Advanced Electrochemical Impedance Spectroscopy

Transcription

1 Advanced Electrochemical Impedance Spectroscopy 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. More on Development of Process Models Chapter 7. Constant Phase Elements Chapter 8. Regression Analysis Chapter 9. Error Structure Chapter 10. Kramers-Kronig Relations Chapter 11. Use of Measurement Models Chapter 12. Conclusions Chapter 13. Suggested Reading Chapter 14. Notation

3 Chapter 1. Introduction page 1: 1 Advanced Electrochemical Impedance Spectroscopy Chapter 1. Introduction How to think about impedance spectroscopy EIS as a generalized transfer function Objective of course Mark E. Orazem, All rights reserved.

4 Chapter 1. Introduction page 1: no logo

5 Chapter 1. Introduction page 1: 3 June 16-21, 2013 Okinawa, Japan

6 Chapter 1. Introduction page 1: 4 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!...

7 Chapter 1. Introduction page 1: 5 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

8 Chapter 1. Introduction page 1: 6 Interrogate Electrochemical Cell V Electrochemical Cell + V I + I V Z ω = = Z + jz I ( ) r j M. E. Orazem and B. Tribollet, Electrochemical Impedance Spectroscopy, John Wiley & Sons, Hoboken, New Jersey, 2008

9 Chapter 1. Introduction page 1: 7 Low Frequencies IV ( ) : slope = 1 R p IV ( ) di 1 = dv Z I V V lim Z( f) = R p f 0

10 Chapter 1. Introduction page 1: 8 Moderate Frequencies IV ( ) I V V ( f ) = exp( jφ ) = Zr + j j Z Z Z

11 Chapter 1. Introduction page 1: 9 Applications of EIS Electrode-Electrolyte Interface Electrical Double Layer Diffusion Layer Reaction Mechanism Electrochemical Reactions Corrosion Electrodeposition Human Skin Batteries Fuel Cells Materials Characterization

12 Chapter 1. Introduction page 1: 10 Course Objectives Benefits and advantages of impedance spectroscopy Methods to improve experimental design Global and local impedance Interpretation of data graphical representations equivalent circuits process models regression error analysis

13 Chapter 1. Introduction page 1: 11 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. More on Development of Process Models Chapter 7. Constant Phase Elements Chapter 8. Regression Analysis Chapter 9. Error Structure Chapter 10. Kramers-Kronig Relations Chapter 11. Use of Measurement Models Chapter 12. Conclusions Chapter 13. Suggested Reading Chapter 14. Notation

14 Chapter 1. Introduction page 1: 12 Comment on Notation IUPAC Convention: Present Work: Z' and Z Z and r Z '', i j, j

15 Chapter 1. Introduction page 1: 13

16 Chapter 2. Motivation page 2: 1 Advanced 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.

17 Chapter 2. Motivation page 2: 2 Steady-State Polarization Curve R 0 C 1 R 1 (V) C 2 R Current, ma Potential, V

18 Chapter 2. Motivation page 2: 3 Steady-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

19 Chapter 2. Motivation page 2: 4 Transient Response to a Step in Potential R1( V) R2 Current / ma } V=10 mv R 0 C 1 C 2 R ( V) 1 potential input R 2 Current / ma I = R V ( + R 0 + R1 V ) 2 current response (t-t 0 ) / ms (t-t 0 ) / s

20 Chapter 2. Motivation page 2: 5 Transient Response to a Step in Potential R1( V) R2 1.5 } V=10 mv potential input Short times/high frequency C 1 C 2 Current / ma R 0 C 1 R ( V) 1 C 2 R 2 R 0 R 1 (V) R 2 current response long times/low frequency (t-t 0 ) / ms

21 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

22 Chapter 2. Motivation page 2: 7 Sinusoidal Perturbation ( ω ) { } V( t) = V + V cos t 0 i( t) = i exp( bv) exp( bv) 0 a c dv ic = C0 dt if = ( ) f V Vt ()

23 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

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

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

26 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.

27 Chapter 2. Motivation page 2: 12 Impedance Spectroscopy vs. Step-Change Transients Information sought is the same Increased sensitivity stochastic errors frequency range consistency check Better decoupling of physical phenomena

28 Chapter 2. Motivation page 2: 13 System with Large Ohmic Resistance C R 0 R 1 Circuit R 0, Ω R 1, Ω C, µf R 0 /R 1 1 1, M. E. Orazem, T. El Moustafid, C. Deslouis, and B. Tribollet, Journal of The Electrochemical Society, 143 (1996), f RC = Hz 2π RC = 1

29 Chapter 2. Motivation page 2: 14 Impedance Spectrum Hz -40 -Z j / Ω Z r / Ω

30 Chapter 2. Motivation page 2: 15 Error Structure

31 Chapter 2. Motivation page 2: 16

32 Chapter 3. Impedance Measurement page 3: 1 Advanced 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.

33 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.

34 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

35 Chapter 3. Impedance Measurement page 3: 4 Lissajous Analysis Current B Vt ( ) = Vsin( ωt) V It ( ) = sin( ωt+ φ) Z A' D' O D A B' Potential V OA A'A Z = = = I OB B'B OD D'D sin( φ) = = OA A'A

36 Chapter 3. Impedance Measurement page 3: 5 General Signal Reference Signal A Phase Sensitive Detection n= 0 A= A sin( ) 0 ωt+ φa 4 1 S = sin ( 2n+ 1) ωt+ φs π 2n + 1 4A 1 = si ( ) sin ( 2 1) 2 1 n A n+ t+ S π n + 0 S ωt + φ ω φ n= 0 ω 2π 2π 0 ω ASdt = 2A π 0 cos ( φ φ ) A S Has maximum value when φ A = φ S

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

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

39 Chapter 3. Impedance Measurement page 3: 8 Comparison single-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

40 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

41 Chapter 3. Impedance Measurement page 3: 10 Experimental Considerations Electrode Dimension 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

42 Chapter 3. Impedance Measurement page 3: 11 Sinusoidal Perturbation ( ω ) V( t) = V + V cos t 0 i( t) = i exp( bv) 0 a dv ic = C0 dt if = ( ) f V Vt ()

43 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

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

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

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

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

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

49 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 () η s () t

50 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

51 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

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

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

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

55 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)

56 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

57 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

58 Chapter 3. Impedance Measurement page 3: 27 Improve Experimental Design Question How can we isolate the role of positive electrode, negative electrode, and separator in a battery? Answer - Develop a very sophisticated process model. - Use a four-electrode configuration. Working Electrode Ref 1 Ref 2 Counterelectrode

59 Chapter 3. Impedance Measurement page 3: 28 Improve Experimental Design Question How can we reduce stochastic noise in the measurement? Answer - 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.

60 Chapter 3. Impedance Measurement page 3: 29 Improve Experimental Design Question How can we determine the cause of variability between experiments? Answer - Use the Kramers-Kronig relations to see if data are selfconsistent. - Change mode of modulation to variable-amplitude galvanostatic to ensure that the base-line is not changed by the experiment.

