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Influence of current distribution on the interpretation of the impedance spectra collected for a rotating disk electrode

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Influence of current distribution on the interpretation of the impedance spectra collected for a rotating disk electrode
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Durbha, Madhav
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English
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xx, 310 leaves : ill. ; 29 cm.

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Chemical Engineering thesis, Ph.D ( lcsh )
Corrosion and anti-corrosives ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF ( lcsh )
Electrodes, Copper ( lcsh )
Seawater ( lcsh )
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Thesis:
Thesis (Ph.D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 300-309).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Madhav Durbha.

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University of Florida
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Full Text
INFLUENCE OF CURRENT DISTRIBUTIONS ON THE INTERPRETATION OF
THE IMPEDANCE SPECTRA COLLECTED FOR A ROTATING DISK ELECTRODE
By
MADHAV DURBHA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1998


Dedicated
To
My Parents


ACKNOWLEDGMENTS
I would like to sincerely thank Prof. Mark Orazem for his constant support,
encouragement, and guidance through the course of this research work. Apart from
educating me in several aspects of electrochemical engineering, he provided me with the
unique opportunity of working for a semester with the electrochemical research group at
the CNRS, Paris. I would also like to thank him for equipping the laboratory with state of
the art computing facilities without which this gigantic piece of work would not have been
finished in this time span.
My heartfelt thanks to Prof. Luis Garcia Rubio of University of South Florida, and
to Drs. Claude Deslouis, Bernard Tribollet, and Hisasi Takenouti of CNRS, Paris, for their
valuable suggestions during various stages of this research work. I would like to thank
Profs. Oscar Crisalle, Chang Park, Raj Rajagopalan of Chemical Engineering, and Prof.
C.C.Hsu of the University of Florida for serving on my dissertation committee.
Thanks are due to my colleagues Steven Carson and Michael Membrino, not just
for numerous intellectual exchanges I had with them which thoroughly enhanced my
understanding of the subject, but for also educating me in various aspects of American
culture. Douglas Riemers help with software and computers is invaluable. I would like to
acknowledge the financial support of the Office of Naval Research. I would also like to
thank CNRS, Paris, for their supporting my stay in Paris.


On a more personal level there are a number of people who contributed to my
success and it would be impossible thank each of them individually. However I would like
to make a special mention of my friend and colleague Basker Varadharajan for his
wonderful friendship and for being an outstanding roommate through my graduate studies.
I would like to thank my sisters Dr.Padma and Mrs.Sujana for leading the path
towards an advanced degree in engineering, and for creating the academic ambience at
home. The constant care and encouragement of my sisters and brothers-in-law has been
instrumental for my success.
I would like to take this opportunity to thank a very special person, Dr.Apama, for
all the love, care, and encouragement that she provided during the final phase of my
research work. I cherish all the sweet moments that we shared and look forward to an
exciting future with her.
I would not have been where I am without all the sacrifices made by my mother
and father in providing me with every possible opportunity at every phase of my life. Both
of them being in academic positions was of tremendous help towards my academic
achievements. With their genuine concern for others and with their extremely likable
personalities, they serve as my role models in shaping up my overall personality. I owe
everything to them for what I was, for what I am, and for what I am going to be. This
work is dedicated to them as a small token of my gratitude.
IV


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
LIST OF TABLES ix
LIST OF FIGURES xi
ABSTRACT xix
CHAPTERS
1 INTRODUCTION 1
1.1 Rotating Disk Electrode 2
1.2 Frequency Domain Techniques 4
1.3 Motivation for this Work 5
1.4 Approach to the Problem 6
2 THE SCHMIDT NUMBER FOR FERRICYANIDE IONS: EXPERIMENTAL
DESIGN AND DATA ANALYSIS 13
2.1 Measurement Model 14
2.1.1 Importance of identifying the Stochastic Noise Level 14
2.1.2 Classification of Errors 16
2.1.3 Kramers-Kronig Relations 17
2.1.4 Identification of Noise Level in the Measurement and Consistency
Check 18
2.2 Process Model 19
2.2.1 Process Model for EIS 20
2.2.2 Process Model for EHD 24
2.3 Experimental Design 26
2.3.1 Choice of Surface Treatment 26
2.3.2 Experimental Setup 28
2.4 Data Analysis: Measurement Model 30
2.4.1 Initial Regressions of the Data 30
2.4.2 Obtaining the Error Structure 32
2.4.3 Identification of the Self-Consistent Part of the Impedance Spectra 34
2.5 Data Analysis: Process Model 36
2.6 Results from the Steady State Measurements 38
v
I


2.7 Discussion 40
2.8 Conclusions 42
3 INFLUENCE OF SURFACE PHENOMENA ON THE IMPEDANCE
RESPONSE OF A ROTATING DISK ELECTRODE 82
3.1 Experimental Protocol 83
3.2 Results and Discussion 84
3.3 Conclusions 87
4 STEADY STATE MODEL FOR A ROTATING DISK ELECTRODE BELOW
THE MASS-TRANSFER LIMITED CURRENT 97
4.1 Theoretical Development 99
4.1.1 Diffusion Layer 99
4.1.2 Outer Region: Laplaces Equation 103
4 .1.3 Diffuse Part of the Double Layer 104
4.1,3a Solution of Poissons equation 105
4.1.3b Calculation of double-layer capacitance 108
4.2 Numerical Procedure 109
4.2.1 Solution to the Convective Diffusion Equation 109
4.2.2 Algorithm for Implementation of the Model 111
4.3 Application to Experimental Systems 112
4.3.1 Electrodeposition of Copper 112
4.3.2 Reduction of Ferricyanide on Pt 113
4.3.2a Current, Potential, and Charge Distributions 113
4.3.2b Zero Frequency Asymptotes of Local Impedance 115
4.4 Conclusions 116
5 A MATHEMATICAL MODEL FOR THE RADIALLY DEPENDENT
IMPEDANCE OF A ROTATING DISK ELECTRODE 132
5.1 Theoretical Development 134
5.1.1 Convective Diffusion 134
5.1.2 Conditions on Current 141
5.1.2a Mass transport 141
5.1.2b Kinetics 142
5.1.3 Potential 143
5.2 Numerical Procedure 145
5.3 Results and Discussion 148
5.3.1 Uniform Current Distribution 148
5.3.2 Non-Uniform Current Distribution 149
5.4 Conclusions 152
6 CHEBYSHEV POLYNOMIAL SOLUTION FOR THE STEADY STATE
CONVECTIVE DIFFUSION FOR A ROTATING DISC ELECTRODE 170
6.1 Transformation of the Convective Diffusion Equation 173
6.1.1 Series Approximations 175
vi


6.1.2 Recursion Relation for X*y' 177
6.1.3 Substitution into the Convective Diffusion Equation 179
6.1.4 Non Homogeneous Equations 180
6.2 Results and Discussion 181
6.3 Conclusions 182
7 SPECTROSCOPY APPLICATIONS OF THE KRAMERS-KRONIG
TRANSFORMS: IMPLICATIONS FOR ERROR STRUCTURE
IDENTIFICATION 186
7.1 Experimental Motivation 187
7.2 Application of the Kramers-Kronig Relations 189
7.3 Absence of Stochastic Errors 190
7.4 Propagation of Stochastic Errors 192
7.4.1 Transformation from Real to Imaginary 193
7.4.2 Transformation from Imaginary to Real 197
7.5 Experimental Verification 199
7.6 Implications for the Error Structure 200
7.7 Conclusions 202
8 COMMON FEATURES FOR FREQUENCY DOMAIN MEASUREMENTS 208
8.1 Similarity in Terms of Line Shapes 210
8.2 Similarity in Terms of Transfer Function 211
8.2.1 Electrochemical Impedance Spectroscopy 211
8.2.2 Rheology of Viscoelastic Fluids 211
8.2.3 Optical Spectroscopy 212
8.2.4 Acoustophoretic Spectroscopy 214
8.3 Similarity in Terms of the Kramers-Kronig Relations 215
8.4 Similarity in Terms of Error Structure 215
8.5 Experimental Results and Discussion 217
8.5.1 Electrochemical Impedance Spectroscopy 217
8.5.2 Test Circuit 220
8.5.3 Electrohydrodynamic Impedance Spectroscopy 222
8.5.4 Rheology of Viscoelastic Fluids 224
8.5.5 Acoustophoretic spectroscopy 225
8.6 Conclusions 225
9 CONCLUSIONS 244
10 SUGGESTIONS FOR FUTURE WORK 246
APPENDICES
A STEADY STATE MODEL FOR THE ROTATING DISK ELECTRODE 247
B FREQUENCY DOMAIN MODEL FOR THE ROTATING DISK
ELECTRODE 269
vii


LIST OF REFERENCES 300
BIOGRAPHICAL SKETCH 310
vm


LIST OF TABLES
Table page
2 .1. The conditions and the number of repeated measurements chosen for
establishing the error structure for treatment 1 43
2.2. The conditions and the number of repeated measurements chosen for
establishing the error structure for treatment 2 43
2.3. Results from the process model regression for EIS data collected using
treatment 1, without CPE correction 44
2.4. Selected results from the process model regression for EIS data collected
using treatment 1, without CPE correction 46
2.5. Results from the process model regression for EIS data collected using
treatment 1, with CPE correction 47
2.6. Selected results from the process model regression for EIS data collected
using treatment 1, with CPE correction 48
2.7. Results from the process model regression for EIS data collected using
treatment 2, without CPE correction 49
2.8. Selected results from the process model regression for EIS data collected
using treatment 2, without CPE correction 51
2.9. Results from the process model regression for EIS data collected using
treatment 2, with CPE correction 52
2.10. Results from the process model regression for EIS data collected using
treatment 2, with CPE correction 53
2.11 Results from the process model regression for EHD data collected at the
mass-transfer-limited current using treatment 1 54
2.12. Selected results from the process model regression for EHD data collected
at the mass-transfer-limited current using treatment 1 54
2.13. Limiting current values obtained for treatment 2 at different rotation speeds 55
IX


2.14. Schmidt numbers obtained from the ium values presented in Table 2.14 55
4.1. Polynomial coefficients in the expansion for 0'm(0) resulting from the
solution of the convective diffusion equation. The number of significant
digits reported are based on the respective confidence intervals from the
regression 118
4.2. Input parameters used for the ferri/ferro cyanide in 1M KC1 system reacting
on the Pt disc electrode 119
5.1. One-dimensional frequency domain process model regressions for the two-
dimensional model calculations for a Sc value of 1100 and for an exchange
current density of 50mA/cm2. Dimensionless parameter J= 3.7445 154
6.1. Comparison between the Chebyshev approximation and the FDM scheme
for m = 0 in the convective diffusion equation. Value of 0'm(o) obtained
by extrapolation to a step size of zero value is -1.119846522021 183
6.2. Comparison between the Chebyshev approximation and the FDM scheme
for m = 5 in the convective diffusion equation. Value of 0'm (o) obtained by
extrapolation to a step size of zero value is -2.340450747254 183
6.3. Comparison between the Chebyshev approximation and the FDM scheme for
m = 10 in the convective diffusion equation. Value of 0'm (o) obtained by
extrapolation to a step size of zero value is -2.901505452807 184
x


LIST OF FIGURES
Figure page
1.1. Flow near a rotating disk electrode 10
1.2. Small-signal analysis of an electrochemical non-linear system 11
1.3. Flow diagram of the research work performed. Major contributions from this
work are italicized 12
2.1. The DC polarization curves for various surface treatments. The results are
presented for the cathodic region, as this is the region of interest for this
work. Measurements were made at 600 rpm 56
2.2. Experimental setup for the impedance measurements 57
2.3. Regression of a measurement model with 8 Voigt elements to the EIS data
obtained for rotating disk electrode at 120 rpm, l/4th of the limiting current.
Solid line in the figures is the measurement model fit and circles represent
the data 58
2.4. Regression of a measurement model with 8 Voigt elements to the EIS data
obtained for rotating disk electrode at 120 rpm, l/4th of the limiting
current, corresponding to the condition in Figure 2.3. (a) real part as a
function of frequency and (b) imaginary part as a function of frequency 59
2.5. Normalized residual errors in the (a)real and (b)imaginary parts as functions of
frequency for the regression of a measurement model with 8 Voigt elements
to the EIS data obtained for rotating disk electrode at 120 rpm, l/4th of
the limiting current 60
2.6. Standard deviations in the real (O) and imaginary (A) parts calculated using
measurement models with modulus weighting for the 3 replicates of EIS
data collected at 120rpm, 1/4* of limiting current case for the rotating disk
electrode system 61
2.7. Regression of a measurement model with 2 Voigt elements to the EHD data
obtained for rotating disk electrode at 120 rpm, mass-transfer-limited current
using treatment 1. Solid line in the figures is the measurement model fit and
circles represent the data 62
xi


