ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY FOR THE CHARACTERIZATION OF CORROSION AND CATHODIC PROTECTION OF
BURIED PIPELINES
By
KENNETH E. JEFFERS
A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE
UNIVERSITY OF FLORIDA
1999
ii
ACKNOWLEDGMENTS
Sincere thanks and appreciation go to Professor Mark Orazem for accepting me
into his research group and giving me the chance to learn about electrochemical
engineering. His technical and professional advisement were integral to the success of
this work. Thanks also to supervisory committee members Professor Ranganathan
Narayanan and Professor John Ambrose for their contributions to the completion of this
work. I would also like to thank Peter Zory for hearing my defense after very short notice.
I wish to acknowledge and express appreciation to the Pipeline Research
Committee International and the Gas Research Institute whose funding supported this
work.
Thanks to colleagues Steve Carson, Doug Riemer, Mike Membrino, and Kerry
Allahar for knowing more than me and sharing their experience and skills with me.
Thanks for allowing me to pick your brains. Additional thanks to Doug Riemer for
maintaining the group’s computing power.
Finally, and most importantly, special thanks to my wonderful wife, Beth, who
was willing to delay settling down into family life while I pursued this opportunity to
complete an advanced degree. None of this could have been possible without her
encouragement and support. She is truly a blessing, and I would be lost without her.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
CHAPTERS
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Steel Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Current-Potential Behavior of Steel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Surface Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Principles of EIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Statistical EIS Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Process Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.1 Reaction Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.2 Current Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 Cell Electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.4 Corrosion Cell Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.5 Instrumentation and Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Applied DC Bias and Frequency Range . . . . . . . . . . . . . . . . . . . . . . . . 393.3.2 Variable Amplitude Galvanostatic Modulation . . . . . . . . . . . . . . . . . . 413.3.3 Initial Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
iii
4 EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1 Corrected Cell Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Cylinder Electrode Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Experiment 1 - Modulation About the Corrosion Potential . . . . . . . . . 594.2.2 Experiment 2 - Modulation About 1.6 µA/cm2 . . . . . . . . . . . . . . . . . . 614.2.3 Experiment 3 - Modulation about 2.5 µA/cm2 . . . . . . . . . . . . . . . . . . . 624.2.4 Experiment 4 - Modulation about 4.0 µA/cm2 . . . . . . . . . . . . . . . . . . . 63
4.3 Discrete Holiday Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.1 Holiday Experiment 1 - Modulation About the Corrosion Potential . . 704.3.2 Holiday Experiment 2 - Modulation About 5.0 µA/cm2 . . . . . . . . . . . 71
5 DATA ANALYSIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Process Model Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.2 Quality of Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2.3 Regression Parameter Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.2.4 Parameter Values as a Function of Applied Current Density . . . . . . . 101
5.3 Estimation of Polarization Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.4 Link to Polarization Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7 SUGGESTIONS FOR FUTURE WORK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
APPENDICES
A FORTRAN CODE FOR BEM SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . 130
B LABVIEW CONTROL OF EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 141
C MEASUREMENT MODEL APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
D REGRESSION PARAMETER RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
iv
v
LIST OF TABLES
Table page
2-1. Parameter values used to calculate the polarization curve for steel in neutral to slightly basic, oxygenated soil electrolytes. Potentials werereferenced to the copper-copper sulfate (Cu/CuSO4) electrode. . . . . . . . . . . . . . . 8
3-1. Chemical analysis of the supplied pipeline grade, 5LX52, steel coupons. . . . . . . . 46
3-2. Results for the total current integrated on the electrode surface determined from the current distribution resulting from the BEM simulations. Also included is the calculated electrolyte resistance for both electrode types while accounting for the porosity of the solid matrix. The porosity or void fraction assumed for the calculation was 0.40. Also included are the results from impedance measurements and from using the anoderesistance formula, equation (3-7), for the cylinder electrode. . . . . . . . . . . . . . . . 46
3-3. Calculated concentrations of ionic species included in simulated soil electrolyte. Molarity units are in moles/liter. The calculatedconductivity is also included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3-4. Masses of salts in g/L added to water to prepare simulated soil electrolyte.The solution pH is included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3-5. Experimental outline including electrode type and applied current density. . . . . . . 48
5-1. Process model parameter values, at selected times, used to extrapolate the impedance response of the cylinder electrode maintained at thecorrosion potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5-2. Parameter values, at selected times, used to extrapolate the impedance response of the cylinder electrode at an applied cathodic current densityof 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5-3. Process model parameter values, at selected times, used to extrapolate the impedance response of the holiday electrode maintained at the corrosionpotential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5-4. Extrapolated values for Re, Z(0), and Rp from regression of the process modeland measurement models to the impedance data presented in Figure 5-43.. . . . 112
. 50
. . 51
. 52
. . 53
. . 54
. 55
. . 56
LIST OF FIGURES
Figure page
2-1. Calculated polarization curve of steel with the potential as a function of the applied current density. Current-potential curves are included for eachreaction contributing to the total current density. . . . . . . . . . . . . . . . . . . . . . . . . . 9
2-2. Schematic diagram of a circuit containing a resistor in series with a Voigtcircuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2-3. Nyquist plot for the circuit in Figure 2-2 with the parameter valuesRe = 10 Ω, Cd = 10 µF, and Rt = 250 Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2-4. Geometry for the diffusion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3-1. Schematic of the simulated holiday electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49
3-2. Axisymmetric plane, including boundary conditions, of the 1/8” holidayelectrode for BEM simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3. Current density and potential distributions, generated from BEM simulation, as a function of axial position on the 1/8” holiday electrode. The center of the holiday or conductive metal band was located 3” fromthe end of the bottom acrylic insulating piece. . . . . . . . . . . . . . . . . . . . . . . . .
3-4. Current density and potential distributions, generated from BEMsimulation, as a function of axial position on the cylinder electrode. . . . . . . . .
3-5. Schematic of corrosion cell body showing position of electrodes. . . . . . . . . . . .
3-6. Schematic of corrosion cell top cover piece.. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7. Corrosion cell flow diagram including instrumentation.. . . . . . . . . . . . . . . . . . . .
3-8. Preliminary experimental polarization curve for pipeline grade steel, generated from a galvanodynamic sweep from anodic to cathodic current densities at a rate of 0.3 µA/cm2 per minute. The closed circles correspond to the applied conditions listed in Table 3-5 for the cylinderelectrode experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
3-9. Preliminary impedance spectrum in Nyquist form to identify high frequency instrumental artifacts. The response is from the cylinder electrode in liquid electrolyte only, with κ = 0.00122 Ω-1cm-1, to variable amplitude galvanostatic modulation about the corrosion potential. The tested frequency range was 1000 Hz to 0.01 Hz. The calculated spectrum was generated from measurement model regressionparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4-1. The corrosion potential, measured with respect to a calomel referenceelectrode, as a function of time for the cylinder electrode. . . . . . . . . . . . . . . . . . 64
4-2. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about zeroapplied current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
4-3. Bode plots of the negative imaginary component as a function of frequency, at selected times, for the cylinder electrode in response tovariable amplitude galvanostatic modulation about zero applied current. . . . . . .65
4-4. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the cylinder electrode maintained at an appliedcathodic current density of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
4-5. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about anapplied cathodic DC current density bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . 66
4-6. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the cylinder electrode maintained at an appliedcathodic current density of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4-7. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about anapplied cathodic DC current density bias of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . 68
4-8. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the cylinder electrode maintained at an appliedcathodic current density of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
4-9. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about anapplied cathodic DC current density bias of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . 69
4-10. The corrosion potential, measured with respect to a calomel referenceelectrode, as a function of time for the holiday electrode.. . . . . . . . . . . . . . . . . . 72
vii
4-11. Nyquist plots at selected times for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about zeroapplied current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4-12. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the holiday electrode maintained at an applied cathodic current density of 5 µA/cm2. The increase in the potential at the end of the trace occurred after resetting the appliedcurrent to 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4-13. Nyquist plots at selected times for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about anapplied cathodic DC current density bias of 5.0 µA/cm2. . . . . . . . . . . . . . . . . . . 73
5-1. The impedance response in Nyquist form of the cylinder electrode to variable amplitude galvanostatic modulation about the corrosion potential, including the results for the process model regression using modulus weighting. The error bars represent the 95.4% confidence intervals for the model estimation for both the real and imaginary components. The data were generated 24 hours after the WE wasexposed to the electrolytic environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5-2. Both the normalized real and imaginary component residual errors, as a function of frequency, resulting from process model regression to thedata of Figure 5-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5-3. The normalized real component residual errors, as a function of frequency, resulting from process model regression to the data of Figure 5-1. Theestimated stochastic noise limits are included. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5-4. The normalized imaginary component residual errors, as a function of frequency, resulting from process model regression to the data of Figure5-1. The estimated stochastic noise limits are included.. . . . . . . . . . . . . . . . . . . 81
5-5. The normalized real component residual errors, as a function of frequency, resulting from process model regression to data generated from modulation about an applied DC current density bias of 1.6 µA/cm2. The impedance response was measured from the cylinder electrode after 24 hours of exposure. The estimated stochastic noise limits are includedwith the 95.4% confidence intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
viii
5-6. The normalized imaginary component residual errors, as a function of frequency, resulting from process model regression to data generated from modulation about an applied DC current density bias of 1.6 µA/cm2. The impedance response was measured from the cylinder electrode after 24 hours of exposure. The estimated stochastic errorstructure limits are included with the 95.4% confidence intervals. . . . . . . . . . . . 83
5-7. The diffusion time constant for the film and cell potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . . 90
5-8. The bulk layer diffusion time constant and WE potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . . 90
5-9. The ratio of the diffusivities of oxygen in the bulk to the film and WE potential as functions of time for the cylinder electrode with the appliedcurrent equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5-10. The calculated film thickness in microns and WE potential as functionsof time for the cylinder electrode with the applied current equal to zero. . . . . . . 92
5-11. The calculated bulk diffusion layer thickness in microns and WE potential as functions of time for the cylinder electrode with the appliedcurrent equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5-12. The calculated film thickness in microns as a function of potential for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . . 93
5-13. The calculated bulk diffusion layer thickness in microns as a function ofpotential for the cylinder electrode with the applied current equal to zero. . . . . 93
5-14. The effective charge transfer resistance and WE potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . . 94
5-15. The effective charge transfer resistance as a function of potential for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . . 94
5-16. The charge transfer resistance for oxygen reduction and WE potential as functions of time for the cylinder electrode with the applied currentequal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5-17. The charge transfer resistance for oxygen reduction as a function ofpotential for the cylinder electrode with the applied current equal to zero. . . . . 95
5-18. The diffusion impedance coefficient and WE potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . . 96
ix
5-19. The diffusion impedance coefficient as a function of potential for the cylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . 96
5-20. The cell capacitance and WE potential as functions of time for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . . 97
5-21. The cell capacitance as a function of potential for the cylinder electrodewith the applied current equal to zero.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5-22. The electrolyte resistance as a function of time for the cylinder electrodewith the applied current equal to zero.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5-23. The diffusion time constant for the film and WE potential as functions of time for the cylinder electrode with an applied DC current density biasof 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5-24. The diffusion time constant for the film as a function of potential for thecylinder electrode with an applied DC current density bias of 1.6 µA/cm2. . . . . 99
5-25. The effective charge transfer resistance and WE potential as functions of time for the cylinder electrode with an applied DC current density biasof 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5-26. The effective charge transfer resistance as a function of potential for thecylinder electrode with an applied DC current density bias of 4.0 µA/cm2. . . . 100
5-27. The diffusion time constant for the film after 4 days of exposure plottedas a function of applied current density for the cylinder electrode.. . . . . . . . . . 102
5-28. The bulk layer diffusion time constant after 4 days of exposure plotted asa function of applied current density for the cylinder electrode. . . . . . . . . . . . . 102
5-29. The ratio of the diffusivities of oxygen in the bulk to the film after 4 days of exposure plotted as a function of applied current density for thecylinder electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5-30. The calculated film thickness in microns after 4 days of exposure plottedas a function of applied current density for the cylinder electrode.. . . . . . . . . . 103
5-31. The calculated bulk diffusion layer thickness in microns after 4 days of exposure plotted as a function of applied current density for the cylinderelectrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5-32. The effective charge transfer resistance after 4 days of exposure plottedas a function of applied current density for the cylinder electrode.. . . . . . . . . . 104
x
5-33. The diffusion impedance coefficient after 4 days of exposure plotted as afunction of applied current density for the cylinder electrode. . . . . . . . . . . . . . 105
5-34. The cell capacitance after 4 days of exposure plotted as a function ofapplied current density for the cylinder electrode. . . . . . . . . . . . . . . . . . . . . . . 105
5-35. The electrolyte resistance after 4 days of exposure plotted as a functionof applied current density for the cylinder electrode. . . . . . . . . . . . . . . . . . . . . 106
5-36. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about zeroapplied current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5-37. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about anapplied cathodic DC current density bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . 113
5-38. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about anapplied cathodic DC current density bias of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . 114
5-39. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about anapplied cathodic DC current density bias of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . 114
5-40. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about zeroapplied current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5-41. The cell potential, measured with respect to the calomel reference electrode, and the applied current density as functions of time for theholiday electrode.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5-42. Cathodic polarization curve, generated from a galvanodynamic sweep performed using the holiday electrode. The sweep rate was 0.33 µA/cm2 per minute. The measured cell potential is plotted as a function of the applied current density including the experimental pointscorresponding to the step changes in Figure 5-41. . . . . . . . . . . . . . . . . . . . . . . 116
xi
146
147
. 148
149
. 154
5-43. Nyquist plots for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about several applied current densities. The collected data and extrapolated spectra using theprocess model are included.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5-44. The slope of the polarization curve, calculated from the data presented in Figure 5-42, as a function of applied current density for the holiday electrode. Extrapolated polarization resistance values, using both the process model and the measurement model approach, are included forthe experiments in Figure 5-43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5-45. The natural logarithm of the effective charge transfer resistance plotted as a function of potential including the equation for the fitted line. Values were obtained from process model regression to impedance response data collected for the cylinder electrode maintained at thecorrosion potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122
5-46. The corrosion current as a function of time calculated from the Tafel slope for iron dissolution determined for the cylinder electrodemaintained at the corrosion potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5-47. The natural logarithm of the effective charge transfer resistance plotted as a function of potential, including the equation for the fitted line. Values were obtained from process model regression to impedance response data collected for the cylinder electrode with an appliedcathodic DC current density of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5-48. The hydrogen evolution current density as a function of time calculated from the Tafel slope determined from the cylinder electrode with anapplied cathodic DC current density of 4.0 µA/cm2.. . . . . . . . . . . . . . . . . . . . . 123
A-1. The holiday electrode cell boundary including the x and y coordinates foreach vertex. The edges are numbered for a total of 8. . . . . . . . . . . . . . . . . . . . . 131
B-1. Flow chart for operation of main control, ‘1260/273_main_8/98.vi’. . . . . . . . . . .
B-2. Flow chart for operation of ‘I_V monitor.vi’. . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-3. Flow chart for operation of ‘run impedance scan.vi’.. . . . . . . . . . . . . . . . . . . . .
B-4. Flow chart for operation of ‘poll 1260 for data.vi’. . . . . . . . . . . . . . . . . . . . . . . .
C-1. The impedance response and the measurement model prediction, in Nyquist form, from a preliminary scan conducted on the cylinder electrode using variable amplitude galvanostatic modulation about thecorrosion potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
C-2. The normalized real component residual errors with confidence intervals, as a function of frequency, resulting from measurement model regression, using modulus weighting, to the real component of the datain Figure C-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
C-3. The normalized residual errors with confidence intervals, as a function of frequency, between the imaginary data of Figure C-1 and the predicted imaginary component resulting from measurement model regression tothe real component of the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
C-4. The data of Figure C-1, in Nyquist form, including the full set of replicatescans, after rejecting the high frequency artifacts. . . . . . . . . . . . . . . . . . . . . . . . 156
C-5. The residual errors, as a function of frequency, for the real component of the impedance resulting from measurement model regression of 5 line shapes, using modulus weighting, to the complex data of each individualscan of Figure C-4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C-6. The residual errors, as a function of frequency, for the imaginary component of the impedance resulting from measurement model regression of 5 line shapes, using modulus weighting, to the complexdata of each individual scan of Figure C-4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C-7. The standard deviation, as a function of frequency, of the real and imaginary stochastic errors calculated from the real and imaginary residual errors of Figure C-5 and Figure C-6, respectively. The model for the standard deviation includes the parameters, with values,β = 0.0017792 and δ = 0.021916. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
C-8. The normalized residual errors, as a function of frequency, between the imaginary data of scan 1, shown in Figure C-4, and the predicted values resulting from measurement model regression of 3 line shapes, using error structure weighting, to the real component of the data. Error structure weighting was used. The plot includes the confidence intervalsand the limits of the stochastic error structure model. . . . . . . . . . . . . . . . . . . . . 159
C-9. The normalized residual errors, as a function of frequency, between the real data of scan 1, shown in Figure C-4, and the predicted values resulting from measurement model regression of 2 line shapes, using error structure weighting, to the imaginary component of the data. The plot includes the confidence intervals and the limits of the stochasticerror structure model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
xiii
C-10. The normalized residual errors, as a function of frequency, for the imaginary component of scan 1, shown in Figure C-4, resulting from measurement model regression of 2 line shapes, using error structure weighting, to the imaginary component of the data. The plot includesconfidence intervals and the limits of the stochastic error structure model.. . . . 161
C-11. The normalized real component residual errors, as a function of frequency, resulting from a complex fit of 5 line shapes, using error structure weighting, after rejecting inconsistent high and low frequency points from the data shown in Figure C-1. The confidence intervals andstochastic error structure limits are included. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
C-12. The normalized imaginary component residual errors, as a function of frequency, resulting from a complex fit of 5 line shapes, using error structure weighting, after rejecting inconsistent high and low frequency points from the data presented in Figure C-1. The confidence intervalsand stochastic error structure limits are included. . . . . . . . . . . . . . . . . . . . . . . . 163
D-1. The electrolyte resistance as a function of time for the cylinder electrodewith the applied current equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
D-2. The diffusion time constant for the film and cell potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . 166
D-3. The diffusion time constant for the film as a function of potential for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . 166
D-4. The bulk layer diffusion time constant and cell potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . 167
D-5. The bulk layer diffusion time constant as a function of potential for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . 167
D-6. The ratio of the diffusivities of oxygen in the bulk to the film and cell potential as functions of time for the cylinder electrode with the appliedcurrent equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
D-7. The ratio of the diffusivities of oxygen in the bulk to the film as a functionof potential for the cylinder electrode with the applied current equal to zero. . . 168
D-8. The calculated film thickness in microns and cell potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . 169
D-9. The calculated film thickness in microns as a function of potential for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . 169
xiv
D-10. The calculated bulk diffusion layer thickness in microns and cell potential as functions of time for the cylinder electrode with the appliedcurrent equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
D-11. The calculated bulk diffusion layer thickness in microns, as a function ofpotential for the cylinder electrode with the applied current equal to zero. . . . 170
D-12. The effective charge transfer resistance and cell potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . 171
D-13. The effective charge transfer resistance as a function of potential for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . 171
D-14. The charge transfer resistance for oxygen reduction and cell potential as functions of time for the cylinder electrode with the applied currentequal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
D-15. The charge transfer resistance for oxygen reduction as a function ofpotential for the cylinder electrode with the applied current equal to zero. . . . . 172
D-16. The diffusion impedance coefficient and cell potential as functions oftime for the cylinder electrode with the applied current equal to zero. . . . . . . . 173
D-17. The diffusion impedance coefficient as a function of potential for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . 173
D-18. The cell capacitance and cell potential as functions of time for thecylinder electrode with the applied current equal to zero. . . . . . . . . . . . . . . . . . 174
D-19. The cell capacitance as a function of potential for the cylinder electrodewith the applied current equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
D-20. The electrolyte resistance as a function of time for the cylinder electrodewith an applied DC current density bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . 175
D-21. The diffusion time constant for the film and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
D-22. The diffusion time constant for the film as a function of potential for thecylinder electrode with an applied DC current density bias of 1.6 µA/cm2. . . . 176
D-23. The bulk layer diffusion time constant and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
xv
D-24. The bulk layer diffusion time constant as a function of potential for thecylinder electrode with an applied DC current density bias of 1.6 µA/cm2. . . . 177
D-25. The ratio of the diffusivities of oxygen in the bulk to the film and the cell potential as functions of time for the cylinder electrode with an appliedDC current density bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
D-26. The ratio of the diffusivities of oxygen in the bulk to the film as a function of potential for the cylinder electrode with an applied DCcurrent density bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
D-27. The calculated film thickness and cell potential as functions of time for the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
D-28. The calculated film thickness as a function of potential for the cylinderelectrode with an applied DC current density bias of 1.6 µA/cm2. . . . . . . . . . . 179
D-29. The calculated bulk diffusion layer thickness and cell potential as functions of time for the cylinder electrode with an applied DC currentdensity bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
D-30. The calculated bulk diffusion layer thickness as a function of potential for the cylinder electrode with an applied DC current density bias of1.6 µA/cm2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
D-31. The effective charge transfer resistance and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
D-32. The effective charge transfer resistance as a function of potential for thecylinder electrode with an applied DC current density bias of 1.6 µA/cm2. . . . 181
D-33. The diffusion impedance coefficient and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
D-34. The diffusion impedance coefficient as a function of potential for thecylinder electrode with an applied DC current density bias of 1.6 µA/cm2. . . . 182
D-35. The cell capacitance and cell potential as functions of time for thecylinder electrode with an applied DC current density bias of 1.6 µA/cm2. . . . 183
D-36. The cell capacitance as a function of potential for the cylinder electrodewith an applied DC current density bias of 1.6 µA/cm2. . . . . . . . . . . . . . . . . . . 183
xvi
D-37. The electrolyte resistance as a function of time for the cylinder electrodewith an applied DC current density bias of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . 184
D-38. The diffusion time constant for the film and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
D-39. The diffusion time constant for the film as a function of potential for thecylinder electrode with an applied DC current density bias of 2.5 µA/cm2. . . . 185
D-40. The bulk layer diffusion time constant and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
D-41. The bulk layer diffusion time constant as a function of potential for thecylinder electrode with an applied DC current density bias of 2.5 µA/cm2. . . . 186
D-42. The ratio of the diffusivities of oxygen in the bulk to the film and the cell potential as functions of time for the cylinder electrode with an appliedDC current density bias of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
D-43. The ratio of the diffusivities of oxygen in the bulk to the film as a function of potential for the cylinder electrode with an applied DCcurrent density bias of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
D-44. The calculated film thickness and cell potential as functions of time for the cylinder electrode with an applied DC current density bias of2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
D-45. The calculated film thickness as a function of potential for the cylinderelectrode with an applied DC current density bias of 2.5 µA/cm2. . . . . . . . . . . 188
D-46. The calculated bulk diffusion layer thickness and cell potential as functions of time for the cylinder electrode with an applied DC currentdensity bias of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
D-47. The calculated bulk diffusion layer thickness as a function of potential for the cylinder electrode with an applied DC current density bias of2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
D-48. The effective charge transfer resistance and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
xvii
D-49. The effective charge transfer resistance as a function of potential for thecylinder electrode with an applied DC current density bias of 2.5 µA/cm2. . . . 190
D-50. The diffusion impedance coefficient and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
D-51. The diffusion impedance coefficient as a function of potential for thecylinder electrode with an applied DC current density bias of 2.5 µA/cm2. . . . 191
D-52. The cell capacitance and cell potential as functions of time for thecylinder electrode with an applied DC current density bias of 2.5 µA/cm2. . . . 192
D-53. The cell capacitance as a function of potential for the cylinder electrodewith an applied DC current density bias of 2.5 µA/cm2. . . . . . . . . . . . . . . . . . . 192
D-54. The electrolyte resistance as a function of time for the cylinder electrodewith an applied DC current density bias of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . 193
D-55. The diffusion time constant for the film and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
D-56. The diffusion time constant for the film as a function of potential for thecylinder electrode with an applied DC current density bias of 4.0 µA/cm2. . . . 194
D-57. The bulk layer diffusion time constant and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
D-58. The bulk layer diffusion time constant as a function of potential for thecylinder electrode with an applied DC current density bias of 4.0 µA/cm2. . . . 195
D-59. The ratio of the diffusivities of oxygen in the bulk to the film and the cell potential as functions of time for the cylinder electrode with an appliedDC current density bias of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
D-60. The ratio of the diffusivities of oxygen in the bulk to the film as a function of potential for the cylinder electrode with an applied DCcurrent density bias of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
D-61. The calculated film thickness and cell potential as functions of time for the cylinder electrode with an applied DC current density bias of4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
xviii
D-62. The calculated film thickness as a function of potential for the cylinderelectrode with an applied DC current density bias of 4.0 µA/cm2. . . . . . . . . . . 197
D-63. The calculated bulk diffusion layer thickness and cell potential as functions of time for the cylinder electrode with an applied DC currentdensity bias of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
D-64. The calculated bulk diffusion layer thickness as a function of potential for the cylinder electrode with an applied DC current density bias of4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
D-65. The effective charge transfer resistance and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
D-66. The effective charge transfer resistance as a function of potential for thecylinder electrode with an applied DC current density bias of 4.0 µA/cm2. . . . 199
D-67. The diffusion impedance coefficient and cell potential as functions of time for the cylinder electrode with an applied DC current density biasof 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
D-68. The diffusion impedance coefficient as a function of potential for thecylinder electrode with an applied DC current density bias of 4.0 µA/cm2. . . . 200
D-69. The cell capacitance and cell potential as functions of time for thecylinder electrode with an applied DC current density bias of 4.0 µA/cm2. . . . 201
D-70. The cell capacitance as a function of potential for the cylinder electrodewith an applied DC current density bias of 4.0 µA/cm2. . . . . . . . . . . . . . . . . . . 201
xix
xx
Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY FOR THE CHARACTERIZATION OF CORROSION AND CATHODIC PROTECTION OF
BURIED PIPELINES
By
Kenneth E. Jeffers
August 1999
Chairman: Mark E. OrazemMajor Department: Chemical Engineering
An electrochemical cell was constructed to simulate a steel pipeline buried in low
ionic strength soil and connected to a cathodic protection system for corrosion prevention.
