ETH - Ceramicshttp://ceramics.ethz.ch Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich Institute for Nonmetallic Materials Head: Prof. L.J. Gauckler

ETH - Ceramicshttp://ceramics.ethz.ch Solid Oxide Fuel Cell (SOFC) cathode O 2- e-e- e-e- H2OH2OH2H2 O2O2 O2O2 anode electrolyte O 2 reduction O2O2 O 2- ½O 2 + 2e -  O 2- mechanism? rate-limiting steps? fuel oxidation O 2- H 2 + O 2-  H 2 O + 2e - mechanism? rate-limiting steps? H2OH2O H2H2 overall: H 2 + ½O 2 → H 2 O

ETH - Ceramicshttp://ceramics.ethz.ch Electrochemical Impedance Spectroscopy and State-Space Modelling Oxygen reduction at electronic conducting SOFC cathodes Oxygen reduction at mixed conducting SOFC cathodes Outline

ETH - Ceramicshttp://ceramics.ethz.ch Impedance spectroscopy I = Z(jω) 1 E ~~ Small amplitude (5-10 mV) input signal → Linearization Potential (V) Current (A) E ~ I ~ steady-state operating point Admittance Transfer Function complex ( j 2 = -1) frequency dependant ( ω = 2π f ) Z = impedance Principle of impedance spectroscopy

ETH - Ceramicshttp://ceramics.ethz.ch EIS spectra and equivalent circuits Re (Z) Im (Z) R Re (Z) / Ω Im (Z) /Ω f (Hz) experimental EIS spectra How to interpret the experimental equivalent circuit ?? Re (Z) Im (Z) R C f = 1 2π RC R Re (Z) Im (Z) R2R2 C2C2 R1R1 Re (Z) Im (Z) R2R2 C2C2 R1R1 R3R3 C3C3 R1R1 R 2 +R 3

ETH - Ceramicshttp://ceramics.ethz.ch State-Space Model electrode potential E ( input) faradaic current I F ( output) O 2(g)  O ads  O 2- electrochemical system KbKb KfKf K des K ads K = model parameters (K ads, K des, K f, K b …)  = state variable (O ads concentration) = f (  , E , K)  state equation I F = g (  , E , K )  output equation dd dt State-Space Model calculating the faradaic impedance:

ETH - Ceramicshttp://ceramics.ethz.ch I E State-Space Modelling θ = A θ +B E I F = C θ +D E. linearization θ = 0. steady-state analysis time domain Z F (jω) Laplace transform frequency domain varying ω Re(Z F ) * * * * * * * * * * * Im(Z F ) θ = K ads (1- θ ) I F = -K f θ e -fE + …. state-space model* Simulink ® : easy implementation of the model. Matlab ® : state-space calculations and computing

ETH - Ceramicshttp://ceramics.ethz.ch Oxygen reduction at electronic conducting SOFC cathodes

ETH - Ceramicshttp://ceramics.ethz.ch Electronic conducting cathodes No O 2 -reduction through the bulk of the electrode. Electrode = electron supplier triple phase boundary (tpb) Electrolyte = O 2- conductor (V o and O o ).. x Typically YSZ (Y 2 O 3 - ZrO 2 ) O ads O2O2 O2O2 electrolyte electrode e-e- x O 2- ( O o ) Typical material: La x Sr 1-x Mn y O 3 (LSM).

ETH - Ceramicshttp://ceramics.ethz.ch Oxygen reduction reaction models Diffusion processes 2 nd Fick‘s law: → Finite difference approach to estimate time and space derivatives → state variable θ (θ 1, θ 2, θ 3 ) = vector θ 1 ≤ θ 2 ≤ θ 3 tpb O ads reservoir θ eq electrolyte diffusion layer θ1θ1 θ2θ2 θ3θ3 O 2- ( O o ) x O ads O2O2 O2O2 O2O2 O2O2

ETH - Ceramicshttp://ceramics.ethz.ch Model implementation in Simulink ® Block diagrams: as many as compartments in the diffusion layer. E input IFIF output θ1θ1 θ1θ1 θ3θ3 θ2θ2 θ2θ2 θ2θ2 θ3θ3 tpb (1) (2) (3) in-portout-port

ETH - Ceramicshttp://ceramics.ethz.ch Model Implementation in Simulink ® u2u2 K des state variable u2u K ads p O2 Block diagram n°2 θ2θ2 out-port to block diagrams (1) and (3) K dif / /3 2/3 θ1θ1 θ3θ3 in-ports θ2θ2

