ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY FOR THE ...ufdcimages.uflib.ufl.edu/UF/00/10/06/97/00001/jeffers.pdf · electrochemical impedance spectroscopy for the characterization of corrosion

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

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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


4-8. 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


4-10. 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



5-40. 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-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-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-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


xvii











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-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-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







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



-1000

-900

-800

-700

-600

-500

0 20 40 60 80 100 120 140

t, hr

Pot

entia

l, m

V (

SC

E)

68



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





-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


-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


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


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


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


Figure 5-26. The effective charge transfer resistance as a function of potential for the


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


2.5 µA/cm2.


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




phi = 1d0do i = 1,n6 write(12,*) nebc,phienddo



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


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


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


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


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


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


-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)


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)


-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)


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)


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


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


Figure D-22. The diffusion time constant for the film as a function of potential for the


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)


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


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


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)


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


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)


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


Figure D-32. The effective charge transfer resistance as a function of potential for the


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


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


Figure D-34. The diffusion impedance coefficient as a function of potential for the


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)


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


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



0

20

40

60

80

100

1 10 100 1000t, hr

Re,

Ω

185





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)


0

2

4

6

8

10

12

-950 -900 -850 -800 -750 -700Potential, mV (SCE)

τf , s

186





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)


0

200

400

600

800

1000

1200

-950 -900 -850 -800 -750 -700Potential, mV (SCE)

τb, s

187




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)


0

0.2

0.4

0.6

0.8

1

-950 -900 -850 -800 -750 -700Potential, mV (SCE)

Db /

Df

188





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)


0

50

100

150

200

-950 -900 -850 -800 -750 -700

Potential, mV (SCE)

δf, µ

m

189





-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)


-200

0

200

400

600

800

1000

-950 -900 -850 -800 -750 -700

Potential, mV (SCE)

δb, µ

m

190





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)


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





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)


1.0E+03

1.0E+04

1.0E+05

-950 -900 -850 -800 -750 -700

Potential, mV (SCE)

Zd,

0, Ω

cm

2

192





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



0

20

40

60

80

100

0.1 1 10 100t, hr

Re,

Ω

194





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


0

0.5

1

1.5

2

-1000 -950 -900 -850 -800 -750

Potential, mV (SCE)

τf , s

195





0

500

1000

1500

2000

0.1 1 10 100t, hr

τb, s

-1000

-900

-800

-700

Pot

entia

l, m

V (

SC

E)


0

500

1000

1500

2000

-1000 -950 -900 -850 -800 -750Potential, mV (SCE)

τb, s

196




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





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)


0

20

40

60

80

100

-1000 -950 -900 -850 -800 -750

Potential, mV (SCE)

δf, µ

m

198





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)


0

500

1000

1500

2000

2500

-1000 -950 -900 -850 -800 -750

Potential, mV (SCE)

δb, µ

m

199





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


0

20000

40000

60000

80000

100000

-1000 -950 -900 -850 -800 -750Potential, mV (SCE)

Ref

f, Ω

cm

2

200





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)


0

1000

2000

3000

4000

-1000 -950 -900 -850 -800 -750Potential, mV (SCE)

Zd,

0, Ω

cm

2

201





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


0

100

200

300

400

-1000 -950 -900 -850 -800 -750Potential, mV (SCE)

Cd,

µF

/cm

2

e

s

lly

ve

se

REFERENCES

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[4] Orazem, M. E., Esteban, J. M., Kennelley, K. J., and Degerstedt, R. M. “Mathematical Models for Cathodic Protection of an Underground Pipeline withCoating Holidays: Part 2 - Case Studies of Parallel Anode Cathodic ProtectionSystems.” Corrosion 53, 6 (1997): 427-436.

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[29] Wojcik, Paul T., Agarwal, Pankaj, and Orazem, Mark E. “A Method for Maintaina Constant Potential Variation During Galvanostatic Regulation of ElectrochemImpedance Measurements.” Electrochimica Acta 41, 7/8 (1996): 977-983.

[30] Wojcik, P. T. and Orazem, M. E. “Variable-Amplitude Galvanostatically ModulatImpedance Spectroscopy as a Tool for Assessing Reactivity at the Corrosion Potential Without Distorting Temporal Evolution of the System.” Corrosion 54, 4 (1998): 289-298.

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[33] Macdonald, J. R. and Garber, J. A. “Analysis of Impedance and Admittance Dfor Solids and Liquids.” Journal of the Electrochemical Society 124, 7 (1977): 1022-1030.

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[35] Agarwal, Pankaj. Application of Measurement Models to Impedance Spectrosc. Ph.D. dissertation, University of Florida, Department of Chemical Engineering,1994.

[36] Reid, Robert C., Prausnitz, John M., and Thomas K. Sherwood. The PropertieGases and Liquids. New York: McGraw-Hill, 1977.

[37] Grimm, R. D., West, A. C., and Landolt, D. “AC Impedance Study of AnodicallyFormed Salt Films on Iron in Chloride Solution.” Journal of the ElectrochemicaSociety 139, 6 (1992): 1622-1629.

[38] Wojcik, Paul T. Electrochemical Behavior of Copper and Copper Nickel Alloys Synthetic Sea Water. Ph.D. dissertation, University of Florida, Department of Chemical Engineering, 1997.

[39] Marcus P. and J. Oudar, eds. Corrosion Mechanisms in Theory and Practice. New York: M. Dekker, 1995.

[40] Simpson, L. J. and Melendres, C. A. “Surface-Enhanced Raman Spectroelectrochemical Studies of Corrosion Films on Iron in Aqueous CarbonSolution.” Journal of the Electrochemical Society 143, 7 (1996): 2146-2152.

[41] deFaria, D. L. A., Silva, S. V. and deOliveira, M. T. “Raman MicrospectroscopySome Iron Oxides and Oxyhydroxides.” Journal of Raman Spectroscopy 28, 11 (1997): 873-878.

206

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.

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