61 Chapter 3. Impedance Measurement page 3: 30 Facilitate Interpretation Question How can we be certain that the instrument is not corrupting the data? Answer - Use the Kramers-Kronig relations to see if data are selfconsistent. - 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).

62 Chapter 3. Impedance Measurement page 3: 31 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

63 Chapter 3. Impedance Measurement page 3: 32 Can We Perform Impedance on Transient Systems? Timeframes for measurement Individual frequency Individual scan Multiple scans

64 Chapter 3. Impedance Measurement page 3: 33 EHD Experiment -0.2 Imaginary Impedance, µa/rpm Data Set #2 Data Set #4 Data Set # Real Impedance, µa/rpm

65 Chapter 3. Impedance Measurement page 3: 34 Time per Frequency for 200 rpm Time per Measurement, s Filter On Filter Off Frequency, Hz

66 Chapter 3. Impedance Measurement page 3: 35 Impedance Scans -15 Z j / Ω cm Time Interval s s s Z r / Ω cm 2

67 Chapter 3. Impedance Measurement page 3: 36 Impedance Scans 4000 Elapsed Time / s f / Hz

68 Chapter 3. Impedance Measurement page 3: 37 Time per Frequency Measured 10 2 t f / s 3 cycles 5 cycles seconds f / Hz

69 Chapter 3. Impedance Measurement page 3: 38

70 Chapter 3. Impedance Measurement page 3: 39

71 Chapter 4. Representation of Impedance Data page 4: 1 Advanced 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.

72 Chapter 4. Representation of Impedance Data page 4: 2 Impedance is a Complex Number Z = Z + jz r i -Z j / Ω Z 200 φ Z r / Ω Z= Ze jφ ( φ) sin ( φ) Z= Zcos + jz 2 ( ) ( ) 2 Z = Z + Z Z j φ = arctan Zr r j

73 Chapter 4. Representation of Impedance Data page 4: 3 Circuit Elements ZR = R ZL jlω Z = C = 1 1 = j jcω Cω

74 Chapter 4. Representation of Impedance Data page 4: 4 Addition of Impedance Z 1 Z Y Y Y Z Z Z + = + = Z 1 Z Y Y Y Z Z Z + = + =

75 Chapter 4. Representation of Impedance Data page 4: 5 Circuit Components: Resistor -5 R e Z R e Z j / Ω cm 2 0 = e R Z r / Ω cm 2

76 Chapter 4. Representation of Impedance Data page 4: 6 Circuit Components: Capacitor R e C dl 1 1 = + = j ωc Z Re R e jωcdl j= 1 dl R e

77 Chapter 4. Representation of Impedance Data page 4: 7 C dl R e Simple RC Circuit Z j / Ω cm 2 R t Hz f RC = 1 2π RC t dl Z = R + R e R t 1 R = e t + 1 jωc R t jωrc t dl R ωr C = R dl 2 t t dl e j 2 2 ( ωrc ) ( ωrc ) t dl t dl R Z r / Ω cm 2 e Re + Rt Note: ω = ω f 2π f -1 rad / s or s cycles / s or Hz

78 Chapter 4. Representation of Impedance Data page 4: 8 R e Impedance C dl R t Z r or Z j / Ω cm 2 Slope = f / Hz real imaginary 10 2 real imaginary Z j / Ω cm Hz Z r or Z j / Ω cm Slope = -1 Z r / Ω cm f / Hz

79 Chapter 4. Representation of Impedance Data page 4: 9 Bode Representation -90 Z / Ω cm Z = Z + jz Z Z r j r = Z cos = Z sin j ( φ ) ( φ ) φ / degrees f / Hz

80 Chapter 4. Representation of Impedance Data page 4: 10 Input & output are in phase Modified Phase Angle φ adj Z 1 j = tan Zr Re Z / Ω cm Z adj Z φ adj φ f / Hz -75 Note: -60 The modified phase angle yields excellent-45 insight, but we need an accurate estimate for the solution resistance φ / degrees φ Z tan 1 j = Zr Input & output are out of phase

81 Chapter 4. Representation of Impedance Data page 4: 11 Constant Phase Element R e Z j / R t CPE R t Z R t R e = j 1 ( ω) α RQ t α=1 α=0.9 α=0.8 α=0.7 α=0.6 α= (Z r R e ) / R t

82 Chapter 4. Representation of Impedance Data page 4: 12 Slope = + α 10 0 CPE: Log Imaginary Slope = α 10-1 Z j / R t α=1 α=0.9 α=0.8 α=0.7 α=0.6 α= ω ωτ/ τ Note: With a CPE (α 1), the asymptotic slopes are no longer ±1.

83 Chapter 4. Representation of Impedance Data page 4: CPE: Modified Phase Angle φ adj / degrees α=1 α=0.9 α=0.8 α=0.7 α=0.6 α=0.5 Note: 0 The high-frequency asymptote for the modified phase angle depends on the CPE coefficient α ω / τ ωτ

84 Chapter 4. Representation of Impedance Data page 4: 14 Example for Synthetic Data Z( f) Rt + zd( f ) α ( π f ) Q R z ( f ) = Re j2 dl t + d ( ) z f = z f d ( ) ( ) d tanh j2π fτ j2π fτ M. Orazem, B. Tribollet, and N. Pébère, J. Electrochem. Soc., 153 (2006), B129-B136.

85 Z j / Ω cm 2 Chapter 4. Representation of Impedance Data page 4: 15 Traditional Representation 400 α = 1 α = 0.7 α = Hz 100 Hz 1 Hz 0.1 Hz 10 mhz 1 mhz Z r / Ω cm 2 Z / Ω cm α = 1 α = 0.7 α = 0.5 φ / degree α = 1 α = 0.7 α = f / Hz f / Hz

86 Chapter 4. Representation of Impedance Data page 4: 16 R e -Corrected Bode Phase Angle φ adj Z 1 j = tan Zr R e,est φ ( ) = 90( α) adj φ adj / degree -90 α = 1-75 α = 0.7 α = f / Hz

87 Chapter 4. Representation of Impedance Data page 4: 17 R e -Corrected Bode Magnitude adj ( ) 2 2 Z = Z R + Z r e,est j Slope = α 10 3 Z adj / Ω cm α = 1 α = 0.7 α = 0.5 slope = α f / Hz

88 Chapter 4. Representation of Impedance Data page 4: 18 Slope = α Log( Z j ) Z j / Ω cm α = slope=α slope= α α = 0.7 α = 0.5 Slope = α f / Hz

89 Chapter 4. Representation of Impedance Data page 4: 19 Effective Capacitance or CPE Coefficient 1 απ 1 Z Qeff = sin CPE = α α 2 Q j2π f ( 2 π f ) Zj( f) CPE ( ) Q eff / Q dl 100 α = 1 α = 0.7 α = f / Hz

