electrochemical impedance spectroscopy with application …che.ufl.edu/orazem/pdf-files/eis for fuel...

Electrochemical Impedance Spectroscopy Electrochemical Impedance Spectroscopy with Application to Fuel Cellswith Application to Fuel Cells

Mark E. OrazemDepartment of Chemical Engineering

University of FloridaGainesville, 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

ContentsContents• Chapter 1. Introduction • Chapter 2. Motivation • Chapter 3. Impedance Measurement • Chapter 4. Representations of Impedance Data• Chapter 5. Development of Process Models• Chapter 6. Time-Constant Dispersion• Chapter 7. Regression Analysis• Chapter 8. Error Structure• Chapter 9. Kramers-Kronig Relations• Chapter 10. Application to PEM Fuel Cells• Chapter 11. Conclusions• Chapter 12. Suggested Reading• Chapter 13. Notation

Chapter 1. Introduction page 1: 1

Electrochemical Impedance Spectroscopy Electrochemical Impedance Spectroscopy with Application to Fuel Cellswith Application to Fuel Cells

Mark E. OrazemDepartment of Chemical Engineering

University of FloridaGainesville, Florida 32611

[email protected]

352-392-6207


Course sponsored for the Fuel Cell Seminar by The Electrochemical Society


Electrochemical Impedance Electrochemical Impedance SpectroscopySpectroscopyChapter 1. IntroductionChapter 1. Introduction

• How to think about impedance spectroscopy• EIS as a generalized transfer function• Overview of applications of EIS• Objective and outline of course



1992 – no logo


June 6 - 11, 2010 • Hotel Tivoli CarvoeiroCarvoeiro, Algarve, Portugal


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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!” ...


Electrochemical Impedance Electrochemical Impedance SpectroscopySpectroscopy

• Electrochemical technique– steady-state– transient– impedance spectroscopy

• Measurement in terms of macroscopic quantities– total current– averaged potential

• Not a chemical spectroscopy• Type of generalized transfer-function measurement


-10

-5

5

10

-0.2 -0.1 0.1 0.2 0.3 0.4

Potential, V

CurrentDensity, A/cm2

~I

V~

Impedance SpectroscopyImpedance Spectroscopy

r jVZ Z jZI



• Electrochemical systems– Corrosion– Electrodeposition– Human Skin– Batteries– Fuel Cells

• Materials

• Dielectric spectroscopy• Acoustophoretic

spectroscopy• Viscometry• Electrohydrodynamic

impedance spectroscopy

Applications Fundamentals


Physical DescriptionPhysical Description

• Electrode-Electrolyte Interface– Electrical Double Layer– Diffusion Layer

• Electrochemical Reactions• Electrical Circuit Analogues


Electrochemical ReactionsElectrochemical Reactions

eU V iR

F ddVi i Cdt

2 22

2

2 2

OO O O OO expF

Fi i n Fk V

RTc V

- -2 2O +2H O+ 4e 4OH

Total current density = Faradaic + charging

Faradaic current density

Cell potential = electrode potential + Ohmic potential drop


Electrical AnaloguesElectrical Analogues


Electrical AnalogueElectrical Analogue

Simple electrochemical reaction

Simple electrochemical reaction with mass transfer


Course ObjectivesCourse Objectives

• Benefits and advantages of impedance spectroscopy• Methods to improve experimental design• Interpretation of data

– graphical representations– regression– error analysis– equivalent circuits– process models


ContentsContents• Chapter 1. Introduction • Chapter 2. Motivation • Chapter 3. Impedance Measurement • Chapter 4. Representations of Impedance Data• Chapter 5. Development of Process Models• Chapter 6. Time-Constant Dispersion• Chapter 7. Regression Analysis• Chapter 8. Error Structure• Chapter 9. Kramers-Kronig Relations• Chapter 10. Application to PEM Fuel Cells• Chapter 11. Conclusions• Chapter 12. Suggested Reading• Chapter 13. Notation


NotationNotation

• IUPAC Convention: Z’ and Z’’; i

• Present Work: Zr and Zj; j

Chapter 2. Motivation page 2: 1


Chapter 2. MotivationChapter 2. Motivation

• Comparison of measurements– steady state– step transients– single-sine impedance

• In principle, step and single-sine perturbations yield same results

• Impedance measurements have better error structure



SteadySteady--State State Polarization Polarization

CurveCurve

0

0.1

0.2

0.3

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Potential, V

Cur

rent

, mA


SteadySteady--State TechniquesState Techniques

• Yield information on state after transient is completed• Do not provide information on

– system time constants– capacitance

• Influenced by– Ohmic potential drop– non-stationarity– film growth– coupled reactions


Transient Response to a Step in PotentialTransient Response to a Step in Potential

0 2 4 6 8 10

0.5

1.0

1.5

R0

R1(V)

C1

R2

C2

Cur

rent

/ m

A

(t-t0) / ms

current response

potential inputV=10 mV}

10-5 10-4 10-3 10-2 10-1 100

10-2

10-1

100

Cur

rent

/ m

A

(t-t0) / s

210 )( RVRRVI


Transient Response to a Step in PotentialTransient Response to a Step in Potential

long times/low frequency

Short times/high frequency

0 2 4 6 8 10

0.5

1.0

1.5

R0

R1(V)

C1

R2

C2

Cur

rent

/ m

A

(t-t0) / ms

current response

potential inputV=10 mV}


Transient Techniques: Transient Techniques: potential or current stepspotential or current steps

• Decouples phenomena– characteristic time constants

• mass transfer • kinetics

– capacitance• Limited by accuracy of measurements

– current– potential– time

• Limited by sample rate– <~1 kHz


Sinusoidal PerturbationSinusoidal Perturbation

0CdVi Cdt

fi f V

( )V t

0

0

( ) cos

( ) exp( ) exp( )a c

V t V V t

i t i b V b V



-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

1 mHz100 Hz

(i-i 0) /

max

(i-i 0)

(V-V0) / V0

10 kHz


LissajousLissajous RepresentationRepresentation

| |

sin( )

VZI

OAOB

ODOA

( ) cos( )

( ) cos( )

V t V tVI t tZ

-1.0 -0.5 0 0.5 1.0

-1.0

-0.5

0

0.5

1.0

Y(t)/

Y 0

X(t)/X0

O D A

B


Impedance ResponseImpedance Response

0 10 20 30 40 50

0

-5

-10

-15

-20

-25

Z j /

cm

-2

Zr / cm-2

100 Hz


Impedance SpectroscopyImpedance Spectroscopy• Decouples phenomena

– characteristic time constants• mass transfer • kinetics

– capacitance• Gives same type of information as DC transient.• Improves information content and frequency range by

repeated sampling.• Takes advantage of relationship between real and

imaginary impedance to check consistency.