2.8. Regression of a measurement model with 2 Voigt elements to the EHD data
obtained for rotating disk electrode at 120 rpm, mass-transfer-limited current.
The corresponds to that in Figure 2.7. Solid line in the figures is the
measurement model fit. (a) real and (b) imaginary parts as functions of
frequency 63
2.9. Normalized residual errors in the (a) real and (b) imaginary parts as functions of
frequency from the regression of a measurement model with 2 Voigt elements
to the EHD data obtained (corresponding to Figure 2.7) for rotating disk
electrode at 120 rpm, mass-transfer-limited current 64
2.10. The solid line represents the error structure model obtained by accounting
for various conditions for the rotating disk electrode, using treatment 1.
The (O)s and the (A)s represent the standard deviations of the stochastic
noise obtained by using the measurement model approach applied to
120rpm, 1/4th of limiting current 65
2.11. The solid line represents the error structure model obtained by
accounting for various conditions for the rotating disk electrode, using
treatment 1. The (O)s and the (A)s represent the standard deviations of the
stochastic noise obtained by using the measurement model approach applied
to 1200 rpm, 1/2 of limiting current 66
2.12. The solid line represents the error structure model obtained by accounting
for various conditions for the rotating disk electrode, using treatment 1.
The (O)s and the (A)s represent the standard deviations of the stochastic noise
obtained by using the measurement model approach applied to 3000 rpm,
1/2 of limiting current 67
2.13. Checking for consistency with the Kramers-Kronig relations. EIS data
collected for 120 rpm, 1/4* of the limiting current case for the rotating disk
electrode system. Measurement model was regressed to the (a)real part and
(b)imaginary part was predicted based on the 10 lineshape parameters
obtained. The outer lines represent the 95.4% confidence limits and the
line through the data is the measurement model fit 68
2.14. Normalized residual errors in (a) real and (b) imaginary parts corresponding
to the regression results presented in Figure 2.13. The outer lines represent
the 95.4% confidence limits and the line through the data is the measurement
model fit 69
2.15. Checking for consistency with the Kramers-Kronig relations. 120 rpm, 1/4th of
the limiting current case for the rotating disk electrode system.
Measurement model was (a)regressed to the imaginary part and (b)real part is
predicted based on the 11 lineshape parameters obtained. The outer lines
represent the 95.4% confidence limits and the line through the data is the
Xll


measurement model fit.
70
2.16. Normalized residual errors in (a) real and (b) imaginary parts corresponding to
the regression results presented in Figure 2.15. The outer lines represent
the 95.4% confidence limits and the line through the data is the
measurement model fit 71
2 17. Process model regression (with error structure weighting) for 120 rpm,
1/4* of the limiting current case for the rotating disk electrode system.
Error structure was used to fit the data to the model. The solid line
represents fit of the model to the data 72
2.18. Process model regression (with error structure weighting) for EIS data
collected at 120 rpm, 1/4* of the limiting current case for the rotating
disk electrode system, corresponds to Figure 2.17. Error structure was
used to fit the data to the model. The solid line represents fit of the model
to the data. Outer lines represent the 95.4% confidence limits 73
2.19. Normalized residual errors in (a) real and (b) imaginary parts corresponding
to the regression results presented in Figure 2.17. The outer lines
represent the 95.4% confidence limits 74
2.20. Process model regression (with error structure weighting) accounting for
CPE correction for the EIS data collected for 120 rpm, 1/4* of the
mass-transfer-limited current case for the rotating disk electrode system
using treatment 1. The solid line represents fit of the model to the data 75
2.21. Process model regression (with error structure weighting) accounting for
CPE correction for the EIS data collected for 120 rpm, 1/4* of the
mass-transfer-limited current case for the rotating disk electrode system
using treatment 1, corresponding to Figure 2.20. The solid line
represents fit of the model to the data, (a) real and (b) imaginary parts
as functions of frequency 76
2.22.Normalized residual errors in (a) real and (b) imaginary parts corresponding
to the regression results presented in Figure 2.20. The dashed lines
represent the normalized noise level 77
2.23. Process model regression (with error structure weighting) for the EHD data
collected for 120 rpm, at the mass-transfer-limited current case for the
rotating disk electrode system using treatment 1. The solid line represents
fit of the model to the data 78
2.24. Process model regression (with error structure weighting) for the EHD
data collected for 120 rpm, at the mass-transfer-limited current case for
the rotating disk electrode system using treatment 1, corresponds to
Figure 2.23. The solid line represents fit of the model to the data.
xm


(a) Real part and (b) Imaginary part as functions of frequency 79
2.25. Residual errors in (a) real and (b) imaginary parts corresponding to the
regression results presented in Figure 2.23. The dashed lines represent
the noise level 80
2.26. The square root of the rotation speed plotted against the mass-transfer-
limiting current value. The line passing through is regressed ignoring
the 3000rpm case 81
3.1. Imaginary part of the impedance for reduction of ferricyanide on a Pt disk
rotating at 120 rpm and at 174th of the limiting current. The time
trending between the spectra can be seen very clearly 88
3.2. Real part of the impedance for the repeated measurements with time as a
parameter 89
3.3. Error structure for the data presented in Figure 3.1 and Figure 3.2: filled
symbols represent the statistically calculated standard deviations of
repeated measurements; open symbols are the standard deviations of
the stochastic noise calculated using the measurement model approach 90
3.4. Normalized residual sum of squares for regression of a process model to the
data presented in Figure 3.1 and Figure 3.2. The inner and outer dashed
lines correspond to the 0.05 and 0.01 levels of significance for the F-test 91
3.5. Schmidt number obtained by regression of process model to the data 92
3.6. Charge transfer resistance obtained by regression of process model to the data 93
3.7. Mass transfer resistance obtained by regression of a process model to the data 94
3.8. Double layer capacitance obtained by regression of process model to the data 95
3.9. Exponent in the CPE element obtained by regression of process model to the
data 96
4.1. Determination of the accurate value for for infinite Schmidt number,
making use of the values obtained from the FDM scheme using varying
step-sizes 120
4.2. A sixth degree polynomial fit for 7(0) vs. Sc"1/3 121
4.3. Errors in 6^(0) values between polynomial fits and the values calculated from
the FDM scheme 122
4.4. Calculated (a) concentration and (b) current distribution on the surface of
xiv


the disk electrode for deposition of copper under the condition
corresponding to figures (6) and (7) of reference (41) with N=50. Adjacent
infinite Sc (dashed lines) and finite Sc (solid lines) are for same applied
potential. In the order of decreasing concentration, the applied potentials
(V-Oref) used were -0.08V, -0.28 V, -0.68V, -0.98V, -1.28V, and -1.58V 123
4.5. Calculated current distributions for the reduction of ferricyanide on a Pt disk
electrode rotating at (a)120rpm and (b)3000 rpm. System properties are
given in Table 4.2 124
4.6. Calculated overpotentials for the case of Figure 4.5a (120rpm) at (a)l/4th of ium,
(b)l/4 of iiimOn an enlarged scale to show the distributions of tjs and £. 125
4.7. Calculated overpotentials for the case of Figure 4.5a (120rpm) at 3/4th of in 126
4.8. Calculated overpotentials for the case of Figure 4.5b (3000rpm) at (a)l/4th of ium,
(b)l/4 of iiimOn an enlarged scale to show the distributions of r¡* and £. 127
4.9. Calculated overpotentials for the case of Figure 4.5b (3000rpm) at 3/4th of ium 128
4.10. Calculated charge distributions for the cases of (a) Figure 4.5a (120rpm) and
(b) Figure 4.5b (3000rpm) 129
4.11. Calculated local impedance distributions corresponding to Figure 4.5a
(120 rpm) for (a)l/4th of ii, and (b)3/4th of ium 130
4.12. Calculated local impedance distributions corresponding to Figure 4.5b
(3000 rpm) for (a)l/4th of ium, and (b)3/4th of ium 131
5.1. (a) Comparison between one-dimensional and two-dimensional models for
the slow kinetics case at 174th of /iim and Q=120rpm with /'0 = 3 mA/cm2,
D = 0.3095xl05 cm2/sec, J = 0.225, N= 0.0695, and Sc = 2730. In this
case steady-state distributions tend to be highly uniform, (b) Differences
between the calculations from two-dimensional and one-dimensional
model normalized with respect to the two-dimensional model as a function
of frequency 155
5.2. Comparison between the impedance spectra generated by ID and 2D models
for 120rpm, 174th of ium, /'0 = 30 mA/cm2, D = 0.3095x10'5 cm2/sec, J = 2.247,
N= 0.0695, and Sc = 2730. Results presented for impedance plane plot 156
5.3. Comparison between the impedance spectra generated by ID and 2D models
for 120rpm, 174th of in, z0 = 30 mA/cm2, D = 0.3 095x10'5 cm2/sec, J= 2.247,
N= 0.0695, and Sc = 2730 (corresponds to Figure 5.2). Results presented
for (a) real part as a function of frequency (b) imaginary part as a function
of frequency 157
xv


5.4. Comparison between the impedance spectra generated by ID and 2D model for
3000rpm, 1/4* of in, i0 = 100 mA/cm2, D = 0.3195xl0'5 cm2/sec, J= 7.489,
N= 0.3552, and Sc = 2650 158
5.5. Comparison between the impedance spectra generated by ID and 2D model
for 3000rpm, lM* of ilim, i0 = 100 mA/cm2, D = 0.3195x10'5 cm2/sec,
J= 7.489, N = 0.3552, and Sc = 2650 (corresponds to Figure 5.4).
Results presented for (a) real part as a function of frequency (b) imaginary
part as a function of frequency 159
5.6. Comparison between the impedance spectra generated by ID and 2D model
for 120rpm, 3/4th of ium, io = 100 mA/cm2, D = 0.5095xl0'5 cm2/sec,
J= 7.489, N = 0.0970, and Sc = 1660. Results presented for impedance
plane plot 160
5.7. Comparison between the impedance spectra generated by ID and 2D model for
120rpm, 31^ of ium, io = 100 mA/cm2, D = 0.5095xl0'5 cm2/sec, J= 7.489,
N = 0.0970, and Sc = 1660 (corresponds to Figure 5.6). Results presented
for (a) real part as a function of frequency (b) imaginary part as a function
of frequency 161
5.8. Comparison between the impedance spectra generated by ID and 2D model
for 3000rpm, 3/4* of ium, io = 75mA/cm2, D = 0.6795xl05 cm2/sec, J= 5.617,
N = 0.5874, and Sc = 1250. Results presented for impedance plane plot 162
5.9.Comparison between the impedance spectra generated by ID and 2D model
for 3000rpm, 3/4* of ium, io = 75mA/cm2, D = 0.6795xl0*5 cm2/sec, J- 5.617,
N= 0.5874, and Sc = 1250 (corresponds to the condition of Figure 5.8).
Results presented for (a) real part as a function of frequency (b) imaginary
part as a function of frequency 163
5.10.Distributions for local impedance values for a dimensionless frequencies
of K=1 and K=2.8. The parameter values are those given in Figure 5.6
and Figure 5.8 164
5.11. Results for regression of the ID model to a 2D model simulation for
120rpm, 3/4*** of iim. An input value of Sc = 1660 for 2D model resulted
in a regressed Sc of 1780 for the ID model case, (a) complex plane plot
(b) real impedance as a function of frequency (c) imaginary impedance
as a function of frequency 166
5.12. Results for regression of the ID model to a 2D model simulation for
3000rpm, 3/4^ of ium. An input value of Sc = 1250 for 2D model resulted
in a regressed Sc of 1530 for the ID model case, (a) complex plane plot
(b) real impedance as a function of frequency (c) imaginary impedance
as a function of frequency 168
xvi


6.1.Chebyshev polynomials as functions of x
185
7.1. Hierarchical representation of spectroscopic measurements. The shaded
boxes represent measurement strategies for which the real and imaginary
parts of Kramers-Kronig-transformable impedance were found to have the
same standard deviation. Following completion of the analysis reported here,
an experimental investigation was begun which showed that the real and
imaginary parts of complex viscosity also have the same standard deviation
if the spectra are consistent with the Kramers-Kronig relations 204
7.2. Path of integration for the contour integral in the complex-frequency plane 205
7.3. Weighting factor for Eq. (7.17) as a function of m normalized to show relative
contributions to the integral 206
7.4. Real (a) and imaginary (b) parts of a typical electrochemical impedance spectrum
as a function of frequency. The normal probability distribution function,
shown at a frequency of 0.03 Hz, shows that one consequence of the equality
of the standard deviations for real and imaginary components is that the level
of stochastic noise as a percentage of the signal can be much larger for
one component than the other 207
8.1. Line-shape models yielding the same mathematical structure for spectroscopic
response: a) Voigt model for electrochemical systems; b) Kelvin-Voigt
model for rheology of viscoelastic fluids 227
8.2. (a)The impedance response obtained under potentiostatic modulation for
reduction of ferricyanide on a Pt disk electrode rotating at 120 rpm, at
174th of mass-transfer limited current in a 1M KC1 aqueous solution.
Closed symbols represent the impedance values and open symbols represent
the corresponding standard deviation. O) Real part and A) Imaginary part.
(b) F-test parameters. The inner dashed lines represent the 95% confidence
limits for the F-test parameter and the outer lines represent the 99%
confidence limits. Circles represent the F-test parameters for the raw
standard deviations, (c) F-test parameters after deleting the point close to
50Hz and 100Hz. (d) Histogram with 7-test results 228
8.3. (a)The impedance response obtained under galvanostatic modulation for a
parallel RiCi circuit in series with a resistor Ro (Ro/Ri=10). Closed
symbols represent the impedance values and open symbols represent
the corresponding standard deviation. The line represents the model for
the error structure given as equation (8.18). O) Real part and A) Imaginary
part, (b) F-test corresponding to the variances of stochastic noise
(c) Histogram with 7-test results corresponding to the variance of
stochastic noise 232
XVII


8.4. (a)The EHD impedance response obtained for reduction of ferricyanide on a
Pt disk electrode rotating at 200 rpm in a 1M KC1 aqueous solution.
Closed symbols represent the electro-hydrodynamic impedance values and
open symbols represent the corresponding standard deviation. The line
represents the model for the error structure given as equation (8.18).
O) Real part and A) Imaginary part, (b) Statistical F-test to verify the
equality of standard deviations in the stochastic noise (c) Histogram with
/-test results corresponding to the variance of stochastic noise 235
8.5. (a)The complex viscosity for high density polyethylene melt. Closed symbols
represent the viscosity values and open symbols represent the
corresponding standard deviation. O) Real part and A) Imaginary part.
(b) F-test corresponding to the variances of stochastic noise (c) Histogram
with /-test results corresponding to the variance of stochastic noise 238
8.6. (a)The complex mobility for a suspension of polyacrylic acid (PAA) with a
density of 0.062 g/L, a pH of 10, and a molecular weight of 5000. Closed
symbols represent the mobility values and open symbols represent the
corresponding standard deviation. O) Real part and A) Imaginary part.
(b) F-test corresponding to the variances of stochastic noise (c) Histogram
with /-test results corresponding to the variance of stochastic noise 241
XVlll


