Electrochemical Impedance Spectroscopy Electrochemical Impedance Spectroscopy with Application to Fuel Cells with Application to Fuel Cells Mark E. Orazem Department of Chemical Engineering University of Florida Gainesville, Florida 32611 [email protected]352-392-6207 Mark E. Orazem, 2000-2009. All rights reserved. Course sponsored for the Fuel Cell Seminar by The Electrochemical Society
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Electrochemical Impedance Spectroscopy Electrochemical Impedance Spectroscopy with Application to Fuel Cellswith Application to Fuel Cells
• How to think about impedance spectroscopy• EIS as a generalized transfer function• Overview of applications of EIS• Objective and outline of course
Mark E. Orazem, 2000-2009. All rights reserved.
Chapter 1. Introduction page 1: 3
1992 – no logo
Chapter 1. Introduction page 1: 4
June 6 - 11, 2010 • Hotel Tivoli CarvoeiroCarvoeiro, Algarve, Portugal
Chapter 1. Introduction page 1: 5
The Blind Men and the ElephantThe Blind Men and the ElephantJohn Godfrey SaxeJohn Godfrey Saxe
It was six men of IndostanTo learning much inclined,Who went to see the Elephant(Though all of them were blind),That each by observationMight satisfy his mind.
The First approached the Elephant,And happening to fallAgainst his broad and sturdy side,At once began to bawl:“God bless me! but the ElephantIs very like a wall!” ...
– digital transfer function analyzer– fast Fourier transform
D. Macdonald, Transient Techniques in Electrochemistry, Plenum Press, NY, 1977.J. Ross Macdonald, editor, Impedance Spectroscopy Emphasizing Solid Materials and Analysis, John Wiley and Sons, New York, 1987.C. Gabrielli, Use and Applications of Electrochemical Impedance Techniques, Technical Report, Schlumberger, Farnborough, England, 1990.
Chapter 3. Impedance Measurement page 3: 3
AC BridgeAC Bridge
Generator
Z2Z1
Z3 Z4
D
• Bridge is balanced when current at D is equal to zero
• Time consuming• Accurate
1 4 2 3Z Z Z Z
10 Hzf
Chapter 3. Impedance Measurement page 3: 4
| |
sin( )
VZI
OA A'AOB B'B
OD D'DOA A'A
( ) sin( )
( ) sin( )
V t V tVI t tZ
Potential
Current
ADO
B
B'
D'A'
LissajousLissajous AnalysisAnalysis
Chapter 3. Impedance Measurement page 3: 5
Phase Sensitive DetectionPhase Sensitive Detection0 sin( )AA A t
• Reduce time for measurement– shorter integration (fewer cycles)
• accept more stochastic noise to get less bias error– fewer frequencies
• more measured frequencies yields better parameter estimates• fewer frequencies takes less time
– avoid line frequency and harmonic (±5 Hz)• takes a long time to measure on auto-integration• cannot use data anyway
– select appropriate modulation technique• decide what you want to hold constant (e.g., current or potential)• system drift can increase measurement time on auto-integration
Chapter 5. Development of Process ModelsChapter 5. Development of Process Models
• Use of Circuits to guide development• Develop models from physical grounds• Model case study• Identify correspondence between physical models and
electrical circuit analogues• Account for mass transfer
Mark E. Orazem, 2000-2009. All rights reserved.
Chapter 5. Development of Process Models page 5: 2
Use circuits to create frameworkUse circuits to create framework
Chapter 5. Development of Process Models page 5: 3
Addition of PotentialAddition of Potential
Chapter 5. Development of Process Models page 5: 4
Addition of CurrentAddition of Current
Chapter 5. Development of Process Models page 5: 5
Equivalent Circuit at the Corrosion PotentialEquivalent Circuit at the Corrosion Potential
Chapter 5. Development of Process Models page 5: 6
Equivalent Circuit for a Partially Blocked Equivalent Circuit for a Partially Blocked ElectrodeElectrode
Chapter 5. Development of Process Models page 5: 7
Equivalent Circuit for an Electrode Coated Equivalent Circuit for an Electrode Coated by a Porous Layerby a Porous Layer
Chapter 5. Development of Process Models page 5: 8
Equivalent Electrical Circuit for an Electrode Equivalent Electrical Circuit for an Electrode Coated by Two Porous LayersCoated by Two Porous Layers
1402 0
2 2 2
; 8.8452 10 F/cm
R /
C
Chapter 5. Development of Process Models page 5: 9
Use kinetic models to determine Use kinetic models to determine expressions for the interfacial impedanceexpressions for the interfacial impedance
Chapter 5. Development of Process Models page 5: 10
ApproachApproachidentify reaction mechanismwrite expression for steady state current contributionswrite expression for sinusoidal steady statesum current contributionsaccount for charging currentaccount for ohmic potential dropaccount for mass transfercalculate impedance
Chapter 5. Development of Process Models page 5: 11
General Expression for Faradaic CurrentGeneral Expression for Faradaic Current
,0 ,
,
,0, ,0 , ,
,
i k j j i k
j j j k
f iic i V c
kk k V c
f fi V cV c
f
,0, ,f i ii f V c
tjeiii ~Re
Chapter 5. Development of Process Models page 5: 12
Reactions ConsideredReactions Considered
• Dependent on Potential• Dependent on Potential and Mass Transfer• Dependent on Potential, Mass Transfer, and Surface
Coverage• Coupled Reactions
Chapter 5. Development of Process Models page 5: 13