System with Large Ohmic ResistanceSystem with Large Ohmic Resistance

• R0 =10,000 • R1 =1,000 • C1 = 10.5 F

– s (15 Hz)

M. E. Orazem, T. El Moustafid, C. Deslouis, and B. Tribollet, J. Electrochem. Soc.,143 (1996), 3880-3890.


Impedance SpectrumImpedance Spectrum-600

-400

-200

09800 10000 10200 10400 10600 10800 11000 11200

Zr,

Z j,

0.1

1

10

100

1000

10000

100000

0.1 1 10 100 1000 10000 100000

Frequency, Hz

Impe

danc

e,

Zr,

-Zj,


Experimental DataExperimental Data

1

10

100

1000

10000

100000

0.1 1 10 100 1000 10000 100000

Frequency, Hz

Impe

danc

e,

3% of |Z|

1% of |Z|

6 Repeated Measurements

Real

Imaginary


Impedance Spectroscopy Impedance Spectroscopy vs. Stepvs. Step--Change TransientsChange Transients

• Information sought is the same• Increased sensitivity

– stochastic errors– frequency range– consistency check

• Better decoupling of physical phenomena

Chapter 3. Impedance Measurement page 3: 1


Chapter 3. Impedance MeasurementChapter 3. Impedance Measurement

• Overview of techniques– A.C. bridge– Lissajous analysis– phase-sensitive detection (lock-in amplifier)– Fourier analysis

• Experimental design



Measurement TechniquesMeasurement Techniques

• A.C. Bridge• Lissajous analysis• Phase-sensitive detection (lock-in amplifier)• Fourier analysis

– 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.


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


| |

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


Phase Sensitive DetectionPhase Sensitive Detection0 sin( )AA A t

0

4 1 sin 2 12 1 S

n

S n tn

0

0

4 1 si sin 2 12 1

n Sn

A n tn

A A tS

General Signal

Reference Signal

SAAdtAS

cos22

02

0

Has maximum value when

A S


Fourier Analysis:Fourier Analysis:singlesingle--frequency inputfrequency input

0( ) cos( )II t I t

0( ) cos( )V t V t

0

0

0

0

1 ( ) cos( )

1 ( )sin( )

1 ( )cos( )

1 ( )sin( )

T

r

T

j

T

r

T

j

I I t t dtT

I I t t dtT

V V t t dtT

V V t t dtT

( ) Re

( ) Im

r jr

r j

r jj

r j

V jVZ

I jI

V jVZ

I jI


-10

0

10

0

TimeSign

al S

treng

th

Fourier Analysis:Fourier Analysis:multimulti--frequency inputfrequency input

Z

Z()

Fast Fourier

Transform

-10

0

10

0

TimeSig

nal S

treng

th


ComparisonComparisonsinglesingle--sine input multisine input multi--sine inputsine input

• Good accuracy for stationary systems

• Frequency intervals of f/f– economical use of

frequencies• Used for entire frequency

domain• Kramers-Kronig inconsistent

frequencies can be deleted

• Good accuracy for stationary systems

• Frequency intervals of f– dense sampling at high

frequency required to get good resolution at low frequency

• Often paired with Phase-Sensitive-Detection (f>10 Hz)

• Correlation coefficient used to determine whether spectrum is inconsistent with Kramers-Kronig relations


Measurement TechniquesMeasurement Techniques

• A.C. bridge– obsolete

• Lissajous analysis– obsolete– useful to visualize impedance

• Phase-sensitive detection (lock-in amplifier)– inexpensive– accurate– useful at high frequencies

• Fourier analysis techniques– accurate


Experimental ConsiderationsExperimental Considerations

• Frequency range– instrument artifacts– non-stationary behavior– capture system response

• Linearity– low amplitude perturbation– depends on polarization curve for system under study– determine experimentally

• Signal-to-noise ratio



0CdVi Cdt

fi f V

( )V t

0

0

( ) cos( ) exp( )a

V t V V ti t i b V


LinearityLinearity

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0 f = 1 mHz 1 mV 20 mV 40 mV

(I-I 0)/m

ax(I-

I 0)

(V-V0)/V-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0 f = 10 Hz 1 mV 20 mV 40 mV

(I-I 0)/m

ax(I-

I 0)

(V-V0)/V

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0 f = 100 Hz 1 mV 20 mV 40 mV

(I-I 0)/m

ax(I-

I 0)

(V-V0)/V-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

f = 10 kHz 1 mV 20 mV 40 mV

(I-I 0)

/max

(I-I 0)

