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 - Ceramics
http://ceramics.ethz.ch
Solid Oxide Fuel Cell (SOFC)
O2 reduction
O2
½O2 + 2e- D O2-
O2
O2
O2mechanism?
rate-limiting steps?
e-
cathode
electrolyte
fuel oxidation
O2-
O2anode
e-
H2
H2
H2O
H2 + O2- D H2O
+ 2eH2O
mechanism?
rate-limiting steps?
overall: H2 + ½O2 → H2O
ETH - Ceramics
http://ceramics.ethz.ch
Outline
Electrochemical Impedance Spectroscopy and State-Space Modelling
Oxygen reduction at electronic conducting SOFC cathodes
Oxygen reduction at mixed conducting SOFC cathodes
ETH - Ceramics
http://ceramics.ethz.ch
Impedance spectroscopy
Principle of impedance spectroscopy
Small amplitude (5-10 mV) input signal
→ Linearization
Current (A)
steady-state
operating point
~
I
~
I=
1
Z(jω)
~
E
Admittance Transfer Function
~
E
Potential (V)
Z = impedance
complex ( j2 = -1)
frequency dependant ( ω = 2π f )
ETH - Ceramics
http://ceramics.ethz.ch
EIS spectra and equivalent circuits
Im (Z)
experimental EIS spectra
R
Re (Z)
-6
Im (Z)
f=
1
2π RC
Im (Z) /Ω
f (Hz)
C
-4
103
102
-2
104
Re (Z)
R
10
1
0
R
0
2
4
6
8
10
12
Re (Z) / Ω
C2
Im (Z)
Re (Z)
R1
R2
C2
Im (Z)
C3
How to interpret
the experimental equivalent circuit ??
R1
R2+R3
ETH - Ceramics
Re (Z)
R1
R2
R3
http://ceramics.ethz.ch
State-Space Model
calculating the faradaic impedance:
electrochemical system
Kads
Kb
O2(g) D Oads D O2Kdes
E (input)
electrode potential
Kf
K = model parameters (Kads, Kdes, Kf , Kb …)
IF (output)
faradaic current
q = state variable (Oads concentration)
State-Space Model
dq = f (q , E , K)  state equation
dt
IF = g (q, E , K)  output equation
ETH - Ceramics
http://ceramics.ethz.ch
State-Space Modelling
time domain
.
linearization
.
θ = Kads(1- θ)2 +...
θ = Aθ +B E
IF = -Kf θ e-fE + …
IF = Cθ +D E
varying ω
Laplace transform
ZF(jω)
Im(ZF)
state-space model*
frequency domain
*
*
**
* * **
**
*
Re(ZF)
I
.
θ=0
E
Simulink®: easy implementation of the model.
Matlab®: state-space calculations and computing
steady-state analysis
ETH - Ceramics
http://ceramics.ethz.ch
Oxygen reduction at electronic conducting SOFC cathodes
ETH - Ceramics
http://ceramics.ethz.ch
Electronic conducting cathodes
O2(g)  2s
O2
K ads
K des
2Oads
(adsorption)
Oads
Kdif
Oads  Oads
(surface diffusion towards the tpb)
electrode
O2
Oads 
Vo
 2e
-
Kf
Kb
Oox  s
(charge transfer at the tpb)
Oads
e-
x
O2- ( Oo )
No O2-reduction through the bulk of the electrode.
Electrode = electron supplier
Typical material: LaxSr1-xMnyO3 (LSM).
ETH - Ceramics
electrolyte
triple phase boundary (tpb)
x
..
Electrolyte = O2- conductor (Vo and Oo)
Typically YSZ (Y2O3 - ZrO2)
http://ceramics.ethz.ch
Oxygen reduction reaction models
O2(g)  2s
K ads
K des
2Oads
(adsorption)
O2
Oads
O2
Oads
Oads reservoir
θeq
Kdif
Oads  Oads
Oads  Vo  2e-
(surface diffusion towards the tpb)
Kf
Kb
Diffusion processes
2nd Fick‘s law:
Oox  s
θ3
(charge transfer at the tpb)
θ  2θ
 2
t z
→ Finite difference approach to
estimate time and space derivatives
diffusion
layer
O2
Oads
θ2
O2
Oads
θ1
x
O2- ( Oo )
tpb
electrolyte
→ state variable θ (θ1, θ2, θ3) = vector
θ1 ≤ θ 2 ≤ θ 3
ETH - Ceramics
http://ceramics.ethz.ch
Model implementation in Simulink®
Block diagrams: as many as compartments
in the diffusion layer.