Irreversible Reaction: Irreversible Reaction: Dependent on PotentialDependent on Potential
A A z ne
z+
• Potential-dependent heterogeneous reaction• Two-dimensional surface• No effect of mass transfer
Chapter 5. Development of Process Models page 5: 14
Current DensityCurrent Density
A,At
ViR
,A
1expt
A A A
RK b b V
AA A A exp Fi n Fk V
RT
steady-state
A AexpA Ai K b b V V
oscillating component
A A exp Ai K b V
Chapter 5. Development of Process Models page 5: 15
f dl
f dl
dVi i Cdt
i i j C V
Charging CurrentCharging Current
low frequency
high frequency
,A
,A
1
dlt
dlt
Vi j C VR
V j CR
,A
,A1t
t dl
RVi j R C
Chapter 5. Development of Process Models page 5: 16
Ohmic ContributionsOhmic Contributions
VRiU
ViRU
e
e~~~
A
,A
,A1+
e
te
t dl
U VZ Ri i
RR
j R C
Chapter 5. Development of Process Models page 5: 17
A A Aexp 2.303i K V
A A2.303/ b
A
,AA
1exp 2.303t
A A A
RK b b V i
AA
,A
A ,A A
2.303
2.303t
t
iR
R i
A A exp Ai K b V
Steady Currents in Terms of Steady Currents in Terms of RRt,At,A
Chapter 5. Development of Process Models page 5: 18
Irreversible Reaction: Irreversible Reaction: Dependent on Potential and Mass TransferDependent on Potential and Mass Transfer
O Rne
• Irreversible potential-dependent heterogeneous reaction• Reaction on two-dimensional surface• Influence of transport of O to surface
Chapter 5. Development of Process Models page 5: 19
Current DensityCurrent Density
,OO O,0 O
1exptR
K c b V
O O O,0 Oexpi K c b V
steady-state
O O O O,0 O O O O,0
O O O,0,O
exp exp
expt
i K b c b V V K b V c
V K b V cR
oscillating component
Chapter 5. Development of Process Models page 5: 20
Mass TransferMass Transfer
OO O O
0
O O ORe j t
dci n FDdy
i i i e
OO O O
0
dci n FDdy
O,0O O O
O
0c
i n FD
Chapter 5. Development of Process Models page 5: 21
Combine ExpressionsCombine Expressions
O O O O,0,O
expt
Vi K b V cR
O O
O,0O O 0
icn FD
Chapter 5. Development of Process Models page 5: 22
Current DensityCurrent Density
OO
,OO O O,0 O
,O ,O
1 1(0)t
t d
ViR
n FD c b
VR z
O,O
O O O,0 O
1 1(0)dz
n FD c b
,OO O,0 O
1exptR
K c b V
Chapter 5. Development of Process Models page 5: 23
Calculate ImpedanceCalculate Impedance
O
f dl
dl
dVi i Cdt
i i j C V
,O ,O
,O ,O
1
dlt d
dlt d
Vi j C VR z
V j CR z
VRiU
ViRU
e
e~~~
O
,O ,O
,O ,O1+
e
t de
dl t d
U VZ Ri i
R zR
j C R z
Chapter 5. Development of Process Models page 5: 24
Comparison to Circuit AnalogComparison to Circuit Analog
Cdl
Re
Rt,O
zd,O
,O ,O
O,O ,O1+
t de
dl t d
R zZ R
j C R z
Chapter 5. Development of Process Models page 5: 25
Irreversible Reaction: Irreversible Reaction: Dependent on Potential and Adsorbed IntermediateDependent on Potential and Adsorbed Intermediate
1
2
-
-
B X+eX P+e
k
k
• Potential-dependent heterogeneous reactions• Adsorption of intermediate on two-dimensional surface• Maximum surface coverage
B
X X XX
P
Chapter 5. Development of Process Models page 5: 26
SteadySteady--State Current DensityState Current Density
1 1 1 11 expi K b V V
reaction 1: formation of X
reaction 2: formation of P
2 2 2 2expi K b V V
total current density
1 2i i i
Chapter 5. Development of Process Models page 5: 27
Chapter 5. Development of Process Models page 5: 41
Capacitance and Ohmic ContributionsCapacitance and Ohmic Contributions
ff
ff
ii
ii~~
Faradaic
VCjiidtdVCii
df
df
~~~
Faradaic and Charging
VRiU
ViRU
e
e~~~
Ohmic
Chapter 5. Development of Process Models page 5: 42
ddt
ddtt,t,
jr
CjzRR
CjzRRR
jZZiUZ
22
222
O,O,eff
O,O,HFe
111
1111
~~
2eff Fe H
1 1 1
t, t,R R R
Process ModelProcess Model
Rt,O2
Rt,H2Rt,Fe
Cdl
Zd,O2
Chapter 5. Development of Process Models page 5: 43
Development of Impedance ModelsDevelopment of Impedance Models
identify reaction mechanism write expression for steady state current contributions write expression for sinusoidal steady state sum current contributions account for charging current account for ohmic potential drop• account for mass transfer calculate impedance
Chapter 5. Development of Process Models page 5: 44
Mass TransferMass Transfer
Chapter 5. Development of Process Models page 5: 45
Film DiffusionFilm Diffusion2
2i i
ic cDt z
,
,0
as
at 0i i f
i i
c c z
c c z
,0 , ,0i i i if
zc c c c
steady state
Chapter 5. Development of Process Models page 5: 46
Film DiffusionFilm Diffusion2
2i i
ic cDt z
2 2
2 2
2
2
2
2
j t j ti ii i
j t j tii
ii
d c d cj ce D D ed z d zd cj ce D ed zd cj c Dd z
tjiii eccc ~Re
2
(0)
fi
i
ii
i
KDc
c
i
z
2
2 0ii i
d jKd
Chapter 5. Development of Process Models page 5: 47
Warburg ImpedanceWarburg Impedance
2
2 0
exp exp
ii i
i i i
d jKd
A jK B jK
2
2
0 at 1
1 at 0
tanh1(0)
tanh1(0)
i
i
i
i i
i
i
i
jKjK
jD
jD
2
0 at
1 at 01 1(0)
1 1(0)
i
i
i i
i
i
jK
jD
Chapter 5. Development of Process Models page 5: 48
ConcentrationConcentration
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2
c()
/c(
)
t=0, 2n/K
t=n/K
t=0.5n/K
t=1.5n/K
K=100
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2
c()
/c(
)
t=0, 2n/K
t=n/K
t=0.5n/K
t=1.5n/K
K=1-40
-30
-20
-10
00 10 20 30 40 50 60
Zr, Zr,D,
Z j, Z
j,D,
Diffusion Impedance Only (infinite)Diffusion Impedance Only (finite)Complete Impedance
Chapter 5. Development of Process Models page 5: 49
Rotating DiskRotating Disk
Chapter 5. Development of Process Models page 5: 50
Convective DiffusionConvective Diffusion
2
21zc
rrc
rrD
zcv
rcv
tc ii
ii
zi
ri
...
