Electrochemical Impedance Spectroscopy - …ww2.che.ufl.edu/orazem/pdf-files/Orazem EIS Spring...

Electrochemical Impedance Electrochemical Impedance SpectroscopySpectroscopy

Mark E. OrazemDepartment of Chemical Engineering

University of FloridaGainesville, Florida 32611

[email protected]

352-392-6207

© Mark E. Orazem, 2000-2008. All rights reserved.

mailto:[email protected]

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. Regression Analysis• Chapter 7. Error Structure• Chapter 8. Kramers-Kronig Relations• Chapter 9. Use of Measurement Models• Chapter 10. Conclusions• Chapter 11. Suggested Reading• Chapter 12. Notation

Chapter 1. Introduction page 1: 1


Mark E. OrazemDepartment of Chemical Engineering

University of FloridaGainesville, Florida 32611

[email protected]

352-392-6207


mailto:[email protected]


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


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!” ...



• 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, μA/cm2

~ΔI

ΔV~

( )

Impedance SpectroscopyImpedance Spectroscopy

r jVZ Z jZI

ω Δ= = +Δ


Impedance SpectroscopyImpedance Spectroscopy

• 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

α⎧ ⎫= = −⎨ ⎬

⎩ ⎭

Faradaic current density

→- -2 2O +2H O+ 4e 4OH

Total current density = Faradaic + charging

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. Regression Analysis• Chapter 7. Error Structure• Chapter 8. Kramers-Kronig Relations• Chapter 9. Use of Measurement Models• Chapter 10. Conclusions• Chapter 11. Suggested Reading• Chapter 12. Notation

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 inputΔV=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 inputΔV=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

ω= + Δ

= − −



-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

φ

Δ= =

Δ

= −

OAOB

ODOA

( ) cos( )

( ) cos( )

V t V tVI t tZ

ω

ω φ

= ΔΔ

= +

-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 / Ω

cm

-2

Zr / Ω 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 Ω• R1 =1,000 Ω • C1 = 10.5 μF

– τ = 0.0105 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, Ω

Z j, Ω

0.1

1

10

100

1000

10000

100000

0.1 1 10 100 1000 10000 100000

Frequency, Hz

Impe

danc

e, Ω

Zr, Ω

-Zj, Ω


Experimental DataExperimental Data

0.01

0.1

1

10

100

1000

10000

100000

0.1 1 10 100 1000 10000 100000Frequency, Hz

Impe

danc

e, Ω


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

φ

Δ= = =Δ

= − = −

OA A'AOB B'B

OD D'DOA A'A

( ) sin( )

( ) sin( )

V t V tVI t tZ

ω

ω φ

= ΔΔ

= +

Potential

Current

ADO

B

B'

D'A'

LissajousLissajous AnalysisAnalysis


Phase Sensitive DetectionPhase Sensitive Detection0 sin( )AA A tω φ= +

( )0

4 1 sin 2 12 1 S

n

S n tn

ω φπ

∞

=

= + +⎡ ⎤⎣ ⎦+∑

( ) ( )0

0

4 1 si sin 2 12 1

n Sn

A n tn

A A tS ω ω φπ

φ∞

=

+ + ⎦++= ⎡ ⎤⎣∑

General Signal

Reference Signal

( )SAAdtAS φφππ

ω ωπ

−=∫ cos22

02

0

Has maximum value when

A Sφ φ=


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

0( ) cos( )II t I tω φ= +

0( ) cos( )V t V tω=

( )

( )

( )

( )

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

ω ω

ω ω

ω ω

ω ω

=

= −

=

= −

∫

∫

∫

∫

( ) Re

( ) Im

r jr

r j

r jj

r j

V jVZ

I jI

V jVZ

I jI

ω

ω

⎧ ⎫+⎪ ⎪= ⎨ ⎬+⎪ ⎪⎩ ⎭⎧ ⎫+⎪ ⎪= ⎨ ⎬+⎪ ⎪⎩ ⎭


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

Time

Sign

al O

utpu

t

Time

Sign

al In

put

Z

Z(ω)

Fast Fourier

Transform


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

• Good accuracy for stationary systems

• Frequency intervals of Δf/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 Δf– 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

ω= + Δ

=


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)/ΔV-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)/ΔV

-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)/ΔV-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)/ΔV


0 10 20 30 40 50

0

-10

-20

model 1 mV 20 mV 40 mV

Z j / Ω

cm

2

Zr / Ω 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| /

Ω 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

ΔV b

Guideline for LinearityGuideline for Linearity0.2b VΔ ≤


Influence of Ohmic ResistanceInfluence of Ohmic Resistance

0CdVi Cdt

=

( )fi f V=

( )V t( )s tη

eR

( ) ei t R

( )0.2 1 /e tb V R RΔ ≤ +


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 tη

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

α

α α

⎡ ⎤⎛ ⎞= − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎛ ⎞−⎜ ⎟⎝ ⎠

⎡ ⎤= + +⎢ ⎥

⎢ ⎥⎣

⎛ ⎞−⎜⎝ ⎠ ⎦

⎟ …

( ) ( )/I V Z V I Zω ωΔ = Δ Δ = Δ


Experimental StrategiesExperimental Strategies

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

I

ΔVWE 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


Measurement Case StudyMeasurement Case Study


Impedance Data from a BatteryImpedance Data from a Battery

-20

-10

00 10 20 30Zr / Ω

Z j / Ω

0.75 hours9.5 hours168 hours169 hours170 hours171 hours

-3

-2

-1

00 1 2 3 4 5

potentiostatic modulation2-electrodespiral-wound Alkaline AA


Improve Experimental DesignImprove Experimental Design

• Question– How can we isolate the role of positive electrode, negative

electrode, and separator?• Answer

Ref 1

Ref 2

WorkingElectrode

Counterelectrode

- Develop a very sophisticated process model.

- Use a four-electrode configuration.



• Question– How can we reduce stochastic noise in the measurement?

- Use more cycles during integration. - Add at least a 2-cycle delay.- Ensure use of optimal current measuring resistor.- Check wires and contacts.- Avoid line frequency and harmonics.- Use a Faraday cage.

• Answer



• Question– How can we determine the cause of variability at long times?

- Use the Kramers-Kronig relations to see if data are self-consistent.- Change mode of modulation to variable-amplitude galvanostatic to ensure that the base-line is not changed by the experiment.

• Answer


Facilitate InterpretationFacilitate Interpretation

• Question– How can we be certain that the instrument is not corrupting

the data?

