
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00018888/00001
Material Information
 Title:
 Influence of current distribution on the interpretation of the impedance spectra collected for a rotating disk electrode
 Creator:
 Durbha, Madhav
 Publication Date:
 1998
 Language:
 English
 Physical Description:
 xx, 310 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Chemical Engineering thesis, Ph.D ( lcsh )
Corrosion and anticorrosives ( lcsh ) Dissertations, Academic  Chemical Engineering  UF ( lcsh ) Electrodes, Copper ( lcsh ) Seawater ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph.D.)University of Florida, 1998.
 Bibliography:
 Includes bibliographical references (leaves 300309).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Madhav Durbha.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 029539680 ( ALEPH )
40074509 ( OCLC )

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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 brothersinlaw 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 KramersKronig 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 SelfConsistent 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 MASSTRANSFER 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 doublelayer 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 NonUniform 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 KRAMERSKRONIG
TRANSFORMS: IMPLICATIONS FOR ERROR STRUCTURE
IDENTIFICATION 186
7.1 Experimental Motivation 187
7.2 Application of the KramersKronig 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 KramersKronig 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
masstransferlimited current using treatment 1 54
2.12. Selected results from the process model regression for EHD data collected
at the masstransferlimited 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. Onedimensional 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. Smallsignal analysis of an electrochemical nonlinear 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, masstransferlimited 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, masstransferlimited 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, masstransferlimited 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 KramersKronig 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 KramersKronig 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
masstransferlimited 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
masstransferlimited 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 masstransferlimited 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 masstransferlimited 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 masstransfer
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 Ftest 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
stepsizes 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
(VOref) 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 Â£. 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 rÂ¡* and Â£. 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 onedimensional and twodimensional 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 steadystate distributions tend to be highly uniform, (b) Differences
between the calculations from twodimensional and onedimensional
model normalized with respect to the twodimensional 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 KramersKronigtransformable 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 KramersKronig relations 204
7.2. Path of integration for the contour integral in the complexfrequency 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. Lineshape models yielding the same mathematical structure for spectroscopic
response: a) Voigt model for electrochemical systems; b) KelvinVoigt
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 masstransfer 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) Ftest parameters. The inner dashed lines represent the 95% confidence
limits for the Ftest parameter and the outer lines represent the 99%
confidence limits. Circles represent the Ftest parameters for the raw
standard deviations, (c) Ftest parameters after deleting the point close to
50Hz and 100Hz. (d) Histogram with 7test 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) Ftest corresponding to the variances of stochastic noise
(c) Histogram with 7test 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 electrohydrodynamic 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 Ftest 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) Ftest 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) Ftest 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 inhouse, 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 steadystate 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 frequencydomain process model for the system
xix
of interest. It is shown here that this frequency domain model provides a significant
improvement over the onedimensional 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 KramersKronig 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 nonelectrochemical 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 semiconductor process engineers are interested in submicron
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 indepth 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 metalinsulator 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 masstransferlimited current density.
Due to the imposed rotation rate, the effects associated with the free convection can
be ignored.
By increasing the rotation speed, the masstransferlimited 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. Steadystate techniques
include simple currentvoltage 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 steadystate. Frequency domain data are more complicated to analyze
compared to the steadystate 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
multisine and singlesine techniques. Fast measurement time and mild perturbation of the
system under investigation are stated to be the strengths of the multisine technique,
though it has been shown that by using a fast frequency response analyzer, the difference
in the measurement time for a singlesine and a multisine technique is small [1], One of
the disadvantages of this technique is that it has a small frequency range. The multisine
technique is also sensitive to harmonic distortion. In this work impedance data was
collected using the singlesine technique.
The use of singlesine 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
(cot4)) 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 nonlinear in nature, higher order derivatives do
exist. However, for a very small perturbation in potential, terms of order
f d2I^
(AÂ£) and higher can be neglected and only the linear terms need to be
Eo.^o
5
retained. This process is known as quasilinearization and is widely used in nonlinear
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 masstransfer 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 flowinduced
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
onedimensional 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 nonuniform distribution of current and potential on the surface of the disk
electrode, which points to the inadequacy of Tribollet and Newmans onedimensional
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 masstransfer/reaction rate. The primary focus for this work is
in explaining the influence of nonuniform 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. [711] 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 KramersKronig 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 KramersKronig 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 nonuniformity 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 twodimensional frequencydomain 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 onedimensional model and the two
dimensional model developed here. In this chapter it is shown that the nonuniformities 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 twodimensional
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 KramerKronig 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. Smallsignal analysis of an electrochemical nonlinear 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 onedimensional
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 [1517], However in chapter 7 it would be
illustrated that such an assumption is incorrect for the systems that are consistent with the
KramersKronig 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 (Â£res) that arise due to the regression of a model (Z) to
experimental data (Z^) can be of two types, systematic errors (Â£sys) and random or
stochastic errors (Â£i,oc).
Zexp ~Z + Â£n
(2.3)
Â£res ~ Â£sys + Â£stoc
(2.4)
The systematic errors can arise due to the lack of fit of the model to the data (Â£//) or due
to an experimental bias (Skias), that is,
Â£ ays Â£lof + Â£bias
(2.5)
Experimental bias errors can arise from nonstationary behavior corresponding to a
changing baseline during the course of the impedance scan or from instrumental artifacts.
Thus,
Â£.. = Â£ + Â£ (2.6)
Most electrochemical systems are inherently nonstationary 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, KramersKronig 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 KramersKronig Relations
KramersKronig relations are selfconsistent integral relations which apply to
systems that are linear, causal, stable, and stationary. By using the quasilinear 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 timeinvariance of the system. KramersKronig relations can be
expressed in a number of different ways as presented in [12], Through the form of
KramersKronig 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 KramersKronig 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 RC
circuit elements or the Voigt elements used in these models satisfy the conditions
19
associated with the KramersKronig 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 KramersKronig relations. The procedure
for using measurement models to identify the noise level in the measurement and to check
for consistency with the KramersKronig relations is illustrated in a subsequent section.
Once the selfconsistent 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 onedimensional
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 masstransferlimited current or under kinetic control. As EIS (Electrochemical
Impedance Spectroscopy) measurements cannot be conducted at the masstransferlimited
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 masstransfer 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 onedimensional 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 rÂ¡ 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 VT, 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 onedimensional expression for conservation of
species / (equation (2.8)) yields
22
d2c, de, ... ~
+v'HrJKici=
d? z dÂ£
(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 (Â£) as
ci. o =ci + Re^,,o^i()}
(2.18)
(2.19)
Thus:
dc.
dz
2=0
= MLÂ¡'(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=RjfR
J
Ci,0
From equation (2.20)
S' V> 8,
nFD: ff'(6)
Ci,0 ~
Equation (2.25) becomes
= RjfR,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 masstransferlimited 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 masstransfer 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 onedimensional 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 highspeed lowinertia 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 masstransferlimited current for all the rotation speeds, using treatments 1 and 2.
The EHD measurements were conducted at the masstransferlimited 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 startup 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 inhouse 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 selfconsistent part of the
impedance spectra is illustrated with the example of EIS data sets collected for 120rpm at
174th of masstransferlimited current for the system of interest. The data were analyzed
employing the userfriendly MATLABbased visual interface created inhouse 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 masstransferlimited 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 nonstationary (Â£) 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 masstransferlimited current, and using 2 Voigt
elements, is presented in Figure 2.7. A noweighting 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\ + yjZ + >
Km
(2.39)
where a, /?, y and 8 are constants determined by regression analysis, RsoÂ¡ 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 KramersKronig 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 2Â¡ 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 SelfConsistent Part of the Impedance Spectra
The use of measurement models to identify the selfconsistent portion of the
impedance spectra takes advantage of the fact that the KramersKronig 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 MonteCarlo 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 masstransfer
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 KramersKronig 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 KramersKronig 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 masstransferlimited current, and the portions of the
spectra which are inconsistent with KramersKronig relations were identified and deleted.
36
For the EHD data collected for treatment 1, the consistency check with KramersKronig
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 onedimensional 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 masstransferlimited 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 masstransferlimited current
39
measurements. The masstransferlimiting 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 masstransferlimiting
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 masstransferlimited 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
transferlimiting 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 masstransport
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 masstransfer. 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 2.32.6. 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 masstransferlimited 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 onedimensional
model is adequate, as the current density is uniform at the masstransferlimited 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 onedimensional 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: nonuniform 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 physicochemical impedance model
accounting for the nonuniform 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 twodimensional frequency domain
model, it is necessary to understand the steadystate 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 KramersKronig
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 masstransfer 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 iiÂ¡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 iim
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 iim
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
masstransferlimited 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 masstransferlimited 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, masstransferlimited 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, masstransferlimited 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,
masstransferlimited 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 KramersKronig 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 KramersKronig 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 masstransferlimited
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 masstransferlimited
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 masstransferlimited 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 masstransferlimited 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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INGEST IEID EQZ6XPBRR_QJS4M9 INGEST_TIME 20140429T21:48:50Z PACKAGE AA00018888_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
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 Riemerâ€™s 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 brothersinlaw 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 KramersKronig 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 SelfConsistent 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 MASSTRANSFER LIMITED CURRENT 97
4.1 Theoretical Development 99
4.1.1 Diffusion Layer 99
4.1.2 Outer Region: Laplaceâ€™s Equation 103
4 .1.3 Diffuse Part of the Double Layer 104
4.1,3a Solution of Poissonâ€™s equation 105
4.1.3b Calculation of doublelayer 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 NonUniform 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 KRAMERSKRONIG
TRANSFORMS: IMPLICATIONS FOR ERROR STRUCTURE
IDENTIFICATION 186
7.1 Experimental Motivation 187
7.2 Application of the KramersKronig 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 KramersKronig 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
masstransferlimited current using treatment 1 54
2.12. Selected results from the process model regression for EHD data collected
at the masstransferlimited 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 G'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. Onedimensional 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. Smallsignal analysis of an electrochemical nonlinear 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, masstransferlimited 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, masstransferlimited 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, masstransferlimited 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 KramersKronig relations. EIS data
collected for 120 rpm, 1/4â€œ* 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 KramersKronig 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
masstransferlimited 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
masstransferlimited 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 masstransferlimited 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 masstransferlimited 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 masstransfer
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 1/4* 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 Ftest 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 6^(o) for infinite Schmidt number,
making use of the values obtained from the FDM scheme using varying
stepsizes 120
4.2. A sixth degree polynomial fit for Â¿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
(VOref) 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 rjs and Â£. 125
4.7. Calculated overpotentials for the case of Figure 4.5a (120rpm) at 3/4th of inâ„¢ 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 rÂ¡* and Â£. 127
4.9. Calculated overpotentials for the case of Figure 4.5b (3000rpm) at 3/4th of iun, 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 iiâ„¢, 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 onedimensional and twodimensional models for
the slow kinetics case at 174th of iym and Q=120rpm with /'0 = 3 mA/cm2,
D = 0.3095xl0â€˜5 cm2/sec, J= 0.225, N= 0.0695, and Sc = 2730. In this
case steadystate distributions tend to be highly uniform, (b) Differences
between the calculations from twodimensional and onedimensional
model normalized with respect to the twodimensional 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 = 0.3095xl0'5 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 inâ„¢, i0 = 30 mA/cm2, D = 0.3095x10â€˜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, 1/4â€œ* of inâ„¢, 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, 1/4* of iIim, i0 = 100 mA/cm2, D = 0.3195xl0'5 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, 3/4* 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 = 0.6795xl0â€˜5 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, /'0 = 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 ihm. 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 KramersKronigtransformable 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 KramersKronig relations 204
7.2. Path of integration for the contour integral in the complexfrequency 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. Lineshape models yielding the same mathematical structure for spectroscopic
response: a) Voigt model for electrochemical systems; b) KelvinVoigt
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 masstransfer 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) Ftest parameters. The inner dashed lines represent the 95% confidence
limits for the Ftest parameter and the outer lines represent the 99%
confidence limits. Circles represent the Ftest parameters for the raw
standard deviations, (c) Ftest parameters after deleting the point close to
50Hz and 100Hz. (d) Histogram with 7test 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) Ftest corresponding to the variances of stochastic noise
(c) Histogram with 7test 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 electrohydrodynamic 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 Ftest 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) Ftest 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) Ftest 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 inhouse, 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 steadystate 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 frequencydomain process model for the system
xix
of interest. It is shown here that this frequency domain model provides a significant
improvement over the onedimensional 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 KramersKronig 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 nonelectrochemical 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 semiconductor process engineers are interested in submicron
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 indepth 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 metalinsulator 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:
â€¢ The fluid flow for this system is very well defined and the uniform axial velocity yields
a uniform masstransferlimited current density.
â€¢ Due to the imposed rotation rate, the effects associated with the free convection can
be ignored.
â€¢ By increasing the rotation speed, the masstransferlimited current can be increased
and this results in an improved signal to noise ratio in measuring the current.
â€¢ 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. Steadystate techniques
include simple currentvoltage 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 steadystate. Frequency domain data are more complicated to analyze
compared to the steadystate 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
multisine and singlesine techniques. Fast measurement time and mild perturbation of the
system under investigation are stated to be the strengths of the multisine technique,
though it has been shown that by using a fast frequency response analyzer, the difference
in the measurement time for a singlesine and a multisine technique is small [1], One of
the disadvantages of this technique is that it has a small frequency range. The multisine
technique is also sensitive to harmonic distortion. In this work impedance data was
collected using the singlesine technique.
The use of singlesine 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
(a>t4>) is observed to be superimposed on the dc current. The Taylor series expansion for
the current is given by
A1 =
rdi^
\dE j
E.J,
AE + 
2
f J* T\
d2I
dE2
(AE)2+
(1.1)
' EqJo
As the system under consideration is nonlinear in nature, higher order derivatives do
exist. However, for a very small perturbation in potential, terms of order
f d2I^
ydE7j
(AÂ£)â€œ and higher can be neglected and only the linear terms need to be
â– Eo.^o
5
retained. This process is known as quasilinearization and is widely used in nonlinear
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 masstransfer 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 flowinduced
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
onedimensional 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 nonuniform distribution of current and potential on the surface of the disk
electrode, which points to the inadequacy of Tribollet and Newmanâ€™s onedimensional
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 masstransfer/reaction rate. The primary focus for this work is
in explaining the influence of nonuniform 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. [711] 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 KramersKronig 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 KramersKronig 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 nonuniformity 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 twodimensional frequencydomain 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 Newmanâ€™s onedimensional model and the two
dimensional model developed here. In this chapter it is shown that the nonuniformities 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 twodimensional
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 KramerKronig 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. Smallsignal analysis of an electrochemical nonlinear 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 onedimensional
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=zâ€œ+?w <2Â»
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
(2.2)
where Zr,k 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 Ã¡r k 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 [1517], However in chapter 7 it would be
illustrated that such an assumption is incorrect for the systems that are consistent with the
KramersKronig 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 (Sm) that arise due to the regression of a model (Z) to
experimental data (Z^) can be of two types, systematic errors (Â£sys) and random or
stochastic errors (Â£stoc)
^ exp â€” % + Â£n
(2.3)
Â£res ~ Â£sys + Â£stoc
(2.4)
The systematic errors can arise due to the lack of fit of the model to the data (Â£/â€ž/) or due
to an experimental bias (Â£bÂ¡as), that is,
Â£sys â€” Â£lof + Â£bias
(2.5)
Experimental bias errors can arise from nonstationary behavior corresponding to a
changing baseline during the course of the impedance scan or from instrumental artifacts.
Thus,
Â£.. = Â£ + Â£ (2.6)
Most electrochemical systems are inherently nonstationary 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, KramersKronig 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 â€œtruly
replicatedâ€. 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 KramersKronig Relations
KramersKronig relations are selfconsistent integral relations which apply to
systems that are linear, causal, stable, and stationary. By using the quasilinear 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 timeinvariance of the system. KramersKronig relations can be
expressed in a number of different ways as presented in [12], Through the form of
KramersKronig 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 KramersKronig 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 RC
circuit elements or the Voigt elements used in these models satisfy the conditions
19
associated with the KramersKronig 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 KramersKronig relations. The procedure
for using measurement models to identify the noise level in the measurement and to check
for consistency with the KramersKronig relations is illustrated in a subsequent section.
Once the selfconsistent 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 onedimensional
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 masstransferlimited current or under kinetic control. As EIS (Electrochemical
Impedance Spectroscopy) measurements cannot be conducted at the masstransferlimited
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 masstransfer 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 onedimensional analysis is employed for the impedance response of the
rotating disk electrode, the unsteady state convective diffusion can be written as
dci dct 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
'YjsÂ¡MizÃ 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, â€”Â» cja> as z â€”> oo (2.10)
and
A
dc, = Sjif
dz nF
(2.11)
where cit0O is the bulk concentration of species i, if is the Faradic current density, n is the
number of electrons transferred, and F is the Faradayâ€™s constant (96,487C/eq). The
Faradic current density is expressed as a function of surface overpotential rÂ¡ and
concentration as
21
if=f(v,c,) (212)
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
where the overbar represents the steady value, j is the imaginary number VT, 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
Ã±
\X
\a2 J
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
S. =
\av J
(2.16)
is a characteristic distance for mass transport of species Substitution of the definition for
concentration (equation (2.13)) into the onedimensional expression for conservation of
species / (equation (2.8)) yields
22
d2ct de, ... ~
72 + vz ~rt ~ jKjCÂ¡ = o
d? z dÂ£
(2.17)
A solution to equation (2.17), dl(E,)=cJci0 can be found which satisfies the boundary
conditions
dt Â» 0 as E, â€”> oo
9i =1 at  = 0
The concentration at the surface of the disk is given in terms of 9i (Â£) as
ci. o = ci +
(2.18)
(2.19)
Thus:
dc.
dz
2=0
= ^L0'(O) = ^
S. â€™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 + Câ€”
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
*/ =
rct
J7,
7+Z
Q.o
'jr'
Kdci,oj
Ci.O
(2.22)
The charge transfer resistance R, is defined to be
23
R< =
\dTlJct,
Thus;
~ 1 ~ ^
v=r+?
df'
Ci,0
Wj.jti
Equation (2.24) can be expressed in terms of overpotential as
v=RjfR,Yj
A;
Ci,0
â€™I'Cj.jm
From equation (2.20)
y *!>/ p,
nFD, ff'(6)
Ci,0 ~
Equation (2.25) becomes
V = R,TfR,Y,
yKoj
s.if Ji
nFDi <9/(o)
or
Tl=7f(Rt + ZD)
where
zd=r,i:
f A
v^.oy
VCjj*t
nFD, e;{o)
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 + 7]
(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 â€” if + jcÃ¼Crj
(2.31)
(2.32)
where C is the double layer capacitance. Equations (2.28), (2.31), and (2.32) result in
Z. = Z=Re+ R'+Zd ^
/' 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 masstransferlimited 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
n0 The frequency of perturbation Â© 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
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
2A0Zp(U)
R
\zd(ti)zd{u)
+ jcoReC
R,
R,
\
Zd{0)zd(u) + l) Zd{0)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/ZJO) and R/ZJO) 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/ZÂ¿(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
I = 2A0zp(u)
Q ja>ReC +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 masstransfer 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 â€œpartial blockingâ€ 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
â€œmirrorlikeâ€ 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 onedimensional 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 highspeed lowinertia 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 Â± 0.1Â°C. 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 masstransferlimited current for all the rotation speeds, using treatments 1 and 2.
The EHD measurements were conducted at the masstransferlimited 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 startup transient often influenced the
value of impedance reported by the instrumentation. Also the data points within Â±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 inhouse 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 selfconsistent part of the
impedance spectra is illustrated with the example of EIS data sets collected for 120rpm at
174th of masstransferlimited current for the system of interest. The data were analyzed
employing the userfriendly MATLABbased visual interface created inhouse 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 rk 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 1/4^ of the masstransferlimited 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 (Â£Â¡0f) 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 nonstationary (eâ€žs) 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 masstransferlimited current, and using 2 Voigt
elements, is presented in Figure 2.7. A noweighting 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 ar
and o] were regressed to the model
cr, = cr,  a
Zj\ + P\Zr Rsol\ + yjZ + Â¿>
(2.39)
where a, Â¡3, y and 8 are constants determined by regression analysis, Rsoi is the solution
resistance or the high frequency asymptote and Rm is the current measuring resistor. This
model is also referred to as the â€œerror structure modelâ€ 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 KramersKronig 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 ZÂ¡ 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 =
3.07652x1 O'5 pA/rpm.
2,4,3 Identification of the SelfConsistent Part of the Impedance Spectra
The use of measurement models to identify the selfconsistent portion of the
impedance spectra takes advantage of the fact that the KramersKronig 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 (o) 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 MonteCarlo 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 masstransfer
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 KramersKronig 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 KramersKronig 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 masstransferlimited current, and the portions of the
spectra which are inconsistent with KramersKronig relations were identified and deleted.
36
For the EHD data collected for treatment 1, the consistency check with KramersKronig
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 onedimensional 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 1073Â±33 as opposed to 811Â±172 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 masstransferlimited current is presented in the form of an impedance plane
plot in Figure 2.23. The Schmidt number obtained in this case was 1147Â±13. 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 masstransferlimited current
39
measurements. The masstransferlimiting current density (/um) 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 Faradayâ€™s constant, D is the diffusion coefficient of the masstransferlimiting
species, Coo is the bulk concentration of the reacting species, t is the transference number, a
is the coefficient in the Cochranâ€™s velocity expansion, v is the kinematic viscosity, Q is the
rotation speed in rad/sec, and Ã©?Ã³(o) is related to the concentration gradient in the axial
direction. Three values for the masstransferlimited 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
transferlimiting 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 masstransport
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 masstransfer. 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 2.32.6. 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 masstransferlimited 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 onedimensional
model is adequate, as the current density is uniform at the masstransferlimited 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 onedimensional 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: nonuniform 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 physicochemical impedance model
accounting for the nonuniform 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 twodimensional frequency domain
model, it is necessary to understand the steadystate 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 KramersKronig
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 masstransfer 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.
Ã2, rpm
Fraction of iiÂ¡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.
Q, 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, fÃ
NRSSQ
120, 1/4 is*
1
160.57 Â±5.38
811Â±172
8.04 Â±0.13
7.54 Â±0.02
4.81 Â±0.10
357.60
2
162.15 Â±6.76
850 Â±222
7.09 Â±0.12
7.43 Â±0.02
4.21 Â±0.09
437.04
3
162.33 Â±6.35
880 Â±212
7.07 Â±0.11
7.44 Â±0.02
4.08 Â±0.09
371.74
120, 1/2 iUm
1
192.82 Â±0.57
1304 Â±24
17.40 Â±0.65
7.41 Â±0.01
0.57 Â±0.004
9.73
600, 1/4 ium
1
72.04 Â±0.09
1445 Â±12
24.80 Â± 1.48
7.57 Â±0.02
0.41 Â±0.01
4.48
600, 1/2 i!im
1
86.72 Â±0.14
1402 Â± 14
18.60 Â± 1.12
7.72 Â±0.02
0.59 Â±0.01
10.82
2
86.60 Â±0.25
1375 Â±25
15.04 Â±0.74
7.51Â±0.02
0.55Â±0.01
11.14
3
86.53 Â±0.26
1385 Â±27
15.38 Â±0.73
7.46 Â±0.02
0.56Â±0.01
12.52
600, 3/4 iiim
1
160.2 Â± 1.29
1441 Â±76
6.44 Â±0.11
7.46 Â± 0.02
2.37 Â±0.02
13.80
1200, 1/4 i,â„¢
1
52.08 Â±0.09
1383 Â± 15
20.40 Â±2.36
7.54 Â±0.06
0.70 Â±0.02
10.78
1/2 ilim
1
67.50 Â± 1.76
868 Â±156
8.58 Â±0.14
7.42 Â±0.02
3.67 Â±0.09
590
2
67.79 Â±1.70
940Â±165
8.43Â±0.14
7.42 Â±0.02
3.72 Â±0.09
499.83
3
67.69 Â±1.13
1102 Â±127
9.74 Â±0.16
7.53 Â±0.02
3.97 Â±0.07
256.84
3/4 ilim
1
106.44 Â±0.51
1764 Â±54
6.71 Â±0.14
7.41 Â±0.02
1.72 Â±0.01
7.63
2400, 1/4 ijim
1
35.90 Â±0.07
1550 Â± 19
12.9 Â± 1.00
7.23 Â±0.02
0.36 Â±0.01
1.42
2
36.08Â±0.06
1597Â±18
17.18Â±0.92
7.26Â±0.012
0.35Â± 0.005
2.51
3
36.16 Â±0.04
1643Â±13
28.49Â±1.10
7.37Â±0.01
0.40Â±0.01
2.87
1/2 ilim
1
48.94Â±1.36
646Â±130
8.55Â±0.15
7.57Â±0.03
5.50Â±0.14
1005.7
2
49.32 Â± 1.32
712 Â± 138
8.41 Â±0.15
7.56 Â±0.03
5.38 Â±0.13
891.46
3
49.59Â±1.55
675 +155
7.71Â±0.13
7.48Â±0.02
5.06Â±0.14
1106.8
3/4 ilim
1
80.01 Â± 1.18
1749Â±164
9.16 Â± 0.13
7.54 Â±0.02
3.36 Â±0.05
11.53
2
81.52Â±1.33
1877Â±197
8.25Â±0.14
7.48 Â±0.02
3.03Â±0.04
15.08
Table 2.3â€”continued
3
82.41 Â± 1.32
1976 Â±199
7.91 Â±0.14
3000, 1/4 ilim
1
32.69 Â±0.04
1762Â±15
0.34 Â± 0.12
1/2 ilim
1
39.79 + 0.05
1859 Â± 16
1.21 Â±0.45
2
40.08 Â± 0.06
1935 Â± 17
7.18 Â± 1.46
3
40.19 Â±0.05
1965 Â± 15
8.50 Â± 1.29
4
40.42 Â± 0.06
2015 Â±19
4.44 Â±0.58
5
40.45 Â±0.05
2044 Â± 16
7.98 Â±0.91
6
40.50 Â±0.05
2071Â±16
11.13 Â± 1.43
3/4 ilim
83.10 Â± 1.85
1663 Â±237
5.68 Â± 0.13
7.46Â±0.02
2.93Â±0.04
15.08
5.61 Â±0.34
1.64 Â±0.33
1.95
6.40 Â±0.23
0.89 Â±0.22
2.26
7.04 Â±0.05
0.33 Â±0.03
2.44
7.08 Â±0.04
0.33 Â±0.01
2.00
6.92 Â±0.04
0.41 Â±0.03
2.44
7.06 Â±0.03
0.34 Â±0.01
1.44
7.14 Â±0.03
0.33 Â±0.01
1.94
7.26 Â±0.02
2.66 Â± 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 ihm
160.57 + 5.38
811 1 172
8.0410.13
7.5410.02
4.81 10.10
357.60
*/2 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
34 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, n
NRSSQ
120, Vi ilim
1
158.11 Â±0.55
1047 Â±24
4.59 Â±0.05
6.92 Â±0.01
7.85 Â±0.05
0.28 Â±0.003
8.26
2
159.00 Â±0.76
1073 Â±33
4.19 Â± 0.05
6.88 Â±0.01
7.06 Â± 0.06
0.29 Â± 0.004
2.35
3
159.40 Â±0.77
1087 Â±33
4.17 Â±0.06
6.88 Â±0.01
6.73 Â±0.06
0.29 Â± 0.004
2.43
600, 3/4 ilim
1
160.45 Â±0.41
1040 Â±30
2.59 Â±0.10
6.71 Â±0.03
4.90 Â±0.10
0.35 Â±0.008
27.68
1200, */2 ilim
1
65.19 Â± 0.21
1113 Â±26
5.05 Â±0.07
6.86 Â±0.01
7.41 Â±0.08
0.32 Â±0.004
7.56
2
65.61 Â±0.22
1192 Â±30
5.02 Â±0.07
6.87 Â±0.05
7.13 Â±0.07
0.30 Â±0.004
7.40
3
66.00 Â±0.13
1243 Â± 18
5.00 Â±0.07
6.86 Â±0.01
6.93 Â±0.05
0.30 Â±0.004
6.15
1200,y4 i,im
1
106.84 Â±0.38
1242 Â±51
2.51 Â±0.20
6.72 Â±0.05
3.47 Â± 0.10
0.35 Â±0.014
28.21
2400, >/2 ilim
1
45.94 Â±0.12
1108 Â±22
4.75 Â±0.04
6.89 Â±0.01
10.49 Â±0.07
0.29 Â± 0.003
8.14
2
46.49 Â±0.12
1156 Â±22
4.64 Â± 0.04
6.88 Â±0.01
10.07 Â±0.06
0.29 Â± 0.002
7.78
3
46.57 Â±0.17
1187 Â±33
4.54 Â±0.05
6.87 Â±0.01
9.90 Â±0.08
0.30 Â±0.003
7.42
2400, y4 i,im
1
78.39 Â±0.20
1691 Â±30
6.08 Â±0.07
7.12 Â± 0.01
5.65 Â±0.06
0.23 Â±0.004
1.66
2
79.87 Â±0.26
1718 Â± 40
5.18 Â± 0.08
7.03 Â±0.01
5.52 Â±0.07
0.27 Â±0.005
1.36
3
80.84 Â±0.24
1757 Â±37
4.75 Â±0.07
6.97 Â±0.01
5.50 Â±0.07
0.28 Â±0.005
1.63
3000, Vi ilim
2
39.82 Â±0.05
1737 Â±27
16.1 Â± 1.38
7.17 Â±0.02
0.47 Â±0.02
0.10 Â±0.008
1.88
3
39.97 Â±0.05
1796 Â±27
15.9 Â± 1.29
7.17 Â±0.02
0.46 Â±0.02
0.09 Â± 0.008
1.73
5
40.34 Â±0.05
1949 Â±27
10.9 Â± 1.18
7.10 Â±0.02
0.41 Â±0.02
0.07 Â±0.014
1.35
6
40.23 Â±0.05
1866 Â±25
19.8 Â± 1.13
7.22 Â±0.01
0.52 Â±0.02
0.09Â± 0.007
1.62
3000, % ilim
1
81.59 Â±0.56
1258 Â±69
3.08 Â±0.09
6.72 Â±0.02
6.30 Â±0.18
0.35 Â±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), D
Sc
C, UF
Re? n
Rt, Q
NRSSQ
120, 14 ilim
159.00 Â±0.76
1073 Â±33
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 Â±0.03
4.90 1 0.09
0.3510.01
27.68
1200, 1/2 ilim
3/4 ilim
66.0010.13
106.8410.38
1243 1 18
1242151
5.0010.07
2.51 Â±0.20
6.8610.01
6.72 1 0.05
6.93 Â±0.05
3.4710.10
0.3010.004
0.35 Â±0.01
6.15
28.21
2400, 1/2 ilim
y* ilim
46.5710.17
79.8710.26
1187133
1718140
4.54 1 0.05
5.18 Â±0.08
6.8710.01
7.03 Â±0.01
9.9010.08
5.5210.07
0.3010.003
0.2710.01
7.42
1.36
3000,14 iiim
3/4 ilim
40.3410.05
81.5910.56
1949127
1258 169
10.901 1.18
3.08 Â±0.09
7.1010.02
6.7210.02
0.41 Â±0.02
6.3010.18
0.0710.01
0.35 Â±0.01
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, Yi i,im
1
200.78 Â±0.87
1161 Â±32
11.15 Â±0.20
7.31 Â±0.008
0.85 Â±0.005
4.24
120, Vi ilim
1
383.13 Â±4.45
1120 Â±77
12.77 Â±0.30
7.36 Â±0.010
0.89 Â±0.031
27.27
2
379.17 Â±2.43
1138 Â± 43
10.43 Â±0.17
7.45 Â±0.011
1.40 Â±0.016
4.32
3
379.06 Â±3.51
1134 Â±62
8.64 Â±0.15
7.41 Â±0.012
1.46 Â±0.018
8.60
600, Vi ilim
1
68.78 Â±0.22
882 Â± 18
12.42 Â±0.60
7.71 Â±0.023
0.82 Â±0.012
14.12
2
68.77 Â±0.38
852 Â±30
7.97 Â±0.35
7.52 Â±0.021
0.87 Â±0.013
32.93
3
68.42 Â±0.24
878 Â± 20
14.68 Â±0.65
7.75 Â±0.020
0.81 Â±0.010
13.23
600, Vi ilim
1
90.66 Â±0.69
1099 Â±52
6.58 Â±0.23
7.36 Â±0.016
0.91 Â±0.010
45.83
2
89.35 Â±0.36
1087 Â±28
12.46 Â±0.43
7.61 Â±0.017
0.91 Â±0.009
16.38
3
89.65 Â±0.32
1115 Â±25
13.68 Â±0.46
7.67 Â±0.018
0.94 Â± 0.009
10.71
600, Vi ilim
1
153.63 Â±0.44
1262 Â±22
6.02 Â±0.17
7.34 Â±0.014
0.86 Â±0.006
2.80
2
154.35 Â±0.78
1208 Â±38
4.41 Â±0.14
7.22 Â±0.014
0.84 Â±0.007
2.85
3
154.05 Â±0.45
1230 Â±22
6.00 Â±0.16
7.33 Â±0.014
0.90 Â± 0.006
1.70
1200, */4 i,im
1
51.27 Â±0.45
980 Â± 58
5.36 Â±0.06
7.49 Â±0.018
4.05 Â± 0.028
105.73
2
51.48 Â±0.76
969 Â± 97
4.40 Â±0.04
7.35 Â±0.013
4.19 Â±0.043
142.11
1200, V4 ilim
1
61.60 Â±0.21
1141 Â±24
14.30 Â±0.52
7.50 Â±0.016
0.77 Â±0.008
15.47
2
61.77 Â±0.27
1141 Â±32
12.19 Â±0.44
7.45 Â±0.015
0.75 Â±0.008
26.69
3
61.78 Â±0.24
1169 Â±30
13.75 Â±0.49
7.51 Â±0.015
0.76 Â± 0.008
21.17
Table 2.7â€”continued
1200, 3/4 ilim
1
106.66 Â±0.40
1230 Â±30
9.03 Â± 0.22
7.47 Â±0.017
1.26 Â±0.012
13.98
2
107.18 Â±0.89
1175 Â±63
5.64 Â±0.16
7.29 Â±0.015
1.13 Â±0.016
59.79
3
107.03 Â±0.63
1221 Â±46
6.67 Â±0.18
7.39 Â±0.016
1.21 Â±0.014
32.17
2400, 1/4 i,im
1
35.42 Â±0.28
972 Â± 50
7.72 Â±0.53
7.46 Â±0.024
0.66 Â±0.016
41.53
2
35.11 Â±0.19
1009 Â±35
20.16 Â±1.10
7.72 Â±0.018
0.63 Â±0.014
24.73
3
35.18 Â± 0.21
1018 Â± 39
10.19 Â± 0.12
6.88 Â±0.007
1.34 Â±0.009
24.45
2400, 1/2 iiim
1
44.11 Â±0.12
1269 Â± 22
15.00 Â±0.62
7.78 Â± 0.022
0.92 Â± 0.009
18.20
2
44.08 Â±0.11
1265 Â±21
15.51 Â±0.63
7.81 Â±0.021
0.89 Â±0.009
17.99
3
44.32 Â±0.21
1230 Â±38
8.29 Â±0.32
7.54 Â±0.020
0.95 Â±0.011
46.65
2400, 3/4 ilim
1
79.49 Â±0.70
1401 Â±75
6.71 Â±0.19
7.42 Â±0.016
1.15 Â± 0.017
6.71
2
79.14 Â±0.58
1391 Â±62
7.29 Â±0.20
7.48 Â±0.016
1.13 Â±0.015
6.33
3
79.36 Â±0.67
1392 Â±72
6.65 Â±0.20
7.45 Â±0.016
1.12 Â± 0.016
7.45
3000, 1/4 iiim
1
31.51 Â±0.13
1103 Â±28
29.53 Â± 1.38
7.80 Â±0.017
0.66 Â±0.015
15.34
2
31.69 Â± 0.19
1068 Â±42
15.71 Â±0.91
7.61 Â±0.017
0.55 Â±0.013
27.38
3
31.29 Â± 0.13
1092 Â±28
30.82 Â± 1.38
7.77 Â±0.016
0.66 Â±0.016
15.83
3000, 1/2 iiim
1
42.72 Â±0.52
1101 Â±92
4.86 Â±0.05
7.66 Â±0.016
5.21 Â±0.05
116.92
2
42.63 Â±0.58
1127Â±105
4.38 Â±0.05
7.52 Â±0.021
5.77 Â±0.06
106.23
3
42.47 Â±0.43
1176 Â±84
4.57 Â±0.04
7.63 Â±0.016
5.98 Â±0.04
80.58
3000, 3/4 ilim
1
78.31 Â±2.12
1167 Â±205
4.86 Â±0.06
7.54 Â±0.022
6.26 Â±0.10
24.17
2
78.22 Â± 1.63
1261 Â±165
4.69 Â±0.05
7.59 Â±0.019
7.22 Â±0.08
15.68
3
78.23 Â± 1.50
1306Â±164
4.60 Â±0.04
7.61 Â±0.019
7.76 Â±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(o), n
Sc
C, tiF
Re, O
Rt, Q
NRSSQ
120, 1/2 ilim
200.78 Â±0.87
1161 Â±32
11.1 Â±0.20
7.31 Â±0.01
0.85 Â±0.01
4.2
3/4 ilim
379.17 Â±2.43
1138 Â± 43
10.4 Â±0.17
7.45 Â±0.01
1.40 Â±0.02
4.3
600, 1/4 ilim
68.42 Â±0.24
878 Â± 20
14.7 Â±0.65
7.75 Â±0.02
0.81 Â±0.01
13.2
1/2 iim
89.65 Â±0.32
1115 Â± 25
13.7 Â±0.46
7.67 Â±0.02
0.94 Â±0.01
10.7
3/4 ilim
154.05 Â±0.45
1230 Â±22
6.00 Â±0.16
7.33 Â±0.01
0.90 Â±0.01
1.7
1200, 1/4 isÂ»
51.27 Â±0.45
980 Â±58
5.36 Â±0.06
7.49 Â±0.02
4.05 Â±0.03
105.7
1/2 ilim
61.60 Â±0.21
1141 Â±24
14.3 Â±0.52
7.50 Â±0.02
0.77 Â±0.01
15.5
3/4 iHm
106.66 Â±0.40
1230 Â±30
9.03 Â±0.22
7.47 Â±0.02
1.26 Â±0.01
14.0
2400, 1/4 iiim
35.18 Â± 0.21
1018 Â± 39
10.2 Â±0.12
6.88 Â±0.01
1.34 Â±0.01
24.5
1/2 ilim
44.08 Â±0.11
1265 Â±21
15.5 Â±0.63
7.81 Â±0.02
0.89 Â±0.01
18.0
3/4 ilim
79.14 Â±0.58
1391 Â±62
7.29 Â± 0.20
7.48 Â±0.02
1.13 Â±0.02
6.3
3000, 1/4 ilim
31.51 Â±0.13
1103 Â±28
29.5 Â± 1.38
7.80 Â±0.02
0.66 Â±0.02
15.3
1/2 ilim
42.47 Â± 0.43
1176 Â±84
4.57 Â±0.04
7.63 Â±0.02
5.98 Â±0.04
80.6
3/4 ilim
78.23 Â± 1.50
1306 Â±164
4.60 Â± 0.04
7.61 Â±0.02
7.76 Â±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
Re9 fi
râ€ž n
4>
NRSSQ
120, 1/2 i,im
1
200.66 Â±0.77
1108 Â±31
9.69 Â±0.36
7.21 Â±0.023
1.15 Â±0.063
0.14 Â±0.026
4.38
120, 3/4 ilim
3
378.86 Â±3.22
1041 Â±64
7.91 Â±0.28
7.29 Â± 0.028
2.33 Â±0.184
0.16 Â±0.026
15.95
600, 1/2 i]im
1
94.97 Â±0.69
677 Â± 26
1.17 Â± 0.18
6.61 Â±0.054
2.25 Â±0.039
0.49 Â±0.009
232.28
1200, l/4i,im
1
50.88 Â±0.13
1074 Â± 19
3.61 Â±0.05
7.05 Â±0.016
4.95 Â±0.030
0.15 Â±0.004
25.90
2
50.94 Â±0.22
1120 Â±32
3.37 Â±0.03
7.04 Â±0.010
5.10 Â±0.032
0.15 Â±0.004
15.19
3
51.07 Â±0.22
1111 Â±32
3.32 Â±0.03
7.04 Â±0.011
5.04 Â±0.033
0.14 Â±0.004
18.03
2400, 1/4 ium
3
35.04 Â± 0.18
1024 Â±32
9.15 Â± 0.18
6.80 Â±0.013
1.56 Â±0.033
0.10 Â± 0.012
16.29
2400, 3/4 i,im
1
82.59 Â±0.99
519 Â± 49
4.44 Â± 0.23
7.04 Â± 0.027
3.55 Â±0.137
0.40 Â±0.012
38.57
2
78.70 Â±0.54
1186 Â±105
6.59 Â±0.34
7.35 Â±0.043
1.83 Â±0.213
0.18 Â±0.043
7.18
3
82.64 Â± 1.01
504 Â± 44
4.43 Â±0.24
7.07 Â± 0.028
3.49 Â±0.132
0.40 Â±0.011
39.70
3000, 1/2 i,im
1
42.14 Â±0.11
1282 Â±25
3.79 Â±0.03
7.35 Â±0.008
6.24 Â±0.027
0.13 Â±0.003
14.64
2
41.98 Â±0.15
1337 Â±33
3.11 Â±0.04
7.08 Â±0.014
7.04 Â±0.041
0.14 Â±0.004
7.47
3
41.98 Â±0.08
1332 Â±19
3.64 Â±0.02
7.33 Â±0.007
6.88 Â±0.021
0.10 Â±0.002
11.03
3000, 3/4 iiim
1
76.43 Â±0.63
1476 Â± 79
3.65 Â±0.04
7.20 Â±0.013
8.50 Â±0.088
0.16 Â±0.005
2.45
2
76.82 Â±0.36
1543 Â±47
3.73 Â±0.02
7.29 Â±0.008
8.93 Â± 0.047
0.12 Â±0.003
1.27
3
76.87 Â±0.32
1565 Â±41
3.67 Â±0.02
7.30 Â±0.008
9.38 Â±0.043
0.11 Â±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 ilim
200.66 + 0.77
1108 Â±31
9.69 Â±0.36
7.21 Â±0.02
1.15 Â±0.06
0.14 Â±0.03
4.38
3/4 ilim
378.86 Â±3.22
1041 Â±64
7.91 Â±0.23
7.29 Â± 0.03
2.33 Â±0.18
0.16 Â±0.03
15.95
600, 1/2 ilim
94.97 Â± 0.69
677 Â± 26
1.17 Â±0.18
6.61 Â±0.05
2.25 Â±0.04
0.49 Â±0.01
232.28
1200, 1/4 ilim
50.94 Â±0.22
1120 Â±32
3.37 Â±0.03
7.04 Â±0.01
5.10 Â±0.03
0.15 Â±0.004
15.19
2400, 1/4 iiim
35.04 Â±0.18
1024 Â±32
9.15 Â±0.18
6.80 Â±0.01
1.56 Â±0.03
0.10Â±0.01
16.29
3/4 ilim
78.70 Â±0.54
1186 Â±105
6.59 Â±0.34
7.35 Â±0.04
1.83 Â± 0.21
0.18 Â±0.04
7.18
3000, 1/2 i,im
41.98 Â± 0.15
1337 Â±33
3.11 Â±0.04
7.08 Â±0.01
7.04 Â± 0.04
0.14 Â±0.004
7.47
3/4 ilim
76.82 Â±0.36
1543 Â±47
3.73 Â±0.02
7.29 Â±0.01
8.93 Â±0.05
0.12 Â±0.003
1.27
54
Table 2.11. Results from the process model regression for EHD data collected at the
masstransferlimited current using treatment 1.