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
INFLUENCE OF CURRENT DISTRIBUTIONS ON TILE INTERPRETATION OF
THE IMPEDANCE SPECTRA COLLECTED FOR A ROTATING DISK ELECTRODE
By
Madhav Durbha
August, 1998
Chairman: Prof. Mark E. Orazem
Major Department: Chemical Engineering
The influence of current distributions on the interpretation of the frequency domain
data collected for a rotating disk electrode is presented in this work. Numerous steady
state and frequency domain measurements were conducted for ferricyanide reduction on a
platinum disk electrode. The error characteristics of the data were established using the
measurement model software developed in-house, and this information was used in further
regressions of the process model to the experimental data. The frequency domain process
model currently used to describe the physics of the system was found to be inadequate. A
very sophisticated steady-state model, accounting for a finite Schmidt number and for
charge distribution within the diffuse part of the double layer, was developed here to
create the base line values needed to study the frequency perturbations. This model was
then used to develop a two dimensional frequency-domain process model for the system
xix


of interest. It is shown here that this frequency domain model provides a significant
improvement over the one-dimensional model used for the preliminary analysis.
One surprising result from this work is the equality of standard deviations in the
real and imaginary parts of the stochastic noise in the frequency domain data analyzed. An
analytical proof for this result, based on Kramers-Kronig relations, is presented here. This
result was very general and was found to be true even in the case of frequency domain
data collected for non-electrochemical systems. Supportive evidence based on a number of
diverse spectroscopic techniques is presented in this work.
xx


CHAPTER 1
INTRODUCTION
Proper understanding of the current and potential distributions in electrochemical
systems is of great practical importance. The predominant industrial applications are in
cathodic protection, electrodeposition, and transdermal delivery of drugs. Underlying
principles being the same, the main focus of these applications is in understanding and
controlling the distribution of current on the surface of interest. One of the classic
examples is the cathodic protection of underground or underwater pipelines by the
placement of sacrificial anodes. These anodes are made of metals that are less noble or
more vulnerable to corrosion compared to the metal used in making the pipeline. As an
example, zinc is used in protecting stainless steel pipes. As the local current density on the
pipeline is an indication of the protection to the surface in that location, it is important to
know the distribution of current on the pipeline in order to identify the regions that are
overprotected or under protected. Knowledge of current distribution is of paramount
importance to the corrosion engineer as this facilitates a judicious choice of one cathodic
protection system over the other or a choice of one electrode configuration over the other.
One of the recent breakthroughs in semiconductor device fabrication is in the use
of copper as an interconnect in integrated circuits. Due to the superior properties of
copper over Aluminium which is traditionally used in device fabrication, a large number of
integrated circuit manufacturers are expressing increased amount of interest in using
copper. Various techniques were attempted for the deposition of copper, such as vapor
1


2
deposition, electroless deposition and so on, and electrodeposition of copper is found to
be the most viable technique. The use of copper for interconnects is expected to enhance
the performance of various appliances such as microprocessors and memory circuits.
Uniformity of deposition is a very important issue here and excess of copper should be
avoided at all costs, as the semi-conductor process engineers are interested in sub-micron
feature sizes. In the context of electrodeposition, the amount of metal being deposited is
directly related to the local current density. Hence it is very important to identify and
control the local current density distribution in order to achieve layers of desirable quality
in specified locations. This is a complicated task, as understanding the current distributions
in an electrochemical system requires an in-depth knowledge of the effects of ohmic drop,
kinetic contribution, and mass transfer related issues. Recent past has seen a growing
amount of literature pertaining to the theoretical and experimental aspects of the current
and potential distributions for a number of electrochemical systems. Two electrode setups
which are very commonly used are rotating disk electrode and impinging jet disk
electrode. These two setups share similarities with respect to the associated fluid
mechanics. Models developed for one system can very easily be extended to the other by
properly accounting for various velocity components in modeling the convection.
1.1 Rotating Disk Electrode
The metal electrode is made in the form of a cylinder and surrounding it is an
insulating material so that the circular face of the disk is exposed to the electrolyte, as
shown in Figure 1.1. The metal-insulator assembly is arranged concentrically and is
rotated about the center with the help of a rotor. Due to this design, the wall and edge


3
effects for the electrode can be ignored. This system is very effective for the identification
of mechanisms and associated rate constants for electrode reactions, for studying
homogeneous reactions accompanied by electrode processes, and for the measurement of
diffusion coefficients of dissolved species. Here are some of the very important features of
this system:
The fluid flow for this system is very well defined and the uniform axial velocity yields
a uniform mass-transfer-limited current density.
Due to the imposed rotation rate, the effects associated with the free convection can
be ignored.
By increasing the rotation speed, the mass-transfer-limited current can be increased
and this results in an improved signal to noise ratio in measuring the current.
The fluid mechanics associated with this system are very well understood.
Because of these features, the rotating disk electrode system attracted the attention
of a number of researchers interested in the current and potential distribution studies. The
introductory parts of chapters 4 and 5 summarize the major contributions in steady state
and in frequency domains for the rotating disk electrode system. Steady-state techniques
include simple current-voltage measurements with no time dependence, that is, for an
imposed value of potential a value of current is obtained and vice versa, after allowing the
system to attain a steady-state. Frequency domain data are more complicated to analyze
compared to the steady-state data, as the information content is more. Before proceeding
further, it is important to gain some basic understanding of the frequency domain
techniques.


4
1.2 Frequency Domain Techniques
The fundamental approach of all frequency domain techniques is to apply a small
amplitude sinusoidal excitation signal (such as voltage or current) to the system under
investigation and to measure the response (such as current or voltage). The excitation
signal can be applied in several ways. The two most commonly employed methods are
multi-sine and single-sine techniques. Fast measurement time and mild perturbation of the
system under investigation are stated to be the strengths of the multi-sine technique,
though it has been shown that by using a fast frequency response analyzer, the difference
in the measurement time for a single-sine and a multi-sine technique is small [1], One of
the disadvantages of this technique is that it has a small frequency range. The multi-sine
technique is also sensitive to harmonic distortion. In this work impedance data was
collected using the single-sine technique.
The use of single-sine technique used for electrochemical impedance spectroscopy
is illustrated in Figure 1.2. As shown in this figure, a low amplitude sine wave AE sin cot is
superimposed on the dc polarization voltage Eo. Hence, a low amplitude sine wave AI sin
(cot-4)) is observed to be superimposed on the dc current. The Taylor series expansion for
the current is given by
A1 =
rdi^
ydEj
E.J,
AE + -
2
( 2 T\
d2I
dE2
(AE)2+-
(1.1)
' EqJo
As the system under consideration is non-linear in nature, higher order derivatives do
exist. However, for a very small perturbation in potential, terms of order
f d2I^
(A£) and higher can be neglected and only the linear terms need to be
Eo.-^o


5
retained. This process is known as quasi-linearization and is widely used in non-linear
system analysis.
In the rotating disk electrode system, a number of variables can influence the
output. In recent years generalized impedance techniques have been introduced in which a
nonelectrical quantity such as pressure, temperature, magnetic field, or light intensity is
modulated to give a current or potential response [2,3], Electrohydrodynamic impedance
(EHD) is one such generalized impedance technique in which sinusoidal modulation of the
disk rotation rate drives a sinusoidal current or potential. One attractive feature of EHD is
that this technique can be used under mass-transfer limitation. More details about this
technique will be presented in a later chapter.
1.3 Motivation for this Work
The importance of the current and potential distributions in case of various
engineering problems is highlighted in the beginning of this chapter. The motivating factor
for this study is to understand the issue of current distributions in the flow-induced
corrosion of copper in seawater. The importance of understanding the current
distributions in this case was illustrated earlier through experiments conducted in the
steady state as well as in the frequency domain [4], Images captured through video
microscopy during the corrosion process revealed that a number of films were being
formed on the surface of the copper disk electrode. However the large number of salts
that are present in the synthetic seawater makes the characterization of this system very
difficult. Especially, impedance data interpretation is very complicated as the information
content obtained from frequency domain techniques is more than what is typically


6
obtained from the steady state calculations. In order to gain a proper understanding of this
system, one needs to possess great expertise in the interpretation of the experimental data
based on the current distributions obtained. The interpretation of the data collected in the
impedance domain based on the current and potential distributions can be said to be the
central theme of this work. A simple model system, which is explained in the next section,
is chosen for the purpose of this study. The interpretations can later be extended to the
phenomena occurring in more complex systems such as the corrosion of copper in
seawater.
1.4 Approach to the Problem
For the purpose of studying the influence of the current distributions on the
interpretation of the impedance data, a model system is chosen that is simpler in nature in
comparison with the copper in seawater system. This is a platinum rotating disk electrode
immersed in an electrolyte containing 0.01M potassium ferricyanide, 0.01M potassium
ferrocyanide, and 1M potassium chloride as the indifferent or supporting electrolyte. The
reaction that is occurring is the reduction of ferricyanide (the operation is in the cathodic
regime), that is,
Fe(CN)t + e o Fe(CN)64'
This system was thought of as an easier system to understand because
1. The surface of the electrode is relatively inactive in highly cathodic regions,
2. Presence of excess supporting electrolyte undermines the influence of the ohmic
contributions, and


7
3. The reaction is very fast and hence the effects of mass transport phenomena are
predominant.
However, during the course of this work, the system at hand which was supposed
to be easier to understand was observed be more difficult than what was thought. The
impedance data for this system collected at CNRS, Paris, France, was analyzed using the
one-dimensional frequency domain model proposed by Tribollet and Newman [5] in order
to obtain the Schmidt number values for the ferricyanide ions. This analysis resulted in
some anomalous observations. The regressed values for Schmidt number increased with an
increase in rotation speed. These values are observed to be quite close to the actual
Schmidt number (around 1100 in case of the system of interest) at low rotation speeds
(120rpm) and progressively increased to quite high values (as high as 1500) at high
rotation speeds (3000rpm). For high rotation speed cases where the Schmidt number
values are closer to 1100, the quality of regressions is poor. These observations are in
agreement with the results obtained by Deslouis and Tribollet [6], There are two major
issues that could cause the disparity in the determined value of the Schmidt number. The
first one is the non-uniform distribution of current and potential on the surface of the disk
electrode, which points to the inadequacy of Tribollet and Newmans one-dimensional
model. The other issue is the partial blocking of the electrode surface, which could be a
result of reactant, product, or intermediate ionic species adhering to the surface of the disk
electrode, thus reducing the mass-transfer/reaction rate. The primary focus for this work is
in explaining the influence of non-uniform current distributions on the interpretation of
impedance spectra.


8
During the experimental stages of this work, repeated impedance measurements
were made for this system, in order to facilitate the use of the measurement model
approach of Agarwal et al. [7-11] in determining the contribution of the stochastic noise in
the measurement. From this analysis, and from a number of other frequency domain
techniques studied, it was observed that the standard deviations of the real and imaginary
parts of the stochastic noise were equal for the system of interest, when the real and
imaginary parts were obtained in a single measurement. This led to investigating a
theoretical basis for such an observation. One of the premises the measurement model is
based on is that the Kramers-Kronig relations [12] are applicable for the particular system
of interest. These are integral transforms, which relate the real and imaginary parts of the
complex quantity measured. It was found that the equality of the standard deviations of
the real and imaginary stochastic components is a direct consequence of the applicability
of Kramers-Kronig relations [13], An analytical proof is presented in chapter 7.
The research work that was proposed and accomplished for this work is presented
in the form of a flow diagram in Figure 1.3. The research work performed in order to
analyze the problem is organized and presented in the subsequent chapters in a logical
manner. In chapter 2, the experimental procedure and the data analysis of the impedance
experiments is presented and the need for studying the non-uniformity and surface
blocking issues is established. The evidence for the presence of surface blocking from the
electrochemical impedance spectroscopy measurements is established in chapter 3. In
chapter 4, a steady state model for the current distributions on the rotating disk electrode
surface is proposed. This model explicitly accounts for a finite Schmidt number correction
and the charge distribution on the electrode surface. This is used as a building block for


9
the development of a two-dimensional frequency-domain model that is presented in
chapter 5. In this chapter, a comparison is made between the experimental data and the
simulated results from Tribollet and Newmans one-dimensional model and the two-
dimensional model developed here. In this chapter it is shown that the non-uniformities do
play an important role in the data interpretation. However there still is a significant
disparity between the experimental and the simulated values from the two-dimensional
model, which suggests that surface blocking phenomena play a significant role. Chapter 6
provides an alternative method for solving the steady state convective diffusion equation
presented in chapter 4. One could proceed to chapter 7 from chapter 5, without sacrificing
the continuity in the flow of text.
An equality of standard deviations in the real and imaginary parts of the stochastic
noise in the measured impedance was observed during the course of this work. An
analytical proof for this observation is provided in chapter 7. The results presented in
Chapter 8 illustrate that the equality of standard deviations in the stochastic noise is valid
for a large variety of frequency domain techniques for which Kramer-Kronig relations are
applicable. Conclusions from this work and suggestions for future work are provided in
chapters 9 and 10 respectively.


10
Figure 1.1. Flow near a rotating disk electrode.


11
i
V
Figure 1.2. Small-signal analysis of an electrochemical non-linear system.