K dif 2θi-1
θi
θ
 K ads pO2 (1  θi )2  K des θi2  2i-2
(
 θi  i1 ) ( + chg transfer kinetics)
t
3
3
2
in-port
out-port
θ1
θ2
θ3
θ2
θ3
(2)
θ2
θ1
(3)
IF
E
input
output
tpb (1)
ETH - Ceramics
http://ceramics.ethz.ch
Model Implementation in Simulink®
Block diagram n°2
K
θ
θ2
2θ
 Kads pO2 (1  θ2 )2  K des θ22  dif ( 1  θ2  3 )
t
4
3
3
u2
Kads pO2
Kdes
u2
1
Kdif/4
+
+
+
-
θ2
state variable
-
in-ports
1/3
+
θ3
2/3
+
θ1
θ2
ETH - Ceramics
out-port to
block diagrams (1) and (3)
http://ceramics.ethz.ch
Im(ZF) / W
-0.6
Modelling results
O2
Oads
-0.4
R1
electrode
O2
-0.2
rds = charge transfer
rds = rate-determining step(s)
Oads
0
2
2.5
Im(ZF) / W
-0.6
Re(ZF) / W
3
electrolyte
3.5
O2O2
C2
Oads
-0.4
R1
-0.2
electrode
O2
R2
Oads
rds = adsorption and charge
transfer
0
2
2.5
Re(ZF) / W
3
electrolyte
3.5
O2-
Im(ZF) / W
-0.6
O2
Oads
-0.4
electrode
O2
-0.2
Oads
45°
rds = diffusion and charge
transfer
0
2
2.5
Re(ZF) / W
3
electrolyte
3.5
O2-
Im(ZF) / W
-0.6
O2
Oads
-0.4
electrode
O2
-0.2
Oads
rds = adsorption, diffusion and
charge transfer
0
2
ETH - Ceramics
2.5
Re(ZF) / W
3
3.5
electrolyte
O2-
http://ceramics.ethz.ch
Modelling results
Im(ZF) / W
-0.6
O2
C2
Oads
-0.4
R1
-0.2
electrode
O2
R2
Oads
rds = adsorption and charge
transfer
0
2
2.5
Re(ZF) / W
3
3.5
electrolyte
O2-
charge transfer
rate constants
(potential dependant)
R2 
R1 and R2 are not
independant
(K f  K b ) R1
2(K ads pO2  K des )x1  2K ads pO2
adsorption/desorption
rate constants
→ Necessity of a modelling approach to comprehensively interpret experimental impedance
even for relatively simple reaction models
ETH - Ceramics
http://ceramics.ethz.ch
Experimental
reference
potentiostat electrode
(DC)
working
electrode
frequency
response
analyzer
counter
(AC)
electrode
~
E+E
~
I+I
CDL
-3
CF
ZTOT =
RΩ
R1
R2
ZF
ETH - Ceramics
Im (ZTOT) / Ω
CDL high
CDL moderate
CDL=0
(ZF)
-2
-1
0
0
1
2
3
4
5
Re (ZTOT) / Ω
http://ceramics.ethz.ch
Experimental
porous
~10-30 mm
„real“ electrodes
dense
~100 nm -1 mm
Geometrically well-defined
electrodes
microstructured
~20 mm -100 mm
top view
ETH - Ceramics
http://ceramics.ethz.ch
Comparison modelling - experiments
Example of the LSM/YSZ interface
Im(Z) [W]
-1.5
La0.85Sr0.15MnO3 @ 800°C
I =170 mA.cm-2 in air.