90 Chapter 4. Representation of Impedance Data page 4: 20 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

91 Chapter 4. Representation of Impedance Data page 4: 21 Application

92 Chapter 4. Representation of Impedance Data page 4: 22 Mg alloy (AZ91) in 0.1 M NaCl Hz Z j / Ω cm khz time / h Hz Hz Z r / Ω cm 2

93 Chapter 4. Representation of Impedance Data page 4: 23 k Mg 1 ( + ) Mg + e ads Proposed Model + k ( 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

94 Chapter 4. Representation of Impedance Data page 4: 24 Normalized Plot G. Baril, G. Galicia, C. Deslouis, N. Pébère, B. Tribollet, and V. Vivier, J. Electrochem. Soc., 154 (2007), C108-C113

95 Chapter 4. Representation of Impedance Data page 4: 25 k Mg 1 + ( Mg ) + e ads ( Mg ) Mg + e ads k k 3 Physical Interpretation diffusion of Mg Z j / Ω cm khz time / h Hz Hz Hz Z r / Ω cm 2 relaxation of (Mg + ) ads. intermediate

96 Chapter 4. Representation of Impedance Data page 4: 26 Log( Z j 10 2 Slope = α Z j / Ω cm time / h f / Hz

97 Chapter 4. Representation of Impedance Data page 4: 27 Effective CPE Coefficient Q eff απ 1 = sin α 2 2 Z ( f) ( π f ) j Q eff / (MΩ -1 cm -2 ) s α 10 5 time / h f / Hz

98 Chapter 4. Representation of Impedance Data page 4: 28 R e -Corrected Magnitude φ ( ) = 90α 1 adj φadj tan 10 3 Z j = Zr R e,est Z adj / Ω cm time / h f / Hz

99 Chapter 4. Representation of Impedance Data page 4: 29 Physical Parameters Z j / Ω cm khz time / h Hz Hz Hz Z r / Ω cm 2

100 Chapter 4. Representation of Impedance Data page 4: 30 Application for Local EIS layer Z local V = i applied local

101 Chapter 4. Representation of Impedance Data page 4: 31 Global Impedance Mg alloy (AZ91) in Na 2 SO M at the corrosion potential after 1 h of immersion. Electrode radius 5500 µm. J.-B. Jorcin, M. E. Orazem, N. Pébère, and B. Tribollet, Electrochimica Acta, 51 (2006), 1473.

102 Chapter 4. Representation of Impedance Data page 4: 32 Local Impedance Local impedance α = 1 to 0.92

103 Chapter 4. Representation of Impedance Data page 4: 33 Results of Graphical Analysis 1.0 α resistance capacitance r / r 0 The local impedance has a pure RC behavior at center. Current distribution affects value at edge.

104 Chapter 4. Representation of Impedance Data page 4: 34 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

105 Chapter 4. Representation of Impedance Data page 4: 35 Guidelines for Nyquist Plots -40 Use symbols, not lines Label characteristic frequencies Use same scale for real and imaginary axes Z j / Ω cm Z j / Ω cm Z r / Ω cm Hz Z r / Ω cm 2

106 Chapter 4. Representation of Impedance Data page 4: 36 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

107 Chapter 4. Representation of Impedance Data page 4: 37

108 Chapter 5. Development of Process Models page 5: 1 Advanced 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.

109 Chapter 5. Development of Process Models page 5: 2 Electrical Circuits

110 Chapter 5. Development of Process Models page 5: 3 Method to Combine Impedances

111 Chapter 5. Development of Process Models page 5: 4 Electrically Equivalent Circuits

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

113 Chapter 5. Development of Process Models page 5: 6 Addition of Potential

114 Chapter 5. Development of Process Models page 5: 7 Addition of Current

115 Chapter 5. Development of Process Models page 5: 8 Equivalent Circuit at the Corrosion Potential

116 Chapter 5. Development of Process Models page 5: 9 Equivalent Circuit for a Partially Blocked Electrode Z R Z F = e ( ) 1 γ j ω γc (1 γ) C Z dl F

117 Chapter 5. Development of Process Models page 5: 10 Implications: Normalized Nyquist

118 Chapter 5. Development of Process Models page 5: 11 Implications: Normalized Impedance

119 Chapter 5. Development of Process Models page 5: 12 Equivalent Circuit for an Electrode Coated by a Porous Layer

120 Chapter 5. Development of Process Models page 5: 13 Equivalent Electrical Circuit for an Electrode Coated by Two Porous Layers C εε = ; ε 0 = F/cm δ 2 R = δ / κ Z R F R 1+ jωc 2 dlz F = Re jωr 2C 2 ZF 1 γ + jωγ C R + 1+ jω dl + C ZF + Z

121 Chapter 5. Development of Process Models page 5: 14 Scaling and Corrosion of a Steel Electrode L Bousselmi, C. Fiaud, B. Tribollet, and E. Triki, Electrochimica Acta, 44 (1999),

122 Chapter 5. Development of Process Models page 5: 15 Use kinetic models to determine expressions for the interfacial impedance

123 Chapter 5. Development of Process Models page 5: 16 Approach identify reaction mechanism write expression for steady state current contributions write expression for sinusoidal steady state sum current contributions account for mass transfer calculate impedance

124 Chapter 5. Development of Process Models page 5: 17 General Expression for Faradaic Current (,, ) i = f Vc γ f i,0 i i = i + Re ~ { j ω i e t } f f f i = V + c + f i,0 k V c, γ i ci,0 k γ Vc,, γ k Vc, γ i,0 k j, j i k j j, j k γ

125 Chapter 5. Development of Process Models page 5: 18 Reactions Considered Dependent on Potential Dependent on Potential and Mass Transfer Dependent on Potential, Mass Transfer, and Surface Coverage Coupled Reactions

126 Chapter 5. Development of Process Models page 5: 19 Irreversible Reaction: Dependent on Potential A z+ + - A A +e A A A A A A A A A A A A A A A A A A A A A A A A A A A A Potential-dependent heterogeneous reaction Two-dimensional surface No effect of mass transfer

127 Chapter 5. Development of Process Models page 5: 20 Impedance Model Development steady-state α F = RT A ia nafka exp V oscillating component i 1 = = exp = ( A ) A ia V Kb A A bv V V V Rt,A R t,a = K b 1 exp ( bv) A A A β A = 2.303/ ba βa ia = 2.303R β A ( ) i = K bv A A exp A t,a = 2.303R i t,a A

128 Chapter 5. Development of Process Models page 5: 21

129 Chapter 5. Development of Process Models page 5: 22 Significance of R t I

130 Chapter 5. Development of Process Models page 5: 23 Simple Approach for Model Development A ia V V V Rt,A Z F i = = V = i A = R t,a 1

131 Chapter 5. Development of Process Models page 5: 24 Impedance Model Development dv i = ia + Cdl dt jωt i = i + Re i { e } jωt V = V + Re{ V e } { jωt} { jωt Re e } d jωt i + i = ia + Re i Ae + Cdl ( V + Re{ V e }) dt j t j t j t i + Re i e ω = i + Re i e ω + C Re{ jωv e ω } { } { } A A Re i e = Re i e + C Re{ jωv e } { } { } A jωt jωt jωt dl i = i + jωcv A dl dl

