eth - ceramics identification of reaction rate-determining steps at sofc electrodes using...
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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
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
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
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
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)
/Ω
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
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:
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
ETH - Ceramics http://ceramics.ethz.ch
Oxygen reduction at electronic conducting SOFC cathodes
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
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
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
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
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
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
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)
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
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
ETH - Ceramics http://ceramics.ethz.ch
Oxygen reduction at mixed ionic-electronic conducting SOFC cathodes
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
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
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
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
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
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…
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
ETH - Ceramics http://ceramics.ethz.ch
END
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
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
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
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
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
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
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
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
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
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).