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ETH - Ceramics http://ceramics.ethz. 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

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Page 1: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Identification of reaction rate-determining steps

at SOFC electrodes using State-Space Modelling

Michel Prestat

ETH-ZürichInstitute for Nonmetallic Materials

Head: Prof. L.J. Gauckler

Page 2: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Solid Oxide Fuel Cell (SOFC)

cathode

O2-

e-

e-

H2OH2

O2 O2

anode

electrolyte

O2 reduction

O2

O2-

½O2 + 2e- O2-

mechanism?rate-limiting steps?

fuel oxidation

O2-

H2 + O2- H2O + 2e-

mechanism?rate-limiting steps?

H2OH2

overall: H2 + ½O2 → H2O

Page 3: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://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

Page 4: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Impedance spectroscopy

I =Z(jω)

1E~~

Small amplitude (5-10 mV) input signal

→ Linearization

Potential (V)

Current (A)

E~

I~

steady-stateoperating point

Admittance Transfer Function

complex ( j2 = -1)

frequency dependant ( ω = 2π f )

Z = impedance

Principle of impedance spectroscopy

Page 5: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

EIS spectra and equivalent circuits

Re (Z)

Im (Z)

R-6

-4

-2

0

0 2 4 6 8 10 12

Re (Z) / Ω

Im (

Z)

1

10

102103

104

f (Hz)

experimental EIS spectra

How to interpret the experimental equivalent circuit ??

Re (Z)

Im (Z)

R

Cf = 1 2π RC

R

Re (Z)

Im (Z)

R2

C2

R1

Re (Z)

Im (Z)

R2

C2

R1 R3

C3

R1 R2+R3

Page 6: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

State-Space Model

electrode potential

E (input)

faradaic current

IF (output)O2(g) Oads O2-

electrochemical system

Kb

KfKdes

Kads

K = model parameters (Kads, Kdes, Kf , Kb …)

= state variable (Oads concentration)

= f (, E, K) state equation

IF = g (, E, K) output equation

ddtState-Space Model

calculating the faradaic impedance:

Page 7: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

I

E

State-Space Modelling

θ = Aθ +B E

IF = Cθ +D E

.

linearization

θ = 0.

steady-state analysis

time domain

ZF(jω)

Laplace transform

frequency domain

varying ω

Re(ZF)

**

* * ** **

**

*

Im(Z

F)θ = Kads(1- θ)2 +...

IF = -Kf θ e-fE + …

.

state-space model*

Simulink®: easy implementation of the model.

Matlab®: state-space calculations and computing

Page 8: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Oxygen reduction at electronic conducting SOFC cathodes

Page 9: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Electronic conducting cathodes

No O2-reduction through the bulk of the electrode.

Electrode = electron supplier

triple phase boundary (tpb)

Electrolyte = O2- conductor (Vo and Oo).. x

Typically YSZ (Y2O3 - ZrO2)

Oads

O2

Oads

O2

electrolyte

electrode

e-

xO2- ( Oo )

Typical material: LaxSr1-xMnyO3 (LSM).

ads

des

2(g) ads

K

KO 2s 2O (adsorption)

dif

ads ads

K

O O (surface diffusion towards the ) tpb

f

b

- xads o o

K

KO V 2e O s (charge transfer at the ) tpb

Page 10: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Oxygen reduction reaction models

ads

des

2(g) ads

K

KO 2s 2O (adsorption)

dif

ads ads

K

O O (surface diffusion towards the ) tpb

f

b

- xads o o

K

KO V 2e O s (charge transfer at the ) tpb

2

2

θ θ

t

zDiffusion processes

2nd Fick‘s law:

→ Finite difference approach to estimate time and space derivatives

→ state variable θ (θ1, θ2, θ3) = vector θ1 ≤ θ2 ≤ θ3

tpb

Oads reservoir

θeq

electrolyte

diffusionlayer

θ1

θ2

θ3

O2- ( Oo )x

OadsO2

OadsO2

OadsO2

OadsO2

Page 11: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Model implementation in Simulink®

Block diagrams: as many as compartments in the diffusion layer.