Electrochemical impedance spectroscopy measurements were performed to monitor
surface film formation and changes in charge transfer reaction kinetics. Statistical models
were regressed to impedance data to identify nonstationary behavior and to estimate the
stochastic error structure of the measurements. A process model for the impedance
response was developed by considering contributions to the total current flow within the
cell and diffusion of reacting species. Regression of the process model to data yielded
parameter values changing with time. The regression parameters were used to extrapolate
asymptotic resistances and were linked to the polarization behavior of steel. The
experimental and analytical methods developed were useful for monitoring the time
dependent electrochemical behavior of steel at different levels of cathodic protection.
CHAPTER 1INTRODUCTION
The motivation for this work stems from the need to gain insight into the design
and operation of cathodic protection (CP) systems for networks of buried pipelines in
service for the transmission of petroleum. Currently, sophisticated boundary element
models are being developed to characterize CP systems to determine if they provide an
adequate level of corrosion prevention for given environmental conditions. Such models
can account for the influence of discrete coating holidays, multiple pipelines in a right-of-
way with single or multiple CP systems, the use of mixed anodes, and variations in
coating properties along a length of pipe [1, 2].
The complications associated with the modeling efforts arise from the need to
employ nonlinear boundary conditions for determining the current and potential
distributions along a given length of pipe and at points within the surrounding soil [1- 5].
The boundary conditions are often characterized by curve fits of experimental data. Thus,
it is important to generate representative data from well-designed experiments which
consider general field conditions. Important physical effects include the role of soil
chemistry and the formation of films on exposed metal surfaces, oxygen diffusion through
porous media including coatings and films, and charge transfer reaction kinetics.
In previous work, the role of film formation and the time dependent polarization
behavior of steel have been investigated. Mathematical models and experimental
procedures were developed and based on real time, current and potential measuring
1
2
methods [6, 7]. Kinetic and transport parameters were regressed from potential-time data
and related to the polarization behavior of steel. The results gave insight into the time
scales necessary to polarize metal structures to assure a desired level of cathodic
protection has been achieved.
The objective of the present work was to develop an experimental method using
electrochemical impedance spectroscopy (EIS) to measure the frequency response of
pipeline grade steel subjected to soil environments and cathodic protection. EIS has been
shown to be a sensitive technique for monitoring non-stationary behavior, which proved to
be ideal for exploring corrosion systems where film formation contributes to influence
reaction kinetics and transport properties over time.
In previous work EIS has been used to monitor corrosion processes and has been
used extensively for characterizing the performance of protective polymer coatings [8].
However, the difficulty associated with EIS is data interpretation. Typically, EIS spectra
are analyzed by fitting equivalent electrical circuits, containing elements such as resistors
and capacitors, to the data [8, 9, 12]. This allows determination of trends in diffusion time
constants and changes in high and low frequency resistance limits. However, it may be
possible to fit several different circuit models to the data, which may or may not explain
the physical phenomena occurring within an electrochemical cell. Others have
endeavored to analyze EIS spectra by fitting models developed from a knowledge of
faradaic and transport processes associated with charge transfer reactions and diffusion of
reacting species from the bulk electrolyte to the metal surface [9-11]. For this work, a
process model was developed by considering the anodic and cathodic electrochemical
3
reactions of pipeline grade carbon steel exposed to oxygenated electrolytic soils and by
solving the governing equations of diffusion for the reacting species.
Well-controlled experiments were designed to assure the generation of reliable
data suitable for regression of process models. To guarantee symmetric current and
potential distributions, an electrochemical cell was designed and constructed with a
cylindrical geometry. A stationary cylindrical coupon, machined from pipeline grade
steel, served as the working electrode. The coupon was embedded in a solid sand matrix,
saturated with low ionic strength electrolyte containing species present in typical soils,
and surrounded by a counter electrode consisting of a platinum-rhodium alloy mesh
screen. Experiments were conducted by controlling the net current flow between the
working and counter electrodes. Appropriate current values were applied to simulate the
conditions of cathodic protection.
The impedance spectra generated for this work were analyzed by regressing
statistical and process models to the data. Application of statistical models allowed
estimation of the measurement error structure, which gave insight to the reliability of the
data. Application of the process model yielded time dependent parameters useful for
monitoring changes in reaction kinetics and diffusion properties with time as films formed
on the steel surface. The thicknesses of the film and bulk diffusion layers were estimated,
and asymptotic spectral values were calculated.
The results from the EIS work showed that the electrochemical system exhibited
non-stationary behavior long after the measured potential had reached a steady state. The
experimental method was useful for extracting parameters changing with time, and such
parameters could then be linked to the polarization behavior of steel.
CHAPTER 2THEORY
2.1 Steel Corrosion
The present work considers pipelines buried in moist, oxygenated soils with
neutral to slightly basic pH. External corrosion of the pipeline surface occurs via chemical
attack from the surrounding medium. Electrons held by the metal are transferred to
electrolyte species during electrochemical oxidation-reduction, half-cell reactions. Metal
dissolution is the anodic reaction as iron, the major component in the pipeline metal alloy,
is oxidized to ferrous ions according to
(2-1)
Reduction of oxygen is the cathodic reaction according to
(2-2)
The rate of oxygen reduction is limited to the rate at which oxygen diffuses to the steel
surface from the surroundings. As the pipeline is polarized to more cathodic potentials,
for example, when connected to a CP system, hydrogen evolution occurs at increasing
rates as water is reduced according to
(2-3)
The current density of the metal surface can be related to the rate of the
electrochemical reactions according to Faraday’s law
Fe Fe2+
2e-+→
O2 2H2O 4e- 4OH-→+ +
2H2O 2e-+ H2 2OH-+→
4
5
(2-4)
where n is the number of electrons transferred during the reaction, and F is the Faraday
constant. The total current density on the pipeline is the sum of the current contributions
from the anodic and cathodic reactions (2-1), (2-2), and (2-3) according to
(2-5)
The sign convention for equation (2-5) is that anodic currents are positive and cathodic
currents are negative.
2.2 Current-Potential Behavior of Steel
The current-potential behavior of a metal in a given electrolyte is shown by the
polarization curve, which can be generated experimentally by performing dynamic sweeps
controlling one electrical quantity through a sequenced range of values and measuring the
other at each point in the sweep. Galvanodynamic sweeps are performed by ramping or
stepping the current and measuring potential; whereas, in potentiodynamic experiments
potential is controlled. The data are usually presented on a plot with potential on the
vertical axis and current density plotted on the horizontal axis on a logarithmic scale. The
polarization curve is often used as a boundary condition for cathodic protection modeling
[1-5], and is a tool for determining the corrosion potential, Tafel slopes, and corrosion
rates [12, 13]. Typical current-potential behavior for steel is presented in Figure 2-1. The
curve was generated by calculating current density values at applied potentials using an
equation adapted by Orazem et al. [3, 4] from that developed by Nisancioglu [7]
according to
inF------- r=
itot iFe iO2iH2
+ +=
6
(2-6)
The terms in equation (2-6) correspond to the current contributions from reactions (2-1),
(2-2), and (2-3), respectively, and V is the potential of the steel measured with respect to a
reference electrode located in the soil electrolyte adjacent to the pipe. The parameters
, , and are the Tafel slopes for each reaction, and is the mass transfer
limited current density due to oxygen reduction. The parameters , , and
are effective equilibrium potentials that include the influence of exchange current
densities, temperature, and concentrations. For example,
(2-7)
where is the exchange current density for iron dissolution [5]. The equilibrium
potential, , can be determined from the Nernst equation according to
(2-8)
where is the standard potential for iron dissolution and is the concentration
of ferrous ions [12]. Figure 2-1 includes curves for the contribution of each separate
reaction as well as the net total current density. The values used for the calculation are
listed in Table 2-1. All potentials were referenced to the copper-copper sulfate
( ) electrode.
Because of limitations with placement of the reference electrode, the polarization
curve must be corrected for the ohmic drop due to current flow through the cell
electrolyte. Many available potentiostats are equipped with options for performing IR
itot 10V V∗Fe–
βFe
--------------------- 1ilim O2,--------------- 10
V V∗O2–( )
βO2
--------------------------
+
1–
10
V V∗H2–( )–
βH2
-----------------------------
––=
βFe βO2βH2
ilim O2,
V∗Fe V∗O2V∗H2
V∗Fe EFe βFe i0 Fe,( )log–=
i0 Fe,
EFe
EFe E°Fe βFe Fe2+[ ]log+=
E°Fe Fe2+[ ]
Cu CuSO4⁄
7
compensation routines to correct the potential measurements. One such method is the
current interrupt technique [14, 15]. The current interrupt routine begins by measuring the
potential and abruptly turning off the cell current. Then, two or more off-potential
measurements are made separated by short time delays usually on the order of 50
milliseconds. A line is fit through the points, and a potential value is extrapolated back to
the point of the current interrupt initiation. The difference between the extrapolated value
and the potential before the interrupt is the ohmic drop. After the routine is finished, the
cell current is resumed to its previous setting or stepped to a new desired level.
Many factors lead to uncertainty in determination of the ohmic drop by use of the
current interrupt method. Errors arise because the time delay between the potential
measurements after the current interrupt is somewhat arbitrary. Increasing or decreasing
the time delay and number of potential measurements recorded can change the slope of the
line fit to the points, thus leading to uncertainty when extrapolating back to the start of the
interrupt. Also, as surface films begin to form, a sudden drastic change in the current can
disrupt the surface causing oscillations or spikes in the potential transient. To avoid
uncertainty in the determination of the cell ohmic resistance, electrochemical impedance
spectroscopy can be employed. As discussed in section 2.4, the ohmic resistance is the
high frequency limit of the impedance.
8
Table 2-1. Parameter values used to calculate the polarization curve for steel in neutral to slightly basic, oxygenated soil electrolytes. Potentials were referenced to the copper-copper sulfate (Cu/CuSO4) electrode.
Parameter Value
-526 mV
-104 mV
-955 mV
59 mV/decade
59 mV/decade
118 mV/decade
1.02 (0.95 mA/ft2)
V∗Fe
V∗O2
V∗H2
βFe
βO2
βH2
ilim O2, µA cm2⁄
9
Figure 2-1. Calculated polarization curve of steel with the potential as a function of the applied current density. Current-potential curves are included for each reaction contributing to the total current density.
-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
0.01 0.1 1 10 100
Current Density, mA/ft2
Pot
entia
l, V
(C
u/C
uSO
4 )
H2 Evolution
Corrosion
O2 Reduction
Total Current
10
2.3 Surface Films
Surface films form due to the localized increase in alkalinity at the steel surface
resulting from the cathodic production of hydroxide ions. To maintain charge neutrality, a
concentration gradient of cations develops near the surface, and several precipitation
reactions can occur. Ferrous ions react with hydroxide ions to form ferrous hydroxide,
Fe(OH)2, which is then further oxidized to ferric hydroxide, Fe(OH)3, also known as rust.
Calcareous deposits have been observed to form as the applied potential becomes
more cathodic. For example, carbonate ions, present from mineral sources or dissolved
carbon dioxide, can react with calcium and magnesium cations to from carbonate
precipitates. At more significant cathodic potentials, the rate of hydroxide ion production
may be enough to increase the pH, near the surface, to levels high enough such that
magnesium hydroxide can also precipitate.
Calcareous deposits reduce corrosion rates by acting as resistive coatings which
inhibit transport of oxygen. Films have been reported to exhibit both blocking and porous
behavior [12]. Carson and Orazem calculated large Tafel slopes, approximately one order
of magnitude larger than literature values, for steel in saturated soils [6]. They attributed
the behavior to calcareous film formation, noting that a large potential shift was required
to reduce the corrosion current.
2.4 Principles of EIS
In situ analysis of an electrochemical system is often performed using electrical
methods, since the electrical response can be attributed to the kinetics of the surface
reactions. Electrochemical impedance spectroscopy is a frequency response technique
where a sinusoidal potential is applied to an electrochemical system, and the responding
11
and
to
ely.
d on
sinusoidal current signal is measured. Typically, a spectrum is generated by sweeping a
range of frequencies and measuring the impedance at each point. Since the polarization of
electrochemical systems can exhibit highly nonlinear behavior (as shown in Figure 2-1),
impedance measurements are normally conducted using small amplitude perturbation
signals. This approach allows confinement to an approximately linear segment of the
polarization curve. In this pseudo-linear system, the response will oscillate at the same
frequency as the input, will be phase shifted, and will be free of harmonics.
Analysis of the input and output signals leads to determination of the cell
impedance. The input potential signal can be expressed in cartesian and polar variables as
(2-9)
where is the oscillating potential at time t, is the signal amplitude, and ω is the
angular frequency. The responding current signal has amplitude and phase shift φ
according to
(2-10)
where the imaginary number . The complex impedance follows Ohm’s law
is found as the transfer function relating the potential and current signals according
(2-11)
where Zr and Zj are the real and imaginary parts of the complex impedance, respectiv
Impedance data is typically presented in a Nyquist plot where the negative of the
imaginary component is plotted on the vertical axis, and the real component is plotte
the horizontal axis.
V t( ) V0 ωt( )cos V0 jωt exp= =
V t( ) V0
I0
I t( ) I0 ωt φ–( )cos I0 j ωt φ–( ) exp= =
j 1–=
ZV t( )I t( )----------- Z
ωt( )cosωt φ–( )cos
----------------------------- Z jφ( )exp Z φcos j φsin+( ) Zr jZj+= = = = =
12
f
ore
Often, EIS data is interpreted using equivalent circuit models made up of resistors,
capacitors, and other elements. For example, Figure 2-2 presents a circuit with a resistor
in series with a Voigt circuit containing a resistor and capacitor in parallel. The circuit is a
simple model for the impedance response of an electrode process. It includes an
electrolyte resistance, Re, the double layer capacitance, Cd, and a charge transfer
resistance, Rt. The impedance for Randle’s cell is calculated as
(2-12)
where τ is the time constant associated with the RC circuit. The nyquist plot for the circuit
with an electrolyte resistance of 10 Ω, double layer capacitance of 10 µF, and charge
transfer resistance of 250 Ω is shown in Figure 2-3. At high frequency, the denominator o
the last term in equation (2-12) becomes very large making the whole term negligible
compared to the first term. Thus, the high frequency limit for the impedance is the
electrolyte resistance, Re, as shown in Figure 2-3 where the left end of the semicircle
intersects the real axis. At very low frequencies, jτω in equation (2-12) approaches zero,
and the result for the impedance is Re + Rt, shown in Figure 2-3 where the right end of the
semicircle intersects the real axis. The Voigt circuit is a starting point for developing m
complex equivalent circuit models. EIS data can also be interpreted using models
developed from a knowledge of physical processes occurring within the cell. Such a
modeling approach is developed in section 2.6.
Z Re1
1Rt
----- jωCd+------------------------+ Re
Rt
1 jτω+------------------+= =
13
Figure 2-2. Schematic diagram of a circuit containing a resistor in series with a Voigt circuit.
Figure 2-3. Nyquist plot for the circuit in Figure 2-2 with the parameter values Re = 10 Ω, Cd = 10 µF, and Rt = 250 Ω.
Cd
Rt
Re
0
50
100
150
0 50 100 150 200 250 300
Zr, Ω
-Zj ,
14
2.5 Statistical EIS Data Analysis
Electrochemical systems involving time dependent film formation usually exhibit
non-stationary behavior during the time required to generate an impedance spectrum.
Impedance data collected under non-stationary conditions will fail to satisfy the Kramers-
Kronig relations. Since most process models applied to impedance spectra assume the
steady-state, it is important to determine whether the collection time was short enough to
model the system as stationary. A statistical technique of regressing measurement models
to impedance spectra has been developed for filtering out non-stationary behavior [16-21].
The measurement model takes the form of the line shape based on the Kramers-Kronig-
consistent Voigt circuit (see Figure 2-2) with impedance given by
(2-13)
where Z0 represents the high frequency impedance or electrolyte resistance, is a
resistance parameter, and τk is an RC time constant. The technique follows an iterative
procedure of adding successive line shapes to the model followed by regression to the
data. The confidence intervals for the parameter estimates are calculated, and the number
of parameters, necessary to fit the spectra, is constrained by the requirement that the
95.4% confidence intervals for each parameter must not include zero.
The measurement model regression technique is also used to determine the nature
of the experimental errors. The residual errors between the data and the model consist of
systematic and stochastic contributions, and , respectively. The systematic
errors consist of lack of fit errors, , due to inadequacies of the model, and bias errors,
Z ω( ) Z0
Rk
1 jτkω+--------------------
k∑+=
Rk
εsyst εstoch
εlof
15
, associated with nonstationary behavior, , and instrumental artifacts, . Thus,
the experimental errors at any frequency can be expressed as
(2-14)
where is the model value for the complex impedance Z [19]. The approach is to collect
consecutive pseudo-replicate impedance spectra and to regress the measurement model to
each scan separately. By fitting the same number of line shapes to each replicate, the non-
stationary error contribution is effectively filtered out as the regressed parameter values
are adjusted for each individual scan. The errors due to instrumental artifacts are assumed
to be constant from one experiment to another, and since one model was regressed to each
replicate data set, the lack of fit error contribution is also constant. Another assumption
stipulates that the stochastic errors, , are normally distributed
with mean . The standard deviations for the real and imaginary components of the
errors at each frequency can be estimated from the deviations of the residual errors from
the mean value by
(2-15)
where and are the calculated variances for the real and imaginary components of
the residual errors, respectively, N is the number of data points at each frequency, and
(2-16)
εbias εns εins
Z Z– εlof εns εins+( ) εstoch+ +=
Z
εstoch εstoch r, jεstoch j,+=
ε 0=
σr2 εres r k, , εres r,–( )2
N 1–-------------------------------------------
k 1=
N
∑=
σj2 εres j k, , εres j,–( )2
N 1–------------------------------------------
k 1=
N
∑=
σr2 σj
2
εres mean εlof εins+( )=
16
Since the standard deviations of the stochastic errors are functions of frequency, a model
for the error structure was developed assuming that the standard deviations of the real and
imaginary components are equal [19] according to
(2-17)
Parameters α, β, γ, and δ are constants, and Rm represents the current measuring resistor.
In summary, the measurement model technique is used to estimate the noise level
associated with stochastic errors for individual measurements and to identify Kramers-
Kronig-consistent data. Knowledge of the error structure can be utilized when regressing
nonlinear models to impedance data. Regression by a weighted least squares strategy,
including errors in the real and imaginary components of the data, is given by
minimization of
(2-18)
where Zr,k and Zj,k are the real and imaginary components, respectively, while and
are the real and imaginary components of the variance at each frequency,
respectively. Variance weighting ensures emphasis and de-emphasis of data with low-
noise and high-noise contents, respectively, and increases the quality of information
obtained from impedance measurements [19, 22].
2.6 Process Model Development
This section outlines the development of a mathematical impedance model. The
model was developed for a pipeline grade steel electrode, covered by a thin porous film or
σr σj σ α Zj β Zr γ Z2
Rm
-------- δ+ + += = =
JZr k, Zr k,–( )2
σr k,2
--------------------------------k
∑Zj k, Zj k,–( )2
σj k,2
-------------------------------k∑+=
σr k,2
σj k,2
17
coating, immersed in dilute electrolytic solution. The contributions to the total current
flow and transport processes associated with oxygen diffusion from the electrolyte to the
electrode surface are explained in detail. The electrical quantities, current density and
potential, as well as concentration are written as sums of steady and oscillating terms:
(2-19)
where X is the variable of interest, the overbar represents the steady value, the tilde
distinguishes the oscillating value, and ω is the frequency of oscillation. The geometry
assumed for solving the transport equations is presented in Figure 2-4. The development
follows principles reported by previous authors [9-11, 23].
2.6.1 Reaction Kinetics
The total current density is given as the sum of the faradaic current and the current
associated with charging of the double layer:
(2-20)
Substitution of V for X in equation (2-19) yields the time derivative
(2-21)
Substituting equation (2-21) into equation (2-20), writing the total current in the form of
equation (2-19), and cancelling the exponential term, yields an expression for the total
current density in terms of oscillating variables according to
(2-22)
The faradaic current density is expressed as a function of the potential and concentration
according to
X X Re X jωt exp[ ]+=
i if Cd tddV
+=
tddV
jωV jωt exp=
i i f jωCd V+=
18
(2-23)
If the magnitude of the oscillating terms is sufficiently small, equation (2-23) can be
linearized according to
(2-24)
The charge transfer resistance is expressed as
(2-25)
Combining equations (2-25) and (2-24) yields
(2-26)
As previously stated by equation (2-5), the total faradaic current density is the sum
of the current contributions from the anodic and cathodic reactions. By assuming Tafel
kinetics, the current contribution from iron dissolution, reaction (2-1), is written in terms
of the potential measured with respect to a reference electrode, V, according to
(2-27)
where VFe is the equilibrium potential for iron dissolution, kFe is the reaction rate constant,
and αFe is the apparent transfer coefficient. Following the form of equation (2-24) where
the amplitude of the potential perturbation is small, equation (2-27) can be linearized:
(2-28)
if f V ci,( )=
i f V∂∂f
ci 0,
Vci 0,∂
∂ f
V cj j i≠,,ci 0,
i∑+=
Rt1
V∂∂f
ci 0,
------------------=
i fVRt
-----˜
ci 0,∂∂ f
V cj j i≠,,ci 0,
i∑+=
i Fe nFeFkFe
αFeF
RT------------- V VFe–( )
exp=
i Fe nFekFe
αFeF 2
RT----------------
αFeF
RT------------- V VFe–( )
Vexp=
19
which can be expressed in terms of the charge transfer resistance as demonstrated by
equation (2-25):
(2-29)
Similarly, the current contribution from hydrogen evolution, reaction (2-3), is
(2-30)
which can be linearized and expressed in terms of the charge transfer resistance to obtain
an expression for the oscillating current density given by
(2-31)
The contribution from the reduction of oxygen, reaction (2-2), can be expressed in terms
of potential and concentration according to
(2-32)
which can be linearized with respect to potential and concentration given by
(2-33)
Also, the flux of oxygen away from the electrode surface is
(2-34)
i FeV
Rt Fe,------------=
iH2nH2
– FkH2
αH2F
RT------------– V VH2
–( )
exp=
i H2nH2
kH2
αH2F 2
RT----------------
αH2F
RT------------– V VH2
–( )
Vexp=V
Rt H2,------------=
i O2nO2
– FkO2cO2 0,
αO2F
RT------------– V VO2
–( )
exp=
iO2nO2
kO2c O2 0,
αO2F 2
RT----------------
αO2F
RT------------– V VO2
–( )
Vexp=
nO2FkO2
αO2F
RT------------– V VO2
–( )
cO2 0,exp–
i O2nO2
– FkO2DO2 f, yd
d cO2
y 0=
nO2FkO2
DO2 f,
cO2 0,
δO2 f,----------- θ '˜ 0( )–==
20
where is the dimensionless flux at the surface. Equation (2-34) yields
(2-35)
The charge transfer resistance is expressed according to
(2-36)
After substituting equations (2-35) and (2-36) into equation (2-33) and solving for , the
contribution due to oxygen reduction is simplified to
(2-37)
where the diffusion impedance is given by
(2-38)
The dimensionless flux term, , will be developed in the next section. After
substituting the sum of equations (2-29), (2-30), and (2-37) into equation (2-20) for the
faradaic current density, the total current density is
(2-39)
The cell potential is the sum of the ohmic drop, due to current flow through the cell
electrolyte, and the surface overpotential given by
(2-40)
θ’˜ 0( )
cO2 0,
i O2δO2 f,
nO2FDO2 f,
------------------------- 1
θ'˜ 0( )------------–
=
1Rt O2,------------ nO2
kO2cO2 0,
αO2F 2
RT----------------
αO2F
RT------------– V VO2
–( )
exp=
iO2
iO2
V
Rt O2, Rt O2,
kO2δO2 f,
DO2 f,------------------- 1
θ'˜ 0( )------------–
αO2F
RT------------– V VO2
–( )
exp+
------------------------------------------------------------------------------------------------------------------------------------- VRt O2, ZD O2,+--------------------------------= =
ZD O2,
δO2 f,
nO2F 2DO2 f, cO2 0,
------------------------------------------ RTαO2
-------- 1
θ’˜ 0( )------------–
=
1
θ’˜ 0( )------------–
i V1
Rt Fe,------------ 1
Rt O2, ZD O2,+-------------------------------- 1
Rt H2,------------ jωCd+ + +
=
U i Re V+=
21
By solving equation (2-40) for , substituting the result into equation (2-39), and
rearranging, the complex impedance is given by
(2-41)
Equation (2-41) can be equally expressed by a series-parallel combination of electrical
circuit elements. However, the advantage of the development is that the model parameters
are explicit impedance response measures in terms of specific proposed kinetic and
transport processes.