Q, rpm
Set no.
A0, pA/rpm
Sc
ReC, sec
NRSSQ
120
1
0.459 Â±0.001
1147 Â±13
0.0129 Â±0.0007
31.32
2
0.458 Â±0.002
1162 Â±15
0.0092 Â± 0.0007
32.45
3
0.459 Â± 0.002
1176 Â±19
0.0112 Â±0.0009
38.455
200
1
0.352 Â±0.001
1220 Â± 16
0.0051 Â±0.0004
25.338
3
0.345 Â±0.002
1286 Â±24
0.0061 Â±0.0006
44.037
600
1
0.199 Â±0.001
1334 Â± 17
0.0012 Â±0.0002
29.522
2
0.199 Â±0.001
1359Â±18
0.0010 Â±0.0002
31.072
3
0.198 Â±0.001
1377Â±19
0.0009 Â± 0.0002
34.577
1200
2
0.137 Â±0.0003
1384 Â±11
0.0006 Â±0.0001
15.674
3
0.139 Â±0.0004
1434 Â±21
0.0002 Â±0.0001
28.567
Table 2.12. Selected results from the process model regression for EHD data collected at
the masstransferlimited current using treatment 1.
O, rpm
A0, pA/rpm
Sc
ReC, sec
NRSSQ
120 rpm
0.459 Â±0.001
1147 Â±13
0.0129 Â±0.0007
31.32
200 rpm
0.352 Â±0.001
1220 Â± 16
0.0051 Â±0.0004
25.34
600 rpm
0.199 Â±0.001
1334 Â± 17
0.0012 Â±0.0002
29.52
1200 rpm
0.137 Â±0.0003
1384 Â±11
0.0006 Â±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 iaâ„¢ 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
*Ao
0.1 r
 0.01 Ã
0.001 ~
11 nuil i m miq i 111 Im i i i mui I 111 mi i 111 in
A Â° AO A,
O
O
O 3
O
0 0001 I limn I I lllllll l 11 lililÃ l mini i limn i i mml in
0.001 0.1 10 1000 100000
CD, Hz
Figure 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, 174th of limiting current case for the rotating disk electrode system.
Zj, iA/rpm
62
Zn 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, masstransferlimited current using
treatment 1. Solid line in the figures is the measurement model fit and circles represent the
data.
63
to, Hz
0.01 0.1 1 10
o, 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, masstransferlimited 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,
masstransferlimited 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 (0)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
Â©, Hz
0.001 0.1 10 1000 100000
co, Hz
Figure 2.13. Checking for consistency with the KramersKronig relations. EIS data
collected for 120 rpm, 174th 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 KramersKronig 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,
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
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
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 masstransferlimited
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
co, 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 masstransferlimited
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
CD, 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 masstransferlimited 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
CD, Hz
Figure 2.24. Process model regression (with error structure weighting) for the EHD data
collected for 120 rpm, at the masstransferlimited 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
80
co, Hz
0.01 0.1 1 10
co, Hz
Figure 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.
81
n1/2, (rad/sec)1'2
Figure 2.26: The square root of the rotation speed plotted against the masstransfer
limiting current value. The line passing through is regressed ignoring the 3000rpm case.
CHAPTER 3
INFLUENCE OF SURFACE PHENOMENA ON THE IMPEDANCE RESPONSE OF
A ROTATING DISK ELECTRODE
In chapter 2 it was shown that the onedimensional frequency domain model
proposed by Tribollet and Newman is inadequate to describe the system of interest for
most of the cases studied, either because the quality of regressed fits is low or because the
determined Schmidt numbers are very high. High Sc values may be resulting from the
partial blocking of the electrode surface to the mass transport phenomena, as a large Sc
value can be caused by a reduced apparent diffusion coefficient.
The role of the surface blocking for the system of interest was discussed earlier by
a number of researchers. Stieble and JÃ¼ttner reported that, depending on the defined
prepolarization conditions, a partial blocking of the electrode surface was observed as
indicated by distinct changes of the impedance spectra [29], They reported blocking index
values as high as 30 in some cases, which results in just about 3% of the total surface area
of the electrode being active. The blocking index is defined as the ratio of the area of
inactive portion of the surface to the area of active portion. Voltammetric techniques were
applied to study the effect of surface blocking [30,31,32], It was observed that the
irreversibility of the heterogeneous kinetics increased with the surface coverage, and the
apparent charge transfer resistance was found to be higher. The film formation was
attributed to the adsorption of certain compounds onto the surface of the Pt disc
electrode. However there is no agreement between the authors about the composition and
82
83
structure of the compounds adsorbed onto the electrode surface. Some suggested that
ferri and ferrocyanide ions are adsorbed on the Pt surface [33,34,35], Others suggested
that iron cyanide complexes are decomposed resulting in the formation of species that are
adsorbed on the electrode surface (e.g. CN' and Fe(CN)3) [36,37,38], It was also believed
that coexistance of different oxidation states of iron as in ferri and ferrocyanide may lead
to a bonding between the two species thus forming a prussian blue layer on the electrode
surface.
The influence of the blocking effect is evident on the charge transfer resistance as
well as the associated masstransfer phenomena. In the present work, application of a
process model to one of the data sets with 27 repeated measurements revealed a temporal
evolution which is attributed to the surface blocking.
3,1 Experimental Protocol
The electrolyte for this study consisted of consisted of 0.01M K3Fe(CN)6 and
0.01M K4Fe(CN)6 in 1M KC1. Roughness of the surface can influence the charge transfer
process and could be strongly associated with the frequency dispersion behavior. Hence
the preparation of electrode surface before each experiment is a very important factor. It is
also believed that the partial blocking of the electrode surface could be a function of the
surface treatment used. Hence the best possible surface treatment is needed in order to
obtain reliable information from the system under consideration. The treatment that
yielded the maximum value for the limiting current, as illustrated in Figure 2.1, was chosen
for the data analysis presented in this section. The electrode was pretreated with wet
polishing on a 1200 grit emery cloth, washed in deionized water, polished using alumina
84
paste and then placed in an equivolume mixture of water and ethanol and subjected to
ultra sound cleaning. Following the ultrasound cleaning, the electrode was washed in deÂ¬
ionized water. Under microscopic observation the surface of the electrode appeared to be
significantly different from one surface treatment to the other. For the case presented in
this chapter, the most â€œmirrorlikeâ€ surface was obtained. The data sets considered in this
chapter were collected at 120rpm and at l/4lh of limiting current. Repeated measurements
were made in order to facilitate the use of measurement model to determine the stochastic
contribution of errors. However the spectra were collected overnight with no polishing in
between measurements, thus making this set of measurements very interesting to study the
evolution of surface blocking effects.
3 2 Results and Discussion
Impedance spectroscopy results presented in Figure 3.1 and Figure 3.2 for 1/4* of
the mass transfer limited current and for a disk rotation speed of 120rpm demonstrate the
timedependent poisoning of the Pt surface. From these figures it can readily be seen that
the high frequency phenomena which correspond to the charge transfer become more and
more significant with an increased experimental time. The error structure for these
measurements as given by equation (2.39) was obtained using the measurement model
approach [9,10], The corresponding error structure parameters are /?= 1.44138x1 O'3, y =
1.47696x1o4, and 8= 1.96453xl03. The regression strategy used in this work took full
account of the error structure of the measurements following the procedure presented in
chapter 2. The quality of the regression and the value obtained for the Schmidt number
was found to be a strong function of the mathematical treatment used to describe
85
convective diffusion. The model that accounted for a 3term expansion of the axial
velocity in the convective diffusion equation and CPE correction provided the smallest
residual sum of squares for the regression of the impedance data.
The standard deviations of the repeated measurements given in Figure 3.3 reflect
contributions from bias errors caused by transient changes in electrode conditions. The
real and imaginary parts of the standard deviation calculated directly were correlated but
are not equal. The standard deviations of the stochastic part of the measurement, obtained
by use of the measurement model to filter bias errors, were much smaller, and the real and
imaginary parts of the standard deviation were statistically equal. This result is found to be
a direct consequence of the way KramersKronig relations transform the stochastic noise
in the measurement, as presented in chapter 7. The standard deviation of the stochastic
part of the measurement (open symbols in Figure 3.3) ranged from 0.2 to 0.01 percent of
the modulus. The stochastic noise level of the measurements shown in Figure 3.1 and
Figure 3.2 was therefore much less than the 35 percent of the modulus (often assumed to
be the noise level for impedance measurements). The residual sum of squares normalized
by the sum of variances for the measurement given in Figure 3.4 represents an FTest for
comparison of variance. The dashed lines are limits for accepting the hypothesis that the
residual sum of squares differs from the sum of variances for the measurement. In spite of
the low noise levels evident in Figure 3.3, with the exception of the first two
measurements the process model yielded residual errors that fell within the noise level of
the measurement. The larger residual errors seen for the first two scans were found to be
the result of inconsistency of the data with the KramersKronig relations. The success of
the regression for the subsequent measurements allowed determination of the role of the
86
poisoning reaction evident in Figure 3.1 and Figure 3.2. The Schmidt number obtained by
regression is presented in Figure 3.5. For the first two hours of the sequential impedance
scans, the Schmidt number obtained was in good agreement with the value of 1100
obtained by DC techniques given by the dashed line in Figure 3.5. After this, the Schmidt
number increased with time. As the residual errors for these regressions (Figure 3.4) were
of the order of the noise of the measurement, the increase of Schmidt number suggests
that the effective diffusion coefficient decreases with time. The charge transfer resistance
also increased with time, as shown in Figure 3.6. The increase of the mass transfer
resistance shown in Figure 3.7 is consistent with the decrease of effective diffusion
coefficient or increase in the value of the Schmidt number suggested by Figure 3.5. The
double layer capacitance, shown in Figure 3.8, and the exponent for the constant phase
element shown in Figure 3.9 approached a constant value. These results show that the
poisoning of platinum which occurs with prolonged experimentation, contributes to both a
decrease of rate constant that hinders electrode kinetics and a blocking of the electrode
surface that hinders mass transfer. This necessitates accounting for the blocking effects in
modeling impedance spectroscopy.
Another possible factor, as mentioned earlier, is the radial dependence of the
current distributions. The process model that is currently used is onedimensional in nature
and this may yield regressed parameters that are significantly different from what is
expected when the current distributions are highly nonuniform in nature. The main focus
of this work is in understanding the issues associated with the nonuniform current
distributions.
87
3.3 Conclusions
The results presented in this chapter are in agreement with the previous
observations regarding the surface blocking phenomena. The resistance to the mass
transport and the charge transfer were observed to be increasing with the duration of the
experiment. Surface treatment in between measurements is very important in order to
obtain reliable information from the impedance spectroscopy data.
N
88
80
60
G 40
KT
20
0
0.001 0.1 10 1000 100000
Frequency, Hz
Figure 3.1. Imaginary part of the impedance for reduction of ferricyanide on a Pt disk
rotating at 120 rpm and at 1/4Ãœ1 of the limiting current. The time trending between the
spectra can be seen very clearly.
89
1000
100
a
N
10
1
0.001 0.1 10 1000 100000
Frequency, Hz
TTTTTTl 1 I I lililÃ 1 I II Mill 1 I I I Hill 1 I I I lllll 1 I llllll 1 I I I Mill 1 I I I MU
miniâ€”l jiiimiâ€”i 11mm i 111mil l mum i_LimnJ i i muu â– â– nun
Figure 3.2. Real part of the impedance for the repeated measurements with time as a
parameter.
90
100 FT nrinnâ€”i i miniâ€”i 11mmâ€”i i miniâ€”r r itmu i i mini i i mimâ€”r i nnm
0.01
0.001 0.1 10 1000 100000
Frequency, Hz
Figure 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.
NRSSQ
91
Figure 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 Ftest.
92
Figure 3.5. Schmidt number obtained by regression of process model to the data.
93
Figure 3.6. Charge transfer resistance obtained by regression of process model to the data.
94
Figure 3.7. Mass transfer resistance obtained by regression of a process model to the data.
d1"1 'o
95
4.3E6
4.2E6
4.1 E6
.OE6
3.9E6
3.8E6
0 4 8 12
Time, h
Figure 3.8. Double layer capacitance obtained by regression of process model to the data.
96
Figure 3.9. Exponent in the CPE element obtained by regression of process model to the
data.
CHAPTER 4
STEADY STATE MODEL FOR A ROTATING DISK ELECTRODE BELOW THE
MASSTRANSFER LIMITED CURRENT
The need for a better understanding of the current distributions is presented in
chapter 2. The development of a steady state model for current distributions accounting
for a finite Schmidt number correction and the charge distribution within the diffuse part
of the double layer is presented in this chapter.
The popularity of the rotating disk for experimental electrochemistry has motivated
development of numerous mathematical models that describe the physics of the system.
The Levich equation for masstransferlimited current was obtained under the assumption
that the Schmidt number is infinitely large [39], As the axial velocity is uniform for the
rotating disk electrode there is no radial distribution of current under the mass transfer
limitation. Newman provided a correction to the Levich equation, which accounted for a
finite value of the Schmidt number [40], This correction amounted to a 3 percent
reduction in the value of the masstransferlimited current obtained for a Schmidt number
of 1000, typical of electrolytic systems. In subsequent work, he relaxed the assumption of
a masstransferlimited condition by coupling convective diffusion in radial and axial
directions in an inner diffusionlayer region with Laplaceâ€™s equation for potential in an
outer domain [41], The cathodic Tafel limit of a ButlerVolmer expression was used to
account for metal deposition kinetics. Appel extended this model to account for redox
reactions as, in principle, Newmanâ€™s model was developed for deposition reactions [42],
97
98
The focus in later years was on frequencydomain techniques. Significant effort has
been expended on developing analytic formulae for the impedance response of a rotating
disk electrode [4354], A comparative study of the application of these models to
interpretation of impedance spectra has been presented by Orazem et al. [55], A oneÂ¬
dimensional numerical model for the impedance response of a rotating disk electrode
which accounted for the influence of a finite Schmidt number was presented by Tribollet
and Newman [5], The first two terms in the Cochran expansion for the axial component of
fluid velocity were included in the convective diffusion equation [56], Tribollet et al.
reported that for a Schmidt number of 1000 the errors caused by neglecting the second
term in the axial velocity expansion could be as high as 24 percent [53], This result was
confirmed by regression of various masstransfer models to impedance data [55],
Mathematical models have been developed which account for frequency dispersion
associated with the nonuniform potential distribution on the disk electrode [5760], but,
to date, no comparable model has been developed for the influence of nonuniform mass
transfer on the impedance response. Appel and Newman provided a preliminary
mathematical development that they proposed would be used to develop a model for the
influence of radiallydependent convective diffusion on the impedance response under the
assumption that the Schmidt number is infinitely large [61], Later Appel provided a model
to calculate the radial distribution of impedance under the assumption that the Schmidt
number is infinitely large [42], However he was able to generate the radial distribution of
impedances over a very limited frequency range due to numerical difficulties.
Though the errors resulting from neglecting the finite Schmidt number correction
are not so significant for steady state calculations, they have been shown to be significant
99
in the frequency domain [53,55], The objective of the work presented in this chapter is to
develop a steady state treatment of current and potential distributions on a rotating disk
electrode that accounts for a finite Schmidt number and for the distribution of charge in
the diffuse part of the double layer. This provides a foundation for the development of a
mathematical model that accounts for the influence of nonuniform mass transfer on the
impedance response of a rotating disk electrode (presented in the next chapter). This work
also provides a means of interpreting anomalous values of Schmidt numbers obtained from
impedance data collected for the reduction of ferricyanide on a Pt disk electrode, as shown
in chapter 2, in terms of the competing roles of surface poisoning and the influence of non
uniform current distribution [6,45],
4,1 Theoretical Development
The domain of interest was divided into an outer region where the concentration
was assumed to be uniform, an inner boundarylayer region where electroneutrality and
the convective diffusion equation for the reacting species were assumed to apply, and an
inner diffuse part of the double layer where the assumption of electroneutrality was
relaxed. The entire electrode surface was assumed to be active; thus, local passivation and
partial blocking phenomena were not included in the model.
4,1.1 Diffusion Layer
The development in this part follows that presented by Newman with the exception
that correction is made for a finite Schmidt number by incorporating 3term expansions
appropriate near the electrode surface for axial and radial velocity [41], A critical
assumption in the development of this work was that the current densities at the inner limit
100
of the diffusion layer and at the inner limit of the outer region are equal to the current
density at the electrode surface. This assumption is valid if the diffusion layer is thin
compared to the electrode radius. The validity of this assumption is implicit for an infinite
Schmidt number because the diffusion layer is infinitely thin.
For a finite Schmidt number, the diffusion layer is still small compared to typical
disk dimensions. The thickness of the boundary layer Â¿>is given by
(3Di)
1/3
\av )
where D, is the diffusion coefficient of the species of interest, v is the kinematic viscosity
coefficient, Q is the rotation speed of the disk electrode in radians/sec, and a is the
coefficient in the expansion for the radial velocity term in equation (4.6) and has a value of
0.51023. The boundary layer thickness for the copper deposition system considered by
Newman [41] is determined by the diffusivity of the cupric ion
(DCu2+ = 0.642 xlO5 cm2/s) and by the kinematic viscosity (v = 0.94452xl0'2 cm2/s). For
a rotation rate of 300 rpm, <5has a value of 13.76 pm which is small compared to the disc
radius of 0.25cm, yielding an aspect ratio 8/r0 = 0.0055, where r0 is the radius of the disk.
The corresponding value for the ferricyanide system treated in a later section of this
chapter is 0.012. The solution of the convective diffusion equation can, in these cases, be
decoupled from the solution of Laplaceâ€™s equation for potential in the region of uniform
composition.
Under the assumption of a steady state, the convective diffusion equation is given
by
101
de
deR
â€”â€” + v
dr 1 dz
'S = DI
d2cB
dz2
(4.2)
where vr and vÂ¿ are the radial and axial components of velocity, cR is the concentration of
the reacting species, and Dr is the diffusion coefficient of the reacting species. A
separation of variables can be applied as shown below:
cr â€” c<*
i+Zk.O'/'oFXte))
(4.3)
where cx is the bulk concentration of the reacting species, Am are the coefficients which
correspond to the radial dependent part of the concentration, and 6m(Â£) is the axial
dependent part, where Â£ is the scaled axial distance given by
f = JtfavHiDR))jafi <4 4)
For 0m=l at Â£=0, the radial concentration distribution on the electrode surface can be
written as
c0 =câ€ž
1 +
m0
(4.5)
The expressions for the radial and axial components of the velocity resulting from
Cochranâ€™s 3term expansion [56] are given by
v,  arz
Q3/2 1 2 Q2 b 3 Q5/2
â€”rz rz
v1/2 2
v 3
.3/2
(4.6)
and
v7 = az
il3/2 1 3Q2 b 4Q5/2
 + z + z
vU2 3
v 6 v
3/2
(4.7)
respectively, where b is a constant with a value of 0.61592. Application of equations
(4.3), (4.4), (4.6), and (4.7) in equation (4.2) yields
102
m0
3^2
a
iaÃ
3
* LÃ
a 2
v av/ y
3 D
< av ,
2/3"
1/3
2 ^3De ^2/3
m=0
V <11/ y
Y,2mAâ€ž(r/rJe.(4)=0
m=0
(4.8)
By equating the coefficients of (r/rÂ¿)2m equation (4.8) can be rewritten as
o;â€”e
a
1/3
6
3
Sc
2/3
6mÂ¿;
a 
U
Ã
'3n
2/3
Sc2*
a
2
Va;
3
KÃ¼J
o=o
(4.9)
The primes in the superscripts refer to derivatives with respect to Â£ Equation (4.9) can be
compared to
e:+3?K6m40.=O (4.10)
obtained by Newman for an infinite Schmidt number. The boundary conditions for
equation (4.9) are Om = 1 at Â£ = 0 and Om = 0 at Â£ = oo. For the numerical calculation
presented here, a value of Â£ = 20 was used for approximation of the boundary condition at
oo. The choice of this value is justified as a value of Â£ = 10 resulted in the same values of
6m to within 15 significant digits and the derivative of On, with respect to Â£ at Â£ = 20
yielded a value identically equal to 0. Equation (4.9) exhibits functional dependence on the
Schmidt number. The numerical procedure that was used to solve the convective diffusion
equation is presented in a subsequent section.
The current distribution on the electrode surface at steady state depends only on
the dimensionless concentration gradient at electrode surface 0'm{0) and not on the 6m
values at the other axial positions, that is,
103
(4.11)
where tR is the transference number of the reacting species, n is the number of electrons
produced when one reactant ion or molecule reacts, F is the Faraday's constant. The
interfacial overpotential resulting from convective diffusion is the concentration
overpotential given by
(4.12)
\co) V cÂ® J
where rÂ¡c is the concentration overpotential, R is the universal gas constant, T is the
temperature of the electrolyte in Kelvin, and Z is the number of equivalents of reactant
used up per one electron produced/consumed.
4.1.2 Outer Region: Laplaceâ€™s Equation
The approach taken in this region followed that developed by Newman [41], In the
outer region, concentrations were assumed to be uniform, and, under the assumption that
there is no charge distribution within the bulk of the solution, the potential in this region
satisfies Laplaceâ€™s equation. Hence
V2d> = 0
(4.13)
where is the potential referenced to infinity. The solution of equation (4.13) after
transformation to rotational elliptic coordinates where z = r0/u rÂ¡ and
r = r0 j(l + yr)(l  //2) and after applying appropriate boundary conditions is
(4.14)
104
where O0 is the ohmic drop on the electrode surface as a function of radial position,
P2n(rj) is a Legendre polynomial of order 2n on the electrode surface (/r = 0),
Z = z+z_/(z+ z_) for single salt and Z = n with supporting electrolyte, and Bâ€ž's are
the constant coefficients to be determined. The current density obtained from the
derivative of the potential just outside the diffusion layer is assumed to be equal to that
obtained from equation (4.11); thus a relationship between coefficients Bâ€ž and Am is found
as
b. = ^nY,{q,..a.}
m= 0
(4.15)
where
Qn,m =(4Â« + l)
4^(0)
n M'in (Â°)
\vi}v2)m P2n{v)dv
ML(o)=
2
n
(4.16)
(4.17)
and
finfavV/3 nZF2Dcâ€ž
V v v 3DJ RT(\  t)Ka
(4.18)
The Am coefficients were determined using an iterative procedure. A more detailed
discussion of the solution procedure can be found in [41],
4,1.3 Diffuse Part of the Double Laver
The method used to account for the twodimensional diffuse part of the double
layer followed the approach suggested by Frumkin [62] in which the kinetic expression
was written in terms of a surface overpotential adjusted by the zeta potential and the
105
concentrations were replaced by the concentrations at the inner limit of the diffuse double
layer.
4.1.3 a Solution of Poissonâ€™s equation
Under the assumption that the double layer is unaffected by the passage of current (see,
e.g., [63]) the concentration at the inner limit of the diffuse part of the double layer is
given by
(4.19)
where c,fi is the concentration of species / at the outermost part of the diffuse part of the
double layer (the innermost part of the diffusion layer), Cyaai is the concentration of species
at the innermost part of the diffuse part of the double layer (the plane of closest approach
to the metal surface), Â£ is the zeta potential, and z, is the charge number of species i. The
values of concentration at the inner limit of the diffusion layer c;,0 were obtained under the
assumptions that the solution at this location is electrically neutral, that the diffusion
coefficients of the reactant and product ions are equal, and that the diffusion coefficients
of the cation and anion from the supporting electrolyte are equal. Thus, a decrease in the
concentration of the reactants was offset by an equal increase in the product
concentration. The above assumptions are appropriate for the case of the reduction of
ferricyanide on Pt electrode discussed in a subsequent section. The diffusion coefficients
of the ferri and ferrocyanide are 0.896xl0'5 cm2/sec and 0.739xl0'5 cm2/sec respectively,
and the diffusion coefficients for K+ and Cf ions are 1.957x1 O'5 cm2/sec and 2.032x1 O'5
cm2/sec, respectively. The above assumptions could be relaxed following the ordinary
perturbation approach of Levich [39], [64],
106
Under the assumption that the diffuse part of the double layer is thin, Poissonâ€™s equation
(4.20)
d2w f z.Fy/^
~7T = LzcÂ»o exp
az
Â£ T
RT
can be solved in the axial dimension subject to the boundary conditions
. . dur q7
Y > 0 as z > oo and â€”â€” = â€” at z = z2
dz Â£
(4.21)
where ^is the potential within the diffuse part of the double layer measured relative to the
outermost region of this diffuse part, q2 is the charge held within the diffuse part of the
double layer at a given radial position, and s is the permittivity of the electrolyte. The use
of equation (4.21) requires introduction of a geometric parameter z2 corresponding to the
distance between the electrode surface and the plane of closest approach for ionic species.
Solution of equation (4.20) yields
q2=\2RTeÂ£ck
exp
\ RT
i'i 1/2
1
(4.22)
where C, is the zeta potential given by the value of y at z=z2.
The total applied potential
0+V.+Ãš+C
(4.23)
is sum of the ohmic potential drop (3>0), concentration overpotential (qc), surface
overpotential (rÂ¡]), and zeta potential (^). The ButlerVolmer expression for the normal
current density as a function of surface overpotential and concentration was modified to
account for the potential and concentration of reacting species at the plane of closest
approach; thus,
107
_ . \co exp( zrFÂ£/RT)}"
/ = /,
J _expt
[aZF .]