631 4
23
2/32 zbzzavz
...
323
2/322/1
zbzzarvr
...
31 3
2/32/1
zazbrv
z
Chapter 5. Development of Process Models page 5: 51
Convective Diffusion in oneConvective Diffusion in one--DimensionDimension
2
2
zcD
zcv
tc i
ii
zi
neMsi
zii
i
zcc ii as,
0at znFis
zcD fii
i
if cfi ,
Chapter 5. Development of Process Models page 5: 52
2
2
2
2 0j t j t j ti ii iz iz id c d cv Dd z d
d c d cj ce v e D edz d zz
Sinusoidal Steady StateSinusoidal Steady State tj
iii eccc ~Re
1/3i
31
2
31
2 Sc99
aDa
Ki
i
i
z
1/3i
31
31
Sc133
aaDi
i
0~~~2
2
iii
zi cjK
dcdv
dcd
Chapter 5. Development of Process Models page 5: 53
Chapter 6. Time Constant DispersionChapter 6. Time Constant Dispersion
• CPEs can arise from surface or axial distributions
• CPE parameters can be interpreted in terms of capacitance, depending on type of distribution
• Time-constant dispersions can be modeled explicitly.
Mark E. Orazem, 2000-2009. All rights reserved.
Chapter 6. Time-Constant Dispersion page 6: 2
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Chapter 6. Time-Constant Dispersion page 6: 3
Page deliberately left blank.
Chapter 6. Time-Constant Dispersion page 6: 4
Types of DistributionsTypes of Distributions
• Surface
• Axial
Chapter 6. Time-Constant Dispersion page 6: 5
Surface DistributionsSurface Distributions
Chapter 6. Time-Constant Dispersion page 6: 6
Axial DistributionsAxial Distributions
Chapter 6. Time-Constant Dispersion page 6: 7
Relationship between Q and CRelationship between Q and C
• Surface Distribution
• Axial Distribution
G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, J. H. Sluyters, J. Electroanal. Chem., 176 (1984), 275-295.
C. H. Hsu, F. Mansfeld, Corrosion, 57 (2001), 747-748.
B. Hirschorn, M. E. Orazem, B. Tribollet, V. Vivier, I. Frateur and M. Musiani, Determination of Effective Capacitance and Film Thickness from CPE Parameters, submitted to Electrochim. Acta, 2009.
Chapter 6. Time-Constant Dispersion page 6: 8
Determination of Film ThicknessDetermination of Film Thickness
Chapter 6. Time-Constant Dispersion page 6: 9
Constant Phase ElementsConstant Phase Elements
• Origin is ambiguous• Can arise from surface or axial distributions• CPE parameters can be interpreted in terms of
capacitance, depending on type of distribution
Chapter 6. Time-Constant Dispersion page 6: 10
Porous ElectrodesPorous Electrodes
Chapter 6. Time-Constant Dispersion page 6: 11
deLeviedeLevie ModelModel
Under the assumption that Z0and R0 are independent of x
intermediate intermediate Shallow pores Two combinations of r , n , and ; i.e.,
3/ 2r n and / r
Chapter 6. Time-Constant Dispersion page 6: 13
2+ -Fe Fe +2e2+ + 3+ -
22Fe +HOCl+H 2Fe +Cl +H O
Cast Iron Pipes
I. Frateur, C. Deslouis, M.E. Orazem and B. Tribollet, Electrochimica Acta, 44 (1999), 4345.
No Free Chlorine
2 mg/l Free Chlorine
+ -2 2
1O +2H +2e H O2
Consumption of Free Consumption of Free Chlorine in Municipal Chlorine in Municipal
Water SuppliesWater Supplies
Chapter 6. Time-Constant Dispersion page 6: 14
• Coupled electrochemical reactions
• Surface films• Convective diffusion• History and time-dependent
parameters• Identification of corrosion rate
Cast Iron
Black Rust(Fe3O4)
Green Rust & Carbonates(Fe2+ and Fe3+)
Red Rust (-FeOOH, -Fe2O3)
Evian WaterIssuesIssues
Chapter 6. Time-Constant Dispersion page 6: 15
Model DevelopmentModel DevelopmentNo Free Chlorine
2 mg/l Free Chlorine
Chapter 6. Time-Constant Dispersion page 6: 16
Rta
Cdla
Za =(b)Z =Za
ZcRe(a)
Cf
Rf Rtc ZD
Cdlc
Z0 =(c)
Z R ZL
c
0 0 coth
ZR
0
0
Model for Impedance ResponseModel for Impedance Response
Chapter 6. Time-Constant Dispersion page 6: 17
Impedance DataImpedance Data
3 days
7 days
28 days
Chapter 6. Time-Constant Dispersion page 6: 18
TimeTime--Constant DispersionConstant Dispersion
• While use of a CPE may lead to improved regressions, the meaning can be ambiguous, and the physical system may not follow the specific distribution implied in the CPE model.
