Electrochemical Impedance Spectroscopy with Application to Fuel Cells

Transcription

1 Course sponsored for the Fuel Cell Seminar by The Electrochemical Society Electrochemical Impedance Spectroscopy with Application to Fuel Cells Mark E. Orazem Department of Chemical Engineering University of Florida Gainesville, Florida Mark E. Orazem, All rights reserved.

2 Contents 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

3 Chapter 1. Introduction Course sponsored for the Fuel Cell Seminar page 1: 1 by The Electrochemical Society Electrochemical Impedance Spectroscopy with Application to Fuel Cells Mark E. Orazem Department of Chemical Engineering University of Florida Gainesville, Florida meo@che.ufl.edu Mark E. Orazem, All rights reserved.

4 Chapter 1. Introduction page 1: 2 Electrochemical Impedance Spectroscopy Chapter 1. Introduction How to think about impedance spectroscopy EIS as a generalized transfer function Overview of applications of EIS Objective and outline of course Mark E. Orazem, All rights reserved.

5 Chapter 1. Introduction page 1: no logo

6 Chapter 1. Introduction page 1: 4 June 6-11, 2010 Hotel Tivoli Carvoeiro Carvoeiro, Algarve, Portugal

7 Chapter 1. Introduction page 1: 5 The Blind Men and the Elephant John Godfrey Saxe It was six men of Indostan To learning much inclined, Who went to see the Elephant (Though all of them were blind), That each by observation Might satisfy his mind. The First approached the Elephant, And happening to fall Against his broad and sturdy side, At once began to bawl: God bless me! but the Elephant Is very like a wall!...

8 Chapter 1. Introduction page 1: 6 Electrochemical Impedance Spectroscopy 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

9 Chapter 1. Introduction page 1: 7 Impedance Spectroscopy Current Density, A/cm 2 ~ I 10 5 V Z Z jz I r j Potential, V -5 V ~ -10

10 Chapter 1. Introduction page 1: 8 Impedance Spectroscopy Applications Electrochemical systems Corrosion Electrodeposition Human Skin Batteries Fuel Cells Materials Fundamentals Dielectric spectroscopy Acoustophoretic spectroscopy Viscometry Electrohydrodynamic impedance spectroscopy

11 Chapter 1. Introduction page 1: 9 Physical Description Electrode-Electrolyte Interface Electrical Double Layer Diffusion Layer Electrochemical Reactions Electrical Circuit Analogues

12 Chapter 1. Introduction page 1: 10 Electrochemical Reactions O + 2H O+ 4e 2 2 4OH - - Faradaic current density O F RT 2 i exp F io n 2 O Fk 2 O c 2 O V V 2 O2 Total current density = Faradaic + charging dv i i F Cd dt Cell potential = electrode potential + Ohmic potential drop U V ir e

13 Chapter 1. Introduction page 1: 11 Electrical Analogues

14 Chapter 1. Introduction page 1: 12 Electrical Analogue Simple electrochemical reaction Simple electrochemical reaction with mass transfer

15 Chapter 1. Introduction page 1: 13 Course Objectives Benefits and advantages of impedance spectroscopy Methods to improve experimental design Interpretation of data graphical representations regression error analysis equivalent circuits process models

16 Chapter 1. Introduction page 1: 14 Contents 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

17 Chapter 1. Introduction page 1: 15 Notation IUPAC Convention: Z and Z ; i Present Work: Z r and Z j ; j

18 Chapter 1. Introduction page 1: 16

19 Chapter 2. Motivation page 2: 1 Electrochemical Impedance Spectroscopy Chapter 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 Mark E. Orazem, All rights reserved.

20 Chapter 2. Motivation page 2: 2 Steady-State State Polarization Curve 0.4 Current, ma Potential, V

21 Chapter 2. Motivation page 2: 3 Steady-State State 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

22 Chapter 2. Motivation page 2: 4 Transient Response to a Step in Potential Current / ma } V=10 mv R 0 C 1 C 2 R 1 (V) potential input R 2 Current / ma I R V ( R 0 R1 V ) 2 current response (t-t 0 ) / ms (t-t 0 ) / s

23 Chapter 2. Motivation page 2: 5 Transient Response to a Step in Potential 1.5 } V=10 mv potential input Short times/high frequency Current / ma R 0 C 1 R 1 (V) C 2 R 2 current response long times/low frequency (t-t 0 ) / ms

24 Chapter 2. Motivation page 2: 6 Transient Techniques: potential 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

25 Chapter 2. Motivation page 2: 7 Sinusoidal Perturbation V() t V V cost 0 it ( ) i exp( bv) exp( bv) 0 a c dv ic C0 dt i f f V Vt ()

26 Chapter 2. Motivation page 2: 8 Sinusoidal Perturbation (i-i 0 ) / max(i-i 0 ) khz 100 Hz 1 mhz (V-V 0 ) / V 0

27 Chapter 2. Motivation page 2: 9 Lissajous Representation Y(t)/Y B O D A V() t V cos( t) V It () cos( t ) Z V OA Z I OB OD sin( ) OA X(t)/X 0

28 Chapter 2. Motivation page 2: 10 Impedance Response Z j / cm Hz Z r / cm -2

29 Chapter 2. Motivation page 2: 11 Impedance 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.

30 Chapter 2. Motivation page 2: 12 System with Large Ohmic Resistance R 0 =10,000 R 1 =1,000 C 1 = 10.5 F s (15 Hz) M. E. Orazem, T. El Moustafid, C. Deslouis, and B. Tribollet, J. Electrochem. Soc., 143 (1996),

31 Chapter 2. Motivation page 2: 13 Impedance Spectrum Zj, Z r, Z r, Impedance, Z j, Frequency, Hz

32 Chapter 2. Motivation page 2: 14 Experimental Data Repeated Measurements Impedance, Real 3% of Z 1% of Z Imaginary Frequency, Hz

33 Chapter 2. Motivation page 2: 15 Impedance Spectroscopy vs. Step-Change Transients Information sought is the same Increased sensitivity stochastic errors frequency range consistency check Better decoupling of physical phenomena

34 Chapter 2. Motivation page 2: 16

35 Chapter 3. Impedance Measurement page 3: 1 Electrochemical Impedance Spectroscopy Chapter 3. Impedance Measurement Overview of techniques A.C. bridge Lissajous analysis phase-sensitive detection (lock-in amplifier) Fourier analysis Experimental design Mark E. Orazem, All rights reserved.

36 Chapter 3. Impedance Measurement page 3: 2 Measurement 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, J. Ross Macdonald, editor, Impedance Spectroscopy Emphasizing Solid Materials and Analysis, John Wiley and Sons, New York, C. Gabrielli, Use and Applications of Electrochemical Impedance Techniques, Technical Report, Schlumberger, Farnborough, England, 1990.