RT
Is r ~ exPi
(TZF
RT
Vs
(4.24)
where zr is the charge number of the reacting species, z'o is the exchange current density, a
and /? are the anodic and cathodic transfer coefficients, and y is the exponent for
composition dependence of the exchange current density. In principle such an expression
for the reaction kinetics is valid only in case of deposition reactions. However Appel
showed that this expression can still be used for redox reactions when the current is a
significant fraction of the masstransfer limiting current [42],
Within this approach, q2, cl0, c^ddi, i, C 7c, fA,hm,c and rfs are functions of radial
position. Assumption that the interface taken as a whole is electrically neutral yields
Vm ~({2
(4.25)
where qm is the charge density associated with the concentration of electrons on the metal
surface. Because the diffuse double layer is thin in comparison to electrode dimensions,
equation (4.25) was assumed to apply at each radial position. Under the assumption that
specific adsorption can be neglected, charge is not present between the metal surface and
the plane of closest approach of the solvated ions. The potential profile in this region can
be assumed to be linear with respect to the axial position; hence,
(4.26)
The development presented to this point constitutes the mathematical model for
convective diffusion to a rotating disk electrode that accounts for finite Schmidt numbers,
the influence of the diffuse double layer, and currents below the mass transfer limited
108
current. Solution of this set of equations can be used to obtain such measurable properties
as the doublelayer capacitance.
4.1.3 b Calculation of doublelaver capacitance
The double layer capacitance (Cdi) is given by
^ _ dqm
1
+ {dÂ£/dqm) (4.27)
(VQl+WoJ
d[Vs+C)
WJdqm)
where C2 is the capacitance between the electrode surface and the plane of closest
approach and Cddi is the capacitance across the diffuse part of the double layer. From
equation (4.26),
(4.28)
r  dclm _ g
dris y2
dq
Cddi can be calculated as Cddl = â€”â€”. The double layer capacitance can be expressed as a
function of zeta potential by introducing equations (4.22) and (4.25)
CM ={2RTeY1Â¡L
1/2
= F
2>,.cI0exp<
i
\z,FÃ
[ RT
!E2cu
exp
RT ]
1
(4.29)
An explicit treatment of the double layer in the model facilitates the calculation of
charge distribution. This information can be used to determine the influence of charge
distribution on the steady state and frequency domain response of disk electrode.
109
4.2 Numerical Procedure
In the following sections the numerical procedure employed to solve the
convective diffusion equation and a stepbystep algorithm for the implementation of the
steady state model are presented.
4,2,1 Solution to the Convective Diffusion Equation
Calculation of the steady state current and potential distributions on the electrode
surface requires accurate values for 0'm (0). A finite difference scheme using central
difference formulae accurate to the order of the square of the element length was used to
solve equation (4.9) for m=0 to 10 at different values of Schmidt number. A tridiagonal
system of equations was obtained and solved by employing the Thomas algorithm [65], To
reduce the error caused by the choice of the step size, calculated values of 0'm{0) for
different element lengths were extrapolated to zero element length by linear regression.
One such regression for an infinite Schmidt number and m=0 is presented in Figure 4.1.
The 95 percent confidence interval for the intercept was used to estimate the precision of
the result. In all cases, 9 significant digits were obtained. For example, the value for 6^(0)
obtained by extrapolation was 1.119846522. This result is consistent with the
corresponding value of 1.11984652 reported by Newman [41], The same results were
obtained by approximating the solution by a series of Chebyshev polynomials, as
illustrated in chapter 6. For the number of significant digits reported here, 0'm(6) was not
sensitive to the use of more decimal places for constants a and b in Cochranâ€™s velocity
expansions [56],
110
Similar extrapolations were made for m=l to 10 and for various values of Schmidt
number. The 6*^(0) values obtained for m=0 are given as functions of Sc'13 in Figure 4.2.
The Â¿9^(0) values obtained for different values of m were regressed to a sixth degree
polynomial as given by
Â«:(0) = Â±aâ€ž ScM (4.30)
k=0
where Sc is the Schmidt number of the reacting ions and amÂ¡k are the polynomial
coefficients to be determined from the polynomial regression. A sixthorder expansion was
used because the 95% confidence intervals for the regressed parameters did not include
zero and because smaller order expansions did not provide a sufficiently accurate
representation of the numerical results. The residual errors for the expansion were
randomly distributed and had magnitudes of roughly 10'11 as shown in Figure 4.3.
The masstransfer limited current density is uniform and is given by
lUm 
nFD,c,â€ž f a
1/3
1/
\3DJ
e'a(o)
(4.31)
Equation (4.30) yields a value for #0'(0) at a Schmidt number of 1000 of1.085880341 as
compared to 1.119846522 for an infinite Schmidt number. Assumption of an infinite
Schmidt number results in a relative error of 3.1% which is in agreement with the value
reported by Newman [40], The largest error expected for aqueous electrolytes may be
seen for hydrogen ion which yields a Schmidt number of the order of 100, resulting in a
relative error of 7.1% in the calculated value of the masstransferlimited current. While
the influence of the finite Schmidt number correction on the calculated masstransfer
Ill
limited current is small, the errors associated with neglecting this correction are more
significant in the frequency domain [53,55],
4.2.2 Algorithm for Implementation of the Model
The numerical procedure employed in this work to handle the diffusion layer and
the outer region with the Ohmic drop is similar to the one implemented by Newman [41];
however, introduction of the diffuse part of the double layer required modification of the
kinetic boundary condition in order to accommodate the zetapotential. An algorithm for
implementation of the numerical scheme is provided below:
1. The values for 6^(0) resulting from the convective diffusion equation were
determined for a given value of Schmidt number from the polynomial expression in
equation (4.30) and by using Table 4.1 for the values of amÂ¡k.
2. A value for Am coefficients was assumed, or, in other words, the concentration
distribution was assumed on the electrode surface as per equation (4.5).
3. The current distribution resulting from masstransport was calculated as per equation
(4.11).
4. The current distribution obtained from the derivative of the potential just outside the
diffusion layer was equated to that from step (3) to obtain the Bn coefficients. The
relation used to obtain these coefficients was provided in equation (4.15).
5. The
6. A value for rfs was substituted in terms of C, from the equations (4.22), (4.25), and
(4.26), and the value of ^was obtained by using a NewtonRaphson technique to solve
the nonlinear equation (4.24).
112
7. The rfs distribution was calculated from the Â«''potential distribution obtained from step
(5) by employing equations (4.22), (4.25), and (4.26).
8. The rÂ¡c distribution was calculated from equation (4.23).
9. The concentration distribution was calculated from equation (4.12).
10. The Am coefficients were obtained again from the concentration distribution from step
(8). The new values for the Am coefficients are the weighted sum of those obtained
from the present iteration and from the previous iteration.
11. Steps (3) through (10) were repeated until convergence was achieved.
12. Once the convergence criteria were met, the charge distribution was obtained from
equation (4.22), and the double layer capacitance was obtained from equations (4.27),
(4.28), and (4.29).
A FORTRAN program used in implementing this algorithm is presented in Appendix A.
4.3 Application to Experimental Systems
The systems treated in this work include electrodeposition of copper and reduction
of ferricyanide on platinum.
4.3.1 Electrodeposition of Copper
To validate the numerical approach used in this work, calculations were performed
for the 0.1M copper sulfate system studied by Newman [41], This part of the study did
not incorporate the diffuse part of the double layer. Newmanâ€™s results for an infinite
Schmidt number were reproduced. The concentration and current distributions obtained
for infinite and finite Schmidt number for this system are compared in Figure 4.4 with
applied potential as a parameter. For a given value of applied potential, the distributions
113
resulting from both these cases differed most significantly as the masstransferlimited
condition was approached. The largest deviation was seen at the center of the disk. The
differences seen between the calculated distributions for infinite and finite Schmidt number
are due to differences in the corresponding values of the masstransferlimited current.
When distributions were presented at the same fraction of the masstransferlimited
current, the differences were not perceptible.
4.3,2 Reduction of Ferricvanide on Pt
The motivation for this study was to explore the influence of the nonuniform
current and potential distributions on the interpretation of the impedance data obtained for
the reduction of ferricyanide on a Pt rotating disk electrode. The electrolytic solution
consisted of 0.01 M K3Fe(CN)6, 0.01 M K4Fe(CN)6, and 1M KC1. Experimental details
are presented in chapter 2. The onedimensional impedance model developed by Tribollet
and Newman was used to analyze the data obtained [5], It was observed that the regressed
values for Schmidt number increased with an increase in rotation speed. As the Schmidt
number is an electrolytic property independent of rotation speed, this result points to the
inadequacy of the onedimensional model used. To motivate development of a two
dimensional model, current, potential, and charge distributions for this system were
obtained as functions of rotation speed and as fractions of the masstransferlimited
current.
4,3,2a Current. Potential, and Charge Distributions
The numerical values of various parameters used for this particular system are
listed in Table 4.2. The rate constant for reduction of ferricyanide on Pt is very large [37],
[66]; thus, the surface overpotential can be expected to be small. Due to the use of excess
114
supporting electrolyte, the contribution of ohmic drop can also be expected to be small.
Thus, mass transport being the predominate factor, the current distribution is expected to
be uniform. Calculated current distributions are presented in Figure 4.5 for rotation rates
of 120 and 3000 rpm. A uniform calculated current distribution was indeed observed at
low rotation speeds, but at large rotation speeds the current distribution was more
nonuniform. These distributions can be contrasted with the extreme case of a primary
current distribution where isur/iaVg=0.5 at the center of the electrode and is equal to oo at the
periphery. As shown in Figure 4.5(b) for 1/4* of the limiting current at 3000rpm, the
current distribution varies from a value of 0.88 at the center to 1.23 at the edge of the
electrode.
The nonuniformity in calculated current distribution shown in Figure 4.5b was
obtained in spite of the use of excess supporting electrolyte. From the calculated
overpotentials shown in Figure 4.6 and Figure 4.8 it can be seen that at both 120 and
3000rpm, the surface overpotential, and the zeta potential are small. The applied potential,
by virtue of the large metal conductivity, is uniform. Thus, a nonuniform distribution in
ohmic potential is compensated by an opposing distribution in concentration overpotential.
The potential distributions are more uniform for the 120rpm case and are less so for the
3000rpm case. A large rotation speed increases the value of the masstransfer limited
current and, therefore, increases the influence of the ohmic potential drop. The
concentration overpotential increases to be a more significant fraction of the applied
potential as current approaches masstransfer limitation. The ohmic potential drop,
therefore, becomes less significant. Thus, the current distribution is more uniform at 3/4
of the masstransferlimited current than at 1/4 (see Figure 4.5).
115
Although, as seen in Figure 4.6 and Figure 4.8 the zeta potential obtained does not
contribute significantly to the overall potential, calculation of the zeta potential allows
determination of the charge distribution from equation (4.22). As can be seen from Figure
4.10, the charge distribution calculated for this system is more nonuniform at 3000rpm
than at 120rpm. The charge distribution follows the distribution of surface overpotential
(see equation (4.26)).
The charge distribution thus obtained was used to calculate the radial distribution
of the double layer capacitance from equations (4.27) and (4.29). The calculated
capacitance was found to be independent of rotation rate and to be uniformly distributed
with a value of 55pF/cm2 for y2 = 10AÂ°. This value is in reasonable agreement with the
experimental value of 35iF/cm2 obtained by Deslouis and Tribollet for the same system
[67], They obtained the same value of capacitance from measurements on disk and ring
disk electrodes. This result is in agreement with the calculations from the present work
which showed that the capacitance has no radial distribution. The value of the double layer
capacitance is sensitive to the choice of y2. By choosing y2 = 17A0 a value of 35pF/cm2
was obtained for the double layer capacitance.
4.3,2b Zero Frequency Asymptotes of Local Impedance
As a preliminary step towards formulating a detailed twodimensional model for
the impedance analysis explicitly accounting for various phenomena, the zero frequency
asymptotes for the local impedances for the ferri/ferrocyanide system were obtained from
steady state distributions calculated at two applied potentials separated by 5mV. The ratio
of the difference in overpotential with respect to the difference in current density
calculated locally provides the contribution of the respective phenomena to overall
116
impedance response. The radial distributions of these values are presented in Figure 4.11
and Figure 4.12 for 120 and 3000rpm respectively. The zero frequency asymptotes for the
local impedance are more uniform for the case of 120rpm and are less so for 3000rpm.
This is a strong indication that a twodimensional model is necessary to obtain reliable
information from impedance data. At higher fractions of the limiting current, the local
impedance resulting from the concentration overpotential dominates. The local impedance
values resulting from the ohmic drop are estimated to be about 20cm2, which
corresponds to 10Q for a disc with a surface area of 0.2cm2, in agreement with
experiment. Appel reported the radial distributions of the impedance calculations for a
single dimensionless frequency of 0.1 applied to 0.0001 equimolar solution of sodium
ferrocyanide and sodium ferricyanide with a 0.1M sodium flouride supporting electrolyte
[42], Similar nonuniformities were observed from his calculations.
4,4 Conclusions
In this chapter, a finite Schmidt number correction was applied to the steady state
model for the rotating disc electrode below the mass transfer limited current. The result of
the convective diffusion equation accounting for this correction was expressed as a
polynomial expansion of Sc'1/3, and the coefficients of the polynomial expansion were
tabulated. The model was then extended to incorporate the diffuse part of the double
layer. Results provided by Newman for deposition of copper were reproduced. The
distributions of current, potential, and charge were assessed for the reduction of
ferricyanide on Pt. The double layer capacitance and the zero frequency asymptotes of the
local impedances were evaluated. Surface nonuniformities were observed for the
117
distribution of various parameters. The zero frequency asymptotes of the local impedances
were calculated and nonuniformities were observed. These asymptotic calculations show
that nonuniformity effects are significant, especially at higher rotation rates. These nonÂ¬
uniformities in fact could cause the anomalous Schmidt number values that were observed
in chapter 2. This points to the need for a twodimensional impedance model accounting
for a finite Schmidt number correction. Development of such a twodimensional
impedance model is presented in the next chapter.
m=l
1
2
3
4
5
6
7
8
9
10
.1. Polynomial coefficients in the expansion for 6'm( 0) resulting from the solution of the convective diffusion equation,
liber of significant digits reported are based on the respective confidence intervals from the regression.
ao
ai
a2
a3
a4
a5
a6
1.119846522
0.333723494
0.0630655
0.02483
0.1009
0.1539
0.361
1.532987928
0.348508169
0.0563987
0.01244
0.0557
0.0760
0.184
1.805490584
0.351502095
0.0501466
0.008443
0.03678
0.0463
0.1034
2.015723734
0.352526040
0.0456708
0.006567
0.02707
0.0321
0.0647
2.189982747
0.352984907
0.0423604
0.005472
0.02132
0.0240
0.0441
2.340450747
0.353227157
0.0398003
0.004743
0.01758
0.0187
0.0323
2.473842754
0.353369827
0.0377463
0.004220
0.01493
0.0153
0.0247
2.594287242
0.353460658
0.03605028
0.003823
0.01295
0.0128
0.0194
2.704520850
0.353521951
0.03461769
0.0035106
0.01141
0.01114
0.0154
2.806460253
0.35356522
0.0333854
0.00326
0.0102
0.010
0.013
2.901505453
0.35359688
0.03230943
0.003038
0.00927
0.00841
0.0109
119
Table 4.2. Input parameters used for the ferri/ferro cyanide in 1M KC1 system reacting on
the Pt disc electrode
v = 0.95lxlO'2 cm2/sec
D = 0.896xl0'5 cm2/sec
F = 96487 coul/eq
Coo = 10'5 mol/c.c.
Z= 1
T = 298K
a = 0.5
0 = 0.5
e = 6.933xlO'12C/V/cm
n = 1
r0 = 0.25 cm
II
o
o
R = 8.314 J/mol/K
i0 â€” 0.3 amp/cm2
Kco= 0.1 ohm'1 cm'1
y2 = 10 A0
o
iâ€”H
II
120
h2 (X104)
Figure 4.1. Determination of the accurate value for 0'm{6) for infinite Schmidt number,
making use of the values obtained from the FDM scheme using varying stepsizes.
121
O 0.05 0.1 0.15 0.2
Sc1'3
Figure 4.2. A sixth degree polynomial fit for Ã©? (Ã“) vs. Sc'13
122
Sc"1/3
Figure 4.3. Errors in ffm(0) values between polynomial fits and the values calculated from
the FDM scheme
123
O 0.2 0.4 0.6 0.8 1
r/r0
0 0.2 0.4 0.6 0.8 1
r/r0
Figure 4.4. Calculated (a) concentration and (b) current distribution on the surface of 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 (VOref) used were 0.08V, 0.28 V, 0.68V, 0.98V, 1.28V, and 
1.58V.
124
r/r0
r/r0
Figure 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.
125
r/r0
r/r0
Figure 4.6. Calculated overpotentials for the case of Figure 4.5 a (120rpm) at (a)l/4th of
iiim, (b) 1/4 of ii,m on an enlarged scale to show the distributions of rj* and Â£.
126
r/iQ
Figure 4.7. Calculated overpotentials for the case of Figure 4.5a (120rpm) at 3/4th of iim.
127
O 0.2 0.4 0.6 0.8 1
r/r0
r/r0
Figure 4.8. Calculated overpotentials for the case of Figure 4.5b (3000rpm) at (a)l/4th of
iiim, (b)l/4 of iim, on an enlarged scale to show the distributions of rj* and Â£
128
O 0.2 0.4 0.6 0.8 1
r/r0
Figure 4.9. Calculated overpotentials for the case of Figure 4.5b (3000rpm) at 3/4th of iHm.
129
r/r0
r/r0
Figure 4.10. Calculated charge distributions for the cases of (a) Figure 4.5a (120rpm) and
(b) Figure 4.5b (3000rpm).
130
0 0.2 0.4 0.6 0.8 1
r/r0
Figure 4.11. Calculated local impedance distributions corresponding to Figure 4.5a (120
rpm) for (a)l/4th of i]im, and (b)3/4th of iiim.
131
O 0.2 0.4 0.6 0.8 1
r/r0
0 0.2 0.4 0.6 0.8 1
r/r0
Figure 4.12. Calculated local impedance distributions corresponding to Figure 4.5b (3000
rpm) for (a)l/4th of i**, and (b)3/4th ofw
CHAPTER 5
A MATHEMATICAL MODEL FOR THE RADIALLY DEPENDENT IMPEDANCE
OF A ROTATING DISK ELECTRODE
The need for a twodimensional frequency domain model accounting for a finite
Schmidt number correction is established in the previous chapter. The development of
such a model, application to the ferricyanide reduction on platinum electrode, and a
comparison with the existing onedimensional frequency domain model is presented in this
chapter.
Frequency domain techniques are commonly employed in the study of electrode
kinetics and mass transfer in electrolytic solutions. While the information obtained is
similar to that from transient techniques, better resolution of physical processes can be
achieved because the noise level in frequencydomain measurements can be made to be
extremely small [10,19,68], A second advantage of the frequency domain techniques over
transient techniques is that the KramersKronig relations can be used to identify
instrumental artifacts or changes in baseline properties [11], [6971],
Due to the growing popularity of the rotating disk electrode, significant effort has
been expended on developing analytic formulae for the impedance response of this system
[4354], A comparative study of the application of these models to interpretation of
impedance spectra has been presented by Orazem et al. [55], A onedimensional numerical
model for the impedance response of a rotating disk electrode which accounted for the
influence of a finite Schmidt number was presented by Tribollet and Newman [5], This
132
133
model accounts for effects associated with kinetics, masstransport, and ohmic potential
drop within the bulk of the solution under the assumption of uniform radial distributions of
current, concentration, and overpotentials. The first two terms in the Cochran expansion
[56] for the axial component of fluid velocity were included in the convective diffusion
equation. Tribollet et al. reported that the errors caused by neglecting the second term in
the axial velocity expansion could be as high as 24 percent for a Schmidt number of 1000
[53], This result was confirmed by regression of various masstransfer models to
impedance data [55],
Mathematical models have been developed which account for frequency dispersion
associated with the nonuniform potential distribution on the disk electrode [5760], But,
to date, no comparable model has been developed for the influence of nonuniform mass
transfer on the impedance response. Appel and Newman provided a preliminary
mathematical development [61 ], valid for an infinite Schmidt number, that they proposed
would be used to develop a model for the influence of radiallydependent convective
diffusion on the impedance response. Later, Appel used this approach to calculate the
radial distribution of impedance, but results were presented only for a single dimensionless
frequency due to numerical difficulties associated with implementation of the model [42],
For some systems, even in the presence of excess supporting electrolyte, the
distributions of current and overpotential on the electrode surface can be significantly non
uniform in the steady state domain as illustrated in the previous chapter (refer to Figure
4.5 (b)). For such cases, the available onedimensional model can be inadequate to
describe the physics of the system. The objective of this work is to provide a development
of a twodimensional model for electrochemical impedance response of a rotating disk
134
electrode that accounts for nonuniform ohmic potential drop, surface and concentration
overpotentials.
5.1 Theoretical Development
The formulation for the twodimensional impedance model results from applying a
sinusoidal perturbation about the steadystate solution presented in the previous chapter.
The capacitance was assumed to be uniformly distributed across the surface of the disk
electrode. This assumption is justified by the results of steady state calculations valid in the
absence of specific adsorption. Assumption of uniform capacitance is also consistent with
the experimental results presented by Deslouis and Tribollet [67], The development of a
twodimensional frequency domain model is presented in this chapter.
5,1.1 Convective Diffusion
For a small sinusoidal perturbation of frequency co in potential, the response can be
assumed to be linear, and all associated variables will oscillate at the same frequency ca
Thus; each variable can be written in the form
j = J +Regexp (jot)} (5.1)
where x's the variable under consideration, % is the corresponding steady state or
baseline value of x> x is the complex amplitude of perturbation, t is the time, co is
frequency, and j = V T. As defined, ^ is a function only of position. The same notation
is used for the other parameters considered in this chapter.
On the basis of equation (5.1), the concentration of the reacting species can be
written as
135
c(r, z, t) = c(r, z) + Re{c (r, z) exp(/
where c and c are functions or radial and axial position. Incorporation of concentration
perturbation terms in the unsteady state convective diffusion equation and cancellation of
the steady state terms yields
. ~ dc dc d2c
j(o c + vr â€”â€”h v. â€”â€” = D
(5.3)
dr 2 dz dz2
where vr and vz are the radial and axial components of the velocity of the fluid and D is the
diffusion coefficient of the species of interest. The velocity components, vr and vz, account
for the first three terms of Cochran's expansion as given by
v, = arz
n3/2 1 , Q b , Q
5/2
â€”rz
v1/2 2
â– â€”rz
v 3
.3/2
(5.4a)
and
v. = az
Ãœ3'2
vm 3
1 3Q2 b 4Q
+ z â€” + z
5/2
,3/2
(5.4b)
v 6 \r
where a and b are the coefficients from Cochran's expansion for the radial and axial
velocities for the rotating disk electrode system, Q is the rotation rate of the disk
electrode, and v is the kinematic viscosity of the electrolyte. For the steady state case, a
variableseparable form was used for the expression of concentration of the reacting
species, c, that is,
00 Ã / _ v2m ]
(5.5)
00
'
r \
2m
c = c00
â€˜+1
Am
m
r
0,(4)
>
m=0
where c* is the bulk concentration of the reacting species, Am values are the coefficients to
be determined from the steady state calculations, and Om(Â£) is the axial dependent term
136
which can also be determined from the steady state values. The dimensionless axial
distance Â£ is given by
â– â– (sTS
(5.6)
The resulting perturbation in concentration c from equation (5.5) can be written as
m=0
[4A(?)+4Afe)]i
. 2m
roy
(5.7)
Substitution of equations (5.4a), (5.4b), and (5.6) in equation (5.3) yields
jKc+\3rÂ£ÃrÂ¿;2BScU3 2Cr^Sc~2'3 ^ q
dc d2c
+ ( 3Â£2 + Bg3Sc~V3 +CÂ£4Sc2,3)â€” =
d{2
(5.8)
where Sc is the Schmidt number of the species controlled by mass transfer, and
n
r
9v
a2D.
,1/3
(5.9)
r ? "\
1/3
B =
J
(5.10)
and
r _ b_(l)
6{a,
5/3
(5.11)
where K is the dimensionless frequency of perturbation. Combination of equations (5.7)
and (5.8), cancellation of terms resulting from the steady state part of the convective
diffusion,
137
0'm +(3
(5.12)
2/3 a =
yields
0; + fe2  B?Sc~m  C4*Sc~in
+ 2m(34 + 42BScV3 +2C43Sc~2l3\emjK0m
v 2
/
f â€œ jKAmOn  0 (5.13)
where primes in the superscripts refer to derivative with respect to 4 The real and
imaginary parts of equation (5.13) provide two ordinary differential equations which must
be solved simultaneously.
Solution of equation (5.13) requires that boundary conditions on 9m be
established. From the steady state case, the concentration at the surface is given by
(5.14)
m=0
Hence, the perturbation in concentration of reacting species on the electrode surface is
given by
c0=cxY,^Arlro)2m
(5.15)
m=0
However, from equation (5.7),
(5.16)
m=0
As 0m(o)=l from steady state calculations, comparison of equations (5.15) and (5.16)
reveals the boundary conditions for equation (5.13) to be
em = 0 at 4 = 0
(5.17)
138
and
dm = 0 at Â£ = oo
(5.18)
The boundary condition at Â£ = oo results from the observation that the imposed
perturbation does not have any effect on the parameter values far away from the electrode
surface. The simultaneous equations resulting from the equation (5.13) can be solved by
the reductioninorder method. The solution for equation (5.13) can be written as
f. C
Qm = Omh
jKAmemet
exp
mh â€” ^ ^â€”Sc1/3  * Sc'2'3
\
y
&L exp
gi_*L
l4
Scm^Sc2'3
d?
(5.19)
+ K\Omh
6Ik exp
o BrScu3_cr
dg + K1BÂ¡
r
Sc
2/3
2^ mh
where Â®mh is the solution to the homogeneous part of the differential equation, and KÂ¡ and
K2 are arbitrary constants that need to be determined based on the boundary conditions.
Equation (5.19) can be written as
9. =0,
m w mh
K2 +
Ã p fr'4
'2 ~~ z"+^_scm
l 4
9mi exp
+
O;
t5
Sc
2/3
(',fe')Vr
(5.20)
where
?
r
>,(?) = K,+J
KA.0A
f
mh
4.
exp
3 B? ^3 Cg
r .
v 4
Sc~Ui ^â€”Sc~2J31 dg
(5.21)
139
From the boundary condition 6m = 0 at Â£ = 0 and equation (5.20), K2 = 0. From the
boundary condition Gm  0 at Â£= Â°Â°,
Kx = ~j
KA
QJmk exp
:3 ,/3 CÂ£5
^2Scin 
Sc
2/3
(5.22)
Equation (5.22) is a direct consequence of the observation that the factor
exp
D /~vprs
Â£'3 +^â€”Scm +^â€”Sc2,i
4 5
approaches infinity as becomes large (the
rate at which G2h approaches O+yO is much faster than the rate at which
exp
p fr'4 pt5
#'3 + â€”â€”Sc~m +^â€”Sc~2n
4 5
approaches 0). Hence, in order for a stable
solution to exist, tx{g') must be zero. One of the parameters of great interest is the
derivative of Gm with respect to Â£ at the electrode surface, as this will be used at a later
stage to calculate the perturbation in current resulting from mass transport:
*:(<>)=
deâ€ž
dt
= Kl= j
KA.
4=o
djmh exp
â€œ3 1/3 C^5
Sc
2/3
df (5.23)
The homogeneous part of the convective diffusion equation is given by
+ &'  B?Scm  C?Scfc,
( 3 V ~ =0 (5.24)
+ 2n^~+ â€”Â¿;2BSc~13 +2C^Sc2/3jemh jKGmh
and the boundary conditions to solve this equation can be chosen to be
Â£*= lati = 0 (5.25)
140
and
0mh = 0 at Â£ Â» oo
(5.26)
The boundary conditions of Equations (5.25) and (5.26) are used for solving the
homogeneous part of the convective diffusion equation, and the boundary conditions for
the overall solution are provided by equations (5.17) and (5.18). Following Levart and
Schuhmann's approach, the solution to the homogeneous part can be expressed as a
polynomial series of Sc1B [72]. Hence, by retaining two terms accounting for the finite Sc
correction, 0mh can be represented by
 #0,mh + Q\,mhSC 1/3 + #2,mh^C
2/3
Thus, equation (5.24) can be rewritten as
(5.27)
+ (3 ?B?Scâ„¢ Ct'Sc^^+S^Sc'â€ +Â§USc~2n)
+ 2^ 3 4 +4'BSc^ + 2 CfSc^y^ + 0,MScm +
 + 0Â¡MSc'â€œ + S2MScm)
= 0 (5.28)
Collection of real and imaginary terms of the same order of Sc yields six simultaneous
ordinary differential equations:
ft"
u\,mh,r
ft"
+ 3
+3
 6m^00 mhtr + K&o,mhJ = 0
(5.29a)
6m%Kmh,J K0O mhr=O
(5.29b)
+ K0hmhJ =B?eUr 3mfBO^
(5.29c)
~K0lmKr = B?0Â¿mhJ 3m?B0OmhJ
(5.29d)
141
OUr+tfOU, +K02,mhJ
= B?eu,  3m^2BOl mh r + C^GUr  4mCZ\mh,r
(5.29e)
(5.29f)
OUi ~Ke2M,r
= BtXmhJ 3m^2B0hmhJ +CZ%mhJ4mCt%mhJ
This system was solved using Newmanâ€™s BAND algorithm. The boundary conditions are
(5.30a)
&0,mâ€ž = 1 at Â£= 0 and d0Â¡mh > 0 as Â£ > oo
0i,mh = 0 at Â£= 0 and 0hmh > 0 as Â£ oo
(5.30b)
and
02,mh = 0 at Â£= 0 and 02>Ilrt > 0 as Â£ > oo
(5.30c)
5,1,2 Conditions on Current
The current density on the electrode surface is constrained by mass transfer and
kinetic considerations.
5,1.2a Mass transport
The normal Faradic current density //must satisfy
r> / \l/3
Z)câ€ž av
D dc
nF l/fi dz
z=0
1 tR\3Dj
n
z^Wr.r^:(o)
v ZZ
(5.31)
m=0
where t is the transference number of the reacting species, n is the number of electrons
produced when one reactant ion or molecule reacts, and F is Faraday's constant. The
oscillation in normal current density is related to the derivative of oscillation in the
concentration by
142
D de
nF 1 tR dz
Dcâ€ž ( av
r=0
V/3
â€¢ 'Â«UD
Â£Â±ÃÂ¡.0L(O)+JÂ¿:(oirtrcr}
(5.32)
The interfacial overpotential resulting from convective diffusion is the
concentration overpotential ( rjc) given by
RT
Vo=~
ZF
In
(c \
Vco J
U
\ cÂ°Â° J
(5.33)
where R is the universal gas constant, T is the temperature of the electrolyte in Kelvin, and
Z is the number of equivalents of reactant used up per one electron produced/consumed.
For the case of sinusoidal perturbation
Vc =
RT
ZF
1 U
"0 oo
(5.34)
5,1,2b Kinetics
The current resulting from the kinetic contribution can be equated to the current
resulting from the mass transport, as was done for the steadystate case. The steadystate
expression for the current due to kinetic contribution is
lf=l o
rc v
Â£
VcÂ® J
exp
aZF
RT
Vs
exp
PZF
RT
Vs
(5.35)
where /'o is the exchange current density, a and ft are the anodic and cathodic transfer
coefficients, y is the composition dependence of the exchange current density, and rjs is the
surface overpotential. The oscillation in Faradic current is given by
143
T Y'f *
lf =
C0 + ,0
f c v
C'Q
vcÂ®y
ZF
RT
i
a exp
aZF \
RT
Vs
n Ã iâ„¢ ^
+ /?exp â€”â€”V>
\ K1
Vs
(5.36)
The capacitive contribution to the current density is given by
Tc = ja>C(rjc+Tjs)
(5.37)
where C is the double layer capacitance.
5,1,3 Potential
Under the assumptions of the absence of concentration gradients within the bulk of the
solution,
V2O = 0 (5.38)
where is the ohmic potential. Hence when a sinusoidal perturbation is imposed, the
equation is transformed to
V20> = 0 (539)
It should be noted that Laplaceâ€™s equation does not have time dependence, even for the
unsteadystate case. The rotational elliptic coordinate transformation applied for the
steady state problem can be applied to equation (5.39), that is,
_d_
d ju
(l + /r)
00
0 fJ.
drj
(,â€žq
00
0//
= 0
(5.40)
where y = rQ/u/j and r  r0^ rÂ¡2) â– The boundary conditions for equation (5.40)
were that ^0/^/7 = 0 at 7 = 0 (on the annulus), 0 = 0 at n = 00 (far from the disk),
and 0 is well defined at 77= l(on the axis of disk). The solution of equation (5.40)
satisfying these boundary conditions can be expressed as,
144
ZF ti
(5.41)
where Bn are coefficients to be determined, P2n(rj) are Legendre polynomials of order
2n, and M2n{/j) are Legendre polinomials of an imaginary argument satisfying
d_
dfj.
(i
+ M
2\dM
2n
= 2n(2n + l)M2n
(5.42)
with the boundary condition M2n =1 at ju 0 and M2n =0 at p  oo. On the surface
of the disk electrode, that is, at p0,
DT co
ZF â€ž=0
(5.43)
The current density at steady state is related to the derivative of the potential just
outside the diffusion layer by
/ = *â– 
ao
dz
50
20
roV
(5.44)
//=0
In terms of oscillating variables,
/ = tcâ€ž
8 O
dz
50
2=0
r0rj dp
(5.45)
n=0
or, from equation (5.41),
r0rj ZF n=0
From the properties of the Legendre functions it can be seen that
2 (24
(5.46)
ML(0) = 
'M'
(5.47)
145
The total current density is given by
T = TC+Tf (548)
where ic and iF are the oscillations in capacitive and Faradic currents and
V=Sâ€™a+rj,+K (S'Â»)
where V is the oscillation in potential applied to the system.
5.2 Numerical Procedure
The numerical procedure adopted for solving for the current and potential
distributions in the frequency domain is discussed in this section. The steps provided in
this section were used to obtain distributions for a given frequency, and calculations were
repeated over a range of frequencies to provide a spectrum:
1. The steady state distributions for concentration, current, various overpotentials, and
other parameters of interest such as Am, 0'm (o) and so on, were calculated using the
steady state model without accounting explicitly for the charge distribution. Correction
for a finite Sc was used in obtaining these results.
2. The solution to the homogeneous part of the convective diffusion equation was
obtained adopting Newmanâ€™s BAND algorithm for a given set of frequencies. A
Simpsonâ€™s l/3rd rule in combination with Simpsonâ€™s 3/8* rule was used to evaluate the
integrals involved in equation (5.23) and a value of n = 10.0 was used to represent the
value for qo, as the perturbations dissipate at this distance. The accuracy of the solution
could be increased using an adaptive integration technique. Use of adaptive integration
should be especially important for calculations at high frequencies.