2.6.2 Transport
This section develops the impedance associated with diffusion of oxygen due to
concentration gradients within the bulk soil electrolyte surrounding the steel electrode.
The development follows the work of DesLouis and Tribollet for transport of a reacting
species through a porous film to the surface of a rotating disc electrode [11]. As will be
explained in CHAPTER 3, a cylindrical steel coupon served as the working electrode.
After imposing several approximations, as will be demonstrated, the system geometry was
modeled in rectangular coordinates as presented in Figure 2-4. Two regions of stagnant
diffusion were proposed to exist: a bulk diffusion layer and a porous film adsorbed onto
the metal surface. The film was allowed to be rust layers, calcareous deposits, or resistive
polymer coatings.
Model development begins by considering, for dilute solutions, the concentration
within a diffusion region to be governed by
V
Ui
------ Zr jZj+ Re1
1Rt Fe,------------ 1
Rt O2, ZD O2,+-------------------------------- 1
Rt H2,------------ jωCd+ + +
------------------------------------------------------------------------------------------+= =
22
(2-42)
for cylindrical coordinates where and are the concentration and diffusivity of
species i, respectively. Since the direction of electrolyte flow is parallel to the metal
surface, as shown in Figure 2-4, . Also, the concentration is assumed to vary in the
r-dimension only. Thus, equation (2-42) reduces to
(2-43)
which governs stagnant diffusion of i in the r-dimension to and from the metal surface. If
the radius of the cylindrical electrode is large compared to the thickness of the film or
diffusion layer, i.e., , the term can be assumed to be negligible. By replacing r
by y, equation (2-43) reduces to
(2-44)
A coordinate system is imposed, as shown in Figure 2-4, where y represents the distance
from the surface, the film-metal interface is at with film thickness , and the
bulk diffusion layer begins at with thickness . Equation (2-44) can
be written for the two diffusion regions, and the appropriate boundary conditions are
(2-45)
(2-46)
t∂∂ci vr r∂
∂ci vz z∂∂ci+ + Di r2
2
∂∂ ci 1
r---
r∂∂ci
z2
2
∂∂ ci+ +
=
ci Di
vr 0=
t∂∂ci Di r2
2
∂∂ ci 1
r---
r∂∂ci+
=
r δ»1r---
r∂∂ci
t∂∂ci Di y2
2
∂∂ ci=
y 0= δi f,
y∗ 0= y δi f,=( ) δi b,
y 0= c i f, c i 0,=
y 0= Di y∂∂ci
y 0=
kci 0, V β⁄( )exp=
23
(2-47)
(2-48)
(2-49)
For the electrochemical system with oscillating voltage and current signals, also
oscillates and can be expressed in the form of equation (2-19) as the sum of steady and
oscillating contributions,
(2-50)
Substitution of equation (2-50) into equation (2-44) yields
(2-51)
The steady term of equation (2-51) can be written for both regions as
(2-52)
Solving equation (2-52) and applying the steady boundary conditions yields equations for
the concentration of i in the film and the bulk, respectively:
(2-53)
y∗ 0= y δi f,=( ) ci f, ci b,=
c i f, c i b,=
y∗ 0= y δi f,=( ) Di f, y∂∂ci f, Di b, y∂
∂ci b,=
Di f, y∂∂c i f, Di b, y∗∂
∂c i b,=
y∗ δi b,= y ∞→( ) ci b, ci ∞,→
c i b, 0→
ci
ci c i Re ci jωt exp[ ]+=
jωc iejωt Di y2
2
d
d c ie jωt– Di y2
2
d
d ci– 0=
y2
2
d
d ci f, 0=y∗2
2
d
d ci b, 0=
ci f,Di b,Di f,----------
ci ∞, ci f 0, ,–
δi b,----------------------------
yDi f,
k V β⁄( )exp----------------------------+=
ci b,ci ∞, ci b 0, ,–
δi b,-----------------------------
y∗ ci b 0, ,+=
24
where the concentration at the film-bulk diffusion layer interface is given by
(2-54)
By imposing the dimensionless variables
(2-55)
the oscillating part of equation (2-51) can be written for both regions according to
(2-56)
where the time constants associated with diffusion of i through the film and the bulk,
respectively, are given by
(2-57)
The dimensionless forms of the boundary conditions for the oscillating concentrations are
(2-58)
(2-59)
(2-60)
(2-61)
(2-62)
The general forms of the solutions to (2-56) for the film and bulk, respectively, are
ci b 0, ,Di b, ci ∞, δi b,⁄
Di f,δi f, Di f, k V β⁄( )exp⁄[ ]+------------------------------------------------------------
Di b,δi b,----------+
------------------------------------------------------------------------------=
θf
ci f,
ci f 0, ,------------= ξ y
δi f,-------=
θb
c i b,ci b 0, ,-------------= ξ∗ y∗
δi b,--------=
ξ2
2
d
d θf jωτi f, θf– 0=ξ∗2
2
d
d θb jωτi b, θf– 0=
τi f,δi f,
2
Di f,---------= τi b,
δi b,2
Di b,----------=
ξ 0= θf 1=
ξ∗ 0= ξ 1=( ) ci f 0, , θf ci b 0, , θb=
ξ∗ 0= ξ 1=( )Di f,δi f,--------- c i f 0, , ξd
dθf Di b,δi b,---------- ci b 0, , ξ∗d
d θf=
ξ∗ 0= θb 1=
ξ∗ 1= θb 0=
25
(2-63)
The constants Mf, Nf, Mb, and Nb are determined by applying the boundary conditions
yielding the solutions for the oscillating concentration profile of i in the bulk as
(2-64)
and through the film as
(2-65)
Taking the first derivative of equation (2-65) with respect to ξ, evaluating at ξ = 0, and
taking the reciprocal gives the dimensionless flux term of equation (2-38) as
(2-66)
Replacement of i with O2 in equation (2-66) yields the reciprocal of the dimensionless
flux of oxygen to the surface of the working electrode, necessary for equation (2-38).
θf Mf jωτi f, ξ exp Nf jωτi f,– ξ exp+=
θb Mb jωτi b, ξ∗ exp Nb jωτi b,– ξ∗ exp+=
θb
jωτi b, ξ∗ 1–( ) sinh
jωτi b,( )sinh---------------------------------------------------------–=
θf
jωτi b,( )tanh jωτi f, ξ 1–( ) coshDi b,Di f,---------- jωτi f, ξ 1–( ) sinh–
jωτi b,( )tanh jωτi f,( )coshDi b,Di f,---------- jωτi f,( )sinh+
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------=
1
θ’˜ 0( )------------–
jωτi b,( )tanhDi b,Di f,---------- jωτi f,( )tanh+
jωτi f, jωτi f,( )tanh jωτi b,( )tanhDi b,Di f,----------+
-------------------------------------------------------------------------------------------------------------------------=
26
Figure 2-4. Geometry for the diffusion model.
2.7 Application
Here, the theories governing methods for characterizing the electrochemical
behavior of pipeline steel have been presented. In the proceeding chapters, the application
of process models to the understanding of impedance data will be demonstrated. By
regression of process models to impedance data, parameters can be extracted which lead
not only to an understanding of the polarization behavior of steel, but also to an
understanding of the physics of surface charge transfer reactions and transport of reacting
species through resistive media. Since information can be obtained from individual scans,
the temporal evolution of the system can also be explored. EIS will be shown to be a
useful alternative to DC current and potential measurement techniques.
δf δbMetal
Surface
y=0y*=0y=δi,f y*=δi,b
y, y*
c∞
c(0)
y, y*→ ∞
FilmBulkDiffusionLayer
BulkElectrolyte
ElectrolyteFlow
z
y
CHAPTER 3EXPERIMENTAL METHODS
3.1 Summary
The purpose of this work was to measure the electrochemical impedance response
of pipeline grade steel exposed to a typical soil environment. Two field conditions were
simulated: (1) bare steel and (2) coated steel containing a discrete holiday. To ensure
symmetrical current distributions on the conductive surfaces, a cylindrical electrochemical
cell was designed to contain stationary cylindrical electrodes. Uniform or symmetrical
current distributions allowed for simpler model development and measurement of
averaged electrical quantities. Pipeline-grade steel served as the working electrode, and
platinum-rhodium alloy screens served as the counter electrodes. Potentials were
measured in reference to a saturated calomel electrode. Electrolyte was prepared
containing species known to contribute to the formation of calcareous films. Tests were
performed under galvanostatic control at the open circuit or corrosion potential and at
applied DC cathodic current densities. Two forms of data were collected: potential-time
traces and impedance data sets at snap shot intervals over the course of an experiment.
The time duration for a typical experiment was 4 to 7 days. Models were regressed to the
impedance data to determine the measurement errors and to extract parameters describing
the effect of film formation on transport processes and reaction kinetics.
27
28
3.2 Experimental Apparatus
Steps were taken to design well-controlled experiments. Since electrochemical
reaction rates are strong functions of temperature, the testing apparatus was isolated
within a controlled environment. To avoid localized activity at specific points on the
electrode surface, the corrosion cell was designed to assure symmetrical current
distributions. To maintain constant chemistry, fresh electrolyte was continuously
delivered to the cell. Galvanostatic control, i.e., current control, proved to be
advantageous over potentiostatic, i.e., potential, control. The cell potential, measured with
respect to a reference electrode, is the sum of the surface potential and the IR drop due to
current flow through the resistive electrolyte between the working and counter electrodes
according to
(3-1)
When holding constant, film formation will cause the current to change as the
resistance of the surface increases. In addition to changing current, which changes the IR
drop, the surface potential adjusts accordingly. Thus, no surface electrical quantities are
held constant. Use of current-controlled experiments guaranteed that one electrical
quantity was held constant over the course of the experiment. From equation (3-1), at the
corrosion potential, where is equal to zero, the measured cell potential, , is equal to
the surface potential, .
3.2.1 Electrodes
Test sample coupons were supplied by Metal Samples. The coupons were
machined from pipeline grade API5LX52 steel to cylindrical rods eight inches in length
V η iRe+=
V
i V
η
29
ss.
h. To
and
eads
m
d with
the
and half an inch in diameter. The chemical analysis and component weight percents for
the steel are given in Table 3-1. The coupons were used as received without any
metallurgical pretreatment such as annealing.
As previously stated, two working electrode types were used simulating buried
uncoated or bare steel and buried coated steel with a discrete holiday with surface area
small in comparison to the overall area of the electrode. The cylindrical rod coupons
served as the bare steel case. A schematic for the holiday electrode is presented in Figure
3-1. It consisted of two end pieces, fabricated from acrylic rods, sandwiching a thin metal
band. Several bands were cut from the supplied 5LX52 steel rods to 1/8” in thickne
The diameter of the acrylic pieces and the metal bands were machined to 0.485 inc
assemble the holiday electrode, the acrylic pieces and the band were center-drilled
tapped. A threaded metal rod was inserted through the top acrylic piece with the thr
protruding out the bottom end. The band was then screwed on followed by the botto
acrylic piece. Before assembly, the top and bottom surfaces of the disc were covere
a thin layer of silicon grease to seal the acrylic-metal crevices.
Platinum-rhodium alloy mesh screens, supplied by Engelhard-Clal, served as
counter electrodes. Two screens were fabricated to 235 mm by 150 mm with the
following specifications:
• 95% platinum, 5% rhodium alloy
• 80 mesh gauze with wire diameter 0.003”
• 0.5 mm diameter Pt border wire with 50 mm extension.
30
’s
l
[
n, for
The screens were pliable and designed to be formed along the inside wall of the cell body.
The fact that the counter electrode fully circled a cylindrical working electrode assured
uniform radial current on the working electrode surface.
3.2.2 Current Distribution
Since the cylinder electrode extended the length of the cell with the counter
electrodes, the current distribution on the surface was uniform, and the electrolyte
resistance could be estimated by a simple analytic formula. The current distribution on the
holiday electrode, however, is nonuniform, and an analytic formula is not available for the
electrolyte resistance. Thus, numerical simulations were performed to determine the
current and potential distributions for the cell arrangement with the holiday electrode.
Under the assumption of electroneutrality, uniform concentration gradients and
constant conductivity, the potential field of the corrosion cell is governed by Laplace
equation [23]
(3-2)
Using the boundary element method (BEM), equation (3-2) was transformed to an integra
equation for axisymmetric geometries describing the boundaries of the corrosion cell24].
The FORTRAN code, developed for numerical solution of equation (3-2) using BEM, is
given in APPENDIX A. By discretizing the cell boundaries into constant elements a
numerical solution was obtained by specifying either an essential boundary conditio
Φ, or a natural boundary condition, for , which was constant over the length of each
element. The solution results yielded both a value for Φ and for each individual
Φ∇2 0=
∇Φ
∇Φ
31
is
ich
ry
e
ndary
c
re
m
of the
nder
nter
e
teel
element. The current density, scaled by the electrolyte conductivity, is equivalent to
by rearrangement of Ohm’s law according to
(3-3)
The axisymmetric plane for the cell geometry containing the holiday electrode
presented, including the imposed boundary conditions, in Figure 3-2. The holiday surface
is shown as being slightly recessed due to machining errors or surface polishing, wh
can impact current distributions significantly [25]. For the electrode surface, the bounda
condition was specified as Φ = 1. The boundary condition for the counter electrode, th
opposite extent of the geometry, was specified as Φ = 0. For the remaining nonconducting
surfaces, the current density, , was specified to be 0. By imposing constant bou
conditions, the scaled primary current distribution, which accounts only for geometri
influences, was obtained [23].
The results for the current density and potential along the holiday electrode a
shown in Figure 3-3. Whereas the current density was radially uniform, the results in
Figure 3-3 show a high degree of axial nonuniformities with the location of highest
current density being the ends of the holiday. The results for the potential are unifor
along the surface and shown to decrease rapidly when moving away from the ends
conducting metal band.
The current density and potential distributions were also calculated for the cyli
electrode. The boundary conditions were similarly specified for the working and cou
electrode surfaces and the insulating surfaces of the top and bottom cell covers. Th
results for the current density and potential as a function of axial position along the s
∇Φ
iκ--- ∇Φ–=
∇Φ
32
rod are plotted in Figure 3-4. The results show that the potential was uniform the full
length of the electrode and the current was uniform except for slight variation at the
extreme edges.
From the solution conductivity and the results from the BEM calculations for the
current distribution on the electrode, the ohmic or electrolyte resistance, Re, was
calculated. First, the total scaled current, , was calculated by integrating the distribution
over the electrode surface. Since constant elements were used for the BEM calculations,
the current density value was constant over the length of each element. This allowed for
the integration to be simplified to a summation of rectangle areas with height and width
∆x, the element length. Over the length of the electrode the total current was thus:
(3-4)
Since the corrosion cell was filled with a solid matrix, the solution conductivity was
corrected for the porosity of the matrix. The effective conductivity was approximated
according to
(3-5)
where κ0 is the conductivity of the solution outside any porous structure and ε is the
porosity or void fraction of the solid matrix [23]. Finally, Re was calculated by Ohm’s
law according to
(3-6)
Iκ---
iκ---
Iκ---
ij
κ---- ∆xj
j 1=
N
∑=
κ κ0ε1.5=
Re∆Φ
I--------=
33
where , the potential difference between the working and counter electrodes as
specified by the boundary conditions. The results for the calculated value of Re for the
two electrode types are listed in Table 3-2, including values determined from the high
frequency limit of the impedance measurements (the impedance results will be discussed
in CHAPTER 4 and CHAPTER 5). For comparison with the BEM results, Table 3-2
includes values for Re calculated for the cylinder electrode from the anode resistance
formula [12]
(3-7)
where L is the length of the electrode and d is the diameter.
3.2.3 Cell Electrolyte
Typical soil conditions were simulated by filling the cell with inert silica sand and
feeding electrolyte containing species known to participate in precipitation reactions that
form calcareous deposits. Electrolyte containing Ca2+, Mg2+, and HCO3- was prepared by
dissolving reagent grade CaCl2, MgSO4, and NaHCO3 in water. Table 3-3 is a list of the
desired concentrations for the charged species included in the electrolyte prepared for this
work. The actual masses of the salts added to produce 50 L of solution are given in Table
3-4. The concentrations of Ca2+, Mg2+, and HCO3- were in agreement with values
typically reported in the field [26, 27].
3.2.4 Corrosion Cell Design
Since measuring or controlling the current at a specific point on the working
electrode was not possible, the overall cell current was controlled. Thus, it was desired to
∆Φ 1=
Re1
2πκL------------- 8L
d------ 1–ln
=
34
design a cell to exhibit symmetric current and potential distributions. Thus, the cell was
designed with cylindrical geometry to guarantee uniform distributions axially and radially
with the cylinder electrode and uniform radial distribution for the holiday electrode. Both
electrode types were oriented in the cell vertically with the ends at right angles to the top
and bottom cell boundaries. For the cylinder electrode, the active surface extended the
entire length of the cell. For the holiday electrode, nonuniform current density was
present along the length of the metal band due to edge effects at the acrylic-metal
boundaries and the differing lengths between the working and counter electrodes.
The cell was fabricated from plexiglass and consisted of a tubular cell body with
flanged ends and two circular cover pieces. A schematic for the cell body is shown in
Figure 3-5. The top and bottom covers were sealed using O-rings and fastened to the
flanged ends of the cell body with 8 screws each. A schematic for the top cover piece is
shown in Figure 3-6. The active portion of the cell measured six inches in internal
diameter and six inches in length.
The working electrode (WE) was inserted in the cell perpendicular to the bottom
cover. The bottom cover had a recessed seat with a small piece of rubber tubing (see
Figure 3-5). To prevent exposure of its bottom circular surface, the WE was pressed into
the rubber seal in the seat. The top one and a quarter inches of the WE protruded out the
top of the cell to facilitate connection of lead wires from the instrumentation. For the
cylinder electrode, the length exposed to the soil environment was 6 inches making the
total active surface area 60.8 . For the holiday electrode, the metal band was centered
in the cell body and fully exposed to the soil environment. The total active metal surface
cm2
35
area for the holiday electrode was 1.23 . The top cover had a drilled hole fitted with
an O-ring to seal around the top of the WE to prevent electrolyte leaks (see Figure 3-6).
To assure O2 saturation and maintain constant chemistry over the course of the
experiment, fresh electrolyte was fed to the bottom of the cell. From the top of the cell,
the electrolyte overflowed into the side of a flask containing a calomel reference electrode
inserted through a rubber stopper sealing the top of the flask. The arrangement for the
reference electrode essentially placed it at infinity since most of the change in potential
occurred at a small distance from the WE surface. Electrolyte then overflowed the flask
through tubing inserted in the stopper. The flow then dripped into a funnel to prevent
siphoning and was subsequently discarded through the drain. Tygon tubing was used for
the electrolyte feed and overflow lines. All the lines between the cell and the reference
electrode flask were primed with electrolyte and purged of air to guarantee continuous
electrical contact. In order to neglect the effects of forced convection associated with
flowing fluid, the electrolyte feed was pumped at a slow rate, approximately 7 liters per
day, using a peristaltic pump.
For electrical connection to measuring instrumentation, a wire was soldered into a
hole drilled in the top of the cylinder electrode. For the holiday electrode, the center rod
protruded out the top and was machined thin enough to allow fastening the lead wire using
an alligator clip. For connection to the counter electrodes, two holes were drilled and
tapped into the cell top cover piece. Small diameter tubing adapters were screwed into the
holes. The counter electrode extension wires were then inserted through the fittings to
allow them to protrude out the top of the cell after fastening the cover piece. The small
cm2
36
spaces between the fittings and the extension wires were sealed by packing putty in the
gaps and wrapping with teflon tape.
A schematic of the corrosion cell is presented in Figure 3-7 including electrolyte
lines, electrical connections, and instrumentation. For temperature control, the cell was
placed inside a plexiglass booth with a 2 ft. X 2 ft. square base and measuring 3 ft. in
height. The booth enclosure was fitted with a drain fitting and had holes cut into the sides
and bulkhead fittings installed for insertion of lead wires, power cords, heat exchange
tubing, and electrolyte feed tubing. The temperature inside the booth was controlled to
C by blowing the air over heat exchange coils containing water at C supplied
by a water chiller. The air inside the booth was circulated using a PC panel mount fan.
Several feet of tubing connected to the electrolyte feed pump were coiled around the cell
to allow the electrolyte to equilibrate to the ambient temperature inside the booth.
During initial testing, it was observed that the fluorescent ceiling lights inside the
laboratory added a significant amount of noise to the current and potential signals. To
prevent the influence of outside electric fields on measurements, the inside of the booth
was wrapped with aluminum foil. The plexiglass booth had dual roles of maintaining a
controlled temperature environment and serving as a Faraday cage.
3.2.5 Instrumentation and Data Collection
As previously stated, the experiments were performed under galvanostatic control.
A three-electrode cell arrangement was used for potential measurement between the
working and reference electrode with the current between the working electrode and the
counter electrode controlled using an EG&G Instruments, PAR 273 Potentiostat/
24 1°± 15°
37
galvanostat. Connections to the electrodes were made via lead wires from the
electrometer [14].
The PAR 273 had an array of current measuring resistors ranging from 1 Ω to 1000
kΩ for controlling or measuring the current in the range of 2 A down to 15 nA. Each
measuring resistor had a range of 15% to 190% of the full scale value. For example, the
100 µA current range (10 kΩ resistor) was suitable for currents ranging from 15 µA to 190
µA. An optimization technique was used for selecting the appropriate current range for a
desired applied current condition. A switch-over factor ranging from 1.55 to 1.85 could
be imposed by the user to initiate selection of an adjacent current range. For example, let
the desired applied current be 175 µA. Normally, the 100 µA current range would be
selected. However, by having a switch-over factor of 1.7, a restriction is imposed on the
maximum allowable current for a given current range of 170%. Thus the next higher
current range of 1 mA (or smaller measuring resistor of 1 kΩ) would be selected.
For oscillating signal generation and impedance measurements, a Solartron
Instruments 1260 Gain Phase Analyzer was used. For impedance determination, the
Solartron 1260 employs frequency response analysis of two voltage signals [9, 28]. The
connections of the 1260 to the PAR 273 are presented in Figure 3-7. The 1260 generator
output is connected to the external input on the front panel of the PAR 273. The I monitor
and E monitor connections of the PAR 273 were connected to the V1 and V2 inputs on the
1260, respectively. The 1260 generator superimposed a sinusoid, additively, on the
applied DC signal controlled by the PAR 273 via the connection at the external input. Two
sinusoidal voltage signals were then output to the 1260 from the I monitor and the E
monitor. The signal from the I monitor was the voltage across the current measuring
38
ls
e
was
e on
talled
ls,
al
the
were
een
y files
ment.
tlined
resistor according to Ohm’s Law. The 1260 performed the analysis of the two signa
while integrating on the measured signal. In the case of galvanostatic control, V1, th
current, was the controlled signal coming from the I monitor while V2, the potential,
the measured signal coming from the E monitor. Thus, the 1260 was set to integrat
V2. The result for the impedance was the ratio of the two signals, i.e., V2/V1.
Both the 1260 and the PAR 273 were connected in parallel to a GPIB card ins
in a personal computer. Virtual Instruments (VIs) were developed using LabVIEW 5.0
graphical programming software for controlling experiments and data acquisition.
LabVIEW VIs have a front panel display which includes all the control settings, dia
and buttons. The front panel serves as the interface between the user and the actu
instruments. The VIs also have a rear panel wiring diagram which maps out all
information flow, decision making, and calculations necessary to execute control of
physical instruments and data acquisition. All initial settings and control parameters
made from the VI front panels using the PC, avoiding any instrumental front panel
executions other than powering on and off. Experimental data was output to the scr
and displayed using virtual strip charts and graphs. Data was also saved to memor
for later analysis. The VIs initiated the PAR 273 to record the WE potential at a rate
determined by the user, typically one measurement per minute. The user could also
initiate and stop individual or sets of impedance scans at any time during the experi
The virtual instruments used to control the 1260 in tandem with the PAR 273 are ou
and described in more detail in APPENDIX B.