37 Chapter 3. Impedance Measurement page 3: 3 AC Bridge Z 1 Z 2 Bridge is balanced when current at D is equal to zero D ZZ ZZ Time consuming Accurate f 10 Hz Z 3 Z 4 Generator

38 Chapter 3. Impedance Measurement page 3: 4 Lissajous Analysis Current B Vt () Vsin( t) V It () sin( t) Z A' D' O D A B' Potential V OA A'A Z I OB B'B OD D'D sin( ) OA A'A

39 Chapter 3. Impedance Measurement page 3: 5 General Signal Reference Signal A Phase Sensitive Detection n0 A A0 sin( ta) 4 1 S sin 2n1tS 2n 1 4A 1 S 0 si sin n t A n n n 1t S ASdt 2A 0 cos A S Has maximum value when A S

40 Chapter 3. Impedance Measurement page 3: 6 Vt () Vcos( t) It () Icos( t ) 0 0 I Fourier Analysis: single-frequency input T 1 Ir It ()cos( tdt ) T 0 1 I j It ()sin( tdt ) T T 0 T 0 1 Vr V()cos( t t) dt T T 1 Vj V()sin( t t) dt T 0 Z Z r j Vr jv j ( ) Re Ir ji j Vr jv j ( ) Im Ir ji j

41 Chapter 3. Impedance Measurement page 3: 7 10 Fourier Analysis: multi-frequency input 10 Signal Strength 0 0 Time Z Signal Strength 0 0 Time Z() Fast Fourier Transform

42 Chapter 3. Impedance Measurement page 3: 8 Comparison single-sine sine input multi-sine 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

43 Chapter 3. Impedance Measurement page 3: 9 Measurement 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

44 Chapter 3. Impedance Measurement page 3: 10 Experimental 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

45 Chapter 3. Impedance Measurement page 3: 11 Sinusoidal Perturbation V() t V V cost 0 it () iexp( bv) 0 a dv ic C0 dt i f f V Vt ()

46 Chapter 3. Impedance Measurement page 3: 12 Linearity (I-I 0 )/max(i-i 0 ) 1.0 f = 1 mhz 1 mv 20 mv mv (I-I 0 )/max(i-i 0 ) 1.0 f = 10 Hz 1 mv 20 mv mv (V-V 0 )/V (V-V 0 )/V (I-I 0 )/max(i-i 0 ) 1.0 f = 100 Hz 1 mv 20 mv mv (I-I 0 )/max(i-i 0 ) f = 10 khz 1 mv 20 mv 40 mv (V-V 0 )/V (V-V 0 )/V

47 Chapter 3. Impedance Measurement page 3: 13 Influence of Nonlinearity on Impedance Z j / cm model 1 mv 20 mv 40 mv Z r / cm 2

48 Chapter 3. Impedance Measurement page 3: 14 Influence of Nonlinearity on Impedance Z r / R t model 1 mv 20 mv 40 mv Z j / cm model 1 mv 20 mv mv f / Hz f / Hz

49 Chapter 3. Impedance Measurement page 3: 15 Guideline for Linearity bv (R t -Z(0))/R t V b

50 Chapter 3. Impedance Measurement page 3: 16 Influence of Ohmic Resistance bv R / R e t R e dv ic C0 dt if f V itr () e Vt () s () t

51 Chapter 3. Impedance Measurement page 3: 17 Influence of Ohmic Resistance

52 Chapter 3. Impedance Measurement page 3: 18 Overpotential as a Function of Frequency R e dv ic C0 dt if f V itr () e Vt () s () t

53 Chapter 3. Impedance Measurement page 3: 19 Cell Design Use reference electrode to isolate influence of electrodes and membranes 2-electrode 3-electrode 4-electrode Working Electrode Working Electrode Working Electrode Ref 1 Ref 1 Ref 2 Counterelectrode Counterelectrode Counterelectrode

54 Chapter 3. Impedance Measurement page 3: 20 Current Distribution Counter Electrode Lead Wires Seek uniform current/potential distribution simplify interpretation reduce frequency dispersion Electrolyte Out Electrolyte Out Cover Flange Working Electrode Cell Body ID = 6 in. L = 6 in. Flange Cover Electrolyte In Electrolyte In Working Electrode Seat

55 Chapter 3. Impedance Measurement page 3: 21 Primary Current Distribution

56 Chapter 3. Impedance Measurement page 3: 22 Potentiostatic Modulation Technique standard approach linearity controlled by potential M in io exp V i n Galvanostatic good for nonstationary systems 2 FeF 1 FeF io V V corr V Vco rr corrosion drug delivery requires variable perturbation amplitude to maintain linearity F V 2 corr 1 RT 2 RT 2 RT D I = DV / Z D V = DI Z ( w) ( w)

57 Chapter 3. Impedance Measurement page 3: 23 Experimental Strategies Faraday cage Short leads Good wires Shielded wires I Oscilloscope WE V Ref CE

58 Chapter 3. Impedance Measurement page 3: 24 Reduce 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)

59 Chapter 3. Impedance Measurement page 3: 25 Reduce Non-Stationary 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

60 Chapter 3. Impedance Measurement page 3: 26 Reduce Instrument Bias Errors Faster potentiostat Short shielded leads Faraday cage Check results against electrical circuit against independently obtained parameters

61 Chapter 3. Impedance Measurement page 3: 27 Experimental 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

62 Chapter 3. Impedance Measurement page 3: 28 Can We Perform Impedance on Transient Systems? Timeframes for measurement Individual frequency Individual scan Multiple scans

63 Chapter 3. Impedance Measurement page 3: 29 EHD Experiment -0.2 Imaginary Impedance, A/rpm Data Set #2 Data Set #4 Data Set # Real Impedance, A/rpm

64 Chapter 3. Impedance Measurement page 3: 30 Time per Frequency for 200 rpm Time per Measurement, s Filter On Filter Off Frequency, Hz

65 Chapter 3. Impedance Measurement page 3: 31 Impedance Scans -15 Z j / cm Time Interval s s s Z r / cm 2

66 Chapter 3. Impedance Measurement page 3: 32 Impedance Scans 4000 Elapsed Time / s f / Hz

67 Chapter 3. Impedance Measurement page 3: 33 Time per Frequency Measured 10 2 t f / s cycles 5 cycles 5 seconds f / Hz

68 Chapter 3. Impedance Measurement page 3: 34

69 Chapter 3. Impedance Measurement page 3: 35

70 Chapter 4. Representation of Impedance Data page 4: 1 Electrochemical Impedance Spectroscopy Chapter 4. Representation of Impedance Data Electrical circuit components Methods to plot data standard plots subtract electrolyte resistance Constant phase elements Mark E. Orazem, All rights reserved.

71 Chapter 4. Representation of Impedance Data page 4: 2 Addition of Impedance Addition of Impedance Z 1 Z Y Y Y Z Z Z Z 1 Z Y Y Y Z Z Z

72 Chapter 4. Representation of Impedance Data page 4: 3 Circuit Components -Zj R e Z R e R e Z r R e C dl -Zj Z R e 1 jc dl R e 1 j C dl j 1 Re Z r

73 Chapter 4. Representation of Impedance Data page 4: 4 R e C dl Simple RC Circuit R t 2 -Zj 1 t R C dl Z R e 1 R Re 1 t R f 1 jc R Rt Re 1 1 t jrc t dl dl 2 t dl j 2 2 RC RC t dl R C t dl R Z e r Re Rt Note: f 2 f rad / s or -1 s cycles / s or Hz

74 Chapter 4. Representation of Impedance Data page 4: 5 Impedance C dl Zror -Zj real imaginary R e 200 R f Slope = Hz Frequency / Hz Zj, Hz 1 Hz Zror -Zj real imaginary Slope = -1 Z r / Frequency / Hz

75 Chapter 4. Representation of Impedance Data page 4: 6 Z Bode Representation magnitude phase angle Z Z jz Z Z r j r Z cos Z sin j / degrees Frequency / Hz

76 Chapter 4. Representation of Impedance Data page 4: 7 Input & output are in phase Modified Phase Angle 1 Z j * tan Zr Re Z magnitude modified magnitude phase angle modified phase angle Note: -60 / degrees The modified phase angle yields excellent insight, but we need an accurate estimate for the solution -30 resistance Z j tan Zr Frequency / Hz Input & output are out of phase

77 Chapter 4. Representation of Impedance Data page 4: 8 CPE R e R t Constant Phase Element Z Re 1 R 1 j RQ t t Z j / R t =1 =0.9 =0.8 =0.7 =0.6 = (Z r R e ) / R t

78 Chapter 4. Representation of Impedance Data page 4: 9 Slope = 10 0 CPE: Log Imaginary Slope = 10-1 Z j / R t =1 =0.9 =0.8 =0.7 =0.6 = / Note: With a CPE (1), the asymptotic slopes are no longer 1.