146
3. Guess values for Am were introduced. The power series was truncated at m = 10.
4. The values for c0 were calculated from equation (5.15) for a specified number of
radial positions r/r0. The values of r/r0 were chosen to be the values of abscissa ranging
between 0 and 1 used for the GaussLegendre quadrature.
5. The values for O^io) were determined from equation (5.23).
6. The if distribution was obtained from equation (5.32) using values for Am and Q'J^O).
7. The rjs distribution was calculated using equation (5.36).
8. The total current density / was determined from equation (5.48).
9. The Bn coefficients were calculated by expressing / resulting from the above step as
a linear combination of Legendre polynomials of rÂ¡, the rotational elliptic
transformation variable, that is,
oo
^ = S^2;fo) (5.50)
1=0
where / is varied from 0 to 10 and r, are complex coefficients. By making use of the
orthogonal properties of Legendre polynomials, r, coefficients were obtained as
i
r,=(4l + \)\7P2l(rj)dT] (5.51)
0
From equations (5.46) and (5.50)
*â€¢Â«, RT
r0rj ZF
n=0
1=0
(n)
(5.52)
From equation (5.52)
147
~ r0T] ZF 1
â– k.RTMâ€™M
(5.53)
10.From the values of Bncoefficients thus obtained, the O0 values were found from the
equation (5.43).
11. The?7c distribution was obtained from (5.49).
12. The c0 distribution was recalculated from equation (5.34). At convergence, the
distribution obtained during this step should match with that obtained from step 4.
13. The c0 distribution resulting from the above step was expressed as a linear
combination of Legendre polynomials of 77, the rotational elliptic transformation
variable, that is,
(5.54)
1=0
where / is varied from 0 to 10 and a, are complex coefficients. By making use of the
orthogonal properties of Legendre polynomials, a, coefficients were obtained as
(5.55)
0
14. From equations (5.15) and (5.50), a relationship between Am and a, was established
and a new set of Am values were obtained.
m
15. The new set of Rvalues were averaged with the Rvalues from previous iteration by
using a weighting factor.
16.Steps 5 through 15 were repeated until convergence was obtained.
148
A FORTRAN program used in implementing the frequency domain calculations
using this algorithm is presented in Appendix B. Using a 300 MHz Digital Equipment
Corporation (DEC)â€™s Alpha processor (model no. DEC221164), the CPU time needed to
generate a typical spectrum with about 60 frequency points obtained within a tolerance
limit of 1 in 108 is about 5 minutes. However the calculations can be made faster by
compromising on the tolerance limit. An improvement in the integration scheme will result
in significant decrease in the time needed to generate the spectrum.
5 3 Results and Discussion
Direct comparison can be made between the onedimensional and twodimensional
models for a specified condition. Agreement between the two models is expected under
conditions for which the current distribution is uniform (slow kinetics or near mass
transfer limitation). The onedimensional model is expected to be inadequate for non
uniform current distributions.
5,3,1 Uniform Current Distribution
From steady state calculations, an a priori assessment of the uniformity of current
distribution can be obtained from dimensionless parameters
J =
i0r0ZF
RTkk
(5.56)
which provides the relative value of kinetic and ohmic contributions to the system
resistance, and
r02Q
1/3 nZF2Dcx
1 y
ODj
rt(Ã
N = 
(5.57)
149
which provides the relative value of mass transfer and ohmic contributions. The influence
of these parameters on the uniformity of distributions can be seen in Figure 8 of [41], The
current distribution is uniform for small values of J and N.
Calculated results are presented in Figure 5.1 for an exchange current density of
3mA/cm2 and a diffusion coefficient value of 0.3095x10'5cm2/sec. Other parameters used
in these simulations are those corresponding to the values for the reduction of ferricyanide
system as reported in chapter 4. Equations (5.56) and (5.57) yield J  0.225 and N =
0.0695. From the results presented in Figure 5.1, it can be seen that a significant high
frequency loop results due to the kinetic effect. In this case, steady state calculations show
that the current distribution has less than 1% variation across the electrode surface.
In this case it is expected that the twodimensional model that is developed here
should be in agreement with the calculations performed for the onedimensional case.
Comparison between the one and twodimensional models are presented Figure 5.1a and
Figure 5.1b. The discrepancy between the two models at high frequencies could be
attributed to the numerical instabilities in the twodimensional model at high frequencies.
However, for most part of the frequency spectrum the normalized differences are less than
3%. This result illustrates that one dimensional model is adequate for cases where kinetic
effects play a dominant role. Similar agreement between the one and twodimensional
models is seen for larger exchange current densities (larger values of J) as the mass
transferlimited current density is approached.
5,3,2 NonUniform Current Distribution
The onedimensional model does not provide an adequate representation of the
impedance response for a disk electrode below the masstransferlimited current for large
150
exchange current densities. The values for the exchange current densities and the diffusion
coefficients used in this section are those required to match the values of experimentally
obtained impedance spectra.
A comparison between the two models is presented in Figure 5.2 and Figure 5.3
for a rotation speed of 120rpm, at i/iiim=0.25 (J = 2.25 and N= 0.0695). Even at this low
rotation speed, where the steadystate current density at the center of the disk is 97
percent of the average value, considerable disparity is observed between the oneÂ¬
dimensional and twodimensional models. This is consistent with the observation that even
small nonuniformities in the steadystate domain could become significantly important in
the frequency domain. The onedimensional model does not adequately describe the
system behavior as reaction kinetics become fast. A similar comparison is presented in
Figure 5.4 and Figure 5.5 for 3000rpm at l/4th of the masstransferlimited current (J =
7.489 and N = 0.3552). The discrepancy is larger due to the increased value of A.
For the same values of J and N as was used in Figure 5.4, the disparity between the
two models for a rotation speed of 120rpm was reduced significantly when calculations
were obtained at 3/4* of the masstransferlimited current, as shown in Figure 5.6 and
Figure 5.7. The agreement is observed because the distributions are more uniform near
masstransfer limitation. At 3000 rpm, however, the two models do not agree, even near
the mass transferlimited current (see Figure 5.8 and Figure 5.9). In this case, J = 5.617,
and N = 0.5874. The discrepancy between the two models can be attributed to the larger
value of N. The influence of nonuniformities becomes more significant with an increase in
rotation speed.
151
Another perspective on the twodimensional character of the disk impedance
response at large N can be seen from the radial distributions of the real and imaginary parts
of local impedance given in Figure 5.10 for the conditions of Figure 5.6 and Figure 5.8.
The distributions are shown for dimensionless frequencies of 1 and 2.8. The increased
nonuniformity of the local impedance at 3000rpm results in the discrepancy seen between
the one and twodimensional models.
A surprising result of the work presented by Orazem et al. [55] was that the oneÂ¬
dimensional model of Tribollet and Newman for impedance of a rotating disk fit the data
to within the noise level of the measurement, even for conditions where the present work
would indicate that a twodimensional model would be required. Their result can be
explained by regressing the onedimensional model to impedance spectra calculated using
the twodimensional model presented in the present work. The results of the regression
are presented in Figure 5.11 for the 120rpm, 374th of mass transfer limited current
comparison shown in Figure 5.6. The quality of the fit is extremely good with residual sum
of square to noise ratio of about 2.0. The quality of the fit is such that it would be difficult
to use regression to experimental data to distinguish between the twodimensional and
onedimensional models. The principal difference between the two models is that the oneÂ¬
dimensional model provided a Schmidt number 7 percent higher (1780 as opposed to 1660
for the twodimensional model). Results from the onedimensional model regression for
the two dimensional model calculations for 3000rpm, 374th of the masstransferlimited
current are presented in Figure 5.12. In this case the onedimensional model provided a
Schmidt number 22 percent higher (1530 as opposed to 1250 for the two dimensional
model). Similar results were obtained for other cases
152
Similar analysis was performed for the impedance spectra generated using the two
dimensional model for a Schmidt number of 1100 and for an exchange current value of
50mA/cm2 for various rotation rates and for various fractions of masstransfer limited
current. Regressions were done using a no weighting strategy. The Schmidt number values
obtained through this analysis are presented in table 5.1. The value of the parameter J is
constant as the exchange current density value is fixed for generating these values.
The kinetic effects are more prominent in the high frequency region, and the two
dimensional model results presented in this work encountered numerical instabilities at
very high frequencies. This could be a reason for the disagreement between the two
models for the charge transfer resistance. The problem of numerical instabilities at very
high frequencies could be addressed by using adaptive integration schemes for calculating
the integrals involved. The present work illustrates that a good fit for the onedimensional
model does not imply that the model is adequate to describe the physics of the system.
Due importance should be attached to the role of nonuniform current distributions for the
assessment of electrochemical impedance data for cases where J and N are large.
5,4 Conclusions
A twodimensional mathematical model for the impedance response of a disk
electrode was developed on the basis of the coupling among the convective diffusion in a
thin diffusion layer, the Laplaceâ€™s equation for potential in an outer region, and the Butler
Volmer expression for the electrode kinetics. The influence of a finite Schmidt number,
which is critical in the frequency domain, was taken into account by including three terms
in the axial and radial velocity expansions. A uniform distribution of capacitance was
153
assumed in the development of the model, and this assumption was supported by
experimental observations reported in the literature for ringdisk measurements [67] and
by preliminary calculations presented in chapter 4. Reliable results were obtained for
dimensionless frequencies less than 1000. Use of improved integration strategies could
extend the frequency range.
The nonuniform current distribution observed below the masstransferlimited
current has a significant effect on the impedance response of a rotating disk electrode
when electrode kinetics are fast. Onedimensional models apply when kinetic or mass
transfer limitations make the current distribution more uniform. Discrepancies between the
one and twodimensional models were observed under conditions where the steadystate
current density at the center of the disk was 97 percent or less of the average value.
Under conditions yielding a nonuniform current distribution, the onedimensional
model may provide an adequate fit to the data, but the physical parameters obtained by
regression will be incorrect. In such cases, nonuniform current distributions should be
taken into account. The effect of nonuniform current distributions on impedance was seen
at 120rpm and is even greater at larger rotation speeds.
Table 5.1. Onedimensional frequency domain process model regressions for the twodimensional model calculations for a Sc value of
1100 and for an exchange current density of 50mA/cm2 Dimensionless parameter J= 3.7445.
N
Z(w=0), Q
Sc
COiF)
Re, Q
Rt, Q
NRSSQ
120, 5% ilim
0.1276
69.47 + 0.03
1122 Â±3
7.10 Â± 0.15
6.29 Â±0.03
2.71 Â±0.03
2.2257
10% i,im
0.1276
73.42 Â± 0.03
1125 Â±3
7.08 Â±0.14
6.28 Â±0.03
2.86 Â± 0.03
2.2629
25% ilim
0.1276
89.71 Â±0.04
1134 Â± 3
7.02 Â±0.11
6.28 Â±0.03
3.45 Â±0.02
2.3953
50% ilim
0.1276
139.58 Â±0.08
1151 Â±4
9.15 Â± 1.34
7.04 Â± 0.39
4.47 Â±0.37
2.8841
75% ilim
0.1276
301.69 Â±0.20
1166 Â±4
8.29 Â±0.89
7.32 Â±0.62
9.44 Â±0.55
3.2026
90% ilim
0.1276
909.22 Â±0.50
1171 Â±4
6.91 Â±0.03
6.29 Â±0.06
25.73 Â±0.04
3 1211
95% ilim
0.1276
2281.4 Â± 1.25
1168 Â± 4
6.89 Â±0.01
6.25 Â±0.03
49.44 Â±0.13
3.0585
96% ilim
0.1276
3109.8 Â± 1.75
1165 Â±4
6.89 Â±0.02
6.24 Â± 0.05
60.00 Â±0.22
3.0046
600, 25% ium
0.2854
40.67 Â±0.04
1147 Â±3
9.66 Â±3.94
6.87 Â±0.74
2.86 Â±0.71
2.1774
50% ilim
0.2854
65.16 Â±0.06
1179 Â±3
7.51 Â±2.28
6.47 Â±0.94
4.96 Â±0.90
2.5536
75% ilim
0.2854
151.39 Â± 0.13
1209 Â±4
7.03 Â± 0.76
6.35 Â±0.66
10.17 Â± 0.58
2.9478
90% ilim
0.2854
520.67 Â±0.32
1211 Â±4
6.91 Â±0.06
6.29 Â±0.11
24.38 Â±0.06
2.9705
1200, 25% ihm
0.4036
29.15 Â±0.02
1157 Â±3
7.40 Â± 1.02
6.37 Â±0.26
3.34 Â±0.25
1.9813
50% ilim
0.4036
47.59 Â±0.04
1200 Â±3
6.47 Â± 1.08
6.04 Â±0.52
5.35 Â±0.50
2.4255
75% i)im
0.4036
115.91 Â±0.14
1240 Â±4
5.61 Â±0.99
4.87 Â± 1.23
11.37 Â± 1.12
2.7498
2400, 25% ilim
0.5708
20.92 Â± 0.02
1169 Â±3
5.73 Â± 1.24
5.85 Â±0.48
3.83 Â±0.46
1.7698
50% ilim
0.5708
35.19 Â±0.04
1228 Â±4
3.84 Â± 1.16
4.15 Â± 1.25
7.13 Â± 1.21
2.1352
75% ilim
0.5708
89.91 Â±0.08
1265 Â±5
8.49 Â±0.13
7.6 (fixed)
8.74 Â± 0.04
2.855
3000, 25% ijim
0.6382
18.79 Â±0.01
1168 Â±3
15.13 Â±0.24
7.6 (fixed)
2.15 Â±0.01
2.0504
50% ilim
0.6382
31.95 Â±0.04
1237 Â±4
4.16 Â± 0.98
4.51 Â±0.92
6.75 Â±0.89
2.087
75% ilim
0.6382
83.61 Â±0.08
1275 Â± 5
8.43 Â±0.14
7.6 (fixed)
8.67 Â±0.05
2.7159
155
o i Hz
Figure 5.1(a) Comparison between onedimensional and twodimensional models for the
slow kinetics case at l/4m of /um and O=120rpm with /0 = 3 mA/cm2, D = 0.3095xl0'5
cm2/sec, J = 0.225, N = 0.0695, and Sc = 2730. In this case steadystate distributions tend
to be highly uniform, (b) Differences between the calculations from twodimensional and
onedimensional model normalized with respect to the twodimensional model as a
function of frequency.
156
Zr,Q
Figure 5.2. Comparison between the impedance spectra generated by ID and 2D models
for 120rpm, l/4th of iim, /'o = 30 mA/cm2, D = 0.3095x10â€˜5 cm2/sec, J= 2.247, N= 0.0695,
and Sc = 2730. Results presented for impedance plane plot.
157
a), Hz
o, Hz
Figure 5.3. Comparison between the impedance spectra generated by ID and 2D models
for 120rpm, 174th of ium, /0 = 30 mA/cm2, D = 0.3095x1 O'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.
158
Zr,Q
Figure 5.4. Comparison between the impedance spectra generated by ID and 2D model
for 3000rpm, 174th of iiim, /'o = 100 mA/cm2, D = 0.3195xl0â€˜5 cm2/sec, J= 7.489, N=
0.3552, and Sc = 2650.
159
co, Hz
co, Hz
Figure 5.5. Comparison between the impedance spectra generated by ID and 2D model
for 3000rpm, 1/4* of ium, /'o = 100 mA/cm2, D = 0.3195xl0'5 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.
160
zr, n
Figure 5.6. Comparison between the impedance spectra generated by ID and 2D model
for 120rpm, 314th of iu,,,, /'o = 100 mA/cm2, D = 0.5095xl0'5 cm2/sec, J= 7.489, N=
0.0970, and Sc = 1660. Results presented for impedance plane plot.
161
co, Hz
co, Hz
Figure 5.7. Comparison between the impedance spectra generated by ID and 2D model
for 120rpm, 374th of iiâ„¢, /0 = 100 mA/cm2, D = 0.5095x1 O'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.
162
Zr,Q
Figure 5.8. Comparison between the impedance spectra generated by ID and 2D model
for 3000rpm, 374th of i^n, /'o = 75mA/cm2, D = 0.6795xl0'5 cm2/sec, J= 5.617,N=
0.5874, and Sc = 1250. Results presented for impedance plane plot.
163
co, Hz
co, Hz
Figure 5.9. Comparison between the impedance spectra generated by ID and 2D model
for 3000rpm, SM1*1 of inâ„¢, /0 = 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.
164
r/r0
r/r0
Figure 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.
165
r/r0
r/r0
Figure 5.10â€” continued
166
O 100 200 300 400 500
Zr, Q
Figure 5.11. Results for regression of the ID model to a 2D model simulation for 120rpm,
3/4* of iiim. 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
167
o, Hz
168
Zr
Figure 5.12. Results for regression of the ID model to a 2D model simulation for
3000rpm, 3/4th of iim. 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
169
Â©, Hz
Figure 5.12â€”continued.
(a, Hz
CHAPTER 6
CHEBYSHEV POLYNOMIAL SOLUTION FOR THE STEADY STATE
CONVECTIVE DIFFUSION FOR A ROTATING DISC ELECTRODE
In this chapter, the applicability of the Chebyshev polynomials in solving
differential equations is illustrated in the case of convective diffusion equation for the
rotating disc electrode system for the infinite Schmidt number assumption. A comparison
is drawn here between the solution obtained by this approach and that from a finite
difference scheme. It is shown that the Chebyshev polynomial approximation provides
results with good accuracy and fewer calculations.
Chebyshev polynomials are the solutions of the differential equation
(6.1)
dx dx
where Tn(x) is a Chebyshev polynomial in x of Arth degree given by
7^(x) = cos [Â«Ã¡rceos x]
(6.2)
and x varies between 1 and +1, as the value of arccos is defined for arguments with
absolute value of less than 1. Chebyshev polynomials of various orders as functions of x
are shown in Figure 6.1. Let x = cos 0. Then equation (6.2) can be written as
Tn(cos 0) = cos nO
(6.3)
Thus,
T0(x) = 1,7Â¡ (x) = x,T2(x) = 2x2 1,7;(*) = 4x3 ~ 3x,T4(x) = 8x4  8x2 +1,... (6.4)
170
171
One of the most important properties of Chebyshev polynomials is that they satisfy the
orthogonality relation
(6.5)
where c0=2, câ€ž=l for n>0, and 8 nm is the Dirac delta function given by
1 if n = m
0 if n^m
(6.6)
Extensive use was made of the orthogonal property of Legendre polynomials in the steady
state and frequency domain modeling (see chapters 4 and 5). These polynomials were used
in approximating various distributions across the electrode surface. In this chapter the use
of the orthogonal property of Chebyshev polynomials is illustrated.
The use of Chebyshev polynomials in the series approximation of various analytic
functions is well established. The particular method of solution using Chebyshev
polynomials described in this work is also known as the tau method, initially devised by
Lanczos [73], The procedure used for this work is a variation of this original method [74],
In the remaining part of this section a more detailed discussion on the use of various
solution techniques is provided.
The tau method in turn belongs to the class of spectral methods [75], Other than
the tau method, spectral methods consist of Galerkin and collocation methods. In all these
methods the solution to the differential equation of interest is approximated by a linear
combination of globally smooth functions. This is the most important distinction of these
methods in comparison with finite difference and finite element methods. In finite
difference and finite element schemes the basis functions are local in nature. Very often it
172
is convenient to choose the basis functions for spectral methods to be orthogonal functions
so that the orthogonal property can be exploited in the solution process. Hence the choice
of basis functions in the tau method is not restricted to Chebyshev polynomials. Other
family of orthogonal functions such as Legendre polynomials, Fourier series, and Laguerre
series can also be used.
Spectral method can be viewed to be a development of the method of weighted
residuals (MWR) [76], The key elements of the MWR are the trial functions (basis
functions used for approximating the solution) and the test functions (also known as
weight functions). The test functions are used to enforce the minimization of the residual.
The choice of weighting function distinguishes Galerkin, collocation, and tau methods. In
the Galerkin approach, the test functions and the trial functions are the same. In the
method of collocation, the test functions are the Dirac delta functions. To state it more
explicitly, in case of the Galerkin methods, the governing differential equation is valid
through the whole domain in an averaged sense where as in case of the collocation
techniques, the differential equation should be satisfied exactly at the collocation points.
Tau methods are closer to Galerkin methods but they differ in the treatment of the
boundary conditions. The tau approach is the most difficult to rationalize within the
context of the MWR.
Clenshaw proposed the use of Chebyshev polynomials in the solution of linear
differential equations [77], In this work he provided various examples for solving first and
second order linear equations and an eigenvalue problem by using Chebyshev polynomial
approach. Later Clenshaw and Norton extended the Chebyshev series approximation to a
broader class of boundary value problems [78], by using the Picard iteration scheme [79],
173
This allowed them to solve linear differential equations with nonpolynomial coefficients
as well as nonlinear differential equations.
6,1 Transformation of the Convective Diffusion Equation
To provide a clear understanding of the solution process of differential equations
with the use of spectral Chebyshev techniques, Johnson treated ordinary differential
equations with constant coefficients and documented the approach in a detailed manner
[80], In case of many engineering problems, the governing equations contain variable
coefficients. In order to demonstrate the applicability of Chebyshev polynomials in such
cases, the steady state convective diffusion equation in two dimensions for a rotating disk
electrode is considered in this chapter. The solution to convective diffusion is important in
case of several electrochemical systems with forced convection where the goal is to
calculate the current distribution. There are very efficient finite difference codes with
excellent convergence, such as the BAND algorithm developed by Newman, to address
the boundary value problems [20], For problems of the nature considered in this work, the
Chebyshev polynomial approximation has a much faster convergence compared to the
traditional finite difference methods. This feature is of great use in problems where the
boundary value problems need to be solved for a number of times. An example is the non
uniform current distribution calculations on rotating disk electrode in the frequency
domain shown in chapter 5, where the homogeneous part of the convective diffusion
needs to be solved for a number of frequencies.
In the Chebyshev spectral tau method, the solution to the differential equation is
written as a summation of Chebyshev polynomials multiplied by unknown weighting
174
coefficients which need to be determined. This summation is also assumed to satisfy the
boundary conditions. (This is an important difference between the tau method and the
Galerkin method. In the Galerkin approach each of the basis functions must satisfy the
boundary conditions whereas in the tau approach it is sufficient if the approximated
function satisfies the boundary conditions.) The unknown coefficients are determined
using the orthogonality of the Chebyshev polynomials.
The convective diffusion for the rotating disk electrode is given by
d c d c d 2c
v. + v. = D
r dr
d z
d z
(6.7)
where vr and v2 are the radial and axial velocity components, r and z are the radial and
axial components. A scaled axial distance is defined to be
r \1/3
' av'
v3 Dj
Â£
(6.8)
The concentration in the diffusion layer can be expressed as a series solution
c0 = C=o
m=0
(6.9)
where c0 is the concentration of the reactant on the electrode surface, Am are constant
coefficients, <9m(Q are the axial dependent terms in the concentration series. Using this
expression in equation (6.7) the axial dependence can be separated as
^ + X2^6mC0m=0 (6.10)
Primes in the superscripts of equation (6.10) refer to the derivatives with respect
to Â£ The boundary conditions for this equation are given by Gm = 1 at Â£ = 0 and 6m = 0 at
175
Â£ Â» co. For the purpose of numerical calculations, as shown in chapter 4, Â£=20 can be
used instead of Â£â€”> Â°o. Hence the boundary conditions transform to
& = / at Â£= 0 and & = 0 at Â£= 20 (6.11)
The domain of interest is Â£e[0,20\. In order to apply the Chebyshev polynomial
approximation, this domain should be transformed to [1,1]. Hence, a linear
transformation
x = 0.1Cl (6.12)
is applied to the system of interest, where x is the transformed variable. 6m(Q is
transformed to y(x). The convective diffusion equation in terms of x and y is given by
0.01y" + 30(x + l)2/  60m(x + l)y = 0 (613)
or
0.0 l.y" + 30x2y' + 60xy' + 30yâ€™  60/w(x + l)y = 0 (6.14)
and the boundary conditions are given by
x(l)= 1 andx(l) = 0 (6.15)
The domain of integration as well as the governing differential equation is transformed to
fit into the domain of interest. At this stage series approximations in terms of Chebyshev
polynomials are introduced for various terms involved in equation (6.14).
6,1,1 Series Approximations
Let
y = TJâ€œnTn (x)
n= 0
(6.16a)
176
where aâ€ž represents the associated weighting factor which needs to be determined. Other
terms resulting from (6.14) can also be expressed as summations of Chebyshev
polynomials as given below.
oo
y=Z^Tn (x)
n= 0
(6.16b)
y" =
n= 0
(6.16c)
'V=iv.W
n=0
(6.16d)
xy'= ÃenPn(x)
nâ€”0
(6.16e)
(6.16f)
n= 0
The coefficients a(^ ,a(2\bâ€ž, dn , and eâ€ž need to be expressed in terms of aâ€ž. These
relations can be obtained in a recursive fashion and the formulae for the above mentioned
functions are provided in literature [75], In order to validate the expressions provided in
this reference, these recursion relations were rederived and the relation provided for
x2y' was found to be incorrect. These relations are provided in the following set of
equations including the one corrected for x2y'.
cna? = 2 2>a, (6.17a)
p=n+1
p+nodd
= 2 Tp{p2"2)ap
p=n+2
p+n even
(6.17b)
177
cb_  â€”
(Â» ~ ]K1 + (" + OÃ1 + Cn+dn K+1 + 4
/>=n+3
p+nodd
(6.17c)
cnen =Â«<*,,+2
p=n+2
p+n even
(6.17d)
Cnfn = (Cnl<*nl +Â«n+1)
(6.17e)
where câ€ž0 (Â«<0), Co=2, câ€ž=l(Â«>0), 0). As an illustration,
the derivation of the recursion relation for x2y' is provided in the next subsection.
6.1.2 Recursion Relation for xV
The key to deriving the recursion relations depends upon identifying the equivalent
trigonometric function for the term that is under consideration. Since
x = cos# (6.18)
x can be substituted by cos 6. Also, since
oo
yHLanTn(x) and Tn(x) = cosnG (6. Â¡9)
n= 0
The derivative of Tâ€ž(x) with respect to x yields
n sin nd
r.'W=
sin 6
(6.20)
Similarly, obtaining the derivatives of Tâ€ž(x) and Tâ€ž(x), and application of the principles of
trigonometry yields
C,W cw
n + \
n1
(6.21)
and
178
x2y = JEâ€œnxX W
n1
A 2 .Â« sin n6
= > aâ€ž cos Ã³'
tT sin 6
where
cos 0
2 â€ž sinÂ«0 1
sin#
sin (n + 2)Q 2 sin nO sin(Â«2)#
sin 6 sin 6 sin 0
Hence from equation (6.20)
**tf*)f
KM , 2r;(x)  Tjjxj
n+2 n n2
(6.22)
(6.23)
(6.24)
However, since = ^anTn{x)
n=0
*V = 2>XM
n=0
=iv2TX*)
n=1
_ynaâ€ž
Â¿1 4
n=1 H
From equations (6.21) and (6.25)
co
2>.
n=0
C,(*) , 2r;(Â»)  c,(x)'
Â« + 2 n n2
KM
C,M]
ynan
CiW
, 2rw,
TU*) 1
n + 1
n 1
n=l ^
Â« + 2
n
n2
It should be noted that 7L'm(jc) = 7^(x) and lim â€”^ = lim S*n/â€^ = 0
m>Â° w m>0 sin#
(6.25)
(6.26)
Expansion of these series and regrouping various terms multiplying various Tâ€ž(x) terms
and simplifying yields equation (6.17c).
179
613 Substitution into the Convective Diffusion Equation
Substitution of equations (6.16) into the convective diffusion equation given by
(6.13) yields
0.01Â¿Â«f r.(x) + 30Â¿ b,T,(x) + 60Â¿ e.T.(x)
n= 0 n= 0 n 0
+ 30Â¿a^Tn(x)  60fjn{x) ~ ajn(x)
â€ž n=0 n=0 n=0
(6.27)
In a practical situation the series has to be truncated to a finite number of terms.
Hence, if y is being estimated by a truncated Chebyshev polynomial series with the highest
power of value N (an N Ã¡ degree polynomial), then y" can be estimated only by a
polynomial of degree N2. An algebraic relationship for the unknown coefficients is
obtained from the inner product of Tn(x) and Tp(x) where Tp(x) is defined in accordance
with equation (6.5).
(Tâ€ž (4 Tp (*)) = â€” { Tn (x)Tp (xXl x2Yndx = Snp (6.28)
UCn 1
where Snp= 1 where n = p and is zero when n * p.
By taking the inner product of equation (6.27) with respect to Tp(x)
0.01 a{2) + 306â€ž + 60eâ€ž + 30^  60m{fn + an) = 0 (6.29)
where n = 0, 1,2, , n3, n2. The inner product is taken only for n varying from 0 to n
2. IfyÂ» is an Â«th degree polynomial, then y" is of degree n2.
Coefficients a
recursive relations (6.17), yeilding n1 equations. However, there are n+1 unknowns (a0
to aâ€ž) to be determined. The two additional equations can be obtained from the boundary
180
conditions x(l) = 1 and x(l) = 0. This method of writing the solution as a series even at
the boundary is known as the Tau method. This is different from the Galerkin method in
the sense that the basis functions Tn(x) are not individually required to satisfy the boundary
conditions. Hence,
ZaÂ«rÂ«(_1) = 1 (6.30a)
n=0
1^0) =0 (6.30b)
n=0
From the properties of Chebyshev polynomials, Tâ€ž(1) = (1)â€ and Tâ€ž( 1) = 1. Hence from
(6.30a) and (6.30b)
ZaÂ«(_1)â€=1 (6.31a)
n=0
N
ZaÂ»=0 (6.31b)
n=0
6,1,4 Non Homogeneous Equations
The differential equation considered for this work was homogeneous. Non
homogeneous equations can also be solved by the use of Chebyshev polynomials. The
added term can be expressed as a linear combination of Chebyshev polynomials. For
example
l = T0(x) (6.32a)
x = Tx(x) (6.32b)
*2=\[>;(*)*â– rÂ«Â«]
(6.32c)
181
xâ€™ = I[r3(*)+37;(*)]
x4 = ^[r4(x)+47;(x)+37;(x)]
(6.3 2d)
(6.32e)
and so on. After this inner products can be taken on the right hand side also.
6 2 Results and Discussion
Equations (6.29),(6.3la), and (6.31b) need to be solved for simultaneously to
obtain aâ€ž coefficients and hence the solution. For the purpose of the present study, the
convective diffusion is solved for using the Chebyshev polynomial expansions as well as a
finite difference scheme with varying step sizes for the case of infinite Schmidt numbers. A
comparison is made in order to emphasize the superiority of the Chebyshev polynomial
approximation over the finite difference scheme in case of the present problem. The
parameter that is of practical importance is 0'm(o) as this signifies the rate of mass
transport. This is the parameter that is obtained for this comparative study. The values of
this parameter for m=0, 5, and 10 are presented in Table 6.1, Table 6.2, & Table 6.3
respectively. As it can be seen from these results, the convergence is much faster in case of
Chebyshev polynomials.
From this it was observed that the convergence is much faster by the application of
Chebyshev polynomials. One drawback of this approach is that the convergence criteria
for this approach is not well established. When higher order cross product terms such as
x3y" are involved the resulting algebraic equations are ill conditioned in nature which
results in lack of accuracy of the solution. This problem was encountered when the
approach was applied to solving the convective diffusion equation accounting for finite
182
Schmidt number. However by choosing an efficient numerical scheme for the solution of
the linear system of equations that results from this approach, this method can be used
even for complicated linear differential equations.
6.3 Conclusions
The applicability of Chebyshev polynomials in solving the steady state convective
diffusion equation for a rotating disc electrode was illustrated. For the problem that was
considered in this work, the convergence obtained by using Chebyshev approximation is
found to be very fast in comparison with that from the finite difference scheme.
183
Table 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
No. of terms in
Chebyshev approx.
Value of 0'm (o)
No. of nodes in
FDM scheme
Value of 6'm (o)
11
1.03094588111193
101
1.13553825010888
26
1.11909234991382
1001
1.11988421505475
51
1.11984784688562
1251
1.11986892628523
101
1.11984652169427
2501
1.11985126312467
151
1.11984652172219
5001
1.11984759958694
201
1.11984652172219
10001
1.11984677775223
Table 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
No. of terms in
Chebyshev approx.
Value of 0â€™m (o)
No. of nodes in
FDM scheme
Value of 0'm (o)
11
2.09070385994810
101
2.57827669933820
26
2.34308047335845
1001
2.34499997552230
51
2.34045102991164
1251
2.34339455441446
101
2.34045075950235
2501
2.34120286825501
151
2.34045075950242
5001
2.34064080790497
201
2.34045075950242
10001
2.34049852431772
184
Table 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
No . of terms in
Chebyshev approx.
Value of 0'm{0)
No. of nodes in
FDM scheme
Value of 0'm(6)
11
2.47551374048620
101
3.25312335973968
26
2.90563022533799
1001
2.91044576289627
51
2.90150518991599
1251
2.90731168783412
101
2.90150549289233
2501
2.90299930892293
151
2.90150549289233
5001
2.90188423570831
201
2.90150549289233
10001
2.90160083994728
185
Figure 6.1. Chebyshev polynomials as functions of x.
CHAPTER 7
SPECTROSCOPY APPLICATIONS OF THE KRAMERSKRONIG TRANSFORMS:
IMPLICATIONS FOR ERROR STRUCTURE IDENTIFICATION
It is well established that the use of weighting strategies that account for the
stochastic error structure of measurements enhances the information that can be extracted
from regression of spectroscopic data [15][l 8],[23],[81][86], Hence the independent
assessment of the error structure is needed. The error structure for most radiationbased
spectroscopic measurements such as absorption spectroscopy and light scattering can be
readily identified [87],[88], The error analysis approach has been successful for some
optical spectroscopy techniques because these systems lend themselves to replication and,
therefore, to the independent identification of the different errors that contribute to the
total variance of the measurements. In contrast, the stochastic contribution to the error
structure of electrochemical impedance spectroscopy measurements cannot generally be
obtained from the standard deviation of repeated measurements because even a mild nonÂ¬
stationary behavior introduces a nonnegligible timevarying bias contribution to the error.