39
3.3 Experimental Procedures
Experiments were set up to allow the working electrode to reach a steady level of
polarization in response to an applied current condition and to measure the impedance at
snap shot intervals over the course of the experiment. It was desired to see the effect of
film growth and polarization on the impedance over time. Impedance scans were run in
sets of 3 to 4 consecutive scans. The replicate scans were necessary for statistical analysis
to determine the error structure of the measurements (see section 2.5). All experiments
were begun using a WE with a clean and polished surface. Typical experiments lasted 100
to 150 hours. A summary of the experiments conducted for this work is outlined in Table
3-5.
3.3.1 Applied DC Bias and Frequency Range
Experiments were conducted at the corrosion potential (zero net current) and at
applied cathodic currents. For the cathodic experiments, it was desired to control the
current to a point lying on the oxygen reduction plateau (see Figure 2-1). Preliminary
galvanodynamic scans were performed to determine the appropriate range of values. An
experimentally generated cathodic polarization curve for pipeline grade steel is presented
in Figure 3-8. Oxygen reduction appeared to be the dominant reaction between -680 and
-900 mV (SCE), corresponding to a range of current densities from 1 to 6 .
As previously stated, the sinusoid generator superimposes a signal on the DC bias
controlled by the PAR 273. Small amplitude signals were used to assure restriction within
an approximately linear range of the polarization curve. Even though the experiments
were performed galvanostatically, the generator applies an AC potential signal resulting
from the product of the applied current perturbation and the optimized measuring resistor.
µA cm2⁄
40
The resulting signal was superimposed on the cell potential measured with respect to a
reference electrode, and the current signal oscillated between the limits of the desired
amplitude.
Since corrosion reactions are typically slow and generating impedance spectra
requires low frequency measurements [9, 12], preliminary scans were performed to
determine the appropriate testing frequency range. The results of the impedance response
for the cylinder electrode in liquid electrolyte only, without the sand matrix, are presented
in Figure 3-9 including the calculated spectrum (application of models for predicting
impedance spectra will be discussed in CHAPTER 5). The tested frequency range was
1000 Hz to 0.01 Hz in increments of 7 frequency steps per decade. The data in Figure 3-9
show the imaginary component of the impedance to be large at 1000 Hz and then decrease
to a minimum at 24 Hz with a corresponding real component of approximately 29 to 30 Ω.
Using parameters obtained from regression of models to the data led to the extrapolation
of Re to be approximately 0. Using the current distribution calculated from the BEM
simulations and the solution conductivity, Re was calculated to be 33.4 Ω. The calculation
for Re agreed with the value determined from the impedance data in the high frequency
range where the imaginary component was at a minimum value.
The initial high frequency data with capacitive behavior in Figure 3-9 were
considered to be the results of an instrumental artifact. Similar high frequency artifacts
were also observed when performing preliminary experiments using steel rotating disc
electrodes. The fluid mechanics for the rotating disc electrode (RDE) have been shown to
be well defined, and the electrolyte resistance is easily calculated knowing the solution
conductivity [23]. The results from calculating the electrolyte resistance of the RDE cell
41
also led to rejection of high frequency data and narrowing of the tested frequency range.
To avoid collecting artifact data, the high frequency limit for testing was set to 100 Hz.
Typically, measurements were collected from low frequency values of 0.01 Hz to 0.001
Hz.
The 1260 analyzer needed many cycles to obtain a converged result for the
impedance at a given frequency. Typically, the analyzer required three to six cycles at the
low frequencies, 1 to 10 mHz. Since a sinusoid with frequency close to 1 mHz has a
period on the order of 1000 seconds, the number of points per scan was optimized to
reduce the time duration to complete each scan. All impedance scans were conducted by
sweeping down from high to low frequency. Frequency transitions were made in log
steps. Usually, 7 or 8 log steps per decade were used. A typical scan sweeping from 100
Hz to 0.001 Hz included 35 to 40 points and required approximately 3 to 4 hours to
complete.
3.3.2 Variable Amplitude Galvanostatic Modulation
Over the range of frequency values swept for a given scan, the impedance can
change by several orders of magnitude. If operating under constant amplitude
galvanostatic control, the resulting amplitude of the potential signal will also change by
several orders of magnitude, and the signal will likely be oscillating outside a linear
segment of the polarization curve. Small amplitude oscillating signals allowed for
simplification of modeling equations by linearization as demonstrated in CHAPTER 2.
Also, since current density is a function of potential, the reaction kinetics will be greatly
influenced by large potential fluctuations. Large potential fluctuations and changes in
42
reaction kinetics can disrupt surface films and upset the natural time dependent behavior
of the system.
A predictive method for adjusting the amplitude of the applied current signal while
maintaining the amplitude of the potential signal at some target value was developed and
dubbed variable amplitude galvanostatic (VAG) modulation [29, 30]. The method
prevents large perturbations in the potential signal at low frequencies. The algorithm
calculates the applied current amplitude according to
(3-8)
At the first frequency in the sweep, an initial guess is used as the estimated impedance.
For impedance scans swept from high to low frequency, a good guess was the electrolyte
resistance. If Re was unknown or could not be easily calculated, a value was obtained by
conducting a high frequency scan using constant amplitude galvanostatic control and
extrapolating the real component where the nyquist plot intersects the real axis. At the
second frequency, the initial guess is again used as the estimate. At the third frequency, a
2-point prediction is made from
(3-9)
From the fourth point to the end of the sweep, the impedance is estimated using a 3-point
prediction according to
(3-10)
The algorithm was incorporated into LabVIEW virtual instrument controls to
automatically set the current signal and optimize selection of the appropriate measuring
IVtarget
Z ω( ) estimated---------------------------------=
Z ωk( ) 2Z ωk 1–( ) Z ωk 2–( )–=
Z ωk( ) 3Z ωk 1–( ) 3Z ωk 2–( ) Z ωk 3–( )+–=
43
resistor. For the experiments of this work, the potential target amplitude was usually set to
10 mV.
3.3.3 Initial Preparation
Electrolyte was prepared in quantities of 50 L which would last approximately 7
days. Oxygen saturation was achieved by bubbling air through the solution, using an
aquarium air pump, for about 24 hours prior to use. The air was first dried and scrubbed
of CO2 by feeding it to the bottom of a column up through a layer of drierite crystals and
through a layer of ascarite II crystals before entering the solution jug. The drierite served
to dry the air while the ascarite scrubbed out any CO2 present.
The working electrode was prepared by polishing the surface to a near mirror
finish and cleaning with ethanol. Polishing was accomplished by using a lathe to spin the
electrode and buffing using silicon carbide grit papers of roughness varying from course to
extremely fine. The electrode was buffed to a shine using a cloth soaked with alumina
slurries. The holiday electrodes were spun for polishing by threading a screw through the
metal band and inserting the screw into the chuck on the lathe.
Before preparing the cell, the chiller was started to circulate temperature controlled
water through the heat exchange coils in the isolation booth. The fan was also started to
circulate the air. The cell was then prepared by first inserting the electrode into the rubber
seal in the recessed seat in the bottom cover of the cell housing. If the holiday electrode
was used, it was first assembled as described in section 3.2.1. After inserting the WE, the
counter electrodes were installed by forming the screens around the inside contour of the
cell, arranged to completely circle the WE. The cell was then filled with all purpose silica
44
top
nted
es for
sion
with
taltic
ive
ence
he
scape.
n and
sand. The active surface area for the bare steel electrode was 60.8 and 1.23 for
the 1/8” band holiday electrode.
After filling the cell with sand, the top cover was placed by pressing it over the
portion of the WE allowing it to slide through the O-ring seal. The top cover was orie
to assure the following:
• the flange O-ring seal was seated properly in its groove
• the extension wires of the counter electrode protruded through the threaded hol
connection to instrumentation lead wires
• the holes for the fastening screws in the cover lined up with those in the flange.
After fastening the top cover, fittings were installed over the counter electrode exten
wires and sealed as described in section 3.2.4 by packing putty in the gaps and wrapping
with teflon tape. Finally, the reference electrode was placed in the flask and sealed
the rubber stopper, and all necessary tubing lines were connected.
After the cell was completely assembled and sealed, it was filled using a peris
pump with a variable speed control. The fill pump was connected in parallel with the
continuous flow feed pump. This arrangement allowed rapid filling of the cell. Air
bubbles were purged from the cell by throttling the fill pump and tipping the cell to dr
the bubbles out the top of the cell. The line connected the top of the cell to the refer
electrode flask had a teed line running to the top of the isolation booth and open to t
atmosphere. This provided a pressure head and allowed any air bubbles to easily e
Once the cell was full, the fill pump was stopped and the feed pump started.
Provided there were no electrolyte leaks, the electrodes were connected to the
electrometer. The controlling PC was also prepared ahead of time for data acquisitio
cm2 cm2
45
initialization of the instruments. The experiment was begun by engaging the cell enable
switch on the PAR 273, starting the main LabVIEW VI, and setting the desired applied
current bias. The LabVIEW VIs automatically measured and stored potential
measurements at the specified data collection rate. Impedance scans were conducted at
any time with the push of a virtual button from the front panel of the VI. Current-potential
data files included the potential in mV, the applied current bias in µA, and the time in
seconds from the beginning of the experiment at which the measurement was taken.
Impedance data files included the initial parameters including the date and time of the
scan, the target potential amplitude, the applied current bias, and the initial impedance
guess value. The experimental point by point quantities of the impedance spectrum
included the frequency, the real component, the imaginary component, instrumental error
codes, the selected measuring resistor, the amplitude of the applied current signal, the
modulus of the impedance, and the phase angle.
46
Table 3-1. Chemical analysis of the supplied pipeline grade, 5LX52, steel coupons.
Chemical Component Weight Percent
Al 0.040%
C 0.090%
Fe 98.487%
Mn 1.070%
P 0.007%
S 0.009%
Si 0.250%
V 0.047%
Table 3-2. Results for the total current integrated on the electrode surface determined from the current distribution resulting from the BEM simulations. Also included is the calculated electrolyte resistance for both electrode types while accounting for the porosity of the solid matrix. The porosity or void fraction assumed for the calculation was 0.40. Also included are the results from impedance measurements and from using the anode resistance formula, equation (3-7), for the cylinder electrode.
ElectrodeI/κ,
A/Ω-1cm-1
Re determined from:
BEM eq. (3-7) EIS
Cylinder 24.57611 81.6 74.6 65
Holiday 2.95274 679.5 --- 650
47
Table 3-3. Calculated concentrations of ionic species included in simulated soil electrolyte. Molarity units are in moles/liter. The calculated conductivity is also included.
Species moles/L ppm
Ca2+ 0.004994 90
Mg2+ 0.002220 40
Na+ 0.000277 5
HCO3- 0.000277 5
SO42- 0.002220 40
Cl- 0.009989 180
κ, 0.00197
Table 3-4. Masses of salts in g/L added to water to prepare simulated soil electrolyte. The solution pH is included.
Salt Mass, g/L
CaCl2-2H2O 0.734
MgSO4 0.267
NaHCO3 0.023
pH 8.0
Ω 1– cm 1–
48
Table 3-5. Experimental outline including electrode type and applied current density.
Electrode typeApplied Current,
µAElectrode Area,
cm2
iapp,
cylinder 0.0 60.33 0.0
cylinder 100.0 60.80 1.6
cylinder 150.8 60.33 2.5
cylinder 241.3 60.30 4.0
holiday 0.0 1.22 0.0
holiday 6.1 1.23 5.0
µA cm2⁄
49
Figure 3-1. Schematic of the simulated holiday electrode.
Acrylic End Pieces
Steel Band Electrode
Threaded Connection Rod
50
Figure 3-2. Axisymmetric plane, including boundary conditions, of the 1/8” holiday electrode for BEM simulation.
Axis of Symmetry
Φ = 1 Φ = 0
∇Φ = 0
∇Φ = 0
CounterElectrodeSurface
Holiday (WE)
51
y or iece.
Figure 3-3. Current density and potential distributions, generated from BEM simulation, as a function of axial position on the 1/8” holiday electrode. The center of the holidaconductive metal band was located 3” from the end of the bottom acrylic insulating p
-0.005
0.000
0.005
0.010
0.015
0.020
2 3 4
Position, in
i/k, A
/(in
Ω-1
in-1
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Pot
entia
l, V
Current Density Potential
52
Figure 3-4. Current density and potential distributions, generated from BEM simulation, as a function of axial position on the cylinder electrode.
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0 1 2 3 4 5 6Position, in
i/k, A
/(in
Ω-1
in-1
)
0.0
0.5
1.0
1.5
2.0
Pot
entia
l, V
Current Potential
53
Figure 3-5. Schematic of corrosion cell body showing position of electrodes.
Working Electrode Seat
Counter Electrode Lead Wires
Cell BodyID = 6 in.L = 6 in.
Working Electrode
Electrolyte In Electrolyte In
ElectrolyteOut
ElectrolyteOut
Flange
Cover
Cover
Flange
54
Figure 3-6. Schematic of corrosion cell top cover piece.
Cell Cover Piecediameter = 8 in.
Fastener Screw (8)
O-Ring Seal
Working ElectrodeO-Ring Seal
Internal Wall ofCell Body
Electrolyte Ports
55
Figure 3-7. Corrosion cell flow diagram including instrumentation.
To Drain
Electrometer
Electrolyte SupplyCoupon(W.E.)
C. E.
R. E.
GPIB
Electrolyte Pump
FRA
Potentiostat
56
Figure 3-8. Preliminary experimental polarization curve for pipeline grade steel, generated from a galvanodynamic sweep from anodic to cathodic current densities at a
rate of 0.3 µA/cm2 per minute. The closed circles correspond to the applied conditions listed in Table 3-5 for the cylinder electrode experiments.
-1200
-1100
-1000
-900
-800
-700
-600
-500
-400
0.01 0.1 1 10 100
iapp, µA/cm2
Pot
entia
l, m
V (
SC
E)
Polarization Curve
Experimental Points
57
Figure 3-9. Preliminary impedance spectrum in Nyquist form to identify high frequency instrumental artifacts. The response is from the cylinder electrode in liquid electrolyte
only, with κ = 0.00122 Ω-1cm-1, to variable amplitude galvanostatic modulation about the corrosion potential. The tested frequency range was 1000 Hz to 0.01 Hz. The calculated spectrum was generated from measurement model regression parameters.
0
20
40
0 20 40 60 80 100 120
Zr, Ω
-Zj,
Ω
Data Calculated Spectrum
1000 Hz
24 Hz 0.01 Hz
CHAPTER 4EXPERIMENTAL RESULTS
4.1 Corrected Cell Potential
This chapter presents the results for the experiments outlined in Table 3-5.
Experiments were conducted over a period of several days with the current flow between
the working and counter electrodes controlled to simulate various levels of cathodic
protection. For each experiment, the DC potential of the WE was measured with respect
to a reference electrode, and impedance scans were conducted, in sets of 3 or 4 replicates,
at various times during system evolution. Impedance data were generated from variable
amplitude galvanostatic modulation about the applied DC bias to prevent large
fluctuations in the output potential signal (see section 3.3.2).
The total cell potential had to be corrected for the ohmic resistance resulting from
the influence of the cell geometry and the current flow through the resistive electrolyte
between the working and counter electrodes. The total cell potential can be expressed by
(4-1)
where is the potential of WE measured with respect to a reference electrode. For
the instrumentation used, cathodic currents were positive in sign. As an example for
determining IR drop, Figure 4-6 and Figure 4-7 present the measured potential and the
impedance response, respectively, for an experiment conducted on the cylinder electrode
with the applied current density equal to 2.5 . From the impedance plots in
Vcorrected Vmeas IRe+=
Vmeas
µA cm2⁄
58
59
Figure 4-7, Re was approximately 50 Ω and the corresponding IR drop for the cell was
calculated to 7.5 mV. The resulting corrected potential was slightly more positive and
hardly noticeable when compared to the data for the measured potential in Figure 4-6. For
experiments conducted at the corrosion potential by setting the total current to zero, IR
compensation was not necessary.
EIS proved to be more reliable for determining the ohmic resistance than using
current interrupt techniques. In previous work employing similar cell arrangements and
electrolytes, current interrupt techniques led to determination of the IR drop to be on the
order of 100 mV [6, 26, 27].
4.2 Cylinder Electrode Experiments
Experiments were conducted with the cylinder electrode at several applied current
densities as indicated in Table 3-5 and in Figure 3-8. Operation at the corrosion potential
was accomplished under galvanostatic control by setting the applied DC current to 0 A.
4.2.1 Experiment 1 - Modulation About the Corrosion Potential
The potential-time data for the cylinder electrode maintained at the corrosion
potential is presented in Figure 4-1. The plot provides evidence of non-stationary
behavior during film formation with the initial potential transient occurring within the first
20 hours of the experiment. During the transient, the WE potential shifted by as much as
several hundred millivolts in the negative direction before reaching a steady value. The
large potential shift was consistent with the increased resistance and blocking effects due
to the formation of surface films [6, 7, 27].
60
The gaps in the trace of Figure 4-1 correspond to times when impedance scans
were performed and the DC potential was not recorded. Between gaps, no peaks in the
potential measurement were observed, verifying that the VAG modulation technique was
noninvasive and did not disrupt surface characteristics. Such was the case for all other
experiments. The impedance data generated during this experiment are presented as
Nyquist plots in Figure 4-2. Unlike the potential-time trace, where the system appeared to
reach steady state within about 20 hours, the impedance results demonstrated that the
system was still evolving after several days. Increases in the magnitude of the impedance,
after the potential steadied, are consistent with films making the surface more resistive to
charge transfer reactions and to diffusion of oxygen, leading to decreases in the rates of
iron dissolution and oxygen reduction.
The semicircle observed in the Nyquist plot represents the capacitive behavior of
the cell. Upon inspection of the Bode plot for the negative of the imaginary component as
a function of frequency, presented in Figure 4-3, it was observed that the characteristic
frequency, where the magnitude of the imaginary component was a maximum, decreased
with time. The reciprocal of the characteristic frequency has units of time according to
(4-2)
The time constant, τ, was proportional to a characteristic diffusion length or layer
thickness. Increases in the characteristic time constant, over the course of the experiment,
give evidence supporting the evolution of film growth.
Each individual impedance scan was considered to be a snap shot of the state of
the system at the time the scan was conducted. At early times in the experiment, during
the initial potential transient, generation of complete spectra could not be achieved since
τ 1f---=
61
sweeping down to the 1 mHz range required several hours. To observe changes in the
impedance during this time of highly non-stationary behavior, shorter scans were
conducted by sweeping to the 10 mHz range, which required about 20 minutes to
complete. As the potential stabilized, sweeps to lower frequencies were accomplished.
The beginnings of a second semicircle or capacitive loop were observed to develop
in the low frequency range of the spectra presented in the Nyquist plots of Figure 4-2. A
complete semicircle would correspond to a second local maximum in the Bode plot. This
result suggested the presence of two diffusion regions from the bulk of the electrolyte to
the surface of the WE.
4.2.2 Experiment 2 - Modulation About 1.6 µA/cm2
The purpose of this experiment was to measure the impedance response when the
working electrode was polarized to a slightly cathodic level. The potential-time trace of
the WE in response to an applied current density bias of 1.6 is presented in
Figure 4-4. The initial transient lasted approximately 10 hours before reaching a steady
potential, similar to the experiment conducted at the corrosion potential. The IR drop was
determined to be approximately 6 mV using the high frequency impedance results. Upon
resetting the applied current to 0 at the end of the experiment, the potential would relax in
the positive direction to the corrosion potential.
The impedance response, presented in Figure 4-5, was observed to increase more
significantly over time than for the experiment conducted at the corrosion potential. The
increases in the measured impedance were consistent with film growth causing reductions
in corrosion current and oxygen reduction current. Both the real and negative imaginary
µA cm2⁄
62
impedance components were increasing after several days, and the beginnings of a second
capacitive loop, or semicircle, were observed in the low frequency range.
4.2.3 Experiment 3 - Modulation about 2.5 µA/cm2
The purpose of this experiment was to measure the impedance response of
cathodically protected steel. With the applied current density at 2.5 , oxygen
reduction became the dominant electrochemical reaction, with the WE polarized to a
position on the oxygen reduction plateau [12, 13, 23]. The potential-time trace for this
experiment is presented in Figure 4-6. As indicated by the plot, the initial potential
transient occurred within the first 20 hours of exposure. However, after the 4th set of
impedance scans, started after approximately 24 hours of exposure, the potential
continued to decrease with further cathodic polarization. After approximately 60 hours
the potential began to increase. The erratic behavior of the potential-time trace could have
been caused by film formation changing the mass transfer limited current density. At such
a level of applied cathodic current, the majority of the total current density was due to
oxygen reduction, and small changes in the mass transfer limited current density resulted
in large changes in the cell potential.
The impedance response is presented in Figure 4-7. Significant increases in the
impedance were observed over the course of the experiment. Although impedance data
were generated for low frequencies, as low as 1 mHz, complete capacitive semicircles
were not observed.
µA cm2⁄
63
4.2.4 Experiment 4 - Modulation about 4.0 µA/cm2
The purpose of this experiment was to measure the impedance response of steel at
a higher level of cathodic protection. The potential-time trace of the WE in response to an
applied current density of 4.0 is presented in Figure 4-8. The initial transient
lasted approximately 12 hours before reaching a steady potential, which was much more
negative than the value measured for the same applied current density in the polarization
curve shown in Figure 3-8. The IR drop was determined to be approximately 6 mV using
the high frequency impedance results. Upon resetting the applied current to 0, the
potential would relax in the positive direction to the corrosion potential.
The impedance response, presented in Figure 4-9, was observed to decrease over
the course of the experiment with most of the change occurring within the first day of
exposure. The decrease in the measured impedance, after reaching a steady WE potential,
is consistent with increasing rates of hydrogen evolution, the dominating reaction at
higher applied cathodic currents. Hydrogen bubbles forming at and diffusing away from
the WE surface can disrupt film formation. As the impedance was observed to be steady
after one day of exposure, it follows that a lack in the presence of films caused the surface
to be less resistive to charge transfer reactions and oxygen transport.
µA cm2⁄
64
Figure 4-1. The corrosion potential, measured with respect to a calomel reference electrode, as a function of time for the cylinder electrode.
Figure 4-2. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about zero applied current.
-900
-800
-700
-600
-500
0 20 40 60 80 100 120t, hr
Pot
entia
l, m
V (
SC
E)
0
50
100
0 50 100 150 200 250 300
Zr, Ω
-Zj,
Ω
1 hr 6 hr 21 hr 46 hr 72 hr
65
Figure 4-3. Bode plots of the negative imaginary component as a function of frequency, at selected times, for the cylinder electrode in response to variable amplitude galvanostatic modulation about zero applied current.
0
20
40
60
80
0.001 0.01 0.1 1 10 100Frequency, Hz
-Zj,
Ω
1 hr 6 hr 21 hr 46 hr 72 hr
66
Figure 4-4. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the cylinder electrode maintained at an applied cathodic current
density of 1.6 µA/cm2.
Figure 4-5. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about an applied cathodic DC
current density bias of 1.6 µA/cm2.
-900
-800
-700
-600
-500
0 20 40 60 80 100 120 140
t, hr
Pot
entia
l, m
V (
SC
E)
0
100
200
0 100 200 300 400
Zr, Ω
-Zj,
Ω
1 hr 24 hr 45 hr 70 hr 129 hr
67
Figure 4-6. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the cylinder electrode maintained at an applied cathodic current
density of 2.5 µA/cm2.
-1000
-900
-800
-700
-600
-500
0 20 40 60 80 100 120 140
t, hr
Pot
entia
l, m
V (
SC
E)
68
Figure 4-7. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about an applied cathodic DC
current density bias of 2.5 µA/cm2.
0
200
400
600
800
0 200 400 600 800 1000Zr, Ω
-Zj,
Ω
2 hr 6 hr 24 hr
72 hr 119 hr
69
Figure 4-8. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the cylinder electrode maintained at an applied cathodic current
density of 4.0 µA/cm2.
Figure 4-9. Nyquist plots at selected times for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about an applied cathodic DC
current density bias of 4.0 µA/cm2.