ETH - Ceramicshttp://ceramics.ethz.ch Modelling results O ads O2O2 O2O2 O 2- electrode electrolyte rds = adsorption, diffusion and charge transfer Re(Z F ) /  Im(Z F ) /  O ads O2O2 O2O2 O 2- electrode electrolyte rds = diffusion and charge transfer Re(Z F ) /  Im(Z F ) /  45° O ads O2O2 O2O2 O 2- electrode electrolyte rds = adsorption and charge transfer Re(Z F ) /  Im(Z F ) /  R2R2 C2C2 R1R1 O ads O2O2 O2O2 O 2- electrode electrolyte rds = charge transfer rds = rate-determining step(s) Re(Z F ) /  Im(Z F ) /  R1R1

ETH - Ceramicshttp://ceramics.ethz.ch Modelling results O ads O2O2 O2O2 O 2- electrode electrolyte rds = adsorption and charge transfer Re(Z F ) /  Im(Z F ) /  R2R2 C2C2 R1R1 → Necessity of a modelling approach to comprehensively interpret experimental impedance even for relatively simple reaction models charge transfer rate constants (potential dependant) adsorption/desorption rate constants R 1 and R 2 are not independant

ETH - Ceramicshttp://ceramics.ethz.ch ZFZF Re (Z TOT ) / Ω Im (Z TOT ) / Ω Experimental E+E ~ I + II + I ~ potentiostat (DC) working electrode frequency response analyzer (AC) counter electrode reference electrode Z TOT = R2R2 CFCF R1R1 RΩRΩ C DL C DL highC DL moderate C DL =0 (Z F )

ETH - Ceramicshttp://ceramics.ethz.ch Experimental porous microstructured dense „real“ electrodes Geometrically well-defined electrodes top view ~10-30  m ~100 nm -1  m ~20  m -100  m

ETH - Ceramicshttp://ceramics.ethz.ch Comparison modelling - experiments One unique vector (K ads, K des K dif, k f ) has to describe at least 4 different impedance spectra Z exp → practical identification - (R Ω, C DL ) La 0.85 Sr 0.15 MnO 800°C I =170 mA.cm -2 in air. Example of the LSM/YSZ interface rds = adsorption, diffusion and charge transfer O ads O2O2 O2O2 O 2- electrode electrolyte

ETH - Ceramicshttp://ceramics.ethz.ch Oxygen reduction at mixed ionic-electronic conducting SOFC cathodes

ETH - Ceramicshttp://ceramics.ethz.ch Mixed ionic-electronic electrodes (MIEC) electrolyte O 2- O ads O2O2 electrode O 2- Competition between two reaction pathways for O 2 reduction: surface and bulk. → the fastest pathway is rate-determining Typical material: La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3-δ (LSCF). Potential (V) Current (A) EeqEeq LSCF LSM ideal electrode Why is LSCF a better electrocatalyst than LSM for oxygen reduction? Intermediate T° ( °C) → the rate-detemining pathway influences the microstructure of the electrode

ETH - Ceramicshttp://ceramics.ethz.ch Mixed ionic-electronic electrodes bulk pathway is rate-limiting: → thin dense electrodes, large electrolyte coverage, low tpb. surface pathway is rate-limiting: → porous electrodes, large tpb bulk and surface pathways are rate-limiting: → Optimization of microstructure (a, b) top view a b

ETH - Ceramicshttp://ceramics.ethz.ch gwd electrodes LSCF electrode CGO electrolyte tpb S S and l tpb constant S=0.25 cm 2, l tpb = 2cm d is varied d ~ 100 nm - 1 µm → Z sim (E, p O2, d) Model system: geometrically well-defined (gwd) electrodes d electrolyte O 2- O2O2 d

ETH - Ceramicshttp://ceramics.ethz.ch Experimental 300 nm LSCF CGO gwd LSCF layers are prepared by pulsed laser deposition: dense, crack-free. → Z exp (E, p O2, d) = Z sim (E, p O2, d) ? LSCF electrode CGO electrolyte tpb S

ETH - Ceramicshttp://ceramics.ethz.ch Summary  electrochemical reactions yield complex impedance behavior  necessity of a modelling approach (analytical or numerical)  use of geometrically well-defined electrodes  State-Space Modelling (with modern computation tools) enables to simulate the electrochemical behavior of multistep reactions  faradaic impedance  I-U curves  electroactive species concentration profiles

ETH - Ceramicshttp://ceramics.ethz.ch Outlook  SOFC electrode reactions with mixed conductors  Single gas chamber SOFC  State-Space Modelling of entire cells: - with ionic conducting electrolyte (YSZ) - with mixed ionic-electronic conducting electrolyte (GdO-CeO 2 )  SSM approach can be applied to any other field of electrochemistry  PEM-FC, batteries, corrosion, electrodeposition…

ETH - Ceramicshttp://ceramics.ethz.ch Many thanks to... ETH-Zürich S. Rey-Mermet, Dr. Paul Muralt (EPF-Lausanne, Lab. de Céramique) Dr. J.-F. Koenig (Université de Strasbourg, Lab. d‘éléctrochimie) S. Schlumpf SOFC group of ETH-Zürich