• Distributed time-constant systems can be modeled explicitly.
• Not all depressed semi-circles correspond to a CPE behavior.
• Regression response surfaces– noise– incomplete frequency range
• Adequacy of fit– quantitative– qualitative
Mark E. Orazem, 2000-2009. All rights reserved.
Chapter 7. Regression Analysis page 7: 2
Test Circuit 1: 1 Time ConstantTest Circuit 1: 1 Time Constant
• R0 = 0• R1 = 1 cm2
• 1 = RC = 1 sR0
R1
C1
(1)
Chapter 7. Regression Analysis page 7: 3
Linear optimization surface roughly Linear optimization surface roughly parabolicparabolic
0.6 0.8 1.01.2
1.4
0.00
0.01
0.02
0.03
0.04
0.05
0.6
0.81.0
1.21.4
Sum
of S
quar
es
/ s
R / cm2
0.5 1.0 1.50.5
1.0
1.5
R / cm2
/ s
0
0.02000
0.04000
0.06000
0.08000
0.1000
dat dat22
,dat ,mod,dat ,mod2 2
1 1( )
N Nj jr r
i kr j
Z ZZ Zf
p
Chapter 7. Regression Analysis page 7: 4
Nonlinear RegressionNonlinear Regression
0 0
2
01 1 1
1( ) ( )2
p p pN N N
j j kj j kj j kp p
f ff f p p pp p p
p p
β α p
dat
21
( | ) ( | )Ni i i
ki ki
Z Z Zp
p p
dat
, 21
( | ) ( | )1Ni i
j ki j ki
Z Zp p
p p
dat 22
21
( ( | ))Ni i
i i
Z Z
p
Variance of data
Derivative of function with respect to parameter
Chapter 7. Regression Analysis page 7: 5
Methods for RegressionMethods for Regression
• Evaluation of derivatives– method of steepest descent– Gauss-Newton method– Levenberg-Marquardt method
• Evaluation of function– simplex
Chapter 7. Regression Analysis page 7: 6
Effect of Noisy DataEffect of Noisy Data
0.5 1.0 1.50.5
1.0
1.5
R / cm2
/ s
1.000
75.75
150.5
225.2
300.0
• response surface remains parabolic• value at minimum increases
0.6
0.8
1.0
1.2
1.40.6
0.81.0
1.21.4
0
100
200
300
Sum
of S
quar
es
R / cm2
/ s
Add noise: 1% of modulus
Chapter 7. Regression Analysis page 7: 7
Test Circuit 2: 3 Time ConstantsTest Circuit 2: 3 Time Constants
R0
R1
C1
(1)
R2
C2
(2)
R3
C3
(3)
• R0 = 1 cm2
• R1 = 100 cm2
• 1 = 0.001 s• R2 = 200 cm2
• 2 = 0.01 s• R3 = 5 cm2
• 3 = 0.05 s
Note: 3rd Voigt element contributes only 1.66% to DC cell impedance.
Chapter 7. Regression Analysis page 7: 8
Effect of Noisy DataEffect of Noisy Data
All parameters fixed except R3 and 3
-4-3
-2-1
01
2-2-1
01
23
410-2
10-1
100
101
102
103
Sum
of S
quar
es log10(R/ cm2) log10( / s)
noise: 1% of modulus
-4-3
-2-1
01
2-2-1
01
23
410-2
10-1
100
101
102
103
Sum
of S
quar
es
log10(R/ cm2) log10( / s)
no noise
Note: use of log scale for parameters
Chapter 7. Regression Analysis page 7: 9
Resulting SpectrumResulting Spectrum
0 50 100 150 200 250 300 3500
-50
-100
-150
Z j /
c
m2
Zr / cm2
Model with 1% noise added Model with no noise
Chapter 7. Regression Analysis page 7: 10
Test Circuit 3: 3 Time ConstantsTest Circuit 3: 3 Time Constants
R0
R1
C1
(1)
R2
C2
(2)
R3
C3
(3)
• R0 = 1 cm2
• R1 = 100 cm2
• 1 = 0.01 s• R2 = 200 cm2
• 2 = 0.1 s• R3 = 100 cm2
• 3 = 10 s
Note: 3rd Voigt element contributes 25% to DC cell impedance. The time constant corresponds to a characteristic frequency 3=0.1 s-1 or f3=0.016 Hz.
Chapter 7. Regression Analysis page 7: 11
Resulting Test SpectraResulting Test Spectra
0 100 200 300 4000
-100
Z j /
cm
2
Zr / cm2
0.01 Hz to 100 kHz1 Hz to 100 kHz
Chapter 7. Regression Analysis page 7: 12
Effect of Truncated DataEffect of Truncated Data
All parameters fixed except R3 and 3
-2-1
01
23
4-1
01
23
45100
101
102
103
Sum
of S
quar
es
log10(R / cm2)
log10( / s)
0.01 Hz to 100 kHz
-2-1
01
23
4-1
01
23
4510-1
100
101
102
103
Sum
of S
quar
es log10(R / cm2)
log10( / s)
1 Hz to 100 kHz
Chapter 7. Regression Analysis page 7: 13
Conclusions from Test Spectra Conclusions from Test Spectra
• The presence of noise in data can have a direct impact on model identification and on the confidence interval for the regressed parameters.
• The correctness of the model does not determine the number of parameters that can be obtained.
• The frequency range of the data can have a direct impact on model identification.
• The model identification problem is intricately linked to the error identification problem. In other words, analysis of data requires analysis of the error structure of the measurement.
Chapter 7. Regression Analysis page 7: 14
When Is the Fit Adequate?When Is the Fit Adequate?