12
Figure 1.3. Flow diagram of the research work performed. Major contributions from this
work are italicized.


CHAPTER 2
THE SCHMIDT NUMBER FOR FERRICYANIDE IONS: EXPERIMENTAL DESIGN
AND DATA ANALYSIS
The importance of current distributions in electrochemical systems is emphasized
in chapter 1. In the present chapter the need for studying the current distributions in
interpreting the impedance data is established. Interpretation of
electrochemical/electrohydrodynamic impedance spectroscopy data requires an
appropriate model that describes the underlying processes occurring in the system under
study. Electric circuit analogue models consisting of resistors, capacitors, inductors, and
specialized distributed elements are commonly used to represent the impedance response
of an electrochemical system. These models can be classified into process models and
measurement models. Process models are used to predict the response of the system
accounting for physical phenomena that are hypothesized to be important. Regression of
process models to data allows identification of physical parameters based on the original
hypothesis. In contrast, measurement models are used to identify the characteristics of the
data set that could facilitate selection of an appropriate process model. It should be noted
that the measurement models are used mostly for the statistical validation of data rather
than to identify the physics of the process. However it is more appropriate to use a
process model to gain a deeper insight into the physics of the system. The steady state and
frequency domain process models developed for a typical electrochemical system are
presented in chapters 4 and 5. In the present chapter, the focus is on the use of
13


14
measurement models for impedance data analysis, on the determination of Schmidt
number of ferricyanide ions through the application of an available one-dimensional
process model, and on identification of the need to understand the current distributions for
the system under consideration.
2,1 Measurement Model
The measurement model selected for this work was the Voigt model given by
zW=z+?w <2
where Z(co) is the quantity being measured, Z0 is the high frequency asymptotic limit of the
impedance, k denotes the number of frequency dependent processes associated with the
system under consideration, Ak is the gain factor, rk is the time constant associated with
each of the relaxation processes, and co is the applied frequency. The above approach can
also be viewed as a series of resistance and capacitance elements connected in series as
illustrated later in Figure 8.1(a). In this case the time constant rk for element k is
equivalent to RkCk, and Ak is equivalent to Rk, where Rk and Ck are resistance and
capacitance respectively. A detailed discussion on measurement models can be found in
reference 14. The major contribution of measurement models has been in identifying the
stochastic and bias errors in impedance measurements. Such information has been used to
enhance the information content that can be obtained from experimental data.
2.1.1 Importance of identifying the Stochastic Noise Level
The choice of a proper weighting strategy for regression of models to data is
facilitated by identification of the stochastic noise in the experimental data. The data with


15
more noise should be assigned less weight towards the regression parameter and vice
versa. The regression parameter is given by
where Znk and ZJik are the real and imaginary parts of the measured impedance at a given
frequency (Ok, Zrk and Z] k are the model values corresponding to either a measurement
or a process model, and ork and ajk are the standard deviations in the real and
imaginary parts of the stochastic noise in the measurement, also referred to as the noise
level in the real and imaginary parts. Different techniques are in existence for weighting
the regressions. One commonly used weighting strategy found in the literature is
proportional weighting, where it is assumed that ar k and ajk are proportional to the
magnitudes of the respective components [15-17], However in chapter 7 it would be
illustrated that such an assumption is incorrect for the systems that are consistent with the
Kramers-Kronig relations. Another form of weighting strategy is the one where it is
assumed that the noise level in the real and imaginary parts is proportional to the modulus
of the complex quantity being measured [18], Assumptions are commonly made that the
noise level in the measurement is about 3% or 5% of the modulus of the quantity being
measured. Overestimation of the noise level may lead to significant loss of information
content [19], For this work, the measurement model approach developed by Agarwal et
al. is used to assess the noise level. At this stage it is necessary to classify errors in the
measurement.


16
212 Classification of Errors
The residual errors (£res) that arise due to the regression of a model (Z) to
experimental data (Z^) can be of two types, systematic errors (£sys) and random or
stochastic errors (£i,oc).
Zexp ~Z + £n
(2.3)
£res ~ £sys + £stoc
(2.4)
The systematic errors can arise due to the lack of fit of the model to the data (£//) or due
to an experimental bias (Skias), that is,
£ ays £lof + £bias
(2.5)
Experimental bias errors can arise from non-stationary behavior corresponding to a
changing baseline during the course of the impedance scan or from instrumental artifacts.
Thus,
£.. = £ + £ (2.6)
Most electrochemical systems are inherently non-stationary and can change during the
time required to conduct an impedance measurement. This is one of the limitations in
choosing the frequency limits for conducting the impedance measurements. The
experimental time should be limited in order to not introduce a significant amount of bias
into the system. Identification of the part of the spectrum which is not corrupted by the
bias errors is very important. In order to address this issue, Kramers-Kronig relations are
used.
For spectroscopic techniques such as optical spectroscopy the noise in the
measurement can easily be assessed by calculating the raw standard deviations of the


17
measurements, as a number of repeated measurements can be made in a very short span of
time without changing the system baseline. In other words optical spectra can be truly
replicated. However this is not the case for impedance measurements. The surface
properties of the electrode may change significantly during the course of the measurement.
The inability to replicate impedance scans motivates the use of measurement models for
filtering lack of replicacy.
2.1,3 Kramers-Kronig Relations
Kramers-Kronig relations are self-consistent integral relations which apply to
systems that are linear, causal, stable, and stationary. By using the quasi-linear approach,
the condition of linearity is satisfied. Causality requires that the response of input cannot
precede the input. Stability refers to the boundedness of the output perturbation, and
stationarity refers to the time-invariance of the system. Kramers-Kronig relations can be
expressed in a number of different ways as presented in [12], Through the form of
Kramers-Kronig relations chosen for this work, the real and imaginary parts of the
complex variable being measured (impedance, in the context of present work) are related
as
oo
(2.7a)
o
and
o
(2.7b)


18
These relations have a number of implications. Given the imaginary part of the spectrum,
the real part can be obtained and vice versa. For the impedance spectroscopy where both
the real and imaginary parts are available, these relations can be used as a consistency
check to obtain the part of the spectrum which is not corrupted by bias errors, that is, the
part of the spectrum where the time dependent variations and instrumental artifacts are not
significant. However using these relations in the form of the above equations is not
feasible because:
1. The limits of integration vary from 0 to oo. However, there is a lower limit for the
frequency range in order to minimize the time dependent variation for a given
impedance scan (At lower frequencies measurements take longer times). Also, there is
an upper limit on the frequency due to the limitations imposed by the instrumentation.
Hence only a finite frequency range is available for the integration.
2. Choice of numerical integration scheme is of high importance in using the relations as
they appear. There is a point of singularity in the domain of integration at x = co and it
should be handled with extreme care. An improper choice of an integration scheme
may result in significant errors in the calculations performed.
These issues are the limiting factors for direct application of the Kramers-Kronig relations.
However by applying measurement models, these relations can be used to identify the self-
consistent part of the spectrum without actually performing the integration.
2,1,4 Identification of Noise Level in the Measurement and Consistency Check
One of the inherent benefits of using the measurement models is that the R-C
circuit elements or the Voigt elements used in these models satisfy the conditions


19
associated with the Kramers-Kronig relations. This eliminates the need for the numerical
integration of these relations.
Frequency scans are repeated in order to facilitate the assessment of the noise level
in the measurement. An error structure model is obtained for the standard deviations in the
noise levels and this model is used in weighting the subsequent regressions. At this stage
the measurement models were used to regress the real part of the data to the model and
predict the imaginary part and vice versa, in order to eliminate the portion of the
experimental data that are inconsistent with the Kramers-Kronig relations. The procedure
for using measurement models to identify the noise level in the measurement and to check
for consistency with the Kramers-Kronig relations is illustrated in a subsequent section.
Once the self-consistent part of the spectrum is identified, this can be used for the process
model regressions in order to determine the parameters of interest.
2.2 Process Model
The process model used for data analysis in this chapter is a one-dimensional
model for impedance spectroscopy developed by Tribollet and Newman [5], Their model
assumes uniform distributions on the surface of the electrode, and this assumption is valid
only at the mass-transfer-limited current or under kinetic control. As EIS (Electrochemical
Impedance Spectroscopy) measurements cannot be conducted at the mass-transfer-limited
current, the existing process model is used both for the EIS data collected below the mass-
transfer limited condition and for the EHD (Electrohydrodynamic Impedance
Spectroscopy) data collected at the mass-transfer limited condition. A brief overview of
the process model for EIS and EHD is presented in this section.


20
2.2.1 Process Model for EIS
When a one-dimensional analysis is employed for the impedance response of the
rotating disk electrode, the unsteady state convective diffusion can be written as
dci dci [ d2ci
dt dz 1 dz
(2.8)
where t is the time, z is the axial coordinate, c, and D, are the concentration and diffusion
coefficient of species /, and v* is the axial component of the fluid velocity. The boundary
conditions for the steady state form of equation (2.8) are that the concentration
approaches a bulk value far from the disk and that the flux at the disk surface is related to
the current density. The heterogeneous reaction can be expressed symbolically as
'^siMizi =ne (2.9)
where the stoichiometric coefficient s has a positive value for a reactant, has a negative
value for a product, and is equal to zero for a species that does not participate in the
reaction. Thus; the boundary conditions are
c, cja> as z > oo (2.10)
and
A
dc, = Sjif
dz nF
(2.11)
where cit0O is the bulk concentration of species if is the Faradic current density, n is the
number of electrons transferred, and F is the Faradays constant (96,487C/eq). The
Faradic current density is expressed as a function of surface overpotential r¡ and
concentration as


21
if=/(rj,ci) (2.12)
Thus, the concentration at the surface is dependent on applied potential through a reaction
mechanism leading to equation (2.12). The concentration perturbation is given by
ct = ct + Re{c e;ffli j (2.13)
where the overbar represents the steady value, j is the imaginary number V-T, co is the
frequency, and the tilde denotes a complex variable which is a function only of position.
Similar definitions are used for all dependent variables.
The dimensionless form of the equation governing the contribution of mass
transfer to the impedance response of the disk electrode is developed here in terms of
dimensionless frequency
K. =
co
Q
9v
CO

\X
V2
Sc
.1/3
(2.14)
and dimensionless position
4
(2.15)
where Q is the rotation rate of the disk in cycles per second, v is the kinematic viscosity in
cm2/s, and
(2.16)
is a characteristic distance for mass transport of species Substitution of the definition for
concentration (equation (2.13)) into the one-dimensional expression for conservation of
species / (equation (2.8)) yields


22
d2c, de, ... ~
+v'-Hr-JKici=
d? z d£
(2.17)
A solution to equation (2.17), dl{E,)=cJci0 can be found which satisfies the boundary
conditions
dt - 0 as E, oo
0,=\ at | = 0
The concentration at the surface of the disk is given in terms of 0t (£) as
ci. o =ci + Re^,,o^i()}
(2.18)
(2.19)
Thus:
dc.
dz
2=0
= ML¡'(0 ) = J-
S. W nFD,
(2.20)
where the Faradic current density is expressed as the sum of a steady and oscillating term.
The current density consists of contributions from Faradic reactions and charging of the
double layer as
i = if+C
f dt
(2.21)
where C is the double layer capacitance. Under the assumption that the magnitude of the
oscillating terms is sufficiently small as to allow linearization of the governing equations
lf =
7+Z
ci.o
Kdci,oj
Ci.O
(2.22)
The charge transfer resistance R, is defined to be


23
R< =
\dTUc,
Thus;
~ 1 ~ ^
v=r+?
df'
Ci, 0
Equation (2.24) can be expressed in terms of overpotential as
V=Rjf-R J
Ci,0
From equation (2.20)
S' V> 8,
nFD: ff'(6)
Ci,0 ~
Equation (2.25) becomes
= Rjf-R,Y
si'f Ji
nFDt <9/(o)
or
Tl=7f(Rt+ZD)
where
zd=-r,i:
f A
v^-oy
7.cjj*i
nFDi 0/(6)
Equation (2.29) provides the Warburg impedance. The cell
respect to the potential of a reference electrode), given by
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
potential (measured with
V = RJ + T]
(2.30)


24
can be written in terms of oscillating variables as
V =Rj + rj
Equation (2.21) can be written in terms of oscillating variables as
i if + jcCrj
(2.31)
(2.32)
where C is the double layer capacitance. Equations (2.28), (2.31), and (2.32) result in
= Z=Re+ R'+Z -
/' 1 + jooC(R,+ZD)
(2.33)
Equation (2.33) represents a generalized form of the impedance response of a disk
electrode.
For the data analysis presented in this chapter, the impedance Z given by equation
(2.33) is regressed to the experimental impedance in order to determine various
parameters such as Re, Rt, Zd, C, and Sc (Schmidt number).
2.2.2 Process Model for EHD
In the usual application of EIS, a complex impedance is calculated as the ratio of
potential to current under a small perturbation of current (galvanic regulation) or potential
(potential regulation). EHD is a generalized impedance technique in which sinusoidal
modulation of disk rotation rate drives a sinusoidal current or potential. EHD has an
advantage over EIS for measurement of the transport properties of ionic species because
measurements can be made at the mass-transfer-limited current plateau, whereas an EIS
measurement must be made on the slope below this plateau.
The angular velocity of the rotating disk electrode in case of EHD is given by
f2 = O0 + AQ cos(iy/) (2.34)