79 Chapter 4. Representation of Impedance Data page 4: 10 CPE: Modified Phase Angle ( ) 90( ) adj -90 * / degrees =1 =0.9 =0.8 =0.7 =0.6 =0.5 0 Note: The 1 10 high-frequency asymptote for the modified phase angle depends / on the CPE coefficient.

80 Chapter 4. Representation of Impedance Data page 4: 11 Example for Synthetic Data Z( f) Rt zd f j f Q R z f Re 1 2 dl t d d tanh j 2 z f z f d f j2 f M. Orazem, B. Tribollet, and N. Pébère, J. Electrochem. Soc., 153 (2006), B129-B136.

81 Chapter 4. Representation of Impedance Data page 4: 12 Traditional Representation Z j cm Hz 100 Hz 1 Hz 0.1 Hz 10 mhz 1 mhz Z r cm 2 Z cm / degree f / Hz f / Hz

82 Chapter 4. Representation of Impedance Data page 4: 13 R e -Corrected Bode Phase Angle adj Z 1 j Zr R e,est ( ) 90( ) adj tan adj / degree f / Hz

83 Chapter 4. Representation of Impedance Data page 4: 14 R e -Corrected Bode Magnitude 2 2 Z Z R Z adj r e,est j 10 3 Slope = Z adj cm slope = f / Hz

84 Chapter 4. Representation of Impedance Data page 4: 15 Slope = Log( Z j ) Z j cm slope= slope= = 1 = 0.7 = 0.5 Slope = f / Hz

85 Chapter 4. Representation of Impedance Data page 4: 16 Effective Capacitance or CPE Coefficient Z CPE CPE 1 Q j2 f Q eff 1 = sin 2 2 f Z ( f ) j Q eff Q dl f / Hz

86 Chapter 4. Representation of Impedance Data page 4: 17 R e -Corrected Bode Plots (Phase Angle) Shows expected high-frequency behavior for surface High-Frequency limit reveals CPE behavior R e -Corrected Bode Plots (Magnitude) High-Frequency slope related to CPE behavior Log Z j Slopes related to CPE behavior Peaks reveal characteristic time constants Effective Capacitance Alternative Plots High-Frequency limit yields capacitance or CPE coefficient

87 Chapter 4. Representation of Impedance Data page 4: 18 Application

88 Chapter 4. Representation of Impedance Data page 4: 19 Mg alloy (AZ91) in 0.1 M NaCl Hz Z j cm khz time / h Hz Hz Z r cm 2

89 Chapter 4. Representation of Impedance Data page 4: 20 k Mg 1 Mg e ads Proposed Model k2 2 1 Mg H2O Mg OH H2 Mg 2 ads 2 Mg Mg e ads 2OH k3 k 3 k4 k 4 Mg(OH) k5 Mg(OH) 2 MgO H2O k negative difference effect (NDE) G. Baril, G. Galicia, C. Deslouis, N. Pébère, B. Tribollet, and V. Vivier, J. Electrochem. Soc., 154 (2007), C108-C113

90 Chapter 4. Representation of Impedance Data page 4: 21 k Mg 1 Physical Interpretation Mg e ads 3 2 Mg Mg e ads k k diffusion of Mg Hz Z j cm khz time / h Hz Hz Z r cm 2 relaxation of (Mg + ) ads. intermediate

91 Chapter 4. Representation of Impedance Data page 4: 22 Log( Z j 10 2 Slope = Z j cm time / h f / Hz

92 Chapter 4. Representation of Impedance Data page 4: 23 Effective CPE Coefficient Q eff 1 = sin 2 2 f Z ( f ) j Q eff / (M -1 cm -2 ) s time / h f / Hz

93 Chapter 4. Representation of Impedance Data page 4: 24 R e -Corrected Phase Angle ( ) 90 1 adj adj tan 10 3 Z j Zr R e,est Z adj cm time / h f / Hz

94 Chapter 4. Representation of Impedance Data page 4: 25 Physical Parameters Hz Z j cm khz time / h Hz Hz Z r cm 2

95 Chapter 4. Representation of Impedance Data page 4: 26 Experimental device for LEIS layer Z local V i applied local

96 Chapter 4. Representation of Impedance Data page 4: 27 Mg alloy (AZ91) in Na 2 SO 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),

97 Chapter 4. Representation of Impedance Data page 4: 28 Local impedance = 1 to 0.92

98 Chapter 4. Representation of Impedance Data page 4: 29 Mg alloy (AZ91) in Na 2 SO M at the corrosion potential after 1 h of immersion. Electrode radius 5500 µm The local impedance has a pure RC behavior. Local impedance The CPE is explained by a 2d distribution of the resistance as shown in the figure.

99 Chapter 4. Representation of Impedance Data page 4: 30 Comparison to Theory

100 Chapter 4. Representation of Impedance Data page 4: 31 Graphical Representation of Impedance Data Expanded Range of Plot Types Facilitate model development Identify features without complete system model Suggested Plots R e -Corrected Bode Plots (Phase Angle) Shows expected high-frequency behavior for surface High-Frequency limit reveals CPE behavior R e -Corrected Bode Plots (Magnitude) High-Frequency slope related to CPE behavior Log Z j Slopes related to CPE behavior Peaks reveal characteristic time constants Effective Capacitance High-Frequency limit yields capacitance or CPE coefficient

101 Chapter 4. Representation of Impedance Data page 4: 32 Choice 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

102 Chapter 4. Representation of Impedance Data page 4: 33

103 Chapter 5. Development of Process Models page 5: 1 Electrochemical Impedance Spectroscopy Chapter 5. Development of Process Models Use of Circuits to guide development Develop models from physical grounds Model case study Identify correspondence between physical models and electrical circuit analogues Account for mass transfer Mark E. Orazem, All rights reserved.