Recent advances in the use of measurement models for filtering lack of replicacy have
made possible experimental determination of the stochastic and bias contributions to the
error structure for impedance measurements [711],
The measurement model approach for identification of error structures is widely
used (see, for example, [87] and [88] for applications to optical spectroscopies and [89]
for general application to spectroscopy). The measurement model approach has recently
186
187
been applied to identify the error structures of impedance spectra collected for a large
variety of electrochemical systems [711], Use of measurement models in identifying the
noise level in the impedance measurements for the reduction of ferricyanide is illustrated in
chapter 2. One striking result of the application of measurement models to impedance
spectroscopy has been that the standard deviation of the real and imaginary components of
the impedance spectra were found to be equal, even where the two components differed
by several orders of magnitude. The only exception was found when the data did not
conform to the KramersKronig relations or when the precision of the measurement did
not allow calculation of the standard deviation of one of the components (i.e., all
significant digits reported by the instrumentation for the replicated measurements were
equal [19]). The equality of standard deviations in the stochastic noise will be illustrated
with examples from a number of frequency domain techniques in chapter 8. The objective
of the present chapter is to explore whether the equality of the noise levels in the real and
imaginary parts of electrochemical impedance spectra can be attributed to the manner in
which errors propagate through the KramersKronig relations when both real and
imaginary components are obtained from the same measurement.
7,1 Experimental Motivation
Spectroscopic measurements, which yield complex variables, are illustrated in
hierarchical form in Figure 7.1. Spectrophotometric techniques such as absorption
spectroscopy and light scattering record the light intensity as a function of the wavelength
of the incident radiation used to interrogate the sample. The frequency dependence arises
from the wavelength of light employed. Electrochemical and mechanical spectroscopic
188
techniques employ a modulation of a system variable such as applied potential, and the
frequency dependence arises from the frequency of the modulation. In electrochemical,
mechanical, and some spectrophotometric techniques, both real and imaginary (or
modulus and phase angle) components are obtained from a single measured variable (e.g
the impedance).
To date, the equality of the standard deviations for real and imaginary components
was observed for a number of such systems which are highlighted in Figure 7.1. The
equality of the standard deviation for real and imaginary components was observed for:
â€¢ ElectroHydrodynamic Impedance Spectroscopy (EHD): ElectroHydrodynamic
Impedance Spectroscopy is a coupled mechanical/electrochemical measurement in
which the rotation rate of a disk electrode is modulated about a preselected value.
The impedance response at a fixed potential is given by A//AQ, where I is the current
through the disk electrode and Q is the rotation speed of the disk [10],
â€¢ Electrochemical Impedance Spectroscopy: The database now includes measurements
for electrochemical and solidstate systems under both potentiostatic and galvanostatic
modulation [79], The equality of the standard deviation for real and imaginary
components was observed, even for systems with a very large solution resistance [19],
â€¢ OpticallyStimulated Impedance Spectroscopy: Impedance measurements were
obtained for solidstate systems under monochromatic illumination [90],
The contention that the variances of the real and imaginary parts of KramersKronig
transformable complex variables are equal is supported by experimental evidence and has
been validated statistically, as discussed in a subsequent section.
189
In contrast to results found for electrochemical and mechanical spectroscopies, the
standard deviations for the real and imaginary components of the complex refractive index
from spectrophotometric measurements were found to be correlated but not necessarily
equal [91], However, for measurement of optical properties (e.gthe real and imaginary
components of the complex refractive index) over a sufficiently broad range of
frequencies, different instruments with their particular error structures have to be used;
whereas, in electrochemical and mechanical spectroscopies a single instrument is used to
measure simultaneously both the real and imaginary components.
For electrochemical and mechanical/electrochemical spectroscopies, the
experimental evidence for the equality of the standard deviation of real and imaginary
components is compelling and suggests that there may be a fundamental explanation for
the observed relationship between the noise level of real and imaginary components of the
impedance response.
7 2 Application of the KramersKronig Relations
The KramersKronig relations are integral equations which constrain the real and
imaginary components of complex quantities for systems that satisfy conditions of
causality, linearity, and stability [9295], The KramersKronig transforms arise from the
constitutive relations associated with the Maxwell equations for the description of an
electromagnetic field at interior points in matter. Bode extended the concept to electrical
impedance and tabulated various forms of the KramersKronig relations [95],
An application of the KramersKronig relations to variables containing stochastic
noise was presented by Macdonald, who showed, through a Monte Carlo analysis with
190
synthetic data and an assumed error structure, that the standard deviation of the
impedance component predicted by the KramersKronig relations was equal to that of the
input component [96], The analysis was incomplete because it did not identify correctly
the conditions under which the variances of the real and imaginary components of
experimental impedance data are equal, and the author continued to use a weighting
strategy in his regressions based upon a modified proportional error structure for which
the variances of the real and imaginary components of experimental impedance data are
different [96,98],
The objective of the present work is to identify the error structure for frequency
dependent measurements. Herein an explicit relationship between the variances of the real
and imaginary components of the error is reported without a priori assumption of the
error structure. The only requirements are that the KramersKronig relations be satisfied,
that the errors be stationary in the sense of replication at the measurement frequency, that
the derivative of the variance with respect to frequency exists, and that the errors be
uncorrelated with respect to frequency. In the subsequent section, stochastic error terms
are incorporated into the derivation of KramersKronig relations in order to examine how
the errors for the real and imaginary terms are propagated.
7 3 Absence of Stochastic Errors
The KramersKronig transforms can be derived under the assumptions that the
system is linear, stable, stationary, and causal. The system is assumed to be stable in the
sense that response to a perturbation to the system does not grow indefinitely and linear in
the sense that the response is directly proportional to an input perturbation at each
191
frequency. Thus, the response to an arbitrary perturbation can be treated as being
composed of a linear superposition of waves. The response is assumed to be analytic at
frequencies of zero to infinity. The statement that the response must be analytic in the
domain of integration may be viewed as being a consequence of the condition of primitive
causality, that is, that the effect of a perturbation to the system cannot precede the cause
of the perturbation [99],
The starting point in the analysis is that the integral around the closed loop (see
Figure 7.2) must vanish by Cauchyâ€™s integral theorem [100],
f(Z(x)Zro)dx = 0 (7.1)
where Z is the impedance response and x is the complex frequency. Equation (7.1) yields
(see, for example, [95])
Z/(* (7.2a)
1 n J xeo
o
and
Zr (Â©) = Zr (Â®) +
2d)
n
xZj(x) + a> Zj(o))
2 2
x Ãœ)
dx
(7.2b)
where co is the frequency of interest, r and j in the subscripts refer to real and imaginary
parts respectively. Only the principal value of the respective integrals is considered. The
terms Zr(a>) and Ã¼jZj(cÃ³) in equations (7.2a) and (7.2b), respectively, facilitate numerical
evaluation of the singular integrals, but do not contribute to the numerical value of the
oo
integrals as J â€”â€”dx = 0 for Â©*0 [101],
192
A similar development cannot be used to relate the real and imaginary parts of
stochastic quantities because equation (7.1) is not satisfied except in an expectation sense.
7.4 Propagation of Stochastic Errors
The stochastic error can be defined by
Zob (
where Z0b((a) is the observed value of the impedance at any given frequency co, Z(co),
Zr(co), and Z/o) represent the errorfree values of the impedance which conform exactly
to the KramersKronig relations, and j is the imaginary number V~T The measurement
error s((o) is a complex stochastic variable such that s(co) = sr(co) + je^co). Clearly, at
any frequency co
E(z(o))ob) = Z(co) (7.4)
only if
Â£(ffr(Â®)) = 0 (7.5a)
and
e{sj(co)) = 0 (7.5b)
where E(.) refers to the expectation of the argument. E(x) represents the expected value of
random variable x where by definition
Â£(*)= ^76^
Allx
where P(x) is the probability for the random variable to have a value of x. Equations (7.5a)
and (7.5b) are satisfied for errors that are stochastic and do not include the effects of bias.
193
7 4 1 Transformation from Real to Imaginary
The KramersKronig relations can be applied to obtain the imaginary part from the
real part of the impedance spectrum only in an expectation sense, that is,
(7.7)
o
It is evident from equation (7.7) that, for the expected value of the observed imaginary
component to approach its true value in the KramersKronig sense, it is necessary that
equation (7.5a) be satisfied and that
(7.8)
For the first condition to be met, it is necessary that the process be stationary in the sense
of replication at every measurement frequency. The second condition can be satisfied in
two ways: In the hypothetical case where all frequencies can be sampled, the expectation
can be carried to the inside of the integral, and equation (7.8) results directly from
equation (7.5a). In the more practical case where the impedance is sampled at a finite
number of frequencies, sr(x) represents the error between an interpolated function and the
â€œtrueâ€ impedance value at frequency x. This term is composed of contributions from the
quadrature and/or interpolation errors and from the stochastic noise at the applied
frequency co. In the latter case, equation (7.8) represents a constraint on the integration
procedure. In the limit that quadrature and interpolation errors are negligible, the residual
errors sr(x) should be of the same magnitude as the stochastic noise sr(co).
194
Under the conditions that equations (7.5a) and (7.8) are satisfied and for a given
evaluation of equation (7.7)
* 2(0
Z (co) + e (co) = â€”*
1 i n
f4
(x)  Zr (co) + er (x)  sr (co)
2 2
x  CO
dx +
&
(x)
dx
co
(7.9)
where e (co) represents the error in the evaluation of the KramersKronig relations
caused by the second integral on the righthand side. The variance of the transformed
imaginary variable can be shown by the following development to be equal to the variance
of the real variable. From equation (7.9)
var^Â£ (&>)) =
r2cos
\ n >
N
1 N
1 k= 1
er(x)
J x co J x 
dx
co
(7.10)
where N is the number of replicate measurements which is assumed to be large. Under the
assumption that equation (7.8) is satisfied
f 00
2co 
\ SrjÃX)
n]
1 x2Â«2
dx
(7.11)
As only the principal values of the integrals are considered, it is appropriate to approach
the point of singularity at x=co equally from both sides. In terms of the principal value, for
specific values of k, the integral in equation (7.11) becomes
2co f e,'k(x) 2co [ srJe(x) 2co [ sr,k{x)
\â€”i. 7dx =â€” â€”r ax H â€” dx
7i J x â€”co n J x co n J x co
(7.12)
Under the transformations x=ay in the domain [0,of] and x =co/y in the domain [co+,oo]
(reference [96]),
195
r
2(0 I
f erAX)
n J
1 x2  CO2
dx =
n
(Â£rk(Ãº)y)erk(co Ay))
iy
dy
(7.13)
Equation (7.13) can be expressed in a summation sense as
2
n
{Sr,ÃCÂ°y)Â£rA(01 y))
1 y2
dy = 
r) M
fz
n m= 1
Â£rA(Â°yn)Â£rACÂ°l ym)
1 y\
w(ym)
(7.14)
where Mis large, Ml represents the number of intervals for the domain of integration, or
M is the number of nodes used for numerical evaluation of integrals, and fV(ym) is the
weighing factor which can be a function of the integration procedure chosen. From
equations (7.11), (7.13), and (7.14)
â– \2
Â£rA(Â°yAÂ£rAcÂ°l ym)
iyl
W(ym)
(7.15)
A general expression for the errors is given by Â£rk(x)= Pk{0,\)crr(x) where ^(0,1) is
the k* observation of a symmetrically distributed random number with a mean value of
zero and a standard deviation of 1, and cxr(x) is the standard deviation for the errors
which is representative of the error structure for the spectroscopy measurements and is
assumed to be a continuous function of frequency. Under the assumption that the errors
are uncorrelated with respect to frequency,
M
^J2 (Â®) = Z {am [ym) + Â°V2 (a> Â¡ ym )]}
m= 1
(7.16)
where am is the weighting factor given by
196
a_ =
W{ym)
iyl
(7.17)
In the limit that A/>oo, the trapezoidal rule yields
4 1
a_ =
;r2 (2m1)2
(7.18)
and
2X = 05
m=l
(7.19)
As shown in Figure 7.3, the am coefficients decay rapidly away from_y=l Beyond the first
five terms, the individual contribution of each term is less than 1% of the first term. The
series approaches its limiting value to within 1 percent when 20 terms are used. The error
associated with using a finite number of terms in equation (7.16) can be made therefore to
be negligibly small.
Under the assumption that
can be written in terms of frequency x which, when expressed in terms of the
transformation variable x = coym (valid for x
Â°v2 (*)=
xco
Similarly, for the variable transformation x=Â©/ym valid for x>o,
0â€™r(*)=0â€™r(Â®) +
(7.20)
(daHx(\
f 1 ) Ã
V
f i Y
CO
â€” 1+0
CO
â€” â€” l
\ dx )
xco
V
VTm ).
' J
(7.21a)
In the vicinity of jv=l, equation (7.21a) can be expressed in a form similar to equation
(7.20), that is,
197
(7.21b)
Equation (7.21b) is justified because the major part of the contribution to the integral
occurs within the range of y=\ to roughly ^=11 O'6. The assumption that higher order
terms in the expansion for a2 (x) can be neglected is justified because the region over
which linearization is assumed to apply extends only 1/1000 of the frequency co, e.g., 1 Hz
at a frequency Â© of 1000 Hz.
Substitution of equations (7.20) and (7.21) into equation (7.16) yields
(7.22)
where pÂ«M and accounts for intervals in the vicinity of y=l. Following equation (7.19)
(7.23)
This result is consistent with the results of Monte Carlo calculations for assumed error
structures but was obtained here without explicit assumption of an error structure
[91], [96], The only requirements are that the KramersKronig relations be satisfied, that
the errors be stationary in the sense of replication at each measurement frequency, that the
errors be uncorrelated with respect to frequency, and that the derivative of the variance
with respect to frequency exists.
7 4 2 Transformation from Imaginary to Real
The KramersKronig relations for obtaining the real part from the imaginary part
of the spectrum can be expressed as equation (7.2b), which, in terms of expectations
becomes
198
Â£(Zr(ffl)Zr (Â»)) = Â£
2 fxZ}(x) + aZj(co)xÂ£;(x) +1oej(co)
2 2
x co
dx
(7.24)
Following the discussion in the earlier section, the necessary conditions for Kramers
Kronig transformability become equation (7.5b) and
2
n,
XEj (x)
2 2
x  cy
dx
= 0
(7.25)
The variance of the error in the evaluation of equation (7.24) is given by
v4*)=iÂ¿tÃ
( CO
2_
n
~XÂ£]JC(X)
2 2
X 0)
\2
dx
(7.26)
Under the transformation used in the earlier section,
xÂ£]k(x)
2 2
x  CO
dx 
yÂ£],k(cÂ°y) ,
TT"**
(i/>>)*,.* (o/y) , Ã
iy
>â– 2
â– ^+
*(y2)
ty
(7.27)
where a point of singularity at ym=0 (at Â©=oo) introduced by the transformation x = co/y
was avoided by further subdivision of the integral. Since yi can be chosen such that the
third integral of equation (7.27) has negligible value and that the first integral has a
negligible contribution from the range y=0 to yi, equation (7.27) can be solved by
following a procedure similar to what was adopted in the earlier section. A form similar
to equation (7.15) is obtained,
ym Â£j.k (<Â»ym)  (i / ym )Â£j ,k (Â«/ ym)
In the region of interest (ym> 1),
iyl
My.)
(7.28)
ym
iyl
and
ym{x~yl)
tend toward
iyl
Hence,
199
(7.29)
where pÂ«M and accounts for the intervals in the region of 1. Equation (7.29) is
directly analogous to equation (7.15). Following the discussion in the previous section,
oâ€™2(co) =
(7.30)
Thus, the variance of the real part of the impedance is equal to the variance of the
imaginary part of the impedance, independent of the direction of the transformation, if the
KramersKronig relations are satisfied in an expectation sense. In order for the Kramers
Kronig relations to be satisfied, the conditions stated through equations (7.5), (7.8), and
(7.25) must be satisfied. Equations (7.5a) and (7.5b) represent the usual constraints on the
experimental stochastic errors; whereas, equations (7.8) and (7.25) represent constraints
on the integration procedure.
As summarized in the following section, the theoretical development presented
here is supported by experimental observations for various physical systems that satisfy the
conditions of KramersKronig relations.
7,5 Experimental Verification
The equality of the standard deviations for real and imaginary parts of the
impedance has been observed for the impedance response of solid state systems (GaAs
Schottky diodes, ZnO varistors, ZnS electroluminescent panels) [8,9,14], corrosion of
copper in seawater (under either potentiostatic or galvanostatic modulation) [9,14,69],
electrochemistry at metal hydride electrodes (LaNi5 and Mischmetal) [14,70], the electroÂ¬
hydrodynamic impedance response for reduction of ferricyanide and oxidation of
200
ferrocyanide on Pt rotating disks [10,102], the impedance response of membranes for
which the solution resistance is large [19], and the impedance response of electrical
circuits with a large leading resistance [19], Some examples are illustrated in detail in the
next chapter.
7.6 Implications for the Error Structure
The result of the development presented here is that, for data that are consistent
with the KramersKronig relations in an expectation sense, the standard deviation of the
real part of a complex spectrum at a given frequency must be equal to the standard
deviation of the imaginary part. The development did not require any assumptions
concerning analyticity of the stochastic noise with respect to frequency; therefore, the
result applies to spectra in which data are collected sequentially as well as to spectra in
which a single observation is used to resolve a spectrum, as is done, for example, by
Fourier transformation of transient data.
The implications of this result are illustrated in Figure 7.4, where the real and
imaginary parts of an impedance spectrum are presented as functions of frequency. The
probability distribution function for the data, corresponding to an equal standard deviation
for the real and imaginary parts, is shown at a frequency of 0.03 Hz. The real part of the
impedance at this frequency is roughly 100 O as compared to 3 O for the imaginary part,
and the noise level therefore represents a much larger percentage of the imaginary signal
than the real. This result, which has also been observed in a large number of experimental
systems [711,14,19,69,70,102] is now shown to have a fundamental basis.
201
While the development presented here shows that the standard deviations for the
real and imaginary parts of a KramersKronigtransformable complex quantity are equal,
the instantaneous realizations of the stochastic errors for the respective components has
been shown experimentally to be uncorrelated [9,69], The errors were also found to be
uncorrelated with respect to frequency [9,69], an observation that supports a key
assumption made in the present work. The assumption of the existence of the first
derivative of the variance is supported by the identification of error structures presented in
the figures for the error structures for various systems, presented in the next chapter.
The need to identify an appropriate form for the impedance Z(x) in the presence of
stochastic errors (see, for example, equations (7.9) and (7.24)) supports the use of
measurement models composed of lineshapes that themselves satisfy the KramersKronig
relations. The use of such measurement models is superior to the use of polynomial fitting
because fewer parameters are needed to model complex behavior. Experimental data
seldom contain a frequency range sufficient to approximate the range of integration of 0 to
oo required to evaluate the KramersKronig integrals; therefore, extrapolation of the data
set is required. Measurement models can be used to extrapolate the experimental data set,
and the implications of the extrapolation procedure are quite different than from
extrapolations with polynomials. The extrapolations done with measurement models are
based on a common set of parameters for the real and imaginary parts and on a model
structure that has been shown to represent adequately the observations. The confidence in
the extrapolation using measurement models is, therefore, higher. In addition, as the
lineshapes used satisfy the KramersKronig relations, experimental data can be checked for
consistency with the KramersKronig relations without actually integrating the equations
202
over frequency, avoiding the concomitant quadrature errors [11,69,70,71,98,102], The
correlation and bias errors introduced by quadrature errors are discussed further in [91],
The results presented here also indicate clearly the importance of identifying the nature of
the error structure prior to application of the KramersKronig transforms.
The analytic approach presented here establishes an explicit relationship between
Ur and o} with the only requirement that the errors be stationary in the sense of replication
at each measurement frequency, that errors be uncorrelated with respect to frequency, that
the derivative of the variance with respect to frequency exists, and that the Kramers
Kronig relations be satisfied. In addition, the conditions for the applicability of the
KramersKronig relations to experimental data have been identified, that is, equations
(7.5), (7.8), and (7.25) be satisfied.
The observation regarding the equality of the standard deviations in the real and
imaginary parts of the stochastic noise received some criticism in the recent past based on
the work performed by Macdonald and Pieterbarg [103], Their conclusions are based only
on numerical simulations conducted on synthetic data. However, the result presented in
this chapter is very general in nature and is applicable for all spectroscopic measurements
obeying KramersKronig transforms with real and imaginary components obtained
simultaneously using the same instrumentation. Huge amount of experimental evidence is
available to support this result, some examples of which are presented in chapter 8.
7.7 Conclusions
Knowledge of the error structure plays a critical role in interpreting spectroscopic
measurements. An assessment of the stochastic (or noise) component of the errors allows
203
refinement of regression strategies and can guide design of experiments to improve signal
tonoise ratios. Assessment of consistency of data with the KramersKronig relations is
also important because inconsistencies can be attributed to experimental bias errors that
must be accounted for during interpretation of measurements.
In this work, the experimental observation that real and imaginary parts of the
impedance have the same standard deviation was found to have a fundamental basis for
stationary error structures. The propagation of stochastic errors through the Kramers
Kronig relations in both directions (realtoimaginary and imaginarytoreal) yielded
standard deviations that were equal. Thus, the observations concerning the error structure
of electrochemical, opticalelectrochemical, and mechanicalelectrochemical impedance
spectra should apply as well to purely mechanical spectroscopic measurements.
This work suggests also that evaluation of stochastic errors could provide insight
into the degree of consistency with the KramersKronig relations. The concept that there
exists a relationship between stochastic error structure and bias errors is supported by
repeated experimental observation that the standard deviation of real and imaginary parts
of the impedance were equal except for spectra that were found to be inconsistent with the
KramersKronig relations.
204
V:
m:
P:
Potential
Electrode Mass
Light Intensity
Figure 7.1. Hierarchical representation of spectroscopic measurements. The shaded boxes
represent measurement strategies for which the real and imaginary parts of Kramers
Kronigtransformable 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 KramersKronig relations.
205
A
I
I
I
â–¼
Figure 7.2. Path of integration for the contour integral in the complexfrequency plane.
Normalized Weighting Factor
206
m
Figure 7.3. Weighting factor for Eq. (7.17) as a function of m normalized to show relative
contributions to the integral.
207
Frequency, Hz
Figure 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.
CHAPTER 8
COMMON FEATURES FOR FREQUENCY DOMAIN MEASUREMENTS
Spectroscopy techniques involving measurement of a frequencydependent
complex quantity are commonly used to identify material properties. Spectrophotometric
techniques such as absorption spectroscopy and light scattering record the light intensity
as a function of the wavelength of the incident radiation used to interrogate the sample.
The frequency dependence arises from the wavelength of light employed. Electrochemical
and mechanical spectroscopic techniques employ modulation of a system variable such as
applied potential, and the frequency dependence arises from the frequency of the
modulation. Interpretation of the resulting spectra in terms of material properties usually
requires regression of a process model that describes the physics of the system.
Knowledge of the error structure plays a critical role in interpreting spectroscopic
measurements. A discussion on error structure can be found in chapter 2. Use of
weighting strategies that account for the stochastic error structure of measurements
enhances the information that can be extracted from regression of a model to the
spectroscopic data. The advantage of weighting the regressions with respect to the error
structure is that part of the spectrum where the noise level is high is assigned lesser weight
and part of the spectrum where noise level is low is assigned more weight in the
regressions. This enhances the information content that can be obtained from the
impedance measurements. In addition, through identification of the error structure,
inconsistencies can be attributed to experimental bias errors that might otherwise be
208
209
interpreted as needing refinement of the process model. However, for many spectroscopic
measurements, (e.gimpedance spectroscopy) the long time required to collect a
spectrum interferes with identification of the error structure because even a mild nonÂ¬
stationary behavior introduces a nonnegligible timevarying bias contribution to the error.
The central idea of this chapter is to illustrate the extension of the measurement
model concept from electrochemical impedance spectroscopy to other spectroscopy
measurements. There are several features of impedance spectra that should be seen as well
in other spectroscopy measurements. It was shown in the previous chapter that the
variances of the real and imaginary components of impedance spectra are equal if the
KramersKronig relations are satisfied. This result has been observed experimentally for
electrochemical impedance measurements. It should be observed as well for any
spectroscopy measurement in which a single instrument is used to obtain both real and
imaginary components of a complex quantity simultaneously and for which the system
under study satisfies the constraints of causality, linearity, and stability.
The spectroscopic techniques discussed in this chapter share similarities in terms of
lineshapes which can be used to describe the frequency dependence of the measured
quantity, in terms of the KramersKronig relations which constrain the real and imaginary
parts of the measured quantity, and the error structure. The common features exploited by
measurement model analysis will be supported with experimental data from the
electrochemical impedance spectroscopy (electrochemical), test circuits (electrical),
rheology of viscoelastic fluids (mechanical), electrohydrodynamic impedance spectroscopy
(mechanical/electrical), and acoustophoretic spectroscopy (mechanical/electrical). It is
210
worth noting that the similarities are seen despite the inherently different physical
principles upon which these different techniques are based.
8,1 Similarity in Terms of Line Shapes
Spectroscopic measurements which yield complex variables are illustrated in
hierarchical form in Figure 7.1. In electrochemical, mechanical, and some
spectrophotometric techniques, both real and imaginary components are obtained from a
single measured variable (e.gimpedance).
The measurement model technique was widely used for identification of error
structure associated with spectroscopic measurements [8789], It has been applied to
optical spectroscopy [88], electrochemical [79,11,69,104] and electrohydro dynamic
impedance spectroscopy [10], and recently, to rheology of viscoelastic fluids [105], The
similarity between different spectroscopic techniques is evident from the fact that line
shapes containing the same mathematical structure can be applied to each of the
spectroscopies.
The measurement model [711,69,104] used to determine the error structure of
electrochemical impedance data was given by a generalized Voigt model, that is,
Z(a) = Z0 +
z
(l + jrkco)
(8.1)
Equation (8.1) represents a superposition of line shapes (/. e., a series arrangement of
circuits involving dissipative elements Ak in parallel with oscillating elements with time
constants Tk) that has been applied in the fields of optics, solid mechanics, rheology and in
211
electrochemistry (where, for AC impedance, Akâ€™s are resistances and Z0 is the resistance
associated with ohmic drop).
8.2 Similarity in Terms of Transfer Function
In this section, the similarity between various frequency domain techniques is
established based on the transfer function used to describe the frequency response of the
systems considered.
8 2.1 Electrochemical Impedance Spectroscopy
The Voigt model as used in electrochemistry is presented in Figure 8.1a. The
currentvoltage relationship for a resistor is given by
(8.2)
where 1 is the current, R is the resistance, and V is the potential drop across the resistor.
The corresponding relationship for the capacitor is
/cÂ£
dt
(8.3)
where C is the capacitance. The response of the Voigt model (Figure 8. la) to a sinusoidal
current or potential perturbation is given in terms of impedance Z = AV / A/ by (8.1).
8,2.2 Rheology of Viscoelastic Fluids
The rheological model related to the Voigt model used in electrochemistry is the
KelvinVoigt model presented in Figure 8.1b [106], The stressstrain relationship for a
spring is given by
S = Gy
(8.4)
212
where S is the stress, G is the Hookian spring constant, and y is the strain. The
corresponding relationship for the dashpot is
S=rjy
(8.5)
where ij is the viscosity for the dashpot and y is the rate of strain (e g., dv/dy). The
response of the KelvinVoigt model shown in Figure 8. lb to a sinusoidal stress is given by
â€¢'<Â®)7r+It
<^0 k *
1/ Gu.
+ jco\
(8.6)
where J(co) is the complex compliance of the system, Gk is the Hookian spring constant for
spring k, and Ak is the time constant given by Ak= rji/Gk. The complex compliance is
related to the complex viscosity rj by
J =
COT]
(8.7)
The lineshape represented by equation (8.1) can be applied to the complex viscosity as
well as the complex compliance.
8 2 3 Optical Spectroscopy
The form of equation (8.1) applied to optical spectroscopies is the Debye model,
which represents a special case of the Lorentz oscillator [87], The Lorentz oscillator treats
charged particles in the medium as harmonic oscillators;
s{.cd) = e0 +X
0)
P.k
TG>lk
(8.8)
where e is the dielectric permittivity and So, co2pk, co2k and Yk are adjusted parameters.
The natural frequency for the undamped oscillator is co2k and copk is proportional to the
concentration of charged particles per unit volume. The damping constant Yk depends on
213
the width of the ground state which is narrow, and the width of state k, which depends on
the transition probabilities to all other states. Summation over the different oscillators
accounts for multiple discrete energy levels, and provides agreement with quantum
mechanical results.
The Drude model applies to free electrons (metals) and appears as a special case of
the Lorenz oscillator for which the spring constant of the oscillator is equal to zero:
*(Â®) = *b ~Z'
co
p,k
â€ž (8.9)
k G> +J0}/k
Combined models could, in principle, be developed to account for bound and free
electrons. An example of a combined model is given by
s(co) = Â£0Yj
CO
p.â€˜
CO
p,k
7 co2 + jcoye , col  0)2 ~ j^k
(8.10)
The Debye model accounts for a linear response of a charged particle to a timedependent
field. The multiple Debye model is given by
A.
f(iÂ») = f0+Slâ€”7
Ir 1 /
JCOTk
(8.11)
where A* describes the maximum value of the real part of the function (seen at co^0) and
Tk is the characteristic time constant for element k. The Debye model could be regarded to
be a low frequency limit to the Lorenzian model. Equation (8.1) can be written as
co,
P.k
Â£(â€œ>) = Â£0 +Z
Kk 0)2
k ljoy
Yk
(8.12)
(0o,k  (0
which approaches equation (8.11) under the assumption that col ))
214
8 2 4 Acoustophoretic Spectroscopy
Acoustophoretic spectroscopy capitalizes on two types of electroacoustic effects.
One is the generation of sound waves by an alternating electric field. As an applied
electric field is alternated, the polar attractions of the charged particles will oscillate, that
is, the particles will at one moment be attracted to one pole and in the next moment be
attracted to the opposite pole. This particle movement creates pressure disturbances,
although small, which propagate out from the surface. The second electroacoustic effect is
the generation of an alternating electric field by the application of an ultrasonic beam. The
propagating sound waves impart motion to the particles and surrounding liquid. The
relative motion between the phases creates alternating dipoles, the sum of which creates a
detectable alternating electric field.
The acoustosizer can measure both types of electroacoustic effects although the
instrument is usually operated to measure the generated sound waves resulting from an
alternating applied electric field, as the signal to noise ratio is larger for this mode of
operation. The electroacoustic signal is related to the size and charge of the suspended
particles. The determination of particle size and charge is a two step process in which the
dynamic mobility is first calculated and subsequently the size and shape are determined.
There exist multiple techniques for the determination of these properties however all are
limited to systems of rather dilute suspensions. Acoustophoresis has the advantage in that
it is possible to measure electroacoustic effects in suspensions of any concentration [107,
1HX], Acoustophoresis data collected [109] for a suspension of polyacrylic acid, 0.062g/L,
pH of 10 and molecular weight 5,000 (Polyacrylic Acid) is analyzed by the measurement
model to support the presence of the common features in acoustophoretic spectra.
215
8 3 Similarity in Terms of the KramersKronig Relations
The KramersKronig transforms, as discussed in chapter 7, are integral equations
that constrain the real and imaginary components of complex quantities for systems that
satisfy conditions of causality, linearity, and stability [9295],[99],[110], These transforms
arise from the constitutive relations associated with the Maxwell equations for description
of an electromagnetic field at interior points in matter. A contour integral approach can
also be used to derive the KramersKronig relations [95], In effect, the KramersKronig
relations constrain the complex properties associated with propagation of a wave through
matter under assumptions of primitive causality, linearity, and stability [99], As the
constraints required for satisfaction of the KramersKronig relations are not very stringent,
the KramersKronig relations can be expected to apply to each of the spectroscopic
techniques described here.
8,4 Similarity in Terms of Error Structure
In addition to similarities in various aspects shared by a number of transfer
functions, a significant similarity exists in terms of the stochastic error structure of the
measurements.
The error structure for most radiationbased spectroscopic measurements such as
absorption spectroscopy and light scattering can be readily identified [87,88], The error
analysis approach has been successfully used for some optical spectroscopy techniques
because these systems lend themselves to replication and, therefore, to the independent
identification of the different errors that contribute to the total variance of the
216
measurements. In contrast, the stochastic contribution to the error structure of
electrochemical impedance spectroscopy measurements cannot generally be obtained from
the standard deviation of repeated measurements because even a mild nonstationary
behavior introduces a nonnegligible timevarying bias contribution to the error. Recent
advances in the use of measurement models for filtering lack of replicacy have made
experimental determination of the stochastic and bias contributions to the error structure
for impedance measurements possible [710,69,104],
The generalized measurement model increases stepwise the number of line shapes
used to regress the experimental data until a maximum is obtained. The data are regressed
using a weighted least squares technique which minimizes the residual errors in the real
and imaginary components of the measurement according to
r,k
(8.13)
where Zr,k and ZJfk are the real and imaginary components of the klh data point, respectively
and Zr k and Zj k are the corresponding model values. Once the model is regressed to fit
the data, residual errors are calculated as
^ ^ ^residual ^lof ^bias ^stochastic
(8.14)
where Z is the experimental value and Zis the model value. The residual errors are
composed of errors from lack of fit, lof, systematic bias and stochastic noise. Lack of fit
errors are minimized by maximizing the number of line shapes to obtain a good fit to the
experimental data. Specific criterion for a â€œgood fitâ€ cannot be explicitly stated, as the
217
quality of a fit depends on the noise level of the measurement itself [9], One can say that a
fit is â€œgoodâ€ if the resulting residual errors are within the noise level of the data.
One striking result of application of measurement models to impedance
spectroscopy has been that the standard deviation of the real and imaginary components of
the impedance spectra were found to be equal, even where the two components differed
by several orders of magnitude. The only exception was found when the data did not
conform to the KramersKronig relations or when the precision of the measurement did
not allow calculation of the standard deviation of one of the components (i.eall
significant digits reported by the instrumentation for the replicated measurements were
equal [19]). In chapter 7 it was shown that, in addition to constraining the values of
complex properties, the KramersKronig relations constrain the error structure such that
the variances of the real and imaginary components are equal.
The experimental support that is presented in the next section illustrates the
common features shared by various spectroscopic techniques in terms of the applicability
of measurement models and in terms of the measured error structure.
8 5 Experimental Results and Discussion
8,5,1 Electrochemical Impedance Spectroscopy
Potential was controlled by a Solartron 1286 potentiostat and driven by a
Solartron 1250 frequency response analyzer. Impedance measurements were collected
frequency by frequency from high frequency to low. The longintegration feature of the
frequency response analyzer was used, which terminated measurement at a given
frequency when a 1% closure error was achieved on the measured channel (potential when
218
galvanostatic modulation is employed and current when potentiostatic modulation is
employed).