25
where AO is a fixed amplitude of perturbation in angular velocity around a mean value of
O0 The frequency of perturbation is varied to obtain the widest possible range. The
resulting current /, at fixed potential, can be expressed as
/ = 70 + AI cos (cot + ) (2.35)
where Io is the average current, A/ is the amplitude of the sinusoidal current response, and
(J) is the phase shift. These equations can be introduced into the differential equations that
describe the physics and chemistry of the system to calculate explicitly the transfer
function for the system, as illustrated in case of EIS in the previous section. Details of the
development of this model are presented in [5], From this development a theoretical
transfer function for this system can be written as
I_
Q
2Aozp M
R
\zd(Q)zd{u)
+ j)ReC
R,
R,
\
Zd{0)zd(u) + l) Zd{6)zd{u)
(2.36)
+ 1
where
u = Scm
(2.37)
zp(u) and z^u) are tabulated functions for the electrohydrodynamic impedance and the
convective diffusion impedance, respectively, which include corrections for a finite
Schmidt number, 2A0 and Zj(0) are the respective moduli for the impedance at zero
frequency, and Sc is the Schmidt number. The terms R/Zj(0) and R/ZJfi) correspond to
corrections for kinetic and ohmic resistance, respectively, normalized by the zero
frequency limit for the convective Wauburg impedance. The surface capacitance appears
in the lumped parameter ReC. The correction terms R/Z(0), RReC are


26
generally considered to be negligible as compared to unity if the electrode kinetics are fast.
In this work, only one correction term of R/ZJO), R/Z^O), or ReC, could be resolved for
any given regression. Order of magnitude of analysis by Tribollet and Newman [5]
suggested that ReC should be the most important correction term to account for high
frequency processes on the limiting current plateau. Thus, the process model for this
system was given by
/ ^ 2A0zp(u)
Q jcoReC +1
(2.38)
Equation (2.38) was regressed to the experimental data to obtain the values for A0, ReC,
and Sc.
2.3 Experimental Design
The system used for this work, potassium ferri/ferrocyanide redox couple reacting
on the Platinum rotating disk, is a classic system to study mass-transfer related issues, as
the redox reaction is very fast. Also, the effect of migration on this system is very small
(Figure 19.3 of reference 20), because the product ion is always present at the electrode
surface. In this chapter the experimental procedure and the data analysis for EIS and EHD
measurements are presented along with an interpretation of the results obtained. In this
section the experimental procedure for conducting these measurements is explained.
2,3,1 Choice of Surface Treatment
Roughness of the surface can influence the charge transfer process and could be
strongly associated with the frequency dispersion behavior [21,22], Hence proper
preparation of electrode surface before each measurement is very important. It is also


27
believed that partial blocking of the electrode surface could be related to the surface
treatment used. Hence the treatment that results in the least amount of surface blocking
among the available surface treatments is needed to obtain reliable information from the
system under consideration. The best surface treatment, or the treatment that provides
least amount of surface blocking, was considered to be the one that provided the highest
value for the limiting current. DC data were collected using four surface treatments:
1. The electrode was polished using a 1200 grit emery cloth and washed with
deionized water.
2. The electrode was prepolished with 1200 grit emery cloth, washed in deionized
water, polished with alumina paste and then was subjected to ultrasound
cleaning in a 1:1 solution of water and ethyl alcohol.
3. The electrode was polished using 1200 grit emery cloth, washed with deionized
water, and then prepolarized by sweeping from -0.5V to +0.5V measured
against saturated calomel reference electrode, and back to -0.5V at 10mV/sec.
4. The treatment procedure described in treatment 2 was used, followed by a
prepolarization sweep from -0.5V to +0.5V and back to -0.5V at 10mV/sec.
The polarization curves in the cathodic region are shown in Figure 2.1. The cathodic
region of operation was used for this work because platinum is relatively inactive when
compared to the anodic region. The measurements were made at a disk rotation rate of
600 rpm. Treatment 1 yielded the smallest limiting current value, and treatment 2 yielded
the largest value. Under microscopic observation, the surface of the electrode appeared to
be significantly different from one surface treatment to the other. With treatment 2, a more
mirrorlike surface was obtained. EIS and EHD measurements were conducted for both


28
these treatments in order to facilitate a comparative study of the influence of surface
treatment employed on the interpretation of impedance data. The data from treatment 2
can be used to validate the existing one-dimensional frequency domain model proposed by
Tribollet and Newman with greater confidence as this treatment results in a smaller
amount of blocking.
2.3,2 Experimental Setup
A schematic of a typical impedance experimental setup is presented in Figure 2.2.
The electrode was rotated using a high-speed low-inertia rotating disk apparatus
developed at the CNRS [23], The rotator was rated at a power of 115W. This high power
is necessary to obtain modulation at high frequencies (up to 100Hz). The rise time of the
rotator between 0 and lOOOrpm when a stepwise potential is applied is less than 2ms
(indicating a low inertia). Such low inertia is of extreme importance in case of the
electrohydrodynamic impedance (EHD) measurements. The long term stability and
accuracy of the rotation speed is 0.2% from 100 to lOOOrpm. The tachometer has 24
poles, and the electromechanical constant was 3mV/rpm.
The potentials and currents were measured and controlled by a Solartron 1286
potentiostat. A Solartron 1250 frequency response analyzer (FRA) was used to apply the
sinusoidal perturbation and to calculate the resulting transfer function. A matched two-
channel Kemo type VBF8 48 low pass Butterworth analog filter was used to reduce the
noise level of the input signals to the FRA. However from the experimental observations
made during the course of this work as well as from the observations made by Agarwal et
al. [10] use of the filter resulted in a slightly longer experimental time in case of EHD
measurements. Hence EHD measurements were conducted without employing a filter.


29
Use of filter is necessary for EHD experimentas conducted below the mass transfer limited
plateau.
The electrolyte consisted of equimolar (0.01M) concentrations of potassium
ferricyanide and potassium ferrocyanide and a 1M potassium chloride solution. The
electrode diameter is 5 mm, yielding a surface area of 0.1963cm2. The temperature was
controlled at 25.0 0.1C. Temperature control is very important as the transport
properties of various species exhibit a very strong functional dependence on the
temperature of the electrolyte.
A series of EIS measurements were conducted on a platinum rotating disk
electrode for rotation rates of 120, 600, 1200, 2400, and 3000rpm, at lM*, 1/2, and 374th
of the mass-transfer-limited current for all the rotation speeds, using treatments 1 and 2.
The EHD measurements were conducted at the mass-transfer-limited condition at rotation
rates of 120, 200, 600, and 1200rpm in case of treatment 1. For each of these conditions,
repeated measurements were made in order to establish a structure for the stochastic noise
in the measurements. The EHD data collected using treatment 2 were found to be
corrupted due to the overpolishing of electrode which resulted in the depletion of platinum
working electrode.
The EIS data were collected from the high frequency to the low frequency with 12
logarithmically spaced frequencies per decade. The first measurement in each spectrum
obtained was discarded in the analysis because the start-up transient often influenced the
value of impedance reported by the instrumentation. Also the data points within 5 Hz of
the line frequency of 50Hz (60Hz if the experiments were to be conducted in U S A.) and
its first harmonic of 100Hz (120Hz in U.S.A.) were discarded. The influence of these


30
points on the error structure is illustrated in chapter 8. The EHD data were collected from
low frequency to high frequency with 20 logarithmically spaced frequencies per decade.
As the signal to noise ratio is quite low at high frequencies in case of EHD, measurements
take longer time at higher frequencies. The long (1% closure error) autointegration option
of the frequency response analyzer was used, and the channel used for integration was that
corresponding to current. The electrolyte was not deaerated but experiments were
conducted at a polarization potential of 0V (SCE) in order to minimize the influence of
oxygen on the reduction of ferricyanide. FraCom software developed in-house at CNRS
by H.Takenouti [24] was used for data acquisition.
2.4 Data Analysis: Measurement Model
In this section the use of measurement models for the assessment of stochastic
noise in frequency domain measurements and in identifying the self-consistent part of the
impedance spectra is illustrated with the example of EIS data sets collected for 120rpm at
174th of mass-transfer-limited current for the system of interest. The data were analyzed
employing the user-friendly MATLAB-based visual interface created in-house by Mark
Orazem.
2.4.1 Initial Regressions of the Data
The initial regressions to the measurement model were performed using modulus
weighting for the EIS data and using no weighting approach for the EHD data, as the
error structure was yet to be determined. In accordance with equation (2.1), a
measurement model was constructed by sequentially adding k Voigt elements with
parameters Ak and zjk until the fit was no longer improved by addition of yet another


31
element. The best fit was obtained for a model containing the maximum number of
lineshapes that satisfied the requirement that the 95.4% confidence intervals of all the
regression parameter estimates, calculated under the assumption that the model could be
linearized about the trial solution, do not include zero. The results of regression are
presented in Figure 2.3 with real part of the impedance plotted against the imaginary part
for one of the three replicates for the 120rpm at lM* of the mass-transfer-limited current
using treatment 1. In this case, 8 line shapes were obtained. In Figure 2.4 (a) and (b) these
results are presented with the real and imaginary parts of the impedance plotted as
functions of frequency. The normalized residual errors obtained from the real and the
imaginary parts are shown in Figure 2.5 (a) and (b). Similar regressions were performed
for the rest of the data sets collected for this condition. For the sake of consistency, 8 line
shapes were used for the other two replicates also, though more line shapes could be
obtained for these cases. Such an approach results in obtaining lack of fit (Sbf) errors that
are representative of same quality fit for all the replicates and hence do not contribute to
the standard deviations. The measurement model parameters for each of the replicates are
different because the system changed from one experiment to the other. Hence, by
regressing a new measurement model to each individual data set, the changes of the
experimental conditions are incorporated into the measurement model parameters. As a
consequence non-stationary (£) errors are equal to zero for each separate regression.
Standard deviations of the residual errors obtained by using measurement model approach
for the repeated measurements were calculated and are presented in Figure 2.6. These
provide estimates for the standard deviation of the stochastic part of the impedance


32
response. The standard deviations were also calculated for all other experimental
conditions.
The impedance plane plot from the preliminary regression for EHD data, collected
for a rotation speed of 120 rpm at mass-transfer-limited current, and using 2 Voigt
elements, is presented in Figure 2.7. A no-weighting strategy was used for the preliminary
regression of EHD data. The same regression results are represented as real and imaginary
parts as functions of frequency in Figure 2.8 (a) and (b). The normalized residual errors
from this initial regression are presented in Figure 2.9 (a) and (b).
2.4,2 Obtaining the Error Structure
The standard deviations obtained from the EIS measurements conducted for
various rotation rates and various fractions of limiting current were grouped together, and
a common model for these standard deviations was obtained. The standard deviations or
and o] were regressed to the model
cr, = cr, a
Zj\ + P\Zr -Rsol\ + y-jZ- + >
Km
(2.39)
where a, /?, y and 8 are constants determined by regression analysis, Rso¡ is the solution
resistance or the high frequency asymptote and Rm is the current measuring resistor. This
model is also referred to as the error structure model in this work. The equality of
standard deviations of the real and imaginary parts of the stochastic noise is found to be
true when the real and imaginary parts of the complex quantity are measured using the
same instrument for a system that is consistent with the Kramers-Kronig relations. An
analytical proof for this observation can be found in Chapter 7, and Chapter 8 illustrates
this result for a number of spectroscopic techniques such as electrohydrodynamic


33
impedance, viscoelastic measurements, and acoustophoretic measurements. This result is
found to be true even for systems where the real and imaginary parts differ by several
orders of magnitude. The impedance spectroscopy applied to polyaniline (PANI)
membranes is an example for such cases [19].
The standard deviations obtained under different operating conditions for the
rotating disk electrode were obtained using the measurement model approach. A
generalized error structure model was obtained for all the conditions. The conditions and
the number of replicates used for treatment 1 are listed in Table 2.1. Only /? and ^values
could be extracted for the error structure as the confidence intervals for a and 8 included
zero, and these values are given by /? = 1.00249 x 10'3 and y= 2.77789 x 10'4. It could be
seen from Figure 2.10, Figure 2.11, and Figure 2.12 that the error structure model
describes the noise level in the measurement in a satisfactory manner. Similar analysis was
performed for the data sets collected using treatment 2 to obtain the error structure. The
conditions and the number of replicates used for this treatment are listed in Table 2.2. The
error structure parameters that could be obtained for this case are /? = 1.00587 x 10'3 and
y= 2.53830 x 10'4. The error structure of the impedance measurements was not affected
by polishing technique. A common model could be found that described the error structure
for both sets of experiments.
The model for the stochastic contribution of the error structure for EHD data is
given by
(2.40)


34
where Zr and 2¡ are the real and imaginary parts of the EHD transfer function,
respectively, and a, /?, and 8 are parameters which were found by regression to the set of
standard deviations obtained using the measurement model approach. For the set of EHD
measurements conducted for treatment 1 it was found that (8 = 9.87004x1 O'4 and 8 =
3.07652xl0'5 pA/rpm.
2,4,3 Identification of the Self-Consistent Part of the Impedance Spectra
The use of measurement models to identify the self-consistent portion of the
impedance spectra takes advantage of the fact that the Kramers-Kronig transforms relate
the real part to the imaginary and vice versa. Once the error structure is obtained, it can be
used to weigh the subsequent regressions. This strategy assigns less weight to more noisy
data and vice versa. The measurement model is regressed to the real (or imaginary) part of
the spectrum, and the regression parameters are used to predict the imaginary (or real)
part. Experimental data inevitably contain stochastic errors associated with the
measurement. The presence of these errors gives rise to an uncertainty in the prediction of
parameters in regression. The uncertainty in the parameter estimation is quantified by the
standard deviation (d) of the parameters, that is, one can say with 95.4% certainty that the
parameter estimates lie within 2cr of the value calculated by the regression. Due to this
uncertainty in parameter estimation, there is uncertainty in any prediction that is made
using these parameters. The Monte-Carlo simulation technique is used in determining the
95.4% confidence interval for the prediction. Calculation of this interval takes the
stochastic component of measurement error into account. Hence it could be said with
95.4% confidence that the data points which lie outside this predicted confidence interval


35
are corrupted by systematic error, that is, they represent the inconsistent portion of the
spectrum.
The measurement model regression performed for real part of the impedance
spectrum obtained for a rotating disk electrode at 120rpm and 174th of mass-transfer
limited current is shown in Figure 2.13(a). In this case 10 lineshapes were obtained. The
imaginary part of the spectrum obtained using these 10 lineshape parameters is shown
along with the 95.4% confidence intervals in Figure 2.13(b). The scale of this plot
obscures the inconsistent portion of the imaginary part. Normalized residual errors are
presented in Figure 2.14(a) and (b). In this case it could clearly be seen that 4 points at the
high frequency end are inconsistent with the Kramers-Kronig relations. These data were
assumed to be corrupted by instrumental artifacts. In Figure 2.15 (a) and (b), prediction of
the real part based on imaginary part is shown, and the corresponding normalized residual
errors are presented in Figure 2.16 (a) and (b). It could be seen that at higher frequencies
(above 30 Hz) the real part of the spectrum is not predicted properly. However this cannot
be attributed to the bias errors or the inconsistency in spectrum. The imaginary part
approaches the asymptotic limit at these frequencies and hence is incapable of capturing
the changes that occur in the real part.
From this analysis it is found that 4 points at the high frequency end of the
spectrum fell outside the confidence interval for the model and were therefore assumed to
be inconsistent with the Kramers-Kronig relations. These points were deleted for further
regressions. Similar analysis was performed for the spectra collected at 600, 1200, 2400,
and 3000rpm, at 1/4, 1/2, 374th of mass-transfer-limited current, and the portions of the
spectra which are inconsistent with Kramers-Kronig relations were identified and deleted.