104 Chapter 5. Development of Process Models page 5: 2 Use circuits to create framework

105 Chapter 5. Development of Process Models page 5: 3 Addition of Potential

106 Chapter 5. Development of Process Models page 5: 4 Addition of Current

107 Chapter 5. Development of Process Models page 5: 5 Equivalent Circuit at the Corrosion Potential

108 Chapter 5. Development of Process Models page 5: 6 Equivalent Circuit for a Partially Blocked Electrode

109 Chapter 5. Development of Process Models page 5: 7 Equivalent Circuit for an Electrode Coated by a Porous Layer

110 Chapter 5. Development of Process Models page 5: 8 Equivalent Electrical Circuit for an Electrode Coated by Two Porous Layers C R / ; F/cm 2 2 2

111 Chapter 5. Development of Process Models page 5: 9 Use kinetic models to determine expressions for the interfacial impedance

112 Chapter 5. Development of Process Models page 5: 10 Approach 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

113 Chapter 5. Development of Process Models page 5: 11 General Expression for Faradaic Current i i f V, c, f i,0 i i Re ~ i e j t f f i f V c V c, i,0,, ci,0 k i Vcj, ji f k k Vc, j k j, jk i,0 k

114 Chapter 5. Development of Process Models page 5: 12 Reactions Considered Dependent on Potential Dependent on Potential and Mass Transfer Dependent on Potential, Mass Transfer, and Surface Coverage Coupled Reactions

115 Chapter 5. Development of Process Models page 5: 13 Irreversible Reaction: Dependent on Potential z+ A A z ne Potential-dependent heterogeneous reaction Two-dimensional surface No effect of mass transfer

116 Chapter 5. Development of Process Models page 5: 14 steady-state Current Density exp AF ia nafka V RT i K b V A A exp A oscillating component i K b exp b V V A A A A i A V R t,a R t,a Kb 1 exp bv A A A

117 Chapter 5. Development of Process Models page 5: 15 Charging Current high frequency i if Cdl dt f dv i i jc V dl low frequency V i R V t,a 1 R t,a jc V dl jc V Rt,A i 1 j R C t,a dl dl

118 Chapter 5. Development of Process Models page 5: 16 Ohmic Contributions U ~ U ire V ~ ~ i R V e Z A U V Re i i Rt,A Re 1+ j R C t,a dl

119 Chapter 5. Development of Process Models page 5: 17 Steady Currents in Terms of R t,a i exp A KA bav 2.303/ b A A i K exp 2.303V A A A R t,a 1 A Kbexp bv 2.303i A A A A i A A A 2.303R t,a 2.303R i t,a A

120 Chapter 5. Development of Process Models page 5: 18 Irreversible Reaction: Dependent on Potential and Mass Transfer O ne R Irreversible potential-dependent heterogeneous reaction Reaction on two-dimensional surface Influence of transport of O to surface

121 Chapter 5. Development of Process Models page 5: 19 steady-state Current Density i K c exp b V O O O,0 O oscillating component exp i K bc exp bv VK bv c O O O O,0 O O O O,0 V K expb Vc R t,o O O O,0 R t,o K c 1 exp b V O O,0 O

122 Chapter 5. Development of Process Models page 5: 20 Mass Transfer O io nofdo dy 0 j t io io Re i Oe dc dc O io nofdo dy 0 c O,0 io nofdo O 0

123 Chapter 5. Development of Process Models page 5: 21 Combine Expressions V i K exp b V c c O O O O,0 Rt,O O,0 i nfd O O O O 0

124 Chapter 5. Development of Process Models page 5: 22 i O R R t,o Current Density V O nfdc V z t,o d,o O O O,0 O 1 1 b (0) z d,o O nfdc O O O,0 O 1 1 b (0) R t,o K c 1 exp b V O O,0 O

125 Chapter 5. Development of Process Models page 5: 23 Calculate Impedance dv i if Cdl dt i R V z t,o d,o jc V dl i i jc V O dl V R 1 z t,o d,o jc dl U ~ U ire V ~ ~ i R V e Z O U V Re i i Rt,O zd,o Re 1+ jc R z dl t,o d,o

126 Chapter 5. Development of Process Models page 5: 24 Comparison to Circuit Analog Z O R e R t,o d,o 1+ jc R z z dl t,o d,o C dl R e z d,o R t,o

127 Chapter 5. Development of Process Models page 5: 25 Irreversible Reaction: Dependent on Potential and Adsorbed Intermediate B X k1 k2 X+e P+e - - B X P X X X Potential-dependent heterogeneous reactions Adsorption of intermediate on two-dimensional surface Maximum surface coverage

128 Chapter 5. Development of Process Models page 5: 26 Steady-State State Current Density reaction 1: formation of X reaction 2: formation of P total current density 1 exp i K b V V i K exp b V V i i i 1 2

129 Chapter 5. Development of Process Models page 5: 27 Steady-State State Surface Coverage balance on d i1 i2 dt F F 0 steady-state value for K 1exp b1 V V1 exp K exp b V V K b V V

130 Chapter 5. Development of Process Models page 5: 28 Steady-State State Current Density i K 1 exp b V V K exp b V V where K 1exp b1 V V1 exp K exp b V V K b V V

131 Chapter 5. Development of Process Models page 5: 29 Oscillating Current Density i K 1 exp b V V K exp b V V i V K exp b V V K b V V Rt,1 R t,2 exp exp R K b b V V t, R K b exp b V V t,

132 Chapter 5. Development of Process Models page 5: 30 balance on Need Additional Equation j V K exp b V V K b V V F Rt,1 R t,2 exp Rt,1 Rt,2 V F j K exp b V V K b V V exp

133 Chapter 5. Development of Process Models page 5: 31 Impedance exp K2exp b2 V V2 K1 b1 V V 1 Rt,1 R t,2 Z R Fj F K exp b V V K b V V t 1 A R j B t exp where R R R t t, M t, X

134 Chapter 5. Development of Process Models page 5: 32

135 Chapter 5. Development of Process Models page 5: 33

136 Chapter 5. Development of Process Models page 5: 34 i f Corrosion of Steel i Fe i i H 2 O 2 Potential, V (Cu/CuSO4) H 2 O + 2e - H 2 +2OH - Fe Fe e - O 2 +2H 2 O + 4e - 4OH Current Density, ma/ft 2

137 Chapter 5. Development of Process Models page 5: 35 Corrosion: Fe Fe e - i K exp b V Fe Fe Fe i K b exp b V V Fe Fe Fe Fe ~ i Fe ~ V R t,fe R t,fe K b 1 exp b V Fe Fe Fe

138 Chapter 5. Development of Process Models page 5: 36 Steady Currents in Terms of R t i K exp b V Fe Fe Fe 2.303/ b Fe Fe i K exp 2.303V Fe Fe Fe R t,fe 1 Fe K b exp b V 2.303i Fe Fe Fe Fe i Fe Fe 2.303R t,fe Important: Note the relationship among steady-state current density, Tafel slope, and charge transfer resistance.

139 Chapter 5. Development of Process Models page 5: 37 H 2 Evolution: 2H 2 O + 2e - H 2 +2OH - i K exp b V H H H i K b exp b V V H H H H ~ i H 2 ~ V R t,h 2 R t,h 2 1 K b exp b V H H H 2 2 2

140 Chapter 5. Development of Process Models page 5: 38 O 2 Reduction: O 2 +2H 2 O + 4e - 4OH - i K c exp b V O O O,0 O i K b c exp b V V O O O O,0 O K exp O O O,0 b V c 2 2 2

141 Chapter 5. Development of Process Models page 5: 39 O 2 Evolution: O 2 +2H 2 O + 4e - 4OH - mass transfer: in terms of dimensionless gradient at electrode surface dc O 2 io n 2 O FD 2 O2 dy c O,0 2 no FD 2 O 2 O 2 0 0

142 Chapter 5. Development of Process Models page 5: 40 O 2 Evolution: O 2 +2H 2 O + 4e - 4OH - i O 2 R R z d,o 2 coupled expression t,o 2 V O no FD 2 O c 2 O 2,0 b O (0) 2 V z t,o d,o 2 2 O no FDO co,0 b O (0) R t,o 2 1 K b c exp b V O O O,0 O

