rea)^3); nec = (1-nic); Zia = 3+(Zcoord-3)*(ria/rea)^2/(nea+(1-nea)*(ria/rea)^2); Zea = 3+(Zcoord-3)            /(nea+(1-nea)*(ria/rea)^2); Ziia = nia * Zia^2/6; Zeea = nea * Zea^2/6; Ppia = (1-((4.236-Ziia)/2)^2.5)^0.4; Ppea = (1-((4.236-Zeea)/2)^2.5)^0.4; Zic = 3+(Zcoord-3)*(ric/rec)^2/(nec+(1-nec)*(ric/rec)^2); Zec = 3+(Zcoord-3)            /(nea+(1-nec)*(ric/rec)^2); Ziic = nic * Zic^2/6; Zeec = nec * Zec^2/6; Ppic = (1-((4.236-Ziic)/2)^2.5)^0.4; Ppec = (1-((4.236-Zeec)/2)^2.5)^0.4;  torc = 3; % cathode tortuosity tora = 3; % anode tortuosity porc = 0.4; % cathode porosity pora = 0.4; % anode porosity  % Geometry elyt = 150E-6; % Ely Thickness [m] anot = 40e-6;       % Anode thickness [m] catt = 40e-6; % Cathode thickness [m] tT = anot+catt+elyt;% Total thickness  % Electrochemistry kinc = 25; % cathode local exchange current density kina = 50;  % cathode local exchange current density kYSZ = 100 * exp(0.0127*T-17.463);  % Conductivity of YSZ ki = kYSZ;  % Charge transfer coefficients alphaaa = 1.2; % anode anodic direction % Gonzalez-Cuenca alphaac = 0.8; % anode cathodic direction % Gonzalez-Cuenca alphaca = 1.5; % cathode anodic direction % Kenney - Karan alphacc = 0.5; % cathode cathodic direction % Kenney - Karan  % Conductivities kLSM = 100 * (0.244*T-8.412);       % S/m electronic conductivity of LSM kNi = 100*1/(10*1e-6*(1+5*(Temp-20)/1000)); % S/m electronic conductivity Ni D. % Pollock, Physical Properties of Materials for Engineers, 2nd ed., CRC Press, % p210 (1993) Cdla = 0.4;  % Double layer capacitance anode Cdlc = 90;   % Double layer capacitance cathode  % Probability of percolation approximated as fraction of phase kia = ki * (1-pora) * xia * xia; % Anode ionic conductivity 152 kic = ki * (1-porc) * xic * xic; % Cathode ionic conductivity kea = kNi * (1-pora)* xea * xea; % Anode electronic conductivity kec = kLSM * (1-porc)*xec * xec; % Cathode electronic conductivity  % Diffusion parameters % Bulk diffusion coefficients were calculated using the Lennard Jones % relatioship sigmaO2N2  = 0.5*(3.433+3.681); sigmaH2H2O = 0.5*(2.968+2.649); omegaO2N2  = 0.74; omegaH2H2O = 0.74; DObulk = 1.86E-7*T^1.5*(1/16+1/28)^0.5/(sigmaO2N2^2*omegaO2N2); DHbulk = 1.86E-7*T^1.5*(1/2+1/18)^0.5/(sigmaH2H2O^2*omegaH2H2O); DO = DObulk * porc / torc; % Oxygen effective conductivity DH = DHbulk * pora / tora; % Hydrogen effective conductivity  % Interconnect geometry p = 2E-3; %  landing width q = 2E-3; %  channel width r = 2E-3; %  landing height cca = 2E-3; %  interconnect vertical part anode ccc = 2E-3; %  interconnect vertical part cathode  % Uniform design experiment MUD=0; if UD == 1     auxvar = 20  % 21 for delamination because includes delam thickness     MUD=csvread(['C:\Uniform Design\UD_',num2str(auxvar),'.csv'],0,0,[0 0 29 auxvar-1]);     Nud = 30;     cnt = 1;  %% Anode S %     if nrun == 1 %         Sastp = linspace(1e5,1e6,Nud); Sa = Sastp(MUD(k,cnt)); varia(k,cnt) = Sa; cnt = cnt+1; %     else %         Sastp = linspace(1e5,1e6,Nud); Sa = 0.1*Sastp(MUD(k,cnt)); varia(k,cnt) = Sa;  cnt = cnt+1; %     end     Sastp = linspace(1e5,1e6,Nud); Sa = Sastp(MUD(k,cnt)); varia(k,cnt) = Sa; cnt = cnt+1; %% Cathode S     Scstp = linspace(1e5,1e6,Nud); Sc = Scstp(MUD(k,cnt)); varia(k,cnt) = Sc; cnt = cnt+1; %% Anode - anodic alpha     alphastp = linspace(0.5,2,Nud); alphaaa = alphastp(MUD(k,cnt)); varia(k,cnt) = alphaaa; cnt = cnt+1; %% Anode - cathodic alpha     alphastp = linspace(0.2,1,Nud); alphaac = alphastp(MUD(k,cnt)); varia(k,cnt) = alphaac; cnt = cnt+1; %% Cathode - anodic alpha     alphastp = linspace(0.5,2,Nud); alphaca = alphastp(MUD(k,cnt)); varia(k,cnt) = alphaca; cnt = cnt+1; %% Cathode - cathodic alpha     alphastp = linspace(0.2,1,Nud); alphacc = alphastp(MUD(k,cnt)); varia(k,cnt) = alphacc; cnt = cnt+1; %% Anode porosity     porstp =   linspace(0.1,0.7,Nud);  pora = porstp  (MUD(k,cnt)); varia(k,cnt) = pora;    cnt = cnt+1; %% Cathode porosity     porstp =   linspace(0.1,0.7,Nud);  porc = porstp  (MUD(k,cnt)); varia(k,cnt) = porc;    cnt = cnt+1; 153 %% Anode and cathode tortuosity     torstp =   linspace(2,7,Nud);      tora = torstp  (MUD(k,cnt)); varia(k,cnt) = tora;    cnt = cnt+1;                                        torc = torstp  (MUD(k,cnt)); varia(k,cnt) = torc;    cnt = cnt+1; %% Electrolyte thickness     elytstp=linspace(150e-6,150e-6,Nud);   elyt =    elytstp(MUD(k,cnt)); varia(k,cnt) = elyt;      cnt = cnt+1; %     elytstp=linspace(150e-6,150e-6,Nud);   elyt =    elytstp(MUD(k,cnt)); varia(k,cnt) = elyt;      cnt = cnt+1; %% Anode thickness     anotstp=linspace(10e-6,100e-6,Nud);  anot = anotstp(MUD(k,cnt)); varia(k,cnt) = anot;      cnt = cnt+1; %% Cathode thickness     cattstp=linspace(10e-6,100e-6,Nud); catt =    cattstp (MUD(k,cnt)); varia(k,cnt) = catt;      cnt = cnt+1; %% Double layer capacitances     Cdlstp=logspace(-1,2,Nud);          Cdla =    Cdlstp  (MUD(k,cnt)); varia(k,cnt) = Cdla;      cnt = cnt+1;                                     Cdlc =    Cdlstp  (MUD(k,cnt)); varia(k,cnt) = Cdlc;      cnt = cnt+1; %% Partial pressures at the inlet     p0cstp=linspace(0.1,0.21,Nud);      pO0 =     p0cstp  (MUD(k,cnt)); varia(k,cnt) = pO0;       cnt = cnt+1;     p0astp=linspace(0.1,0.97,Nud);      pH0 =     p0astp  (MUD(k,cnt)); varia(k,cnt) = pH0;       cnt = cnt+1; %     p0cstp=linspace(0.21,0.21,Nud);      pO0 =     p0cstp  (MUD(k,cnt)); varia(k,cnt) = pO0;       cnt = cnt+1; %     p0astp=linspace(0.97,0.97,Nud);      pH0 =     p0astp  (MUD(k,cnt)); varia(k,cnt) = pH0;       cnt = cnt+1; %% Vcell and exchange current densities     vstp = linspace(0.2,0.99,Nud);      v =       vstp    (MUD(k,cnt)); varia(k,cnt) = v;         cnt = cnt+1;     istp = logspace(1,3,Nud);           kina =    istp    (MUD(k,cnt)); varia(k,cnt) = kina;      cnt = cnt+1;                                         kinc =    istp    (MUD(k,cnt)); varia(k,cnt) = kinc;      cnt = cnt+1; % if auxvar == 21 %     tstp = linspace(0.1e-6,5e-6,Nud);   t =       tstp    (MUD(k,cnt)); varia(k,cnt) = t;         cnt = cnt+1; % end % % Repetition of the variables that depend on UD variables     Vrefc = E0T+R*T/2/F*log(pH0 * pO0^0.5 / (1-pH0));     Vcell = Vrefc * v;     kia = ki * (1-pora) * xia * xia;     kic = ki * (1-porc) * xic * xic;     kea = kNi * (1-pora)* xea * xea;     kec = kLSM * (1-porc)*xec * xec;     DO = DObulk * porc / torc;     DH = DHbulk * pora / tora;     tT = anot+catt+elyt;     pH2O0 = 1-pH0;     cH0 = pH0*101300/R/T;     cO0 = pO0*101300/R/T;     cH2O0 = pH2O0*101300/R/T; end  kair = 1E-10; % conductivity of air epsair = 8.85E-12;  % permittivity of air (turn to zero when calculating using 10^10 Hz) epsely = 30*epsair; % permittivity of zirconia PhiAC = 0.02; % perturbation size (V)  154 L = 0.010;  % button cell radius a = L * (0.3)^0.5; % delam radius a = L-a;            % flip needed to respect the geometry % definition kit = 8e5;          % interconnect conductivity  oxide = 1 % oxide layer (0) or detachment (1)  if oxide == 0     oxlyra = 100*10^-2.6;  % anode interlayer conductivity     oxlyrc = 1;            % cathode interlayer conductivity elseif oxide == 1     oxlyra = 1E-15;  % anode interlayer conductivity     oxlyrc = 1E-15;  % cathode interlayer conductivity end kila = kit;   % anode interlayer conductivity kilc = kit;   % cathode interlayer conductivity epsila  = 30*epsair; % interlayer permittivity anode epsilc  = 30*epsair; % interlayer permittivity cathode  if interl == 0     kairc = kit;     kaira = kit;     epsaira = epsila; epsairc = epsilc; elseif interl == 1  % cathode detachment or oxide layer growth     kairc = oxlyrc;     kaira = kit;     epsairc = epsair; epsaira = epsila;  % permittivity of air F/m elseif interl == 2  % anode detachment or oxide layer growth     kairc = kit;        kaira = oxlyra;     epsairc = epsilc; epsaira = epsair;  % permittivity of air F/m end  z0stp = [0 0.1 0.21]; z0 = 0.21;  % Oxygen volume fraction around which the expansion will be made A1 = (1-z0)^-1-z0*(1-z0)^-2; A2 = (1-z0)^-2;  % Constants need to be loaded in the fem structure for COMSOL  fem.const={ 'F',    '96485',...             'Vrefa',num2str(Vrefa),...             'i0a',num2str(kina), ...             'i0c',num2str(kinc), ...             'R','8.314',...             'Vrefc',num2str(Vrefc),...             'Sa',num2str(Sa),...             'Sc',num2str(Sc),...             'ki',num2str(ki),...             'Vcell',num2str(Vcell),...             'kiano',num2str(kia),...             'kicat',num2str(kic),...             'kea',num2str(kea),...             'kec',num2str(kec),...             'kit',num2str(kit),...             'kila',num2str(kila),...             'kilc',num2str(kilc),...             'kaira',num2str(kaira),...             'kairc',num2str(kairc),...             'kair',num2str(kair),...             'T',num2str(273+Temp),...             'PhiAC',num2str(PhiAC),...             'f','0',...             'alphaaa',num2str(alphaaa),...             'alphaac',num2str(alphaac 155             'alphaca',num2str(alphaca             'alphacc',num2str(alphacc             'epsair',num2str(epsair),...             'epsaira',num2str(epsaira),...             'epsairc',num2str(epsairc),...             'epsely',num2str(epsely),...             'epsila',num2str(epsila),...             'epsilc',num2str(epsilc),...             'epsely',num2str(epsely),...             'Cdla',num2str(Cdla),...             'Cdlc',num2str(Cdlc),...             'p',num2str(p),...             'q',num2str(q),...             'r',num2str(r),...             