(V-V0)/V


0 10 20 30 40 50

0

-10

-20

model 1 mV 20 mV 40 mV

Z j /

cm

2

Zr / cm2

Influence of Nonlinearity on ImpedanceInfluence of Nonlinearity on Impedance


Influence of Nonlinearity on ImpedanceInfluence of Nonlinearity on Impedance

10-3 10-2 10-1 100 101 102 103 104 105 106

0.0

0.2

0.4

0.6

0.8

1.0

model 1 mV 20 mV 40 mVZ r /

Rt

f / Hz10-3 10-2 10-1 100 101 102 103 104 105 106

10-4

10-3

10-2

10-1

100

101 model 1 mV 20 mV 40 mV

|Zj| /

c

m2

f / Hz


0.01 0.1 0.2 1

0.01

0.1

0.51

10

10

0(R

t-Z(0

))/R

t

V b

Guideline for LinearityGuideline for Linearity0.2b V


Influence of Ohmic ResistanceInfluence of Ohmic Resistance

0CdVi Cdt

fi f V

( )V t( )s t

eR

( ) ei t R

0.2 1 /e tb V R R


Influence of Ohmic ResistanceInfluence of Ohmic Resistance


Overpotential as a Function of FrequencyOverpotential as a Function of Frequency

0CdVi Cdt

fi f V

( )V t( )s t

eR

( ) ei t R


Cell DesignCell Design• Use reference electrode to isolate influence of electrodes and

membranes

WorkingElectrode

Counterelectrode

2-electrode

Ref 1

WorkingElectrode

Counterelectrode

3-electrode

Ref 1

Ref 2

WorkingElectrode

Counterelectrode

4-electrode


Current DistributionCurrent Distribution

• Seek uniform current/potential distribution

– simplify interpretation– reduce frequency dispersion

Working Electrode Seat

Counter Electrode Lead Wires

Cell BodyID = 6 in.L = 6 in.

Working Electrode

Electrolyte In Electrolyte In

ElectrolyteOut

ElectrolyteOut

FlangeCover

CoverFlange


Primary Current DistributionPrimary Current Distribution


Modulation TechniqueModulation Technique• Potentiostatic

– standard approach– linearity controlled by

potential

• Galvanostatic– good for nonstationary

systems• corrosion• drug delivery

– requires variable perturbation amplitude to maintain linearity

2

2

MO corr

Fecorr

2Fe

coO rr

exp

12

1n

nF V V

RT

Fi i V

F

V

T

RT

i Vi VR

( ) ( )/I V Z V I Zw wD = D D = D


Experimental StrategiesExperimental Strategies

• Faraday cage• Short leads• Good wires• Shielded wires• Oscilloscope

I

VWE CERef


Reduce Stochastic NoiseReduce Stochastic Noise

• Current measuring range• Integration time/cycles

– long/short integration on some FRAs• Delay time• Avoid line frequency and harmonics (±5 Hz)

– 60 Hz & 120 Hz– 50 Hz & 100 Hz

• Ignore first frequency measured (to avoid start-up transient)


Reduce NonReduce Non--Stationary EffectsStationary Effects

• 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


Reduce Instrument Bias ErrorsReduce Instrument Bias Errors

• Faster potentiostat• Short shielded leads• Faraday cage• Check results

– against electrical circuit– against independently obtained parameters


Experimental ConsiderationsExperimental Considerations

• Frequency range– instrument artifacts– non-stationary behavior

• Linearity– low amplitude perturbation– depends on polarization curve for system under study– determine experimentally

• Signal-to-noise ratio


Can We Perform Impedance on Transient Can We Perform Impedance on Transient Systems?Systems?

• Timeframes for measurement– Individual frequency– Individual scan– Multiple scans


EHD ExperimentEHD Experiment

-0.2

-0.15

-0.1

-0.05

0

0.05-0.05 0 0.05 0.1 0.15 0.2

Real Impedance, A/rpm

Imag

inar

y Im

peda

nce,

A/rp

mData Set #2Data Set #4Data Set #1


Time per Frequency for 200 rpmTime per Frequency for 200 rpm

1

10

100

1000

0.01 0.1 1 10 100

Frequency, Hz

Tim

e pe

r Mea

sure

men

t, s

1

10

100

10000.01 0.1 1 10 100

Filter OnFilter Off


Impedance ScansImpedance Scans

0 5 10 15 20 25 30 35 400

-5

-10

-15

Time Interval 0-1183 s 1183-2366 s 2366-3581 s

Z j /

cm

2

Zr / cm2


Impedance ScansImpedance Scans

10-2 10-1 100 101 102 103 104 105

0

1000

2000

3000

4000

E

laps

ed T

ime

/ s

f / Hz


Time per Frequency MeasuredTime per Frequency Measured

10-2 10-1 100 101 102 103 104 105

101

102

t f /

s

f / Hz

5 cycles

3 cycles

5 seconds

Chapter 4. Representation of Impedance Data page 4: 1


Chapter 4. Representation of Impedance DataChapter 4. Representation of Impedance Data

• Electrical circuit components• Methods to plot data

– standard plots– subtract electrolyte resistance

• Constant phase elements



Addition of ImpedanceAddition of Impedance

Z1 Z2

21

21

111

YY

Y

ZZZ

Z1

Z2

21

21

111

YYYZZ

Z


Circuit ComponentsCircuit Components

Re

Zr

-Zj

eRZ

1

11

dldl

j

CjR

CjRZ ee

eR

eR Zr

-Zj

CdlRe


Zr

-Zj

Simple RC CircuitSimple RC Circuit

dl

dl

2dl

2 2dl dl

11

1

1 1

e

t

te

t

t te

t t

Z Rj C

RRR

j R C

R R CR jR C R C

eR e tR R

2tR

1t dlR C

Rf

Cdl

Re

-12rad / s or scycles / s or Hz

f

f

Note:


1

10

100

1000

10000

0.001 0.01 0.1 1 10 100 1000 10000

Frequency / Hz

Z ro

r -Z j

realimaginary

ImpedanceImpedance

Slope = 1

Slope = -1

Rf

Cdl

Re

-600

-400

-200

00 200 400 600 800 1000 1200

Zr /

Z j,

10 Hz

1 Hz

100 Hz

0

200

400

600

800

1000

1200

0.001 0.01 0.1 1 10 100 1000 10000

Frequency / Hz

Z ro

r -Z j

realimaginary


Modified Phase AngleModified Phase Angle

0.01

0.1

1

10

100

1000

10000

0.001 0.01 0.1 1 10 100 1000 10000

Frequency / Hz

|Z|

-90

-60

-30

0

/

degr

eesmagnitude

modified magnitudephase anglemodified phase angle

1* tan j

r e

ZZ R

Input & output are in phase

Input & output are out of phase

1tan j

r

ZZ

Note:

The modified phase angle yields excellent insight, but we need an accurate estimate for the solution resistance.