143 Chapter 5. Development of Process Models page 5: 41 Capacitance and Ohmic Contributions Capacitance and Ohmic Contributions f f f f i i i i ~ ~ Faradaic C V j i i dt dv C i i d f d f ~ ~ ~ Faradaic and Charging V R i U V ir U e e ~ ~ ~ Ohmic

144 Chapter 5. Development of Process Models page 5: 42 Z ~ U ~ i 1 R 1 R eff t, Fe Z r R 1 R t,o jz t, H 2 j z R d,o 2 1 Process Model t,o 2 1 z jc d,o d 2 jc d C dl R t,fe R t,h2 Z d,o R R R eff t, Fe t, H 2 R t,o2

145 Chapter 5. Development of Process Models page 5: 43 Development 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

146 Chapter 5. Development of Process Models page 5: 44 Mass Transfer

147 Chapter 5. Development of Process Models page 5: 45 Film Diffusion c t i c z 2 i Di 2 steady state c c as z i i, f c c at z 0 i i,0 z c c c c i i,0 i, i,0 f

148 Chapter 5. Development of Process Models page 5: 46 Film Diffusion c t c i i c c z 2 i Di 2 i Re c~ e i j t dc dz 2 2 jt i i jt jce Di D 2 i e 2 2 jt dci jt jce Di e 2 jc dz 2 dci Di dz 2 dc dz z K i i D i i i 2 f c i c (0) 2 d jk d i 0 2 i i

149 Chapter 5. Development of Process Models page 5: 47 Warburg Impedance d 2 d i jk 0 2 ii exp Aexp jk B jk i i i i 0 at 1 1 at 0 i 1 (0) i 1 (0) i tanh jk 2 j tanh Di 2 j D i jk i i i 0 at 1 at 0 i 1 1 (0) jk i 1 1 (0) 2 i j D i i

150 Chapter 5. Development of Process Models page 5: Concentration c()/c() t=0, 2n/K t=n/k t=0.5n/k t=1.5n/k K=1-30 Diffusion Impedance Only (infinite) Diffusion Impedance Only (finite) Complete Impedance Zj, Zj,D, Z r, Z r,d, c()/c() t=0, 2n/K t=0.5n/k t=1.5n/k t=n/k K=

151 Chapter 5. Development of Process Models page 5: 49 Rotating Disk

152 Chapter 5. Development of Process Models page 5: 50 Convective Diffusion Convective Diffusion z c r rc r r D z c v r c v t c i i i i z i r i / 2 z b z z a v z / 2 2 1/ z b z z a r v r / 2 1/ z a z b r v z

153 Chapter 5. Development of Process Models page 5: 51 Convective Diffusion in one Convective Diffusion in one-dimension Dimension 2 2 z c D z c v t c i i i z i ne M s i z i i i z c c i i as, at 0 z nf s i z c D f i i i f c i f i,

154 Chapter 5. Development of Process Models page 5: j t j t j t i i i i z i z i dc d c v D dz d dc d c j ce v e D e dz d z z Sinusoidal Steady State Sinusoidal Steady State t j i i i e c c c ~ Re 1/3 i Sc 9 9 a D a K i i i z 1/3 i Sc a a D i i 0 ~ ~ ~ 2 2 i i i z i c jk d dc v d c d

155 Chapter 5. Development of Process Models page 5: 53 Finite Length Warburg Impedance v z ci z 0 z d z d D D 13 / 23 / i j ( ) tanh 0 j i Sc 2 13 / 1 (0) i tanh 2 j D 2 j D i i

156 Chapter 5. Development of Process Models page 5: 54 Impedance Impedance 0 ~ ~ ~ 2 2 i i i z i jk d d v d d 0, ~ ~ ~ i i i c c 0 at 1 ~ as 0 ~ i i 0 ~,, 0, i i i i c i i t D nfd s c f R Z i j j 0 ~ (0) i D D Z Z

157 Chapter 5. Development of Process Models page 5: 55 One Term v z z Numerical Solutions 2 a zd zd( 0) 0 psc 1/3 ;. a Infinite Sc p 1 Two Terms v z a Finite Sc z 1 z ( 0) d d 3 1/3 psc 0 Z 1 psc Sc 1/3 1/3 Three Terms v b z a b z d 1/3 1/3 Sc Sc 1 Z1 p Z2 p zd( 0) 1/3 1/3 psc Sc Sc 0 2/3

158 Chapter 5. Development of Process Models page 5: 56 Coupled Diffusion Impedance z d z D i, outer i, inner d, outer d, inner Di, inner i, outer zd, inner zd, outer j D 2 inner z D D i, outer i, inner iinner, iinner, iouter,

159 Chapter 5. Development of Process Models page 5: 57 Interpretation Models for Impedance Spectroscopy 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.

160 Chapter 5. Development of Process Models page 5: 58

161 Chapter 6. Time-Constant Dispersion page 6: 1 Basic Impedance Spectroscopy Chapter 6. Time Constant Dispersion CPEs can arise from surface or axial distributions CPE parameters can be interpreted in terms of capacitance, depending on type of distribution Time-constant dispersions can be modeled explicitly. Mark E. Orazem, All rights reserved.

162 Chapter 6. Time-Constant Dispersion page 6: 2 Page deliberately left blank.

163 Chapter 6. Time-Constant Dispersion page 6: 3 Page deliberately left blank.

164 Chapter 6. Time-Constant Dispersion page 6: 4 Types of Distributions Surface Axial

165 Chapter 6. Time-Constant Dispersion page 6: 5 Surface Distributions

166 Chapter 6. Time-Constant Dispersion page 6: 6 Axial Distributions

167 Chapter 6. Time-Constant Dispersion page 6: 7 Relationship 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), C. H. Hsu, F. Mansfeld, Corrosion, 57 (2001), 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.

168 Chapter 6. Time-Constant Dispersion page 6: 8 Determination of Film Thickness

169 Chapter 6. Time-Constant Dispersion page 6: 9 Constant 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

170 Chapter 6. Time-Constant Dispersion page 6: 10 Porous Electrodes

171 Chapter 6. Time-Constant Dispersion page 6: 11 delevie Model Under the assumption that Z 0 and R 0 are independent of x

172 Chapter 6. Time-Constant Dispersion page 6: 12 Limiting Behavior 2 / rz eq coth 2 / rzeq description Parameters obtainable from regression large 1 Semi-infinite pores 3/2 small 2 / rz 1 eq Planar surface r n 2 nr intermediate intermediate Shallow pores Two combinations of r, n, and ; i.e., 3/2 r n and / r

173 Chapter 6. Time-Constant Dispersion page 6: 13 No Free Chlorine Consumption of Free Chlorine in Municipal Water Supplies Cast Iron Pipes 2 mg/l Free Chlorine 2+ - Fe Fe +2e Fe +HOCl+H 2Fe +Cl +H2O 1 O+2H + +2e - 2 HO 2 2 I. Frateur, C. Deslouis, M.E. Orazem and B. Tribollet, Electrochimica Acta, 44 (1999), 4345.