- Use the Kramers-Kronig relations to see if data are self-consistent.- Build a test circuit with the same impedance response and see if you get the correct result.- Compare results to independently obtained data (such as electrolyte resistance).

• Answer


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, μA/rpm

Imag

inar

y Im

peda

nce,

μA/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 / Ω

cm

2

Zr / Ω 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

Δt 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 ωω

eR

eR Zr

-Zj

ω

CdlRe


Zr

-Zj ω

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

ω

ω

ωω ω

= ++

= ++

= + −+ +

eR e tR R+

2tR

1t dlR C

ω=

Rf

Cdl

Re

-12rad / s or scycles / s or Hz

f

f

ω πω

=≡≡

Note:


1

10

100

1000

10000

0.001 0.01 0.1 1 10 100 1000 10000

Frequency / Hz

Z r o

r -Z j

/ Ω

realimaginary

ImpedanceImpedance

Slope = 1

Slope = -1

Rf

Cdl

Re

-600

-400

-200

00 200 400 600 800 1000 1200

Zr / Ω

Z j, Ω ω

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 r o

r -Z j

/ Ω

realimaginary


Bode RepresentationBode Representation

1

10

100

1000

10000

0.001 0.01 0.1 1 10 100 1000 10000

Frequency / Hz

|Z| /

Ω-90

-60

-30

0

φ / d

egre

es

magnitudephase angle

( )( )

| | cos

| | sin

r j

r

j

Z Z jZ

Z Z

Z Z

φ

φ

= +

=

=


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| /

Ω

-90

-60

-30

0

φ∗ /

degr

eesmagnitude

modified magnitudephase anglemodified phase angle

1* tan j

r e

ZZ R

φ − ⎛ ⎞= ⎜ ⎟−⎝ ⎠Input & output are in phase

Input & output are out of phase

1tan j

r

ZZ

φ − ⎛ ⎞= ⎜ ⎟

⎝ ⎠

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 Qαω−

= ++

Rt

Re

CPE

0.0 0.2 0.4 0.6 0.8 1.00.0

-0.2

-0.4

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

Z j / R t

(Zr−Re) / 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

α=1 α=0.9 α=0.8 α=0.7 α=0.6 α=0.5

|Zj| /

Rt

ω / τ

CPE: Log ImaginaryCPE: Log Imaginary

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

Slope = − α

Slope = + α


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

α=1 α=0.9 α=0.8 α=0.7 α=0.6 α=0.5

φ* / de

gree

s

ω / τ

Note:

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


Example for Synthetic DataExample for Synthetic Data

( )( ) ( )( )dl

( )1 2

t de

t d

R z fZ f R

j f Q R z fαπ

+= +

+ +

( ) ( ) tanh 22d d

j fz f z f

j fπ τ

π τ=

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 α = 1 α = 0.7 α = 0.5

1 mHz100 Hz10 Hz

1 Hz 0.1 Hz

−Zj /

Ω c

m2

Zr / Ω cm2

10 mHz

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

102

103

α = 1 α = 0.7 α = 0.5

| Z |

/ Ω c

m2

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

0

-15

-30

-45

-60

-75

-90 α = 1 α = 0.7 α = 0.5

φ / d

egre

e

f / Hz


RRee--Corrected Bode Phase AngleCorrected Bode Phase Angle1

adj,est

tan j

r e

ZZ R

φ − ⎛ ⎞= ⎜ ⎟⎜ ⎟−⎝ ⎠ adj ( ) 90( )φ α∞ = −

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

0

-15

-30

-45

-60

-75

-90 α = 1 α = 0.7 α = 0.5

φ ad

j / de

gree

f / Hz


RRee--Corrected Bode MagnitudeCorrected Bode Magnitude( )2 2

,estadj r e jZ Z R Z= − +

Slope = −α

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

10-1

100

101

102

103

α = 1 α = 0.7 α = 0.5

| Z

| adj /

Ω c

m2

f / Hz

slope = −α


Log(|ZLog(|Zjj|)|)

Slope = −α

Slope = α

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

10-2

10-1

100

101

102

103

α = 1 α = 0.7 α = 0.5

slope=α

slope=−α

| Z

j | / Ω

cm

2

f / Hz


Effective Capacitance or CPE CoefficientEffective Capacitance or CPE Coefficient

( )effj

1 = sin2 2 ( )

Qf Z fα

αππ

⎛ ⎞− ⎜ ⎟⎝ ⎠( )CPE

CPE

12

ZQ j f απ

=

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

1

10

100 α = 1 α = 0.7 α = 0.5

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 / Ω

cm

2

Zr / Ω 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 / Ω

cm

2

Zr / Ω 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 = −α

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

101

102

time / h 0.5 3.0 6.0

|Z

j| / Ω

cm

2

f / Hz


Effective CPE CoefficientEffective CPE Coefficient

( )effj

1 = sin2 2 ( )

Qf Z fα

αππ

⎛ ⎞− ⎜ ⎟⎝ ⎠

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 /

(MΩ

-1 c

m-2) s

α

f / Hz


RRee--Corrected Phase AngleCorrected Phase Angle

adj ( ) 90φ α∞ = − 1adj

,est

tan j

r e

ZZ R

φ − ⎛ ⎞= ⎜ ⎟⎜ ⎟−⎝ ⎠

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 /

Ω c

m2

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 / Ω

cm

2

Zr / Ω cm2

65.5 Hz


Experimental device for LEISExperimental device for LEIS

appliedlocal

local

VZ

iΔ

=Δ

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 (α = 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

α = 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


Plots based on Deterministic ModelsPlots based on Deterministic Models


Convective Diffusion to a RDEConvective Diffusion to a RDElow-frequency asymptote

Reduction of Fe(CN)63- on a Pt Disk, i/ilim = 1/2

0 50 100 150 200 2500

-50

-100

120 rpm 600 rpm 1200 rpm 2400 rpm

Z j / Ω

Zr / Ω


Normalized Impedance PlaneNormalized Impedance Plane

0 0.2 0.4 0.6 0.8 1.0 1.20

0.2

0.4

0.6 120 rpm 600 1200 2400

−Zj /

(Zr,m

ax−R

e )

(Zr−Re) / (Zr,max−Re)Note: We again need an accurate estimate for the solution resistance.