-1100
-1000
-900
-800
-700
-600
0 20 40 60 80 100t, hr
Pot
entia
l, m
V (
SC
E)
0
100
200
0 100 200 300 400
Zr, Ω
-Zj,
Ω
1 hr 5 hr 12 hr 24 hr 49 hr 73 hr 94 hr
70
4.3 Discrete Holiday Experiments
To characterize the impedance response of a localized conductive area
surrounded by highly resistive material as is the case for a coating holiday, experiments
were conducted using the simulated holiday electrode. The results from the holiday
experiments were also used to identify experimental high frequency artifacts by
comparing the electrolyte resistance measured by EIS to the results calculated from the
BEM simulations (see section 3.2.2). Similar to the cylinder electrode experiments, the
current flow between the working and counter electrodes was controlled, and the WE
potential was measured with respect to a reference electrode. Impedance scans were
conducted at snap shot intervals over the course of the experiment. Consistent with the
localized activity indicated as occurring on the surface by the plot for the current
distribution in Figure 3-3, corrosion products were observed to be concentrated on the
edges of the metal band upon removing the electrode from the cell.
4.3.1 Holiday Experiment 1 - Modulation About the Corrosion Potential
The purpose of this experiment was to measure the impedance response for a
simulated coating holiday with the electrode maintained at the corrosion potential. The
potential-time trace for this experiment is presented in Figure 4-10. The potential shifted
approximately 150 mV to a steady value within the first 5 hours of the experiment. The
impedance response data are presented in Figure 4-11. Much of the spectrum was
generated with the beginnings of a low frequency loop appearing at later times in the
experiment. The results were consistent with the presence of two diffusion regions.
Similar to the behavior observed using the cylinder electrode, the impedance
response for the holiday electrode appeared to be steady after the first day of exposure.
71
As the experiment progressed, more details were observed in the low frequency features
of the spectra. Because of the geometric effects and the smaller conductive surface area,
the ohmic resistance of the holiday electrode was an order of magnitude larger than that
for the cylinder electrode. The larger impedance values were consistent with the fact that
for a given current density distribution, the total current integrated over the conductive
surface was much smaller for the holiday electrode than the cylinder electrode.
4.3.2 Holiday Experiment 2 - Modulation About 5.0 µA/cm2
The purpose of this experiment was to measure the impedance response for a
simulated cathodically protected coating holiday. The potential-time trace in response to a
DC bias of 5.0 is presented in Figure 4-12. The initial transient lasted about 20
hours before the potential reached a steady value. Using the ohmic resistance determined
from the impedance plots, the IR drop was determined to be approximately 4 mV.
Consistent with other experiments, the impedance response, presented in Figure 4-13,
continued to increase with time, long after the WE potential had stabilized. The
magnitude of the impedance was larger than for the corrosion potential experiment,
suggesting a reduction in the corrosion current due to film formation. Low frequency
loops were observed as the experiment progressed.
µA cm2⁄
72
Figure 4-10. The corrosion potential, measured with respect to a calomel reference electrode, as a function of time for the holiday electrode.
Figure 4-11. Nyquist plots at selected times for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about zero applied current.
-900
-800
-700
-600
-500
0 20 40 60 80 100 120t, hr
Pot
entia
l, m
V (
SC
E)
0
1000
2000
0 1000 2000 3000 4000 5000
Zr, Ω
-Zj,
Ω
6 hr 22 hr 46 hr 99 hr
73
Figure 4-12. The cell potential, measured with respect to a calomel reference electrode, as a function of time for the holiday electrode maintained at an applied cathodic current
density of 5 µA/cm2. The increase in the potential at the end of the trace occurred after resetting the applied current to 0.
Figure 4-13. Nyquist plots at selected times for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about an applied cathodic DC
current density bias of 5.0 µA/cm2.
-900
-800
-700
-600
-500
0 20 40 60 80 100 120t, hr
Pot
entia
l, m
V (
SC
E)
0
2000
4000
0 2000 4000 6000 8000
Zr, Ω
-Zj,
Ω
2 hr 23 hr 73 hr 114 hr
CHAPTER 5DATA ANALYSIS
5.1 Overview
This chapter discusses the analyses performed on the impedance data generated for
this work. Mathematical models for the impedance response were regressed to the data to
assess measurement errors and to describe physical phenomena. Since the models
contained nonlinear terms and complex quantities, regression procedures were performed
using complex nonlinear least-squares (CNLS) algorithms [18, 31-35]. The CNLS
technique allowed for a set of common parameters to be determined from simultaneous
model regression to both the real and imaginary components of the collected data.
Statistical data analysis was performed using the measurement model approach, to
identify non-stationary behavior and inconsistencies with the Kramers-Kronig relations
and to determine an estimate for the stochastic measurement errors. An example for
applying the measurement model approach is presented in APPENDIX C. For detailed
analysis of the physical phenomena, an eight-parameter process model was regressed to
data sets, individually, to obtain values describing the transport and kinetic processes
associated with the electrochemistry of the cell. The regressed parameters were plotted as
a function of exposure time to identify time dependent changes of the WE when subjected
to a particular level of polarization. Finally, the regressed values were related to
polarization parameters and used to extrapolate impedance spectra at frequencies outside
the tested range.
74
75
5.2 Process Model Regression Analysis
A mathematical model for the impedance response of a stationary cylindrical steel
electrode was developed (see section 2.6) and regressed to data generated for this work.
The model accounted for the kinetics of the electrochemical reactions including iron
dissolution, oxygen reduction, and hydrogen evolution, and it accounted for the diffusion
of oxygen from the surrounding bulk electrolyte to the steel electrode surface. The model
was used to obtain values for parameters changing with time.
5.2.1 Model Parameters
The model expresses the transfer function relating an input oscillating current
signal to an output oscillating potential signal as the complex electrochemical impedance
according to
(5-1)
The diffusion impedance, given by equation (2-38), depends on the oscillating flux
of oxygen at the surface. By combining the variables preceding into the lumped
coefficient , the diffusion impedance can be expressed as
(5-2)
where depends on the angular frequency and is given by equation (2-66). By
introducing the substitution
Zr jZj+ Re1
1Rt Fe,------------ 1
Rt O2, ZD O2,+-------------------------------- 1
Rt H2,------------ jωCd+ + +
------------------------------------------------------------------------------------------+=
ZD O2,
1
θ’˜ 0( )------------–
ZD 0,
ZD O2, ZD 0,1
θ’˜ 0( )------------–
=
1
θ’˜ 0( )------------–
76
(5-3)
equation (2-66) can be simplified to
(5-4)
By combining the charge transfer resistances due to iron dissolution and hydrogen
evolution into one effective charge transfer resistance parameter given by
(5-5)
equation (5-1) was simplified for regression analysis as follows:
(5-6)
From equations (5-4) and (5-6), the complex electrochemical impedance response
of the steel WE was described as a function of angular frequency, ω, with eight parameter
constants summarized as follows:
• is the time constant in seconds, given by equation (2-57), associated with
diffusion through a porous film,
• is the time constant in seconds associated with diffusion in the bulk, see
equation (2-57),
• is the ratio of the diffusivity of oxygen in the bulk to that in the film,
• is an effective charge transfer resistance in Ω given by equation (5-5),
Db f⁄DO2 b,
DO2 f,-------------=
1
θ'˜ 0( )------------–
jωτO2 b,( )tanh Db f⁄ jωτO2 f,( )tanh+
jωτO2 f, jωτO2 f,( )tanh jωτO2 b,( )tanh Db f⁄+[ ]-------------------------------------------------------------------------------------------------------------------------------------=
1Reff-------- 1
Rt Fe,------------ 1
Rt H2,------------+=
Zr jZj+ Re1
1Reff-------- 1
Rt O2, ZD 0,1
θ'˜ 0( )------------–
+---------------------------------------------------- jωCd+ +
---------------------------------------------------------------------------------------+=
τO2 f,
τO2 b,
Db f⁄
Reff
77
tion.
ial,
, and
ge
and
ved
• is the charge transfer resistance associated with oxygen reduction in Ω, given by
equation (2-36),
• is the coefficient of the diffusion impedance in Ω, see equation (5-2),
• is the cell capacitance associated with double layer charging in Farads, and
• is the electrolyte or ohmic resistance in Ω.
Depending on the applied current density, could be simplified with
assumptions regarding the comparative rates of iron dissolution and hydrogen evolu
For experiments conducted with the cylinder WE maintained at the corrosion potent
the corrosion current and the oxygen reduction current are assumed to be balanced
the hydrogen evolution current is assumed small enough to be neglected. The char
transfer resistance for hydrogen evolution, , is then large and, by equation (5-5),
is approximately equal to . As the applied current density becomes much more
cathodic, the rate of iron dissolution becomes small compared to hydrogen evolution
becomes large allowing for to be equated to .
In some cases, including the parameter in the regression procedure pro
difficult without constraining its value. Simplifying equation (2-36) with substitution of
from equation (2-32) and solving for yielded
(5-7)
where A is the surface area of the WE. From equation (5-7), a reasonable estimate for
required an appropriate value for and A. At cathodic current densities,
Rt O2,
ZD 0,
Cd
Re
Reff
Rt H2, Reff
Rt Fe,
Rt Fe, Reff Rt H2,
Rt O2,
i O2Rt O2,
Rt O2,RT
αO2F
------------ 1i O2
A------------–=
Rt O2, i O2
78
polarizing the WE to more negative potentials, as applied during the experiments for this
work, the contribution to the current flow due to oxygen reduction was equivalent to the
mass-transfer-limited current, i.e., , as shown by the oxygen reduction
plateau in Figure 2-1. From dynamic sweep data, was estimated to be in the range
of 4 to 6 µA/cm2, as, for example, in the plot in Figure 3-8. Substituting the apparent
surface area for A, approximately 60 cm2 for the cylinder electrode, was calculated
to a value on the order of 100 Ω.
In most cases, an initial guess of 100 Ω for , proved to be large, and
regression of equation (5-6) to impedance data usually failed to converge. In cases where
the regression did converge, the parameter estimation for was much less than 1, and
the calculated confidence interval for the parameter estimation included zero. In such
cases, the regressions were performed by fixing the value of to zero. Values for
were successfully obtained only for the experiment conducted using the cylinder
electrode maintained at the corrosion potential.
5.2.2 Quality of Regression
Results from the regression analyses showed reasonable agreement between the
process model and the measured data. For example, the impedance response of the
cylinder electrode to variable amplitude galvanostatic modulation about the corrosion
potential, after 24 hours of exposure, is presented in Nyquist form in Figure 5-1. The
figure includes the expected response yielded from the process model. Upon inspection of
the Nyquist plot, the model appeared to fit the data well with the curve passing through the
i O2ilim O, 2
=
ilim O, 2
Rt O2,
Rt O2,
Rt O2,
Rt O2,
Rt O2,
79
data points. The residual errors, presented as a function of frequency for both the real and
imaginary parts in Figure 5-2, are on the order of 1 to 2 percent of the model prediction.
In the high frequency region, the fitting errors were larger, approximately 6 to 8 percent of
the model prediction. Though the residual errors appeared to be small, they exhibited
oscillating behavior about the zero line. This result suggests that the fit could be improved
upon with further development of the process model.
The residual errors were also compared to the estimated stochastic noise level,
estimated from the measurement model regression analyses. In some cases, as shown in
Figure 5-3 and Figure 5-4, the residual errors were larger than the estimated noise limits.
In other cases, the errors were on the order of the noise level, as shown in Figure 5-5 and
Figure 5-6. The 95.4% confidence interval, generated from Monte Carlo simulations at
each frequency step, are included in Figure 5-5 and Figure 5-6 [20]. The confidence
interval gave insight to parts of the spectrum where regression uncertainties appeared.
Typically, the confidence intervals were broader in the low frequency range than the high
frequency range. Because of sweep-time limitations, only the first few points of the low
frequency features could be obtained, thus reducing the certainty of the process model
prediction. Severe lack of fit errors were evident where data points lay outside the
confidence intervals.
80
Figure 5-1. The impedance response in Nyquist form of the cylinder electrode to variable amplitude galvanostatic modulation about the corrosion potential, including the results for the process model regression using modulus weighting. The error bars represent the 95.4% confidence intervals for the model estimation for both the real and imaginary components. The data were generated 24 hours after the WE was exposed to the electrolytic environment.
Figure 5-2. Both the normalized real and imaginary component residual errors, as a function of frequency, resulting from process model regression to the data of Figure 5-1.
0
100
0 100 200 300Zr, Ω
-Zj,
ΩData Model
-0.100
-0.050
0.000
0.050
0.100
0.001 0.01 0.1 1 10Frequency, Hz
(Z da
ta -
Z m
odel)/
Z m
odel
Real Errors Imaginary Errors
81
Figure 5-3. The normalized real component residual errors, as a function of frequency, resulting from process model regression to the data of Figure 5-1. The estimated stochastic noise limits are included.
Figure 5-4. The normalized imaginary component residual errors, as a function of frequency, resulting from process model regression to the data of Figure 5-1. The estimated stochastic noise limits are included.
-0.010
-0.005
0.000
0.005
0.010
0.001 0.01 0.1 1 10
Frequency, Hz
(Z r,
data
- Z
r,mod
el)/
Z r,m
odel
Real Errors Error Structure
-0.050
0.000
0.050
0.001 0.01 0.1 1 10Frequency, Hz
(Z j,d
ata
- Z
j,mod
el )/Z
j,mod
el
Imaginary Errors Error Structure
82
Figure 5-5. The normalized real component residual errors, as a function of frequency, resulting from process model regression to data generated from modulation about an
applied DC current density bias of 1.6 µA/cm2. The impedance response was measured from the cylinder electrode after 24 hours of exposure. The estimated stochastic noise limits are included with the 95.4% confidence intervals.
-0.050
-0.025
0.000
0.025
0.050
0.001 0.01 0.1 1 10
Frequency, Hz
(Z r,
data
- Z
r,mod
el)/
Z r,m
odel
Real Errors Error Structure
Confidence Interval
83
Figure 5-6. The normalized imaginary component residual errors, as a function of frequency, resulting from process model regression to data generated from modulation
about an applied DC current density bias of 1.6 µA/cm2. The impedance response was measured from the cylinder electrode after 24 hours of exposure. The estimated stochastic error structure limits are included with the 95.4% confidence intervals.
-0.200
-0.100
0.000
0.100
0.200
0.001 0.01 0.1 1 10
Frequency, Hz
(Z j,d
ata
- Z
j,mod
el )/Z
j,mod
el
Real Errors Error Structure
Confidence Interval
84
5.2.3 Regression Parameter Results
Parameter values, regressed from impedance data, and the corresponding cell
potential measured at the time of each scan, were compared to gain insight into the time
behavior of the system. To explore the effect of polarization, the regression parameter
values were also investigated as a function of the measured cell potential. For
experiments conducted with the WE at applied current densities where Tafel kinetics
dominated, correlation between the parameter values and the cell potential was observed.
For experiments conducted with the WE polarized to a position on the oxygen reduction
plateau, the regression parameters did not appear to be correlated with the cell potential.
Complete regression parameter results and calculated film thicknesses for all the
experiments conducted using the cylinder electrode are presented in APPENDIX D. For
the results presented here and in APPENDIX D, the parameters , , and
were multiplied by the apparent WE surface area and expressed in Ω•cm2. The cell
capacitance, , is given in µF/cm2. Working electrode areas for each experiment are
listed in Table 3-5.
Regressed values for the parameters , , and are plotted as
functions of time in Figure 5-7, Figure 5-8, and Figure 5-9, respectively for the experiment
conducted with the cylinder electrode at zero applied current. The plots also include the
corresponding cell potential, corrected for IR drop, as a function of time. The time
coordinate was plotted on a logarithmic scale to expand the results obtained from early
times during the experiment. Both and appeared to increase with time, while
the results for showed no clearly observable correlation with time or potential.
Reff Rt O2, ZD 0,
Cd
τO2 f, τO2 b, Db f⁄
τO2 f, τO2 b,
Db f⁄
85
Film and diffusion layer thicknesses were estimated from the parameters ,
, and . A value of cm2/s was assumed for the diffusivity of
oxygen in aqueous solution [36]. In the porous sand matrix, however, the effective
diffusivity was estimated by multiplying the assumed value, in the absence of porous
media, by a porosity factor according to [23]
(5-8)
Assuming a porosity or void fraction of 0.40, as previously for effective conductivity
approximation, , was estimated to for early times during an
experiment. The bulk diffusion layer thickness was then calculated by rearranging
equation (2-57) to
(5-9)
To obtain an estimate of the film thickness, was determined by
rearrangement of equation (5-3), and was found from equation (5-9) by inserting
the variables for the film. Over the course of an experiment, it was assumed that
was reduced as loosely adsorbed films, mainly corrosion products, deposited into the sand
surrounding the WE, and remained approximately constant. After long exposure
times, was calculated from and , and the film and bulk diffusion layer
thicknesses were again calculated using the appropriate forms of equation (5-9). The
results for the calculated film thickness are presented in Figure 5-10. It appeared that a
compact film [37] formed early in the experiment as increased rapidly within the
τO2 f,
τO2 b, Db f⁄ 2.41 10 5–×
DO2 b, DO2 0, ε0.5=
DO2 b, 1.52 10 5–×
δO2 b, τO2 b, DO2 b,( )0.5=
DO2 f,
δO2 f,
DO2 b,
DO2 f,
DO2 b, DO2 f, Db f⁄
δO2 f,
86
first 10 hours of WE exposure. As increased, the cell potential became more
negative as was observed in Figure 5-12.
From Figure 5-8, was initially small, on the order of 20 to 30 seconds, and
increased to values on the order of 1000 seconds within the first 24 hours of the
experiment. The results are consistent with oxygen being initially abundant near the
surface and subsequently being consumed as the experiment progressed, causing the
development of a concentration gradient between the bulk of the electrolytic environment
and the surface. Results for the calculated diffusion layer thickness are presented in
Figure 5-11, showing reached values of approximately 800 microns after long
exposure times. From Figure 5-13, appears to be correlated with the cell potential,
by increasing as the potential became more negative.
The error bars in Figure 5-8, Figure 5-11, and Figure 5-13, representing the 2σ
interval for the parameter estimation, are broad enough to include values less than zero
since limited data were obtained in the very low frequency range. As previously
described, at early times when the WE potential shifted through a large transient, the low
frequency part of the impedance spectrum could not be collected. As the WE potential
steadied, more data in lower frequencies could be collected, and the regression procedure
yielded more accurate values for the time constants. However, since the complete low
frequency spectrum was not collected, the confidence intervals for the model fit were
broad enough in some cases to include values less than zero. Though time constant values
less than zero make no physical sense, the level of uncertainty was reported as
δO2 f,
τO2 b,
δO2 b,
δO2 b,
87
demonstrating the limitations encountered when attempting to measure the low frequency
phenomena.
The results for the parameter are presented in Figure 5-14, and correlation
was observed with potential as presented in Figure 5-15. Values for were observed to
increase with time and increase as the potential became more negative. With the cell
maintained near or at the corrosion potential, was assumed to be influenced only by
the rate of iron dissolution, consistent with Figure 2-1. Since the corrosion potential
reached a steady value after the first day of exposure, as shown in Figure 4-1, changes in
indicated changes in the corrosion current as a result of film formation.
Regressed values for are plotted as functions of time in Figure 5-16 and
potential in Figure 5-17. As previously discussed, regressing for was difficult. For
data sets collected at early times of the experiment, preliminary regressions yielded an
estimation much less than 1 for . For these regressions, the value was fixed at zero.
For data sets collected at longer times during the experiment, the regressions yielded
estimations on the order of 20 to 30 Ω. The parameter estimates were significantly lower
than estimated values, on the order of 100 Ω, as previously discussed, raising questions as
to the reliability of the assumed values for and the WE surface area. From Figure
5-16, , given in Ω•cm2, was shown to increase with time when the regression
procedure successfully yielded significant values. From Figure 5-17, was observed
to increase slightly with increasing potential.
Reff
Reff
Reff
Reff
Rt O2,
Rt O2,
Rt O2,
ilim O, 2
Rt O2,
Rt O2,
88
The regression results for the parameter are presented in Figure 5-18. The
most significant changes in occurred in the initial 10 hours consistent with the time
for forming compact films. The behavior of , as a function of potential, is presented
in Figure 5-19, where was observed to increase as the potential became more
negative. The results for and combined suggest that film formation also
influenced the rate of oxygen reduction ( ) after the cell potential reached a steady
value within one day of exposure.
The results for the cell capacitance, , are plotted as functions of time in Figure
5-20 and potential in Figure 5-21. From Figure 5-21, the capacitance was correlated with
potential where was observed to decrease as the potential became more negative. The
values for were uncertain since the actual WE surface area was not known. The actual
values were expected to be lower than those indicated in Figure 5-20 and Figure 5-21. A
degree of surface roughening would cause the actual surface area to be larger than the
apparent area calculated from the dimensions of the electrode.
The results for the electrolyte resistance, are presented in Figure 5-22. The
regressed values for , approximately 60 to 65 Ω, were constant over time and
independent of potential. The results also agreed with the calculated values for the cell
geometry, listed in Table 3-2.
To contrast the results from the experiment conducted at zero applied current
density, the behavior of with time and cell potential is presented in Figure 5-23 and
Figure 5-24, respectively. These are given as examples for regression parameter values
ZD 0,
ZD 0,
ZD 0,
ZD 0,
ZD 0, Rt O2,
ilim O, 2
Cd
Cd
Cd
Re
Re
τO2 f,
89
obtained using the cylinder electrode at an applied DC current density of 1.6 µA/cm2,
corresponding to a position on the oxygen reduction plateau. From Figure 5-23,
increased with time, consistent with early formation of a compact film. However,
did not appear to be correlated with the cell potential as shown in Figure 5-24. None of
the other regression parameters appeared to be correlated with potential for this
experiment. Similar results were observed for the experiment conducted using the
cylinder electrode with the applied current density at 2.5 µA/cm2 (see APPENDIX D).
As applied currents became more cathodic, was assumed to be equal to ,
according to equation (5-5), and to influence only the rate of hydrogen evolution as the
rate of corrosion decreased to negligible values as consistent with Figure 2-1. For the
experiment conducted on the cylinder electrode at 4.0 µA/cm2, the results for as a
function of time and cell potential are presented in Figure 5-25 and Figure 5-26,
respectively. The values for were observed to decrease with time and decreasing
potential. After one day of exposure, the potential reached a steady value, as shown in
Figure 4-8, which indicated that changes in led to changes in the rate of hydrogen
evolution. Increases in the rate of hydrogen evolution could be due to hydrogen bubbles
forming and diffusing away from the surface disrupting films causing the surface to be
less resistive.
τO2 f,
τO2 f,
Reff Rt H2,
Reff
Reff
Reff
90
Figure 5-7. The diffusion time constant for the film and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure 5-8. The bulk layer diffusion time constant and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
0
5
10
15
20
1 10 100t, hr
τf , s
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Time Constant Potential
-500
0
500
1000
1500
2000
2500
3000
3500
1 10 100t, hr
τb, s
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Time Constant Potential
91
Figure 5-9. The ratio of the diffusivities of oxygen in the bulk to the film and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
0
10
20
30
40
50
60
1 10 100t, hr
Db /
Df
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Diffusivity Ratio Potential
92
Figure 5-10. The calculated film thickness in microns and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure 5-11. The calculated bulk diffusion layer thickness in microns and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
0
10
20
30
40
1 10 100t, hr
δf, µ
m
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Film Thickness Potential
-500
0
500
1000
1500
2000
2500
3000
1 10 100t, hr
δb, µ
m
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Diffusion Layer Potential
93
Figure 5-12. The calculated film thickness in microns as a function of potential for the cylinder electrode with the applied current equal to zero.
Figure 5-13. The calculated bulk diffusion layer thickness in microns as a function of potential for the cylinder electrode with the applied current equal to zero.
0
10
20
30
40
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
δf, µ
m
-500
0
500
1000
1500
2000
2500
3000
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
δb, µ
m
94
Figure 5-14. The effective charge transfer resistance and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure 5-15. The effective charge transfer resistance as a function of potential for the cylinder electrode with the applied current equal to zero.
1000
10000
100000
1 10 100t, hr
Ref
f, Ω
cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Effective Resistance Potential
1000
10000
100000
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
Ref
f, Ω
cm
2
95
Figure 5-16. The charge transfer resistance for oxygen reduction and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure 5-17. The charge transfer resistance for oxygen reduction as a function of potential for the cylinder electrode with the applied current equal to zero.
100
1000
10000
1 10 100t, hr
Rt,O
2, Ω
cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Oxygen Reduction Resistance Potential
100
1000
10000
-820 -810 -800 -790 -780
Potential, mV (SCE)
Rt,O
2, Ω
cm
2
96
Figure 5-18. The diffusion impedance coefficient and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure 5-19. The diffusion impedance coefficient as a function of potential for the cylinder electrode with the applied current equal to zero.
0
5000
10000
15000
20000
1 10 100t, hr
Zd,
0, Ω
cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Impedance Coefficient Potential
0
5000
10000
15000
20000
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
Zd,
0, Ω
cm
2
97
Figure 5-20. The cell capacitance and WE potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure 5-21. The cell capacitance as a function of potential for the cylinder electrode with the applied current equal to zero.