174 Chapter 6. Time-Constant Dispersion page 6: 14 Issues Evian Water Coupled electrochemical reactions Surface films Convective diffusion History and time-dependent parameters Identification of corrosion rate Red Rust (-FeOOH, -Fe 2 O 3 ) Green Rust & Carbonates (Fe 2+ and Fe 3+ ) Black Rust (Fe 3 O 4 ) Cast Iron

175 Chapter 6. Time-Constant Dispersion page 6: 15 No Free Chlorine Model Development 2 mg/l Free Chlorine

176 Chapter 6. Time-Constant Dispersion page 6: 16 Model for Impedance Response C dl a Z c (a) Z = R e (b) Z a = Z a R t a C f C dl c (c) Z 0 = R f R t c Z D Z c L R0Z0 coth Z R 0 0

177 Chapter 6. Time-Constant Dispersion page 6: 17 Impedance Data 3 days 7 days 28 days

178 Chapter 6. Time-Constant Dispersion page 6: 18 Time-Constant 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.

179 Chapter 6. Time-Constant Dispersion page 6: 19

180 Chapter 7. Regression Analysis page 7: 1 Electrochemical Impedance Spectroscopy Chapter 7. Regression Analysis Regression response surfaces noise incomplete frequency range Adequacy of fit quantitative qualitative Mark E. Orazem, All rights reserved.

181 Chapter 7. Regression Analysis page 7: 2 Test Circuit 1: 1 Time Constant R 0 = 0 C 1 R 1 = 1 cm 2 1 = RC = 1 s R 0 (1) R 1

182 Chapter 7. Regression Analysis page 7: 3 Linear optimization surface roughly parabolic f ( p) 2 Ndat Z Ndat r,dat Zr,mod Z j,dat Z j,mod 2 2 i1 r k1 j Sum of Squares / s / s R / cm R / cm 2

183 Chapter 7. Regression Analysis page 7: 4 Nonlinear Regression p p p 2 f 1 f f( p ) f( p ) p p p 0 N N N j j k j1pj 2 j1 k1p p jpk p N ( Zi Z( p)) dat 2 i1 i 2 i β αp Variance of data k N dat i1 Z Z( p) Z( p) i i i 2 i pk 1 Z( p) Z( p) Ndat i i jk, 2 i1 i pj pk Derivative of function with respect to parameter

184 Chapter 7. Regression Analysis page 7: 5 Methods for Regression Evaluation of derivatives method of steepest descent Gauss-Newton method Levenberg-Marquardt method Evaluation of function simplex

185 Chapter 7. Regression Analysis page 7: 6 Effect of Noisy Data Add noise: 1% of modulus Sum of Squares R / cm / s / s R / cm response surface remains parabolic value at minimum increases

186 Chapter 7. Regression Analysis page 7: 7 Test Circuit 2: 3 Time Constants R 0 = 1 cm 2 R 1 = 100 cm 2 C 1 C 2 C 3 1 = s R 2 = 200 cm 2 (1) (2) (3) 2 = 0.01 s R 0 R 3 = 5 cm 2 R 1 R 2 R 3 3 = 0.05 s Note: 3 rd Voigt element contributes only 1.66% to DC cell impedance.

187 Chapter 7. Regression Analysis page 7: 8 Effect of Noisy Data no noise noise: 1% of modulus Sum of Squares Sum of Squares log 1 10 (R/ cm 2 ) -1 log 10 ( / s) log (R/ cm 2 ) log ( / s) Note: use of log scale for parameters All parameters fixed except R 3 and 3

188 Chapter 7. Regression Analysis page 7: 9 Resulting Spectrum -150 Z j / cm Z r / cm 2 Model with 1% noise added Model with no noise

189 Chapter 7. Regression Analysis page 7: 10 Test Circuit 3: 3 Time Constants R 0 = 1 cm 2 R 1 = 100 cm 2 1 = 0.01 s R 2 = 200 cm 2 2 = 0.1 s R 0 C 1 (1) C 2 (2) C 3 (3) R 3 = 100 cm 2 3 = 10 s R 1 R 2 R 3 Note: 3 rd Voigt element contributes 25% to DC cell impedance. The time constant corresponds to a characteristic frequency 3 =0.1 s -1 or f 3 =0.016 Hz.

190 Chapter 7. Regression Analysis page 7: 11 Resulting Test Spectra Z j / cm Z r / cm Hz to 100 khz 1 Hz to 100 khz

191 Chapter 7. Regression Analysis page 7: 12 Effect of Truncated Data 0.01 Hz to 100 khz 1 Hz to 100 khz Sum of Squares log 10 (R / cm 2 ) log 10 ( / s) Sum of Squares log 10 (R / cm 2 ) log 10 ( / s) All parameters fixed except R 3 and 3

192 Chapter 7. Regression Analysis page 7: 13 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.

193 Chapter 7. Regression Analysis page 7: 14 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

194 Chapter 7. Regression Analysis page 7: 15 Test Case: Mass Transfer to a RDE Z ( ) Rt zd R e 1 j CR z t d Single reaction coupled with mass transfer. Consider model for a Nernst stagnant diffusion layer: z d z tanh d j j

195 Chapter 7. Regression Analysis page 7: 16 Evaluation of 2 Statistic / Z() /

196 Chapter 7. Regression Analysis page 7: 17 Comparison of Model to Data 80 Impedance Plane (Nyquist( Nyquist) Z j / Z r / Value of 2 has no meaning without accurate assessment of the noise level of the data

197 Chapter 7. Regression Analysis page 7: 18 Modulus 100 Z / f / Hz

198 Chapter 7. Regression Analysis page 7: Phase Angle / degrees f / Hz

199 Chapter 7. Regression Analysis page 7: Real 160 Z r / f / Hz

200 Chapter 7. Regression Analysis page 7: Imaginary 60 Z j / f / Hz

201 Chapter 7. Regression Analysis page 7: Log Imaginary Slope = -1 for RC Z j / f / Hz

202 Chapter 7. Regression Analysis page 7: Modified Phase Angle / degrees Z j * tan Zr Re f / Hz

203 Chapter 7. Regression Analysis page 7: Real Residuals r / Z r noise level f / Hz

204 Chapter 7. Regression Analysis page 7: Imaginary Residuals j / Z j noise level f / Hz

205 Chapter 7. Regression Analysis page 7: 26 Plot 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

206 Chapter 7. Regression Analysis page 7: 27 Test Case: Better Model for Mass Transfer to a RDE Z ( ) R t zd Re 1 j QR z t d Consider 3-term expansion with CPE to account for more complicated reaction kinetics: z d Sc 1/3 Sc 1/3 1 Z1 p Z2 p zd(0) 1/3 Sc Sc 0 psc 2 /=4.86 1/3 2/3

207 Chapter 7. Regression Analysis page 7: 28 Comparison of Model to Data 80 Impedance Plane (Nyquist( Nyquist) Z j / Z r /

208 Chapter 7. Regression Analysis page 7: Phase Angle -30 Z j / f / Hz

209 Chapter 7. Regression Analysis page 7: Modified Phase Angle * / degrees Z j * tan Zr Re f / Hz

210 Chapter 7. Regression Analysis page 7: 31 Log Imaginary 100 Z j / f / Hz

211 Chapter 7. Regression Analysis page 7: Real Residuals r / Z r f / Hz

212 Chapter 7. Regression Analysis page 7: Imaginary Residuals 0.01 j / Z j f / Hz

213 Chapter 7. Regression Analysis page 7: 34 Confidence 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

214 Chapter 7. Regression Analysis page 7: 35 Regression 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.

215 Chapter 7. Regression Analysis page 7: 36

216 Chapter 8. Error Structure page 8: 1 Electrochemical Impedance Spectroscopy Chapter 8. Error Structure Contributions to error structure Weighting strategies General approach for error analysis Experimental results Mark E. Orazem, All rights reserved.