ETH - Ceramicshttp://ceramics.ethz.ch END

ETH - Ceramicshttp://ceramics.ethz.ch Tentative model for oxygen reduction at LSCF O2O2 Electrolyte (CGO) O 2- O ads O2O2 Dense electrode (LSCF) Slow surface diffusion (negligible) O 2- K in K out K des K ads D KfKf KbKb K des K ads K fs K bs triple phase boundary (tpb) gas/electrode/electrolyte Dense electrode: - mixed conducting: e- (h) and O 2- - low potential gradient - well-defined dimension Electrolyte - pure O 2- conductor gas phase: - air - no gas phase diffusion - pO2 constant T = 700°C Competition between surface and bulk pathways

ETH - Ceramicshttp://ceramics.ethz.ch Mixed ionic-electronic electrodes Modelling MIEC is still very controversial - incorpotation of oxygen in the the - extension of space charge region - influence of permittivity - influence of permittivity: presence of a displacement current? - is electroneutrality fulfilled? electrolyte O 2- O ads O2O2 electrode O 2- O ads O2O2 Modelling MIEC is still very controversial

ETH - Ceramicshttp://ceramics.ethz.ch Model Implementation in Simulink ® K ads p O2 (1-  ads )  2 – K des  ads 2 – K f (E)  ads + K b (E) (1-  ads ) I F = K i [– K f (E)  ads + K b (E) (1-  ads )] dθ ads dt = u2u2 K des x -+-+ KiKi IFIF output E input k f exp(-2   f E) k b exp (2 (1-   f E) function K f (E) function K b (E) x state variable θ ads 1-θ ads θ ads K ads p O2 u2u2

ETH - Ceramicshttp://ceramics.ethz.ch Model Implementation in Simulink ® K ads p O2 (1-  ads )  2 – K des  ads 2 – K f (E)  ads + K b (E) (1-  ads ) I F = K i [– K f (E)  ads + K b (E) (1-  ads )] dθ ads dt = K ads p O2 u2u2 K des xx-+-+ integrator E input KiKi IFIF output k f exp(-2   f E) k b exp (2 (1-   f E) function K f (E) function K b (E) 1-θ ads θ ads u2u2

ETH - Ceramicshttp://ceramics.ethz.ch Alternative approch: modeling the impedance Re (Z) Im (Z) Re (Z) Im (Z) experimental impedance reaction model O 2  O ads  O 2- state-space modeling faradaic impedance new model O 2  O 2, ads  O 2- validation of the model assessment of kinetics

ETH - Ceramicshttp://ceramics.ethz.ch O 2(g) + 2s  2O ads K ads K des Model 1 (without surf. diffusion) Dissociative adsorption: Charge transfer: O 2- ( O o ) x O ads O2O2 electrolyte electrode θ ads → state variable θ ads = scalar → consecutive reaction steps O ads + 2e -  O 2- K f (E) K b (E) K ads p O2 (1-  ads )  2 – K des  ads 2 – K f (E)  ads + K b (E) (1-  ads ) I F = K i [– K f (E)  ads + K b (E) (1-  ads )] dθ ads dt = → state-space model

ETH - Ceramicshttp://ceramics.ethz.ch O 2(g) + 2s  2O ads K ads K des O ads + 2e -  O 2- K f (E) K b (E) O ads  O ads K dif Numerical approach Diffusion processes 2 nd Fick‘s law: → Finite difference approach to estimate time and space derivatives tpb O ads reservoir θ eq electrolyte diffusion layer θ1θ1 θ2θ2 θ3θ3 O 2- ( O o ) x O ads O2O2 O2O2 O2O2 O2O2 → state variable θ (θ 1, θ 2, θ 3 ) = vector θ 1 ≤ θ 2 ≤ θ 3

ETH - Ceramicshttp://ceramics.ethz.ch Model implementation in Simulink ® u2u K ads p O2 u2u2 K des 1 s Integrator (θ 2 ) Block diagram n°2 θ2θ2 out-port to block diagrams (1) and (3) K dif / /3 2/3 θ1θ1 θ3θ3 in-ports

ETH - Ceramicshttp://ceramics.ethz.ch Modelling results O ads O2O2 O2O2 O 2- electrode electrolyte rds = adsorption and charge transfer Re(Z F ) /  Im(Z F ) /  R2R2 C2C2 R1R1 amount of adsorbed oxygen → Necessity of a modelling approach to comprehensively interpret experimental impedance charge transfer rate constants (potential dependant) adsorption/desorption rate constants R 1 and R 2 are not independant

ETH - Ceramicshttp://ceramics.ethz.ch Experimental investigation electrolyte O 2- O2O2 electrode O 2- Consequence of parallel pathways Intermediate T° SOFC ( °C). Typical material: La x Sr 1-x Co y Fe 1-y O 3 (LSCF).