• Chi-squared statistic– includes variance of data– should be near the degree of freedom
• Visual examination– should look good– some plots show better sensitivity than others
• Parameter confidence intervals– based on linearization about solution– should not include zero
Chapter 7. Regression Analysis page 7: 15
Test Case: Mass Transfer to a RDETest Case: Mass Transfer to a RDE
Single reaction coupled with mass transfer. Consider model for a Nernst stagnant diffusion layer:
( )1
t de
t d
R zZ R
j C R z
tanhd d
jz z
j
Chapter 7. Regression Analysis page 7: 16
Evaluation of Evaluation of 22 StatisticStatistic
/|Z()| 1 0.1 0.05 0.03 0.01
0.0408 4.08 16.32 45.32 408
2/ 0.00029 0.029 0.12 0.32 2.9
Chapter 7. Regression Analysis page 7: 17
Comparison of Model to DataComparison of Model to Data
• Mathematical form and interpretation• Application to noisy data• Methods to evaluate consistency
Mark E. Orazem, 2000-2009. All rights reserved.
Chapter 9. Kramers-Kronig Relations page 9: 2
ContraintsContraints
• Under the assumption that the system is– Causal– Linear– Stable
• A complex variable Z must satisfy equations of the form:
dxx
ZxZZ rrj
0
22)()(2)(
dxx
ZxxZZZ jj
rr
0
22
)()(2)()(
Chapter 9. Kramers-Kronig Relations page 9: 3
Use of KramersUse of Kramers--Kronig RelationsKronig Relations
• Concept– if data do not satisfy Kramers-Kronig relations, a
condition of the derivation must not be satisfied• stationarity / causality• linearity• stability
– interpret result in terms of • instrument artifact• changing baseline
– if data satisfy Kramers-Kronig relations, conditions of the derivation may be satisfied
Chapter 9. Kramers-Kronig Relations page 9: 4
For real data (with noise)For real data (with noise)
))()(())()(()()()(ob
jjrr ZjZZZ
)()( jr j
)()( ob ZZE 0)( E
where
If and only if
0
22)()()()(2)( dx
xxZxZEZE rrrr
j
0
22
)()()()(2)()( dxx
xxZxxZEZZE jjjj
rr
Kramers-Kronig in an expectation sense
Chapter 9. Kramers-Kronig Relations page 9: 5
The KramersThe Kramers--Kronig relations can be Kronig relations can be satisfied ifsatisfied if
• This means– the process must be stationary in the sense of replication at every
measurement frequency. – As the impedance is sampled at a finite number of frequencies, r(x)
represents the error between an interpolated function and the “true”impedance value at frequency x. In the limit that quadrature and interpolation errors are negligible, the residual errors r(x) should be of the same magnitude as the stochastic noise r().
0)(2
0
22
dxx
xE r
0)( E and
Chapter 9. Kramers-Kronig Relations page 9: 6
10-4 10-3 10-2 10-1 100 101 102 103 104 105
-1000
-800
-600
-400
-200
0
(Zr(x
)-Zr(
)) /
(x2 -
2 )
f / Hz
Meaning ofMeaning of 2 2
0
2 ( ) 0r xE dxx
0.2 0.5 1-400
-300
-200
-100
0
Interpolated Value
Correct Value
(Zr(x
)-Zr(
)) /
(x2 -
2 )
f / Hz
Chapter 9. Kramers-Kronig Relations page 9: 7
Use of KramersUse of Kramers--Kronig RelationsKronig Relations
• Quadrature errors– require interpolation function
• Missing data at low and high frequency
Chapter 9. Kramers-Kronig Relations page 9: 8
Methods to Resolve Problems of Insufficient Methods to Resolve Problems of Insufficient Frequency RangeFrequency Range
• Direct Integration– Extrapolation
• single RC• polynomials• 1/ and asymptotic behavior
– simultaneous solution for missing data• Regression
– proposed process model– generalized measurement model
Chapter 9. Kramers-Kronig Relations page 9: 9
Chapter 10. Application to PEM Fuel Cells page 10: 1
Chapter 10. Application to PEM Fuel Cells page 10: 14
Impedance Data after Measurement Impedance Data after Measurement Model AnalysisModel Analysis
S. K. Roy and M. E. Orazem, “Error Analysis of the Impedance Response of PEM Fuel Cells,” J. Electrochem. Soc., 154 (2007), B883-B891.