Typical results are presented in Figure 8.2a for the impedance response of a Pt disk
electrode rotating at 120rpm and at 1/4* of the limiting current for reduction of
ferricyanide in a 1M KC1 supporting electrolyte at 25Â°C. Impedance measurements were
conducted under potentiostatic modulation. The open circles and triangles represent the
standard deviation of the stochastic noise in the repeated impedance measurements. The
measurement model approach described by Agarwal et al. was used to filter lack of
replicacy [9], The real and imaginary parts of the impedance are seen to have the same
standard deviation, even at frequencies where the two components differ by over one
order of magnitude. Data collected at certain frequencies were found to be corrupted in
the frequency scans collected for this work. The very high frequency end point in each
spectrum was eliminated for the data analysis and subsequent regressions, as this point
was found to be effected by startup transients associated with the measurement. These
impedance data were collected at the CNRS, Paris and the in line frequency for the power
supply is 50Hz. During the data analysis it was observed that the data collected close to
50Hz and 100Hz (a harmonic of 50Hz) were found to be outliers. The measurement
model approach is very sensitive to outliers and it was observed that the inclusion of the
data points close to these two frequencies resulted in significant distortion of error
structure. Hence, the points with in Â±5Hz of these two points were eliminated for the data
analysis conducted.
In order to verify the equality of standard deviations of the stochastic noise in the
measurement a statistical Ftest was conducted, the results of which are presented in
219
Figure 8.2b and Figure 8.2c. The Ftest involves calculating the ratio of the variances of
the two quantities of interest, in this case the real and the imaginary parts, and verifying
the bounds of this ratio over a number of frequencies. This ratio is expected to be
randomly scattered around a value of 1. In Figure 8.2b the circles represent the ratio of
standard deviations of the real and imaginary parts the experimental data without using
any filtering technique. 28 replicates were used for this case. These ratios are either
extremely high or extremely low for most of the frequencies and there is a definite
trending as the bias errors are not filtered. The â€˜xâ€™s represent the ratio of the variances of
the stochastic errors after the bias errors are filtered and the dashed lines represent the
confidence limits for the Ftest. In this case the two data points that fall within Â±5Hz of the
50Hz and 100Hz points are not deleted. This caused visible trending especially in
intermediate to high frequency ranges. The ratio calculated in this case is not randomly
scattered around 1. Figure 8.2c is for the case where these two high frequency end points
are deleted. In this case the errors are randomly scattered around 1. However, one may
question the equality of the standard deviations since for some frequency ranges the points
are scattered beyond the confidence limits. Observation of Figure 8.2a reveals that the
standard deviations of the stochastic noise for parts of the frequencies are less than 3
orders of magnitude as compared to the parameter that is being measured and only 5
decimal places of the real and the imaginary parts were reported by the instrument. This
may lead to uncertainties in the measured errors. A quantitative way of determining
whether the equality of standard deviation is observed is by conducting a twotailed /test.
As the ratio of the variances is expected to be 1, the value of log(crr / a, )2 is
expected to be randomly scattered around a mean value of 0. A histogram for the relative
220
frequency of occurrence of the value of log(crr /
test parameter which is given by (x  ju)/(sx/Vw), where X is the sample mean, that is,
the mean value of log(crr /
mean, sx is the sample variance, and n is the number of available data points, in this case
74 frequency points. The /test parameter in this case yielded a value of 0.63. This is very
well acceptable for the hypothesis that the ratio of variances in the real and imaginary parts
of the stochastic noise is equal to 1, or the standard deviations of the stochastic
components are equal, as this value is within 1.99, the confidence limit for ttest for a level
of significance a=0.05. In this case it can be seen that the equality of standard deviations is
satisfied.
8,5,2 Test Circuit
In the initial stages when the equality of standard deviations was observed for the
electrochemical systems, in order to convince ourselves that this is a fundamental property
of the systems under consideration due to the applicability of KramersKronig relations,
we sought verification by analyzing test circuit measurements conducted by T. El
Moustafid of CNRS, Paris [19], This system is very ideal as there is no timedependent
bias contribution, in other words the system is not evolving with time. Impedance
measurements were collected under galvanostatic modulation at the opencircuit condition
for a circuit composed of a resistor Ro in series with a parallel resistor/capacitor
combination, RiCi. Potential was controlled by a Solartron 1186 potentiostat and driven
by a Solartron 1250 frequency response analyzer. Replication was achieved for these
221
measurements so that the standard deviation of the real and imaginary parts of the
impedance could be calculated directly, without use of the measurement model.
The data obtained are presented in Figure 8.3a. The ratio Ro/Ri was equal to 10,
which meant that the real part of the impedance was at least 20 times larger than the
imaginary part of the impedance at all frequencies and was almost 2,000 times larger at the
lowest frequency measured (0.1Hz). The significant departure from RC circuit behavior
seen at frequencies above 2,000Hz was attributed to potentiostatic limitations. The data
at frequencies above this point were found by the methods discussed in chapter 2 to be
inconsistent with the KramersKronig relations.
In spite of the three orders of magnitude difference between the real and imaginary
parts of the impedance, the standard deviations of the repeated measurements were
indistinguishable except in two frequency regimes. The standard deviations were found to
be different in value in the higher (greater than 2,000Hz) frequency range where
instrumental limitations affected the measurement. The standard deviations were also
found to be different in value in the frequency range between 10 and 200Hz. In this
frequency range, the calculation of the standard deviation of the real part was constrained
by the 5 digits reported by the FRA. The dashed line in Figure 8.3a represents a value
corresponding to 0.5 parts in 105. This line drops an order of magnitude at the frequency
of 180 Hz where the measured real part of the impedance changes from 1000.0 to 999.99
ohms. For data points marked by an x, all 5 digits for six repeated measurements were
identical, and a standard deviation of 0 was calculated. In other words, the standard
deviations for the real and the imaginary parts of the impedance were equal unless the
number of significant digits reported by the FRA were inadequate to calculate ar or unless
222
potentiostatic limitations influenced the result. The results of a statistical Ftest are
presented in Figure 8.3b. The equality of standard deviations was verified by conducting a
ttest as illustrated in Figure 8.3c.
8,5,3 Electrohydrodynamic Impedance Spectroscopy
In the usual application of electrochemical impedance spectroscopy, a complex
impedance is calculated as the ratio of potential to current under a small perturbation of
current (galvanostatic regulation) or potential (potentiostatic regulation). The impedance
is measured as a function of the frequency of the perturbation, and regression of models to
the resulting spectra yields values for physical properties. In recent years generalized
impedance techniques have been introduced in which a nonelectrical quantity such as
pressure, temperature, magnetic field, and light intensity is modulated to give a current or
potential response [2,3], Electrohydrodynamic Impedance Spectroscopy (EHD) is one
such generalized impedance technique in which sinusoidal modulation of disk rotation rate
drives a sinusoidal current or potential. The technique has been applied to surface or
electrode processes that are under mass transport control, and it has been used to obtain
diffusivities of ionic species by the determination of the Schmidt number (Sc=v/D)
[10,54,110122],
Electrohydrodynamic impedance experiments are performed by modulating the
rotation speed fi of the rotating disk electrode with a fixed amplitude of perturbation AO
around a mean rotation speed of fio The perturbation frequency Â© is varied to obtain the
widest possible range. The angular velocity can thus be written as
fi = fi0 + Aficos(iyf) (8.15)
The resulting current /, at fixed potential, can be expressed as
223
I = 70 + AI cos(cot + 0) (816)
where 70 is the average current, AI is the amplitude of the sinusoidal current response, and
(/> is the phase shift. The EHD impedance, given by
M
~ AO
(8.17)
has the structural form of an admittance. Typical EHD data are presented in Figure 8.4 as
a function of frequency. The measurement model approach was used to extract values for
the standard deviation of repeated measurements, given as open symbols in Figure 8.4.
The open circles and triangles represent the real part and the imaginary part
respectively of the standard deviation. The corresponding closed symbols represent the
standard deviations of the repeated measurements. The line in Figure 8.4 corresponds to a
model for the error structure given by
a, = a. = a
i lzl2
ZJ+/)ZrZj + r^L +
(8.18)
where a, /?, y and 8 are parameters determined by regression. The parameter Rm accounts
for the currentmeasuring resistor, and Z* is the highfrequency asymptote for the real part
of the impedance. For this case y=0 and ZÂ»=0. The standard deviations of the real and
imaginary parts of the noise are equal, a result that is consistent with observations made in
the previous section for EIS. The results presented in Figure 8.4a also suggest that the
noise level becomes a significant fraction of the signal at higher frequencies, a
characteristic that influences the regression of models to EHD data. The ratio of variances
of the real and imaginary parts of the noise and corresponding Ftest limits are presented
224
in Figure 8.4b and the equality of standard deviation using ttest is illustrated in Figure
8.4c.
8,5,4 Rheology of Viscoelastic Fluids
The experiments were performed using a parallel plate rheometer [121], In the
parallel plate system, the lower plate is forced to oscillate sinusoidal and the other is
stationary. The amplitude of the oscillation is given by [106], [122]
0 = 0osin(
The torque needed to hold the lower plate in a fixed mean position is measured as
T  T0 sin(ÃyÂ¿ + a) (8.20)
Under the assumptions that inertia can be neglected, the free surface is cylindrical with
radius R, edge and surface tension effects can be neglected, and 0O is small, the complex
viscosity has components
ti'(co)
2HT0 sin(a)
7zR4coÂ®0
and
rj"(co)
2HT0 cos (a)
nR4
(8.21a)
(8.21b)
for the real and imaginary parts, respectively, where H is the gap height and a is the
measured phase lag between T and 0.
An example is given in Figure 8.5a for the complex viscosity of high density
polyethylene melt. The open circles and triangles represent the real part and the imaginary
part respectively of the standard deviation of the complex viscosity. The corresponding
closed symbols represent the standard deviations of the repeated measurements. The line
225
given in Figure 8.5 a is the fitted model for the error structure which is the same as
presented as equation (8.18) with the exception that Rm= 1 and Zco=0. The ratio of
variances of the real and imaginary parts of the noise and corresponding Ftest limits are
presented in Figure 8.5b and the equality of standard deviation using Â¿test is illustrated in
Figure 8.5c. This result is consistent with observations made in the previous sections for
EIS and EHD
8,5,5 Acoustophoretic spectroscopy
Acoustophoretic spectra were collected 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 and the complex
data thus obtained was analyzed using measurement model approach. More details
regarding the acoustophoretic spectrscopy technique and the data analysis can be found in
reference 108. The standard deviations in the stochastic noise obtained through this
analysis are presented in Figure 8.6a and the Ftest results are presented in Figure 8.6b. In
case of this data the ttest results indicated that the equality of standard deviations in the
real and imaginary parts is not satisfied. For analyzing this spectrum only 13 data points
are available. However a more complete spectrum would have resulted in data that would
have yielded to the hypothesis.
8,6 Conclusions
Spectroscopic techniques share similarities in terms of lineshapes which can be
used to describe the frequency dependence of the measured quantity, in terms of the
KramersKronig relations which constrain the real and imaginary parts of the measured
quantity, and the error structure, which is shown here to share common features.
226
Knowledge of the error structure plays a critical role in interpreting spectroscopic
measurements. Use of weighting strategies that account for the stochastic error structure
of measurements enhances the information that can be extracted from regression of a
model to spectroscopic data. In addition, through identification of the error structure,
inconsistencies can be attributed to experimental bias errors that might otherwise be
interpreted as needing refinement of the process model.
227
(a)
Figure 8.1. Lineshape models yielding the same mathematical structure for spectroscopic
response: a) Voigt model for electrochemical systems; b) KelvinVoigt model for rheology
of viscoelastic fluids.
228
co, Hz
Figure 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) Ftest parameters. The inner dashed lines represent
the 95% confidence limits for the Ftest parameter and the outer lines represent the 99%
confidence limits. Circles represent the Ftest parameters for the raw standard deviations,
(c) Ftest parameters after deleting the point close to 50Hz and 100Hz. (d) Histogram
with /test results.
229
0.01 1 100 10000
co, Hz
Figure 8.2â€”continued
230
0.01 1 100 10000 1000000
Figure 8.2â€”continued
Frequency, Hz
1.2
1
9$ 0.8
0.6
o
O"
0
0
03 0.4
0
o:
0.2
0
2
J
1
0
log(crr/crj):
Figure 8.2â€”continued.
T
I' 'i I i I i I i I r i i i i
tTest parameter = 0.63
compare to 1.99 (a=0.05,n=74)I
K>
U)
1
2
232
Frequency, Hz
Figure 8.3. (a)The impedance response obtained under galvanostatic modulation for a
parallel R1C1 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) Ftest corresponding to the variances of stochastic noise (c)
Histogram with ttest results corresponding to the variance of stochastic noise
233
co, Hz
Figure 8.3â€”continued.
>
_CÃœ
CD
O'
3210 1 2 3
logÃCTf/a,)2
Figure 8.3â€”continued.
234
235
Frequency, Hz
Figure 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 electrohydrodynamic 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 Ftest to
verify the equality of standard deviations in the stochastic noise (c) Histogram with /test
results corresponding to the variance of stochastic noise
236
0.01 0.1 1 10 100
CD, Hz
Figure 8.4â€”continued
>
o
c
0
cr
0
1â€”
LL
0
>
0
0
3210123
logÃcj/aj)2
Figure 8.4â€”continued.
237
238
0.001 0.01 0.1 1 10 100
Frequency, Hz
Figure 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) Ftest corresponding to the variances
of stochastic noise (c) Histogram with /test results corresponding to the variance of
stochastic noise
239
Frequency, Hz
Figure 8.5â€”continued.
3
o
0
LL
0
>
0
0
3210123
log(ar/o,)2
Figure 8.5â€”continued.
240
241
co, MHz
Figure 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) Ftest corresponding to the variances
of stochastic noise (c) Histogram with /test results corresponding to the variance of
stochastic noise
242
0.1 1 10 100
to, MHz
Figure 8.6â€”continued.
TO
0
a:
3210123
logÃa/aj)2
to
u>
Figure 8.6â€”continued
CHAPTER 9
CONCLUSIONS
The influence of the current and potential distributions as well as that of the
surface phenomena on the interpretation of impedance data was presented in this thesis. A
model system of potassium ferri/ferrocyanide reacting on the surface of a platinum
rotating disk electrode was chosen for the purpose of this study. Experiments were
conducted in the steady state domain in order to determine the best possible surface
treatment. Repeated electrochemical impedance measurements were made to facilitate the
study of the stochastic error structure. In all the cases it was observed that the standard
deviations in the real and imaginary parts of the stochastic noise were equal. This result
was proved to be a direct consequence of the applicability of KramersKronig transforms
to this system. The equality of standard deviations was also observed for a number of
other frequency domain techniques and the results were reported in this thesis.
The frequency domain model developed by Tribollet and Newman was used to
determine the Schmidt number of the ferricyanide ions using the measurement model
regression strategy that employs the stochastic error structure weighting. It was observed
that the Schmidt number increased with the rotation speed of the electrode. Also, when
the repeated measurements were made with no surface polishing between the
measurements, it was observed that the Schmidt number increased with the duration of the
experiment. This was a strong evidence for the blocking phenomena.
244
245
An impedance model that accounts for the radial dependence of current,
concentration, and overpotentials was developed in order to account for the nonÂ¬
uniformity related issues in the frequency domain. From this model it was observed that
the nonuniform distributions become more significant at high rotation speeds as well as at
high frequencies. A comparison is made between the onedimensional model that was
developed earlier and the twodimensional model developed for this work. Significant
differences were obtained between the result generated using the two models, especially
for cases where nonuniform distributions are more prominent.
From this work it could conclusively be stated that both nonuniform distributions
as well as surface phenomena should be accounted for in interpreting the impedance data.
CHAPTER 10
SUGGESTIONS FOR FUTURE WORK
There are numerous significant contributions in the broad area of electrochemical
engineering made during the course of this work. This work leads to some new exciting
research problems. For example, the frequency domain model developed for this work
should be reformed in order to facilitate the regression of the experimental data to
determine the Schmidt numbers. A twodimensional model accounting for the distributions
in case of electrohydrodynamic impedance conducted below the mass transfer limitation
can be developed based on the model presented for this work. The boundary conditions
can be modified in order to account for the surface passivation reactions occurring in
systems such as the corrosion of copper in seawater environment. It would be interesting
to explicitly account for the charge distribution within the diffuse part of the double layer
to study how this will influence the result, though the impact may not be very significant.
An explicit treatment for the surface blocking phenomena in the model would be an
interesting and challenging problem.
With respect to the statistical aspects, an error structure should be obtained for
transient measurements. Propagation of these errors when the transient data is
transformed into the frequency domain would be an interesting study.
246
APPENDIX A
STEADY STATE MODEL FOR THE ROTATING DISK ELECTRODE
Q ********************************************************
PROGRAM DDLSS3DIS
C
C Program written by Madhav Durbha and Mark E. Orazem, documented
C into this final form on 11th June 1998. The mathematical model
C on which this program is based, is presented in detail in
C The Journal of the Electrochemical Society (June 1998), Vol. 145,
C No. 6, pp 19401949.
C
C Program to simulate the steady state current distributions for a
C rotating disk electrode system. The model on which the computer
C program is based explicitly accounts for the effects of mass
C transfer, kinetics, and ohmic drop within the bulk of the
C solution. This model also exclusively accounts for charge
C distribution in the diffuse part of the double layer and a finite
C schmidt number correction is incorporated into the solution for
C the convective diffusion equation.
C
C This program outputs the radial distributions of current, potential,
C concentration, charge distribution, and double layer cpacitance
C on the surface of the disk electrode.
C Details of various branches of the program are explained in individual
C subroutines. In order to obtain accurate solutions, all the calculations
C were performed using double precision through out the program.
C DDLDATA is the input file, and VOLTS, AMTHP, and SSOUT are the output
C files.
C
INTEGER K,NMAX,MMAX,NO,IZ,IDRAMAX,I,L,IT,ITTOL,NL,LO
C
PARAMETER(LDRAMAX=3 0)
C
C IDRAMAX is the variable corresponding to the number of radial positions
C used for the calculations performed. These dimensionless radial
C coordinates are generated to be the abscissa for the GaussLegendre
C quadrature.
C
247
248
DOUBLE PRECISION CINF, CCENT, AL(20), Q(20,20), XMPR(20), TEMP, T,
+ UPR(20), PI, XN, R, OM, RO, XNU, D, A, F, B(20), XILIM, ALPHA, BETA,
+ XKINF, XN1, XN2, XN3, ETA(IDRAMAX), Cl, XIO, AMTOL, XJIO, GA,
+ PHI(IDRAMAX), V, SUMP(IDRAMAX), AM(20), csurg2(IDRAMAX),
+ XIDEN(IDRAMAX), XN4, CSURGl(IDRAMAX), CSUR(IDRAMAX),
+ RAMTOL, AMST(20), SUMRAM, W2, XINT(20,20), XLEGP,
+ WE(IDRAMAX), DRA(EDRAMAX), RERAM(IDRAMAX), XI, X2, error,
+ XIAV, XJ, SC, X, ETAC(IDRAMAX), ZETA(IDRAMAX), PERM, Y2,
+ CPOT(EDRAMAX), CFERRO(IDRAMAX), CCL(IDRAMAX),
+ CFERRI(IDRAMAX), ETAST(IDRAMAX), Q2(EDRAMAX), STEP, FAKEO,
+ FAKE1, FAKE2, INVCAPD, CAPD(IDRAMAX), Q2AV
C
OPEN(UNIT= 13 ,FILE='DDLD AT A', STATUS='UNKNOWN')
READ(13,38)XNU
38 FORMAT(39X,F19.16)
C
C XNU is kinetic viscosity in (cm)**2/sec
C
READ(13,41)D
41 FORMAT(39X,F19.16)
C
C D is the diffusion coefficient in (cm)**2/sec
C
READ (13,23)LO
23 F0RMAT(33X,I1)
IF (LO EQ. 0) THEN
SC = XNU/D
X = 1/(SC)**(1.0D0/3.D0)
PRINT *,'THE CALCULATIONS ARE DONE WITH FINITE SCHMIDT
+ NUMBER CORRECTION'
ELSE
PRINT *,'THE CALCULATIONS ARE DONE WITH INFINITE SCHMIDT
+ NUMBER ASSUMPTION'
X = 0.D0
END IF
NL = IDRAMAX
XI = 0.0D0
X2 = LODO
C
CALL GAULEG(X1,X2,DRA,WE,NL)
C
C This is the subroutine that is used for generating the necessary
C abscissas and weighting factors used for the GaussLegendre quadrature
C integration procedure.
o o o o o onon
249
NMAX = 11
NMAX is the maximum value of N+l used in Q(N,M)in the steady state
model.
MM AX =11
MMAX is the maximum value of M+l, M is the degree of the polynomial
used to estimate the concentration of the reactant as a function of
the normal distance from the surface of the disc electrode.
UPR(1) = 1.119846522026D0+0.333723493681D0*X
+ +0.063065495796D0*X**2
+ 0.024831364762*X**30.100924424861 *X**4
+ 0.153916634180*X* *50.361110032914*X**6
UPR(2) = 1.532987928427+0.348508168763 *X+0.056398738367*X* *2
+ 0.012443161467*X**30.055731225137*X**4
+ 0.075985761086*X* *50.184336630530*X* *6
UPR(3) = 1.805490583604+0.3 51502095496*X+0.0501466003 84*X* *2
+ 0.008443399975*X**30.036778525256*X**4
+ 0.046258489551*X**50.103363494337*X**6
UPR(4) = 2.015723734147+0.352526039999*X+0.045670783752*X**2
+ 0.006567389500*X**30.027072177649*X**4
+ 0.032082498207*X* *50.064688069153 *X* *6
UPR(5) = 2.189982747231+0.352984907151*X+0.042360393354*X**2
+ 0.005472434784*X* *30.021316136406*X* *4
+ 0.024033 824753 *X* *50.044058591888*X* *6
UPR(6) = 2.340450747257+0.3 53227157058*X+0.039800253199*X* *2
+ 0.004743049979*X**30.017577424448*X**4
+ 0.018741170708*X**50.032338077692*X**6
UPR(7) = 2.473842753935+0.353369826902*X+0.037746320639*X**2
+ 0.004219897577*X**30.014925747091*X* *4
+ 0.015256845657*X* *50.024676089933 *X**6
UPR(8) = 2.594287242379+0.353460658362*X+0.036050283712*X**2
+ 0.003822713947*X* *30.012953 807670*X* *4
+ 0.012811642639*X* *50.019394632191 *X* *6
UPR(9) = 2.704520850202+0.3 53 521950785*X+0.034617693958*X* *2
+ 0.003 510623955 *X**30.011410194860*X**4
+ 0.011135715480*X**50.015353967373*X**6
UPR( 10)= 2.806460252572+0.3 535652153 82*X+0.0333 85436778 *X* *2
+ 0.003255643179*X**30.010205811988*X**4
+ 0.009703 503 554*X* *50.012705306833 *X* *6
UPR(11)= 2.901505452808+0.353596877262*X+0.032309432319*X**2
+ 0.00303 8288308*X* *30.009267906946*X* *4
+ 0.008414271284*X**50.010937287457*X* *6
250
C
C One dimensional array UPR is the required value that is derivative of
C THETA(M) at the surface of the disc electrode. The coefficients for
C the polynomial expansion are provided by the polynomial regressions
C performed as described in the model.
C
PI = 3.141592654D0
C
C One dimensional array MPR is used to store the value of M(2N)PRIME.
C
CALL INTEGRAL(MMAX,NMAX,XINT)
CALL MPRIME(NMAX,PI,XMPR)
CALL CALCQ(MMAX,NMAX,PI,UPR,XMPR,XINT,Q)
C
C CINF is the concentration of the species in the bulk of the solution.
C
READ(13,35)R0
35 FORMAT(39X,F 19.16)
C
C R0 is the radius of the disk electrode in cm
C
READ(13,37)OM
37 FORMAT(39X,F19.16)
OM = OM*PI/30
C
C OM is the angular speed of the disk electrode in rad/sec
C
READ(13,40)A
40 FORMAT(39X,F 19.16)
C
C A is coefficient in series for the expansion of velocity terms
C
READ(13,42)NO
42 FORMAT(39X,I2)
C
C NO is the number of equivalents
C
READ(13,43)IZ
43 FORMAT(35X,I2)
C
C IZ corresponds to the number of equivalents
C
READ(13,44)F
44 FORMAT(39X,F 19.16)
C
251
C F is the Faraday's constant in Coulomb/equivalent
C
READ(13,45)CINF
45 FORMAT(3 9X,F 19.16)
C
C CINF is the concentration of the species at infinite distance in moles/c.c.
C
READ(13,46)R
46 FORMAT(39X,F19.16)
C
C R is universal gas constant in J/mol./Kelvin.
C
READ(13,47)TEMP
47 FORMAT(3 9X,F 19.16)
C
C TEMP is the temperature of the system in Kelvin.
C
READ(13,48)T
48 FORMAT(39X,F 19.16)
C
C T is the transference number of the reactant.
C
READ(13,49)XKINF
49 FORMAT(39X,F19.16)
C
C XKINF is the conductivity of the solution in 1/(ohm. cm)
C
Cl = IZ*F/(R*TEMP)
XN1 = (R0**2*OM/XNU)**0.5
XN2 = (A*XNU/(3*D))**(l./3)
XN3 = NO*Cl*F*D*CINF/((lT)*XKINF)
XN = XN1 *XN2*XN3
C
READ(13,50)V
50 FORMAT(39X,F19.16)
C
C V is the applied potential in volts.
C
READ(13,51)W2
51 F ORMAT (3 9X,F 19.16)
C
C W2 is the weighting factor used in the iterations.
C
READ(13,52)CCENT
52 FORMAT(39X,F19.16)
252
C
DO 85L=1,IDRAMAX
CSUR(L) = CCENT
85 CONTINUE
C
AL(1) = CCENT/CINF  1
C
C AL is the onedimensional array of the coefficients a(l) in the
C Legendre polynomial expansion for concentration terms.
C DRA is the dimensionless radial position, that is, (r/rO)
C EDRAMAX is the total number of DRA values chosen excluding 0.
C
DO 101= 1JDRAMAX
ETA(I) = (1(DRA(I))**2)**0.5
10 CONTINUE
PRINT PROGRAM IS RUNNING '
XILEM = ((NO*F*D*CINF)/(1T))*XN2*((OM/XNU)**0.5)*UPR(1)
C
C XILIM refers to the limiting current density value in amp./(cm)**2
C
READ(13,53)GA
53 FORMAT(33X,F19.16)
C
C GA is the value of Gamma.
C
READ(13,54)XI0
54 FORMAT(38X,F19.16)
C
C XIO is the exchange current density in amp/(cm)**2
C
READ( 13,55) ALPHA
55 FORMAT(3 6X,F 19.16)
RE AD( 13,5 6)BET A
56 FORMAT(22X,F19.16)
AM(1) = AL(1)
XN4 = (NO*F*D*CINF*XN2*(OM/XNU)**0.5)/(1T)
READ( 13,5 7)ITT OL
57 FORMAT(39X, 16)
READ(13,58)RAMTOL
58 FORMAT(39X,F19.16)
READ(13,59)AMTOL
59 FORMAT(38X,F 19.16)
RE AD( 13,60)PERM
60 FORMAT(38X,F19.16)
READ(13,61)Y2
o o o o
253
61 F ORMAT (3 8X,F 19.16)
READ(13,61)STEP
62 FORMAT(38X,F19.16)
C
PRINT *, 'PLEASE NOTE:1
PRINT *, 'THIS PROGRAM IS SPECIFIC TO THE SYSTEM CONSIDERED.â€™
C
DO 16 IT = l,ITTOL
CALL CALCB(XN,Q,AM,MMAX,NMAX,PI,B)
CALL CALPHI(C1,NMAX,IDRAMAX,ETA,B,PHI)
C
C PHI is the potential just "outside" the diffusion layer.
C V is the potential applied to the disk electrode measured with
C respect to a reference electrode. This is constant through out.
C and is equal to the sum of the ohmic potential, surface
C overpotential, zeta potential, and concentration overpotential.
C
CALL CALIDEN(XN4,AM,DRA,IDRAMAX,MMAX,UPR,XIDEN)
C
DO 291= 1,IDRAMAX
CCL(I) = 0.995D3 + 0.5*CSUR(I)
CPOT(I) = 1.075D3  0.5*CSUR(I)
CFERRO(I) = 0.020D3  CSUR(I)
CFERRI(I) = CSUR(I)
29 CONTINUE
C
C Subroutine SPLITETAS is the one that determines the relative
C contributions of surface overpotetial and the potential drop across
C the diffuse part of the double layer.
C
CALL SPLITETAS(CCL, CPOT, CFERRO, CFERRI, IDRAMAX, Cl, F, R, TEMP,
+ IZ, PERM, Y2, CINF, CSUR, ALPHA, BETA, GA, XIO, XEDEN, ETAST,
+ RAMTOL, ITTOL, V, STEP, ZETA, IT, PHI)
C
DO 334 1= 1,IDRAMAX
ETAC(I) = V  PHI(I)  ETAST(I)  ZETA(I)
334 CONTINUE
Calculates the concentration of the reacting species on the surface
of the disk electrode.
CALL CALCSUR(IDRAMAX,C 1 ,T,CINF,ETAC,RAMTOL,ITTOL,CSUR)
C
254
DO 36 L = 1,MMAX
AMST(L) = AM(L)
36 CONTINUE
C
CALL CALAL(CSUR,CINF,DRA,IDRAMAX,WE,MMAX,AL)
C
CALL CALAM(MMAX, AL,AM)
C
SUMRAM = 0
do 39 i = l,mmax
RERAM(I)  ABS((AM(I)AMST(I))/AM(I))
SUMRAM = SUMRAM + RERAM(I)
AM(I) = W2*AM(I) + (1W2)*AMST(I)
39 CONTINUE
IF (SUMRAM LE. AMTOL OR. IT EQ. ITTOL) THEN
XIAV = 0.D0
Q2AV = 0.D0
DO 434 I = 1,IDRAMAX
XIAV = XIAV + XIDEN(I)*DRA(I)*WE(I)
Q2(I) = ETAST(I)*PERM/Y2
Q2AV = Q2AV + Q2(I)*DRA(I)*WE(I)
434 CONTINUE
XIAV = 2.D0*XIAV
Q2AV = 2.D0*Q2AV
D0584I= 1JDRAMAX
FAKEO = F *ZET A(I)/(R* TEMP)
FAKE1 = CPOT (I)*(DEXP(FAKEO)1. DO)+CCL(I)*(DEXP(FAKEO)1 DO)
+ +CFERRI(I)*(DEXP(3.D0*FAKE0)1 DO)
+ +CFERRO(I)*(DEXP(4.DO*FAKEO)1 DO)
FAKE1 = DSQRT(2.D0*FAKE1)
FAKE2 = CPOT(I)*DEXP(FAKEO)CCL(I)*DEXP(FAKEO)
+ 3 D0*CFERRI(I)*DEXP(3 D0*FAKE0)
+ 4.DO*CFERRO(I)*DEXP(4.DO*FAKEO)
FAKE 1 = (FAKE 1/FAKE2)*DSQRT(R*TEMP/PERM)/F
INVCAPD = (Y 2/PERM)+F AKE1
CAPD(I) = 1/INVCAPD
584 CONTINUE
XJ = XIAV*RO*BETA*F/(R*TEMP*XKINF)
XJIO = X30*R0*IZ*F/(R*TEMP*XKINF)
PRINT *,'XILIM',XILIM
print *,'XIAV',XIAV
print *,'xn,,XN
PRINT *,'XJ',XJ
PRINT *,XIO,RO,IZ,F,R,TEMP,XKINF
PRINT *,'XJIO',XJIO
255
print *,'pot. diff.',PHI(IDRAMAX)PHI( 1)
Print *,'error',SUMRAM
PRINT *,'curr @ r=0',XIDEN(l)
print *,'c(0)/c_inf ,CSUR( 1 )/CINF
print *,'i(0)/i_avg.XIDEN(1 )/XIAV
print *,'XIAV/XILIM',XIAV/XILIM
c
OPEN(UNIT = 3,FILEVOLTS',STATUSUNKNOWN')
CLOSE(UNIT = 3, STATUS = 'DELETE')
OPEN(UNIT = 3,FILEVOLTS',STATUSUNKNOWN')
WRITE(3,173)
OPEN(UNIT = 13 ,FILE='AMTHP', STATU SUNKNOWN')
CLOSE(UNIT = 13,STATUS = 'DELETE')
OPEN(UNIT = 13,FILEAMTHP', ST ATU SUNKNOWN')
C
OPEN(UNIT = 33,FILESSOUT',STATUSUNKNOWN')
CLOSE(UNIT = 33,STATUS  'DELETE')
OPEN(UNIT = 33,FILESSOUT',STATUSUNKNOWN')
WRJTE(33,174)
DO 597 K= UDRAMAX
WRITE(3,111 )DRA(K),ET AST (K),ZET A(K),ET AC(K),PHI(K),Q2(K)
+ ,CSUR(K)/CINF,XIDEN(K)/XIAV,XIDEN(K)/XILIM,
+ CAPD(K),Q2(K)/Q2AV
WRITE(3 3,116)DRA(K),ET AST(K),ZET A(K),ET AC(K),PHI(K),Q2(K)
+ ,CSUR(K),XIDEN(K)
597 CONTINUE
C
PRINT *,'Y2 is ',Y2
print *,'IT is ',IT
WRITE(13,*)'The values for Am coefficients are'
112 FORMAT(6X,E20.14)
DOI= 1,11
WRITE( 13,112) AM(I)
ENDDO
WRITE(13,*)
WRITE(13,*)'The values for thmpr coefficients are'
DOI= 1,11
WRITE( 13,112)UPR(I)
ENDDO
WRITE(13,*)
WRITE(13,*)'The values for the Q(N,M) coefficients are'
DO I = 1 ,NMAX
DO J = 1,MMAX
WRITE( 13,112)Q(I, J)
ENDDO
o o
256
ENDDO
WRITE(13,*)M
173 FORMAT(2X,'RADIAL POSITION',9X,'ETAS STAR', 12X,'ZETA POT.'.IOX,'
+CONC OVERPOT.',9X,'OHMIC DROP',14X,'Q2',18X,â€™CSUR/CINF',l IX,
+'ISUR/IAVG', 12X,'ISUR/ILIM', 1 OX,'CAPACITANCE', 10X,'Q2/Q2AVG')
174 FORMAT(2X,'RADIAL POSITION', 12X,'ETAS STAR', 14X,'ZETA POT.',12X,'
+CONC OVERPOT.',12X,'OHMIC DROP',16X,'Q2',20X,'CSUR',19X,'ISUR')
111 FORMAT(E18.12,3X,E18.12,3X,E18.12,3X,E18.12,3X,E18.12,3X,E18.12,
+ 3X,E18.12,3X,E18.12,3X,E18.12,3X,E18.12,3X,E18.12)
116 FORMAT(E20.14,3X,E20.14,3X,E20.14,3X,E20.14,3X,E20.14,3X,E20.14,
+ 3X,E20.14,3X,E20.14)
C
CLOSE(33)
CLOSE(13)
CLOSE(3)
C
STOP
ENDIF
16 CONTINUE
STOP
END
SUBROUTINE GAULEG(X1,X2,X,W,NL)
C
c
C Source: Numerical Recipes in FORTRAN, 2nd ed.