132 Chapter 5. Development of Process Models page 5: 25 Impedance Model Development i = i + jωcv i A V = + R t,a dl jωcv dl 1 1+ jωrt,ac dl = V + jωcdl = V R t,a R t,a V R i 1+ jωr C t, A = = t,a dl Z F

133 Chapter 5. Development of Process Models page 5: 26 Impedance Model Development U = ir + V e jωt jωt jωt U + Re{ U e } = i + Re{ i e } R + V + Re{ V e } ( ) e U = ire + V U = ir e + V U V = Re + = i i Z A

134 Chapter 5. Development of Process Models page 5: 27 Impedance Model Development Z A U V = = Re + i i Rt,A = Re + 1+ jω R C t,a dl R e C dl R t,a

135 Chapter 5. Development of Process Models page 5: 28 Interpretation Models for Impedance Spectroscopy Models can account rigorously for proposed kinetic and mass transfer mechanisms. Circuits can provide a framework for model development. Models can be expressed in terms of circuits.

136 Chapter 5. Development of Process Models page 5: 29

137 Chapter 6. More on Development of Process Models page 6: 1 Advanced Electrochemical Impedance Spectroscopy Chapter 6. More on 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.

138 Chapter 6. More on Development of Process Models page 6: 2 Approach identify reaction mechanism write expression for steady state current contributions write expression for sinusoidal steady state sum current contributions account for mass transfer calculate impedance

139 Chapter 6. More on Development of Process Models page 6: 3 General Expression for Faradaic Current (,, ) i = f Vc γ f i,0 i i = i + Re ~ { j ω i e t } f f f i = V + c + f i,0 k V c, γ i ci,0 k γ Vc,, γ k Vc, γ i,0 k j, j i k j j, j k γ

140 Chapter 6. More on Development of Process Models page 6: 4 Reactions Considered Dependent on Potential Dependent on Potential and Mass Transfer Dependent on Potential, Mass Transfer, and Surface Coverage Coupled Reactions

141 Chapter 6. More on Development of Process Models page 6: 5 Irreversible Reaction: Dependent on Potential A z+ + - A A +e A A A A A A A A A A A A A A A A A A A A A A A A A A A A Potential-dependent heterogeneous reaction Two-dimensional surface No effect of mass transfer

142 Chapter 6. More on Development of Process Models page 6: 6 Impedance Model Development Z A U V = = Re + i i Rt,A = Re + 1+jωR C t,a dl R e C dl R t,a

143 Chapter 6. More on Development of Process Models page 6: 7 Irreversible Reaction: Dependent on Potential and Mass Transfer O R O + ne R Irreversible potential-dependent heterogeneous reaction Reaction on two-dimensional surface Influence of transport of O to surface

144 Chapter 6. More on Development of Process Models page 6: 8 steady-state Current Density oscillating component ( ) i = K c exp bv O O O,0 O ( ) exp( ) i = K bc exp bv V K bv c O O O O,0 O O O O,0 V = K exp( bv) c R t,o O O O,0 R t,o = K c 1 exp ( bv) O O,0 O

145 Chapter 6. More on Development of Process Models page 6: 9 Mass Transfer = O io nofdo dy O O O dc 0 { j t} i = i + Re ie ω dc O io = nofdo dy 0 c O,0 io = nofdo θ δ O 0 ( )

146 Chapter 6. More on Development of Process Models page 6: 10 Combine Expressions V i = K exp bv c c ( ) O O O O,0 Rt,O O,0 i δ = n FD O O O Oθ 0 ( )

147 Chapter 6. More on Development of Process Models page 6: 11 i O = = R R t,o O + V + z t,o d,o δ n FD c V O O O,0 O Current Density 1 1 b θ (0) ZF = Rt,O + zd,o z d,o δ n FD c O = O O O,0 O 1 1 b θ (0) R t,o = K c 1 exp ( bv) O O,0 O

148 Chapter 6. More on Development of Process Models page 6: 12 i O = R Simple Approach for Model Development t,o d,o O V + z V ZF = = R + z i t,o d,o

149 Chapter 6. More on Development of Process Models page 6: 13 Mass Transfer

150 Chapter 6. More on Development of Process Models page 6: 14 Film Diffusion c i t 2 c i = Di 2 y steady state c c ( ) as y δ i i f c = c (0) at y = 0 i i y c = c (0) + c ( ) c (0) ( ) i i i i δf

151 Chapter 6. More on Development of Process Models page 6: 15 Film Diffusion c t i 2 c i = Di 2 y i i i j { } c = c + Re ce ωt 2 2 jωt d ci d c i jωce = Di + D 2 i e 2 d y d y 2 jωt d c i jωce = Di e 2 d y d j c D c ω = d y 2 i i 2 jωt jωt ξ = K i y δ f 2 ωδf = D ci θi = c (0) i i d d θ 2 i K 2 iθi ξ = j 0

152 Chapter 6. More on Development of Process Models page 6: 16 i i Warburg Impedance 2 d θi jk 2 iθi = 0 dξ ( ) ( ) θ = Aexp ξ jk + Bexp ξ jk i i i θi = 0 at ξ = 1 θ = 1 at ξ = 0 1 tanh jk = θ (0) jk 2 jωδf tanh 1 Di = θ 2 i (0) jωδf D i i i θi = 0 at ξ = θ = 1 at ξ = 0 i 1 1 = θ (0) jk i 1 1 = θ i (0) jωδ D i i 2 f

153 Chapter 6. More on Development of Process Models page 6: Concentration c(ξ)/c( ) t=0, 2nπ/K t=nπ/k t=0.5nπ/k t=1.5nπ/k K= ξ/δ c(ξ)/c( ) t=0, 2nπ/K t=0.5nπ/k t=1.5nπ/k t=nπ/k K= ξ/δ

154 Chapter 6. More on Development of Process Models page 6: 18 Rotating disk electrode

155 Chapter 6. More on Development of Process Models page 6: 19 Concentration Profile

156 Chapter 6. More on Development of Process Models page 6: 20 Coupled Diffusion Impedance Z Z + D i f,i D D,f Di,f δ N,i D,net = 2 δ f,i D δ i f,i ZD,f ZD jω D + i,f Di,f δ N,i δ Z

157 Chapter 6. More on Development of Process Models page 6: 21 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

158 Chapter 6. More on Development of Process Models page 6: 22 Steady-State Current Density reaction 1: formation of X reaction 2: formation of P total current density ( 1 γ ) exp ( ) ( ) i = K b V V ( ) ( ) i = K γ exp b V V i = i1+ i2

159 Chapter 6. More on Development of Process Models page 6: 23 Steady-State Surface Coverage balance on γ dγ i1 i2 Γ = dt F F = 0 steady-state value for γ γ = ( ) K ( ) 1exp b1 V V1 ( ( )) + exp( ( )) K exp b V V K b V V