Chapter 10. Application to PEM Fuel Cells page 10: 15
Impedance Process Model Impedance Process Model DevelopmentDevelopment
Chapter 10. Application to PEM Fuel Cells page 10: 16
Proposed ReactionsProposed Reactions
1. Hydrogen oxidation and oxygen reduction
2. Hydrogen oxidation and oxygen reduction with peroxide intermediate
3. Hydrogen oxidation and oxygen reduction with Pt deactivation
Chapter 10. Application to PEM Fuel Cells page 10: 17
Model AssumptionsModel Assumptions
Gas
Diff
usio
n La
yer
Gas
Diff
usio
n La
yer
PEM
CA THODE
ANODE
ii
H
• Assumptions:– Uniform membrane properties– Uniform surface overpotential– No convection– Diffusion through stagnant film of finite thickness– Uniform surface and distribution of reactants and products
O2 H2
Cathode Reaction Kinetics
H
Anode Reaction Kinetics
Chapter 10. Application to PEM Fuel Cells page 10: 18
Steps in Model DevelopmentSteps in Model Development
Reaction Mechanisms
Steady-State Current
Expressions
Polarization Curve
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Pote
ntia
l / V
Current / A
Faradaic Impedance
ApplySinusoidal
Perturbation
Electrolyte Resistance
Double Layer Capacitance
Overall Impedance
0.00 0.05 0.10 0.15 0.20 0.25
0.00
0.05
0.10
Z j /
cm
2
Zr / cm2
Chapter 10. Application to PEM Fuel Cells page 10: 19
Model Development: Case 1Model Development: Case 1
• Oxygen Reduction
• Hydrogen Oxidation+ -
2H 2H + 2e
+ -2 2O + 4H + 4e 2H O
Chapter 10. Application to PEM Fuel Cells page 10: 20
Model 1: Simple Reaction KineticsModel 1: Simple Reaction Kinetics
Chapter 10. Application to PEM Fuel Cells page 10: 21
Model 1: Simple Reaction KineticsModel 1: Simple Reaction Kinetics
Chapter 10. Application to PEM Fuel Cells page 10: 22
SteadySteady--State Current Density: 1State Current Density: 1
2 2 2 2 2O O O O O(0)expi K C b
2 2 2 2 2H H H H H(0)expi K C b
2 2H Oi i
Chapter 10. Application to PEM Fuel Cells page 10: 23
Model Predictions: Case 1Model Predictions: Case 1
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Experimental data Model 1
Pote
ntia
l / V
Current / A
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.05
0.10
Experimental data Model 1
Z j /
cm
2
Zr / cm2
Chapter 10. Application to PEM Fuel Cells page 10: 24
Model Development: Case 2Model Development: Case 2
• Oxygen Reduction
• Hydrogen Oxidation+ -
2H 2H + 2e
+ -2 2 2O + 2H + 2e H O
+ -2 2 2H O + 2H + 2e 2H O
C. F. Zinola, W. E. Triaca and A. J. Arvia, J. Appl. Electrochem. 25 (1995) 740.
Chapter 10. Application to PEM Fuel Cells page 10: 25
Model 2: Peroxide Intermediate Model 2: Peroxide Intermediate 2
2, 2
2
2
a
0,a
e c,1
0,c
c,2
Chapter 10. Application to PEM Fuel Cells page 10: 26
SteadySteady--State Current Density: 2State Current Density: 2
2 2 2 2 2 2 2 2 2 2 2 2H O H O H O H O H O H O(0) expi K C b
2 2 2 2 2 2 2O O O H O O O(0) 1- exp -i K C b
2 2 2 2 2H H H H H(0)expi K C b
2 2 2 2H O H Oi i i
Chapter 10. Application to PEM Fuel Cells page 10: 27
Model 2 Response Compared with DataModel 2 Response Compared with Data
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.05
0.10
Experimental data Model 2
Z j /
cm
2
Zr / cm2
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Experimental data Model 2
Pot
entia
l / V
Current / A
Chapter 10. Application to PEM Fuel Cells page 10: 28
Model 2 Response Compared with DataModel 2 Response Compared with Data
0.00 0.05 0.10 0.15 0.20 0.25
0.00
0.05
0.10
Experimental data Model 2
Z j /
cm
2
Zr / cm2
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Experimental data Model 2
Pot
entia
l / V
Current / A
Chapter 10. Application to PEM Fuel Cells page 10: 29
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Experimental data Model 2
Pot
entia
l / V
Current / A
Model 2 Response Compared with DataModel 2 Response Compared with Data
0.0 0.1 0.2 0.3 0.4 0.5-0.1
0.0
0.1
0.2
Experimental data Model 2
Z j /
cm
2
Zr / cm 2
Chapter 10. Application to PEM Fuel Cells page 10: 30
Model Development: Case 3Model Development: Case 3
• Oxygen Reduction
• Hydrogen Oxidation
R. M. Darling and J. P. Meyers, J. Electrochem. Soc., 150 (2003) A1523.
+ -2 2
+ -2
+ 2+2
O +4H +4e 2H O
Pt +H O 2H +2e +PtO
PtO+2H Pt +H O
+ -2H 2H +2e
Chapter 10. Application to PEM Fuel Cells page 10: 31
Model 3: Pt dissolution and formation of Model 3: Pt dissolution and formation of PtOPtO 2
2, 2
2
2
a
0,a
e c,1
0,c
c,2
Chapter 10. Application to PEM Fuel Cells page 10: 32
SteadySteady--State Current Density: 3State Current Density: 3
2 2 2 2O ,Pt O O O(0)expeffi k C b
PtO Pt,f Pt Pt Pt,b PtO Pt Pt1- exp expPti k b k b
+2
PtO PtO PtO H(0)r k C
Pt PtO Pt PtO- effk k k k
2 2 2 2 2H H H H H(0)expi k C b
Chapter 10. Application to PEM Fuel Cells page 10: 33
Model 3 Response Compared with DataModel 3 Response Compared with Data
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.05
0.10
Experimental data Model 3
Z j /
cm
2
Zr / cm2
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Experimental data Model 3
Pot
entia
l / V
Current / A
Chapter 10. Application to PEM Fuel Cells page 10: 34
Model 3 Response Compared with DataModel 3 Response Compared with Data
0.00 0.05 0.10 0.15 0.20
0.00
0.05
0.10
Experimental data Model 3
Z j /
cm
2
Zr / cm2
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Experimental data Model 3
Pot
entia
l / V
Current / A
Chapter 10. Application to PEM Fuel Cells page 10: 35
Model 3 Response Compared with DataModel 3 Response Compared with Data
0.0 0.1 0.2 0.3 0.4 0.5
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Experimental data Model 3
Z j /
cm
2
Zr / cm2
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Experimental data Model 3
Pot
entia
l / V
Current / A
Chapter 10. Application to PEM Fuel Cells page 10: 36
Model development suggests supporting Model development suggests supporting experimentsexperiments
• Formation of peroxide• Signs of membrane degradation• Formation of PtO• Reduction in electrochemically active area• Dissolved Pt in outflow
Chapter 10. Application to PEM Fuel Cells page 10: 37
Evidence for Evidence for PtOPtOxx
Helena and Jason Weaver, University of Florida
Chapter 10. Application to PEM Fuel Cells page 10: 38
Integrated ApproachIntegrated Approach
Chapter 10. Application to PEM Fuel Cells page 10: 39
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
-0.2
-0.4
-0.6
0.005 Hz1 kHz
0.1 0.3
0.5
0.6
0.7 A cm-2
Z j /
cm
2
Zr / cm2
Flooding and EIS responseFlooding and EIS responseEIS response as a function of current density collected at 4040°C.