C
C Given the lower and upper limits of integration XI and X2, and given
C NL, this routine returns arrays X(1 :NL) and W(1 :NL) of length NL,
C containing the abscissas and weights of the Gauss Legendre NLpoint
C quadrature formula.
C
C
DOUBLE PRECISION XM,XL,Z,P1,P2,P3,PP,EPS
DOUBLE PRECISION X1 ,X2,W(NL),X(NL),Z 1,DUMMY
INTEGER M,NL,I,J
EPS = l.D14
M = (NL+l)/2
XM = 0.5*(X2+X1)
XL = 0.5*(X2X1)
DUMMY=1.0/(NL+0.5)
DO 121= 1,M
Z = COS(3.141592654*(I0.25)*DUMMY)
1 CONTINUE
oooono _ nono
257
P1 = 1.0
P2 = 0.0
DO 11 J= 1,NL
P3 = P2
P2 = P1
P1 = ((2*Jl)*Z*P2(Jl)*P3)/DFLOAT(J)
11 CONTINUE
PP = NL*(Z*P1P2)/(Z*Z1)
Z1 = Z
Z =Z1P1/PP
IF (ABS(ZZl) GT. EPS) GO TO 1
X(I) = XM  XL*Z
X(NL+1I) = XM +XL*Z
W(I) = 2*XL/((1Z*Z)*PP*PP)
W(NL+1I) = W(I)
12 CONTINUE
RETURN
END
DOUBLE PRECISION FUNCTION FACT(N)
Function subprogram to calculate the factorial of a given number
INTEGER N
IF (N . GE. 2) THEN
FACT = 1.
DO 15 1= 1,N
FACT  FACT*DFLOAT(I)
CONTINUE
ELSE
FACT = 1.
END IF
RETURN
END
SUBROUTINE MPRIME(NMAX,PI,XMPR)
Subroutine subprogram that evaluates the value of M(2N)PRIME at 0.
INTEGER NMAX,I,IDRAMAX
DOUBLE PRECISION XMPR(20),PI
DOUBLE PRECISION FACT,PROD,INVPI
258
INVPI= 1.0/PI
DO 101= l.NMAX
XMPR(I) = 2*(2**(2*(Il))*FACT(Il)/PROD(2*(Il),Il))**2*INVPI
10 CONTINUE
RETURN
END
C
C
SUBROUTINE INTEGRAL(MMAX,NMAX,Y)
C
c
C Subroutine subprogram that evaluates the integral that is involved in
C calculating Q(N,M).
C Please note that the values of the the definite integral are exact.
C There is no approximation involved.
C
INTEGER MMAX,NMAX,M,N,K,L,IB,IA,IC
DOUBLE PRECISION D,XNUM,DIN,Y(20,20),SUM,A1
DOUBLE PRECISION FACT,PROD
DO 10 N= l.NMAX
DO 21 M = 1,MMAX
SUM = 0
DO 30 K =0,M1
DO 40 L = 0,N1
A1 = PROD(M1 ,MK1 )/F ACT(K)
XNUM = Al*(PROD(4*(Nl)2*L,2*(Nl)L)/FACT(2*(Nl)2*L))
DIN = FACT(L)
IA = 2**(2*N1)
IB = NL+K
IC = (1)**(L+K)
D = (IC*XNUM/IA)*1/(DIN*IB)
SUM = SUM + D
40 CONTINUE
30 CONTINUE
Y(N,M) = SUM
C
21 CONTINUE
10 CONTINUE
RETURN
END
C
C
DOUBLE PRECISION FUNCTION PROD(M,N)
C
c
259
C Function PROD(M,N) gives the value of FACT(M)/FACT(N).
C
INTEGER M,N
IF (M.GE.l AND. N.LT.M) THEN
PROD = 1.
DO 20 I = N+1,M
PROD = PROD * DFLO AT(I)
20 CONTINUE
ELSE
PROD = 1.
ENDIF
RETURN
END
C
C
SUBROUTINE CALCQ(MMAX,NMAX,PI,UPR,XMPR,XINT,Q)
C
c
C This subroutine calculates the value of Q(N,M) for different values of
C N and M.
C
INTEGER N,M,MMAX,NMAX
DOUBLE PRECISION PI,UPR(20),XMPR(20),XINT(20,20),Q(20,20)
DO 10 N = 1,NMAX
DO 20 M = 1,MMAX
Q(N,M) = (4*N3)*4* UPR(M) * XINT (N, M)/(PI * XMPR(N))
20 CONTINUE
10 CONTINUE
RETURN
END
C
C
SUBROUTINE CALCB(XN,Q,AM,MMAX,NMAX,PI,B)
C
c
C This subroutine calculates the values for B making use of Q and AL.
C
INTEGER MMAX,NMAX,I,J
DOUBLE PRECISION XN,Q(20,20),AM(20),PI,B(20)
DO 101= 1,NMAX
B(I) = 0
DO 20 J = 1,MMAX
B(I) = B(I) + Q(I,J)*AM(J)
20 CONTINUE
B(I) = PI*XN*B(I)*.25
260
10 CONTINUE
RETURN
END
SUBROUTINE CALPHI(C1,NMAX,IDRAMAX,ETA,B,PHI)
C
c
C This subroutine calculates the values of PHI
C
INTEGER NMAX, IDRAMA X, I, J
DOUBLE PRECISION Cl, ETA(EDRAMAX), B(20), PHI(IDRAMAX)
DOUBLE PRECISION XLEGP, INVC1
INVC 1=1.0/Cl
DO 10 I = 1JDRAMAX
PHI(I) = 0
DO 20 J= l.NMAX
PHI (I) = PHI(I)+B(J)*XLEGP(J1 ,ETA(I))
20 CONTINUE
PHI(I) = PHI(I)*INVC1
10 CONTINUE
RETURN
END
C
C
DOUBLE PRECISION FUNCTION XLEGP(N,XIN)
C
c
C This function calculates the Legendre polynomial expansion for P(2*N,X)
C
INTEGER N, I, IA, IB, IC
DOUBLE PRECISION XIN, XNUM, DIN
DOUBLE PRECISION FACT, PROD, SUM
SUM = 0.
NP1=N+1
DO 10 K= l.NPl
I=K1
IA = 4*N2*I
IB = 2*NI
IC = 2*N2*I
IF (N EQ. 0) THEN
XLEGP = 1
RETURN
END IF
261
IF (XIN.EQ.O AND. IC.EQ.O) THEN
XNUM = ((PROD(I A, EB)/F ACT (IC))/F ACT (I))
ELSE
XNUM = ((PROD(I A, IB )/F ACT (IC))/F ACT (I)) * XIN * * IC
END IF
DIN = DFLOAT (2* *(2*N))
SUM = SUM + (1)**I*XNUM/DIN
10 CONTINUE
XLEGP=SUM
RETURN
END
C
C
SUBROUTINE CALIDEN(XN4,AM,DRA,IDRAMAX,MMAX,UPR,XIDEN)
C
c
C This subroutine calculates the current density at various points on the
C disk from using various input parameters.
C
DOUBLE PRECISION XN4, AM(20), DRA(IDRAMAX), UPR(20),
+ XIDEN(IDRAMAX)
INTEGER MMAX,IDRAMAX,I,J
DO 101= UDRAMAX
XIDEN(I) = 0
DO 20 J = 1,MMAX
XIDEN(I) = XIDEN(I) + AM(J)*(DRA(I))**(2*(J1))*UPR(J)
20 CONTINUE
XIDEN(I) = XN4*XIDEN(I)
10 CONTINUE
RETURN
END
C
C
SUBROUTINE SPLITETAS (CCL, CPOT, CFERRO, CFERRI, IDRAMAX, Cl,
+ F, R, TE, IZ, PERM, Y2, CINF, CSUR, ALPHA, BETA, GA, Xioâ€
+ XIDEN, ETAST, RAMTOL, ITTOL, V, STEP, ZETA, IT, PHI)
C
c
C Subroutine to calculate the individual contributions of surface
C overpotential and zeta potential. Uses a combination of NewtonRaphsorv
C and bisection schemes. Source code taken from the 2nd edition of
C numerical recipes in FORTRAN and is adapted for solving the present
C problem.
INTEGER IDRAMAX,IZ,ITTOL,IT
non
262
DOUBLE PRECISION CCL(EDRAMAX), CPOT(IDRAMAX),
+ CFERRO(IDRAMAX), CFERRI(IDRAMAX), Cl, F, R, TE, PERM, Y2, CINF,
+ CSUR(IDRAMAX), XIO, ALPHA, BETA, GA, XIDEN(IDRAMAX),
+ ETAST(EDRAMAX), ZETA(IDRAMAX), DUMO, DUM1, DUM2, DUM3,
+ DUM4, DUM5, SQR, RTSAFE(IDRAMAX), XL, XH, DF, X, XI, X2,
+ RAMTOL, DXOLD, V, TEMP, FL, FH, FI, PHI(IDRAMAX), STEP
DOUBLE PRECISION FUN,FD
C
DO 101= 1,IDRAMAX
COMPUTE SEQUENCE OF POINTS CONVERGING TO THE ROOT
XI =0.
H = STEP*(VPHI(I))
X2 = H
FL=FUN(Cl,IZ,ALPHA,BETA,GA,R,TE,CINF,PERM,Y2,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),CSUR(I),XIO,XIDEN(I),X1)
FH=FUN(C1,IZ, ALPHABET A, GA,R,TE,CINF,PERM, Y2,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),CSUR(I),XIO,XIDEN(I),X2)
DO WHILE (FL*FH GT. 0. AND. DABS(X2) LT. DABS(V))
XI =X1 +H
X2 = XI + H
FL=FUN(C1,IZ, ALPHABET A, GA,R,TE,CINF,PERM, Y2,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),CSUR(I),XIO,XIDEN(I),X1)
FH=FUN(C1,IZ, ALPHABET A, GA,R,TE,CINF,PERM, Y2,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),CSUR(I),XIO,XIDEN(I),X2)
END DO
IF ((FL.GT.O..AND.FH.GT.O.).OR.(FL.LT.O..AND.FH.LT.O.)) THEN
PAUSE 'ROOT MUST BE BRACKETED IN RTSAFE'
END IF
IF (FL.EQ.O.) THEN
RTSAFE(I) = XI
GOTO 10
ELSE IF (FH.EQ.O.) THEN
RTSAFE(I) = X2
GOTO 10
ELSE IF (FL.LT.0.) THEN
XL = XI
XH = X2
ELSE
XH = XI
XL = X2
END IF
RTSAFE(I) = 0.5*(X1+X2)
263
DXOLD = ABS(X2X1)
DX =DXOLD
F1=FUN(C1,IZ, ALPHABET A,GA,R,TE,CINF,PERM, Y2,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),CSUR(I),XIO,XIDEN(I),RTSAFE(I))
DF=FD(C1,IZ, ALPHABET A,GA,R,TE,PERM, Y2,F,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),RTSAFE(I))
DO 11 J= l,ITTOL
IF(((RTSAFE(I)XH)*DFF1)*((RTSAFE(I)XL)*DFF1).GE.0.
+ OR.ABS(2.0*F1).GT ABS(DXOLD*DF)) THEN
DXOLD = DX
DX = 0.5*(XHXL)
RTSAFE(I)= XL+DX
IF(XL.EQ.RTSAFE(I))GO TO 10
ELSE
DXOLD = DX
DX = Fl/DF
TEMP = RTSAFE(I)
RTSAFE(I)= RTSAFE(I)DX
IF (TEMP.EQ RTSAFE(I))GO TO 10
END IF
IF (ABS(DX).LT.RAMTOL) GO TO 10
F1=FUN(C1,IZ, ALPHABET A, GA,R,TE,CINF,PERM, Y2,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),CSUR(I),XIO,XIDEN(I),RTSAFE(I))
DF=FD(C 1 ,IZ, ALPHA,BETA, GA,R,TE,PERM, Y2,F,CPOT(I),CCL(I),
+ CFERRI(I),CFERRO(I),RTSAFE(I))
IF (F1.LT.0.) THEN
XL = RTSAFE(I)
ELSE
XH = RTSAFE(I)
END IF
11 CONTINUE
PAUSE 'RTSAFE EXCEEDING MAXIMUM ITERATIONSâ€™
10 CONTINUE
DO 20 I=1,IDRAMAX
ZETA(I) = RTSAFE(I)
DUM0 = ZETA(I)*C1/IZ
DUM1 = CPOT(I)*(DEXP(DUM0)1.0)
DUM2 = CCL(I)*(DEXP(DUM0)1.0)
DUM3 = CFERRI(I)*(DEXP(3.0*DUM0)1.0)
DUM4 = CFERRO(I)*(DEXP(4.0*DUM0)1.0)
DUM5 = 2.0*R*TE/PERM
SQR = (DUM5 *(DUM1+DUM2+DUM3+DUM4))* *0.5
o o o o o o o o on
264
ETAST(I) = Y2*SQR
20 CONTINUE
RETURN
END
DOUBLE PRECISION FUNCTION FUN(C1, IZ, ALPHA, BETA, GA, R, TEMP,
+ CINF, PERM, Y2, CPOT, CCL, CFERRI, CFERRO, CSUR, XIO, XIDEN,
+ ZETA)
INTEGER IZ
DOUBLE PRECISION Cl, ALPHA, BETA, GA, R, TEMP, PERM, Y2, CPOT,
+ CCL, CINF, CFERRI, CFERRO, ZETA, XIDEN, XIO, CSUR, DUMO, DUM1,
+ DUM3, DUM4, DUM5,DUM6,DUM7,DUM8,SQR
DUMO = ZETA*C1/IZ
DUM1 = DEXP(DUM0*GA*3.0)
DUM3 = CPOT*(DEXP(DUM0)1.0)
DUM4 = CCL*(DEXP(DUM0)1.0)
DUM5 = CFERRI * (DEXP(3.0 * DUMO) LO)
DUM6 = CFERRO * (DEXP(4.0 * DUMO) LO)
DUM7 = 2.0 * R* TEMP/PERM
DUM8 = (XIDEN/XIO)*(CINF/CSUR)**GA
SQR = (DUM7*(DUM3+DUM4+DUM5+DUM6))**0.5
FUN = DUM1*(DEXP(ALPHA*C1*Y2*SQR)DEXP(BETA*C1*Y2*SQR))
+ DUM8
RETURN
END
DOUBLE PRECISION FUNCTION FD(C1, IZ, ALPHA, BETA, GA, R, TEMP,
+ PERM, Y2, F, CPOT, CCL, CFERRI, CFERRO, ZETA)
INTEGER IZ
DOUBLE PRECISION C1,ALPHA,BETA,GA,R,TEMP,PERM, Y2,CPOT,CCL,F,
+ CFERRI,CFERRO,ZETA, VARO, VARI ,VAR2,VAR3,VAR4,VAR5,
+ VARÃ“,VAR7,VAR8,FIRST,SECOND,FIRST1,SEC0ND1,SQR
VARO = ZETA*C1/IZ
C VAR2 = DEXP(VAR0*(BETA*IZGA*3.0))
VAR3 = CPOT*(DEXP(VARO)1.0)
VAR4 = CCL*(DEXP(VAR0)1.0)
VAR5 = CFERRI*(DEXP(3.0*VAR0)1.0)
nono no on
265
VAR6 = CFERRO*(DEXP(4.0*VAR0)1.0)
VAR7 = 2.0*R*TEMP/PERM
SQR = (VAR7*(VAR3+VAR4+VAR5+VAR6))**0.5
FIRST = DEXP(ALPHA*C1*Y2*SQR)
SECOND = DEXP(BETA*C1*Y2*SQR)
FIRST 1 = (3.0*GA*Cl/IZ)*DEXP(3.0*GA*VAR0)*(FIRSTSECOND)
VAR8 = CPOT*DEXP(VARO)CCL*DEXP(VARO)
+ 3.0*CFERRI*DEXP(3.0*VAR0)4.0*CFERRO*DEXP(4.0*VAR0)
VAR8 = F*VAR8/PERM
SECOND 1 = Cl*Y2*(VAR8/SQR)*(ALPHA*FIRST+BETA*SECOND)*
+ DEXP(3.0*GA*VAR0)
FD = FIRST 1 + SECOND 1
RETURN
END
SUBROUTINE CALCSUR (IDRAMAX, Cl, T, CINF, ETAC, RAMTOL, ITTOL,
+ CSUR)
INTEGER IDRAMAX, I,ITTOL
DOUBLE PRECISION Cl, T, CINF, ETAC(IDRAMAX), CSUR(IDRAMAX), X,
+ XI, X2, RTSAFE(IDRAMAX),DX,DXOLD,F,XL,XH,RAMTOL,FL,FH
DOUBLE PRECISION FUN1,FD1
DO 101= 1,IDRAMAX
COMPUTE SEQUENCE OF POINTS CONVERGING TO THE ROOT
XI = 1.0D250
X2 = LODO
FL=FUN 1 (ETAC(I),C 1 ,T,X 1)
FH=FUN 1 (ETAC(I),C 1 ,T,X2)
IF ((FL.GT.0..AND.FH.GT.0.) OR.(FL.LT.O..AND FH LT.O.))
+ PAUSE 'ROOT MUST BE BRACKETED IN RTSAFE'
IF (FL.EQ.O.) THEN
RTSAFE(I) = XI
GOTO 10
ELSE IF (FH.EQ.O.) THEN
RTSAFE(I) = X2
GOTO 10
ELSE IF (FL.LT.O.) THEN
XL = XI
XH = X2
on on
266
ELSE
XH = XI
XL = X2
ENDIF
RTSAFE(I) = 0.5*(X1+X2)
DXOLD = ABS(X2X1)
DX = DXOLD
F=FUN 1 (ETAC(I),C 1 ,T,RTS AFE(I))
DF=FD 1 (C1 ,T,RTSAFE(I))
DO 11 J= LITTOL
IF(((RTSAFE(I)XH)*DFF)*((RTSAFE(I)XL)*DFF).GE.O.
+ OR ABS(2.0*F).GT.ABS(DXOLD*DF)) THEN
DXOLD  DX
DX = 0.5*(XHXL)
RTSAFE(I)= XL+DX
IF(XL.EQ.RTSAFE(I))GO TO 10
ELSE
DXOLD = DX
DX = F/DF
TEMP = RTSAFE(I)
RTSAFE(I)= RTSAFE(I)DX
IF (TEMP.EQ.RTSAFE(I))GO TO 10
ENDIF
IF (ABS(DX).LT RAMTOL) GO TO 10
F=FUN 1 (ETAC(I),C 1 ,T,RTS AFEO))
DF=FD 1 (C1 ,T,RTS AFE(I))
IF (F.LT.O.) THEN
XL = RTSAFE(I)
ELSE
XH = RTSAFE(I)
ENDIF
11 CONTINUE
PAUSE 'RTSAFE EXCEEDING MAXIMUM ITERATIONS'
10 CONTINUE
DO 20 I=1,EDRAMAX
CSUR(I) = CINF*RTSAFE(I)
20 CONTINUE
RETURN
END
DOUBLE PRECISION FUNCTION FUN1(ETAC,C1,T,X)
DOUBLE PRECISION ETAC,C1,T,X
267
FUÃ‘I  (DLOG(X)+T*( 1 X))/C 1  ETAC
RETURN
END
C
C
DOUBLE PRECISION FUNCTION FD1(C1,T,X)
C
c
DOUBLE PRECISION C1,T,X
FD1 = (1/X  T)/C1
RETURN
END
C
C
SUBROUTINE CALAL(CSUR,CINF,DRA,IDRAMAX,WE,MMAX,AL)
C
c
C This subroutine calculates the AL coefficients using trapezoidal rule.
C
DOUBLE PRECISION CSUR(IDRAMAX),CINF,DRA(IDRAMAX),
+ AL(20), SUM, WE(EDRAMAX), A
INTEGER IDRAMAX,I,J,MMAX
DOUBLE PRECISION XLEGP,INYCINF
INVCINF=1 0/CINF
DO 101= 1,MMAX
SUM = 0
DO 20 J= UDRAMAX
A = ((CSUR(J))*INVCINF  1)*XLEGP(I1,DRA(J))
SUM = SUM + A*WE(J)
20 CONTINUE
AL(I) = (4*13)* SUM
10 CONTINUE
RETURN
END
C
C
SUBROUTINE CALAM(MMAX,AL,AM)
C
c
C This subroutine calculates the AM coefficints using AL coefficints.
C
INTEGER MMAX,I,J
DOUBLE PRECISION AL(20),AM(20),SUM
DOUBLE PRECISION COEFF
non nnooooo
268
DO 101= 1.MMAX
SUM = 0
DO 20 J = I,MMAX
SUM = SUM + AL(J)*COEFF(Jl,JI)
20 CONTINUE
AM(I) = SUM
10 CONTINUE
RETURN
END
DOUBLE PRECISION FUNCTION COEFF(N,IR)
This function subprogram calculates the coefficients in the legendre
polynomial expansion.
INTEGER N,IR,IA,IB,IC
DOUBLE PRECISION FACT,PROD
DOUBLE PRECISION XNUM,DIN
IA = 4*N  2*IR
IB = 2*N  IR
IC = 2*N  2*ER
XNUM  (PROD(IA, IB)/FACT(IC))/FACT(IR)
DIN = 2**(2*N)
COEFF = ((1)**IR)*XNUM/DIN
RETURN
END
nono
APPENDIX B
FREQUENCY DOMAIN MODEL FOR THE ROTATING DISK ELECTRODE
PROGRAM CAPIMP8 LDIS
C
C Model formulated and program written by Madhav Durbha and Mark E. Orazem.
C DATE LAST MODIFIED: June 11th, 1998. The theoretical development of the
C Mathematical model is presented in Chapter 5. Model is also presented in the
C manuscript "A Mathematical Model for the Radially Dependent Impedance
C of a Rotating Disk Electrode," to be published in the conference
C proceedings volume of the 193rd Electrochemical Society meeting
C (May 3rd,1998)held in San Diego, California.
C
C
C This program calculates the local impedances for a specified frequency.
C By specifying a number of frequencies, impedance spectrum can be obtained.
C All the computations are performed using double precision. Baseline
C values for impedance calculations are obtained by using the steady state
C program without explicitly accounting for the double layer capacitance.
C Capacitance is assumed to be uniformly distributed across the surface
of the electrode. A finite Schmidt number correction is incorporated.
John Newman's BAND algorithm is employed to solve for the homogenious
part of the convective diffusion equation in the frequency domain.
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BA/ A(6,6),B(6,6),C(6,10001 ),D(6,13),G(6),X(6,6), Y(6,6)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1 ,NFRE
COMMON/AA/ XNU,DIF,FRK(90),BK,CK,MA 1 ,CBULK(6),CONC(6,10001),
+ CONV(6,10001), AMTG, AMS S( 11 ),THMPRSS( 11 ),CINF,TRAN,
+ AAL,OM,NO,FA,QSS(11,11),RO,IZ,TEMP,XN,R,XKINF,WW(90),
+ CDDL
COMMON/BB/ H,DEL
COMMON/CC/ SUMH
COMMON/SIMP/ XI( 10001)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
COMMON/OP/CURFRTIL(60),CURFJTIL(60),THPTR( 11),
+ THPTJ(11),AMTR(11),AMTJ(11),CSURTILR(60),
+ CSURTILJ(60)
COMMON/QTEL/QRTILD A( 11,11),QJTILDA(11,11)
269
270
DIMENSION THMREAP( 11,10001 ),THMIMAP( 11,10001),
+ FRKHSQR(90),THMREAPSS(11,10001),
+ FUNCR( 10001 ),FUNCJ( 10001), AMTRST (11),AMTJST(11),
+ OUTR(l l),OUTJ(l 1 ),ZR(90),ZJ(90),ZLOCR(60),
+ ZLOCJ(60),DUMETC(60),CURPRESR(60),CURPRESJ(60),
+ ALTRST(11),ALTJST(11)
COMMON/BTIL/BRTILDA(l 1),BJTILDA(11),C1
COMMON/PHI/PHITR(60),PHIT J (60)
COMMON/SS/ ETASTSS(60),ZETASS(60),ETACSS(60),PHISS(60),
+ Q2SS(60),CSURSS(60),XIDENSS(60)
COMMON/ETAS/ ALPHA,BETA,GAMMA,
+ XZ(60),YZ(60),VAR0(60),DNRZ(60),XI0,
+ ZETTELR(60),ZETTILJ(60),ETASTILR(60))
+ ETASTILJ(60),BZ(60),Y2
C OMMON/CON C/ ET ACTELR(60), ET ACTIL J (60)
COMMON/AL/ ALTR(11),ALTJ(11)
COMMON/AV/ CURTIL A VR(90), CURTIL A V J(90)
COMMON/CUR/CURTILR(60),CURTILJ(60)
COMMON/CL/ CLTR(11),CLTJ(11)
COMMON/CM/ CMTR(11),CMTJ(11)
C
NMAXP1 = 11
MMAXP1 = 11
IDRAMAX =30
C 30 radial positions are used
PI = 22.DO/7.0D0
C
CALL INIT
C
VTELR = 0.01
VTILJ = 0.
open(unit=96,file='RESULTS', STATU S='UNKN OWN')
close(unit=96, statusdelete')
open(unit=96,file='RESULTS',STATUSUNKNOWN')
C
WRITE (96,107)
C
C1 = IZ*FA/(R*TEMP)
DUMT = TRAN/CINF
C
DO 10 I =1,IDRAMAX
XZ(I) = ALPHA*C1*DEXP(ALPHA*C1*ETASTSS(I))
+ +BETA*C1 *DEXP(BETA*C 1 *ETASTSS(I))
DUMETC(I) = (1/CSURSS(I)  TRAN/CINF)/C1
10 CONTINUE
non o o o
271
C
C
C XNU is the kinematic viscosity and DIF is the diffusion coefficient.
C SC is the schmidt number of the reacting species.
C
C
SC = XNU/DIF
1209 format (i4,3x,I4,3X,l l(E22.14,3x))
XI = 0.0D0
X2 = 1.0D0
CALL GAULEG(X1,X2,IDRAMAX)
C
CALL INTEGRAL
C
UMAX = 3000
AMTOL = l.D8
W2 = 1.D0
W2 is the weighting factor used in performing the iterations.
DO 487 LFREQ =1,NFRE
NFRE is the total number of frequencies in the impedance spectrum.
W2 = W2*0.9
AMTOL = AMTOL*0.9
DO 12 MAI = 1,MMAXP1
AMTR(MAl) = AMTG
AMTJ(MAl) = AMTG
J=0
C
C Initialize coefficient matrices for derivative boundary conditions.
C This is for setting up the matrices that are needed for implementation
C of BAND algorithm.
C
DO I=1,NVAR
DO K=T,NVAR
Y(I,K)=0.0
X(I,K)=0.0
ENDDO
ENDDO
C
SUMH=0.0
C
272
C Initialize all coefficient matrices to zero
C
351 J=J+1
SUMH=SUMH+H
DO I=1,NVAR
G(I)=0.0
DO K=1,NVAR
A(I,K)=0.0
B(I,K)=0.0
D(I,K)=0.0
ENDDO
ENDDO
C
C Boundary conditions for bulk solution
C
IF(J.EQ.l) CALL BC1(J)
C
C EQUATIONS FOR INTERIOR DOMAIN
C
IF ((J.GT.l).AND.(J.LT.NJ)) CALL INNER(J)
C
C Equtions for second Boundary conditions
C
IF (J.EQ.NJ) Call BC2(J)
C
CALL BAND(J)
C
IF (J.NE.NJ) GOTO 351
C
FRKHSQR(LFREQ) = (FRK(LFREQ)/2.D0)**0.5
DO 99 J= 1,NJ
XI(J) = (J1)*H
THMREAP(MA1,J) = C(l,J)+C(3,J)*SC**(l./3.)
+ +C(5,J)*SC**(2./3.)
THMIMAP(MA 1, J) = C(2,J)+C(4,I)*SC**(l./3.)
+ +C(6, J) * S C * * (2. /3.)
99 CONTINUE
IF (LFREQ EQ. 1) THEN
DO 991 J=1,NJ
THMREAPSS(MA1,J) = THMRE AP(MA 1, J)
991 CONTINUE
END IF
C
12 CONTINUE
nono nono
273
IF (LFREQ EQ. 1) THEN
GOTO 487
ENDIF
C
DO 271= 1,IDRAMAX
CSURTILR(I) = 0.
CSURTELJ(I) = 0.
DO 28 K = 1,MMAXP1
CSURTELR(I) = CSURTELR(I)
+ + AMTR(K) * (DRA(I)) * * (2 * (K1))
CSURTILJ(I) = CSURTILJ(I)
+ + AMTJ(K) * (DRA(I)) * * (2 * (K1))
28 CONTINUE
C
27 CONTINUE
C
DO 26MA1 = 1,11
DO J= 1,2001
Depending on the upper limit of J Simpsons rule subroutine should be
adjusted
FUNCR(J) = THMREAPSS(MA1, J)*THMREAP(MA1, J)
+ *DEXP(XI(J)**30.25*BK*XI(J)**4*SC**(l./3.)
+ 0.2*CK*XI(J)**5*SC**(2./3.))
FUNCJ(J) = THMREAP S S(MA 1, J) * THMIMAP(MA 1, J)
+ *DEXP(XI(J)* *30.25 *BK*X3(J)* *4* SC* *(1./3.)
+ 0.2*CK*XI(J)**5*SC**(2./3.))
ENDDO
CALL SIMPSONSRULE(FUNCR,OR)
CALL SIMP SONSRULE(FUNCJ, O J)
Combination of Simpson's 1/3rd and 3/8th rule is used in order to
evaluate the integrals that are involved.
OUTR(MAl) = OR* FRK(LFREQ)
OUTJ(MAl) = OJ*FRK(LFREQ)
C
OUTR(MAl) = OUTR(MA 1)/AM S S (M A1)
OUTJ(MAl) = OUTJ(MAl)/AMSS(MAl)
26 CONTINUE
C
DO 394 ITER = 1,ITMAX
274
DO 291 MAI = 1,11
THPTR(MAl) = AMTJ(MA1)*0UTR(MA1)
+ + AMTR(MA1)*0UTJ(MA1)
THPTJ(MAl) = AMTJ(MA1)*0UTJ(MA1)
+  AMTR(MA 1) * OUTR(MA 1)
C
291 CONTINUE
C
CALL CURFTILDA
C
CALL ETASTILDA
C
DO I = 1,IDRAMAX
CURPRESR(I) = CURTELR(I)
CURPRESJ(I) = CURTILJ(I)
D2R = ET ASTILR(I)+ET ACTILR(I)
D2J = ETASTELJ(I)+ETACTILJ(I)
C
CURTILR(I) = CURFRTEL(I)2. *PI*WW(LFREQ)*CDDL*D2J
CURTELJ(I) = CURFJTIL(I)+2.*PI*WW(LFREQ)*CDDL*D2R
C
ENDDO
CALL CALCLTILDA
CALL CALCMTILDA
DO I = l.IDRAMAX
DUMMR = 0
DUMMJ = 0
DO M = 1,MMAXP1
DUMMR = DUMMR+CMTR(M) * DRA(I) * * (2 * (M1))
DUMMJ = DUMMJ+CMT J(M) *DRA(I) * * (2 * (M1))
ENDDO
C
ENDDO
C
C
CALL BTILDA
C
CALL PHITILDA
C
DO I = 1,EDRAMAX
ET ACTILR(I)=VTILRET AS TELR(I)PHITR(I)
ET ACTIL J(I)=VTELJET ASTIL J(I)PH3T J(I)
C
non
275
ENDDO
CALL CONCTIL
C
DO 36 M = LMMAXPl
AMTRST(M) = AMTR(M)
AMTJST(M) = AMTJ(M)
C
36 CONTINUE
C
CALL CALALTILDA
C
CALL CALAMTILDA
C
SUMRAM = 0
do 39 i = l,mmaxPl
C
RERAMR = AB S(( AMTR(I)AMTRST (I))/ AMTR(I))
RERAMJ = AB S ((AMT J (I)AMT J ST (I))/ AMT J(I))
SUMRAM = SUMRAM + RERAMR + RERAMJ
AMTR(I) = W2*AMTR(I) + (1W2)*AMTRST(I)
AMTJ(I) = W2*AMTJ(I) + (1W2)*AMTJST(I)
39 CONTINUE
C
IF (SUMRAM LE. AMTOL OR. ITER EQ. ITMAX) THEN
PRINT *,'CONVERGENCE MET AT',ITER,SUMRAM,LFREQ
C
CURTILAVR(LFREQ)  0.
CURTEL AVJ (LFREQ) = 0.
DO I â€œ 1,IDRAMAX
CURTELAVR(LFREQ)  CURTILAVR(LFREQ)+CURTILR(I)
+ * DRA(I) * WE(I)
CURTILAVJ(LFREQ) = CURTILAVJ(LFREQ)+CURTILJ(I)
+ *DRA(I)*WE(I)
ENDDO
CURTILAVR(LFREQ) = 2. * CURTIL AVR(LFREQ)
CURTILAVJ(LFREQ)  2.*CURTILAVJ(LFREQ)
'RATCOMP' calculates the ratio of two complex numbers
CALL RATCOMP(VTILR,VTILJ,CURTILAVR(LFREQ),
+ CURTILAVJ(LFREQ),ZR(LFREQ),ZJ(LFREQ))
C
DO I =1,IDRAMAX
C
CALL RATCOMP(VTILR,VTILJ,CURFRTIL(I),
276
+ CURFJTEL(I),ZLOCR(I),ZLOCJ(I))
C
C One should decide between the two WRITE statements that follow. For
C somone interested in studying the radial distributions, first WRITE
C is appropriate to use and the second WRITE is appropriate to generate
C impedance values at different frequencies.
C
WRITE (96,113) DRA(I),CSURTILR(I),CSURTILJ(I),
+ CURTELR(I),CURTELJ(I),PHITR(I),
+ PHIT J(I),ET ACTILR(I),ET ACTIL J(I),
+ ET ASTELR(I),ET ASTILJ(I),ZLOCR(I),
+ ZLOCJ(I)
ENDDO
C
C WRITE (96,115) DRA(20),CSURTILR(20),CSURTILJ(20),
C + CURFRTEL(20),CURFJTIL(20),PHITR(20),
C + PHITJ(20),ETACTILR(20),ETACTILJ(20),
C + ETASTELR(20),ETASTELJ(20),ZR(LFREQ),ZJ(LFREQ)
C + ,WW(LFREQ)
c
107 F ORMAT (2X'RADIAL POSITION', 17X,'CONCR',22X,'CONCJ',22X,'IR,,22X,
+ 'IJ',22X,'PHIR',22X,'PHIJ',22X,'ETACR',22X,'ETACJ',22X,
+ 'ETASR',22X,'ETASJ',22X,'ZR,,22X,'ZJ',22X,'FREQUENCY')
115 FORMAT(E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,
+ E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,
+ E20.14,6X,E20.14,6X,E20.14)
113 FORMAT(E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,
+ E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,E20.14,6X,
+ E20.14,6X,E20.14)
C CLOSE(96)
C
GOTO 487
END IF
394 CONTINUE
487 CONTINUE
CLOSE(96)
STOP
END
C
C
SUBROUTINE GAULEG(X1,X2)
C
iâ€”> >â€”> >â€”* n e"}
277
Subroutine used to generate the weighting factors and abscissas for
the GaussLegendre quadrature. Adapted from the 2nd edition of
"Numerical Recipes in FORTRAN".