Normalized RealNormalized Real

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

10-4

10-3

10-2

10-1

100

120 rpm 600 1200 2400

(Zr−R

e ) /

(Zr,m

ax−R

e )

p

/p ω= Ω


Normalized ImaginaryNormalized Imaginary

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

10-2

10-1

100

120 rpm 600 1200 2400

−Zj /

(Zr,m

ax−R

e )

p


0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1.0 120 rpm 600 1200 2400

(Zr−R

e ) /

(Zr,m

ax−R

e )

−pZj / (Zr,max−Re )

Straight line at low frequencyStraight line at low frequency

Tribollet, Newman, and Smyrl, JES 135 (1988), 134-138.


Slope at low frequency yields DSlope at low frequency yields Dii

0 0.01 0.02 0.03 0.040.4

0.6

0.8

1.0

(Zr−R

e ) /

(Zr,m

ax−R

e )

−pZj / (Zr,max−Re)

{ }{ }

1/ 30

1/ 3 2 / 3

Relim ScIm

1.2261 0.84Sc 0.63Sc

Sc 1090.6

p

d Zs

dp Zλ

λ

→

− −

⎛ ⎞= =⎜ ⎟⎜ ⎟

⎝ ⎠= + +

=


100 101 102 103 104 105

10-1

100

101

102

103

104

105

106

320 K 340 K 360 K 380 K 400 K

−Zj /

Ω

f / Hz

nn--GaAsGaAs Diode:Diode:scaling based on scaling based on

Arrhenius relationshipArrhenius relationship

0 2x106 4x106 6x106 8x106 1x1070

-2x106

-4x106

340 K

320 K

Z j / Ω

Zr / Ω

100 101 102 103 104 105100

101

102

103

104

105

106

107

108

320 K 340 K 360 K 380 K 400 K

Z r / Ω

f / Hz


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

0.0

0.2

0.4

0.6

0.8

1.0 320 K 340 K 360 K 380 K 400 K

Z r / |Z

max

|

f*

Normalized RealNormalized Real

0

* expf Eff kT

⎛ ⎞= ⎜ ⎟⎝ ⎠

Jansen, Orazem, Wojcik, and Agarwal, JES 143 (1996), 4066-4074.


Normalized ImaginaryNormalized Imaginary

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

0.0

0.1

0.2

0.3

0.4

0.5

320 K 340 K 360 K 380 K 400 K

−Zj /

|Zm

ax|

f*


Normalized Log RealNormalized Log Real

10-4 10-3 10-2 10-1 100 101 102 10310-6

10-5

10-4

10-3

10-2

10-1

100

320 K 340 K 360 K 380 K 400 K

Z r / |Z

max

|

f*


Normalized Log ImaginaryNormalized Log Imaginary

10-4 10-3 10-2 10-1 100 101 102 10310-6

10-5

10-4

10-3

10-2

10-1

100

320 K 340 K 360 K 380 K 400 K

−Zj /

|Zm

ax|

f*


SuperlatticeSuperlatticeStructureStructure

-4.0E+08

-2.0E+08

0.0E+000.0E+00 2.0E+08 4.0E+08 6.0E+08

Zr, ΩZ j

, Ω

298.8290285270260

-4.0E+07

-2.0E+07

0.0E+000.0E+00 2.0E+07 4.0E+07 6.0E+07

Zr, Ω

-Zj,

Ω

260292.5297.2297.7297.8297.8298.8


0

0.2

0.4

0.6

0.8

1

1.2

0.0001 0.01 1 100 10000

Z/Z r

,max

Normalized Real PartNormalized Real Part

0

* expf Eff kT

⎛ ⎞= ⎜ ⎟⎝ ⎠


Normalized Imaginary PartNormalized Imaginary Part-0.4

-0.3

-0.2

-0.1

0

0.10.0001 0.01 1 100 10000

Zj /

Zr,m

ax

0

* expf Eff kT

⎛ ⎞= ⎜ ⎟⎝ ⎠


DLTS of DLTS of nn--GaAsGaAs diodediode

0.002 0.003 0.004 0.005 0.006 0.007

1/T / K-1

e/T2 /

s-1

K2

0.83 eV

0.48 eV0.32 eV


nn--GaAsGaAs

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

1.E+11

0.0022 0.0024 0.0026 0.0028 0.003 0.0032 0.0034

1/T / K-1

TR /

K

0.77 eV

0.45 eV

0.25 eV

0.27 eV


MottMott--SchottkySchottky PlotsPlotshigh-frequency asymptote

0

sc

T

qC ∂=

∂Φ

( )( )2

2 d ad e p n N Ndy ε

Φ= − − + −

( )( )

0

2

21 fb

d a

kTe

C F N Nε

Φ − Φ ±= −

−

Ef

Ec

Ev

Et Eg

Position

Elec

tron

Ener

gy

GaAs

BlockingContact

OhmicContact

Dean and Stimming, J. Electroanal. Chem. 228 (1987), 135-151.


C as a function of PotentialC as a function of Potentialintrinsic semiconductor

-0.4 -0.2 0 0.2 0.410-8

10-7

10-6

10-5

10-4

10-3

C

/ μF

cm

-2

(Φ-Φfb) / V


1/C1/C22 as a function of Potentialas a function of Potentialn- or p-doped semiconductor

-3 -2 -1 0 1 2 3

0

1

2

3

4

p-typen-type

−1017 cm-3 +1017 cm-3

(Nd−Na )= +1016 cm-3 (Nd−Na )= −1016 cm-3

C-2 /

108 cm

4 μF-2

(Φ-Φfb) / V


GaAsGaAs--SchottkySchottky DiodeDiode

-3 -2 -1 0 1 20

2

4

6

8

C -2

/ nF

-2

Potential / V


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 εεε

δδ κ

−= = ×

=


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 General Expression for FaradaicFaradaic CurrentCurrent