217 Chapter 8. Error Structure page 8: 2 Contributions to Error Structure Sampling Errors Stochastic Phenomena Bias Errors Lack of Fit Changing baseline (non-stationary processes) Instrumental artifacts

218 Chapter 8. Error Structure page 8: 3 Time-domain Frequency Domain 1 mhz 10 Hz 100 Hz 10 khz

219 Chapter 8. Error Structure page 8: 4 Error Structure ( ) Z ( ) Z ( ) resid obs model fitting error ( ) ( ) ( ) fit bias stochastic inadequate model noise (frequency domain) experimental errors: inconsistent with the Kramers-Kronig relations

220 Chapter 8. Error Structure page 8: 5 J No Weighting Weighting Strategies k Strategy Modulus Weighting Proportional Weighting Experimental Refined Experimental ( Z Z Z Z rk 2 rk) ( jk jk),,,, 2 2 rk, k jk, 2 Implications r = j r = j r j r r r j j j r = j j r + 2 /R m r = j j r -R sol + 2 /R m +

221 Chapter 8. Error Structure page 8: 6 Assumed Error Structure often Wrong Repeated Measurements Real Impedance, Imaginary 3% of Z 1% of Z Frequency, Hz Data obtained by T. El Moustafid, CNRS, Paris, France

222 Chapter 8. Error Structure page 8: 7 Reduction of Fe(CN) 3-6 on a Pt Disk, 120 rpm -300 I/i lim Zj, /4 1/2 3/ Z r,

223 Chapter 8. Error Structure page 8: 8 Reduction of Fe(CN) 3-6 on a Pt Disk, Imaginary 120 rpm, 1/4 i lim Zj, Frequency, Hz

224 Chapter 8. Error Structure page 8: 9 Reduction of Fe(CN) 3-6 on a Pt Disk, 1000 Real 120 rpm, 1/4 i lim 100 Zr, Frequency, Hz

225 Chapter 8. Error Structure page 8: 10 Reduction of Fe(CN) 3-6 on a Pt Disk, 10 Error 120 rpm, 1/4 i lim Standard Deviation, Frequency, Hz

226 Chapter 8. Error Structure page 8: 11 Reduction of Fe(CN) 3-6 on a Pt Disk, 100 Error 120 rpm, 1/4 i lim / Z, Percent Frequency, Hz

227 Chapter 8. Error Structure page 8: 12 Interpretation 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

228 Chapter 8. Error Structure page 8: 13

229 Chapter 9. Kramers-Kronig Relations page 9: 1 Electrochemical Impedance Spectroscopy Chapter 9. Kramers-Kronig Kronig Relations Mathematical form and interpretation Application to noisy data Methods to evaluate consistency Mark E. Orazem, All rights reserved.

230 Chapter 9. Kramers-Kronig Relations page 9: 2 Contraints Contraints Under the assumption that the system is Causal Linear Stable A complex variable Z must satisfy equations of the form: dx x Z x Z Z r r j ) ( ) ( 2 ) ( dx x Z x xz Z Z j j r r ) ( ) ( 2 ) ( ) (

231 Chapter 9. Kramers-Kronig Relations page 9: 3 Use of Kramers-Kronig Kronig 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

232 Chapter 9. Kramers-Kronig Relations page 9: 4 For real data (with noise) For real data (with noise) )) ( ) ( ( )) ( ) ( ( ) ( ) ( ) ob ( j j r r Z j Z Z Z ) ( ) ( j r j ) ( ) ( ob Z Z E 0 ) ( E where If and only if ) ( ) ( ) ( ) ( 2 ) ( dx x x Z x Z E Z E r r r r j ) ( ) ( ) ( ) ( 2 ) ( ) ( dx x x x Z x xz E Z Z E j j j j r r Kramers-Kronig in an expectation sense

233 Chapter 9. Kramers-Kronig Relations page 9: 5 The Kramers-Kronig Kronig relations can be satisfied if E ( ) 0 This means and 2 E 0 r ( x) 2 x dx 2 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

234 Chapter 9. Kramers-Kronig Relations page 9: 6 2 r ( x) Meaning of E dx x 0 0 (Z r (x)-z r ()) / (x 2-2 ) (Z r (x)-z r ()) / (x 2-2 ) Correct Value Interpolated Value f / Hz f / Hz

235 Chapter 9. Kramers-Kronig Relations page 9: 7 Use of Kramers-Kronig Kronig Relations Quadrature errors require interpolation function Missing data at low and high frequency

236 Chapter 9. Kramers-Kronig Relations page 9: 8 Methods to Resolve Problems of Insufficient Frequency Range Direct Integration Extrapolation single RC polynomials 1/ and asymptotic behavior simultaneous solution for missing data Regression proposed process model generalized measurement model

237 Chapter 9. Kramers-Kronig Relations page 9: 9

238 Chapter 10. Application to PEM Fuel Cells page 10: 1 Electrochemical Impedance Spectroscopy Chapter 10. Application to PEM Fuel Cells Error analysis for EIS data Model development in terms of proposed reaction mechanisms Integration with Independent measurements Mark E. Orazem, All rights reserved.

239 Chapter 10. Application to PEM Fuel Cells page 10: 2 Issues for PEM Fuel Cell Impedance Instrument artifacts Nonstationary behavior Nonuniform surfaces Complicated mass transfer geometry Side reactions

240 Chapter 10. Application to PEM Fuel Cells page 10: 3 Sample Impedance Data Pure Inductance Mass Transfer Capacitive Loop Inductive Loop R. Makharia, M. F. Mathias, and D. R. Baker, J. Electrochem. Soc.,152 (2005), A970.

241 Chapter 10. Application to PEM Fuel Cells page 10: 4 Approach for Impedance Analysis Measurement Model Assess stochastic error structure of data Instrument artifacts Nonstationary behavior Process Model Based on proposed reactions

242 Chapter 10. Application to PEM Fuel Cells page 10: 5 Experimental Results Operating Conditions T : 40 C, P : 1 atm. Flow Rate : 0.1 l/m for H 2, 0.5 l/m for O 2 Steady State Measurement Current Range : A/cm 2 Increment : 10 ma/ 30 Sec. EIS Measurement Galvanostatic Mode Frequency : ,000 Hz Amplitude : 10 ma I= 0.5 Amps I= 1.0 Amps I= 1.5 Amps Potential / V Z j / cm Current Density / A cm -2 Z r / cm 2 Polarization Curve Nyquist Plot

243 Chapter 10. Application to PEM Fuel Cells page 10: 6 Replicated Impedance Scans

244 Chapter 10. Application to PEM Fuel Cells page 10: 7 Stochastic Error Structure γ Z σ r = σ j = R m 2 γ = e-001 R = 100 m Z j / cm h 2.73 h 4.96 h 7.19 h 9.42 h 3 KHz Hz r & j / cm Real part Imaginary part Error structure Z r / cm 2 Five Scans of impedance measurements at 0.2 A/cm f / Hz Error Structure of Impedance Data

245 Chapter 10. Application to PEM Fuel Cells page 10: 8 Bias Errors at High Frequency

246 Chapter 10. Application to PEM Fuel Cells page 10: 9 High-frequency artifacts extend to negative imaginary values

247 Chapter 10. Application to PEM Fuel Cells page 10: 10 Bias Error at Low Frequency First Scan of Impedance Measurement

248 Chapter 10. Application to PEM Fuel Cells page 10: 11 Normalized Residual Errors First Scan of Impedance Measurement

249 Chapter 10. Application to PEM Fuel Cells page 10: 12 Bias Errors at Low Frequency Second Scan of Impedance Measurement

250 Chapter 10. Application to PEM Fuel Cells page 10: 13 Normalized Residual Errors Second Scan of Impedance Measurement

251 Chapter 10. Application to PEM Fuel Cells page 10: 14 Impedance Data after Measurement Model 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.