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX
EPS = l.D14
M = (IDRAMAX+1 )/2
XM = 0.5*(X2+X1)
XL = 0.5*(X2X1)
DUMMY= 1.0/(IDRAMAX+0.5)
DO 121= 1,M
Z = COS(3.141592654*(I0.25)*DUMMY)
CONTINUE
PI = 1.0
P2 = 0.0
DO 11 J= LIDRAMAX
P3 = P2
P2 = PI
PI  ((2*J1)*Z*P2(J1)*P3)/DFLOAT(J)
1 CONTINUE
PP = IDRAMAX*(Z*P1P2)/(Z*Z1)
zi =z
Z = Z1P1/PP
IF (ABS(ZZ1) GT. EPS) GO TO 1
DRA(I) = XM  XL*Z
DRA(IDRAMAX+1 I) = XM +XL*Z
WE(I) = 2 * XL/(( 1Z*Z)*PP*PP)
WE(IDRAMAX+1 I)  WE(I)
2 CONTINUE
RETURN
END
Subroutine that reads the input variables
Subroutine INIT
C
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BA/ A(6,6),B(6,6),C(6,10001),D(6,13),G(6),X(6,6),Y(6,6)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1 ,NFRE
COMMON/AA/ XNU,DIF,FRK(90),BK,CK,MA1,CBULK(6),CONC(6,10001),
278
+ C0NV(6,10001),AMTG,AMSS(11),THMPRSS(11),CINF,TRAN,AAL,OM,
+ NO,FA,QSS(l 1,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),CDDL
COMMON/BB/ H,DEL
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
COMMON/SS/ ETASTSS(60),ZETASS(60),ETACSS(60),PHISS(60),
+ Q2SS(60),CSURSS(60),XIDENSS(60)
COMMON/ETAS/ ALPHA,BETA,GAMMA,
+ XZ(60),YZ(60),VAR0(60),DNRZ(60),X30,
+ ZETTILR(60),ZETTILJ(60),ETASTILR(60),
+ ETASTELJ(60),BZ(60),Y2
C
C Input file from which various parameters are being read is called
C 'SSUNDAT'.
C
open(unit= 13 6,filed :\users\DURBHA\S SMODEL\S SUND AT',
+ STATUSUNKNOWN')
C
PI = 3.141592654D0
READ(136,35)XNU
35 FORMAT(3 9X,F 19.16)
C
C XNU is kinetic viscosity in (cm)**2/sec
C
READ(136,35)DIF
C
C D is the diffusion coefficient in (cm)**2/sec
C
READ (136,*)
23 F ORMAT (3 9X, 12)
C
READ(136,23) IDRAMAX
READ(136,35)R0
C
C R0 is the radius of the disk electrode in cm
C
READ(136,35)OM
C
C OM is the angular speed of the disk electrode in rPM
C
READ(136,35)AAL
C
279
C AAL is coefficient in series for the expansion of velocity terms
C
READ( 13 6,42)NO
42 FORMAT(39X,I2)
C
C NO is the number of equivalents
C
READ(136,43)IZ
43 FORMAT(35X,I2)
C
C IZ corresponds to the number of equivalents
C
READ(136,35)FA
C
C F is the Faraday's constant in Coulomb/equivalent
C
READ(136,35)CINF
C
C CINF is the concentration of the species at infinite distance in moles/c.c.
C
READ(136,35)R
C
C R is universal gas constant in J/mol./Kelvin.
C
READ(136,35)TEMP
C
C TEMP is the temperature of the system in Kelvin.
C
READ(136,35)TRAN
C
C T is the transference number of the reactant.
C
READ(136,35)XKINF
C
C XKINF is the solution conductivity in 1/(ohm. cm)
C
READ (136,*)
READ (136,*)
READ (136,*)
READ (136,35) GAMMA
READ (136,35) XIO
READ (136,35) ALPHA
READ (136,35) BETA
READ (136,143) ITTOL
no on noon oooooo
280
143 F ORMAT (3 9X, 14)
READ (136,*)
READ (136,*)
READ (136,*)
READ (136,*)
READ (136,*)
READ (136,35) AMTG
READ (136,35) CDDL
print *,XNU,Dif,IDRAMAX,LO,RO,OM,AAl,NO,IZ,FA,CINF,R, TEMP,TRAN,
+ XKINF,VSS,W2,CCENT,GAMMA, XIO, Alpha,Beta,ITTOL,RAMTOL,AMTOL,
+ PERM,Y2,STEP,AMTG,CDDL
PAUSE
PRINT *,'AMTG',AMTG
CLOSE(136)
N is the NUmber of unknowns^ NUmber of species +1 (Voltage)
NVAR=6
NJ=NUMBER OF NODES
NJ=10001
DOMAIN LENGTH # OF DEBYE LENGTHS
DEL = 10.0
H=Stepsize
H=DEL/(NI1)
WW = frequency of perturbation (Hz)
WW(2) = 1.43
NFRE =2
NFRE is the variable that sets the number of frequency points
that are needed
DO I = 2,NFRE
WW(I) = WW(2)*10.**((I2)/12.)
ENDDO
OMEGA = rotational speed (cycles/s)
OMEGA=OM/60.
OM = OM*PI/30.D0
C XNU =
BBL=0.61592201
DO L = l,nfre
FRK(L)=(WW (L)/OMEGA) * (((9. * XNU)/((AAL * * 2) * DIF)) * * (1. /3.))
ENDDO
noon o o o
281
BK =(3./AAL**4)**(l./3.)
CK = (BBL/6.)*((3./AAL)**(5./3.))
C
OPEN(UNIT=24,FILE='d :\USERS\DURBHA\S SMODELVAMTHP',
+ STATUSUNKNOWN')
READ(24,*)
DOI=l,MMAXPl
READ(24,112)AMSS(I)
ENDDO
READ(24,*)
READ(24,*)
DO I=1,MMAXP1
READ(24,112)THMPRSS(I)
ENDDO
READ(24,*)
READ(24,*)
DO I = 1,NMAXP1
DO J = 1,MMAXP1
READ(24,112)QSS(I, J)
ENDDO
ENDDO
C
112 FORMAT(6X,E20.14)
CLOSE(24)
C
OPEN(UNIT=3 3 ,FILE='D :\USERS\DURBHA\S SMODEL\S SOUT1,
+ STATUSUNKNOWN')
READ(33,*)
DO K=1,IDRAMAX
READ(33,116)DRA(K),ETASTSS(K),ETACSS(K),PfflSS(K),CSURSS(K),
+ XIDENSS(K)
ENDDO
116 FORMAT(E20.14,3X,E20.14,3X,E20.14,3X,E20.14,3X,E20.14,3X,E20.14)
CLOSE(33)
BULK SOLUTION CONCENTRATIONS MOL/LITER
CBULK(1)=1.0
DO I = 2,NVAR
CBULK(I) = 0.
ENDDO
Initialize entire domain of the array by assigning
Bulk Concentration values at all nodes
noon o non
282
DO I=1,NVAR
DO J=1,NJ
C(I,J)=0.0
CONC(I,J)=CBULK(I)
ENDDO
ENDDO
RETURN
END
Subroutine BC1(J)
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BA/ A(6,6),B(6,6),C(6,10001),D(6,13),G(6),X(6,6),Y(6,6)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/AA/ XNU,DEF,FRK(90),BK,CK,MA1 ,CBULK(6),CONC(6,10001),
+ CONV(6,10001),AMSS(11),THMPRSS(1 l),CINF,TRAN,AAL,OM,
+ NO,FA,QSS(ll,l 1),R0,IZ,TEMP,XN,R,XKINF,WW(90),CDDL
COMMON/BB/ H,DEL
Boundary conditions for bulk solution
DO I=1,NVAR
B(I,I)=H
G(I)=(CBULK(I)*H)
ENDDO
RETURN
END
Subroutine Inner(J)
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BA/ A(6,6),B(6,6),C(6,10001),D(6,13),G(6),X(6,6),Y(6,6)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/AA/ XNU,DIF,FRK(90),BK,CK,MAI,CBULK(6),CONC(6,10001),
+ CONV(6,10001), AMS S( 11 ),THMPRS S( 11 ),CINF,TRAN, AAL,OM,
+ NO,FA,QSS(l 1,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),CDDL
COMMON/BB/ H,DEL
DIMENSION AD(6,6),BD(6,6),DD(6,6)
H2=H**2.
XI = (J1)*DEL/(NJ1)
DO I = 1,NVAR
283
DO K = 1,NVAR
AD(I,K) = 0.
BD(I,K) = 0.
DD(I,K) = 0.
ENDDO
ENDDO
DO I = 1,NVAR
AD(I,I) = 1.
BD(I,I) = 3.*XI**2
DD(I,I) = 6.*(MA11)*XI
ENDDO
DO I = 3,NVAR
BD(I,I2) = BK*XI**3
DD(I,I2) = 3.*(MA11)*BK*XI**2
ENDDO
DO I = 5,NVAR
BD(I,I4) = CK*XI**4
DD(I,I4) = 4*(MA11)*CK*XI**3
ENDDO
DO I = 2,2,NYAR
DD(I1,I) = FRK(LFREQ)
DD(I,I1) = FRK(LFREQ)
ENDDO
DO I = 1,NVAR
DO K =1,NVAR
A(I,K) = AD(I,K)  H*BD(I,K)*0.5D0
B(I,K) =2*AD(I,K)+H2*DD(I,K)
D(I,K)  AD(I,K)+H*BD(I,K)*0.5D0
ENDDO
G(I) = 0.
ENDDO
RETURN
END
Subroutine BC2(J)
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BA/ A(6,6),B(6,6),C(6,10001),D(6,13),G(6),X(6,6), Y(6,6)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/AA/XNU,DIF,FRK(90),BK,CK,MAI,CBULK(6),CONC(6,10001),
+ CONV(6,10001),AMSS(11),THMPRSS(11),CINF,TRAN,AAL,OM,
non
284
+ N0,FA,QSS(11,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),CDDL
COMMON/BB/ H,DEL
Boundary conditions at inner boundary
do i=l,NVAR
B(i,i)= H
G(i)=0.0
enddo
RETURN
END
SUBROUTINE MATINV(N,M,DETERM)
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BA/ A(6,6),B(6,6),C(6,10001 ),D(6,13),G(6),X(6,6),Y(6,6)
COMMON/NS/ NTEMP,NJ,LFREQ,NMAXP 1 ,MMAXP 1
DIMENSION ID(6)
DETERM=1.01
DO 1 1=1,N
1 ID(I)=0
DO 18 NN=1,N
BMAX=1.1
DO 6 1=1,N
IF(ID(I).NE.O) GOTO 6
BNEXT=0.0
BTRY=0.0
DO 5 J=1,N
IF (ID(J).NE.O) GOTO 5
IF (DABS(B(I,J)).LE.BNEXT) GOTO 5
BNEXT=DABS(B(I,J))
IF (BNEXT.LE.BTRY) GOTO 5
BNEXT=BTRY
BTRY=DABS(B(I,J))
JC=J
5 CONTINUE
IF(BNEXT.GE.BMAX*BTRY) GOTO 6
BMAX=BNEXT/BTRY
IROW=I
JCOL=JC
6 CONTINUE
285
IF (ED(JC).EQ.O) GOTO 8
DETERM=0.0
RETURN
8 ID(JCOL)=l
IF (JCOL.EQ.IROW) GO TO 12
DO 10 J=1,N
SAVE=B(IROW,J)
B(IROW,J)=B(JCOL,J)
10 B(JCOL,J)=SAVE
DO 11 K=1,M
SAVE=D(IROW,K)
D(IROW,K)=D(JCOL,K)
11 D(JCOL,K)=SAVE
12 FF=1.0/B(JCOL,JCOL)
DO 13 J=1,N
13 B (JCOL, J)=B (JCOL, J) * FF
DO 14 K=1,M
14 D(JCOL,K)=D(JCOL,K)*FF
DO 18 1=1, N
IF (I.EQ.JCOL) GOTO 18
FF=B(I,JCOL)
DO 16 J=1,N
16 B(I,J)HB(I,J)FF*B(JCOL,J)
DO 17 K=1,M
17 D(I,K)=D(I,K)FF*D(JCOL,K)
18 CONTINUE
RETURN
END
SUBROUTINE BAND(J)
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
DIMENSION E(6,7,10001)
COMMON/BA/ A(6,6),B(6,6),C(6,10001),D(6,13),G(6),X(6,6),Y(6,6)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
101 FORMAT(15HDETERM=0 AT J=I4)
IF (J2) 1,6,8
1 NP1=NVAR+1
DO 2 I=1,NVAR
D(I,2*NVAR+1 )=G(I)
DO 2 L=1,NVAR
LPNL+NVAR
2 D(I,LPN)=X(I,L)
CALL MATINV(NVAR,2*NVAR+1,DETERM)
286
IF (DETERM) 4,3,4
3 PRINT 101,J
4 DO 5 K=1,NVAR
E(K,NP 1,1 )=D(K,2*NVAR+1)
DO 5 L=1,NVAR
E(K,L,1)= D(K,L)
LPN=L+NVAR
5 X(K,L)= D(K,LPN)
RETURN
6 DO 7 I=1,NVAR
DO 7 K=1,NVAR
DO 7 L=1,NVAR
7 D(I, K)=D(I, K)+A(I,L) * X(L, K)
8 IF (JNJ) 11,9,9
9 DO 10 I=1,NVAR
DO 10L=1,NVAR
G(I)=G(I)Y (I,L)*E(L,NP 1, J2)
DO 10 M=1,NVAR
10 A(I,L)=A(I,L)+Y(I,M) * E(M, L, J2)
11 DO 12 1=1, NY AR
D(I,NP1)=G(I)
DO 12 L= 1,NVAR
D(I,NP 1 )=D(I,NP 1 )+A(I,L)*E(L,NP 1, J1)
DO 12 K=1,NVAR
12 B (I,K)=B (I,K)+A(I, L) * E(L,K, J1)
CALL MATINV(N VAR, NP1, DETERM)
IF (DETERM) 14, 13, 14
13 PRINT 101,J
14 DO 15 K=1,NVAR
DO 15 M=1,NP1
15 E(K,M,J)= D(K,M)
IF (JNJ) 20,16,16
16 D0 17K=1,NVAR
17 C(K, J)=E(K,NP 1,J)
DO 18 JJ=2,NJ
M=NJJJ+1
DO 18 K=1,NYAR
C(K,M)=E(K,NP 1 ,M)
DO 18 L=1,NVAR
18 C(K,M)=C(K,M)+E(K,L,M) * C(L,M+1)
DO 19 L=1,NVAR
DO 19 K=1,NVAR
19 C(K, 1)=C(K, 1)+X(K,L)*C(L,3)
20 RETURN
END
287
C
c
SUBROUTINE SIMPSONSRULE(FUNC,OUTPUT)
C Combination of Simpson's 1/3rd and 3/8th rule to evaluate the
C integral involved in solving the convective diffusion equation.
C
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BB/ H,DEL
COMMON/SIMP/ XI( 10001)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
DIMENSION FUNC( 10001)
C
NJU = 2001
M = NJU
C M is the upper limit of J where simpson's routine is called in the
C main program
C
OUTPUT = 0.0
IF (MOD(NJU,2).NE.O AND. NJU.GT.l) THEN
OUTPUT = OUTPUT+3 *H*(FUNC(NJU3)+3 *(FUNC(NJU2)
+ +FUNC(NJUl))+FUNC(NJU))/8.0
M = NJU3
END IF
C
IF (M GT. 1) THEN
SUMI =0
DO 20 J= l.M1,2
SUMI = SUMI + FUNC(J+1)
20 CONTINUE
SUM2 = 0
DO 30 K = 2,M2,2
SUM2 = SUM2 + FUNC(K+1)
30 CONTINUE
OUTPUT = OUTPUT + H*(FUNC(l)+4*SUMl+2*SUM2+FUNC(M+l))/3.
END IF
RETURN
END
C
C
C
SUBROUTINE CURFTILDA
C
c
nono no
288
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/AA/XNU,DIF,FRK(90),BK,CK,MAI,CBULK(6),CONC(6,10001),
+ CONV(6,10001), AMTG, AMS S( 11 ),THMPRS S( 11 ),CINF,TRAN,
+ AAL,OM,NO,FA,QSS(11,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),
+ CDDL
COMMON/OP/CURFRTIL(60),CURFJTIL(60),THPTR( 11),
+ THPTJ(11),AMTR(11),AMTJ(11)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
C
DUM = (NO*FA*DIF*CINF/(l.TRAN))*(AAL*XNU/(3.*DIF))**(l./3.)
DUM  DUM*(OM/XNU)**0.5
C
DO 101= 1JDRAMAX
CR = 0.
CJ = 0.
DO Ml = 1,MMAXP1
C
DR = AMTR(M 1 )* THMPRS S(M 1 )+AMS S(M 1) * THPTR(M 1)
DR = DR* (DRA(I)) * * (2 * (M1 1))
CR = CR + DR
C
DJ = AMTJ(M1)*THMPRSS(M1)+AMSS(M1)*THPTJ(M1)
DJ = D J * (DRA(I)) **(2*(M11))
CJ = CJ + DJ
ENDDO
C
CURFRTEL(I) = DUM*CR
CURFJTIL(I) = DUM*CJ
10 CONTINUE
C
RETURN
END
DOUBLE PRECISION FUNCTION FACT(N)
Function subprogram to calculate the factorial of a given number
INTEGER N
IF (N . GE. 2) THEN
FACT = 1.
DO 15 1= 1,N
on k> oooooo
289
FACT = FACT*DFLOAT(I)
15 CONTINUE
ELSE
FACT = 1.
END IF
RETURN
END
DOUBLE PRECISION FUNCTION PROD(M,N)
Function PROD(M,N) gives the value of FACT(M)/FACT(N).
INTEGER M,N
IF (M.GE.l AND. N.LT.M) THEN
PROD = 1.
DO 20 I = N+1,M
PROD = PROD * DFLO AT(I)
CONTINUE
ELSE
PROD = 1.
END IF
RETURN
END
SUBROUTINE INTEGRAL
C
c
C Subroutine subprogram that evaluates the integral that is involved in
C calculating QRTILDA(L,NA1,MA1), QJTILDA(L,NA1,MA1).
C Please note that the values of the the definite integral are exact.
C There is no approximation involved.
C
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
C
DOUBLE PRECISION D,XNUM,DIN,Y(11,11),SUM,A1
DOUBLE PRECISION FACT,PROD
DO 10 N1 = 1,NMAXP1
DO 21 Ml = 1,MMAXP1
SUM = 0
oo o o o on
290
DO 30 K =0,M11
DO 40 L = 0,N11
Al = PR0D(M11,M1K1)/FACT(K)
XNUM = A1*(PR0D(4*(N11)2*L,2*(N11)L)/FACT(2*(N11)2*L))
DIN = FACT(L)
IA = 2**(2*N11)
IB = NlL+K
IC = (1)**(L+K)
D = (IC*XNUM/IA)*1/(DIN*IB)
SUM = SUM + D
40 CONTINUE
30 CONTINUE
YINT(N1,M1) = SUM
21 CONTINUE
10 CONTINUE
RETURN
END
SUBROUTINE QTILDA
This subroutine calculates the value of Q(N,M) for different values of
N and M.
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/OP/CURFRTIL(60), CURFJTIL(60), THPTR( 11),
+ THPTJ(11),AMTR(11),AMTJ(11)
COMMON/QTIL/QRTILDA(l 1,11),QJTILDA(11,11)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
EXTERNAL XMPR
C
PI = 22. DO/7. DO
DO 10 N = 1,NMAXP1
DO 20 M = 1,MMAXP1
DUM = (4*N3)*4*YINT(N,M)/(PI*XMPR(N))
QRTILDA(N,M) = DUM*THPTR(M)
QJTILDA(N,M) = DUM*THPTJ(M)
C
20 CONTINUE
C
10 CONTINUE
nono nono on
291
RETURN
END
DOUBLE PRECISION FUNCTION XMPR(N)
Subroutine subprogram that evaluates the value of M(2N)PRIME at 0.
IMPLICIT DOUBLE PRECISION (AH,0Z)
DOUBLE PRECISION FACT,PROD
PIINV = 7. DO/22. DO
XMPR = 2*(2**(2*(Nl))*FACT(Nl)/PROD(2*(Nl),Nl))**2*PIINV
RETURN
END
SUBROUTINE B TILDA
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/QTIL/QRTILD A( 11,11),QJTILDA(11,11)
COMMON/AA/XNU,DIF,FRK(90),BK,CK,MAI,CBULK(6),CONC(6,10001),
+ CONV(6,10001), AMTG, AMSS( 11 ),THMPRS S( 11 ),CINF,TRAN,
+ AAL,OM,NO,FA,QSS(l 1,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),
+ CDDL
COMMON/OP/CURFRTIL(60),CURFJTIL(60),THPTR( 11),
+ THPTJ(11),AMTR(11),AMTJ(11)
COMMON/N S/NV AR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/BTIL/BRTILD A( 11),BJTILDA(11),C1
COMMON/ETAS/ ALPHA,BETA,GAMMA,
+ XZ(60),YZ(60),VAR0(60),DNRZ(60),XI0,
+ ZETTELR(60),ZETTELJ(60),ETASTILR(60),
+ ETASTILJ(60),BZ(60),Y2
COMMON/CONC/ ETACTILR(60),ETACTILJ(60)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
COMMON/CUR/CURTILR(60),CURTILJ(60)
COMMON/CM/ CMTR(11),CMTJ(11)
on on
292
C
EXTERNAL XMPR,XLEGP
C
XK1 = Cl *R0/XKINF
CALL INTEGRAL
DO 15 J=1,NMAXP1
DUM1 = XK1 * (4 * J3 )/XMPR( J)
DUMR = 0.
DUMJ = 0.
DO 25 1= 1.MMAXP1
DUMR = DUMR+CMTR(I)*YINT(J,I)
DUMJ = DUMJ+CMTJ(I)*YINT(J,I)
25 CONTINUE
C
BRTILDA(J) = DUM1 *DUMR
BJTILDA(J) = DUM1 *DUMJ
C
15 CONTINUE
C
RETURN
END
SUBROUTINE PHITILDA
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/PHI/PHITR(60),PHITJ(60)
COMMON/N S/NV AR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/BTIL/BRTILDA( 11),BJTILDA(11),C1
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
COMMON/AA/XNU,DIF,FRK(90),BK,CK,MA 1 ,CBULK(6),CONC(6,10001),
+ CONV(6,10001),AMTG,AMSS(11),THMPRSS(11),CINF,TRAN,
+ AAL,OM,NO,FA,QSS(l 1,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),
+ CDDL
COMMON/CONC/ ETACTILR(60),ETACTILJ(60)
COMMON/ETAS/ ALPHABETA, GAMMA,
+ XZ(60),YZ(60),VAR0(60),DNRZ(60),XI0,
+ ZETTILR(60),ZETTILJ(60),ETASTILR(60),
+ ETASTILJ(60),BZ(60),Y2
EXTERNAL XMPR,XLEGP
DO 101= 1,IDRAMAX
nono o o o
293
C
PH3TR(I) = 0.
PHITJ(I) = 0.
C
DO 20 J= l.NMAXPl
C
PH3TR(I) = PHITR(I)+BRTILDA(J)*XLEGP(J1 ,ETA(I))
PHITJ(I) = PHITJ(I)+B JTELD A(J) * XLEGP(J1 ,ET A(I))
20 CONTINUE
C
PHITR(I) = PHITR(I)/C1
PHITJ(I) = PHITJ(I)/C1
C
10 CONTINUE
C
RETURN
END
DOUBLE PRECISION FUNCTION XLEGP(N,XIN)
This function calculates the legendre polynomial expansion for P(2*N,X)
INTEGER N,I,IA,IB,IC
DOUBLE PRECISION XIN,XNUM,DIN
DOUBLE PRECISION FACT,PROD,SUM
SUM = 0.
NP1=N+1
DO 10 K = 1,NP1
I=K1
IA = 4*N2*I
IB  2*NI
IC = 2*N2*I
IF (N EQ. 0) THEN
XLEGP = 1
RETURN
END IF
IF (XIN.EQ.O AND. IC.EQ.0) THEN
XNUM = ((PROD(I A, IB)/F ACT (IC))/F ACT(I))
ELSE
XNUM = ((PROD(IA,IB)/FACT(IC))/FACT(I))*XIN**IC
END IF
DIN = DFLO AT (2 * *(2 *N))
o o n o n
294
SUM = SUM + (1)**I*XNUM/DIN
10 CONTINUE
XLEGP=SUM
RETURN
END
C
C
SUBROUTINE ETASTILDA
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/SS/ ETASTSS(60),ZETASS(60),ETACSS(60),PH3SS(60),
+ Q2SS(60),CSURSS(60),XIDENSS(60)
COMMON/BTIL/BRTILDA( 11),BJTILDA(11),C1
COMMON/ETAS/ ALPHA,BETA,GAMMA,
+ XZ(60),YZ(60),VAR0(60),DNRZ(60),XI0,
+ ZETTILR(60),ZETTILJ(60),ETASTILR(60),
+ ETASTILJ(60),BZ(60),Y2
COMMON/OP/CURFRTIL(60),CURFJTIL(60),THPTR( 11),
+ THPTJ(11),AMTR(11),AMTJ(11),CSURTILR(60),
+ CSURTELJ(60)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
COMMON/AA/ XNU,DIF,FRK(90),BK,CK,MA1 ,CBULK(6),CONC(6,10001),
+ CONV(6,10001), AMTG, AMSS( 11 ),THMPRSS( 11 ),CINF,TRAN,
+ AAL,OM,NO,FA,QSS(11,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),
+ CDDL
DO 101= LIDRAMAX
XNUMR = GAMMA* XIDEN S S (I) * C SURTILR(I)/C SURS S (I)
XNUMR = CURFRTIL(I)XNUMR
XNUMJ = GAMMA*XIDENSS(I)*CSURTILJ(I)/CSURSS(I)
XNUMJ = CURFJTIL(I)XNUMJ
XDIN = XIO*((CSURSS(I)/CINF)**GAMMA)*XZ(I)
ETASTILR(I)= XNUMR/XDIN
ETASTILJ(I)= XNUMJ/XDIN
CONTINUE
RETURN
END
SUBROUTINE CONCTIL
C
o o o o on
295
C
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/BTIL/BRTILDA(l 1),BJTILDA(11),C1
COMMON/SS/ ETASTSS(60),ZETASS(60),ETACSS(60),PHISS(60),
+ Q2SS(60),CSURSS(60),XIDENSS(60)
COMMON/AA/ XNU,DIF,FRK(90),BK,CK,MA1,CBULK(6),CONC(6,10001),
+ CONV(6,10001),AMTG,AMSS(11),THMPRSS(11),CINF,TRAN,
+ AAL,OM,NO,FA,QSS(l 1,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),
+ CDDL
COMMON/CONC/ ETACTILR(60),ETACTILJ(60)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/OP/CURFRTEL(60),CURFJTEL(60),THPTR( 11),
+ THPTJ(11),AMTR(11),AMTJ(11),CSURTILR(60),
+ CSURTELJ(60)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
C
DO I = LIDRAMAX
CSURTILR(I)
+ = C1 *ETACTILR(I)/((1/CSURSS(I))(TRAN/CINF))
CSURTILJ(I)
+ = C1 *ETACTILJ(I)/((1/CSURSS(I))(TRAN/CINF))
ENDDO
C
RETURN
END
SUBROUTINE CALALTELDA
This subroutine calculates the AL coefficients using trapezoidal rule.
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
COMMON/AL/ ALTR(11),ALTJ(11)
COMMON/OP/CURFRTLL(60),CURFJTIL(60),THPTR(11),
+ THPTJ(11),AMTR(11),AMTJ(11),CSURTILR(60),
+ CSURTILJ(60)
COMMON/AA/ XNU,DEF,FRK(90),BK,CK,MA1,CBULK(6),CONC(6,10001),
+ CONV(6,10001), AMTG,AMSS(11),THMPRSS(11),CINF,TRAN,
+ AAL,OM,NO,FA,QSS(l 1,11),R0,IZ,TEMP,XN,R,XKINF,WW(90),
+ CDDL
o o o o no
296
C
20
10
DOUBLE PRECISION XLEGP
XINVCINF=1.0/CINF
DO 101= l.MMAXPl
SUMR = 0.
SUMJ = 0.
DO 20 J = 1,IDRAMAX
AR = (CSURTILR(J))*XINVCINF*XLEGP(I1,DRA(J))
SUMR = SUMR + AR*WE(J)
AJ = (CSURTILJ(J))*XINVCINF*XLEGP(I1,DRA(J))
SUMJ = SUMJ + AJ*WE(J)
CONTINUE
ALTR(I) = (4*1  3)*SUMR
ALTJ(I) = (4*1  3)*SUMJ
CONTINUE
RETURN
END
SUBROUTINE CALAMTILDA
This subroutine calculates the AM coefficints using AL coefficints.
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP1
COMMON/AL/ ALTR(11),ALTJ(11)
COMMON/OP/CURFRTIL(60),CURFJTIL(60),THPTR( 11),
+ THPTJ(11),AMTR(11),AMTJ(1 ÃXCSURTILRÃÃ“O),
+ CSURTILJ(60)
EXTERNAL COEFF
DO 101= l.MMAXPl
SUMR = 0.
SUMJ = 0.
DO 20 J = I,MMAXP1
C PRINT *,I,J,COEFF(J1 ,JI),'COEFF'
SUMR = SUMR + ALTR(J)*COEFF(J1, JI)
SUMJ = SUMJ + ALTJ(J)*COEFF(Jl,JI)
20 CONTINUE
AMTR(I) = SUMR
AMTJ(I) = SUMJ
10 CONTINUE
RETURN
non n nnnnnnnnn nnnnn
297
END
C
C
DOUBLE PRECISION FUNCTION COEFF(N,IR)
This function subprogram calculates the coefficients in the legendre
polynomial expansion.
INTEGER N,ER,IA,IB,IC
DOUBLE PRECISION FACT,PROD
DOUBLE PRECISION XNUM,DIN
IA = 4*N  2*IR
IB = 2*N  IR
IC = 2*N  2*IR
XNUM = (PROD(I A, IB)/FACT (IC))/F ACT (IR)
DIN = 2**(2*N)
COEFF  ((1 )* *IR)*XNUM/DIN
RETURN
END
SUBROUTINE RATCOMP(VR,VJ,CURR,CURJ,ZR,ZJ)
This calculates the ratio between two complex numbers VR + j VJ and
CURR + jCURJ
REAL* 8 VR,VJ,CURR,CURJ,ZR,ZJ,DIN
DIN = CURR**2 + CURJ**2
ZR = (VR*CURR + VJ*CURJ)/DIN
ZJ = (VJ*CURR  VR*CURJ)/DIN
RETURN
END
o n n o
298
SUBROUTINE CALCLTILDA
C
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/CL/ CLTR(11),CLTJ(11)
COMMON/CUR/CURTILR(60),CURTILJ(60)
COMMON/GAUQUA/ DRA(60),WE(60),IDRAMAX,YINT(11,11),ETA(60)
DOUBLE PRECISION XLEGP
C
DO 101= LMMAXPl
SUMR = 0.
SUMJ = 0.
DO 20 J= LIDRAMAX
CR = (CURTILR(J))*XLEGP(I1,DRA(J))
SUMR = SUMR + CR*WE(J)
CJ = (CURTILJ(J))*XLEGP(I1,DRA(J))
SUMJ = SUMJ + CJ*WE(J)
20 CONTINUE
CLTR(I) = (4*1  3)*SUMR
CLTJ(I) = (4*1  3)*SUMJ
10 CONTINUE
RETURN
END
C
SUBROUTINE CALCMTILDA
This subroutine calculates the AM coefficints using AL coefficints.
IMPLICIT DOUBLE PRECISION (AH,0Z)
IMPLICIT INTEGER* 8 (IN)
COMMON/NS/ NVAR,NJ,LFREQ,NMAXP 1 ,MMAXP 1
COMMON/CL/ CLTR( 11),CLTJ(11)
COMMON/CM/ CMTR( 11),CMTJ(11)
DOUBLE PRECISION COEFF
DO 101= LMMAXPl
SUMR = 0.
SUMJ = 0.
DO 20 J = I.MMAXP1
SUMR = SUMR + CLTR(J)*COEFF(Jl,JI)
SUMJ = SUMJ + CLTJ(J)*COEFF(J1, JI)
20 CONTINUE
CMTR(I) = SUMR
299
CMTJ(I) = SUMJ
10 CONTINUE
RETURN
END
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BIOGRAPHICAL SKETCH
Madhav Durbha received a Bachelor of Technology in Chemical Engineering from
the Indian Institute of Technology, Madras, in June 1993. He joined the PhD. program in
the Department of Chemical Engineering at the University of Florida in August 1993. He
conducted research at the CNRS, Paris, in the Spring/Summer of 1995 with financial
support from the CNRS laboratories. He received the Ray W. Fahien Teaching award
from the Department of Chemical Engineering at the University Florida for his outstanding
contributions toward undergraduate education in the capacity of Teaching Assistant
during the 1996/97 academic year. He also received Outstanding Academic Achievement
award from the College of Engineering, University of Florida, in 1998. After receiving his
Ph D., Madhav Durbha will be joining i2 Technologies Inc., Irving, TX.
310
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Professor of ^hpmical Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
OaacuJ A^'^aJUU.
Oscar D. Crisalle
Associate Professor of Chemical
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope/and quality, as a
dissertation for the degree of Doctor of Philosophy.
Luis H. GfacciaRikoio
Professor of Chemical Engineering
Umversity of South Florida
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
5.
Chang W+Park
Associate Professor of Chemical
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
pJL/Am
ChenChi Hsu
Professor of Aerospace
Engineering, Mechanics and
Engineering Science
This dissertation was submitted to the Graduate Faculty of the College of Engineering
and to the Graduate School and was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
\ugust 1998
Winfred M. Phillips
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School
LD
1780
1998
â€¢ M S3.
UNIVERSITY OF FLORIDA
I III III Hill III"
3 1262 08555 1025