,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

γ γ

γ

γγ

≠

≠

⎛ ⎞∂ ∂⎛ ⎞= + ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂+ ⎜ ⎟∂⎝ ⎠

∑

∑

( ),0, ,f i ii f V c γ=

{ }tjeiii ω~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

RTα⎛ ⎞= ⎜ ⎟⎝ ⎠

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 Vω

= +

= +

Charging CurrentCharging Current

low frequency

high frequency

,A

,A

1

dlt

dlt

Vi j C VR

V j CR

ω

ω

= +

⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠

,A

,A1t

t dl

RVi j R Cω=

+


Ohmic ContributionsOhmic Contributions

VRiU

ViRU

e

e~~~ +=

+=

A

,A

,A1+

e

te

t dl

U VZ Ri i

RR

j R Cω

= = +

= +


( )A A Aexp 2.303i K V β=

A A2.303/ bβ =

( )A

,AA

1exp 2.303t

A A A

RK b b V i

β= =

AA

,A

A ,A A

2.303

2.303t

t

iR

R i

β

β

=

=

( )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 ω

= −

= +

OO O O

0

dci n FDdy

= −

( )O,0O O O

O

0c

i n FD θδ

′= −


Combine ExpressionsCombine Expressions

( )O O O O,0,O

expt

Vi K b V cR

= −

( )O O

O,0O O 0

icn FD

δθ

= −′



OO

,OO O O,0 O

,O ,O

1 1(0)t

t d

ViR

n FD c b

VR z

δθ

=⎛ ⎞

+ −⎜ ⎟′⎝ ⎠

=+

O,O

O O O,0 O

1 1(0)dz

n FD c bδ

θ⎛ ⎞

= −⎜ ⎟′⎝ ⎠

( ),OO O,0 O

1exptR

K c b V=

−


Calculate ImpedanceCalculate Impedance

O

f dl

dl

dVi i Cdt

i i j C Vω

= +

= +

,O ,O

,O ,O

1

dlt d

dlt d

Vi j C VR z

V j CR z

ω

ω

= ++

⎛ ⎞= +⎜ ⎟⎜ ⎟+⎝ ⎠

VRiU

ViRU

e

e~~~ +=

+=

( )

O

,O ,O

,O ,O1+

e

t de

dl t d

U VZ Ri i

R zR

j C R zω

= = +

+= +

+


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 zω+

= ++


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 Vγ= − −

reaction 1: formation of X

reaction 2: formation of P

( )( )2 2 2 2expi K b V Vγ= −

total current density

1 2i i i= +


SteadySteady--State Surface CoverageState Surface Coverage

1 2

0

i iddt F Fγ

Γ = −

=

balance on γ

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

1 1 1

1 1 1 2 2 2

exp

exp exp

K b V V

K b V V K b V Vγ

−=

− + −

steady-state value for γ


SteadySteady--State Current DensityState Current Density

( ) ( )( ) ( )( )1 1 1 2 2 21 exp expi K b V V K b V Vγ γ= − − + −

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

1 1 1

1 1 1 2 2 2

exp

exp exp

K b V V

K b V V K b V Vγ

−=

− + −

where


Oscillating Current DensityOscillating Current Density

( ) ( )( ) ( )( )1 1 1 2 2 21 exp expi K b V V K b V Vγ γ= − − + −

( )( ) ( )( )( )2 2 2 1 1 1,1 ,2

1 1 exp expt t

i V K b V V K b V VR R

γ⎛ ⎞

= + + − − −⎜ ⎟⎜ ⎟⎝ ⎠

( ) ( )( )( )( )

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

γ

γ

−

−

⎡ ⎤= − − −⎣ ⎦

⎡ ⎤= − −⎣ ⎦


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

ωγ γ⎛ ⎞

Γ = − − − + −⎜ ⎟⎜ ⎟⎝ ⎠

balance on γ

( )( ) ( )( )( )1 1

,1 ,2

1 1 1 2 2 2exp expt tR R

VF j K b V V K b V V

γω

− −−=

Γ + − + −


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

ω

ω

− −⎡ ⎤ ⎡ ⎤− − − −⎣ ⎦⎣ ⎦= +Γ + − + −

= ++

, ,

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 β=

Fe Fe2.303/ bβ =

( )Fe

,FeFe Fe Fe Fe

1exp 2.303tR

K b b V iβ

= =

FeFe

,Fe2.303 t

iR

β=

( )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 θδ

= −

′= −


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

δθ

=⎛ ⎞

+ −⎜ ⎟′⎝ ⎠

=+

( )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 bδ

θ⎛ ⎞

= −⎜ ⎟′⎝ ⎠

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

~~~ ω+=

+=Faradaic and Charging

VRiU

ViRU

e

e~~~ +=

+=Ohmic


ddt

ddtt,t,

jr

CjzRR

CjzRRR

jZZiUZ

ω

ω

++

+=

++

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

+==

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 mechanismwrite expression for steady state current contributionswrite expression for sinusoidal steady statesum current contributionsaccount for charging currentaccount for ohmic potential drop

• account for mass transfercalculate 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

δ∞→ →

= =

( ),0 , ,0i i i if

zc c c cδ ∞= + −

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

ω ω

ω ω

ω

ω

ω

= +

=

=

{ }tjiii eccc ω~Re+=

2

(0)

fi

i

ii

i

KDc

c

ωδ

θ

=

=

i

zδ

ξ =

2

2 0ii i

d jKdθ

θξ

− =


Warburg ImpedanceWarburg Impedance

( ) ( )

2

2 0

exp exp

ii i

i i i

d jKd

A jK B jK

θθ

ξ

θ ξ ξ

− =

= + −

2

2

0 at 1

1 at 0

tanh1(0)

tanh1(0)

i

i

i

i i

i

i

i

jKjK

jD

jD

θ ξ

θ ξ

θ

ωδ

θ ωδ

= =

= =

= −′

= −′

2

0 at

1 at 01 1(0)

1 1(0)

i

i

i i

i

i

jK

jD

θ ξ

θ ξ

θ

θ ωδ

= = ∞

= =

= −′

= −′


ConcentrationConcentration

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

ξ/δ

c(ξ)

/c( ∞

)

t=0, 2nπ/K

t=nπ/K

t=0.5nπ/K

t=1.5nπ/K

K=100

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

ξ/δ

c(ξ)

/c( ∞

)

t=0, 2nπ/K

t=nπ/K

t=0.5nπ/K

t=1.5nπ/K

K=1-40

-30

-20

-10

00 10 20 30 40 50 60

Zr, Zr,D, Ω

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

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎟⎠⎞

⎜⎝⎛ Ω+⎟

⎠⎞

⎜⎝⎛ Ω+

Ω−Ω= ...

631 4

23

2/32 zbzzavz ννν

ν

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎟⎠⎞

⎜⎝⎛ Ω−⎟

⎠⎞

⎜⎝⎛ Ω−⎟

⎠⎞

⎜⎝⎛ ΩΩ= ...

323

2/322/1

zbzzarvr ννν

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎟⎠⎞

⎜⎝⎛ Ω+⎟

⎠⎞

⎜⎝⎛ Ω−Ω= ...

31 3

2/32/1

zazbrvννθ

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 ,η=


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

ω ω ωω + + −− =

Sinusoidal Steady StateSinusoidal Steady State{ }tj

iii eccc ω~Re+=

1/3i

31

2

31

2 Sc99⎟⎠⎞

⎜⎝⎛

Ω=⎟⎟

⎠

⎞⎜⎜⎝

⎛Ω

=aDa

Ki

iωνω

i

zδ

ξ =Ω

⎟⎠⎞

⎜⎝⎛=

Ω⎟⎠⎞

⎜⎝⎛=

ννν

δ 1/3i

31

31

Sc133

aaDi

i

0~~~2

2

=−− iii

zi cjK

dcdv

dcd

ξξ


Finite Length Warburg ImpedanceFinite Length Warburg Impedance

vcz

D

zi

i

∂∂

ν

=

=

0

2598 1 3 2 3. / /

Ω

z zj

j

D

d d

i

=

= =

( )tanh

. /

0

25982 1 3

ωτωτ

τSc

Ω

2

2

tanh1(0)

i

i

i

jD

jD

ωδ

θ ωδ= −

′


ImpedanceImpedance

0~~~2

2

=−− iii

zi jK

ddv

dd θ

ξθ

ξθ ( ) 0,