160 Chapter 6. More on Development of Process Models page 6: 24 Steady-State Current Density ( γ ) ( ) ( ) γ ( ( )) i = K 1 exp b V V + K exp b V V where γ = K ( ( )) 1exp b1 V V1 ( ) exp ( ) + ( ( )) K exp b V V K b V V

161 Chapter 6. More on Development of Process Models page 6: 25 Oscillating Current Density ( γ ) ( ) ( ) γ ( ( )) i = K 1 exp b V V + K exp b V V i = + V + K b V V K b V V Rt,1 R t,2 ( ( )) exp ( ) ( ( )) 2exp ( 1 γ ) exp ( ) ( ) R = Kb b V V t, ( ( )) R = Kbγ exp b V V t, γ

162 Chapter 6. More on Development of Process Models page 6: 26 balance on γ Need Additional Equation Γ jωγ = V K b V V + K b V V F Rt,1 R t,2 ( ( )) exp ( ) ( ) ( ) 1exp Rt,1 Rt,2 γ = V F Γ jω + K b V V + K b V V ( exp ( ( )) ( ( ))) exp 2 2 γ

163 Chapter 6. More on Development of Process Models page 6: Impedance ( ( )) ( ( )) 1 1 exp t t ( ( ( )) ( ( ))) 1exp 1 1 2exp 2 2 K exp b V V K b V V R R ,1,2 = + ZF Rt FΓ jω + F K b V V + K b V V 1 A = + R jω + B t where = + R R R t tm, tx,

164 Chapter 6. More on Development of Process Models page 6: 28

165 Chapter 6. More on Development of Process Models page 6: 29 R e C dl Z f

166 Chapter 6. More on Development of Process Models page 6: 30 Corrosion of Steel i f = i + i + i Fe H 2 O 2 Fe Fe e - O 2 +2H 2 O + 4e - 4OH - 2H 2 O + 2e - H 2 +2OH -

167 Chapter 6. More on Development of Process Models page 6: 31 Corrosion: Fe Fe e - ( ) i = K exp bv Fe Fe Fe i = K b exp bvv ( ) Fe Fe Fe Fe ~ i = Fe ~ V R t,fe R t,fe = K b 1 exp ( bv) Fe Fe Fe

168 Chapter 6. More on Development of Process Models page 6: 32 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

169 Chapter 6. More on Development of Process Models page 6: 33 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 b V c O O O,

170 Chapter 6. More on Development of Process Models page 6: 34 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 θ 2 δ O ( )

171 Chapter 6. More on Development of Process Models page 6: 35 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 δo no FD 2 O c 2 O 2,0 b O θ (0) 2 V + z t,o d,o 2 2 δ 1 1 no FDO co,0 b O θ (0) O2 = R t,o 2 = 1 ( ) K b c exp b V O O O,0 O

172 Chapter 6. More on Development of Process Models page 6: 36 U Z = = Zr + jz i R = e + = R + e j Rt, Fe R t, H 2 Rt,O + z 2 d,o jωcd R R + z eff t,o d,o 2 2 Process Model 1 jωc d C dl R t,fe R e R t,h2 Z d,o = + R R R eff t, Fe t, H 2 R t,o2

173 Chapter 6. More on Development of Process Models page 6: 37 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 mass transfer calculate impedance

174 Chapter 6. More on Development of Process Models page 6: 38 Interpretation Models for Impedance Spectroscopy Systematic method to account for proposed kinetic and mass transfer mechanisms. Model development can be expressed in terms of circuits.

175 Chapter 6. More on Development of Process Models page 6: 39

176 Chapter 7. Constant Phase Elements page 7: 1 Advanced Electrochemical Impedance Spectroscopy Chapter 7. Constant Phase Elements CPEs can arise from surface or axial distributions CPE parameters can be interpreted in terms of capacitance, depending on type of distribution Mark E. Orazem, All rights reserved.

177 Chapter 7. Constant Phase Elements page 7: 2 Constant Phase Element (CPE) -Z j Semi-Circle Z = Re R t jωcr 0 t Depressed Semi-Circle R e Z r R e +R t Z = Re R t ( jω ) α QR t R e R e R e R e R t C 0 R t C 0 R t C 0 R t C 0 CPE caused by distribution of time constants

178 Chapter 7. Constant Phase Elements page 7: 3 Intentionally left blank

179 Chapter 7. Constant Phase Elements page 7: 4 Intentionally left blank CPE: Constant Phase Element Semi-Circle Z = Re R t jωcr Depressed Semi-Circle Z = Re R t ( jω ) 0 t α QR t

180 Chapter 7. Constant Phase Elements page 7: 5 Proposed Origins of a CPE 1. distribution of capacitance 2. electrode roughness 3. distribution of reaction rate constants 4. nonuniform current distributions 5. coupled reactions 6. distribution of resistivity Usual Use of a CPE 1. improve fit for a circuit model 2. explanation added as needed

181 Chapter 7. Constant Phase Elements page 7: 6 Recent Developments in our Understanding of the CPE

182 Chapter 7. Constant Phase Elements page 7: 7 Types of Distributions Surface Axial

183 Chapter 7. Constant Phase Elements page 7: 8 Interpretation in Terms of Capacitance 1. Direct use of Q = Q Ceff,Q 2. Surface Distribution C eff,surf Q RR 1/ α e t = Re + Rt (1 α)/ α 3. Normal (Axial) Distribution C Q R eff,norm 1/ α (1 α )/ α f G. J. Brug, A. L. G. van den Eeden, M. Sluyters- Rehbach, J. H. Sluyters, J. Electroanal. Chem., 176 (1984), 275. = C. H. Hsu, F. Mansfeld, Corrosion, 57 (2001), Power-Law Normal Distribution ( ρ εε ) 1 C = Q g eff,pl δ 0 g α = (1 α) B. Hirschorn, M. E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, J. Electrochem. Soc., 157 (2010), C452.

184 Chapter 7. Constant Phase Elements page 7: 9 Surface Distribution From characteristic frequency for admittance C eff,surf Q RR 1/ α e t = Re + Rt (1 α)/ α B. Hirschorn, M.E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, Electrochim. Acta, 55 (2010), 6218.

185 Chapter 7. Constant Phase Elements page 7: 10 Normal (Axial) Distribution From characteristic frequency for impedance C = Q R eff,surf 1/ α (1 α )/ α f B. Hirschorn, M.E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, Electrochim. Acta, 55 (2010), 6218.

186 Chapter 7. Constant Phase Elements page 7: 11 Power-Law Model ρ δ R ( x) = ρ( x) dx 0 dx i ρ 0 C i = εε dx Z f = δ 0 ρ( x) 1+ j ωρ( x) εε 0 dx Z cpe = 1 (j ω) α Q Assume dielectric constant is uniform Is there a resistivity distribution that leads to a CPE?