0
100
200
300
400
1 10 100t, hr
Cd,
µF
/cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Capacitance Potential
0
100
200
300
400
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
Cd,
µF
/cm
2
98
Figure 5-22. The electrolyte resistance as a function of time for the cylinder electrode with the applied current equal to zero.
0
20
40
60
80
100
1 10 100t, hr
Re,
Ω
99
Figure 5-23. The diffusion time constant for the film and WE potential as functions of
time for the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure 5-24. The diffusion time constant for the film as a function of potential for the
cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
0
20
40
60
80
1 10 100 1000t, hr
τf , s
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)
Time Constant Potential
0
20
40
60
80
-765 -760 -755 -750 -745 -740Potential, mV (SCE)
τfs
100
Figure 5-25. The effective charge transfer resistance and WE potential as functions of
time for the cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure 5-26. The effective charge transfer resistance as a function of potential for the
cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
0
20000
40000
60000
80000
100000
0.1 1 10 100t, hr
Ref
f, Ω
cm
2
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)Effective Resistance Potential
0
20000
40000
60000
80000
100000
-1000 -950 -900 -850 -800 -750Potential, mV (SCE)
Ref
f, Ω
cm
2
101
5.2.4 Parameter Values as a Function of Applied Current Density
To explore the long term polarization effects on WE surface characteristics, the
parameter values were investigated as functions of applied current density. Parameter
values, regressed from impedance data collected for the cylinder electrode after 4 days of
exposure, are plotted as functions of current density in Figure 5-27 to Figure 5-35.
Trends were not observed when inspecting the results for the parameters ,
, and presented in Figure 5-27, Figure 5-28, and Figure 5-29, respectively.
However, as previously demonstrated, these parameters were used to calculate and
, given in Figure 5-30 and Figure 5-31, respectively. The results for showed
that the film thickness increased as the applied current density increased along the oxygen
reduction plateau of the polarization curve. However, the decrease in the value for
at the 4.0 µA/cm2 point suggested that the WE was sufficiently polarized to where the rate
of hydrogen evolution was enough to disrupt some of the surface film structure.
The results for the parameters and , given in Figure 5-32 and Figure 5-34,
respectively, also suggested changes in kinetic behavior occurred as hydrogen evolution
began to dominate the surface reactions. As surface films are disrupted, the diffusion
impedance should also be expected to decrease as was the case for the results of ,
presented in Figure 5-33. Since the electrolytes contained the same compositions of ionic
species for each experiment, the electrolyte resistance did not vary with applied current
density. The small variation observed in the results for , presented in Figure 5-35, are
due to differences in packing and height of the sand matrix in the corrosion cell.
τO2 f,
τO2 b, Db f⁄
δO2 f,
δO2 b, δO2 f,
δO2 f,
Reff Cd
ZD 0,
Re
102
Figure 5-27. The diffusion time constant for the film after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
Figure 5-28. The bulk layer diffusion time constant after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
0
10
20
30
40
50
0 1 2 3 4 5
iapp, µA/cm2
τf , s
0
500
1000
1500
2000
0 1 2 3 4 5
iapp, µA/cm2
τb, s
103
Figure 5-29. The ratio of the diffusivities of oxygen in the bulk to the film after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
Figure 5-30. The calculated film thickness in microns after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
0.1
1
10
100
0 1 2 3 4 5
iapp, µA/cm2
Db /
Df
0
50
100
150
200
0 1 2 3 4 5
iapp, µA/cm2
δf, µ
m
104
Figure 5-31. The calculated bulk diffusion layer thickness in microns after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
Figure 5-32. The effective charge transfer resistance after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
0
500
1000
1500
2000
2500
0 1 2 3 4 5
iapp, µA/cm2
δb, µ
m
1.0E+04
1.0E+05
1.0E+06
0 1 2 3 4 5
iapp, µA/cm2
Ref
f, Ω
cm
2
105
Figure 5-33. The diffusion impedance coefficient after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
Figure 5-34. The cell capacitance after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
100
1000
10000
100000
0 1 2 3 4 5
iapp, µA/cm2
Zd,
0, Ω
cm
2
0
100
200
300
400
500
600
0 1 2 3 4 5
iapp, µA/cm2
Cd,
µF
/cm
2
106
Figure 5-35. The electrolyte resistance after 4 days of exposure plotted as a function of applied current density for the cylinder electrode.
0
20
40
60
80
100
0 1 2 3 4 5
iapp, µA/cm2
Re,
Ω
107
5.3 Estimation of Polarization Resistance
The polarization resistance
(5-10)
is often used to provide a simple measure of corrosion rates. The parameters obtained
from the regression analyses were used to extrapolate spectra outside the tested frequency
range. The parameter values were input into the model as constants and the impedance
was calculated for a broad frequency range. Generation of complete spectra yielded the
details of the low frequency features, such as the presence of additional capacitive loops,
which allowed determination of the impedance as the frequency asymptotically
approached zero. From the two intersections of the Nyquist curve on the real axis at
and the zero-frequency limit, Z(0), the polarization resistance, , was calculated
according to equation (5-10). The polarization resistance is also the slope of the
polarization curve with potential as a function of applied current density. The polarization
resistance can be used to calculate corrosion rates from
(5-11)
where is the applied current density, is the corrosion current density, and the
parameters and are the Tafel slopes for the anodic and cathodic reactions,
respectively [12].
The impedance spectra extrapolated, using the parameter values regressed for the
cylinder electrode experiments, are presented in Figure 5-36 to Figure 5-39. Example
parameter values, at selected times, are listed in Table 5-1 to Table 5-2 for experiments
Rp Z 0( ) Re–=
Re
Rp
Rp∆V
∆iapp
-------------βaβc
2.3icorr βa βc+( )---------------------------------------= =
iapp icorr
βa βc
108
using the cylinder electrode with the applied current density at zero and 1.6 µA/cm2.
Extrapolated values for , presented in Figure 5-36 to Figure 5-38, were observed to
increase with time and as the applied current density was more cathodic. From equation
(5-11), increases in follow decreases in the corrosion current. From Figure 2-1, the
corrosion current is shown to decrease as the WE is more cathodically polarized or as the
cell potential becomes more negative. From Figure 5-39, was observed to decrease
with time. As the cell became more cathodic, the rate of hydrogen evolution began to
increase, consistent with Figure 2-1. The results in Figure 5-39 suggest that hydrogen
evolution became the dominating reaction which contributed to decreases in the
extrapolated values for .
The process model was also used to obtain regression parameters for extrapolating
complete spectra for the holiday electrode experiments. An example is presented in
Figure 5-40 for the holiday electrode maintained at the corrosion potential, and the
parameter values used for calculation are given in Table 5-3. Even though the model did
not account for the nonuniform current density distribution, it could still be employed with
some success for estimating the low frequency behavior of impedance spectra. The
polarization resistance was observed to increase with time, even after the cell potential had
reached a steady value, suggesting a reduction in the corrosion current accompanied film
formation.
In previous work, the measurement model approach had been used to predict the
low frequency impedance response of individual impedance spectra [38]. For this work,
Rp
Rp
Rp
Rp
109
good agreement was obtained between the process model and the measurement model
extrapolations for Z(0) as demonstrated by the results discussed below.
Using the holiday electrode, an experiment was performed where the applied
current density was stepped through a series of cathodic values. The experiment began at
0 current (corrosion potential) to allow the WE to undergo the initial transient potential.
Impedance data were collected during and after the initial transient. After the WE
potential reached a steady value, the current was stepped to a more cathodic value. Upon
the potential reaching a new steady value, impedance data were collected and the current
stepping process was repeated. The data, including the potential and applied current
density plotted as functions of time, are presented in Figure 5-41. The current density
values applied during the experiment were 0, 1.5, 3.0, 5.0, 10.0, and 12.0 µA/cm2. The
applied values included the corrosion potential, slightly cathodic points, points on the
oxygen reduction plateau, and a point on the hydrogen evolution curve. The curve in
Figure 5-42 is the polarization curve for the holiday electrode generated from a
galvanodynamic sweep at a rate of 0.33 µA/cm2 per minute. The experimental points for
the applied current densities and resulting potential are also included in Figure 5-42.
Selected impedance data generated at each level of polarization, including extrapolation of
the full spectrum using the process model regression parameters, are presented as
Nyquist plots in Figure 5-43. Consistent with the slope of the curve in Figure 5-42, the
impedance response and extrapolated values for were observed to increase beginning
at the point of the corrosion potential and continuing along the oxygen reduction plateau.
As the hydrogen evolution curve was encountered, the impedance response and
extrapolation for at such an applied level decreased, as was consistent with the slope of
Rp
Rp
110
the polarization curve. Extrapolated values for , Z(0), and , from both process
model and measurement model regressions, are given in Table 5-4.
For comparison, the slope of the curve in Figure 5-42, calculated from the data
using a centered difference formula, is plotted as a function of applied current density in
Figure 5-44. The extrapolated values of using both the process model and the
measurement model approach, included in Figure 5-44, show good agreement when
compared to each other. The results also demonstrated that the polarization curve could be
characterized using impedance spectra at various levels of WE polarization to determine
.
Re Rp
Rp
Rp
111
Table 5-1. Process model parameter values, at selected times, used to extrapolate the impedance response of the cylinder electrode maintained at the corrosion potential.
Time, hr
, Ω , s,
s
,
Ω, Ω , µF
,
Ω
2 61 8 20 40 190 170 14700 -
5 62 12 200 14 230 200 8700 -
25 62 12 700 22 190 430 6700 -
49 61 7 1200 40 140 800 10000 22
72 62 8 1300 32 140 600 11000 29
Table 5-2. Parameter values, at selected times, used to extrapolate the impedance
response of the cylinder electrode at an applied cathodic current density of 1.6 µA/cm2.
Time, hr , Ω , s , s , Ω , Ω , µF
1 64 9 20 14 100 580 10700
24 63 31 240 30 270 300 10300
48 62 28 260 30 250 520 11200
70 61 26 270 23 230 1200 12500
129 60 37 600 12 280 1400 21300
Re τO2 f,τO2 b, Db f⁄
ZD 0, Reff CdRt O2,
Re τO2 f, τO2 b, Db f⁄ ZD 0, Reff Cd
112
Table 5-3. Process model parameter values, at selected times, used to extrapolate the impedance response of the holiday electrode maintained at the corrosion potential.
Time, hr , Ω , s , s , Ω , Ω , µF
60 690 5 300 900 5100 6600 210
46 650 7 530 1000 5600 8200 240
99 630 6 800 400 5400 8200 260
Table 5-4. Extrapolated values for Re, Z(0), and Rp from regression of the process model and measurement models to the impedance data presented in Figure 5-43.
iapp,
µA/cm2
Process Model Prediction Measurement Model Prediction
, Ω Z(0), Ω , Ω , Ω Z(0), Ω , Ω
0 630 2760 2130 600 2760 2160
1.5 660 3600 2940 640 3830 3190
3.0 660 6600 5940 630 6460 5830
5.0 640 12100 11460 620 14500 13880
10.0 670 18400 17730 620 18010 17390
12.0 670 9000 8330 630 8730 8100
Re τO2 f, τO2 b, Db f⁄ ZD 0, Reff Cd
Re Rp Re Rp
113
Figure 5-36. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about zero applied current.
Figure 5-37. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about an applied cathodic DC current density bias of
1.6 µA/cm2.
0
100
0 100 200 300 400
Zr, Ω
-Zj,
Ω
2 hr 5 hr 25 hr 49 hr 72 hr
0
100
200
0 100 200 300 400 500
Zr, Ω
-Zj,
Ω
1 hr 24 hr 48 hr 70 hr 129 hr
114
Figure 5-38. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about an applied cathodic DC current density bias of
2.5 µA/cm2.
Figure 5-39. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the cylinder electrode to variable amplitude galvanostatic modulation about an applied cathodic DC current density bias of
4.0 µA/cm2.
0
500
1000
0 500 1000 1500 2000
Zr, Ω
-Zj,
Ω
2 hr 6 hr 24 hr 72 hr 119 hr
0
100
200
0 100 200 300 400 500
Zr, Ω
-Zj,
Ω
1 hr 5 hr 12 hr 24 hr 94 hr
115
Figure 5-40. Nyquist plots at selected times, including experimental data and process model extrapolations, for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about zero applied current.
0
1000
2000
0 1000 2000 3000 4000 5000
Zr, Ω
-Zj,
Ω
6 hr 46 hr 99 hr
116
Figure 5-41. The cell potential, measured with respect to the calomel reference electrode, and the applied current density as functions of time for the holiday electrode.
Figure 5-42. Cathodic polarization curve, generated from a galvanodynamic sweep
performed using the holiday electrode. The sweep rate was 0.33 µA/cm2 per minute. The measured cell potential is plotted as a function of the applied current density including the experimental points corresponding to the step changes in Figure 5-41.
-1100
-1000
-900
-800
-700
-600
0 20 40 60 80 100t, hr
Pot
entia
l, m
V (
SC
E)
-5
0
5
10
15
Cur
rent
Den
sity
, A
/cm
2
Potential Current Density
-1200
-1000
-800
-600
-400
0.0001 0.001 0.01 0.1 1 10 100
Current Density, µA/cm2
Pot
entia
l, m
V (
SC
E)
Polarization Curve Experimental Points
117
Figure 5-43. Nyquist plots for the impedance response of the holiday electrode to variable amplitude galvanostatic modulation about several applied current densities. The collected data and extrapolated spectra using the process model are included.
0
5000
10000
0 5000 10000 15000 20000
Zr, Ω
-Zj,
Ω
i = 0 µA/cm² i = 1.5 µA/cm²i = 3.0 µA/cm² i = 5.0 µA/cm²i = 10.0 µA/cm² i = 12.0 µA/cm²
118
Figure 5-44. The slope of the polarization curve, calculated from the data presented in Figure 5-42, as a function of applied current density for the holiday electrode. Extrapolated polarization resistance values, using both the process model and the measurement model approach, are included for the experiments in Figure 5-43.
1000
10000
100000
0.1 1 10 100
Current density, µA/cm2
dV/d
i, R
p: Ω
cm
2
dV/di From Data Rp From Process Model
Rp From Measurement Model
119
5.4 Link to Polarization Parameters
The effective charge transfer resistance, , was used to calculate polarization
parameters for steel. For the experiment conducted with the cylinder electrode maintained
at the corrosion potential, was approximated to according to equation (5-5).
From equations (2-28) and (2-29)
(5-12)
By making the substitution
(5-13)
and taking the natural logarithm of both sides, equation (5-12) can be written as
(5-14)
where is the Tafel slope for iron dissolution [12, 13, 23]. Equation (5-14) is of the
form for a line where , , , and
. The potentials and are given with respect
to the calomel reference electrode used in these experiments. The results for
regressed from the individual impedance scans were presented in Figure 5-14 and Figure
5-15 for the experiment conducted with the WE maintained at the corrosion potential. The
results for fitting a straight line to a plot of as a function of are presented in
Reff
Reff Rt Fe,
1Rt Fe,------------ nFeFkFe
αFeF
RT-------------
αFeF
RT------------- V VFe–( )
exp=
αFeF
RT------------- 2.303
βFe
-------------=
Rt Fe,ln2.303βFe
-------------V–2.303βFe
-------------VFe
2.303nFeFkFe
βFe
----------------------------------
ln–+=
βFe
y mx b+= y Rt Fe,ln= m2.303βFe
-------------= x V–=
b2.303βFe
-------------VFe
2.303nFeFkFe
βFe
----------------------------------
ln–= V VFe
Reff
Reffln V–
120
Figure 5-45. The slope of the line, m, was determined to be 34.5 V-1, and from
, the Tafel slope, , was calculated to be 0.067 V or 67 mV.
An estimation for the corrosion current was determined by multiplying equation
(2-27) for by the reciprocal of equation (5-12) for according to
(5-15)
Substitution of for and for yields
(5-16)
The results for are plotted as a function of time in Figure 5-46. Changes in the
corrosion current were consistent with changes in the effective charge transfer resistance
due to film formation.
The development for determining was followed for estimating a value for the
Tafel slope for hydrogen evolution, . For experiments conducted using the cylinder
electrode with large cathodic applied currents, the corrosion current was assumed to be
negligible. Equation (5-5) was then reduced to as was assumed to be
very large. Following the development of equations (5-12) and (5-13) for hydrogen
evolution (see also equations (2-30) and (2-31)) an expression similar to equation (5-14)
was written according to
(5-17)
m2.303βFe
-------------= βFe
i Fe Rt Fe,
i Fe Rt Fe,⋅ RTαFeF-------------
βFe
2.303-------------= =
icorr i Fe Reff Rt Fe,
icorr
βFe
2.303Reff----------------------=
icorr
βFe
βH2
Reff Rt H2,∼ Rt Fe,
Rt H2,ln2.303βH2
-------------V2.303βH2
-------------VH2–
2.303nH2FkH2
βH2
----------------------------------
ln–=
121
Equation (5-17) is also in the form of a line. Determination of was accomplished
using data collected during the experiment using the cylinder electrode with the applied
current density equal to 4.0 . The results for were presented in Figure 5-25
and Figure 5-26. The plot of as a function of is presented in Figure 5-47. By
fitting a straight line to the points, was estimated from the slope, according to
, to be 205 mV.
Following the same development for equations (5-15) and (5-16) using equation
(2-30) for , the hydrogen evolution current was estimated from
(5-18)
Estimated values for are plotted as a function of time in Figure 5-48. The results
show that the magnitude of increased within the first 10 hours of the experiment
before reaching a steady value. Results from the impedance plots for the same
experiment, given in Figure 5-39, show that the system appeared to reach a steady state
within the first day of exposure.
βH2
µA cm2⁄ Reff
Rln eff V
βH2
m2.303βH2
-------------=
i H2
i H2
βH2
2.303Reff----------------------=
i H2
i H2
122
Figure 5-45. The natural logarithm of the effective charge transfer resistance plotted as a function of potential including the equation for the fitted line. Values were obtained from process model regression to impedance response data collected for the cylinder electrode maintained at the corrosion potential.
Figure 5-46. The corrosion current as a function of time calculated from the Tafel slope for iron dissolution determined for the cylinder electrode maintained at the corrosion potential.
y = 34.501x - 21.472
0
2
4
6
8
10
12
0.76 0.77 0.78 0.79 0.8 0.81 0.82
-V, V (SCE)
ln(R
eff )
0
1
2
3
4
1 10 100
t, hr
icorr, µ
A/c
m2
123
Figure 5-47. The natural logarithm of the effective charge transfer resistance plotted as a function of potential, including the equation for the fitted line. Values were obtained from process model regression to impedance response data collected for the cylinder electrode
with an applied cathodic DC current density of 4.0 µA/cm2.
Figure 5-48. The hydrogen evolution current density as a function of time calculated from the Tafel slope determined from the cylinder electrode with an applied cathodic DC
current density of 4.0 µA/cm2.
y = 11.249x + 16.785
0
2
4
6
8
-0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93 -0.92Potential, V (SCE)
ln(R
eff )
-5
-4
-3
-2
-1
0
1 10 100t, hr
iH2,
µA
/cm
2
CHAPTER 6CONCLUSIONS
The objective of this work was to apply electrochemical impedance spectroscopy
as a measuring technique for investigating corrosion processes and cathodic protection of
buried pipelines. A bench-scale corrosion cell was constructed and operated to simulate
stagnant diffusion of oxygen to the surface of a pipeline grade steel electrode embedded in
a neutral to slightly basic, electrolytic soil environment. Two field conditions were
investigated: (1) bare steel pipeline and (2) coated steel pipeline with a discrete holiday.
The cylindrical rod working electrode and Pt-Rh alloy counter electrode mesh were
arranged as concentric cylinders inside a cylindrical cell housing to guarantee symmetry
and uniform surface kinetics. Although a uniform radial current density distribution was
achieved for the holiday electrode experiments, the axial current density was highly
nonuniform with the edges of the metal band joining the acrylic insulating pieces
experiencing the highest level of surface activity.
EIS experiments were performed under variable amplitude galvanostatic
modulation about an applied DC current density bias to guarantee a unique output
potential signal with small perturbation amplitude. The variable amplitude algorithm
proved to be minimally invasive as large spikes in the steady potential were not measured
before and after impedance scans. Applied DC values included 0 net current, which
guaranteed the WE was maintained at the corrosion potential, and cathodic current
densities, which are consistent with the application of cathodic protection. The transient
124
125
potential, in response to the applied DC bias, was recorded, and EIS spectra were
generated, in replicates of 3 to 4 scans, at snap shot intervals to describe the state of the
system after a given duration of exposure. Though the DC cell potential reached steady
values within the first day of exposure for each experiment, the impedance response was
observed to be still changing after several days of exposure.
EIS spectra were generated by sweeping from high frequency, 100 Hz, to low
frequency, between 0.01 Hz and 0.001 Hz depending on how fast the system was
evolving. Data points, exhibiting unexpected capacitive behavior, were observed at
frequencies higher than 100 Hz. These were identified as instrumental artifacts using the
results from BEM simulations to calculate the cell ohmic resistance and performing
statistical error analyses to determine data inconsistent with the Kramers-Kronig relations.
Only the initial points describing the low frequency diffusion processes could be obtained
because of time limitations. A typical sweep from 100 Hz to 1 mHz in increments of 7
frequency steps per decade required 3 to 4 hours to complete. Sweeping down a full
decade more could have required a day. Generating replicates would be impossible since
the system would evolve significantly in the time required for the completion of just one
scan.
A non-equivalent circuit, steady-state process model was developed to describe the
impedance response of the cylinder electrode. The model accounted for the total current
density to the surface of the WE including the contributions from corrosion, oxygen
reduction, hydrogen evolution, and charging of the double layer. The flux of oxygen to
the surface of the WE was determined by solving the differential equations describing
stagnant diffusion through a bulk diffusion layer and a porous film adsorbed on the WE.
126
The impedance response was then determined from the ratio of the output sinusoidal
potential to the input current density signal and expressed as a complex function of the
frequency of oscillation. The model identified 8 lumped parameter constants describing
the cell ohmic or electrolyte resistance, film thicknesses, diffusion properties, and faradaic
kinetics.
The measurement model approach was used to assess the stochastic measurement
errors and to identify non-stationary behavior in the impedance response. The results
from the statistical analyses verified the non-stationary behavior of the system over the
duration of WE exposure. However, the impedance response could be considered to be
steady during the time windows for collecting sets of replicate scans. Subsequent process
model regressions were successful, and values for 7 parameters were extracted, (one of the
eight was assumed to be negligible, and its value was fixed to zero during regression
procedures). The quality of the process model fits were such that the magnitude of the
residual errors, between the model and the data, were usually on the order of the estimated
noise level determined from the measurement model analyses. However, the fitting errors
oscillated about zero suggesting the need for further model development. Because of
limited data collected at low frequencies, the standard deviations for the regressed values
of the low frequency time constants were large and the included zero in some cases.
Although the surface flux term did not account for the geometry contributing to
nonuniformities in the current density, the process model was regressed to the data
collected from the experiments using the holiday electrode.
The regression analyses allowed for extraction of time-dependent parameters for
monitoring the evolution of the system with time. From estimated film thicknesses, it was
127
observed that an inner layer formed on the steel surface quickly, typically within the first
10 hours of exposure. Increases in the impedance response, over time, in the low
frequency part of spectra were associated with an increasing outer layer or bulk layer
thickness. The ratio of the diffusivity of oxygen in the bulk to that in the film was
observed to change with time as loosely adsorbed corrosion products deposited into the
sand surrounding the steel electrode. Regression of process models to EIS data also
proved to be a superior technique for obtaining kinetic parameters over regressing models
to potential-time data.
The regression parameter values were input into the process model as constants,
and the corresponding impedance response at a given time was calculated for a wide range
of frequencies outside the tested range. The evolution of complete low frequency
diffusion processes was extrapolated, and the polarization resistance could be calculated
from the difference between the zero frequency asymptote and the high frequency
electrolyte resistance limit. Polarization resistance values and Tafel slopes, determined by
relating kinetic resistance parameters to the polarization behavior of steel, could be used to
calculate corrosion rates.
CHAPTER 7SUGGESTIONS FOR FUTURE WORK
This work has demonstrated the utility of electrochemical impedance spectroscopy
to characterize the evolutionary phenomena associated with corrosion and cathodic
protection of buried pipelines. Though the regression analyses yielded actual values for
parameters influencing behavior, the results could be further quantified with alternative
measurement techniques. For example, many assumptions were made concerning the
nature and composition of films, however, no surface analyses were performed. Though
the cell, employed for this work, was well-designed to address current and potential
distribution issues and to simulate stagnant environments, it did not allow for easy in situ
surface analyses since the working electrode was embedded in an opaque solid matrix.
Also, removing the WE from cell, while preserving surface formations for ex situ
analyses, proved to be difficult. Future studies could benefit with a cell design suitable for
performing in situ Raman spectroscopy or microscopy measuring techniques [39-41].