252 Chapter 10. Application to PEM Fuel Cells page 10: 15 Impedance Process Model Development

253 Chapter 10. Application to PEM Fuel Cells page 10: 16 Proposed 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

254 Chapter 10. Application to PEM Fuel Cells page 10: 17 Model Assumptions i i Cathode Reaction Kinetics O 2 Gas Diffusion Layer C A T H O D E PEM H H A N O D E Gas Diffusion Layer Anode Reaction Kinetics H2 Assumptions: Uniform membrane properties Uniform surface overpotential No convection Diffusion through stagnant film of finite thickness Uniform surface and distribution of reactants and products

255 Chapter 10. Application to PEM Fuel Cells page 10: 18 Steps in Model Development Reaction Mechanisms Steady-State Current Expressions Polarization Curve Potential / V Apply Sinusoidal Perturbation Current / A Faradaic Impedance Z j / cm Double Layer Capacitance Electrolyte Resistance Overall Impedance Z r / cm 2

256 Chapter 10. Application to PEM Fuel Cells page 10: 19 Model Development: Case 1 Oxygen Reduction O + 4H + 4e 2H O Hydrogen Oxidation H2 2H + 2e + -

257 Chapter 10. Application to PEM Fuel Cells page 10: 20 Model 1: Simple Reaction Kinetics

258 Chapter 10. Application to PEM Fuel Cells page 10: 21 Model 1: Simple Reaction Kinetics

259 Chapter 10. Application to PEM Fuel Cells page 10: 22 Steady-State State Current Density: 1 i K C (0)exp b H H H H H i K C (0)exp b O O O O O i H i O 2 2

260 Chapter 10. Application to PEM Fuel Cells page 10: 23 Model Predictions: Case Potential / V Experimental data Model 1 Z j / cm Experimental data Model Current / A Z r / cm 2

261 Chapter 10. Application to PEM Fuel Cells page 10: 24 Model Development: Case 2 Oxygen Reduction O + 2H + 2e Hydrogen Oxidation H H O + 2H + 2e 2H + 2e + - H O 2H O C. F. Zinola, W. E. Triaca and A. J. Arvia, J. Appl. Electrochem. 25 (1995) 740.

262 Chapter 10. Application to PEM Fuel Cells page 10: 25 Model 2: Peroxide Intermediate 2 2 0,a 0,c a e c,1 c,2 2 2, 2

263 Chapter 10. Application to PEM Fuel Cells page 10: 26 Steady-State State Current Density: 2 i K C (0) 1- exp -b O O O H O O O i K C (0) exp b HO HO HO HO HO HO i K C (0)exp b H H H H H i i i H O H O

264 Chapter 10. Application to PEM Fuel Cells page 10: 27 Model 2 Response Compared with Data 0.10 Z j / cm Experimental data Model Z r / cm 2 Potential / V Experimental data Model Current / A

265 Chapter 10. Application to PEM Fuel Cells page 10: 28 Model 2 Response Compared with Data 0.10 Z j / cm Experimental data Model Z r / cm 2 Potential / V Experimental data Model Current / A

266 Model 2 Response Compared with Data Chapter 10. Application to PEM Fuel Cells page 10: Z j / cm Experimental data Model Z r / cm 2 Potential / V Experimental data Model Current / A

267 Chapter 10. Application to PEM Fuel Cells page 10: 30 Model Development: Case 3 Oxygen Reduction O+4H+4e Pt +H O Hydrogen Oxidation 2 2HO + - 2H +2e +PtO + 2+ PtO+2H Pt +H2O H 2H +2e R. M. Darling and J. P. Meyers, J. Electrochem. Soc., 150 (2003) A1523.

268 Chapter 10. Application to PEM Fuel Cells page 10: 31 Model 3: Pt dissolution and formation of 2 PtO 2 0,a 0,c a e c,1 c,2 2 2, 2

269 Chapter 10. Application to PEM Fuel Cells page 10: 32 Steady-State State Current Density: 3 i k C (0)exp b O,Pt eff O O O i 1- k exp b k exp b Pt PtO Pt,f Pt Pt Pt,b PtO Pt Pt r k C 2 (0) PtO PtO PtO + H k k k - k eff Pt PtO Pt PtO i k C (0)exp b H H H H H

270 Chapter 10. Application to PEM Fuel Cells page 10: 33 Model 3 Response Compared with Data 0.10 Z j / cm Experimental data Model Z r / cm 2 Potential / V Experimental data Model Current / A

271 Chapter 10. Application to PEM Fuel Cells page 10: 34 Model 3 Response Compared with Data 0.10 Z j / cm Experimental data Model Z r / cm 2 Potential / V Experimental data Model Current / A

272 Chapter 10. Application to PEM Fuel Cells page 10: 35 Model 3 Response Compared with Data Z j / cm Experimental data Model Z r / cm 2 Potential / V Experimental data Model Current / A

273 Chapter 10. Application to PEM Fuel Cells page 10: 36 Model development suggests supporting experiments Formation of peroxide Signs of membrane degradation Formation of PtO Reduction in electrochemically active area Dissolved Pt in outflow

274 Chapter 10. Application to PEM Fuel Cells page 10: 37 Evidence for PtO x Helena and Jason Weaver, University of Florida

275 Chapter 10. Application to PEM Fuel Cells page 10: 38 Integrated Approach

276 Chapter 10. Application to PEM Fuel Cells page 10: 39 Flooding and EIS response EIS response as a function of current density collected at 40 C. Z j / cm khz A cm Hz Z r / cm 2 Effect of flooding is visible

277 Chapter 10. Application to PEM Fuel Cells page 10: EIS Response at 70 C 25 Hz Effect of flooding is visible Z j / cm khz 25 Hz 25 Hz Hz 25 Hz A cm Hz Z r / cm 2

278 Chapter 10. Application to PEM Fuel Cells page 10: 41 Standard deviation of the Impedance 0.10 Hz 0.10 Hz r, obs / cm Hz 100 Hz j, obs / cm Hz 100 Hz Current Density / A cm -2 Current Density / A cm -2

279 Chapter 10. Application to PEM Fuel Cells page 10: 42 Normalized Noise in Impedance Response Hz 10 Hz r,obs / base Hz Shows flooding - 10X more noise Shows need for better base error structure Current Density / A cm -2

280 Chapter 10. Application to PEM Fuel Cells page 10: 43 Calculated Parameters

281 Chapter 10. Application to PEM Fuel Cells page 10: 44 Fractional Surface Coverage of Intermediates Fractional Surface Coverage Hydrogen Peroxide Platinum Oxide Fractional Surfcae Coverage Hydrogen Peroxide Platinum Oxide Potential / V Current / A

282 Chapter 10. Application to PEM Fuel Cells page 10: 45 Low-frequency Inductive Loops in PEM Fuel 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

283 Chapter 10. Application to PEM Fuel Cells page 10: 46

284 Chapter 11. Conclusions page 11: 1 Impedance Spectroscopy Chapter 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 Mark E. Orazem, All rights reserved.