E

input

IF

output

θ1

θ1

θ3

θ2θ2

θ2 θ3

tpb (1)

(2)

(3)

i( ) ( ) 2

i-12 2 dif i 1iads O i des i 2i-2

Kθ 2K 1 θ K θ θ ( + chg transfer kinetics)

t 3 32

θθ

p

in-port out-port

Page 12: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Model Implementation in Simulink®

u2Kdes

state variable

+-

1

u2

-++

Kads pO2

( ) ( )2

2 2 dif 32 1ads O 2 des 2 2

K θθ 2θK 1 θ K θ θ

t 4 3 3

p

Block diagram n°2

θ2

out-port toblock diagrams (1) and (3)

Kdif/4

-

+

+

1/3

2/3θ1

θ3

in-ports

θ2

Page 13: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Modelling results

Oads

Oads

O2

O2

O2-

electrode

electrolyte

rds = adsorption, diffusion and charge transfer

-0.6

-0.4

-0.2

0

2 2.5 3 3.5

Re(ZF) /

Im(Z

F)

/

Oads

Oads

O2

O2

O2-

electrode

electrolyte

rds = diffusion and charge transfer

-0.6

-0.4

-0.2

0

2 2.5 3 3.5

Re(ZF) /

Im(Z

F)

/

45°

Oads

Oads

O2

O2

O2-

electrode

electrolyte

rds = adsorption and charge transfer

-0.6

-0.4

-0.2

0

2 2.5 3 3.5

Re(ZF) /

Im(Z

F)

/

R2

C2

R1

Oads

Oads

O2

O2

O2-

electrode

electrolyte

rds = charge transferrds = rate-determining step(s)

-0.6

-0.4

-0.2

0

2 2.5 3 3.5

Re(ZF) /

Im(Z

F)

/ R1

Page 14: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Modelling results

Oads

Oads

O2

O2

O2-

electrode

electrolyte

rds = adsorption and charge transfer

-0.6

-0.4

-0.2

0

2 2.5 3 3.5

Re(ZF) /

Im(Z

F)

/

R2

C2

R1

→ Necessity of a modelling approach to comprehensively interpret experimental impedanceeven for relatively simple reaction models

2 2

f b 12

ads O des 1 ads O

(K K ) RR

2(K K ) 2K

p x p

charge transfer rate constants

(potential dependant)

adsorption/desorption rate constants

R1 and R2 are not independant

Page 15: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

ZF

0 2 3 4 5

Re (ZTOT) / Ω

1

-1

-2

-3

0

Im (

ZT

OT)

/ Ω

Experimental

E+E~

I + I~

potentiostat(DC)

workingelectrode

frequencyresponseanalyzer

(AC)counter

electrode

referenceelectrode

ZTOT =R2

CF

R1RΩ

CDL

CDL high CDL moderate CDL=0(ZF)

Page 16: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://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

Page 17: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

2 3 4 50.0

-0.5

-1.0

-1.5

Im(Z

) []

Re (Z) []

Zexp

Zsim

Comparison modelling - experiments

One unique vector (Kads, Kdes Kdif, kf) has to describe at least 4 different impedance spectra Zexp

→ practical identification

1.0 1.5 2.0 2.5 3.00.0

-0.5

-1.0

Im(Z

F)

[]

Re (ZF) []

ZF

- (RΩ, CDL)

La0.85Sr0.15MnO3 @ 800°CI =170 mA.cm-2 in air.

Example of the LSM/YSZ interface

rds = adsorption, diffusion and charge transfer

Oads

Oads

O2

O2

O2-

electrode

electrolyte

Page 18: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Oxygen reduction at mixed ionic-electronic conducting SOFC cathodes

Page 19: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Mixed ionic-electronic electrodes (MIEC)

electrolyteO2-

Oads

Oads

O2

electrodeO2-

O2-

O2-

Competition between two reaction pathways for O2 reduction: surface and bulk.→ the fastest pathway is rate-determining

Typical material: La0.6Sr0.4Co0.2Fe0.8O3-δ

(LSCF).

Potential (V)

Current (A)

Eeq

LSCF

LSM

ideal electrode

Why is LSCF a better electrocatalyst than LSM

for oxygen reduction?

Intermediate T° (500-800°C)

→ the rate-detemining pathway influences the microstructure of the electrode

Page 20: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://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

Page 21: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

gwd electrodes

LSCF electrode

CGO electrolyte

tpb

S

S and ltpb constantS=0.25 cm2, ltpb= 2cm

d is variedd ~ 100 nm - 1 µm

→ Zsim (E, pO2, d)

Model system: geometrically well-defined (gwd) electrodes

d

electrolyte

O2-

O2

O2-

d

Page 22: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Experimental

300 nm

LSCF

CGO

gwd LSCF layers are prepared by pulsed laser deposition: dense, crack-free.

→ Zexp (E, pO2, d) = Zsim (E, pO2, d) ?