Several modifications to experimental methods could be considered. For this
work, electrolyte was aerated for 24 hours prior to use. Experiments could be conducted
where the electrolyte is aerated constantly. Dynamic sweeps could be performed before
and after experiments to observe changes in the mass transfer limiting current density.
Any microbiologically influenced corrosion (MIC) issues could be prevented with pre-
experimental sterilization procedures and the use of microbe growth inhibitors.
128
129
The results from this work could also be further quantified with the development
of more sophisticated models for the impedance response. In some cases, the fitting errors
were small and on the order of the estimated noise level for the measurement. However,
they did not scatter randomly about zero, but exhibited sinusoidal behavior about zero.
The process model could be improved upon with considerations including fluxes of
species participating in precipitation reactions, the kinetics of film formation, and
diffusion of dissoluted metal ions away from surface. Further model development could
also consider nonuniformities in the current and potential distributions associated with
more complicated cell geometries such as that of the holiday electrode.
Future efforts could include conducting experiments over a longer period of
exposure, such as several weeks or months, to study long time behavior. An interesting
study would be to extend the information obtained from time dependent parameters,
regressed from EIS data, to develop time dependent polarization curves. Finally, a large
data base could be collected by performing experiments for soil types including differing
clays and ionic species.
the
r the
te for
on.
dge
the
left
r all
dary
are
the
APPENDIX AFORTRAN CODE FOR BEM SIMULATIONS
The code presented here was used to solve the potential and current density
distributions, on the holiday electrode, shown in Figure 3-1 and Figure 3-2. The
axisymmetric cell domain is presented in Figure A-1 showing the coordinates for each
vertex. The coordinates are given in arbitrary units with scale, 1 unit = 0.0005 inches.
Each boundary edge was numbered for a total of 8 edges. Each edge was further divided
into subregions with node spacing as desired by the user. The file ‘nodein’ included
number of subregions followed by a list for each subregion containing the number fo
edge, the x coordinate for endpoint 1, the x coordinate for endpoint 2, the y coordina
endpoint 1, the y coordinate for endpoint 2, and the node spacing within the subregi
The subroutine CELLGEOM reads the file ‘nodein’ and discretizes each boundary e
into elements by considering the endpoints of the subregions within each edge and
node spacing for each subregion. The discretization procedure begins at the lower
corner of the cell boundary and proceeds counterclockwise along each edge.
CELLGEOM outputs the x and y coordinates for the endpoint of each element. Afte
the endpoints have been listed, CELLGEOM outputs the type and value of the boun
condition for each element. The assigned boundary conditions for the cell boundary
shown in Figure 3-2. The element endpoints and boundary conditions are output into
file ‘endpts’.
130
131
Figure A-1. The holiday electrode cell boundary including the x and y coordinates for each vertex. The edges are numbered for a total of 8.
500, 0 6000, 0
6000, 12000500, 12000
500, 6125
500, 5875
0, 0
485, 6125
485, 5875
1
2
3
4
5
6
7
8
1 unit = 0.0005 inches
8 Edges
132
c ****************************************************************c PROGRAM cpcellmainc This program solves an axisymmetric three dimensional potential problemc using constant boundary elements. The executable file is cpcellBEM.exec The x and y coordinates for the element endpoints and the assignedc boundary condition for each element are input from the file ’endpts’.c Eight point Gaussian integration was used to construct the G and Hc matrices. The x-coordinates and the weighting factors were input from thec data file ’gaussi8.txt’. The results for the potential and the potentialc derivative or flux for each element were output to the file ’elemdata’c ****************************************************************
parameter(max=800)integer N, kode(max)double precision G(max,max),H(max,max)double precision x(max),y(max),xm(max),ym(max)double precision fi(max),dfi(max)
c set maximum dimension of the system of equations (nx)c nx = maximum number of nodes = maximum number of elements
nmax=maxwrite(*,*) ’Program Running’
c read data for the position of the node and the c type and value of boundary condition
call inputpc(N,x,y,kode,fi)
c compute G and H matrices and form system (AX = F)call ghmatpc(N,x,y,xm,ym,G,H,fi,dfi,kode,nmax)
c print results for potential and flux at each elementcall outptpc(N,x,y,xm,ym,fi,dfi)
write(*,*)write(*,*) ’Program Finished’stopend
c **************************************************c SUBROUTINE INPUTPCc This subroutine reads in the endpoints and boundary conditions for eachc element. N is the number of boundary elements.c **************************************************
subroutine inputpc(N,x,y,kode,fi)
integer N,kode(1)
133
double precision x(1),y(1)double precision fi(1)
c SUBROUTINE cellgeom discretizes the cell boundary into elements andc outputs the x and y coordinates for the endpoints of each element andc outputs the type and value for the boundary condition assigned to eachc element. The data is output to the file ’endpts’.
call cellgeom(N)
open(unit=14, file=’endpts’, status=’old’)
c Read in element endpoint x and y coordinates from ’endpts’.do i=1,Nread(14,*) x(i),y(i)end do
c Read in element boundary conditions.c If kode(i) = 0, the potential is known.c If kode(i) = 1, the potential derivative is known.
do i=1,N read(14,*) kode(i),fi(i)end do
close(14)returnend
c *************************************************c SUBROUTINE CELLGEOMc This subroutine reads in the number of subregions for each boundaryc edge, reads the endpoint coordinates for each subregion and the node c spacing and discretizes the cell boundary into elements. The coordinatesc for the endpoints and the type and value of the boundary condition for eachc element are output to the file ’endpts’. An essential boundary condition isc type 0 and a natural boundary condition is type 1.c *************************************************
subroutine cellgeom(N)
implicit integer (n)double precision dphi,phi
open(unit=12, file=’endpts’)close(unit=12, status=’delete’)open(unit=12, file=’endpts’)
134
open(unit=11, file=’nodein.txt’, status=’old’)
c Read in number of subregions within the edgesc and divide subregions into nodes with given spacing.
read(11,*)NsubrDo i=1,Nsubr read(11,*)nedge,nx1,nx2,ny1,ny2,nstep select case (nedge) case (1) ndir = 1 call incrx(nx1,nx2,nstep,ny1,ndir,nnode) n1 = n1 + nnode case (2) ndir = 1 call incry(ny1,ny2,nstep,nx1,ndir,nnode) n2 = n2 + nnode case (3) ndir = -1 call incrx(nx1,nx2,nstep,ny1,ndir,nnode) n3 = n3 + nnode case (4) ndir = -1 call incry(ny1,ny2,nstep,nx1,ndir,nnode) n4 = n4 + nnode case (5) ndir = -1 call incrx(nx1,nx2,nstep,ny1,ndir,nnode) n5 = n5 + nnode case (6) ndir = -1 call incry(ny1,ny2,nstep,nx1,ndir,nnode) n6 = n6 + nnode case (7) ndir = 1 call incrx(nx1,nx2,nstep,ny1,ndir,nnode) n7 = n7 + nnode case (8) ndir = -1 call incry(ny1,ny2,nstep,nx1,ndir,nnode) n8 = n8 + nnode end selectend doN = n1+n2+n3+n4+n5+n6+n7+n8
c assign boundary conditions to each node
135
c nebc: essential boundary conditionc nnbc: natural boundary condition
nebc = 0nnbc = 1
dphi = 0d0do i = 1,n1 write(12,*) nnbc,dphienddo
phi = 0d0do i = 1,n2 write(12,*) nebc,phienddo
dphi = 0d0do i = 1,n3 write(12,*) nnbc,dphienddo
dphi = 0d0do i = 1,n4 write(12,*) nnbc,dphienddo
dphi = 0d0do i = 1,n5 write(12,*) nnbc,dphienddo
phi = 1d0do i = 1,n6 write(12,*) nebc,phienddo
dphi = 0d0do i = 1,n7 write(12,*) nnbc,dphienddo
dphi = 0d0do i = 1,n8 write(12,*) nnbc,dphienddo
write(*,*) ’Number of elements = ’,N
136
close(11)close(12)returnend
c SUBROUTINE incrxc Increments the element assignment for horizontal edges.
subroutine incrx(x1,x2,xstep,y,xdir,nxnode)
integer x1,x2,xstep,nxnode,xcd,y,xdirnxnode = 0nxnode = abs(x2-x1)/xstepxcd = x1if (xdir.eq.-1) then do while (xcd.gt.x2) write(12,100) xcd,y xcd = xcd + (xdir*xstep) end doelse do while (xcd.lt.x2) write(12,100) xcd,y xcd = xcd + (xdir*xstep) end doend if
return100 format(5i,5x,5i)
endc SUBROUTINE incryc Increments the element assignment for vertical edges.
subroutine incry(y1,y2,ystep,x,ydir,nynode)
integer y1,y2,ystep,nynode,ycd,x,ydirnynode = 0nynode = abs(y2-y1)/ystepycd = y1if (ydir.eq.-1) then do while (ycd.gt.y2) write(12,101) x,ycd ycd = ycd + (ydir*ystep) end doelse do while (ycd.lt.y2) write(12,101) x,ycd ycd = ycd + (ydir*ystep) end do
137
end if
return101 format(5i,5x,5i)
end
c *************************************************c SUBROUTINE GHMATPCc This subroutine computes the G and H matrices andc forms and solves the system of equations AX = Fc H and G are square N x N matricesc *************************************************
subroutine ghmatpc(N,x,y,xm,ym,G,H,fi,dfi,kode,nmax)INCLUDE ’mathd.fi’
integer i,j,k,ngp,ipath,N,nmax,kode(1)double precision x(1),y(1),xlen,ylen,elemlendouble precision xm(1),ym(1),temp,jacdouble precision G(nmax,nmax),H(nmax,nmax),fi(1),dfi(1),wxi(8,2)double precision Km,Em,zc,rc,zx,rx,a,b,gidouble precision m,m1,ubar,qbar,Qg,dQg,qcoeff,qnr,qnz
open(unit=13, file=’gaussi8.txt’, status=’old’)read(13,*) ngpdo i=1,ngp read(13,*) wxi(i,1), wxi(i,2)end doclose(13)
c Determine the midpoint x and y coordinates for each elementx(N+1) = x(1)y(N+1) = y(1)do i = 1,N xm(i) = (x(i)+x(i+1))/2.0 ym(i) = (y(i)+y(i+1))/2.0end do
do i = 1,N do j = 1,N G(i,j) = 0.0 H(i,j) = 0.0 end doend do
c Compute G and H matricesc The elliptic integrals of the first kind, K(x), are solved using the IMSL math/
138
c library special function DELK for using double precision data types.c The elliptic integrals of the second kind, E(x), are solved using the IMSLc math/library special function DELE for using double precision data types.
do i = 1,N rc = xm(i) zc = ym(i)
do j = 1,N xlen = x(j+1)-x(j) ylen = y(j+1)-y(j) elemlen = sqrt((xlen**2.0) + (ylen**2.0)) jac=elemlen/2.0 nr = ylen/elemlen nz = -1.0*xlen/elemlen
do k = 1,ngp rx = xm(j) + wxi(k,1)*xlen/2.0 zx = ym(j) + wxi(k,1)*ylen/2.0 a = (rc**2.0)+(rx**2.0)+(zc-zx)**2.0 b = 2.0*rc*rx m = 2.0*b/(a+b) Km=DELK(m) Em=DELE(m) if(i.eq.j) then gi=1.0 +((a-b)/b) Qg = -0.5*log((gi-1)/32.0) ubar = ((8.0/b)**0.5)*Qg qbar = 0.0 else ubar=4.0*Km/sqrt((a+b)) qcoeff = 4.0/sqrt((a+b)) qnr =(((rc**2.0)-(rx**2.0)+(zc-zx)**2.0)*Em/(a-b)-Km)/(2*rx) qnz =(zc-zx)*Em/(a-b) qbar = qcoeff*(qnr*nr + qnz*nz) end if G(i,j)=G(i,j)+ wxi(k,2)*rx*ubar*jac H(i,j)=H(i,j)+ wxi(k,2)*rx*qbar*jac end do end do write(*,300)
300 format(’.’,$)end do
c Compute diagonal elements in H matrix, H(i,i)do i=1,N do j=1,N
139
if(i.ne.j) then H(i,i) = H(i,i)-H(i,j) end if end doend do
c Rearrange G and H matrices to form AX=F systemc H matrix becomes A matrix
do j = 1,N if(kode(j).eq.0) then do i = 1,N temp = G(i,j) G(i,j) = -1.0*H(i,j) H(i,j) = -1.0*temp end do end ifend do
c Compute F vectordo i = 1,N dfi(i) = 0.0 do j = 1,N dfi(i) = dfi(i) + G(i,j)*fi(j) end doend do
c Solve the AX=F system, IPATH = 1c DLSLRG is an IMSL math/library function for solving a linear system ofc equations with double precision data types.
ipath = 1call DLSLRG(N,H,nmax,dfi,ipath,dfi)
c Put all potential values in fic put all potential derivatives in dfi
do i=1,N if(kode(i).eq.1) then temp = dfi(i) dfi(i) = fi(i) fi(i) = temp end ifend do
returnend
140
c ********************************************c SUBROUTINE OUTPTPCc This subroutine outputs the results for thec potential and flux of each element into a c data file ’elemdata’c ********************************************
subroutine outptpc(N,x,y,xm,ym,fi,dfi)integer Ndouble precision x(1),y(1),xm(1),ym(1),fi(1),dfi(1),dl
open(unit=16, file=’elemdata’)close(unit=16, status=’delete’)open(unit=16, file=’elemdata’)
x(N+1) = x(1)y(N+1) = y(1)
100 format(’ Xm Ym phi dphi dl ’)write(16,100)
do i=1,N dl=((x(i+1)-x(i))**2.0 + (y(i+1)-y(i))**2.0)**0.5
200 format(e15.6,2x,e15.6,2x,e20.10,2x,e20.10,2x,e15.6) write(16,200) xm(i),ym(i),fi(i),dfi(i),dlend do
close(16)returnend
dth,
int
ial or
APPENDIX BLABVIEW CONTROL OF EXPERIMENTS
LabVIEW, by National Instruments, was used to develop virtual instruments
(VIs) for controlling experiments. Each VI consisted of a front panel and a block diagram.
The front panel, serving as the user interface, contained buttons, dials, and switches for
setting all control parameters, and it contained graphs and other indicators for monitoring
collected data and the progress of the experiment. The block diagram contained the
source code in pictorial form which directed all programming steps, data flow, and
calculations.
For this work, VIs were developed to operate the PAR 273 Potentiostat and the
Solartron 1260 Gain Phase Analyzer and to acquire measurement data including the cell
potential and impedance response. All the main VIs and subVIs were contained in a VI
library called ‘kj SI 1260_PAR 273.llb’. For conducting an experiment, two VI front
panels were opened, ‘1260/273_main_8/98.vi’ (MAIN) and ‘I_V monitor.vi’. The
process flow and necessary user inputs will be further discussed below.
The front panel of MAIN was partitioned into controls for the PAR 273 and
controls for the Solartron 1260. The controls for the PAR 273 had to be set before
executing the MAIN VI. These included the file information, operating mode, bandwi
and measuring resistor switching factor. If operating in potentiostatic mode, the bias
reference point and the current range had to be selected. The DC bias reference po
determined whether a DC potential bias would be set versus the open circuit potent
141
142
ould
ained
nt
d
ed
anel.
lso
reen
s
d idle
the point. The actual value of the DC bias could be changed during an experiment,
and the appropriate measuring resistor, or current range, would be set automatically. The
partition for the PAR 273 controls also included a STOP button. The STOP was used to
turn the cell off and terminate execution of the VI when an experimental run had been
completed. The STOP could not be used to terminate the Solartron 1260 generator.
The controls for the Solartron 1260 could be set during execution of the MAIN VI
prior to starting a set of EIS scans. These controls included the file information, frequency
range, and all other operating controls for conducting a sweep [28]. An individual or set
of impedance scans was initiated by pressing and releasing the run impedance scan
START button. Upon initiating an impedance experiment, two additional front panels
opened automatically. Opening first, the front panel for ‘run impedance scan.vi’
contained charts for plotting the data recorded from the experiment. These charts w
update for each scan. Opening second, the front panel for ‘poll 1260 for data.vi’ cont
only the ‘STOP Current Sweep’ button. This STOP was used to terminate the curre
impedance experiment. If preset replicate scans were remaining, the program woul
prompt the user to continue or cancel the scheduled sweeps.
The front panel of ‘I_V monitor.vi’ contained charts for monitoring the measur
cell potential and applied current. The rate of data collection was also set from this p
The data rate could be set at any time during execution of the VI. The front panel a
contained an indicator light to let the user know if data was being collected or not. G
indicated data collection while red indicated the VI was in idle mode and no data wa
being collected. When the Solartron 1260 generator was operating, the light indicate
mode and no current-potential data was collected.
V 0=
143
lobal
r or
73,
loop
ght
ed
nce
of
d, the
ll was
s’
set to
l charts
rate of
The process flow for the execution of MAIN is presented as a flowchart in Figure
B-1. Upon execution of MAIN, all global variables were initialized, the PAR 273 was
initialized, the DC bias was set to 0, and the cell was turned on. The ‘Chart Status’ g
variable was set to TRUE, which was input to ‘I_V monitor.vi’ and controlled whethe
not a current-potential measurement was performed. After initialization of the PAR 2
the process flow branched to the control of two simultaneous WHILE loops. The left
controlled execution of ‘run impedance scan.vi’ and termination of MAIN while the ri
loop controlled execution of ‘I_V monitor.vi’. If an impedance experiment was initiat
by pressing the START button on the front panel of MAIN, ‘Chart Status’ was set to
FALSE, and ‘run impedance scan.vi’ was executed. When execution of ‘run impeda
scan.vi’ was complete, ‘Chart Status’ was set to TRUE and the condition for iteration
the loop was checked. If the STOP button on the MAIN front panel had been presse
‘Program Status’ boolean was set to FALSE and the experiment was stopped, the ce
turned off, all open files were closed, and the program terminated. If ‘Program Statu
was TRUE, the loop iterated. Iteration of the loop controlling execution of ‘I_V
monitor.vi’ was also determined by ‘Program Status’.
The process flow for the execution of ‘I_V monitor.vi’ is presented in Figure B-2.
The subVI first checked if the DC bias setting had changed. If so, the DC bias was
the new value. Next, the ‘Chart Status’ boolean was read. If the value was TRUE, a
potential measurement was performed and the data was recorded on the front pane
and stored in a file. The subVI then executed a time delay, corresponding to the set
data collection, before returning execution to MAIN.
144
set,
o the
n,
ed if
t the
d if
ion
aused
E and
not to
ron
was
lean
C bias
d not
The process flow for the execution of ‘run impedance scan.vi’ is presented in
Figure B-3. This subVI was executed when initiating an impedance experiment by
pressing START on the front panel of MAIN. First, a file path was prepared for the
collected data. Next, the Solartron 1260 was initialized, all sweep parameters were
and the perturbation signal generator was started. Execution was then transferred t
subVI ‘poll 1260 for data.vi’, which controlled impedance data collection, presentatio
and storage. When execution returned to ‘run impedance scan.vi’, the subVI check
the sweep was complete. If not, execution returned to ‘poll 1260 for data.vi’ to collec
next data point. If sweep generation was complete, ‘run impedance scan.vi’ checke
scheduled replicate scans remained. If no scans remained to be performed, execut
returned to MAIN. If replicate scans remained, the subVI checked if cell potential
measurements were to be performed during the delay before the next scan. If
measurements were to be made, ‘Chart Status’ was set to TRUE. The subVI then p
for the preset delay between scans. After the delay, ‘Chart Status’ was set to FALS
the Solartron 1260 was reset for the next scan. If cell potential measurements were
be made, the pause was performed, ‘Chart Status’ remained FALSE, and the Solart
1260 was reset. This loop was iterated as long as replicate scans remained to be
performed.
The flow chart for ‘poll 1260 for data.vi’ is presented in Figure B-4. The first step
performed by this subVI was to initiate a single measurement. Next, the polling loop
begun by reading the boolean value for the ‘Stop Current Sweep’ button. If the boo
was TRUE, indicating the scan was to be aborted, the generator was stopped, the D
was reset, and execution control returned to ‘run impedance scan.vi’. If the scan ha
145
vi’. If
.vi’ in
been aborted, the status byte of the Solartron 1260 was queried. If the measurement was
not ready, the polling loop was iterated. If the measurement was ready, the data was
transferred from the Solartron 1260, plotted, and filed. The PAR 273 and the Solartron
1260 were prepared for the next frequency step as the appropriate current measuring
resistor and the perturbation signal amplitude were set, respectively. If the status byte
from the 1260 had also indicated that the sweep was complete, the generator was stopped,
the DC bias was reset, and execution control was returned to ‘run impedance scan.
the sweep was not complete, execution control was returned to ‘run impedance scan
preparation for the next frequency step.
146
Figure B-1. Flow chart for operation of main control, ‘1260/273_main_8/98.vi’.
START MAIN
Start EIS = T
Yes
Program Status = T Yes
Chart Status = T
Executerun impedance scan.vi
Chart Status = F
Program Status = TYes
Execute I_V monitor.vi
No
No
STOP MAIN
No
Initialize PAR 273, Set up potential data file,Set 0 bias and turn on cell
Chart Status = T
147
Figure B-2. Flow chart for operation of ‘I_V monitor.vi’.
Execute I_V monitor.vi
Bias SettingChanged?
Yes
Potential MeasurementRecord data to chart and file
Chart Status = T
Return to MAIN
Pause before next datapoint
No
No
Yes
Reset bias
148
Figure B-3. Flow chart for operation of ‘run impedance scan.vi’.
Executerun impedance scan.vi
Set up Impedance data file
Initialize Solartron 1260 set sweep parameters
Start Generator
Executepoll 1260 for data.vi
Sweep Complete? YesReplicate scans
remaining?
Yes
Measure Potentialbetween scans?
Chart Status = T
Delay between scansChart Status = F
Return to MAINNo
Yes
No
No
149
Figure B-4. Flow chart for operation of ‘poll 1260 for data.vi’.
Executepoll 1260 for data.vi
Initiate Measurement
Query Status Byte
MeasurementReady?
Abort Scan?
Collect, plot, and file data;Set 1260 and Par 273 for next
frequency step
SweepComplete?
Stop generatorReset bias
Yes
Return torun impedance scan.vi
No
Yes
No
No
Yes
nt of
ency
ntal
ly 70
g the
onds,
e snap
APPENDIX CMEASUREMENT MODEL APPROACH
This appendix describes an example for conducting statistical analyses, using the
measurement model approach of section 2.5, to identify Kramers-Kronig-inconsistent
impedance data. The example follows closely the methods previously reported [16-22].
The Nyquist plot for the first of 3 replicate scans is presented in Figure C-1 as the filled
circles. The spectrum was generated for the cylinder electrode in response to variable
amplitude galvanostatic modulation about the corrosion potential. The tested frequency
range, high to low plotted from left to right in Figure C-1, was 100 Hz to 1 mHz. The high
and low frequency ‘tails’ at the ends of the spectrum, where the imaginary compone
the impedance was increasing, were questionable in regards to consistency with the
Kramers-Kronig relations. The tails have the appearance of the end of a high-frequ
capacitive loop and the beginning of a low-frequency capacitive loop, respectively.
A possible explanation for the high-frequency tail was the result of an instrume
artifact, since the calculated ohmic resistance for the cell geometry was approximate
Ω. The low-frequency tail appeared to be the result of non-stationary behavior durin
time necessary to complete the scan. The period of oscillation at 1 mHz is 1000 sec
and the total time to generate the spectrum of Figure C-1 was approximately three and a
half hours. This result suggests that the metaphorical shutter speed of the impedanc
shot was not fast enough to capture the short interval behavior of the system.
150
151
The measurement model regression procedure was used to evaluate the
measurement reliability at the high and low frequency ends of the spectrum. The result of
fitting 6 line shapes (see equation (2-13)) to the complex data, while using the modulus of
the impedance to weight the regression, is plotted in Figure C-1 as the solid curve. The
regression treats the high frequency tail as the first capacitive loop and extrapolates the
ohmic resistance to be approximately 34 Ω.
Assuming the data to be consistent with the Kramers-Kronig relations, the
imaginary component of the spectrum could be predicted from the real component. Four
line shapes were fit to the real component of the data only, and the regressed parameters
were used to predict the imaginary part of the spectrum. The residual errors between the
real data and the model prediction, as a function of frequency, are plotted in Figure C-2
including the 95.4% confidence interval generated by performing 5000 Monte Carlo
simulations at each tested frequency step. As shown in Figure C-2, there was good
agreement between the data and the prediction with all but the first high-frequency point
lying within the confidence intervals. The residual errors between the imaginary data and
the model prediction, including the confidence interval, are plotted as a function of
frequency in Figure C-3. Accurate prediction of the imaginary component in the high
frequency part of the spectrum failed as indicated by the first 5 high frequency points
lying outside the confidence limits. A judgement was made to reject them as being
inconsistent with the Kramers-Kronig relations and to exclude them from further analysis.