~~~iii cc=ξθ

0at1~as0~

==

∞→→

ξθ

ξθ

i

i

( )0~,,0, i

i

i

i

ci itD nFD

scfRZ

ijjθδ

η′⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂∂

−=≠

∑

( )0~)0(

i

DD

ZZθ ′

=


Numerical SolutionsNumerical Solutions

One Term

( )v a

z a

z = −

= =

νΩ ζ

ζν

2

051203Ω

; .

( )z zp

p

d d= −′

⎛

⎝⎜⎜

⎞

⎠⎟⎟

=

( )01

0θ

ω

Sc1/3

Ω

Two Termsv az = − +

⎛⎝⎜

⎞⎠⎟νΩ ζ ζ2 31

3 ( )( )

z zp

Z pd d= −

′+

⎛

⎝⎜⎜

⎞

⎠⎟⎟( )0

1

0

1

θ ScSc

Sc1/3

1/3

1/3

Three Termsv a

b

b

z = − + +⎛⎝⎜

⎞⎠⎟

= −

νΩ ζ ζ ζ2 3 413 6

0 616.( )

( ) ( )z z

pZ p Z p

d d= −′

+ +⎛

⎝⎜⎜

⎞

⎠⎟⎟( )0

1

0

1 2

θ 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

δδ

δδωδ

+=

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠


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. Regression Analysis page 6: 1


Chapter 6. Regression AnalysisChapter 6. 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 Ω cm2

• τ1 = τRC = 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

τ / s

R / Ω cm2

0.5 1.0 1.50.5

1.0

1.5

R / Ω cm2

τ / 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

σ σ= =

−−= +∑ ∑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= = =

∂ ∂= + Δ + Δ Δ +

∂ ∂ ∂∑ ∑∑p p …

= ⋅ Δβ α p

dat

21

( | ) ( | )Ni i i

ki i k

y y x y xp

βσ=

− ∂=

∂∑ p p

dat

, 21

1 ( | ) ( | )Ni i

j ki i j k

y x y xp p

ασ=

⎡ ⎤∂ ∂≈ ⎢ ⎥

∂ ∂⎢ ⎥⎣ ⎦∑ p p

dat 22

21

( ( | ))Ni i

i i

y y xχσ=

−= ∑ p

i i

f y Zx ω= =

=

For our purposes:

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 / Ω cm2

τ / 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 / Ω cm2

τ / 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 Ω cm2

• R1 = 100 Ω cm2

• τ1 = 0.001 s• R2 = 200 Ω cm2

• τ2 = 0.01 s• R3 = 5 Ω cm2

• τ3 = 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 τ3

-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/Ω cm2) log10(τ / 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/Ω cm2) log10(τ / 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 /

Ω c

m2

Zr / Ω 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 Ω cm2

• R1 = 100 Ω cm2

• τ1 = 0.01 s• R2 = 200 Ω cm2

• τ2 = 0.1 s• R3 = 100 Ω cm2

• τ3 = 10 s

Note: 3rd Voigt element contributes 25% to DC cell impedance. The time constant corresponds to a characteristic frequency ω3=0.1 s-1 or f3=0.016 Hz.


Resulting Test SpectraResulting Test Spectra

0 100 200 300 4000

-100

Z j / Ω

cm

2

Zr / Ω cm2

0.01 Hz to 100 kHz1 Hz to 100 kHz


Effect of Truncated DataEffect of Truncated Data

All parameters fixed except R3 and τ3

-2-1

01

23

4-1

01

23

45100

101

102

103

Sum

of S

quar

es

log10(R / Ω cm2)

log10(τ / 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 / Ω cm2)

log10(τ / 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 zω

ωω ω

+= +

+ +

( ) ( ) tanhd d

jz z

jωτ

ω ωωτ

=


Evaluation of Evaluation of χχ22 StatisticStatistic

σ/|Z(ω)| 1 0.1 0.05 0.03 0.01

χ2 0.0408 4.08 16.32 45.32 408

χ2/ν 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 /

Ω

Zr / Ω

Impedance Plane (Impedance Plane (NyquistNyquist))

Value of χ2 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| /

Ω

f / Hz


Phase AnglePhase Angle

10-2 10-1 100 101 102 103 1040

-15

-30

-45

φ

/ deg

rees

f / Hz


RealReal

10-2 10-1 100 101 102 103 1040

40

80

120

160

200

Z r /

Ω

f / Hz


ImaginaryImaginary

10-2 10-1 100 101 102 103 1040

20

40

60

80

−Zj /

Ω

f / Hz


Log ImaginaryLog Imaginary

10-2 10-1 100 101 102 103 1041

10

100

Slope = -1 for RC

−Z

j / Ω

f / Hz



10-2 10-1 100 101 102 103 1040

-15

-30

-45

-60

-75

-90φ∗ /

degr

ees

f / Hz

1* tan j

r e

ZZ R

φ − ⎛ ⎞= ⎜ ⎟−⎝ ⎠


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

ε r /

Z r

f / Hz

±2σ noise level


Imaginary ResidualsImaginary Residuals

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

-0.2

-0.1

0

0.1

0.2

0.3ε j /

Z j

f / Hz

±2σ noise 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 zα

ωω

ω ω

+= +

+ +

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

pθ

⎛ ⎞⎜ ⎟= − + +⎜ ⎟′⎝ ⎠

χ2/ν=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 /

Ω

Zr / Ω


Phase AnglePhase Angle

10-2 10-1 100 101 102 103 1040

-15

-30

-45

−Z

j / Ω

f / Hz



10-2 10-1 100 101 102 103 1040

-15

-30

-45

-60

-75

-90φ* /

degr

ees

f / Hz

1* tan j

r e

ZZ R

φ − ⎛ ⎞= ⎜ ⎟−⎝ ⎠


Log ImaginaryLog Imaginary

10-2 10-1 100 101 102 103 1041

10

100

−Z

j / Ω

f / Hz


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

-0.003

0

0.003

0.006ε r /

Z r

f / Hz

Real ResidualsReal Residuals

±2σ


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

-0.01

0

0.01

0.02ε j /

Z j

f / Hz

Imaginary ResidualsImaginary Residuals

±2σ


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 7. Error Structure page 7: 1