Constant Phase ElementConstant Phase Element

1

1e

t t

Z RR j R Q

Rt

Re

CPE

0.0 0.2 0.4 0.6 0.8 1.00.0

-0.2

-0.4

-0.6 =1 =0.9 =0.8 =0.7 =0.6 =0.5

Z j / R t

(ZrRe) / Rt


10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 10510-5

10-4

10-3

10-2

10-1

100

=1 =0.9 =0.8 =0.7 =0.6 =0.5

|Zj| /

Rt

/

CPE: Log ImaginaryCPE: Log Imaginary

Note: With a CPE (1), the asymptotic slopes are no longer 1.

Slope =

Slope =


CPE: Modified Phase AngleCPE: Modified Phase Angle

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105

0

-30

-60

-90

=1 =0.9 =0.8 =0.7 =0.6 =0.5

* / de

gree

s

/

Note:

The high-frequency asymptote for the modified phase angle depends on the CPE coefficient .

adj( ) 90( )


Example for Synthetic DataExample for Synthetic Data

dl

( )1 2

t de

t d

R z fZ f R

j f Q R z f

tanh 22d d

j fz f z f

j f

M. Orazem, B. Tribollet, and N. Pébère, J. Electrochem. Soc., 153 (2006), B129-B136.


Traditional RepresentationTraditional Representation

0 200 400 600 800 1000 1200

0

200

400

1 mHz100 Hz10 Hz

1 Hz 0.1 Hz

Zj

cm

2

Zr cm2

10 mHz

10-3 10-2 10-1 100 101 102 103 104 105 106

102

103

Z

cm

2

f / Hz10-3 10-2 10-1 100 101 102 103 104 105 106

0

-15

-30

-45

-60

-75

-90

/ d

egre

e

f / Hz


RRee--Corrected Bode Phase AngleCorrected Bode Phase Angle1

adj,est

tan j

r e

ZZ R

adj ( ) 90( )

10-3 10-2 10-1 100 101 102 103 104 105 106

0

-15

-30

-45

-60

-75

-90

ad

j / de

gree

f / Hz


RRee--Corrected Bode MagnitudeCorrected Bode Magnitude 2 2

,estadj r e jZ Z R Z

Slope =

10-3 10-2 10-1 100 101 102 103 104 105

10-1

100

101

102

103

Z

adj

cm

2

f / Hz

slope =


Log(|ZLog(|Zjj|)|)

Slope =

Slope =

10-3 10-2 10-1 100 101 102 103 104 105 106

10-2

10-1

100

101

102

103

= 1 = 0.7 = 0.5

slope=

slope=

Z

j

cm

2

f / Hz


Effective Capacitance or CPE CoefficientEffective Capacitance or CPE Coefficient

effj

1 = sin2 2 ( )

Qf Z f

CPE

CPE

12

ZQ j f

10-3 10-2 10-1 100 101 102 103 104 105 106

1

10

100

Q

eff

Qdl

f / Hz


• Re-Corrected Bode Plots (Phase Angle) – Shows expected high-frequency behavior for surface– High-Frequency limit reveals CPE behavior

• Re-Corrected Bode Plots (Magnitude) – High-Frequency slope related to CPE behavior

• Log|Zj|– Slopes related to CPE behavior– Peaks reveal characteristic time constants

• Effective Capacitance– High-Frequency limit yields capacitance or CPE

coefficient

Alternative PlotsAlternative Plots


ApplicationApplication


Mg alloy (AZ91) in 0.1 M Mg alloy (AZ91) in 0.1 M NaClNaCl

0 100 200 300 400 500 600

200

100

0

-100

-200

15.5 kHz

0.011 Hz

0.0076 Hz

time / h 0.5 3.0 6.0

Z j

cm

2

Zr cm2

65.5 Hz


Proposed ModelProposed Model

22 Mg(OH)OH2Mg

4

4

k

k

OHMgOMg(OH) 25-k

5k

2

eMgMg ads1k

2212

2ads HOHMgOHMg 2 k

eMgMg 2ads

3

3

k

k

G. Baril, G. Galicia, C. Deslouis, N. Pébère, B. Tribollet, and V. Vivier, J. Electrochem. Soc.,154 (2007), C108-C113

“negative difference effect”(NDE)


0 100 200 300 400 500 600

200

100

0

-100

-200

15.5 kHz

0.011 Hz

0.0076 Hz

time / h 0.5 3.0 6.0

Z j

cm

2

Zr cm2

65.5 Hz

Physical InterpretationPhysical Interpretationdiffusion of Mg2+

eMgMg ads1k

eMgMg 2ads

3

3

k

k

relaxation of (Mg+)ads.intermediate


Log(|ZLog(|Zjj||

Slope =

10-3 10-2 10-1 100 101 102 103 104 105100

101

102

time / h 0.5 3.0 6.0

Z

j

cm

2

f / Hz


Effective CPE CoefficientEffective CPE Coefficient

effj

1 = sin2 2 ( )

Qf Z f

10-2 10-1 100 101 102 103 104 105

101

102

103

104

105

time / h 0.5 3.0 6.0

Q

eff /

(M

-1 c

m-2) s

f / Hz


RRee--Corrected Phase AngleCorrected Phase Angle

adj ( ) 90 1adj

,est

tan j

r e

ZZ R

10-3 10-2 10-1 100 101 102 103 104 105100

101

102

103

time / h 0.5 3.0 6.0

Z

adj

cm

2

f / Hz


Physical ParametersPhysical Parameters

0 100 200 300 400 500 600

200

100

0

-100

-200

15.5 kHz

0.011 Hz

0.0076 Hz

time / h 0.5 3.0 6.0

Z j

cm

2

Zr cm2

65.5 Hz


Experimental device for LEISExperimental device for LEIS

appliedlocal

local

VZ

i

layer


Mg alloy (AZ91) in Na2SO4 10-3 M at the corrosion potential after 1 h of immersion. Electrode radius 5500 µm.