36
For the EHD data collected for treatment 1, the consistency check with Kramers-Kronig
relations revealed that all the spectra collected for various conditions were completely
consistent. The process model was regressed to the EIS and EHD data using the error
structure weighting.
2,5 Data Analysis: Process Model
In this section the results from the regressions to the one-dimensional process
model developed by Tribollet and Newman are presented. After establishing the self-
consistent portion of the spectrum, the model was regressed to the data taking advantage
of the established error structure for the given set of measurements. The results from one
such regression to the data collected using treatment 1 are presented as impedance plane
plot in Figure 2.17, and corresponding plots of the real and the imaginary parts as
functions of frequency are shown in parts Figure 2.18 (a) and (b) respectively. The
normalized residual errors are represented in Figure 2.19 (a) and (b). From the residual
errors it can be seen that the high frequency data were not very well predicted with this
model, as there is definite trending in the errors and the errors in the imaginary part are as
high as 70% for very high frequencies. This resulted in a Schmidt number value of
8111172 as opposed to the expected value of 1100.
In order to address the disparity between the model and the data, a constant phase
element (CPE) was added to the process model [25], The expression for the impedance
given by the process model in equation (2.33) was
?=Z = R | R<+Zd
/ 1 + \jcoC{Rt+ZD)l*
(2.41)


37
In which \-0, the
form illustrated in equation (2.33) is recovered.
Several qualitative justifications have been advanced in the literature for
incorporating a CPE correction into the process model:
(a) If the electrode surface is rough, the peaks and the valleys will be accessible to a
different degree at different frequencies [26],
(b) If the frequency becomes large relative to the kinetics of ion sorption in the double
layer, then the apparent double layer capacity will depend on the frequency [27],
(c) The occurrence of faradic reactions can cause a frequency dependence of the values in
the equivalent circuit model
(d) The current distribution can be different at different frequencies and this can lead to a
frequency dispersion [28],
In essence, the CPE correction is introduced in order to address the frequency
dispersion behavior. A more satisfactory regression was obtained when a constant phase
element was employed, as can be seen in impedance plane representation in Figure 2.20.
The regression results for the real and imaginary parts as functions of frequency are
presented in Figure 2.21 (a) and (b) respectively, and the corresponding normalized
residual errors are presented in Figure 2.22 (a) and (b). For the same data set that was
considered earlier the normalized residual errors were now at the most 2% as compared to
about 70% in extreme case when a CPE correction was not applied. The Schmidt number
obtained in this case was 107333 as opposed to 811172 obtained through the process
model regression without accounting for the CPE correction, and the expected value of
1100. However, it was not always possible to obtain a CPE correction for a given


38
measurement. During the course of the data analysis performed for this work it was
observed that a CPE correction was obtained whenever there is a significant high
frequency effect which can be attributed to a surface blocking effect. The results of the
regressions for treatment 1 are presented in table 2.3 (without CPE correction). Some
selected results based on the lowest normalized residual sum of squares for a given
condition are presented in table 2.4. Results accounting for CPE correction are presented
in table 2.5 and selected results from table 2.5 are presented in table 2.6. Regressed results
for treatment 2 are presented in tables 2.7 and 2.8(without CPE correction) and in tables
2.9 and 2.10 (with CPE correction).
The process model regression for the EHD data collected for a rotation rate of
120rpm at the mass-transfer-limited current is presented in the form of an impedance plane
plot in Figure 2.23. The Schmidt number obtained in this case was 114713. The
regressions for the real and the imaginary parts as functions of frequency are presented in
Figure 2.24 (a) and (b), respectively. The residual errors in this case are not normalized in
these figures as for an intermediate frequency, the real part of the EHD impedance tends
to 0, which results in a very high value of normalized residual error. These errors are
presented in Figure 2.25 (a) and (b). These results from the EHD measurements and
selected representative measurements are presented in tables 2.11 and 2.12 respectively.
2.6 Results from the Steady State Measurements
The Schmidt numbers obtained from the EIS and EHD measurements can be
compared against those obtained from the steady state mass-transfer-limited current


39
measurements. The mass-transfer-limiting current density (/i,m) for the rotating disk
electrode is given by
(2.42)
where n is the number of electrons produced when one reactant ion or molecule reacts, F
is the Faradays constant, D is the diffusion coefficient of the mass-transfer-limiting
species, Coo is the bulk concentration of the reacting species, t is the transference number, a
is the coefficient in the Cochrans velocity expansion, v is the kinematic viscosity, Q is the
rotation speed in rad/sec, and ?(o) is related to the concentration gradient in the axial
direction. Three values for the mass-transfer-limited current values were obtained at each
of the rotation speeds of 120, 600, 1200, 2400, and 3000 rpm, as presented in table 2.13.
The diffusion coefficient values and the Schmidt number values obtained from the mass-
transfer-limiting current are presented in table 2.14. The Sc values obtained at low
rotation speeds are in reasonable agreement with the value 1100 that is expected for the
system of interest. However the disparities grew larger with the rotation speed. These
results are in agreement with the Sc values obtained from the EHD measurements.
From equation (2.42) it is evident that the relation between i^ and Vo is linear
and the slope of the resulting plot between these two parameters should result in a value
for diffusion coefficient. Such a plot is presented in Figure 2.26. The line passing through
the data points was regressed ignoring the data points corresponding to the 3000rpm case.
From the plot it is clear that the data corresponding to this case do not conform to the
regressed straight line. The Schmidt number calculated from the slope of this straight line
was 1202.


40
2.7 Discussion
As the system chosen for this study is traditionally used to study mass-transport
phenomena, one of the interesting regressed parameters from the process model is the
Schmidt number of ferricyanide ions. The schmidt number (Sc) is defined to be the ratio of
kinematic viscosity (v) to the diffusion coefficient (D) of the ionic species that is
controlled by mass-transfer. The Schmidt number for the ferricyanide ions in the present
system is reported to be about 1100, based on the DC and EHD measurements conducted
by Robertson et al [23],
The regressed results of the process model as applied to the EIS data collected for
the treatment 1 are presented in tables 2.3-2.6. The CPE correction was obtained only in
few cases. The value NRSSQ (Residual Sum of Squares normalized with respect to the
variance in the stochastic errors) is a measure of the quality of the fit. If the model
describes the data adequately the NRSSQ parameter is expected to be about 1. The quality
of the data for this treatment is in question as this treatment yielded the most blocked
surface. The inadequacy of fit is evident here and the regressed Sc values were as much as
80% higher than the reported value of 1100 for some of the cases.
The representative regressed results of the process model based on as applied to
the EIS data collected for treatment 2 are presented in tables 2.8 and 2.10, without and
with a CPE correction respectively. This treatment was expected to provide the least
blocked surface for the electrode based on the DC limiting current values obtained.
Schmidt numbers reasonably close to the reported value of 1100 were obtained with the
CPE correction. However, the quality of fit is not good, as the NRSSQ values are above


41
10 for most of the cases. In general, the quality of the fit is more reasonable when CPE
correction was employed. However, in this case the Schmidt numbers progressively
increased with an increase in rotation speed, whereas the Schmidt number should be
independent of rotation speed.
The regressed values of the process model as applied to the EHD data obtained
using treatment 1 and at the mass-transfer-limited current are presented in table 2.11 for
rotation speeds of 120, 200, 600, and 1200 rpm. Some representative results based on
normalized residual errors are presented in table 2.12. In this case, the one-dimensional
model is adequate, as the current density is uniform at the mass-transfer-limited condition.
However, deviations from the expected Schmidt number of 1100 were observed for these
data sets also, with the extreme variation of about 25% at a rotation speed of 1200rpm.
This trend is consistent with the DC analysis results presented in table 2.14. As the non-
uniform current distribution does not exist in this case, the differences should be attributed
to the surface blocking effects. The ReC values presented in the table 2.11 from the
regressed values from the EHD data were consistently higher by at least an order of
magnitude in all the cases. This anomaly could be attributed to surface blocking effects as
discussed by Orazem et aI [10], EHD data collected using treatment 2 should provide
Schmidt number values that are more acceptable in nature as it is observed that this is a
better polishing technique and hence should provide less blocking.
From the above observations it is clear that the one-dimensional model does not
provide an adequate fit to the experimental data. A more sophisticated model is necessary
to understand the underlying physics of the system. The disparity in the experimental
results can be attributed to two factors: non-uniform surface distributions and surface


42
blocking. Surface blocking effects are discussed in a greater detail in the next chapter. The
main focus of this dissertation is on developing a physico-chemical impedance model
accounting for the non-uniform current distributions in the frequency domain. Once a two-
dimensional frequency domain model is established, the surface blocking effects can be
singled out. Before fully justifying the need for a two-dimensional frequency domain
model, it is necessary to understand the steady-state current distributions for the system of
interest. These distributions are presented in chapter 4.
2,8 Conclusions
The applicability of the measurement model to the EIS and EHD data was
demonstrated in this chapter. Generalized stochastic error structures are obtained for the
two different surface treatments considered for this work. Measurement models were used
to identify part of the impedance spectrum that is consistent with the Kramers-Kronig
relations. A one dimensional process model was used to analyze the EIS data, and it was
observed that this model does not describe the physics of the system adequately. When the
one dimensional process model was applied to EHD data collected at mass-transfer limited
current for various rotation rates, evidence for surface blocking was found. The process
model regressions established the need for a better understanding of the current
distributions in the steady state as well as in the frequency domain.


43
Table 2.1. The conditions and the number of repeated measurements chosen for
establishing the error structure for treatment 1.
2, rpm
Fraction of ii¡m
Number of replicates
120
1/4
3
600
1/2
3
1200
1/2
3
1/4
3
2400
1/2
3
3/4
3
3000
1/2
6
Table 2.2. The conditions and the number of repeated measurements chosen for
establishing the error structure for treatment 2.
O, rpm
Fraction of itim
Number of replicates
120
1/2
2
3/4
3
600
1/4
3
1/2
3
3/4
3
1200
1/4
3
1/2
3
3/4
3
2400
1/4
3
1/2
3
3/4
3
3000
1/4
3
1/2
3
3/4
3


Table 2.3. Results from the process model regression for EIS data collected using treatment 1, without CPE correction.
Condition
Data set no.
z(0), a
Sc
C, UF
Re,Q
Rt, f
NRSSQ
120, 1/4 is*
1
160.57 5.38
811172
8.04 0.13
7.54 0.02
4.81 0.10
357.60
2
162.15 6.76
850 222
7.09 0.12
7.43 0.02
4.21 0.09
437.04
3
162.33 6.35
880 212
7.07 0.11
7.44 0.02
4.08 0.09
371.74
120, 1/2 iUm
1
192.82 0.57
1304 24
17.40 0.65
7.41 0.01
0.57 0.004
9.73
600, 1/4 ilim
1
72.04 0.09
1445 12
24.80 1.48
7.57 0.02
0.41 0.01
4.48
600, 1/2 i!im
1
86.72 0.14
1402 14
18.60 1.12
7.72 0.02
0.59 0.01
10.82
2
86.60 0.25
1375 25
15.04 0.74
7.510.02
0.550.01
11.14
3
86.53 0.26
1385 27
15.38 0.73
7.46 0.02
0.560.01
12.52
600, 3/4 iiim
1
160.2 1.29
1441 76
6.44 0.11
7.46 0.02
2.37 0.02
13.80
1200, 1/4 ilim
1
52.08 0.09
1383 15
20.40 2.36
7.54 0.06
0.70 0.02
10.78
1/2 ilim
1
67.50 1.76
868 156
8.58 0.14
7.42 0.02
3.67 0.09
590
2
67.79 1.70
940165
8.430.14
7.42 0.02
3.72 0.09
499.83
3
67.69 1.13
1102 127
9.74 0.16
7.53 0.02
3.97 0.07
256.84
3/4 ilim
1
106.44 0.51
1764 54
6.71 0.14
7.41 0.02
1.72 0.01
7.63
2400, 1/4 iiim
1
35.90 0.07
1550 19
12.9 1.00
7.23 0.02
0.36 0.01
1.42
2
36.080.06
159718
17.180.92
7.260.012
0.35 0.005
2.51
3
36.16 0.04
164313
28.491.10
7.370.01
0.400.01
2.87
1/2 ilim
1
48.941.36
646130
8.550.15
7.570.03
5.500.14
1005.7
2
49.32 1.32
712 138
8.41 0.15
7.56 0.03
5.38 0.13
891.46
3
49.591.55
675 +155
7.710.13
7.480.02
5.060.14
1106.8
3/4 iiim
1
80.01 1.18
1749164
9.16 0.13
7.54 0.02
3.36 0.05
11.53
2
81.521.33
1877197
8.250.14
7.48 0.02
3.030.04
15.08