Chapter 7. Error StructureChapter 7. Error Structure

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



Error Analysis Increasingly Important Error Analysis Increasingly Important

• Improved Instrumentation– improved signal-to-noise– more data– more detailed interpretation of data possible

• New experimental techniques– more ways to study a system– proper weight must be assigned to each experiment– error analysis used to improve experimental design

• Improved computational speed– increased use of regression– increased use of detailed models


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 Zε ω ω ωε ω ε ω ε ω

= −= + +

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

2σ σ

Strategy ImplicationsNo Weighting σr=σj

σ = αModulus Weighting σr=σj

σ = α|Ζ|Proportional Weighting σr≠σj

σr = αr|Ζr|; σj = αj|Ζj|Experimental σr=σj

σ = α|Ζj| + β|Ζr| + γ|Ζ|2/RmRefined Experimental σr=σj

σ = α|Ζj| + β|Ζr-Rsol| + γ|Ζ|2/Rm+δ


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, Ω

Z j, Ω

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 8. Kramers-Kronig Relations page 8: 1


Chapter 8. KramersChapter 8. 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)(

ωω

πωω

dxx

ZxxZZZ jj

rr ⎮⌡⌠

−

+−−+∞=

∞

0

22

)()(2)()(ω

ωωπωω


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

ωεωωεωωεωω

jjrr ZjZZZ

+++=+=

( ) )()( ωεωεωε jr j+=

( ) )()( ob ωω ZZE = ( ) 0)( =ωεE

where

If and only if

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⎮⌡⌠

−−+−

−=

∞

0

22)()()()(2)( dx

xxZxZEZE rrrr

j ωωεεω

πωω

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⎮⌡⌠

−

+−+−−=∞−

∞

0

22

)()()()(2)()( dxx

xxZxxZEZZE jjjj

rr ωωωεεωω

πω

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, εr(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 εr(x) should be of the same magnitude as the stochastic noise εr(ω).

0)(2

0

22 =⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⎮⌡⌠

−−∞

dxx

xE r

ωε

πω( ) 0)( =ωεE and


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

-1000

-800

-600

-400

-200

0

(Zr(x

)-Zr(ω

)) /

(x2 -ω

2 )

f / Hz

Meaning ofMeaning of 2 2

0

2 ( ) 0r xE dxx

ω επ ω

∞⌠⎮⎮⌡

⎛ ⎞⎜ ⎟− =

−⎜ ⎟⎝ ⎠

0.2 0.5 1-400

-300

-200

-100

0

Interpolated Value

Correct Value

(Zr(x

)-Zr(ω

)) /

(x2 -ω

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/ω and ω asymptotic behavior

– simultaneous solution for missing data• Regression

– proposed process model– generalized measurement model

Chapter 9. Use of Measurement Models page 9: 1


Chapter 9. Use of Measurement ModelsChapter 9. Use of Measurement Models

• Generalized model• Identification of stochastic component of error structure• Identification of bias component of error structure



ApproachApproach• Experimental design

– quality (signal-to-noise, bias errors)– information content

• Process model– account for known phenomena– extract information concerning other phenomena

• Regression– uncorrupted frequency range– weighting strategy

• Error analysis - measurement model– stochastic errors– bias errors


Error StructureError Structure

obs model res

obs model fit bias stoch

Z ZZ Z

εε ε ε

− =− = + +

• Stochastic errors• Bias errors

– lack of fit– experimental artifact

• Kramers-Kronig consistent• Kramers-Kronig inconsistent


Measurement ModelMeasurement Model

• Superposition of lineshapes

01 1

nk

k k k

RZ Rj R Cω=

= ++∑


Identification of Stochastic Component Identification of Stochastic Component of Error Structureof Error Structure

How?• Replicate impedance scans.• Fit measurement model to

each scan:– use modulus or no-

weighting options.– same number of

lineshapes.– parameter estimates do

not include zero.• Calculate standard deviation

of residual errors.• Fit model for error structure.

Why?• Weight regression.• Identify good fit.• Improve experimental

design.


Replicated MeasurementsReplicated Measurements

0

50

100

0 50 100 150 200

Zr / Ω

-Zj /

Ω


1 Voigt Element1 Voigt Element

0

50

100

0 50 100 150 200

Zr / Ω

-Zj /

Ω


2 Voigt Elements2 Voigt Elements

0

50

100

0 50 100 150 200

Zr / Ω

-Zj /

Ω



0

50

100

0 50 100 150 200

Zr / Ω

-Zj /

Ω


Residual Residual ErrorsErrors

-0.005

0

0.005

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

εr /

Z r

-0.03

0

0.03

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hzεj

/ Zj

imaginary

real


Residual Sum of SquaresResidual Sum of Squares

0.1

1

10

100

1000

10000

100000

1000000

1 3 5 7 9 11

# Voigt Elements

Σ(ε2 / σ

2 )


Fit to Repeated Data SetsFit to Repeated Data Sets

0

50

100

0 50 100 150 200

Zr / Ω

-Zj /

Ω


Residual ErrorsResidual ErrorsReal

-0.005

0

0.005

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

εr /

Z r


Standard Deviation of Residual ErrorsStandard Deviation of Residual ErrorsImaginary

-0.02

0

0.02

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

εj / Z

j


Standard DeviationStandard Deviation

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,

ImaginaryReal

direct calculation of standard deviation

standard deviation obtained from measurement model analysis


Comparison to ImpedanceComparison to Impedance

0.001

0.01

0.1

1

10

100

1000

0.01 0.1 1 10 100 1000 10000 100000

Frequency, Hz

Z or

,

RealImaginary3% of |Z|


Identification of Bias Component of Error Structure

How?• Fit measurement model to real

component:– predict imaginary component.– Use Monte Carlo calculations to

estimate confidence interval.• Fit measurement model to

imaginary component:– predict real component.– Use Monte Carlo calculations to

estimate confidence interval.