Global impedance

Global impedance analyzed with a CPE ( = 0.91). Only the HF loop of the diagram is analyzed

J.-B. Jorcin, M. E. Orazem, N. Pébère, and B. Tribollet, Electrochimica Acta, 51 (2006), 1473-1479.


Local impedance

= 1 to 0.92


Mg alloy (AZ91) in Na2SO4 10-3 M at the corrosion potential after 1 h of immersion. Electrode radius 5500 µm

The local impedance has a pure RC behavior.

The CPE is explained by a 2d distribution of the resistance as shown in the figure.Local impedance


Comparison to Comparison to TheoryTheory


Graphical Representation of Impedance Graphical Representation of Impedance DataData

• Expanded Range of Plot Types– Facilitate model development– Identify features without complete system model

• Suggested Plots– Re-Corrected Bode Plots (Phase Angle)

• Shows expected high-frequency behavior for surface• High-Frequency limit reveals CPE behavior

– Re-Corrected Bode Plots (Magnitude) • High-Frequency slope related to CPE behavior

– Log|Zj|• Slopes related to CPE behavior• Peaks reveal characteristic time constants

– Effective Capacitance• High-Frequency limit yields capacitance or CPE coefficient


Choice of RepresentationChoice of Representation

• Plotting approaches are useful to show governing phenomena

• Complement to regression of detailed models• Sensitive analysis requires use of properly weighted

complex nonlinear regression

Chapter 5. Development of Process Models page 5: 1


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



Use circuits to create frameworkUse circuits to create framework


Addition of PotentialAddition of Potential


Addition of CurrentAddition of Current


Equivalent Circuit at the Corrosion PotentialEquivalent Circuit at the Corrosion Potential


Equivalent Circuit for a Partially Blocked Equivalent Circuit for a Partially Blocked ElectrodeElectrode


Equivalent Circuit for an Electrode Coated Equivalent Circuit for an Electrode Coated by a Porous Layerby a Porous Layer


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


Use kinetic models to determine Use kinetic models to determine expressions for the interfacial impedanceexpressions for the interfacial impedance


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


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


Reactions ConsideredReactions Considered

• Dependent on Potential• Dependent on Potential and Mass Transfer• Dependent on Potential, Mass Transfer, and Surface

Coverage• Coupled Reactions


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


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


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


Ohmic ContributionsOhmic Contributions

VRiU

ViRU

e

e~~~

A

,A

,A1+

e

te

t dl

U VZ Ri i

RR

j R C


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


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



,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


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


Combine ExpressionsCombine Expressions

O O O O,0,O

expt

Vi K b V cR

O O

O,0O O 0

icn FD



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


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


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


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


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


SteadySteady--State Surface CoverageState Surface Coverage

1 2

0

i iddt F F

balance on

1 1 1

1 1 1 2 2 2

exp

exp exp

K b V V

K b V V K b V V

steady-state value for


SteadySteady--State Current DensityState Current Density

1 1 1 2 2 21 exp expi K b V V K b V V

1 1 1

1 1 1 2 2 2

exp

exp exp

K b V V

K b V V K b V V

where


Oscillating Current DensityOscillating Current Density

1 1 1 2 2 21 exp expi K b V V K b V V

2 2 2 1 1 1,1 ,2

1 1 exp expt t

i V K b V V K b V VR R

1

,1 1 1 1 1

1

,2 2 2 2 2

1 exp

exp

t

t

R K b b V V

R K b b V V


Need Additional EquationNeed Additional Equation

1 1 1 2 2 2,1 ,2

1 1 1 exp expt t

j V K b V V K b V VF R R

balance on

1 1

,1 ,2

1 1 1 2 2 2exp expt tR R

VF j K b V V K b V V


ImpedanceImpedance

1 12 2 2 1 1 1 ,1 ,2

1 1 1 2 2 2

exp exp1 1exp exp

1

t t

t

t

K b V V K b V V R R

Z R F j F K b V V K b V V

AR j B

, ,

1 1 1

t t M t XR R R

where


-1.2

-1.0

-0.8

-0.6

-0.4

0.01 0.1 1 10 100Current Density, mA/ft2

Pot

entia

l, V

(Cu/

CuS

O4)

Corrosion of Corrosion of SteelSteel

2H2O + 2e- H2+2OH-

O2+2H2O + 4e- 4OH-

Fe Fe2+ + 2e-

22 OHFe iiii f


Fe,Fe

~~tRVi

,FeFe Fe Fe

1exptR

K b b V

Corrosion: Fe Corrosion: Fe FeFe2+ 2+ + 2e+ 2e--

Fe Fe Feexpi K b V

Fe Fe Fe Feexpi K b b V V


Steady Currents in Terms of Steady Currents in Terms of RRtt

Fe Fe Feexp 2.303i K V

Fe Fe2.303/ b

Fe

,FeFe Fe Fe Fe

1exp 2.303tR

K b b V i

FeFe

,Fe2.303 t

iR

Fe Fe Feexpi K b V

Important: Note the relationship among steady-state current density, Tafel slope, and charge transfer resistance.