Effect of flooding is
visible
Chapter 10. Application to PEM Fuel Cells page 10: 40
EIS Response at 70EIS Response at 70°°CC
0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05
0.15
0.00
-0.15
-0.30
-0.45
0.005 Hz
0.30.5
0.7
0.9
1.1 A cm-2
1 kHz
25 Hz
25 Hz25 Hz
25 Hz
25 Hz
Z j /
c
m2
Zr / cm2
Effect of flooding is
visible
Chapter 10. Application to PEM Fuel Cells page 10: 41
Standard deviation of the ImpedanceStandard deviation of the Impedance
0.2 0.4 0.6 0.8 1.0
10-5
10-4
10-3
100 Hz
10 Hz
0.10 Hz
r, ob
s /
cm
2
Current Density / A cm-20.2 0.4 0.6 0.8 1.0
10-5
10-4
100 Hz
10 Hz
0.10 Hz
j, ob
s /
cm
2
Current Density / A cm-2
Chapter 10. Application to PEM Fuel Cells page 10: 42Normalized Noise in Impedance ResponseNormalized Noise in Impedance Response
0.2 0.4 0.6 0.8 1.00.2
1
10
20
100 Hz
10 Hz
0.10 Hz
r,obs
/ba
se
Current Density / A cm-2
Shows flooding -10X more
noise
Shows need for
better base error
structure
Chapter 10. Application to PEM Fuel Cells page 10: 43
Calculated ParametersCalculated Parameters
Chapter 10. Application to PEM Fuel Cells page 10: 44
Fractional Surface Coverage of Fractional Surface Coverage of IntermediatesIntermediates
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8 Hydrogen Peroxide Platinum Oxide
Frac
tiona
l Sur
face
Cov
erag
e
Potential / V0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8 Hydrogen Peroxide Platinum Oxide
Frac
tiona
l Sur
fcae
Cov
erag
e
Current / A
Chapter 10. Application to PEM Fuel Cells page 10: 45
LowLow--frequency Inductive Loops in PEM frequency Inductive Loops in PEM Fuel CellsFuel Cells
• Satisfy the Kramers-Kronig relations• Are consistent with
– Peroxide formation– Pt dissolution
• May provide a means to study reactions that limit lifetime• Demonstrates synergistic approach
– Error structure analysis– Model development– Use of supporting measurements
Chapter 10. Application to PEM Fuel Cells page 10: 46
Impedance SpectroscopyImpedance Spectroscopy• Electrochemical measurement of macroscopic properties• Example of a generalized transfer-function measurement• Can be used to extract contributions of
– electrode reactions– mass transfer– surface layers
• Can be used to estimate – reaction rates– transport properties
• Interpretation of data– graphical representations– regression– process models – error analysis
Chapter 11. Conclusions page 11: 3
AcknowledgementsAcknowledgements• Students
– UF: P. Agarwal, M. Durbha, S. Carson, M. Membrino, P. Shukla, V. Huang, S. Roy
– CNRS: T. El Moustafid, I. Frateur, H. G. de Melo• Collaborators
– USF: L. García-Rubio– UF: O. Crisalle– CNRS: B. Tribollet, C. Deslouis, H. Takenouti, V. Vivier– CIRIMAT: N. Pébère
• General• Process Models• Orazem group work on Fuel Cells• Measurement Models• CPE• Plotting
Mark E. Orazem, 2000-2009. All rights reserved.
Chapter 12. Suggested Reading page 12: 2
Suggested ReadingSuggested ReadingGeneralGeneral
• J. R. Macdonald, editor, Impedance Spectroscopy Emphasizing Solid Materials and Analysis, John Wiley and Sons, New York, 1987.
• E. Barsoukov and J. R. Macdonald, editors, Impedance Spectroscopy: Theory, Experiment, and Applications, 2nd Edition, John Wiley and Sons, New York, 2005.
• C. Gabrielli, Use and Applications of Electrochemical Impedance Techniques, Technical Report, Schlumberger, Farnborough, England, 1990.
• D. Macdonald, Transient Techniques in Electrochemistry, Plenum Press, NY, 1977.
• M. E. Orazem and B. Tribollet, Electrochemical Impedance Spectroscopy, John Wiley and Sons, New York, 2008.
• Other Sources– See instrument vendor websites for for application notes.
• A. Lasia, “Electrochemical Impedance Spectroscopy and Its Applications,” Modern Aspects of Electrochemistry, 32, R. E. White, J. O'M. Bockris and B. E. Conway, editors, Plenum Press, New York, 1999, 143-248.
• C. Deslouis and B. Tribollet, Flow Modulation Techniques in Electrochemistry, Advances in Electrochemical Science and Engineering, 2, H. Gerischer and C. W. Tobias, editors, VCH, Weinheim, 1992, 205-264.
• M. E. Orazem, “Tutorial: Application of Mathematical Models for Interpretation of Impedance Spectra,” Tutorials in Electrochemical Engineering - Mathematical Modeling, PV 99-14, R.F. Savinell, A.C. West, J.M. Fenton and J. Weidner, editors, Electrochemical Society, Inc., Pennington, N.J., 68-99, 1999.