'nlan',num2str(nlan),...             'cH0',num2str(cH0),...             'cO0',num2str(cO0),...             'cH2O0',num2str(cH2O0),...             'cHref',num2str(cHref),...             'cOref',num2str(cOref),...             'cWref',num2str(cWref),...             'DH',num2str(DH),...             'DO',num2str(DO),...             'DObulk',num2str(DObulk),...             'DHbulk',num2str(DHbulk),...             'i00a',num2str(kina), ...             'i00c',num2str(kinc),...             'RTp',num2str(R*T/101300),...             'A1',num2str(A1),...             'A2',num2str(A2),...             'kYSZ',num2str(kYSZ),...             'epsYSZ',num2str(30*epsair),...             };  obj=fem.const;  Main Program flclear fem clear vrsn vrsn.name = 'FEMLAB 3.1'; vrsn.ext = ''; vrsn.major = 0; vrsn.build = 157; vrsn.rcs = '$Name:  $'; vrsn.date = '$Date: 2004/11/12 07:39:54 $'; fem.version = vrsn;  [fem.const R F L a t crack ndel del elyt anot catt Temp T Vcell Vrefc Vrefa Sa Sc kinc kina alphaaa alphaac alphaca alphacc tT ki kLSM kNi pora porc tora torc kia kic kea kec kair kaira kairc Cdla Cdlc epsair epsely expon PhiAC catdel p q r it cca ccc nlan nilr kit kila kilc epsila epsilc DObulk DHbulk DO DH cH0 cO0 varia MUD] = inputparam(k,nrun,varia)  %%% Rectangles specified using dimensions and bottom left corner position %%% g1=rect2(L,anot    ,'base','corner','pos',[0,0]);   % anode g2=rect2(L,catt-del,'base','corner','pos',[0,anot+elyt+del]); % cathode g3=rect2(L,elyt    ,'base','corner','pos',[0,anot]);  % electrolyte  %%% Structure s stores the geometry information %%% 156 clear s s.objs={g1,g2,g3};  %%% The interconnect is generated based on nlan, the number of landing ribs %%% for i=4:2:2*nlan+3     s.objs(i)=rect2(p,it  ,'base','corner','pos',[(i-1-3)*(p+q)/2,tT]); end for i=5:2:2*nlan+3     s.objs(i)=rect2(p,r-it,'base','corner','pos',[(i-1-4)*(p+q)/2,tT+it]); end for i=2*nlan+4:2:4*nlan+3     s.objs(i)=rect2(p,r-it,'base','corner','pos',[(i-1-(2*nlan+3))*(p+q)/2,-r ]); end for i=2*nlan+5:2:4*nlan+3     s.objs(i)=rect2(p,it  ,'base','corner','pos',[(i-1-(2*nlan+4))*(p+q)/2,- it]); end  %%% Declaration of structure s.objs %%% s.objs(4*nlan+4) = rect2(L, cca , 'base','corner','pos' , [0, -r-cca  ]); s.objs(4*nlan+5) = rect2(L, ccc , 'base','corner','pos' , [0, tT+r  ]); for i=1:nlan     s.objs(4*nlan+5+i) = rect2(p, del, 'base','corner','pos' , [(i-1)*(p+q), anot+elyt]); end for i=1:nlan-1     s.objs(4*nlan+5+nlan+i) = rect2(q, del, 'base','corner','pos' , [i*p+(i- 1)*q, anot+elyt]); end for i=1:4*nlan+3+2+2*nlan-1     s.name(i)={['R' num2str(i)]};     s.tags(i)={['g' num2str(i)]}; end %%% Declaration of the geometrical substructures of fem, the basic structure of the code %%% fem.draw=struct('s',s); fem.sdim = {'x','y'}; fem.border = 1; fem=multiphysics(fem); fem.geom=geomcsg(fem);  157 Once the geometry is defined, a vector with values in the range [0,1] defines the rectangular mesh on every side of the rectangles.  This definition needs to be consistent for neighbouring elements, so that every shared edge has the same number of elements. The mesh density depends on the local variability of the function solved for.  Critical regions are the electrode/electrolyte boundaries, and rib/channel/electrode corners. Electrode/electrolyte boundaries need elements of about 1micron.  Mesh density coarsens away from the electrolyte for reasons of computational cost.  ele_size_cat = 1E-6;   % (metres) element size close to the interface length_cat = 4E-6;    % (metres) length along which the size above applies a1 = ele_size_cat/catt;  % normalized value for element size a2 = length_cat/catt;  % normalized distance over which the above applies mcat = [0:a1:a2 a2+0.2:0.2:1]; % electrode meshing [close to interface away from it] ele_size_ano1 = 1E-6;   % (meters) element size close to the interface ele_size_ano2 = 20E-6;   % (meters) element size midway between the interface and the CC(for ASC) length_ano1 = 4E-6;   % (meters) length along which the size above applies length_ano2 = anot/20;   % (meters) length along which the size above applies b1 = ele_size_ano1/anot;  % normalized value for element size b2 = length_ano1/anot;  % normalized distance over which the above b3 = length_ano2/anot;  % normalized distance over which the above mano = 1-fliplr(mcat);  % inversion of element order for anot = catt %%% ASC requires a different meshing strategy than ESC %%% if anot > 200E-6     mano = 1-fliplr([0:b1:b2 b2+0.01:0.01:b2+b3 b2+b3+0.2:0.2:1]); end  s1 = 0.05; % fraction of the landing that determines the vertical mesh refinement at the landing edges s2 = 0.1; % fraction of the landing over which the mesh is refined s5 = 0.2; % equivalent to s1 for the first and last landings s6 = 0.9; % equivalent