2 2 2H H Hexpi K b V

2 2 2 2H H H Hexpi K b b V V

2

2H,

H

~~tRVi

2

2 2 2

t,HH H H

1exp

RK b b V

HH22 Evolution: 2HEvolution: 2H22O + 2eO + 2e-- HH22+2OH+2OH--


2 2 2 2O O O ,0 Oexpi K c b V

2 2 2 2 2

2 2 2

O O O O ,0 O

O O O ,0

exp

exp

i K b c b V V

K b V c

OO22 Reduction: OReduction: O22+2H+2H22O + 4eO + 4e-- 4OH4OH--


OO22 Evolution: OEvolution: O22+2H+2H22O + 4eO + 4e-- 4OH4OH--

mass transfer: in terms of dimensionless gradient at electrode surface

2

2 2 2

2

2 2

2

OO O O

0

O ,0O O

O

0

dci n FD

dyc

n FD


22

2

2 2 2 2

2 2

OO

,OO O O ,0 O

,O ,O

1 1(0)t

t d

ViR

n FD c b

VR z

2

2 2 2 2

t,OO O O ,0 O

1exp

RK b c b V

2

2

2 2 2 2

O,O

O O O ,0 O

1 1(0)dz

n FD c b

OO22 Evolution: OEvolution: O22+2H+2H22O + 4eO + 4e-- 4OH4OH--

coupled expression


Capacitance and Ohmic ContributionsCapacitance and Ohmic Contributions

ff

ff

ii

ii~~

Faradaic

VCjiidtdVCii

df

df

~~~

Faradaic and Charging

VRiU

ViRU

e

e~~~

Ohmic


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


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


Mass TransferMass Transfer


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


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


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


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


Rotating DiskRotating Disk


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


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 ,


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


Finite Length Warburg ImpedanceFinite Length Warburg Impedance

vcz

D

zi

i

0

2598 1 3 2 3

. / /

z zj

j

D

d d

i

( )tanh

. /

0

25982 1 3

Sc

2

2

tanh1(0)

i

i

i

jD

jD


ImpedanceImpedance

0~~~2

2

iii

zi jK

ddv

dd

0,

~~~iii cc

0at1~as0~

i

i

0~,,0, i

i

i

i

ci itD nFD

scfRZ

ijj

0~)0(

i

DD

ZZ


Numerical SolutionsNumerical Solutions

One Term

v a

z a

z

2

051203

; .

z zp

p

d d

( )01

0

Sc1/3

Two Termsv az

2 31

3

z zp

Z pd d

( )0

1

0

1

ScSc

Sc1/3

1/3

1/3

Three Termsv a

b

b

z

2 3 413 6

0 616.

z z

pZ p Z p

d d

( )0

1

0

1 2

ScSc

ScSc

Sc1/3

1/3

1/3

1/3

2/3

Infinite Sc

Finite Sc


Coupled Diffusion ImpedanceCoupled Diffusion Impedance

, ,, ,

, ,

2, ,

, ,, , ,

i outer i innerd outer d inner

i inner i outerd

i outer i innerinnerd inner d outer

i inner i inner i outer

Dz z

Dz

Dz z j

D D


Interpretation Models for Impedance Interpretation Models for Impedance SpectroscopySpectroscopy

• Models can account rigorously for proposed kinetic and mass transfer mechanisms.

• There are significant differences between models for mass transfer.

• Stochastic errors in impedance spectroscopy are sufficiently small to justify use of accurate models for mass transfer.

Chapter 6. Time-Constant Dispersion page 6: 1

Basic Impedance SpectroscopyBasic Impedance Spectroscopy

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.



Page deliberately left blank.


Types of DistributionsTypes of Distributions

• Surface

• Axial


Surface DistributionsSurface Distributions


Axial DistributionsAxial Distributions


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.


Determination of Film ThicknessDetermination of Film Thickness


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


Porous ElectrodesPorous Electrodes


deLeviedeLevie ModelModel

Under the assumption that Z0and R0 are independent of x


Limiting BehaviorLimiting Behavior

2 / eqrZ coth 2 / eqrZ description Parameters obtainable from regression

large 1 Semi-infinite pores 3/ 2r n small 1

2 / eqrZ

Planar surface 2 n r

intermediate intermediate Shallow pores Two combinations of r , n , and ; i.e.,

3/ 2r n and / r


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


• 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


Model DevelopmentModel DevelopmentNo Free Chlorine

2 mg/l Free Chlorine


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


Impedance DataImpedance Data

3 days

7 days

28 days


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.

Chapter 7. Regression Analysis page 7: 1


Chapter 7. Regression AnalysisChapter 7. Regression Analysis

• Regression response surfaces– noise– incomplete frequency range

• Adequacy of fit– quantitative– qualitative



Test Circuit 1: 1 Time ConstantTest Circuit 1: 1 Time Constant

• R0 = 0• R1 = 1 cm2

• 1 = RC = 1 sR0

R1

C1

(1)


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


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


Methods for RegressionMethods for Regression

• Evaluation of derivatives– method of steepest descent– Gauss-Newton method– Levenberg-Marquardt method

• Evaluation of function– simplex


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


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.


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


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


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.


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


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


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.


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


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


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


Comparison of Model to DataComparison of Model to Data

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

Zj /

Zr /

Impedance Plane (Impedance Plane (NyquistNyquist))

Value of 2 has no meaning without accurate assessment of the noise level of the data


ModulusModulus

10-2 10-1 100 101 102 103 104

10

100

|Z| /

f / Hz


Phase AnglePhase Angle

10-2 10-1 100 101 102 103 1040

-15

-30

-45

/ deg

rees

f / Hz


RealReal

10-2 10-1 100 101 102 103 1040

40

80

120

160

200

Z r /

f / Hz


ImaginaryImaginary

10-2 10-1 100 101 102 103 1040

20

40

60

80

Zj /

f / Hz


Log ImaginaryLog Imaginary

10-2 10-1 100 101 102 103 1041

10

100

Slope = -1 for RC

Z

j /

f / Hz



10-2 10-1 100 101 102 103 1040

-15

-30

-45

-60

-75

-90 /

degr

ees

f / Hz

1* tan j

r e

ZZ R


Real ResidualsReal Residuals

10-2 10-1 100 101 102 103 104-0.06

-0.04

-0.02

0

0.02

0.04

0.06

r /

Z r

f / Hz

2noise level


Imaginary ResidualsImaginary Residuals

10-2 10-1 100 101 102 103 104-0.3

-0.2

-0.1

0

0.1

0.2

0.3 j /

Z j

f / Hz

2noise level


Plot Sensitivity to Quality of FitPlot Sensitivity to Quality of Fit• Poor Sensitivity