Zexp
Zsim
-1.0
One unique vector (Kads, Kdes Kdif, kf)
has to describe at least 4 different
impedance spectra Zexp
→ practical identification
-0.5
0.0
2
3
Re (Z) [W]
4
5
- (RΩ, CDL)
O2
Oads
-1.0
Im(ZF) [W]
ZF
electrode
O2
Oads
-0.5
electrolyte
O20.0
1.0
ETH - Ceramics
1.5
2.0
Re (ZF) [W]
2.5
3.0
rds =
adsorption, diffusion and charge transfer
http://ceramics.ethz.ch
Oxygen reduction at mixed ionic-electronic conducting SOFC cathodes
ETH - Ceramics
http://ceramics.ethz.ch
Mixed ionic-electronic electrodes (MIEC)
O2
Current (A)
Oads
O2-
Intermediate T° (500-800°C)
ideal electrode
electrode
LSCF
Oads
O2-
O2-
O2-
LSM
electrolyte
Ee
Typical material: La0.6Sr0.4Co0.2Fe0.8O3-δ
(LSCF).
Competition between two reaction pathways
for O2 reduction: surface and bulk.
→ the fastest pathway is rate-determining
Potential (V)
q
Why is LSCF a better
electrocatalyst than LSM
for oxygen reduction?
→ the rate-detemining pathway influences the microstructure of the electrode
ETH - Ceramics
http://ceramics.ethz.ch
Mixed ionic-electronic electrodes
surface pathway is rate-limiting:
→ porous electrodes, large tpb
top view
bulk and surface pathways are rate-limiting:
→ Optimization of microstructure (a, b)
b
a
bulk pathway is rate-limiting:
→ thin dense electrodes, large electrolyte
coverage, low tpb.
ETH - Ceramics
http://ceramics.ethz.ch
gwd electrodes
Model system: geometrically well-defined (gwd) electrodes
O2
LSCF electrode
d
S
d
CGO electrolyte
electrolyte
tpb
O2-
S and ltpb constant
S=0.25 cm2, ltpb= 2cm
O2-
d is varied
d ~ 100 nm - 1 µm
→ Zsim (E, pO2, d)
ETH - Ceramics
http://ceramics.ethz.ch
Experimental
gwd LSCF layers are prepared by pulsed laser deposition: dense, crack-free.
LSCF electrode
S
CGO electrolyte
tpb
LSCF
300 nm
CGO
→ Zexp (E, pO2, d) = Zsim (E, pO2, d) ?
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
T = 700°C
gas phase:
O2
- air
- no gas phase diffusion
- pO2 constant
Kads
Kdes
Oads
Dense electrode (LSCF)
Kin
Kout
Slow surface diffusion
(negligible)
Dense electrode:
O2Kads
D
O2Electrolyte (CGO)
Kf
Kb
O2-
triple phase boundary (tpb)
gas/electrode/electrolyte
ETH - Ceramics
O2
- mixed conducting: e- (h) and O2- low potential gradient
- well-defined dimension
Kdes
Oads
Kfs Kbs
Electrolyte
- pure O2- conductor
O2-
Competition between surface and
bulk pathways
http://ceramics.ethz.ch
Mixed ionic-electronic electrodes
Modelling MIEC is still very controversial
O2
Oads  Vo  2e-
Oox
Oads
O2-
O2
electrode
Oads
Oads
O2-
O2-
O2-
electrolyte
Modelling MIEC is still very controversial
- incorpotation of oxygen in the the
- extension of space charge region
- influence of permittivity
ETH - Ceramics
- influence of permittivity: presence of a
displacement current?