187 Chapter 7. Constant Phase Elements page 7: 12 Model Development regression with sequential RC circuit conversion in terms of resistivity inferred power-law model ρ ρ δ ρ ρ0 ρ = + 1 ρ δ ξ γ ρ ξ ρ 0 = δ δ γ 1 = δρ g( γ ) (j ωεε ) 1/ γ δ ( γ 1)/ γ 0 = ( 1 jω) α Q α α 0 = Q = ( ) 1 α gδρδ γ 1 γ εε C = Q g ( ρ εε ) 1 eff,pl δ 0 g α = (1 α) 2.375

188 Chapter 7. Constant Phase Elements page 7: 13 Interpretation in Terms of Capacitance 1. Direct use of Q = Q Ceff,Q 2. Surface Distribution C eff,surf Q RR 1/ α e t = Re + Rt (1 α)/ α 3. Normal (Axial) Distribution C Q R eff,norm 1/ α (1 α )/ α f G. J. Brug, A. L. G. van den Eeden, M. Sluyters- Rehbach, J. H. Sluyters, J. Electroanal. Chem., 176 (1984), 275. = C. H. Hsu, F. Mansfeld, Corrosion, 57 (2001), Power-Law Normal Distribution ( ρ εε ) 1 C = Q g eff,pl δ 0 g α = (1 α) B. Hirschorn, M. E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, J. Electrochem. Soc., 157 (2010), C452.

189 Chapter 7. Constant Phase Elements page 7: 14 Impedance of Human Skin Split-thickness human cadaver skin ( µm thick) 7 samples E. White, Characterization of the Skin Barrier to Chemical Permeation by Impedance Spectroscopy, PhD dissertation, Colorado School of Mines, 2011.

190 Chapter 7. Constant Phase Elements page 7: 15 Pin-Hole Experiment Q, α R e R skin Q, α R e R skin R hole R eff 1 1 = + Rskin R hole 1

191 Chapter 7. Constant Phase Elements page 7: 16 Impedance Response 30 before performation after performation characteristic frequency Z j / kω cm f after = 938 Hz f before = 141 Hz Z r / kω cm 2

192 Chapter 7. Constant Phase Elements page 7: 17 Impedance Response Z r / kω cm before perforation after perforation Z j / kω cm before perforation after perforation f before f after f / Hz f / Hz

193 Chapter 7. Constant Phase Elements page 7: 18 Average Regressed Parameters (7 Samples) R t c = 118 ± 54 kωcm α = 0.80 ± 0.01 Q = f Before Pinhole 74 ± 12 nfs / cm = 69 ± 39 Hz 2 α 1 2 f f after before R = 13± 8 t c = 16.2 ± 5 kωcm α = 0.83± 0.01 Q f After Pinhole = ± α nfs / cm = 685 ± 266 Hz 2

194 Chapter 7. Constant Phase Elements page 7: 19 Hypotheses: Skin Unaffected by Pinhole Z j / kωcm Z r / kωcm 2 εε 0 Cskin = Ceff,surf = δ 1 fc = 2π RC fafter CbeforeRbefore = f C R C f f before after after before after before = C = R R after before after M. E. Orazem, B. Tribollet, V. Vivier, S. Marcelin, N. Pébère, A. L. Bunge, E. A. White, D. P. Riemer, I. Frateur, and M. Musiani, ECS Transactions, (2012), in press.

195 Chapter 7. Constant Phase Elements page 7: 20 Hypotheses: Skin Unaffected by Pinhole Z j / kωcm 2 ( ρ εε ) 1 C = Q g eff,pl δ 0 g α = (1 α) Z r / kωcm 2 Constant Q and α f c ( RQ) ( RbeforeQbefore ) 1/ ( ) before after after after before 2π 1 fafter = f R Q f f = Q = α = constant constant R before = Rafte r 1/ α 1/ α 1/ α α

196 Chapter 7. Constant Phase Elements page 7: 21 Test of Hypotheses

197 Chapter 7. Constant Phase Elements page 7: 22 Capacitance obtained from characteristic frequency not a property of the skin

198 Chapter 7. Constant Phase Elements page 7: 23 Hutchinson Technology: Tailoring Stainless Steel Surface Chemistry Application: Floating heads for hard drives Douglas Riemer Principal Scientist Hutchinson Technology Hutchinson, MN

199 Chapter 7. Constant Phase Elements page 7: free machining steel

200 Chapter 7. Constant Phase Elements page 7: 25 Impedance Data: Free-Machining Steel Hz Model parameters Z j / kω cm Hz Q eff = = 11μFs / cm α = 0.91 R = 15.2 Ωcm R e p = α Fs Ωcm / cm α Z r / kω cm 2 Assumed values ε = 12 ρ = 450 Ωcm δ Data courtesy of Douglas Riemer, Principal Scientist, Hutchinson Technology, Hutchinson, MN

201 Chapter 7. Constant Phase Elements page 7: 26 C eff,surf C C C From EIS (CPE) Data From XPS Approach Distribution C eff, µf/cm 2 δ, nm δ, nm = Q Ceff,Q RR 1/ α e t = Q Re + Rt eff,surf eff,norm Qg = Q R 1/ α (1 α)/ α e 1/ α (1 α)/ α f (1 α)/ α surface surface, R R t e = Q R normal ( ) eff,pl = ρ δ 0 1/ α ( α ) g = Analysis of Data normal

202 Chapter 7. Constant Phase Elements page 7: 27 Test of C=Q (normal distributions)

203 Chapter 7. Constant Phase Elements page 7: 28 Test of C=Q (surface distributions)

204 Chapter 7. Constant Phase Elements page 7: 29 Constant Phase Elements Can arise from surface or axial distributions Parameters can be interpreted in terms of capacitance, depending on type of distribution C Q C C eff,norm C= C eff,surf or C= C eff,pl, depending on type of distribution B. Hirschorn, M. E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, J. Electrochem. Soc., 157 (2010) C458-C463. B. Hirschorn, M. E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, J. Electrochem. Soc., 157 (2010) C452-C457.

205 Chapter 7. Constant Phase Elements page 7: 30

206 Chapter 8. Regression Analysis page 8: 1 Advanced Electrochemical Impedance Spectroscopy Chapter 8. Regression Analysis Regression response surfaces noise incomplete frequency range Adequacy of fit quantitative qualitative Mark E. Orazem, All rights reserved.