– Modulus– Real

• Modest Sensitivity– Impedance-plane

• only for large impedance values– Imaginary– Log Imaginary

• emphasizes small values• slope suggests new models

– Phase Angle• high-frequency behavior is counterintuitive due to role of solution resistance

– Modified Phase Angle• high-frequency behavior can suggest new models• needs an accurate value for solution resistance

• Excellent Sensitivity– Residual error plots

• trending provides an indicator of problems with the regression


Test Case: Better Model for Mass Test Case: Better Model for Mass Transfer to a RDETransfer to a RDE

( )1

t de

t d

R zZ R

j Q R z

Consider 3-term expansion with CPE to account for more complicated reaction kinetics:

1/3 1/3

1 21/3 2/31/3

0

Sc Sc1(0)Sc ScScd d

Z p Z pz z

p

2/=4.86


Comparison of Model to DataComparison of Model to DataImpedance Plane (Impedance Plane (NyquistNyquist))

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

Zj /

Zr /


Phase AnglePhase Angle

10-2 10-1 100 101 102 103 1040

-15

-30

-45

Z

j /

f / Hz



10-2 10-1 100 101 102 103 1040

-15

-30

-45

-60

-75

-90* /

degr

ees

f / Hz

1* tan j

r e

ZZ R


Log ImaginaryLog Imaginary

10-2 10-1 100 101 102 103 1041

10

100

Z

j /

f / Hz


10-2 10-1 100 101 102 103 104-0.006

-0.003

0

0.003

0.006 r /

Z r

f / Hz

Real ResidualsReal Residuals

2


10-2 10-1 100 101 102 103 104-0.02

-0.01

0

0.01

0.02 j /

Z j

f / Hz

Imaginary ResidualsImaginary Residuals

2


Confidence IntervalsConfidence Intervals

• Based on linearization about trial solution• Assumption valid for good fits

– for normally distributed fitting errors– small estimated standard deviations

• Can use Monte Carlo simulations to test assumptions


Regression of Models to DataRegression of Models to Data

• Regression is strongly influenced by – stochastic errors in data– incomplete frequency range– incorrect or incomplete models

• Some plots more sensitive to fit quality than others.• Quantitative measures of fit quality require

independent assessment of error structure.• The model identification problem is intricately linked

to the error identification problem.

Chapter 8. Error Structure page 8: 1


Chapter 8. Error StructureChapter 8. Error Structure

• Contributions to error structure• Weighting strategies• General approach for error analysis• Experimental results



Contributions to Error StructureContributions to Error Structure

• Sampling Errors

• Stochastic Phenomena• Bias Errors

– Lack of Fit– Changing baseline (non-stationary processes)– Instrumental artifacts


TimeTime--domain domain Frequency DomainFrequency Domain

1 mHz 10 Hz

100 Hz 10 kHz


Error StructureError Structure

resid obs model

fit bias stochastic

( ) ( ) ( )( ) ( ) ( )

Z Z

inadequate model noise (frequency domain)

experimental errors: inconsistent with the Kramers-Kronig relations

fitting error


Weighting StrategiesWeighting Strategies

JZ Z Z Zr k r k

r kk

j k j k

j kk

( ) ( ), ,

,

, ,

,

2

2

2

2

Strategy ImplicationsNo Weighting r=j

Modulus Weighting r=j

Proportional Weighting rj

rrrj jjExperimental r=j

jr| + 2/RmRefined Experimental r=j

jr-Rsol| + 2/Rm+


Assumed Error Structure often WrongAssumed Error Structure often Wrong

1

10

100

1000

10000

100000

0.1 1 10 100 1000 10000 100000Frequency, Hz

Impe

danc

e,

3% of |Z|

1% of |Z|

6 Repeated Measurements

Real

Imaginary

Data obtained by T. El Moustafid, CNRS, Paris, France


Reduction of Fe(CN)Reduction of Fe(CN)6633-- on a Pt Disk, 120 rpmon a Pt Disk, 120 rpm

-300

-200

-100

00 100 200 300 400 500

Zr,

Z j,

1/41/23/4

I/ilim


Reduction of Fe(CN)Reduction of Fe(CN)6633-- on a Pt Disk, on a Pt Disk,

Imaginary Replicates @ 120 rpm, 1/4 Imaginary Replicates @ 120 rpm, 1/4 iilimlim

-80

-60

-40

-20

00.01 0.1 1 10 100 1000 10000 100000

Frequency, Hz

Z j,



Real Replicates @ 120 rpm, 1/4 Real Replicates @ 120 rpm, 1/4 iilimlim

1

10

100

1000

0.01 0.1 1 10 100 1000 10000 100000

Frequency, Hz

Z r,



Error Structure @ 120 rpm, 1/4 Error Structure @ 120 rpm, 1/4 iilimlim

0.001

0.01

0.1

1

10

0.01 0.1 1 10 100 1000 10000 100000

Frequency, Hz

Stan

dard

Dev

iatio

n,



Error Structure @ 120 rpm, 1/4 Error Structure @ 120 rpm, 1/4 iilimlim

0.01

0.1

1

10

100

0.01 0.1 1 10 100 1000 10000 100000

Frequency, Hz

/|Z|,

Perc

ent


Interpretation of Impedance SpectraInterpretation of Impedance Spectra• Need physical insight and knowledge of error structure

– stochastic component• weighting• determination of model adequacy• experimental design

– bias component• suitable frequency range• experimental design

• Approach is general– electrochemical impedance spectroscopies– optical spectroscopies– mechanical spectroscopies

Chapter 9. Kramers-Kronig Relations page 9: 1


Chapter 9. KramersChapter 9. Kramers--Kronig RelationsKronig Relations

• Mathematical form and interpretation• Application to noisy data• Methods to evaluate consistency



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)()(


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


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


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


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


Use of KramersUse of Kramers--Kronig RelationsKronig Relations

• Quadrature errors– require interpolation function

• Missing data at low and high frequency


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 10. Application to PEM Fuel Cells page 10: 1