to s2 for the first and last landings mlan  = [0:s1:s2 0.2:0.2:0.8 (1-s2):s1:1];  % Locally defined element distribution for a landing mlanf  = [0:s5:s6 (1-s2):s1:1];   % Locally defined element distribution for first landing mlanl  = [0:s1:s2 s5:s5:1];    % Locally defined element distribution for last landing % Subdomain concatenation requires each block to end one element before unity to avoid repetition mlan1 = [0:s1:s2 s5:s5:s6 (1-s2):s1:1-s1]; % Same as above for concatenation mlan1f = [0:s5:s6 (1-s2):s1:1-s1];  % first landing mlan1l = [0:s1:s2 s5:s5:1-s5];   % last landing s3 = 0.05;   % fraction of the gap that determines the vertical mesh refinement at the gap edges s4 = 0.1;   % fraction of the gap over which the mesh is refined mlan2 = [0:s3:s4 0.2:0.2:0.8  (1-s4):s3:1];  % Locally defined element distribution for a gap mlan3 = [0:s3:s4 0.2:0.2:0.8  (1-s4):s3:1-s3]; % Same as above for concatenation 158  % ms is the meshing definition structure.  The # elements is 2 x # edges.  The odd numbered positions are occupied by the numbers from 1 to #edges.  The even numbered % positions give the normalized vectors that define the element distribution on the corresponding % edge  nedge = 8+2*nlan*7+2*(nlan-1)+6+2*(nlan-1)+2*nlan+2*(2*nlan-1)-1;  % total number of edges nsub = 3+4*nlan+2+2*nlan-1; % total number of subdomains aux1 = []; aux2 = []; if nlan>2     aux1 = [48:24:48+24*(nlan-3)];     aux2 = [53:24:53+24*(nlan-3)]; end % vertical edges belonging to the interlayer edgeil = nonzeros([5 24 35:24:35+24*(nlan-2) aux1 nedge-7 15 29 41:24:41+24*(nlan-2) aux2 nedge-2]); % vertical edges belonging to the interconnect aux1 = []; aux2 = []; if nlan>2     aux1 = [46:24:46+24*(nlan-3)]; end edgeic=nonzeros([3 22 33:24:33+24*(nlan-2) aux1 nedge-8 17 31:24:31+24*(nlan-2) 43:24:43+24*(nlan-2) nedge-1]); aux1 = []; aux2 = []; if nlan>2     aux1 = [50:24:50+24*(nlan-3)]; end edgedel = nonzeros([11 26 38:24:38+24*(nlan-2) aux1 nedge-4]);  % initialization of mesh vector ms = cell(1,2*nedge); for i=1:nedge     ms(2*i-1) = {i};     ms(2*i  ) = {1}; end for i=1:nedge     if     i==7 | i==nedge-6         ms(2*i)={mano};   % anode     elseif i==9 | i==nedge-5         ms(2*i)={4};   % electrolyte     elseif i==13 | i==nedge-3         ms(2*i)={mcat};   % cathode     elseif i==1 | i==nedge-9 | i==19 | i==nedge         ms(2*i)={5};   % current collectors     elseif i==10 | i==2 | i==21  % electrolyte, top and bottom current collecting lines         aux1=[];         for kk=1:nlan-1             if kk==1                 aux1=[aux1    mlan1f*p+(kk-1)*(p+q)   kk*p+mlan3*q+(kk-1)*q];             else                 aux1=[aux1    mlan1*p+(kk-1)*(p+q)   kk*p+mlan3*q+(kk-1)*q];             end         end         aux1=[aux1    mlan1l*p+(nlan-1)*(p+q)];         ms(2*i)={[aux1 L]/L};     end     for kk=1:4*nlan         if i==edgeil(kk)             ms(2*i) = {2};         elseif i==edgeic(kk)             ms(2*i) = {5}; 159         end     end     for kk=1:2*nlan         if i==edgedel(kk)             ms(2*i) = {2};         end     end end ms(2*([4 6 8 12 14 16 18 20]))={mlanf};  for i=1:nlan-1     ms(2*([23 25 27 28 30 32]      +24*ones(1,6)*(i-1))) = {mlan2};     ms(2*([34 36 37 39 40 42 44 45]+24*ones(1,8)*(i-1))) = {mlan}; end ms(2*([34 36 37 39 40 42 44 45] +24*ones(1,8)*(nlan-2))) = {mlanl};  %%% Incorporation of the mesh into the fem structure %%% fem.mesh=meshmap(fem,'edgelem',ms);  %%% Plotting of the mesh %%% figure(20) axes('FontName','Helvetica','FontSize',14); axis off meshplot(fem); %ylim([-0.2*(tT+r),1.2*(tT+r)]); xlim([-p/10,L+p/10]);  Each equation needs to be solved separately, specifying the relevant domains and boundary conditions.  As an example, consider the DC electronic potential equations. The information is stored in the fem.appl substructure. clear appl % Clears appl from previous definitions appl.mode.class = 'FlPDEG'; % General PDE formulation (recommended) appl.dim = {'Ve','Ve_t'}; appl.name = 'DCele'; % Name appl.assignsuffix = '_DCele'; clear bnd % Boundary conditions bnd.r = {'-Ve','-Ve','Vcell-Ve'}; bnd.type = {'dir','neu','dir'}; % Type of boundary condition bnd.ind = 2*ones(1,nedge); bnd.ind(edgea) = 1; % edgea contains the anode subdomains bnd.ind(edgec) = 3;  % edgea contains the cathode subdomains appl.bnd = bnd; clear equ %%% Source terms: cathodic and anodic DC Faradaic current equ.f =    {'-ictc','0','-icta','0','0' ,'0','0'}; %%% Flux expression equ.ga = {{{'-kec*Vex'    ;'-kec*Vey'}},...           {{'-kit*Vex'    ;'-kit*Vey'}},...           {{'-kea*Vex'    ;'-kea*Vey'}},...           {{'-kila*Vex'   ;'-kila*Vey'}},...           {{'-kaira*Vex';'-kaira*Vey'}},...           {{'-kilc*Vex' ;'-kilc *Vey'}},...           {{'-kairc*Vex';'-kairc*Vey'}}}; %%% equ.ind(where to apply the equation) = what equation to apply equ.ind(subano)              = 3; % anode equ.ind(suba)                = 2; % anode interconnect equ.ind(subail(1:nilr))      = 5; % anode interlayer delaminated equ.ind(subail(nilr+1:nlan)) = 4; % anode interlayer intact equ.ind(subcat)              = 1; % cathode equ.ind(subc)                = 2; % cathode interconnect 160 equ.ind(subcil(1:nilr))      = 7; % cathode interlayer delaminated equ.ind(subcil(nilr+1:nlan)) = 6; % cathode interlayer intact %%% Initial value to begin the Newton iteration equ.init = {{'Vcell'},0,0,0,0,{'Vcell'},{'Vcell'}}; appl.equ = equ; fem.appl{3} = appl;  COMSOL labels the geometry objects and their edges following a bottom-to-top and left- to-right criterion.  Keeping the subdomain and edge labels in a vector (e.g. subail and edgeil) is very convenient and recommended in order to refer to these labels from anywhere in the code.  The key point is to define the logic of label assignment for a general geometry.  For this task, it is advisable to start with a simple geometry, e.g. three ribs / two channels, let COMSOL label the geometry, and find the general labeling pattern.  The source term definition uses the indexing described above, in essentially the same way: fem.expr = {'faa'   ,'alphaaa*F/R/T',...             'fac'   ,'alphaac*F/R/T',...             'fca'   ,'alphaca*F/R/T',...             'fcc'   ,'alphacc*F/R/T',...             'ictcAC','Sc*i0c*(fca*overAC*exp(fca*overpot) +cO/cOref*fcc*overAC*exp(-fcc*overpot)-cOAC/cOref*exp(fcc*overpot))',...             'ictaAC','Sa*i0a*(cH/cHref*faa*overAC*exp(faa*overpot) +cH2O/cWref*fac*overAC*exp(-fac*overpot)+cHAC/cHref*exp(+faa*overpot)- cWAC/cWref*exp(-fac*overpot))',...             'ictc'  ,'Sc*i0c*(exp(fca*overpot)-cO/cOref*exp(-fcc*overpot))',...             'icta'  ,'Sa*i0a*(cH/cHref*exp(faa*overpot)-cH2O/cWref*exp(- fac*overpot))',...             'tdcsa' ,'Sa*(Cdla*j*2*pi*f) *(VeAC-ViAC)',... % time dependent current source anode             'tdcsc' ,'Sc*(Cdlc*j*2*pi*f) *(VeAC-ViAC)'}; Each expression is defined globally, and the equ.ind structure assigns the source term to the appropriate domain.  The solution mode depends on the equation to be solved.  As explained in Chapter Two, the DC equations are nonlinear because the source term depends on the unknown potential in an exponential form.  The AC equations are linear, because they result from the linearization of the time-dependent conservation equations.  The procedure is as follows: 1- Solution of the DC potentials and DC concentrations 2- Parametric solution of the AC equations, using the DC solution as datum, and using 161 the frequency as the parameter.  Usually, five or six frequencies per decade is a reasonable compromise between resolution and computing time. 3- Integration of the AC current along the current collectors, for each frequency 4- Impedance calculation COMSOL functions for nonlinear and linear solving are femnlin and femlin, respectively. fem.sol=femnlin(fem, ...                 'nullfun','flspnull', ...                 'solcomp',{'Vi','Ve','cH','cO'},... % What variables to solve                 'outcomp',{'Vi','Ve','cH','cO'},... % What variables to output                 'maxiter',35); % Maximum number of Newton iteration fem0 = fem; fem.sol=femlin(fem, ...                 'init',fem0.sol, ...                 'nullfun','flspnull', ...                 'solcomp',{'ViAC','VeAC','cHAC','cOAC'},...                 'outcomp',{'Vi','Ve','ViAC','VeAC','cH','cO','cHAC','cOAC'},...                 'pname','f', ... % Parameter                 'plist',[logspace(-3,6,Nf) 10^10])  % Parameter range   162 II. Appendix Two.  On the linearization of nonlinear PDEs around the steady state point The small size of the AC perturbation imposed on the system in an impedance study allows the linearization of the transport equations, resulting in linear, time- independent, complex-valued equations, largely advantageous from a numerical standpoint.  This simplification is possible for a general nonlinear PDE when: I. The steady state solution exists II. The perturbation from it is small Consider the nonlinear PDE in F: )(2 FfF t F +−∇= ∂ ∂  ( 46 ) where f is any differentiable function, and the time dependence is only the result of a small sinusoidal perturbation of frequency ω.  