207 Chapter 8. Regression Analysis page 8: 2 Nonlinear Regression p p p 2 Z 1 Z Z( p ) = Z( p ) + p + p p + 0 N N N j j k j= 1 pj 2 j= 1 k= 1 p p j pk p χ N ( Zi Z( ω p)) = σ dat 2 i= 1 i 2 i β= α p α β Variance of data k = N dat i= 1 Z Z( x p) Z( ω p) i i i 2 σ i pk 1 Z( ω p) Z( ω p) Ndat i i jk, 2 i= 1 σ i pj pk Derivative of function with respect to parameter

208 Chapter 8. Regression Analysis page 8: 3 Methods for Regression Evaluation of derivatives method of steepest descent Gauss-Newton method Levenberg-Marquardt method Evaluation of function simplex

209 Chapter 8. Regression Analysis page 8: 4 Effect of Noisy Data Add noise: 1% of modulus Sum of Squares R / Ω cm τ / s τ / s R / Ω cm response surface is smooth and parabolic value at minimum greater than zero

210 Chapter 8. Regression Analysis page 8: 5 Effect of Noisy Data: Test Circuit with 3 Time Constants R 0 = 1 Ω cm 2 R 1 = 100 Ω cm 2 C 1 C 2 C 3 τ 1 = s R 2 = 200 Ω cm 2 (1) (2) (3) τ 2 = 0.01 s R 0 R 3 = 5 Ω cm 2 R 1 R 2 R 3 τ 3 = 0.05 s Note: 3 rd Voigt element contributes only 1.66% to DC cell impedance.

211 Chapter 8. Regression Analysis page 8: 6 Effect of Noisy Data no noise noise: 1% of modulus 103 Sum of Squares Sum of Squares log10(r/ω cm ) log10(τ / s) log10(r/ω cm2) Note: use of log scale for parameters All parameters fixed except R3 and τ log10(τ / s)

212 Chapter 8. Regression Analysis page 8: 7 Resulting Spectrum -150 Z j / Ω cm Z r / Ω cm 2 Model with 1% noise added Model with no noise

213 Chapter 8. Regression Analysis page 8: 8 Effect of Frequency Range: Test Circuit with 3 Time Constants R 0 = 1 Ω cm 2 R 1 = 100 Ω cm 2 C 1 C 2 C 3 τ 1 = 0.01 s R 2 = 200 Ω cm 2 (1) (2) (3) τ 2 = 0.1 s R 0 R 3 = 100 Ω cm 2 R 1 R 2 R 3 τ 3 = 10 s Note: 3 rd Voigt element contributes 25% to DC cell impedance. The time constant corresponds to a characteristic frequency ω 3 =0.1 s -1 or f 3 =0.016 Hz.

214 Chapter 8. Regression Analysis page 8: 9 Resulting Test Spectra Z j / Ω cm Z r / Ω cm Hz to 100 khz 1 Hz to 100 khz

215 Chapter 8. Regression Analysis page 8: 10 Effect of Truncated Data 0.01 Hz to 100 khz 1 Hz to 100 khz 103 Sum of Squares Sum of Squares log10(r / Ω cm2) log10(τ / s) log10(r / Ω cm2) All parameters fixed except R3 and τ3-1 4 log10(τ / s)

216 Chapter 8. Regression Analysis page 8: 11 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.

217 Chapter 8. Regression Analysis page 8: 12 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

218 Chapter 8. Regression Analysis page 8: 13 Test Case: Mass Transfer to a RDE Z( ω) Rt + zd ( ω ) ( jωc) R z ( ω) = R + e ( ) t d Single reaction coupled with mass transfer. Consider model for a Nernst stagnant diffusion layer: z d ( ω) = z ( ω) tanh d jωτ jωτ

219 Chapter 8. Regression Analysis page 8: 14 Evaluation of χ 2 Statistic σ/ Z(ω) χ χ 2 /ν

220 Chapter 8. Regression Analysis page 8: 15 Comparison of Model to Data 80 Impedance Plane (Nyquist) Z j / Ω Z r / Ω Value of χ 2 has no meaning without accurate assessment of the noise level of the data

221 Chapter 8. Regression Analysis page 8: 16 Modulus 100 Z / Ω f / Hz

222 Chapter 8. Regression Analysis page 8: Phase Angle φ / degrees f / Hz

223 Chapter 8. Regression Analysis page 8: Real 160 Z r / Ω f / Hz

224 Chapter 8. Regression Analysis page 8: Imaginary 60 Z j / Ω f / Hz

225 Chapter 8. Regression Analysis page 8: Log Imaginary Slope = -1 for RC Z j / Ω f / Hz

226 Chapter 8. Regression Analysis page 8: Modified Phase Angle φ / degrees Z j φ* = tan Zr Re f / Hz

227 Chapter 8. Regression Analysis page 8: Real Residuals ε r / Z r ±2σ noise level f / Hz

228 Chapter 8. Regression Analysis page 8: Imaginary Residuals ε j / Z j ±2σ noise level f / Hz

229 Chapter 8. Regression Analysis page 8: 24 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

230 Chapter 8. Regression Analysis page 8: 25 Test Case: Better Model for Mass Transfer to a RDE Z( ω) R ( ) t + zd ω = Re + α 1 + jω Q R + z ( ω ) ( ) ( ) 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 θ ) Sc Sc 0 psc χ 2 /ν=4.86 1/3 2/3

231 Chapter 8. Regression Analysis page 8: 26 Comparison of Model to Data 80 Impedance Plane (Nyquist) Z j / Ω Z r / Ω

232 Chapter 8. Regression Analysis page 8: Phase Angle Z j / Ω f / Hz

233 Chapter 8. Regression Analysis page 8: Modified Phase Angle φ * / degrees Z j φ* = tan Zr Re f / Hz

234 Chapter 8. Regression Analysis page 8: Log Imaginary Z j / Ω f / Hz

235 Chapter 8. Regression Analysis page 8: Real Residuals ε r / Z r ±2σ f / Hz

236 Chapter 8. Regression Analysis page 8: Imaginary Residuals 0.01 ε j / Z j ±2σ f / Hz

237 Chapter 8. Regression Analysis page 8: 32 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

238 Chapter 8. Regression Analysis page 8: 33 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.

239 Chapter 8. Regression Analysis page 8: 34

240 Chapter 9. Error Structure page 9: 1 Advanced Electrochemical Impedance Spectroscopy Chapter 9. Error Structure Contributions to error structure Weighting strategies General approach for error analysis Experimental results Mark E. Orazem, All rights reserved.

241 Chapter 9. Error Structure page 9: 2 Contributions to Error Structure Sampling Errors Stochastic Phenomena Bias Errors Lack of Fit Changing baseline (non-stationary processes) Instrumental artifacts

242 Chapter 9. Error Structure page 9: 3 Time-domain Frequency Domain 1 mhz 10 Hz 100 Hz 10 khz

243 Chapter 9. Error Structure page 9: 4 Error Structure ε ( ω) = Z ( ω) Z ( ω) resid obs model fitting error = ε ( ω) + ε ( ω) + ε ( ω) fit bias stochastic inadequate model noise (frequency domain) experimental errors: inconsistent with the Kramers-Kronig relations

244 Chapter 9. Error Structure page 9: 5 Weighting Strategies J = No Weighting k Strategy Modulus Weighting Proportional Weighting Experimental Refined Experimental ( Z Z Z Z r k 2 r k ) ( j k j k ),, +,, 2 2 σ σ r, k k j, k 2 Implications σ r =σ j σ = α σ r =σ j σ = α Ζ σ r σ j σ r = α r Ζ r ; σ j = α j Ζ j σ r =σ j σ = α Ζ j + β Ζ r + γ Ζ 2 /R m σ r =σ j σ = α Ζ j + β Ζ r -R sol + γ Ζ 2 /R m +δ