LSCF electrode

CGO electrolyte

tpb

S

Page 23: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://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

Page 24: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://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-CeO2)

SSM approach can be applied to any other field of electrochemistry PEM-FC, batteries, corrosion, electrodeposition…

Page 25: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://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

Page 26: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

END

Page 27: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Tentative model for oxygen reduction at LSCF

O2

Electrolyte (CGO)

O2-

Oads

Oads

O2

Dense electrode (LSCF)Slow surface diffusion

(negligible)

O2-

O2-

O2-

Kin Kout

Kdes Kads

D

Kf Kb

Kdes

Kads

Kfs Kbs

triple phase boundary (tpb) gas/electrode/electrolyte

Dense electrode:

- mixed conducting: e- (h) and O2-

- low potential gradient- well-defined dimension

Electrolyte

- pure O2- conductor

gas phase:

- air- no gas phase diffusion- pO2 constant

T = 700°C

Competition between surface and bulk pathways

Page 28: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://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?

electrolyteO2-

Oads

Oads

O2

electrodeO2-

O2-

O2-

Oads

O2

- xads o oO V 2e O

Modelling MIEC is still very controversial

Page 29: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Model Implementation in Simulink®

Kads pO2(1-ads)2 – Kdesads2 – Kf(E) ads + Kb(E) (1-ads)

IF = Ki [– Kf(E) ads + Kb(E) (1-ads)]

dθads

dt=

u2Kdes

x

-+

Ki IF

outputE

input

kf exp(-2 f E)

kb exp (2 (1-f E)

function Kf (E)

function Kb (E)

x

state variable

θads

1-θads

θads

+-

1

-++

Kads pO2 u2

Page 30: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Model Implementation in Simulink®

Kads pO2(1-ads)2 – Kdesads2 – Kf(E) ads + Kb(E) (1-ads)

IF = Ki [– Kf(E) ads + Kb(E) (1-ads)]

dθads

dt=

+-

1

-++

Kads pO2

u2Kdes

x

x

-+

integrator

E

input

Ki IF

output

kf exp(-2 f E)

kb exp (2 (1-f E)

function Kf (E)

function Kb (E)

1-θadsθads

θads

u2

Page 31: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Alternative approch: modeling the impedance

Re (Z)

Im (Z)

Re (Z)

Im (Z)

experimental impedance

reaction model

O2 Oads O2-

state-space modeling

faradaic impedance

new model

O2 O2,ads O2-

validation of the model assessment of kinetics

Page 32: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

O2(g) + 2s 2Oads

Kads

Kdes

Model 1 (without surf. diffusion)Dissociative adsorption:

Charge transfer:

O2- ( Oo )x

Oads

O2

electrolyte

electrode

θads

→ state variable θads = scalar

→ consecutive reaction steps

Oads + 2e- O2-

Kf(E)

Kb(E)

Kads pO2(1-ads)2 – Kdesads2 – Kf(E) ads + Kb(E) (1-ads)

IF = Ki [– Kf(E) ads + Kb(E) (1-ads)]

dθads

dt=

→ state-space model

Page 33: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

O2(g) + 2s 2Oads

Kads

Kdes

Oads + 2e- O2-

Kf(E)

Kb(E)

Oads Oads

Kdif

Numerical approach

2

2

θ θ

t

zDiffusion processes

2nd Fick‘s law:

→ Finite difference approach to estimate time and space derivatives

tpb

Oads reservoir

θeq

electrolyte

diffusionlayer

θ1

θ2

θ3

O2- ( Oo )x

OadsO2

OadsO2

OadsO2

OadsO2

→ state variable θ (θ1, θ2, θ3) = vector θ1 ≤ θ2 ≤ θ3

Page 34: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Model implementation in Simulink®

+-

1

u2

-++

Kads pO2

u2Kdes

1

s

Integrator (θ2)

( ) ( )2

2 2 dif 32 1ads O 2 des 2 2

K θθ 2θK 1 θ K θ θ

t 4 3 3

p

Block diagram n°2

θ2

out-port toblock diagrams (1) and (3)

Kdif/4

-

+

+

1/3

2/3θ1

θ3

in-ports

Page 35: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Modelling results

Oads

Oads

O2

O2

O2-

electrode

electrolyte

rds = adsorption and charge transfer

-0.6

-0.4

-0.2

0

2 2.5 3 3.5

Re(ZF) /

Im(Z

F)

/

R2

C2

R1

2 2 2

2

2

ads O f b ads O f b ads O des ads b

1

ads O des

2K K K 2K K K 4 K K K K

2 K K

p p px

p

amount of adsorbed oxygen

→ Necessity of a modelling approach to comprehensively interpret experimental impedance

2 2

f b 12

ads O des 1 ads O

(K K ) RR

2(K K ) 2K

p x p

charge transfer rate constants

(potential dependant)

adsorption/desorption rate constants

R1 and R2 are not independant

Page 36: ETH - Ceramics Identification of reaction rate-determining steps at SOFC electrodes using State-Space Modelling Michel Prestat ETH-Zürich

ETH - Ceramics http://ceramics.ethz.ch

Experimental investigation

electrolyteO2-

O2

electrode

O2-

Consequence of parallel pathways

Intermediate T° SOFC (600-800°C).

Typical material: LaxSr1-xCoyFe1-yO3 (LSCF).