Table 2.3continued
3
82.41 1.32
1976 199
7.91 0.14
3000, 1/4 i,jm
1
32.69 0.04
1762 15
0.34 0.12
1/2 ilim
1
39.79 + 0.05
1859 16
1.21 0.45
2
40.08 0.06
1935 17
7.18 1.46
3
40.19 0.05
1965 15
8.50 1.29
4
40.42 0.06
2015 19
4.44 0.58
5
40.45 0.05
2044 16
7.98 0.91
6
40.50 0.05
207116
11.13 1.43
3/4 ilim
83.10 1.85
1663 237
5.68 0.13
7.460.02
2.930.04
15.08
5.61 0.34
1.64 0.33
1.95
6.40 0.23
0.89 0.22
2.26
7.04 0.05
0.33 0.03
2.44
7.08 0.04
0.33 0.01
2.00
6.92 0.04
0.41 0.03
2.44
7.06 0.03
0.34 0.01
1.44
7.14 0.03
0.33 0.01
1.94
7.26 0.02
2.66 0.05
20.57


Table 2.4. Selected results from the process model regression for EIS data collected using treatment 1, without CPE correction.
Condition
z(0), a
Sc
C, HF
Re, D
Rt, fi
NRSSQ
120, Vi ito
160.57 + 5.38
811 1 172
8.0410.13
7.5410.02
4.81 10.10
357.60
V* ilim
192.82 0.57
1304124
17.4010.65
7.41 10.01
0.5710.004
9.73
600, 14 ilim
72.0410.09
1445 1 12
24.801 1.48
7.5710.02
0.41 10.01
4.48
*/2 ilim
86.7210.14
14021 14
18.601 1.12
7.7210.02
0.5910.01
10.82
3/4 him
160.21 1.29
1441 176
6.4410.11
7.46 1 0.02
2.3710.02
13.80
1200, 14 ilim
52.08 10.09
1383 1 15
20.4012.36
7.5410.06
0.7010.02
10.78
14 ilim
67.691 1.13
11021 127
9.7410.16
7.53 10.02
3.9710.07
256.84
34 ilim
106.4410.51
1764154
6.71 10.14
7.41 10.02
1.7210.01
7.63
2400,14 iiim
35.9010.07
15501 19
12.911.00
7.23 1 0.02
0.3610.01
1.42
14 ilim
49.321 1.32
7121 138
8.41 10.15
7.5610.03
5.38 10.13
891.46
3/4 ilim
80.01 1 1.18
17491 164
9.1610.13
7.5410.02
3.3610.05
11.53
3000, 14 ilim
32.6910.04
17621 15
0.3410.12
5.61 10.34
1.6410.33
1.95
14 ilim
40.4510.05
20441 16
7.98 10.91
7.0610.03
0.3410.01
1.44
34 ilim
83.101 1.85
1663 1237
5.68 10.13
7.2610.02
2.6610.05
20.57


Table 2.5. Results from the process model regression for EIS data collected using treatment 1, with CPE correction
File
Set no.
z(0), n
Sc
C, pF
Re, Q
Rt, a
4>
NRSSQ
120, Vi i,im
1
158.11 0.55
1047 24
4.59 0.05
6.92 0.01
7.85 0.05
0.28 0.003
8.26
2
159.00 0.76
1073 33
4.19 0.05
6.88 0.01
7.06 0.06
0.29 0.004
2.35
3
159.40 0.77
1087 33
4.17 0.06
6.88 0.01
6.73 0.06
0.29 0.004
2.43
600,3/4 i)jm
1
160.45 0.41
1040 30
2.59 0.10
6.71 0.03
4.90 0.10
0.35 0.008
27.68
1200, Vi ilim
1
65.19 0.21
1113 26
5.05 0.07
6.86 0.01
7.41 0.08
0.32 0.004
7.56
2
65.61 0.22
1192 30
5.02 0.07
6.87 0.05
7.13 0.07
0.30 0.004
7.40
3
66.00 0.13
1243 18
5.00 0.07
6.86 0.01
6.93 0.05
0.30 0.004
6.15
1200,y4 ilira
1
106.84 0.38
1242 51
2.51 0.20
6.72 0.05
3.47 0.10
0.35 0.014
28.21
2400, >/2 i,im
1
45.94 0.12
1108 22
4.75 0.04
6.89 0.01
10.49 0.07
0.29 0.003
8.14
2
46.49 0.12
1156 22
4.64 0.04
6.88 0.01
10.07 0.06
0.29 0.002
7.78
3
46.57 0.17
1187 33
4.54 0.05
6.87 0.01
9.90 0.08
0.30 0.003
7.42
2400, y4 ilim
1
78.39 0.20
1691 30
6.08 0.07
7.12 0.01
5.65 0.06
0.23 0.004
1.66
2
79.87 0.26
1718 40
5.18 0.08
7.03 0.01
5.52 0.07
0.27 0.005
1.36
3
80.84 0.24
1757 37
4.75 0.07
6.97 0.01
5.50 0.07
0.28 0.005
1.63
3000, Vi ilim
2
39.82 0.05
1737 27
16.1 1.38
7.17 0.02
0.47 0.02
0.10 0.008
1.88
3
39.97 0.05
1796 27
15.9 1.29
7.17 0.02
0.46 0.02
0.09 0.008
1.73
5
40.34 0.05
1949 27
10.9 1.18
7.10 0.02
0.41 0.02
0.07 0.014
1.35
6
40.23 0.05
1866 25
19.8 1.13
7.22 0.01
0.52 0.02
0.09 0.007
1.62
3000, % ilim
1
81.59 0.56
1258 69
3.08 0.09
6.72 0.02
6.30 0.18
0.35 0.009
12.17


Table 2.6. Selected results from the process model regression for EIS data collected using treatment 1, with CPE correction
Condition
z(0), n
Sc
C, UF
Re; O
Rt, O

NRSSQ
120, Va ilim
159.00 + 0.76
1073 133
4.1910.05
6.8810.01
7.06 1 0.06
0.29 1 0.004
2.35
600, Va ilim
160.45 10.41
1040130
2.5910.10
6.71 10.03
4.90 1 0.09
0.3510.01
27.68
1200, Vl ilim
Va ilim
66.0010.13
106.8410.38
1243 1 18
1242151
5.0010.07
2.51 10.20
6.8610.01
6.72 1 0.05
6.93 10.05
3.4710.10
0.3010.004
0.35 10.01
6.15
28.21
2400, */2 ilim
Va ilim
46.5710.17
79.8710.26
1187133
1718140
4.5410.05
5.18 10.08
6.8710.01
7.03 10.01
9.9010.08
5.5210.07
0.3010.003
0.2710.01
7.42
1.36
3000, V2 ilim
Va ilim
40.3410.05
81.5910.56
1949127
1258 169
10.901 1.18
3.08 10.09
7.1010.02
6.7210.02
0.41 10.02
6.3010.18
0.0710.01
0.35 10.01
1.35
12.17


Table 2.7. Results from the process model regression for EIS data collected using treatment 2, without CPE correction.
Condition
Set no.
z(o), n
Sc
C, pF
Re, Q
Rt, Q
NRSSQ
120, Vi ilim
1
200.78 0.87
1161 32
11.15 0.20
7.31 0.008
0.85 0.005
4.24
120, Vi ilim
1
383.13 4.45
1120 77
12.77 0.30
7.36 0.010
0.89 0.031
27.27
2
379.17 2.43
1138 43
10.43 0.17
7.45 0.011
1.40 0.016
4.32
3
379.06 3.51
1134 62
8.64 0.15
7.41 0.012
1.46 0.018
8.60
600, Vi ilim
1
68.78 0.22
882 18
12.42 0.60
7.71 0.023
0.82 0.012
14.12
2
68.77 0.38
852 30
7.97 0.35
7.52 0.021
0.87 0.013
32.93
3
68.42 0.24
878 20
14.68 0.65
7.75 0.020
0.81 0.010
13.23
600, V2 i|im
1
90.66 0.69
1099 52
6.58 0.23
7.36 0.016
0.91 0.010
45.83
2
89.35 0.36
1087 28
12.46 0.43
7.61 0.017
0.91 0.009
16.38
3
89.65 0.32
1115 25
13.68 0.46
7.67 0.018
0.94 0.009
10.71
600, Vi ilim
1
153.63 0.44
1262 22
6.02 0.17
7.34 0.014
0.86 0.006
2.80
2
154.35 0.78
1208 38
4.41 0.14
7.22 0.014
0.84 0.007
2.85
3
154.05 0.45
1230 22
6.00 0.16
7.33 0.014
0.90 0.006
1.70
1200, */4 i,im
1
51.27 0.45
980 58
5.36 0.06
7.49 0.018
4.05 0.028
105.73
2
51.48 0.76
969 97
4.40 0.04
7.35 0.013
4.19 0.043
142.11
1200, V4 ilim
1
61.60 0.21
1141 24
14.30 0.52
7.50 0.016
0.77 0.008
15.47
2
61.77 0.27
1141 32
12.19 0.44
7.45 0.015
0.75 0.008
26.69
3
61.78 0.24
1169 30
13.75 0.49
7.51 0.015
0.76 0.008
21.17


Table 2.7continued
1200, 3/4 ilim
1
106.66 0.40
1230 30
9.03 0.22
7.47 0.017
1.26 0.012
13.98
2
107.18 0.89
1175 63
5.64 0.16
7.29 0.015
1.13 0.016
59.79
3
107.03 0.63
1221 46
6.67 0.18
7.39 0.016
1.21 0.014
32.17
2400, 1/4 i,im
1
35.42 0.28
972 50
7.72 0.53
7.46 0.024
0.66 0.016
41.53
2
35.11 0.19
1009 35
20.16 1.10
7.72 0.018
0.63 0.014
24.73
3
35.18 0.21
1018 39
10.19 0.12
6.88 0.007
1.34 0.009
24.45
2400, 1/2 i,im
1
44.11 0.12
1269 22
15.00 0.62
7.78 0.022
0.92 0.009
18.20
2
44.08 0.11
1265 21
15.51 0.63
7.81 0.021
0.89 0.009
17.99
3
44.32 0.21
1230 38
8.29 0.32
7.54 0.020
0.95 0.011
46.65
2400, 3/4 ilim
1
79.49 0.70
1401 75
6.71 0.19
7.42 0.016
1.15 0.017
6.71
2
79.14 0.58
1391 62
7.29 0.20
7.48 0.016
1.13 0.015
6.33
3
79.36 0.67
1392 72
6.65 0.20
7.45 0.016
1.12 0.016
7.45
3000, 1/4 ilim
1
31.51 0.13
1103 28
29.53 1.38
7.80 0.017
0.66 0.015
15.34
2
31.69 0.19
1068 42
15.71 0.91
7.61 0.017
0.55 0.013
27.38
3
31.29 0.13
1092 28
30.82 1.38
7.77 0.016
0.66 0.016
15.83
3000, 1/2 iiim
1
42.72 0.52
1101 92
4.86 0.05
7.66 0.016
5.21 0.05
116.92
2
42.63 0.58
1127105
4.38 0.05
7.52 0.021
5.77 0.06
106.23
3
42.47 0.43
1176 84
4.57 0.04
7.63 0.016
5.98 0.04
80.58
3000, 3/4 ilim
1
78.31 2.12
1167 205
4.86 0.06
7.54 0.022
6.26 0.10
24.17
2
78.22 1.63
1261 165
4.69 0.05
7.59 0.019
7.22 0.08
15.68
3
78.23 1.50
1306164
4.60 0.04
7.61 0.019
7.76 0.08
12.93


Table 2.8. Selected results from the process model regression for EIS data collected using treatment 2, without CPE correction.
Condition
z(0), a
Sc
C, tiF
Re, O
Rt, Q
NRSSQ
120, 1/2 is
200.78 0.87
1161 32
11.1 0.20
7.31 0.01
0.85 0.01
4.2
3/4 ilia,
379.17 2.43
1138 43
10.4 0.17
7.45 0.01
1.40 0.02
4.3
600, 1/4 ilim
68.42 0.24
878 20
14.7 0.65
7.75 0.02
0.81 0.01
13.2
1/2 i|im
89.65 0.32
1115 25
13.7 0.46
7.67 0.02
0.94 0.01
10.7
3/4 ilim
154.05 0.45
1230 22
6.00 0.16
7.33 0.01
0.90 0.01
1.7
1200, 1/4 him
51.27 0.45
980 58
5.36 0.06
7.49 0.02
4.05 0.03
105.7
1/2 ilim
61.60 0.21
1141 24
14.3 0.52
7.50 0.02
0.77 0.01
15.5
3/4 him
106.66 0.40
1230 30
9.03 0.22
7.47 0.02
1.26 0.01
14.0
2400, 1/4 iiim
35.18 0.21
1018 39
10.2 0.12
6.88 0.01
1.34 0.01
24.5
1/2 ilim
44.08 0.11
1265 21
15.5 0.63
7.81 0.02
0.89 0.01
18.0
3/4 ilim
79.14 0.58
1391 62
7.29 0.20
7.48 0.02
1.13 0.02
6.3
3000, 1/4 ilim
31.51 0.13
1103 28
29.5 1.38
7.80 0.02
0.66 0.02
15.3
1/2 ilim
42.47 0.43
1176 84
4.57 0.04
7.63 0.02
5.98 0.04
80.6
3/4 ilim
78.23 1.50
1306 164
4.60 0.04
7.61 0.02
7.76 0.08
12.9