- is electroneutrality fulfilled?
http://ceramics.ethz.ch
Model Implementation in Simulink®
dθads
2
2
= Kads pO2(1-qads) – Kdesqads – Kf(E) qads + Kb(E) (1-qads)
dt
IF = Ki [– Kf(E) qads + Kb(E) (1-qads)]
u2
Kads pO2
Kdes
u2
1
θads
+
+
θads
+
-
1-θads
state variable
function Kf (E)
kf exp(-2 b f E)
x
+
E
input
kb exp (2 (1-b) f E)
x
Ki
IF
output
function Kb (E)
ETH - Ceramics
http://ceramics.ethz.ch
Model Implementation in Simulink®
dθads
2
2
= Kads pO2(1-qads) – Kdesqads – Kf(E) qads + Kb(E) (1-qads)
dt
IF = Ki [– Kf(E) qads + Kb(E) (1-qads)]
u2
Kads pO2
Kdes
u2
1
θads
+
+
θads
+
-
1-θads
integrator
function Kf (E)
kf exp(-2 b f E)
x
+
E
input
kb exp (2 (1-b) f E)
x
Ki
IF
output
function Kb (E)
ETH - Ceramics
http://ceramics.ethz.ch
Alternative approch: modeling the impedance
new model
reaction model
DOOads
D
OO2-2OO22 D
,
D
2 ads
state-space modeling
experimental impedance
Im (Z)
faradaic impedance
Im (Z)
Re (Z)
Re (Z)
validation of the model
assessment of kinetics
ETH - Ceramics
http://ceramics.ethz.ch
Model 1 (without surf. diffusion)
Dissociative adsorption:
Kads
O2(g) + 2s D 2Oads
Kdes
O2
Charge transfer:
electrode
θads
Oads
Kf(E)
Oads + 2e- D O2-
x
electrolyte
O2- ( Oo )
Kb(E)
→ consecutive reaction steps
→ state variable θads = scalar
→ state-space model
dθads
2
2
= Kads pO2(1-qads) – Kdesqads – Kf(E) qads + Kb(E) (1-qads)
dt
IF = Ki [– Kf(E) qads + Kb(E) (1-qads)]
ETH - Ceramics
http://ceramics.ethz.ch
Numerical approach
Kads
O2(g) + 2s D 2Oads
Kdes
O2
Oads
O2
Oads
Oads reservoir
θeq
Kdif
Oads " Oads
Kf(E)
Oads + 2e- D O2-
θ3
Kb(E)
Diffusion processes
2nd Fick‘s law:
diffusion
layer
θ  2θ
 2
t z
→ Finite difference approach to
estimate time and space derivatives
O2
Oads
θ2
O2
Oads
θ1
x
O2- ( Oo )
tpb
electrolyte
→ state variable θ (θ1, θ2, θ3) = vector
θ1 ≤ θ 2 ≤ θ 3
ETH - Ceramics
http://ceramics.ethz.ch
Model implementation in Simulink®
Block diagram n°2
K
θ
θ2
2θ
 Kads pO2 (1  θ2 )2  K des θ22  dif ( 1  θ2  3 )
t
4
3
3
u2
Kads pO2
Kdes
u2
1
Kdif/4
+
+
+
-
1
s
Integrator (θ2)
-
in-ports
1/3
+
θ3
2/3
+
θ1
θ2
ETH - Ceramics
out-port to
block diagrams (1) and (3)
http://ceramics.ethz.ch
Im(ZF) / W
-0.6
Modelling results
O2
C2
Oads
-0.4
R1
-0.2
rds = adsorption and charge
transfer
electrode
O2
R2
Oads
0
2
2.5
Re(ZF) / W
3
electrolyte
3.5
O2-
charge transfer
rate constants
(potential dependant)
R2 
amount of
adsorbed oxygen
x1 
R1 and R2 are not
independant
(K f  K b ) R1
2(K ads pO2  K des )x1  2K ads pO2
adsorption/desorption
rate constants
2K ads pO2  K f  K b 
 2K
ads pO 2

 Kf  Kb
)
2

)
 4 K ads pO2  K des  K ads  K b )
2 K ads pO2  K des
)
→ Necessity of a modelling approach to comprehensively interpret experimental impedance
ETH - Ceramics
http://ceramics.ethz.ch
Experimental investigation
O2
electrode
O2-
O2-
Consequence of parallel pathways
electrolyte
Intermediate T° SOFC (600-800°C).
Typical material: LaxSr1-xCoyFe1-yO3
(LSCF).
ETH - Ceramics
http://ceramics.ethz.ch