245 Chapter 9. Error Structure page 9: 6 Assumed Error Structure often Wrong Data obtained by T. El Moustafid, CNRS, Paris, France

246 Chapter 9. Error Structure page 9: 7 Reduction of Fe(CN) 6 3- on a Pt Disk, 120 rpm -300 I/i lim Zj, Ω /4 1/2 3/ Z r, Ω

247 Chapter 9. Error Structure page 9: 8 Reduction of Fe(CN) 6 3- on a Pt Disk, Imaginary 120 rpm, 1/4 i lim

248 Chapter 9. Error Structure page 9: 9 Reduction of Fe(CN) 6 3- on a Pt Disk, Real 120 rpm, 1/4 i lim

249 Chapter 9. Error Structure page 9: 10 Reduction of Fe(CN) 6 3- on a Pt Disk, Error 120 rpm, 1/4 i lim

250 Chapter 9. Error Structure page 9: 11 Reduction of Fe(CN) 6 3- on a Pt Disk, Error 120 rpm, 1/4 i lim

251 Chapter 9. Error Structure page 9: 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

252 Chapter 9. Error Structure page 9: 13

253 Chapter 10. Kramers-Kronig Relations page 10: 1 Advanced Electrochemical Impedance Spectroscopy Chapter 10. Kramers-Kronig Relations Mathematical form and interpretation Application to noisy data Methods to evaluate consistency Mark E. Orazem, All rights reserved.

254 Chapter 10. Kramers-Kronig Relations page 10: 2 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 ) ( ω ω π ω ω dx x Z x xz Z Z j j r r + + = ) ( ) ( 2 ) ( ) ( ω ω ω π ω ω

255 Chapter 10. Kramers-Kronig Relations page 10: 3 Use of Kramers-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

256 Chapter 10. Kramers-Kronig Relations page 10: 4 For real data (with noise) )) ( ) ( ( )) ( ) ( ( ) ( ) ( ) ob( ω ε ω ω ε ω ω ε ω ω j j r r Z j Z Z Z = + = ( ) ) ( ) ( ω ε ω ε ω ε j r j + = ( ) ) ( ) ( ob ω ω Z Z E = ( ) 0 ) ( = ω ε E where If and only if ( ) + = ) ( ) ( ) ( ) ( 2 ) ( dx x x Z x Z E Z E r r r r j ω ω ε ε ω π ω ω ( ) + + = ) ( ) ( ) ( ) ( 2 ) ( ) ( dx x x x Z x xz E Z Z E j j j j r r ω ω ωε ε ω ω π ω Kramers-Kronig in an expectation sense

257 Chapter 10. Kramers-Kronig Relations page 10: 5 The Kramers-Kronig relations can be satisfied if E ( ε( ω) ) = 0 This means and 2ω E π 0 ε r ( x) dx 2 x ω 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, ε 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 ε r (x) should be of the same magnitude as the stochastic noise ε r (ω). 0

258 Chapter 10. Kramers-Kronig Relations page 10: 6 2 ω εr ( x) Meaning of E dx = π x ω 0 0 (Z r (x)-z r (ω)) / (x 2 -ω 2 ) (Z r (x)-z r (ω)) / (x 2 -ω 2 ) Correct Value Interpolated Value f / Hz f / Hz

259 Chapter 10. Kramers-Kronig Relations page 10: 7 Use of Kramers-Kronig Relations Quadrature errors require interpolation function Missing data at low and high frequency

260 Chapter 10. Kramers-Kronig Relations page 10: 8 Methods to Resolve Problems of Insufficient Frequency 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

261 Chapter 10. Kramers-Kronig Relations page 10: 9

262 Chapter 11. Use of Measurement Models page 11: 1 Advanced Electrochemical Impedance Spectroscopy Chapter 11. Use of Measurement Models Generalized model Identification of stochastic component of error structure Identification of bias component of error structure Mark E. Orazem, All rights reserved.

263 Chapter 11. Use of Measurement Models page 11: 2 Experimental design Approach 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

264 Chapter 11. Use of Measurement Models page 11: 3 Z Z Error Structure Z = ε obs model res Z = ε + ε + ε obs model fit bias stoch Stochastic errors Bias errors lack of fit experimental artifact Kramers-Kronig consistent Kramers-Kronig inconsistent

265 Chapter 11. Use of Measurement Models page 11: 4 Measurement Model Superposition of lineshapes Z = R + 0 n + Rk jω RC k = 1 1 k k

266 Chapter 11. Use of Measurement Models page 11: 5 Identification of Stochastic Component of Error Structure How? Replicate impedance scans. Fit measurement model to each scan: use modulus or noweighting 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.

267 Chapter 11. Use of Measurement Models page 11: 6 Replicated Measurements 100 -Zj / Ω Z r / Ω

268 Chapter 11. Use of Measurement Models page 11: 7 1 Voigt Element 100 -Zj / Ω Z r / Ω

269 Chapter 11. Use of Measurement Models page 11: 8 2 Voigt Elements 100 -Zj / Ω Z r / Ω

270 Chapter 11. Use of Measurement Models page 11: 9 3 Voigt Elements 100 -Zj / Ω Z r / Ω

271 Chapter 11. Use of Measurement Models page 11: 10 4 Voigt Elements 100 -Zj / Ω Z r / Ω

272 Chapter 11. Use of Measurement Models page 11: 11 5 Voigt Elements 100 -Zj / Ω Z r / Ω

273 Chapter 11. Use of Measurement Models page 11: 12 6 Voigt Elements 100 -Zj / Ω Z r / Ω

274 Chapter 11. Use of Measurement Models page 11: Voigt Elements 100 -Zj / Ω Z r / Ω

275 Chapter 11. Use of Measurement Models page 11: 14 Residual Errors 0.03 εj / Zj 0 imaginary Frequency / Hz real εr / Zr Frequency / Hz

276 Chapter 11. Use of Measurement Models page 11: Residual Sum of Squares Σ(ε 2 /σ 2 ) # Voigt Elements

277 Chapter 11. Use of Measurement Models page 11: 16 Fit to Repeated Data Sets 100 -Zj / Ω Z r / Ω

278 Chapter 11. Use of Measurement Models page 11: Residual Errors Real εr / Zr Frequency / Hz

279 Chapter 11. Use of Measurement Models page 11: 18 Standard Deviation of Residual Errors Imaginary 0.02 εj / Zj Frequency / Hz

280 Chapter 11. Use of Measurement Models page 11: 19 Standard Deviation

281 Chapter 11. Use of Measurement Models page 11: 20 Comparison to Impedance

282 Chapter 11. Use of Measurement Models page 11: 21 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.

283 Chapter 11. Use of Measurement Models page 11: 22 1 st of Repeated Measurements Imaginary

284 Chapter 11. Use of Measurement Models page 11: 23 1 st of Repeated Measurements Real