The two remaining replicate scans were treated in the same manner. The data for each
scan after rejecting the first 5 high frequency data points are presented as Nyquist plots in
Figure C-4. The number of points and frequency steps were the same for each scan.
152
From Figure C-4, bias error is evident in the low frequency part of the spectrum.
Five line shapes were regressed to the complex data of each scan individually using
modulus weighting. Each scan, then, was described as a function of frequency with the
same number of regression parameters. The bias error was effectively filtered because the
values of the corresponding parameters were different from scan to scan. An assessment
of the stochastic errors was obtained by investigating the residual errors between the data
and the model predictions. The residual errors for the real and imaginary components are
plotted as functions of frequency for all three scans in Figure C-5 and Figure C-6,
respectively. The deviations in the errors at each frequency represent the stochastic error
contribution to the total error. By assuming that the residual errors at each frequency were
normally distributed, the standard deviations for the real and imaginary residuals were
calculated. The standard deviations of the real and imaginary residual errors were
assumed to be equal, and a model taking the form of equation (2-17) was fit to the
ensemble. The fitting procedure was iterative in that several models were attempted by
varying the parameters included to obtain the best fit. For the example described here,
including only parameters β and δ yielded the best fit. Thus, the model for the stochastic
error structure was of the form
(C-1)
The results for the standard deviations including the model fit are plotted as functions of
frequency in Figure C-7.
The measurement model fitting procedure was repeated using the error structure
model as the regression weighting factor to identify inconsistencies with the Kramers-
Kronig relations. For this demonstration, the error structure analysis performed on the
σ β Zr δ+=
153
first scan of the example data set was considered. First, three line shapes were fit to the
real component of the data. The residual errors for the imaginary component are plotted
as a function of frequency in Figure C-8 including the confidence interval and the error
structure limits. Upon inspection of the plot, four additional high frequency points, lying
outside of both the confidence intervals and the error structure limits, were deemed to be
inconsistent and rejected. For identification of low-frequency inconsistencies, the model
was fit to the imaginary part of the data. For scan 1, the residual errors for both the
predicted real and the imaginary components are plotted against frequency in Figure C-9
and Figure C-10, respectively. Usually, low-frequency inconsistencies were identified by
observing points lying outside the confidence intervals on the plot of the real residuals as a
function of frequency. However, in this case, four points in the low frequency end of the
plot of the imaginary residuals were observed lying outside the confidence intervals.
These points were subsequently rejected.
After identifying and rejecting points judged to as inconsistent with the Kramers-
Kronig relations, the measurement model was regressed to the remaining complex data.
Plots for the real and imaginary residuals as a function of frequency are presented in
Figure C-11 and Figure C-12, respectively. All of the points lie within the confidence
intervals and most lie within the error structure limits. The results suggested that the
remaining data were consistent with the Kramers-Kronig relations, and the measurement
model was fit to within the estimated noise level of the measured data. Regression of
steady-state process models for extrapolation of spectra outside tested frequency ranges
proved to be easier to accomplish when including only data consistent with the Kramers-
Kronig relations.
154
Figure C-1. The impedance response and the measurement model prediction, in Nyquist form, from a preliminary scan conducted on the cylinder electrode using variable amplitude galvanostatic modulation about the corrosion potential.
0
50
100
0 50 100 150 200 250
Zr, Ω
-Zj,
ΩData Model
155
Figure C-2. The normalized real component residual errors with confidence intervals, as a function of frequency, resulting from measurement model regression, using modulus weighting, to the real component of the data in Figure C-1.
Figure C-3. The normalized residual errors with confidence intervals, as a function of frequency, between the imaginary data of Figure C-1 and the predicted imaginary component resulting from measurement model regression to the real component of the data.
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.001 0.01 0.1 1 10 100
Frequency, Hz
(Z r,
data
- Z
r,mod
el)/
Z r,m
odel
Real Errors
Confidence Interval
-5
0
5
0.001 0.01 0.1 1 10 100
Frequency, Hz
(Z j,d
ata
- Z
j,mod
el )/Z
j,mod
el
Confidence Interval
Imaginary Errors
156
Figure C-4. The data of Figure C-1, in Nyquist form, including the full set of replicate scans, after rejecting the high frequency artifacts.
0
50
100
0 50 100 150 200 250
Zr, Ω
-Zj,
Ωscan 1 scan 2 scan 3
157
Figure C-5. The residual errors, as a function of frequency, for the real component of the impedance resulting from measurement model regression of 5 line shapes, using modulus weighting, to the complex data of each individual scan of Figure C-4.
Figure C-6. The residual errors, as a function of frequency, for the imaginary component of the impedance resulting from measurement model regression of 5 line shapes, using modulus weighting, to the complex data of each individual scan of Figure C-4.
-3
-2
-1
0
1
2
3
0.001 0.01 0.1 1 10 100
Frequency, Hz
Z r,
data
- Z
r,mod
el
scan 1 scan 2 scan 3
-3
-2
-1
0
1
2
3
0.001 0.01 0.1 1 10 100Frequency, Hz
Z j,d
ata
- Z
j,mod
el
scan 1 scan 2 scan 3
158
Figure C-7. The standard deviation, as a function of frequency, of the real and imaginary stochastic errors calculated from the real and imaginary residual errors of Figure C-5 and Figure C-6, respectively. The model for the standard deviation includes the parameters, with values, β = 0.0017792 and δ = 0.021916.
0.01
0.1
1
10
0.0001 0.001 0.01 0.1 1 10 100Frequency, Hz
σr, σ
jReal Imaginary Model
159
Figure C-8. The normalized residual errors, as a function of frequency, between the imaginary data of scan 1, shown in Figure C-4, and the predicted values resulting from measurement model regression of 3 line shapes, using error structure weighting, to the real component of the data. Error structure weighting was used. The plot includes the confidence intervals and the limits of the stochastic error structure model.
-0.5
0.0
0.5
1.0
1.5
2.0
0.001 0.01 0.1 1 10 100
Frequency, Hz
(Z j,d
ata
- Z
j,mod
el )/Z
j,mod
el
Imaginary Errors Error Structure
Confidence Interval
160
Figure C-9. The normalized residual errors, as a function of frequency, between the real data of scan 1, shown in Figure C-4, and the predicted values resulting from measurement model regression of 2 line shapes, using error structure weighting, to the imaginary component of the data. The plot includes the confidence intervals and the limits of the stochastic error structure model.
-0.2
-0.1
0
0.1
0.2
0.001 0.01 0.1 1 10Frequency, Hz
(Z r,
data
- Z
r,mod
el)/
Z r,m
odel
Real Errors Error StructureConfidence Interval
161
Figure C-10. The normalized residual errors, as a function of frequency, for the imaginary component of scan 1, shown in Figure C-4, resulting from measurement model regression of 2 line shapes, using error structure weighting, to the imaginary component of the data. The plot includes confidence intervals and the limits of the stochastic error structure model.
-1
-0.5
0
0.5
1
1.5
0.001 0.01 0.1 1 10Frequency, Hz
(Z j,d
ata
- Z
j,mod
el )/Z
j,mod
el
Imaginary Errors Error Structure
Confidence Interval
162
Figure C-11. The normalized real component residual errors, as a function of frequency, resulting from a complex fit of 5 line shapes, using error structure weighting, after rejecting inconsistent high and low frequency points from the data shown in Figure C-1. The confidence intervals and stochastic error structure limits are included.
-0.10
-0.05
0.00
0.05
0.10
0.001 0.01 0.1 1 10Frequency, Hz
(Z r,
data
- Z
r,mod
el)/
Z r,m
odel
Real Errors Error Structure
Confidence Interval
163
Figure C-12. The normalized imaginary component residual errors, as a function of frequency, resulting from a complex fit of 5 line shapes, using error structure weighting, after rejecting inconsistent high and low frequency points from the data presented in Figure C-1. The confidence intervals and stochastic error structure limits are included.
-0.2
-0.1
0
0.1
0.2
0.001 0.01 0.1 1 10Frequency, Hz
(Z j,d
ata
- Z
j,mod
el )/Z
j,mod
el
Imaginary Errors Error Structure
Confidence Interval
APPENDIX DREGRESSION PARAMETER RESULTS
Complete regression parameter results are presented in this appendix for the
experiments listed in Table 3-5 using the cylinder electrode. Values for the applied current
densities correspond to the experimental points included in Figure 3-8 . Parameter values
are plotted as functions of exposure time and cell potential measured at the beginning of
the corresponding impedance scan. The error bars represent the 2σ interval for the
parameter estimation.
164
165
Figure D-1. The electrolyte resistance as a function of time for the cylinder electrode with the applied current equal to zero.
0
20
40
60
80
100
1 10 100t, hr
Re,
Ω
166
Figure D-2. The diffusion time constant for the film and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-3. The diffusion time constant for the film as a function of potential for the cylinder electrode with the applied current equal to zero.
0
5
10
15
20
1 10 100t, hr
τf , s
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Time Constant Potential
0
5
10
15
20
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
τf , s
167
Figure D-4. The bulk layer diffusion time constant and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-5. The bulk layer diffusion time constant as a function of potential for the cylinder electrode with the applied current equal to zero.
-500
0
500
1000
1500
2000
2500
3000
3500
1 10 100t, hr
τb, s
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Time Constant Potential
-500
0
500
1000
1500
2000
2500
3000
3500
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
τb, s
168
Figure D-6. The ratio of the diffusivities of oxygen in the bulk to the film and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-7. The ratio of the diffusivities of oxygen in the bulk to the film as a function of potential for the cylinder electrode with the applied current equal to zero.
0
10
20
30
40
50
60
1 10 100t, hr
Db /
Df
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Diffusivity Ratio Potential
0
10
20
30
40
50
60
-820 -810 -800 -790 -780 -770 -760 -750Potential, mV (SCE)
Db /
Df
169
Figure D-8. The calculated film thickness in microns and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-9. The calculated film thickness in microns as a function of potential for the cylinder electrode with the applied current equal to zero.
0
5
10
15
20
25
30
1 10 100t, hr
δf, µ
m
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Film Thickness Potential
0
5
10
15
20
25
30
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
δf, µ
m
170
Figure D-10. The calculated bulk diffusion layer thickness in microns and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-11. The calculated bulk diffusion layer thickness in microns, as a function of potential for the cylinder electrode with the applied current equal to zero.
-500
0
500
1000
1500
2000
1 10 100t, hr
δb, µ
m
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Diffusion Layer Potential
-500
0
500
1000
1500
2000
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
δb, µ
m
171
Figure D-12. The effective charge transfer resistance and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-13. The effective charge transfer resistance as a function of potential for the cylinder electrode with the applied current equal to zero.
1000
10000
100000
1 10 100t, hr
Ref
f, Ω
cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Effective Resistance Potential
1000
10000
100000
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
Ref
f, Ω
cm
2
172
Figure D-14. The charge transfer resistance for oxygen reduction and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-15. The charge transfer resistance for oxygen reduction as a function of potential for the cylinder electrode with the applied current equal to zero.
100
1000
10000
1 10 100t, hr
Rt,O
2, Ω
cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Oxygen Reduction Resistance Potential
100
1000
10000
-820 -810 -800 -790 -780
Potential, mV (SCE)
Rt,O
2, Ω
cm
2
173
Figure D-16. The diffusion impedance coefficient and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-17. The diffusion impedance coefficient as a function of potential for the cylinder electrode with the applied current equal to zero.
0
5000
10000
15000
20000
1 10 100t, hr
Zd,
0, Ω
cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)
Impedance Coefficient Potential
0
5000
10000
15000
20000
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
Zd,
0, Ω
cm
2
174
Figure D-18. The cell capacitance and cell potential as functions of time for the cylinder electrode with the applied current equal to zero.
Figure D-19. The cell capacitance as a function of potential for the cylinder electrode with the applied current equal to zero.
0
100
200
300
400
1 10 100t, hr
Cd,
µF
/cm
2
-820
-810
-800
-790
-780
-770
-760
-750
Pot
entia
l, m
V (
SC
E)Capacitance Potential
0
100
200
300
400
-820 -810 -800 -790 -780 -770 -760 -750
Potential, mV (SCE)
Cd,
µF
/cm
2
175
Figure D-20. The electrolyte resistance as a function of time for the cylinder electrode
with an applied DC current density bias of 1.6 µA/cm2.
0
20
40
60
80
100
1 10 100 1000t, hr
Re,
Ω
176
Figure D-21. The diffusion time constant for the film and cell potential as functions of
time for the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure D-22. The diffusion time constant for the film as a function of potential for the
cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
0
20
40
60
80
1 10 100 1000t, hr
τf , s
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)
Time Constant Potential
0
20
40
60
80
-765 -760 -755 -750 -745 -740Potential, mV (SCE)
τfs
177
Figure D-23. The bulk layer diffusion time constant and cell potential as functions of time
for the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure D-24. The bulk layer diffusion time constant as a function of potential for the
cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
0
200
400
600
800
1000
1 10 100 1000t, hr
τb, s
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)Time Constant Potential
0
200
400
600
800
1000
-765 -760 -755 -750 -745 -740
Potential, mV (SCE)
τb, s
178
Figure D-25. The ratio of the diffusivities of oxygen in the bulk to the film and the cell potential as functions of time for the cylinder electrode with an applied DC current density
bias of 1.6 µA/cm2.
Figure D-26. The ratio of the diffusivities of oxygen in the bulk to the film as a function of potential for the cylinder electrode with an applied DC current density bias of 1.6 µA/
cm2.
0
10
20
30
40
50
1 10 100 1000t, hr
Db /
Df
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)
Diffusivity Ratio Potential
0
10
20
30
40
50
-765 -760 -755 -750 -745 -740Potential, mV (SCE)
Db /
Df
179
Figure D-27. The calculated film thickness and cell potential as functions of time for the
cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure D-28. The calculated film thickness as a function of potential for the cylinder
electrode with an applied DC current density bias of 1.6 µA/cm2.
0
10
20
30
40
50
60
1 10 100 1000t, hr
δf, µ
m
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)
Film Thickness Potential
‘
0
10
20
30
40
50
60
-765 -760 -755 -750 -745 -740Potential, mV (SCE)
δf, µ
m
180
Figure D-29. The calculated bulk diffusion layer thickness and cell potential as functions
of time for the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure D-30. The calculated bulk diffusion layer thickness as a function of potential for
the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
0
200
400
600
800
1000
1 10 100 1000t, hr
δb, µ
m
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)
Diffusion Layer Potential
0
200
400
600
800
1000
-765 -760 -755 -750 -745 -740Potential, mV (SCE)
δb, µ
m
181
Figure D-31. The effective charge transfer resistance and cell potential as functions of
time for the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure D-32. The effective charge transfer resistance as a function of potential for the
cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
1.0E+04
1.0E+05
1.0E+06
1 10 100 1000t, hr
Ref
f, Ω
cm
2
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)Effective Resistance Potential
1.0E+04
1.0E+05
1.0E+06
-765 -760 -755 -750 -745 -740
Potential, mV (SCE)
Ref
f, Ω
cm
2
182
Figure D-33. The diffusion impedance coefficient and cell potential as functions of time
for the cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure D-34. The diffusion impedance coefficient as a function of potential for the
cylinder electrode with an applied DC current density bias of 1.6 µA/cm2.
0
5000
10000
15000
20000
25000
1 10 100 1000t, hr
Zd,
0, Ω
cm
2
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)
Impedance Coefficient Potential
0
5000
10000
15000
20000
25000
-765 -760 -755 -750 -745 -740
Potential, mV (SCE)
Zd,
0, Ω
cm
2
183
Figure D-35. The cell capacitance and cell potential as functions of time for the cylinder
electrode with an applied DC current density bias of 1.6 µA/cm2.
Figure D-36. The cell capacitance as a function of potential for the cylinder electrode with
an applied DC current density bias of 1.6 µA/cm2.
0
100
200
300
400
1 10 100 1000t, hr
Cd,
µF
/cm
2
-780
-770
-760
-750
-740
-730
-720
Pot
entia
l, m
V (
SC
E)
Capacitance Potential
0
100
200
300
400
-765 -760 -755 -750 -745 -740Potential, mV (SCE)
Cd,
µF
/cm
2
184
Figure D-37. The electrolyte resistance as a function of time for the cylinder electrode
with an applied DC current density bias of 2.5 µA/cm2.
0
20
40
60
80
100
1 10 100 1000t, hr
Re,
Ω
185
Figure D-38. The diffusion time constant for the film and cell potential as functions of
time for the cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
Figure D-39. The diffusion time constant for the film as a function of potential for the
cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
0
2
4
6
8
10
12
1 10 100 1000t, hr
τf , s
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Time Constant Potential
0
2
4
6
8
10
12
-950 -900 -850 -800 -750 -700Potential, mV (SCE)
τf , s
186
Figure D-40. The bulk layer diffusion time constant and cell potential as functions of time
for the cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
Figure D-41. The bulk layer diffusion time constant as a function of potential for the
cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
0
200
400
600
800
1000
1200
1 10 100 1000t, hr
τb, s
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Time Constant Potential
0
200
400
600
800
1000
1200
-950 -900 -850 -800 -750 -700Potential, mV (SCE)
τb, s
187
Figure D-42. The ratio of the diffusivities of oxygen in the bulk to the film and the cell potential as functions of time for the cylinder electrode with an applied DC current density
bias of 2.5 µA/cm2.
Figure D-43. The ratio of the diffusivities of oxygen in the bulk to the film as a function of potential for the cylinder electrode with an applied DC current density bias of 2.5 µA/
cm2.
0
0.2
0.4
0.6
0.8
1
1 10 100 1000t, hr
Db /
Df
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Diffusivity Ratio Potential
0
0.2
0.4
0.6
0.8
1
-950 -900 -850 -800 -750 -700Potential, mV (SCE)
Db /
Df
188
Figure D-44. The calculated film thickness and cell potential as functions of time for the
cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
Figure D-45. The calculated film thickness as a function of potential for the cylinder
electrode with an applied DC current density bias of 2.5 µA/cm2.
0
50
100
150
200
1 10 100 1000t, hr
δf, µ
m
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Film Thickness Potential
0
50
100
150
200
-950 -900 -850 -800 -750 -700
Potential, mV (SCE)
δf, µ
m
189
Figure D-46. The calculated bulk diffusion layer thickness and cell potential as functions
of time for the cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
Figure D-47. The calculated bulk diffusion layer thickness as a function of potential for
the cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
-200
0
200
400
600
800
1000
1 10 100 1000t, hr
δb, µ
m
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Diffusion Layer Potential
-200
0
200
400
600
800
1000
-950 -900 -850 -800 -750 -700
Potential, mV (SCE)
δb, µ
m
190
Figure D-48. The effective charge transfer resistance and cell potential as functions of
time for the cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
Figure D-49. The effective charge transfer resistance as a function of potential for the
cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1 10 100 1000t, hr
Ref
f, Ω
cm
2
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Effective Resistance Potential
1.0E+04
1.0E+05
1.0E+06
1.0E+07
-950 -900 -850 -800 -750 -700Potential, mV (SCE)
Ref
f, Ω
cm
2
191
Figure D-50. The diffusion impedance coefficient and cell potential as functions of time
for the cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
Figure D-51. The diffusion impedance coefficient as a function of potential for the
cylinder electrode with an applied DC current density bias of 2.5 µA/cm2.
1.0E+03
1.0E+04
1.0E+05
1 10 100 1000t, hr
Zd,
0, Ω
cm
2
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Impedance Coefficient Potential
1.0E+03
1.0E+04
1.0E+05
-950 -900 -850 -800 -750 -700
Potential, mV (SCE)
Zd,
0, Ω
cm
2
192
Figure D-52. The cell capacitance and cell potential as functions of time for the cylinder
electrode with an applied DC current density bias of 2.5 µA/cm2.
Figure D-53. The cell capacitance as a function of potential for the cylinder electrode with
an applied DC current density bias of 2.5 µA/cm2.
0
200
400
600
800
1000
1 10 100 1000t, hr
Cd,
µF
/cm
2
-950
-900
-850
-800
-750
-700
Pot
entia
l, m
V (
SC
E)
Capacitance Potential
0
200
400
600
800
1000
-950 -900 -850 -800 -750 -700
Potential, mV (SCE)
Cd,
µF
/cm
2
193
Figure D-54. The electrolyte resistance as a function of time for the cylinder electrode
with an applied DC current density bias of 4.0 µA/cm2.
0
20
40
60
80
100
0.1 1 10 100t, hr
Re,
Ω
194
Figure D-55. The diffusion time constant for the film and cell potential as functions of
time for the cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure D-56. The diffusion time constant for the film as a function of potential for the
cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
0
0.5
1
1.5
2
0.1 1 10 100t, hr
τf , s
-1100
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)Time Constant Potential
0
0.5
1
1.5
2
-1000 -950 -900 -850 -800 -750
Potential, mV (SCE)
τf , s
195
Figure D-57. The bulk layer diffusion time constant and cell potential as functions of time
for the cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure D-58. The bulk layer diffusion time constant as a function of potential for the
cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
0
500
1000
1500
2000
0.1 1 10 100t, hr
τb, s
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)
Time Constant Potential
0
500
1000
1500
2000
-1000 -950 -900 -850 -800 -750Potential, mV (SCE)
τb, s
196
Figure D-59. The ratio of the diffusivities of oxygen in the bulk to the film and the cell potential as functions of time for the cylinder electrode with an applied DC current density
bias of 4.0 µA/cm2.
Figure D-60. The ratio of the diffusivities of oxygen in the bulk to the film as a function of potential for the cylinder electrode with an applied DC current density bias of 4.0 µA/
cm2.
0
0.5
1
0.1 1 10 100t, hr
Db /
Df
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)Diffusivity Ratio Potential
0
0.5
1
-1000 -950 -900 -850 -800 -750Potential, mV (SCE)
Db /
Df
197
Figure D-61. The calculated film thickness and cell potential as functions of time for the
cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure D-62. The calculated film thickness as a function of potential for the cylinder
electrode with an applied DC current density bias of 4.0 µA/cm2.
0
20
40
60
80
100
0.1 1 10 100t, hr
δf, µ
m
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)
Film Thickness Potential
0
20
40
60
80
100
-1000 -950 -900 -850 -800 -750
Potential, mV (SCE)
δf, µ
m
198
Figure D-63. The calculated bulk diffusion layer thickness and cell potential as functions
of time for the cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure D-64. The calculated bulk diffusion layer thickness as a function of potential for
the cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
0
500
1000
1500
2000
2500
0.1 1 10 100t, hr
δb, µ
m
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)
Diffusion Layer Potential
0
500
1000
1500
2000
2500
-1000 -950 -900 -850 -800 -750
Potential, mV (SCE)
δb, µ
m
199
Figure D-65. The effective charge transfer resistance and cell potential as functions of
time for the cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure D-66. The effective charge transfer resistance as a function of potential for the
cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
0
20000
40000
60000
80000
100000
0.1 1 10 100t, hr
Ref
f, Ω
cm
2
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)Effective Resistance Potential
0
20000
40000
60000
80000
100000
-1000 -950 -900 -850 -800 -750Potential, mV (SCE)
Ref
f, Ω
cm
2
200
Figure D-67. The diffusion impedance coefficient and cell potential as functions of time
for the cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure D-68. The diffusion impedance coefficient as a function of potential for the
cylinder electrode with an applied DC current density bias of 4.0 µA/cm2.
0
1000
2000
3000
4000
0.1 1 10 100t, hr
Zd,
0, Ω
cm
2
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)
Impedance Coefficient Potential
0
1000
2000
3000
4000
-1000 -950 -900 -850 -800 -750Potential, mV (SCE)
Zd,
0, Ω
cm
2
201
Figure D-69. The cell capacitance and cell potential as functions of time for the cylinder
electrode with an applied DC current density bias of 4.0 µA/cm2.
Figure D-70. The cell capacitance as a function of potential for the cylinder electrode with
an applied DC current density bias of 4.0 µA/cm2.
0
100
200
300
400
0.1 1 10 100t, hr
Cd,
µF
/cm
2
-1000
-900
-800
-700
Pot
entia
l, m
V (
SC
E)Capacitance Potential
0
100
200
300
400
-1000 -950 -900 -850 -800 -750Potential, mV (SCE)
Cd,
µF
/cm
2
e
s
lly
ve
se
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in
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BIOGRAPHICAL SKETCH
Kenneth E. Jeffers received a bachelor of science degree in chemical engineering
from the University of Tennessee at Chattanooga in May 1996. He then accepted
employment as a process engineer at Woodbridge Foam Fabricating, Inc., in Chattanooga.
Desiring to pursue an advanced degree and feeling like a displaced Florida native after
residing in Tennessee for six years, he began graduate studies in chemical engineering at
the University of Florida in August 1997. He joined Professor Mark Orazem’s
electrochemical engineering research group to complete a master of science degree. Upon
completion of degree requirements in August 1999, Ken accepted employment with
Motorola in Plantation, Florida.