If F  is the steady state function, known to exist, that satisfies )(0 2 FfF +−∇=  ( 47 ) Then the general solution can be approximated as the sum of the steady state solution plus a small sinusoidal variation, of the same frequency as the perturbation tj eFFF ω~+≈  ( 48 ) Replacing ( 48 ) in ( 46), )~(~~ 22 tjtjtj eFFfeFFFej ωωωω ++∇−−∇=  ( 49 ) The first order expansion of the non-homogeneous term around F is ( ) ( ) ( )tjtj eFFfFfeFFf ωω ~ !1 )~( ′+≈+  ( 50 ) which, replaced in equation ( 49 ) results in ( ) ( )( )tjtjtj eFFfFfeFFFej ωωωω ~~~ 22 ′++∇−−∇=  ( 51 ) Given that the steady state solution exists, the terms without the oscillatory factor vanish, leaving only ( )( )tjtjtj eFFfeFFej ωωωω ~~~ 2 ′+−∇=  ( 52 ) 163 The common time dependent term cancels out as well, ( )FfFFFj ′+−∇= ~~~ 2ω  ( 53 ) F is already known, so ( )Ff ′  is a constant; then, this equation is linear in F~ .  164 III. Appendix Three.  Fitting to an experimental polarization curve. For the purpose of comparison, the DC behaviour of the present model is fitted to the experimentally measured polarization curve presented by Kim et al in J. Electrochem Soc. 146 (1) 69-78 (1999).  The authors of this excellent work measured polarization curves of anode-supported button cells within the 650ºC-800ºC range, using the same materials as those simulated in the present model.  The cell anode consisted of a 750 micron thick Ni-8YSZ cermet, with 38% porosity.  The electrolyte was a thin 10 micron thick layer of 8YSZ.  The cathode was an LSM-8YSZ composite with approximately 40% porosity, 50 microns in thickness.  The button cells were tested in humidified hydrogen / air conditions, and the polarization curve at 800ºC is reproduced in the following graph, along with the result of the fitting of the present model, using the cathode volumetric exchange current density (Scat i0,c = 10E9 A/m3) and the hydrogen effective diffusivity (DH = 0.199E-4 m2/s, calculated by the authors) as the fitting parameters, and considering the anode as reversible (Sano i0,a = 1E14 A/m3 >> Scat i0,c), i.e., the cathode being the main contributor to the cell polarization.  The authors suggest that the anode works in the linear regime, although it is not possible to validate this assumption without individual measurement of each electrode’s polarization.  There is, therefore, an ambiguity in assigning the activation polarization contribution to each electrode when fitting to a full-cell polarization curve.  The open-circuit voltage in the experiment is approximately 0.1V lower than the theoretical OCV, and it was used directly as the OCV.  Finally, the charge transfer coefficients were assumed to be the same as those used at 850ºC in this model.  The cathode polarization curves at different temperatures presented by Co, Xia, and Birss in J. Electrochem. Soc., 152 (3) A570- A576 (2005) (Fig 8, pA574) suggest this assumption to be reasonable.  165 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 1 2 3 4 5 6 i (A/cm2) V C EL L (V ) Kim et al this work  Figure III-1.  Polarization curve fitting of the present model (line) to experimental data (dots) presented by Kim et al. Adapted from Fig 3a, J. Electrochem Soc. 146 (1) 69-78 (1999).   166 IV. Appendix Four.  Uniform design parameter grids The following tables detail the parameter combinations used in the Uniform Design numerical experiments of Chapter Six.  Table 6-1 lists the ranges of the 20 parameters (21 for the case of delamination) varied in the experiments.  In the next two matrices, the 30 rows correspond to each numerical run, and the 20 (or 21) columns correspond to the varied parameters. IV. 1. Delamination: anot catt elyt p0a p0c V a1 a2 a3 a4 Sa Sc pora porc tora torc Cdla Cdlc i0A i0C t # m m m atm atm - - - - - 1/m 1/m - - - - F/m2 F/m2 A/m2A/m2 m 1 3.17E-05 9.38E-05 1.51E-04 0.19 0.10 0.39 0.9 0.3 1.3 0.8 4.72E+05 8.76E+05 0.33 0.16 4.4 3.9 24.0 0.4 16 42 7.76E-07 2 5.03E-05 3.79E-05 1.09E-04 0.73 0.10 0.88 1.8 0.7 0.5 0.7 4.41E+05 5.03E+05 0.66 0.56 6.5 6.7 4.5 2.8 36 12 1.28E-06 3 3.48E-05 7.21E-05 1.28E-04 0.46 0.14 0.55 1.7 0.7 1.4 0.6 1.00E+05 3.79E+05 0.37 0.70 2.2 3.0 0.1 0.1 92 1000 2.69E-07 4 2.86E-05 1.62E-05 1.54E-04 0.91 0.11 0.66 1.1 0.5 1.5 0.5 6.90E+05 1.62E+05 0.14 0.45 6.1 5.4 62.1 0.2 108 728 3.82E-06 5 5.34E-05 5.03E-05 1.96E-04 0.25 0.15 0.45 0.7 0.4 0.6 0.7 8.14E+05 4.41E+05 0.64 0.35 6.7 2.3 0.2 0.5 386 281 3.31E-06 6 6.59E-05 5.97E-05 1.20E-04 0.40 0.20 0.20 0.6 0.6 0.9 0.2 4.10E+05 2.24E+05 0.43 0.37 3.2 6.3 7.3 0.1 30 30 4.16E-06 7 1.62E-05 8.14E-05 1.24E-04 0.88 0.18 0.77 1.4 0.9 0.8 0.9 5.97E+05 9.07E+05 0.45 0.20 5.1 6.1 0.2 0.2 204 240 2.30E-06 8 9.07E-05 2.55E-05 1.36E-04 0.61 0.12 0.74 1.0 0.3 0.8 0.8 1.93E+05 4.10E+05 0.22 0.14 3.4 4.9 0.1 24.0 67 36 3.48E-06 9 7.83E-05 2.86E-05 1.01E-04 0.82 0.13 0.28 1.4 0.4 1.1 0.3 3.79E+05 2.55E+05 0.49 0.18 5.3 3.2 11.7 5.7 853 204 6.07E-07 10 4.10E-05 5.34E-05 9.76E-05 0.13 0.17 0.80 1.5 0.3 1.1 1.0 9.38E+05 1.00E+05 0.16 0.49 2.5 3.4 2.2 1.7 12 127 2.13E-06 11 4.72E-05 4.10E-05 1.89E-04 0.70 0.17 0.34 0.8 0.8 1.8 0.8 2.24E+05 5.34E+05 0.18 0.27 2.9 6.8 5.7 0.9 530 386 1.45E-06 12 8.76E-05 8.45E-05 