• B. Tribollet, Look-up Tables for Rotating Disk Electrode, April 28, 2000, http://www.ccr.jussieu.fr/lple/EnglishVersion.html
Chapter 12. Suggested Reading page 12: 4
Suggested ReadingSuggested ReadingOrazem group work on Fuel CellsOrazem group work on Fuel Cells
• S. K. Roy and M. E. Orazem, “Graphical Estimation of Interfacial Capacitance of PEM Fuel Cells from Impedance Measurements,”Journal of The Electrochemical Society, 156 (2009), B203-B209.
• S. K. Roy and M. E. Orazem, “Analysis of Flooding as a Stochastic Process in PEM Fuel Cells by Impedance Techniques,” Journal of Power Sources, 184 (2008), 212-219.
• S. K. Roy, M. E. Orazem, and B. Tribollet, “Interpretation of Low-Frequency Inductive Loops in PEM Fuel Cells,” Journal of The Electrochemical Society, 154 (2007), B1378-B1388.
• S. K. Roy and M. E. Orazem, “Error Analysis of the Impedance Response of PEM Fuel Cells,” Journal of The Electrochemical Society, 154 (2007), B883-B891.
• P. Agarwal, M. E. Orazem, and L. H. García-Rubio, "Measurement Models for Electrochemical Impedance Spectroscopy: 1. Demonstration of Applicability," Journal of the Electrochemical Society, 139 (1992), 1917-1927.
• P. Agarwal, Oscar D. Crisalle, M. E. Orazem, and L. H. García-Rubio, "Measurement Models for Electrochemical Impedance Spectroscopy: 2. Determination of the Stochastic Contribution to the Error Structure," Journal of the Electrochemical Society, 142 (1995), 4149-4158.
• P. Agarwal, M. E. Orazem, and L. H. García-Rubio, "Measurement Models for Electrochemical Impedance Spectroscopy: 3. Evaluation of Consistency with the Kramers-Kronig Relations,” Journal of the Electrochemical Society, 142 (1995), 4159-4168.
• M. E. Orazem “A Systematic Approach toward Error Structure Identification for Impedance Spectroscopy,” Journal of Electroanalytical Chemistry, 572 (2004), 317-327.
Chapter 12. Suggested Reading page 12: 6
Suggested ReadingSuggested ReadingCPECPE
• G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, and J. H. Sluyters, “The Analysis of Electrode Impedances Complicated by the Presence of a Constant Phase Element,” Journal of ElectroanalyticChemistry, 176 (1984), 275-295.
• C. H. Hsu and F. Mansfeld,Technical Note: “Concerning the Conversion of the Constant Phase Element Parameter Y0 into a Capacitance,” Corrosion, 57 (2001), 747-748.
• V. Huang, V. Vivier, M. Orazem, N. Pébère, and B. Tribollet, “The Apparent CPE Behavior of a Disk Electrode with Faradaic Reactions: A Global and Local Impedance Analysis,” Journal of the Electrochemical Society, 154 (2007), C99-C107.
• M. Orazem, B. Tribollet, and N. Pébère, “Enhanced Graphical Representation of Electrochemical Impedance Data,” Journal of the Electrochemical Society, 153 (2006), B129-B136.
Chapter 13. Notation
Roman a coefficient in the Cochran expansion for velocity, 0.51023 a
b coefficient in the Cochran expansion for velocity, -0.61592 b
ic concentration of reacting species i , mol/cm3
ic steady-state value of the concentration of reacting species i , mol/cm3
ic Oscillating component of the concentration of reacting species i , mol/cm3
,oic concentration of species i on the electrode surface, mol/cm3
,oic steady-state value of the concentration of species i on the electrode surface, mol/cm3
,oic Oscillating component of the concentration of species i on the electrode surface, mol/cm3
c bulk concentration of the reacting species, mol/cm3
dC double layer capacitance, F/cm2
dlC double layer capacitance, F/cm2
iD diffusion coefficient of species i , cm2/s
f arbitrary function, e.g., ,f ii f V c
Chapter 14. Notation page 14: 2
F Faraday’s constant, C/equiv
i Total current density, A/cm2
i Steady-state total current density, A/cm2
i Oscillating component of total current density, A/cm2
fi Faradic current density, A/cm2
fi Steady-state Faradic current density, A/cm2
fi Oscillating component of Faradic current density, A/cm2
0i Exchange current density, A/cm2
j imaginary number, 1
Ak rate constant for reaction identified by index A (units depend on reaction stoichiometry)
dimensionless frequency,
1 13 31/3
2 2
9 9 Sci ii
Ka D a
iM notation for species i
n number of electrons produced when one reactant ion or molecule reacts
R universal gas constant, J/mol/K
eR Ohmic resistance, cm2
,t AR Charge-transfer resistance associated with reaction A, cm2
K
Chapter 14. Notation page 14: 3
r radial coordinate, cm
0r radius of disk, cm
is stoichiometric coefficient for species i , ( 0is for a reactant and 0is for a product)
Sci Schmidt number, Sc /i iD
T electrolyte temperature, K
t time, s
zr vv , radial and axial velocity components, respectively, cm/s
V potential, V
V steady-state potential, V
V oscillating contribution to potential, V
Z impedance, cm2
z axial position, cm
dz diffusion impedance, cm2
iz charge for species i
Greek coefficient used in the exponent for a constant-phase element. When 0 , the element behaves as an ideal capacitor in
parallel with a resistor.
A apparent transfer coefficient for reaction A
Chapter 14. Notation page 14: 4
A Tafel slope for reaction A, V
i Fractional surface coverage by species i
Maximum surface coverage
i characteristic diffusion length for species i
homogeneous solution to the oscillating dimensionless convective diffusion equation
(0) derivative of the solution to the oscillating dimensionless convective diffusion equation evaluated at the electrode surface
surface overpotential, V
solution conductivity, ( cm)-1
characteristic length for a finite-length diffusion layer, 1/ 3 2 / 32.598 /iD