Why?• Identify suitable

frequency range for regression.

• Improve experimental design.


11stst of Repeated Measurementsof Repeated MeasurementsImaginary

-80

-60

-40

-20

00.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

Z j /


11stst of Repeated Measurementsof Repeated MeasurementsReal

1

10

100

1000

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

Z r /


Fit to Imaginary PartFit to Imaginary Part

0

50

100

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

-Zj /

Ω


Fit to Imaginary PartFit to Imaginary Part

-0.03

0

0.03

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

εj / Z

j


Predict Real PartPredict Real Part

0

50

100

150

200

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

Z r / Ω

150

170

190

0.01 0.1


Predict Real PartPredict Real Part

-0.03

0

0.03

0.01 0.1 1 10 100 1000 10000 100000

Frequency / Hz

εr /

Z r


Validation of Measurement Model Validation of Measurement Model ApproachApproach


The error structure identified should The error structure identified should be independent of measurement model be independent of measurement model

usedused

01

( )1 ( )

Ni

i i

RZ Rj

ωω τ=

= ++∑Voigt:

0

0

( )( )

( )

Mni

iiP

njj

j

b jZ

a j

ωω

ω

=

=

=∑

∑Transfer Function:

1/ 2n =

1/2 /20 1 2

1/2 /20 1 2

( ) ( ) ( )( ) ;( ) ( ) ( )

MM

P

b b j b j b jZa a j a j j

ω ω ωωω ω ω

+ + + +=

+ + + +……

; 1PM P a= =


Convective Diffusion to a RDEConvective Diffusion to a RDEReduction of Fe(CN)Reduction of Fe(CN)66

33-- on a Pt Disk: on a Pt Disk: i/ii/ilimlim = = ¼¼; 120 rpm; 120 rpm

0 40 80 120 160 200

0

-20

-40

-60

-80 0 8.93 0.33 9.25 0.66 9.58 1.0 9.91 1.34 1.67 1.99 2.33 0.0.022 Hz

Z j / Ω

Zr / Ω

0.15 Hz


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

0

40

80

120

160

200 0 8.93 0.33 9.25 0.66 9.58 1.0 9.91 1.34 1.67 1.99 2.33

Z r / Ω

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

0

-10

-20

-30

-40

-50

-60

-70

-80

0 8.93 0.33 9.25 0.66 9.58 1.0 9.91 1.34 1.67 1.99 2.33

Z j / Ω

f / Hz

Convective Diffusion to a RDEConvective Diffusion to a RDEReduction of Fe(CN)Reduction of Fe(CN)66

33-- on a Pt Disk: on a Pt Disk: i/ii/ilimlim = = ¼¼; 120 rpm; 120 rpm

real part imaginary part


Data Sets for AnalysisData Sets for Analysis

0 2 4 6 8 10-92

-91

-90

-89

-88

-87

-86 measurement

before impedance scan after impedance scan

3 2

Cur

rent

/ m

A

time / h

1


Regression Results for Set #1: Regression Results for Set #1: 00--1 h1 h

• Data: – 74 frequencies – 148 data points

• Voigt Model: – 10 elements + Re

– 21 parameters• Transfer Function:

– 5 ai + 6 bi

– 11 parameters0 2 4 6 8 10

-92

-91

-90

-89

-88

-87

-86

3 2

Cur

rent

/ m

A

time / h

1


Stochastic Error Structure for Set #1: Stochastic Error Structure for Set #1: VoigtVoigt

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

10-3

10-2

10-1

100

real imag direct VMM

σ Z

r, σZ j /

Ω

f / Hz


Stochastic Error Structure for Set #1: Stochastic Error Structure for Set #1: Voigt & Transfer FunctionVoigt & Transfer Function

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

10-3

10-2

10-1

100

real imag direct VMM

σ Z

r, σZ j /

Ω

f / Hz


Regression Results for Set #3: Regression Results for Set #3: 8.98.9--9.9 h9.9 h

• Data: – 74 frequencies – 148 data points

• Voigt Model: – 9 elements + Re

– 19 parameters• Transfer Function:

– 5 ai + 6 bi

– 11 parameters0 2 4 6 8 10

-92

-91

-90

-89

-88

-87

-86

3 2

Cur

rent

/ m

A

time / h

1


Stochastic Error Structure for Set #3: Stochastic Error Structure for Set #3: VoigtVoigt

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

10-3

10-2

10-1

100

real imag VMM

σ Z

r, σZ j /

Ω

f / Hz


Stochastic Error Structure for Set #3: Stochastic Error Structure for Set #3: Voigt & Transfer FunctionVoigt & Transfer Function

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

10-3

10-2

10-1

100

real imag TMM VMM

σ Z

r, σZ j /

Ω

f / Hz


Identification of Stochastic Component Identification of Stochastic Component of Error Structureof Error Structure

• Direct evaluation of standard deviation includes significant contributions from changes in system behavior

• The error structure identified was independent of measurement model used

• Selection of measurement model– Ease of identification of initial parameters → Voigt– Number of parameters → Transfer Function


Evaluation of Consistency with the Evaluation of Consistency with the KramersKramers--Kronig relationsKronig relations

• Approach– Fit measurement model to imaginary component– Use experimental variance to weight regression– Predict real component– Compare prediction to measured data– Use Monte Carlo calculations to estimate confidence interval

• Assumption– Measurement model satisfies the Kramers-Kronig relations


Voigt Measurement Model: Data 1Voigt Measurement Model: Data 1

10-2 10-1 100 101 102 103 104 105-0.02

-0.01

0

0.01

0.02

(Zj-Z

j,mod

) / Z

j,mod

f / Hz

Fit to Imaginary

10-2 10-1 100 101 102 103 104 105-0.04

-0.02

0

0.02

0.04

• 20 parameters, guessed value for Re

• Tight confidence interval at low frequency• Justify deleting 5 data points at lowest frequency