1.58E-04 0.94 0.20 0.42 1.8 0.4 1.6 0.7 5.03E+05 2.86E+05 0.12 0.47 4.2 2.7 0.3 3.6 329 10 1.96E-06 13 5.66E-05 4.72E-05 9.00E-05 0.49 0.18 0.61 0.5 0.5 1.9 0.9 1.31E+05 9.38E+05 0.41 0.58 7.0 3.7 78.8 78.8 281 92 2.47E-06 14 8.45E-05 4.41E-05 1.92E-04 0.79 0.16 0.72 1.3 0.9 1.0 0.4 2.86E+05 8.14E+05 0.29 0.64 4.9 2.5 18.9 0.3 14 16 3.14E-06 15 7.52E-05 6.90E-05 2.00E-04 0.52 0.16 0.99 2.0 0.4 1.9 0.6 7.83E+05 3.48E+05 0.39 0.22 4.6 6.5 14.9 100.0 19 853 2.80E-06 16 6.90E-05 0.0001 9.38E-05 0.64 0.13 0.31 1.6 0.9 2.0 0.3 9.69E+05 5.66E+05 0.24 0.29 6.8 4.4 1.1 0.7 49 67 2.97E-06 17 1.93E-05 9.07E-05 1.70E-04 0.67 0.18 0.36 1.6 0.2 0.7 0.4 3.17E+05 6.28E+05 0.60 0.53 3.6 4.6 48.9 14.9 57 329 3.65E-06 18 8.14E-05 1.31E-05 1.32E-04 0.34 0.19 0.91 1.2 0.2 1.8 0.3 7.21E+05 6.59E+05 0.62 0.62 4.1 5.1 0.3 0.3 174 108 1.11E-06 19 3.79E-05 7.52E-05 1.05E-04 0.76 0.15 0.53 0.9 0.3 1.3 0.4 8.45E+05 7.83E+05 0.27 0.66 4.8 7.0 0.9 30.4 621 19 4.32E-06 20 2.24E-05 2.24E-05 1.77E-04 0.55 0.19 0.23 1.1 0.9 1.0 0.8 8.76E+05 3.17E+05 0.35 0.60 5.4 4.8 0.7 48.9 26 79 4.38E-07 21 5.97E-05 1.00E-05 1.47E-04 0.16 0.14 0.50 1.9 1.0 0.6 0.3 5.34E+05 8.45E+05 0.10 0.43 2.0 5.8 3.6 62.1 240 174 2.63E-06 22 6.28E-05 3.48E-05 1.39E-04 0.97 0.21 0.58 1.0 0.7 1.4 0.6 9.07E+05 1.00E+06 0.70 0.12 3.0 3.6 2.8 4.5 10 452 5.00E-06 23 2.55E-05 1.93E-05 1.13E-04 0.37 0.15 0.47 1.7 0.8 1.7 0.9 7.52E+05 4.72E+05 0.56 0.24 3.9 2.0 38.6 9.2 149 14 3.99E-06 24 1.31E-05 5.66E-05 1.85E-04 0.85 0.11 0.83 0.6 0.6 1.7 0.2 6.28E+05 6.90E+05 0.47 0.39 2.3 2.2 0.5 18.9 42 57 1.62E-06 25 9.69E-05 6.28E-05 1.43E-04 0.22 0.12 0.25 1.3 0.6 1.6 1.0 3.48E+05 7.52E+05 0.51 0.51 5.6 6.0 0.4 7.3 22 530 4.49E-06 26 9.38E-05 7.83E-05 1.17E-04 0.31 0.19 0.85 0.8 0.8 0.7 0.5 6.59E+05 5.97E+05 0.20 0.33 5.8 2.9 30.4 11.7 79 621 9.45E-07 27 1.00E-05 3.17E-05 1.66E-04 0.28 0.21 0.96 1.9 0.6 1.2 0.5 1.62E+05 7.21E+05 0.31 0.31 6.0 4.2 1.4 1.4 1000 49 4.83E-06 28 7.21E-05 9.69E-05 1.62E-04 0.58 0.12 0.94 0.7 1.0 1.2 0.9 5.66E+05 1.93E+05 0.58 0.68 3.7 4.1 9.2 2.2 728 149 4.66E-06 29 4.41E-05 8.76E-05 1.73E-04 0.10 0.16 0.69 1.2 0.8 1.5 0.4 2.55E+05 1.31E+05 0.68 0.10 6.3 5.3 1.7 38.6 127 22 1.79E-06 30 0.0001 6.59E-05 1.81E-04 0.43 0.13 0.64 1.5 0.5 0.9 0.6 1.00E+06 9.69E+05 0.53 0.41 2.7 5.6 100.0 1.1 452 26 1.00E-07   167 IV. 2. Interconnect oxidation, detachment, and sulfur poisoning, ESC: anot catt elyt p0a p0c V a1 a2 a3 a4 Sa Sc pora porc tora torc Cdla Cdlc i0A i0C # m m m atm atm - - - - - 1/m 1/m - - - - F/m2 F/m2 A/m2A/m2 1 3.48E-05 2.55E-05 1.66E-04 0.13 0.43 0.45 1.0 0.3 1.0 0.9 6.90E+05 8.45E+05 0.68 0.43 3.6 6.3 18.9 0.4 452 36 2 5.34E-05 6.90E-05 1.54E-04 0.20 0.28 0.31 0.7 0.9 1.8 0.2 4.41E+05 7.83E+05 0.29 0.24 3.7 4.8 14.9 1.1 22 12 3 9.07E-05 7.21E-05 1.73E-04 0.13 0.25 0.34 1.6 0.4 1.7 0.7 5.03E+05 2.55E+05 0.41 0.47 6.7 4.9 48.9 0.7 621 853 4 9.38E-05 5.97E-05 1.47E-04 0.17 0.13 0.96 1.4 0.6 1.5 1.0 1.00E+06 6.59E+05 0.49 0.60 2.0 3.0 1.7 5.7 329 10 5 6.59E-05 9.69E-05 1.24E-04 0.16 0.40 0.85 1.4 0.6 0.7 0.3 1.31E+05 8.14E+05 0.53 0.12 6.5 5.6 38.6 11.7 386 67 6 6.28E-05 5.34E-05 1.39E-04 0.19 0.94 0.36 1.1 0.4 0.7 0.6 8.14E+05 7.52E+05 0.33 0.16 2.5 4.6 0.4 38.6 1000 1000 7 5.03E-05 8.45E-05 1.28E-04 0.19 0.37 0.23 1.9 0.3 0.9 0.4 5.66E+05 3.48E+05 0.64 0.58 3.2 2.3 7.3 24.0 49 204 8 5.97E-05 3.48E-05 1.89E-04 0.10 0.22 0.80 1.3 0.4 1.2 0.5 8.45E+05 3.17E+05 0.12 0.18 3.9 4.1 78.8 62.1 26 42 9 3.79E-05 1.31E-05 1.01E-04 0.21 0.88 0.88 1.8 0.3 1.6 0.5 3.79E+05 4.10E+05 0.27 0.41 4.4 3.7 24.0 3.6 728 26 10 1.93E-05 3.79E-05 1.51E-04 0.16 0.91 0.99 1.1 1.0 0.8 1.0 5.34E+05 1.62E+05 0.37 0.14 6.8 2.5 9.2 1.4 67 174 11 5.66E-05 0.0001 1.58E-04 0.15 0.67 0.55 1.2 0.2 1.6 0.2 9.07E+05 1.00E+05 0.20 0.37 5.4 2.0 0.1 0.2 281 92 12 4.41E-05 4.10E-05 1.92E-04 0.19 0.10 0.53 2.0 0.8 1.4 0.9 2.55E+05 6.28E+05 0.22 0.33 5.8 7.0 3.6 30.4 240 386 13 8.14E-05 2.24E-05 1.32E-04 0.13 0.49 0.69 0.6 0.8 1.7 0.4 5.97E+05 1.00E+06 0.62 0.27 4.9 2.2 2.8 100.0 530 281 14 2.55E-05 1.62E-05 1.05E-04 0.15 0.16 0.50 1.5 0.9 0.8 0.3 9.69E+05 5.34E+05 0.51 0.39 4.2 5.3 0.1 2.8 14 728 15 8.76E-05 1.93E-05 1.81E-04 0.17 0.79 0.25 0.8 0.6 0.9 0.4 6.59E+05 3.79E+05 0.16 0.53 6.3 5.8 1.4 7.3 174 19 16 0.0001 5.66E-05 1.43E-04 0.10 0.97 0.47 1.8 1.0 1.2 0.3 2.86E+05 6.90E+05 0.70 0.70 5.6 4.4 2.2 0.5 57 57 17 4.72E-05 8.76E-05 9.38E-05 0.11 0.85 0.28 0.8 0.6 1.4 0.9 9.38E+05 5.97E+05 0.47 0.35 6.1 3.4 30.4 9.2 10 127 18 4.10E-05 6.28E-05 2.00E-04 0.21 0.58 0.83 0.5 0.7 0.5 0.6 8.76E+05 5.03E+05 0.58 0.66 5.1 4.2 100.0 0.3 108 452 19 1.31E-05 5.03E-05 1.09E-04 0.18 0.19 0.42 0.7 0.5 1.3 0.7 1.00E+05 2.86E+05 0.60 0.20 5.3 3.9 0.7 0.1 127 22 20 1.00E-05 8.14E-05 1.13E-04 0.15 0.64 0.39 1.5 0.9 1.9 0.8 7.52E+05 9.07E+05 0.10 0.64 4.8 5.1 4.5 78.8 149 49 21 