Table 2.9. Results from the process model regression for EIS data collected using Treatment 2, with CPE correction.
Condition
Set no.
Z(0), Q
Sc
C, p.F
fi
r n
4>
NRSSQ
120, 1/2 ilim
1
200.66 0.77
1108 31
9.69 0.36
7.21 0.023
1.15 0.063
0.14 0.026
4.38
120, 3/4 i,im
3
378.86 3.22
1041 64
7.91 0.28
7.29 0.028
2.33 0.184
0.16 0.026
15.95
600, 1/2 ilim
1
94.97 0.69
677 26
1.17 0.18
6.61 0.054
2.25 0.039
0.49 0.009
232.28
1200, l/4i,im
1
50.88 0.13
1074 19
3.61 0.05
7.05 0.016
4.95 0.030
0.15 0.004
25.90
2
50.94 0.22
1120 32
3.37 0.03
7.04 0.010
5.10 0.032
0.15 0.004
15.19
3
51.07 0.22
1111 32
3.32 0.03
7.04 0.011
5.04 0.033
0.14 0.004
18.03
2400, 1/4 ium
3
35.04 0.18
1024 32
9.15 0.18
6.80 0.013
1.56 0.033
0.10 0.012
16.29
2400, 3/4 i,im
1
82.59 0.99
519 49
4.44 0.23
7.04 0.027
3.55 0.137
0.40 0.012
38.57
2
78.70 0.54
1186 105
6.59 0.34
7.35 0.043
1.83 0.213
0.18 0.043
7.18
3
82.64 1.01
504 44
4.43 0.24
7.07 0.028
3.49 0.132
0.40 0.011
39.70
3000, 1/2 ium
1
42.14 0.11
1282 25
3.79 0.03
7.35 0.008
6.24 0.027
0.13 0.003
14.64
2
41.98 0.15
1337 33
3.11 0.04
7.08 0.014
7.04 0.041
0.14 0.004
7.47
3
41.98 0.08
1332 19
3.64 0.02
7.33 0.007
6.88 0.021
0.10 0.002
11.03
3000, 3/4 iiim
1
76.43 0.63
1476 79
3.65 0.04
7.20 0.013
8.50 0.088
0.16 0.005
2.45
2
76.82 0.36
1543 47
3.73 0.02
7.29 0.008
8.93 0.047
0.12 0.003
1.27
3
76.87 0.32
1565 41
3.67 0.02
7.30 0.008
9.38 0.043
0.11 0.002
1.35


Table 2.10. Results from the process model regression for EIS data collected using Treatment 2, with CPE correction.
Condition
z(0), a
Sc
C, UF
Re
Rt
d>
NRSSQ
120, 1/2 i,im
200.66 + 0.77
1108 31
9.69 0.36
7.21 0.02
1.15 0.06
0.14 0.03
4.38
3/4 ijim
378.86 3.22
1041 64
7.91 0.23
7.29 0.03
2.33 0.18
0.16 0.03
15.95
600, 1/2 ilim
94.97 0.69
677 26
1.17 0.18
6.61 0.05
2.25 0.04
0.49 0.01
232.28
1200, 1/4 ilim
50.94 0.22
1120 32
3.37 0.03
7.04 0.01
5.10 0.03
0.15 0.004
15.19
2400, 1/4 i,im
35.04 0.18
1024 32
9.15 0.18
6.80 0.01
1.56 0.03
0.100.01
16.29
3/4 iiim
78.70 0.54
1186 105
6.59 0.34
7.35 0.04
1.83 0.21
0.18 0.04
7.18
3000, 1/2 iUm
41.98 0.15
1337 33
3.11 0.04
7.08 0.01
7.04 0.04
0.14 0.004
7.47
3/4 ilim
76.82 0.36
1543 47
3.73 0.02
7.29 0.01
8.93 0.05
0.12 0.003
1.27


54
Table 2.11. Results from the process model regression for EHD data collected at the
mass-transfer-limited current using treatment 1.
Q, rpm
Set no.
Ao, p.A/rpin
Sc
ReC, sec
NRSSQ
120
1
0.459 0.001
1147 13
0.0129 0.0007
31.32
2
0.458 0.002
1162 15
0.0092 0.0007
32.45
3
0.459 0.002
1176 19
0.0112 0.0009
38.455
200
1
0.352 0.001
1220 16
0.0051 0.0004
25.338
3
0.345 0.002
1286 24
0.0061 0.0006
44.037
600
1
0.199 0.001
1334 17
0.0012 0.0002
29.522
2
0.199 0.001
1359 18
0.0010 0.0002
31.072
3
0.198 0.001
137719
0.0009 0.0002
34.577
1200
2
0.137 0.0003
1384 11
0.0006 0.0001
15.674
3
0.139 0.0004
1434 21
0.0002 0.0001
28.567
Table 2.12. Selected results from the process model regression for EHD data collected at
the mass-transfer-limited current using treatment 1.
O, rpm
A0, pA/rpm
Sc
ReC, sec
NRSSQ
120 rpm
0.459 0.001
1147 13
0.0129 0.0007
31.32
200 rpm
0.352 0.001
1220 16
0.0051 0.0004
25.34
600 rpm
0.199 0.001
1334 17
0.0012 0.0002
29.52
1200 rpm
0.137 0.0003
1384 11
0.0006 0.0001
15.67


55
Table 2.13. Limiting current values obtained for treatment 2 at different rotation speeds.
iiim, mA/cm2
O, rpm
Measurement 1
Measurement 2
Measurement 3
Average
120
-1.8397
-1.8285
-1.8440
-1.8374
600
-3.9369
-4.0633
-4.0116
-4.0039
1200
-5.5931
-5.4990
-5.5359
-5.5426
2400
-7.6981
-7.6720
-7.7960
-7.7220
3000
-7.8231
-8.2170
-8.0617
-8.0339
Table 2.14. Schmidt numbers obtained from the ia values presented in Table 2.14.
Sc
Q, rpm
Measurement 1
Measurement 2
Measurement 3
Average
120
1112
1122
1108
1114
600
1187
1132
1154
1158
1200
1179
1210
1198
1195
2400
1228
1234
1205
1222
3000
1417
1317
1355
1362


56
Potential, V
Figure 2.1. The DC polarization curves for various surface treatments. The results are
presented for the cathodic region, as this is the region of interest for this work.
Measurements were made at 600 rpm.


57
Personal Computer
Frequency Response Analyzer
Figure 2.2. Experimental setup for the impedance measurements.


58
O 50 100 150 200
Zr, Q
Figure 2.3. Regression of a measurement model with 8 Voigt elements to the EIS data
obtained for rotating disk electrode at 120 rpm, l/4th of the limiting current. Solid line in
the figures is the measurement model fit and circles represent the data.


59
to, Hz
0.001 0.1 10 1000 100000
co, Hz
Figure 2.4. Regression of a measurement model with 8 Voigt elements to the EIS data
obtained for rotating disk electrode at 120 rpm, l/4th of the limiting current,
corresponding to the condition in Figure 2.3. (a) real part as a function of frequency and
(b) imaginary part as a function of frequency.


60
0.001 0.1 10 1000 100000
co, Hz
0.001 0.1 10 1000 100000
co, Hz
Figure 2.5. Normalized residual errors in the (a)real and (b)imaginary parts as functions of
frequency for the regression of a measurement model with 8 Voigt elements to the EIS
data obtained for rotating disk electrode at 120 rpm, l/4th of the limiting current.


61
a
L I 11 lllll I III
EAo
0.1 r
- 0.01
0.001 ~
11 nuil i m miq i HI Im i i i nuil I 111 mi i 111 in
A AO A,
O
O
O 3
O
0 0001 I limn I I lllllll l 11 lilil l mini i limn i i muH in
0.001 0.1 10 1000 100000
CD, Hz
Figure 2.6. Standard deviations in the real (0) and imaginary (A) parts calculated using
measurement models with modulus weighting for the 3 replicates of EIS data collected at
120rpm, 174th of limiting current case for the rotating disk electrode system.


-Zj, |iA/rpm
62
Zr, pA/rpm
Figure 2.7. Regression of a measurement model with 2 Voigt elements to the EHD data
obtained for rotating disk electrode at 120 rpm, mass-transfer-limited current using
treatment 1. Solid line in the figures is the measurement model fit and circles represent the
data.


63
to, Hz
, Hz
Figure 2.8. Regression of a measurement model with 2 Voigt elements to the EHD data
obtained for rotating disk electrode at 120 rpm, mass-transfer-limited current. The
corresponds to that in Figure 2.7. Solid line in the figures is the measurement model fit. (a)
real and (b) imaginary parts as functions of frequency.


64
Figure 2.9. Normalized residual errors in the (a) real and (b) imaginary parts as functions
of frequency from the regression of a measurement model with 2 Voigt elements to the
EHD data obtained (corresponding to Figure 2.7) for rotating disk electrode at 120 rpm,
mass-transfer-limited current.


65
0.001 0.1 10 1000 100000
co, Hz
Figure 2.10. The solid line represents the error structure model obtained by accounting for
various conditions for the rotating disk electrode, using treatment 1. The (O)s and the (A)s
represent the standard deviations of the stochastic noise obtained by using the
measurement model approach applied to 120rpm, 174th of limiting current.


66
o), Hz
Figure 2.11. The solid line represents the error structure model obtained by accounting for
various conditions for the rotating disk electrode, using treatment 1. The (O)s and the (A)s
represent the standard deviations of the stochastic noise obtained by using the
measurement model approach applied to 1200 rpm, 1/2 of limiting current.


67
o, Hz
Figure 2.12. The solid line represents the error structure model obtained by accounting for
various conditions for the rotating disk electrode, using treatment 1. The (O)s and the (A)s
represent the standard deviations of the stochastic noise obtained by using the
measurement model approach applied to 3000 rpm, 1/2 of limiting current.


68
, Hz
0.001 0.1 10 1000 100000
co, Hz
Figure 2.13. Checking for consistency with the Kramers-Kronig relations. EIS data
collected for 120 rpm, lM* of the limiting current case for the rotating disk electrode
system. Measurement model was regressed to the (a)real part and (b)imaginary part was
predicted based on the 10 lineshape parameters obtained. The outer lines represent the
95.4% confidence limits and the line through the data is the measurement model fit.


69
0), Hz
co, Hz
Figure 2.14. Normalized residual errors in (a) real and (b) imaginary parts corresponding
to the regression results presented in Figure 2.13. The outer lines represent the 95.4%
confidence limits and the line through the data is the measurement model fit.


70
to, Hz
0.001 0.1 10 1000 100000
co, Hz
Figure 2.15. Checking for consistency with the Kramers-Kronig relations. 120 rpm, 174th
of the limiting current case for the rotating disk electrode system. Measurement model
was (a)regressed to the imaginary part and (b)real part is predicted based on the 11
lineshape parameters obtained. The outer lines represent the 95.4% confidence limits and
the line through the data is the measurement model fit.


71
rn. Hz
co, Hz
Figure 2.16. Normalized residual errors in (a) real and (b) imaginary parts corresponding
to the regression results presented in Figure 2.15. The outer lines represent the 95.4%
confidence limits and the line through the data is the measurement model fit.


72
O 50 100 150 200
Zr,Q
Figure 2.17. Process model regression (with error structure weighting) for 120 rpm, 174th
of the limiting current case for the rotating disk electrode system. Error structure was used
to fit the data to the model. The solid line represents fit of the model to the data.


73
0.001 0.1 10 1000 100000
CO, Hz
Figure 2.18. Process model regression (with error structure weighting) for EIS data
collected at 120 rpm, 174th of the limiting current case for the rotating disk electrode
system, corresponds to Figure 2.17. Error structure was used to fit the data to the model.
The solid line represents fit of the model to the data. Outer lines represent the 95.4%
confidence limits.


74
0.001 0.1 10 1000 100000
co, Hz
co, Hz
Figure 2.19. Normalized residual errors in (a) real and (b) imaginary parts corresponding
to the regression results presented in Figure 2.17. The outer lines represent the 95.4%
confidence limits.


75
O 50 100 150 200
Zr,Q
Figure 2.20. Process model regression (with error structure weighting) accounting for
CPE correction for the EIS data collected for 120 rpm, 174th of the mass-transfer-limited
current case for the rotating disk electrode system using treatment 1. The solid line
represents fit of the model to the data.


76
Q, Hz
0.001 0.1 10 1000 100000
to, Hz
Figure 2.21. Process model regression (with error structure weighting) accounting for
CPE correction for the EIS data collected for 120 rpm, 174th of the mass-transfer-limited
current case for the rotating disk electrode system using treatment 1, corresponding to
Figure 2.20. The solid line represents fit of the model to the data, (a) real and (b)
imaginary parts as functions of frequency.


77
0.001 0.1 10 1000 100000
co, Hz
0.001 0.1 10 1000 100000
(D, Hz
Figure 2.22. Normalized residual errors in (a) real and (b) imaginary parts corresponding
to the regression results presented in Figure 2.20. The dashed lines represent the
normalized noise level.


78
-0.1 0 0.1 0.2 0.3 0.4 0.5
Zr, fiA/rpm
Figure 2.23. Process model regression (with error structure weighting) for the EHD data
collected for 120 rpm, at the mass-transfer-limited current case for the rotating disk
electrode system using treatment 1. The solid line represents fit of the model to the data.


79
CD, Hz
0.01 0.1 1 10
co, Hz
Figure 2.24. Process model regression (with error structure weighting) for the EHD data
collected for 120 rpm, at the mass-transfer-limited current case for the rotating disk
electrode system using treatment 1, corresponds to Figure 2.23. The solid line represents
fit of the model to the data, (a) Real part and (b) Imaginary part as functions of frequency


Full Text

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