(Zr-Z

r,mod

) / Z

r,mod

f / Hz

Prediction of Real


Voigt Measurement Model: Data 5Voigt Measurement Model: Data 5

• 20 parameters, guessed value for Re

• Justify keeping all data points

10-2 10-1 100 101 102 103 104 105-0.010

-0.005

0

0.005

0.010

(Zj-Z

j,mod

) / Z

j,mod

f / Hz

Fit to Imaginary

10-2 10-1 100 101 102 103 104 105-0.04

-0.02

0

0.02

0.04

(Zr-Z

r,mod

) / Z

r,mod

f / Hz

Prediction of Real


Interpretation of Impedance SpectraInterpretation of Impedance Spectra

• Physical insight and knowledge of error structure– stochastic component

• weighting• determination of model adequacy• experimental design

– bias component• suitable frequency range• experimental design

• Measurement model approach is general– electrochemical impedance spectroscopies– optical spectroscopies– mechanical spectroscopies

Chapter 10. Conclusions page 10: 1


Chapter 10. ConclusionsChapter 10. Conclusions

• Overview of Course• Credits



Overview of CourseOverview of Course

• Chapter 1. Introduction • Chapter 2. Motivation • Chapter 3. Impedance Measurement • Chapter 4. Representations of Impedance Data• Chapter 5. Development of Process Models• Chapter 6. Regression Analysis• Chapter 7. Error Structure• Chapter 8. Kramers-Kronig Relations• Chapter 9. Use of Measurement Models• Chapter 10. Conclusions• Chapter 11. Suggested Reading• Chapter 12. 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

Chapter 10. Conclusions page 10: 4Integrated ApproachIntegrated Approach


AcknowledgementsAcknowledgements• Students

– University of Florida• P. Agarwal, M. Durbha, S. Carson, M. Membrino, P. Shukla, V. Huang, S. Roy, B. Hirschorn,

S-L Wu– CNRS

• T. El Moustafid, I. Frateur, H. G. de Melo– CIRIMAT

• J-B Jorcin • Collaborators

– University of South Florida• L. García-Rubio

– University of Florida • O. Crisalle

– CNRS, Paris• B. Tribollet, C. Deslouis, H. Takenouti, V. Vivier

– CIRIMAT, Toulouse• N. Pébère

– CNR, Italy• M. Musiani

• Funding– NSF, ONR, SC Johnson, CNRS, NASA

• Other– The Electrochemical Society– The Fuel Cell Seminar

Chapter 11. Suggested Reading page 11: 1


Chapter 11. Suggested Reading Chapter 11. 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


• 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

• S. K. Roy and M. E. Orazem, “Deterministic Impedance Models for Interpretation of Low-Frequency Inductive Loops in PEM Fuel Cells,” in Proton Exchange Membrane Fuel Cells 6, T. Fuller, C. Bock, S. Cleghorn, H. Gasteiger, T. Jarvi, M. Mathias, M. Murthy, T. Nguyen, V. Ramani, E. Stuve, T. Zawodzinski, editors, ECS Transactions, 3:1(2006), 1031-1040.

• S. K. Roy and M. E. Orazem, “Stochastic Analysis of Flooding in PEM Fuel Cells by Electrochemical Impedance Spectroscopy,” in Proton Exchange Membrane Fuel Cells 7, T. Fuller, H. Gasteiger, S. Cleghorn, V. Ramani, T. Zhao, T. Nguyen, A. Haug, C. Bock, C. Lamy, and K. Ota, editors, Electrochemical Society Transactions, 11:1 (2007), 485-495.

• 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.

Suggested ReadingSuggested ReadingOrazem group work on Fuel CellsOrazem group work on Fuel Cells


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.

• V. Huang, V. Vivier, M. Orazem, N. Pébère, and B. Tribollet, “The Apparent CPE Behavior of an Ideally Polarized Blocking Electrode: A Global and Local Impedance Analysis,” Journal of the Electrochemical Society, 154 (2007), C81-C88.

• V. Huang, V. Vivier, M. Orazem, I. Frateur, and B. Tribollet, “The Global and Local Impedance Response of a Blocking Disk Electrode with Local CPE Behavior,” Journal of the Electrochemical Society, 154(2007), C89-C98.

• 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 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 ReadingPlottingPlotting

• M. Orazem, B. Tribollet, and N. Pébère, “Enhanced Graphical Representation of Electrochemical Impedance Data,” Journal of the Electrochemical Society, 153 (2006), B129-B136.

EElleeccttrroocchheemmiiccaall IImmppeeddaannccee

SSppeeccttrroossccooppyy

CChhaapptteerr 1122.. NNoottaattiioonn

• Roman • Greek

Chapter 12. Notation page 12: 2

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 , mol/cm3 i

ic steady-state value of the concentration of reacting species , mol/cm3 i

ic Oscillating component of the concentration of reacting species , mol/cm3 i

,oic concentration of species on the electrode surface, mol/cm3 i

,oic steady-state value of the concentration of species on the electrode surface, mol/cm3 i

,oic Oscillating component of the concentration of species on the electrode surface, mol/cm3 i

∞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 , cm2/s i

f arbitrary function, e.g., ( ),f ii f V c=

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

ω ν ω⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟Ω Ω ⎝ ⎠⎝ ⎠ K

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, Ω cm2

,t AR Charge-transfer resistance associated with reaction A, Ω cm2

r radial coordinate, cm

0r radius of disk, cm

is stoichiometric coefficient for species , ( for a reactant and i 0is > 0is < for a product)


Sci Schmidt number, Sc /i iDν=

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, Ω cm2

z axial position, cm

dz diffusion impedance, Ω cm2

iz charge for species i

Greek α coefficient used in the exponent for a constant-phase element. When 0α = , the element behaves as an ideal capacitor in

parallel with a resistor.

Aα apparent transfer coefficient for reaction A

Aβ Tafel slope for reaction A, V

iγ Fractional surface coverage by species i


Γ Maximum surface coverage

iδ characteristic diffusion length for species i

θ homogeneous solution to the oscillating dimensionless convective diffusion equation

(0)θ ′ derivative of the solution to the oscillating dimensionless convective diffusion equation evaluated at the electrode surface

η surface overpotential, V

∞κ solution conductivity, (Ω cm)-1

characteristic length for a finite-length diffusion layer, 1/ 3 2 / 32.598 /iDν= Ω

μ viscosity, g/cm s

ν kinematic viscosity, /ν μ ρ= , cm2/s

ρ density, g/cm3

τ finite-length diffusion time constant, 2 / iDτ =

ξ normalized axial position, / izξ δ=

Φ ohmic potential, V

ω frequency of perturbation, rad/s

Ω rotation speed, rad/sec

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