Chapter 10. Application to PEM Fuel CellsChapter 10. Application to PEM Fuel Cells

• Error analysis for EIS data• Model development in terms of proposed

reaction mechanisms• Integration with Independent measurements



Issues for PEM Fuel Cell ImpedanceIssues for PEM Fuel Cell Impedance

• Instrument artifacts• Nonstationary behavior• Nonuniform surfaces• Complicated mass transfer geometry• Side reactions


R. Makharia, M. F. Mathias, and D. R. Baker, J. Electrochem. Soc.,152 (2005), A970.

• Pure Inductance • Mass Transfer• Capacitive Loop • Inductive Loop

Sample Impedance DataSample Impedance Data


Approach for Impedance AnalysisApproach for Impedance Analysis• Measurement Model

– Assess stochastic error structure of data– Instrument artifacts– Nonstationary behavior

• Process Model– Based on proposed reactions


0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

-0.05

-0.10

I= 0.5 Amps I= 1.0 Amps I= 1.5 Amps

Z j /

cm

2

Zr / cm2

ExperimentalExperimental ResultsResults

Operating ConditionsT : 40°C , P : 1 atm.Flow Rate : 0.1 l/m for H2 , 0.5 l/m for O2

Steady State MeasurementCurrent Range : 0 – 0.5 A/cm2

Increment : 10 mA/ 30 Sec.

Polarization Curve

EIS MeasurementGalvanostatic ModeFrequency : 0.001 – 3,000 HzAmplitude : 10 mA

0.0 0.1 0.2 0.3 0.4 0.50.2

0.4

0.6

0.8

1.0

Pote

ntia

l / V

Current Density / A cm-2

Nyquist Plot


Replicated Impedance ScansReplicated Impedance Scans


Stochastic Error StructureStochastic Error Structure

m

γ = 6.7855e-001R = 100

Error Structure of Impedance Data

Five Scans of impedance measurements at 0.2 A/cm2

3 KHz 0.001 Hz

2

r jm

γ|Z|σ = σ = R

10-4 10-3 10-2 10-1 100 101 102 103 10410-6

10-5

10-4

10-3

Real part Imaginary part Error structure

r & j /

c

m2

f / Hz

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.210.02

0.00

-0.02

-0.04

-0.06

-0.08

0.50 h 2.73 h 4.96 h 7.19 h 9.42 h

Z j /

cm

2

Zr / cm2


Bias Errors at High FrequencyBias Errors at High Frequency


HighHigh--frequency artifacts extend to negative frequency artifacts extend to negative imaginary valuesimaginary values


First Scan of Impedance Measurement

Bias Error at Low FrequencyBias Error at Low Frequency


Normalized Residual Errors Normalized Residual Errors

First Scan of Impedance Measurement


Second Scan of Impedance Measurement

Bias Errors at Low FrequencyBias Errors at Low Frequency


Normalized Residual Errors Normalized Residual Errors

Second Scan of Impedance Measurement


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.


Impedance Process Model Impedance Process Model DevelopmentDevelopment


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


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


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


Model Development: Case 1Model Development: Case 1

• Oxygen Reduction

• Hydrogen Oxidation+ -

2H 2H + 2e

+ -2 2O + 4H + 4e 2H O


Model 1: Simple Reaction KineticsModel 1: Simple Reaction Kinetics


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


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


Z j /

cm

2

Zr / cm2




• 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.


Model 2: Peroxide Intermediate Model 2: Peroxide Intermediate 2

2, 2

2

2

a

0,a

e c,1

0,c

c,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


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


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


Pot

entia

l / V

Current / A



0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10


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


Pot

entia

l / V

Current / A


0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0


Pot

entia

l / V

Current / A


0.0 0.1 0.2 0.3 0.4 0.5-0.1

0.0

0.1

0.2


Z j /

cm

2

Zr / cm 2




• 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


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



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



0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10


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


Pot

entia

l / V

Current / A



0.00 0.05 0.10 0.15 0.20

0.00

0.05

0.10


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


Pot

entia

l / V

Current / A



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


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


Pot

entia

l / V

Current / A


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


Evidence for Evidence for PtOPtOxx

Helena and Jason Weaver, University of Florida


Integrated ApproachIntegrated Approach


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


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


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


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


Shows flooding -10X more

noise

Shows need for

better base error

structure


Calculated ParametersCalculated Parameters


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


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 11. Conclusions page 11: 1


Chapter 11. ConclusionsChapter 11. Conclusions


• Chapter 1. Introduction • Chapter 2. Motivation • Chapter 3. Impedance Measurement • Chapter 4. Representations of Impedance Data• Chapter 5. Development of Process Models• Chapter 6. Time-Constant Dispersion• Chapter 7. Regression Analysis• Chapter 8. Error Structure• Chapter 9. Kramers-Kronig Relations• Chapter 10. Application to PEM Fuel Cells• Chapter 11. Conclusions• Chapter 12. Suggested Reading• Chapter 13. Notation


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


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

• Funding– NSF, ONR, SC Johnson, CNRS

Chapter 12. Suggested Reading page 12: 1


Chapter 12. Suggested Reading Chapter 12. Suggested Reading

• General• Process Models• Orazem group work on Fuel Cells• Measurement Models• CPE• Plotting



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.


Suggested ReadingSuggested ReadingProcess ModelsProcess Models

• 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


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.


Suggested ReadingSuggested ReadingMeasurement ModelsMeasurement Models

• 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.


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.


Suggested ReadingSuggested ReadingPlottingPlotting

• 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


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


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

viscosity, g/cm s

kinematic viscosity, / , cm2/s

density, g/cm3

finite-length diffusion time constant, 2 / iD

normalized axial position, / iz

ohmic potential, V

frequency of perturbation, rad/s

rotation speed, rad/sec

electrochemical impedance spectroscopy with application …che.ufl.edu/orazem/pdf-files/eis for fuel...

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