8.45E-05 2.86E-05 1.20E-04 0.18 0.70 0.58 1.3 0.7 1.1 0.8 1.62E+05 8.76E+05 0.18 0.49 3.0 2.7 62.1 0.2 16 329 22 2.86E-05 4.41E-05 1.36E-04 0.12 0.76 0.61 0.9 0.5 1.8 0.3 2.24E+05 1.31E+05 0.43 0.62 2.2 6.5 11.7 18.9 92 530 23 7.21E-05 3.17E-05 9.76E-05 0.18 0.34 0.77 1.0 0.4 1.9 0.6 7.21E+05 7.21E+05 0.39 0.68 7.0 6.8 0.3 0.9 30 108 24 6.90E-05 1.00E-05 1.96E-04 0.14 0.61 0.20 1.6 0.7 2.0 0.8 3.17E+05 4.72E+05 0.56 0.10 2.3 3.6 0.3 2.2 79 79 25 1.62E-05 9.07E-05 1.77E-04 0.11 0.31 0.74 0.9 0.9 1.1 0.5 1.93E+05 5.66E+05 0.24 0.51 2.9 3.2 0.5 1.7 853 149 26 3.17E-05 4.72E-05 1.70E-04 0.12 0.55 0.66 1.7 0.3 0.6 0.7 4.10E+05 9.38E+05 0.35 0.56 6.0 2.9 0.2 14.9 19 14 27 7.52E-05 6.59E-05 9.00E-05 0.12 0.52 0.64 1.9 0.8 0.6 0.6 7.83E+05 1.93E+05 0.31 0.29 2.7 6.7 5.7 0.3 204 16 28 9.69E-05 7.52E-05 1.17E-04 0.14 0.46 0.94 0.6 0.2 1.0 0.9 3.48E+05 4.41E+05 0.14 0.31 4.6 5.4 0.9 4.5 42 621 29 2.24E-05 7.83E-05 1.85E-04 0.16 0.82 0.91 1.7 0.5 1.5 0.4 6.28E+05 9.69E+05 0.45 0.22 3.4 6.0 1.1 0.1 36 240 30 7.83E-05 9.38E-05 1.62E-04 0.20 0.73 0.72 1.2 0.8 1.3 0.8 4.72E+05 2.24E+05 0.66 0.45 4.1 6.1 0.2 48.9 12 30   168 IV. 3. Interconnect detachment and sulfur poisoning, ASC: anot catt elyt p0a p0c V a1 a2 a3 a4 Sa Sc pora porc tora torc Cdla Cdlc i0A i0C # m m m atm atm - - - - - 1/m 1/m - - - - F/m2 F/m2 A/m2A/m2 1 6.38E-04 2.55E-05 1.53E-05 0.13 0.43 0.45 1.0 0.3 1.0 0.9 6.90E+05 8.45E+05 0.68 0.43 3.6 6.3 18.9 0.4 452 36 2 7.41E-04 6.90E-05 1.38E-05 0.20 0.28 0.31 0.7 0.9 1.8 0.2 4.41E+05 7.83E+05 0.29 0.24 3.7 4.8 14.9 1.1 22 12 3 9.48E-04 7.21E-05 1.64E-05 0.13 0.25 0.34 1.6 0.4 1.7 0.7 5.03E+05 2.55E+05 0.41 0.47 6.7 4.9 48.9 0.7 621 853 4 9.66E-04 5.97E-05 1.28E-05 0.17 0.13 0.96 1.4 0.6 1.5 1.0 1.00E+06 6.59E+05 0.49 0.60 2.0 3.0 1.7 5.7 329 10 5 8.10E-04 9.69E-05 9.66E-06 0.16 0.40 0.85 1.4 0.6 0.7 0.3 1.31E+05 8.14E+05 0.53 0.12 6.5 5.6 38.6 11.7 386 67 6 7.93E-04 5.34E-05 1.17E-05 0.19 0.94 0.36 1.1 0.4 0.7 0.6 8.14E+05 7.52E+05 0.33 0.16 2.5 4.6 0.4 38.6 1000 1000 7 7.24E-04 8.45E-05 1.02E-05 0.19 0.37 0.23 1.9 0.3 0.9 0.4 5.66E+05 3.48E+05 0.64 0.58 3.2 2.3 7.3 24.0 49 204 8 7.76E-04 3.48E-05 1.84E-05 0.10 0.22 0.80 1.3 0.4 1.2 0.5 8.45E+05 3.17E+05 0.12 0.18 3.9 4.1 78.8 62.1 26 42 9 6.55E-04 1.31E-05 6.55E-06 0.21 0.88 0.88 1.8 0.3 1.6 0.5 3.79E+05 4.10E+05 0.27 0.41 4.4 3.7 24.0 3.6 728 26 10 5.52E-04 3.79E-05 1.33E-05 0.16 0.91 0.99 1.1 1.0 0.8 1.0 5.34E+05 1.62E+05 0.37 0.14 6.8 2.5 9.2 1.4 67 174 11 7.59E-04 0.0001 1.43E-05 0.15 0.67 0.55 1.2 0.2 1.6 0.2 9.07E+05 1.00E+05 0.20 0.37 5.4 2.0 0.1 0.2 281 92 12 6.90E-04 4.10E-05 1.90E-05 0.19 0.10 0.53 2.0 0.8 1.4 0.9 2.55E+05 6.28E+05 0.22 0.33 5.8 7.0 3.6 30.4 240 386 13 8.97E-04 2.24E-05 1.07E-05 0.13 0.49 0.69 0.6 0.8 1.7 0.4 5.97E+05 1.00E+06 0.62 0.27 4.9 2.2 2.8 100.0 530 281 14 5.86E-04 1.62E-05 7.07E-06 0.15 0.16 0.50 1.5 0.9 0.8 0.3 9.69E+05 5.34E+05 0.51 0.39 4.2 5.3 0.1 2.8 14 728 15 9.31E-04 1.93E-05 1.74E-05 0.17 0.79 0.25 0.8 0.6 0.9 0.4 6.59E+05 3.79E+05 0.16 0.53 6.3 5.8 1.4 7.3 174 19 16 1.00E-03 5.66E-05 1.22E-05 0.10 0.97 0.47 1.8 1.0 1.2 0.3 2.86E+05 6.90E+05 0.70 0.70 5.6 4.4 2.2 0.5 57 57 17 7.07E-04 8.76E-05 5.52E-06 0.11 0.85 0.28 0.8 0.6 1.4 0.9 9.38E+05 5.97E+05 0.47 0.35 6.1 3.4 30.4 9.2 10 127 18 6.72E-04 6.28E-05 2.00E-05 0.21 0.58 0.83 0.5 0.7 0.5 0.6 8.76E+05 5.03E+05 0.58 0.66 5.1 4.2 100.0 0.3 108 452 19 5.17E-04 5.03E-05 7.59E-06 0.18 0.19 0.42 0.7 0.5 1.3 0.7 1.00E+05 2.86E+05 0.60 0.20 5.3 3.9 0.7 0.1 127 22 20 5.00E-04 8.14E-05 8.10E-06 0.15 0.64 0.39 1.5 0.9 1.9 0.8 7.52E+05 9.07E+05 0.10 0.64 4.8 5.1 4.5 78.8 149 49 21 9.14E-04 2.86E-05 9.14E-06 0.18 0.70 0.58 1.3 0.7 1.1 0.8 1.62E+05 8.76E+05 0.18 0.49 3.0 2.7 62.1 0.2 16 329 22 6.03E-04 4.41E-05 1.12E-05 0.12 0.76 0.61 0.9 0.5 1.8 0.3 2.24E+05 1.31E+05 0.43 0.62 2.2 6.5 11.7 18.9 92 530 23 8.45E-04 3.17E-05 6.03E-06 0.18 0.34 0.77 1.0 0.4 1.9 0.6 7.21E+05 7.21E+05 0.39 0.68 7.0 6.8 0.3 0.9 30 108 24 8.28E-04 1.00E-05 1.95E-05 0.14 0.61 0.20 1.6 0.7 2.0 0.8 3.17E+05 4.72E+05 0.56 0.10 2.3 3.6 0.3 2.2 79 79 25 5.34E-04 9.07E-05 1.69E-05 0.11 0.31 0.74 0.9 0.9 1.1 0.5 1.93E+05 5.66E+05 0.24 0.51 2.9 3.2 0.5 1.7 853 149 26 6.21E-04 4.72E-05 1.59E-05 0.12 0.55 0.66 1.7 0.3 0.6 0.7 4.10E+05 9.38E+05 0.35 0.56 6.0 2.9 0.2 14.9 19 14 27 8.62E-04 6.59E-05 5.00E-06 0.12 0.52 0.64 1.9 0.8 0.6 0.6 7.83E+05 1.93E+05 0.31 0.29 2.7 6.7 5.7 0.3 204 16 28 9.83E-04 7.52E-05 8.62E-06 0.14 0.46 0.94 0.6 0.2 1.0 0.9 3.48E+05 4.41E+05 0.14 0.31 4.6 5.4 0.9 4.5 42 621 29 5.69E-04 7.83E-05 1.79E-05 0.16 0.82 0.91 1.7 0.5 1.5 0.4 6.28E+05 9.69E+05 0.45 0.22 3.4 6.0 1.1 0.1 36 240 30 8.79E-04 9.38E-05 1.48E-05 0.20 0.73 0.72 1.2 0.8 1.3 0.8 4.72E+05 2.24E+05 0.66 0.45 4.1 6.1 0.2 48.9 12 30       

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