Electrochemical Impedance Spectroscopy

1 - Electrochemical Impedance Spectroscopy [Seite 3]
2 - Contents [Seite 9]
3 - Preface to the Second Edition [Seite 23]
4 - Preface to the First Edition [Seite 25]
5 - Acknowledgments [Seite 29]
6 - The Blind Men and the Elephant [Seite 31]
7 - A Brief Introduction to Impedance Spectroscopy [Seite 35]
8 - History of Impedance Spectroscopy [Seite 43]
9 - I Background [Seite 51]
9.1 - 1 Complex Variables [Seite 53]
9.1.1 - 1.1 Why Imaginary Numbers? [Seite 53]
9.1.2 - 1.2 Terminology [Seite 54]
9.1.2.1 - 1.2.1 The Imaginary Number [Seite 54]
9.1.2.2 - 1.2.2 Complex Variables [Seite 54]
9.1.2.3 - 1.2.3 Conventions for Notation in Impedance Spectroscopy [Seite 55]
9.1.3 - 1.3 Operations Involving Complex Variables [Seite 55]
9.1.3.1 - 1.3.1 Multiplication and Division of Complex Numbers [Seite 56]
9.1.3.2 - 1.3.2 Complex Variables in Polar Coordinates [Seite 59]
9.1.3.3 - 1.3.3 Properties of Complex Variables [Seite 65]
9.1.4 - 1.4 Elementary Functions of Complex Variables [Seite 65]
9.1.4.1 - 1.4.1 Exponential [Seite 65]
9.1.4.2 - 1.4.2 Logarithmic [Seite 67]
9.1.4.3 - 1.4.3 Polynomial [Seite 71]
9.1.5 - Problems [Seite 72]
9.2 - 2 Differential Equations [Seite 75]
9.2.1 - 2.1 Linear First-Order Differential Equations [Seite 75]
9.2.2 - 2.2 Homogeneous Linear Second-Order Differential Equations [Seite 79]
9.2.3 - 2.3 Nonhomogeneous Linear Second-Order Differential Equations [Seite 82]
9.2.4 - 2.4 Chain Rule for Coordinate Transformations [Seite 86]
9.2.5 - 2.5 Partial Differential Equations by Similarity Transformations [Seite 88]
9.2.6 - 2.6 Differential Equations with Complex Variables [Seite 92]
9.2.7 - Problems [Seite 93]
9.3 - 3 Statistics [Seite 95]
9.3.1 - 3.1 Definitions [Seite 95]
9.3.1.1 - 3.1.1 Expectation and Mean [Seite 95]
9.3.1.2 - 3.1.2 Variance, Standard Deviation, and Covariance [Seite 95]
9.3.1.3 - 3.1.3 Normal Distribution [Seite 99]
9.3.1.4 - 3.1.4 Probability [Seite 100]
9.3.1.5 - 3.1.5 Central Limit Theorem [Seite 102]
9.3.2 - 3.2 Error Propagation [Seite 105]
9.3.2.1 - 3.2.1 Linear Systems [Seite 105]
9.3.2.2 - 3.2.2 Nonlinear Systems [Seite 106]
9.3.3 - 3.3 Hypothesis Tests [Seite 109]
9.3.3.1 - 3.3.1 Terminology [Seite 110]
9.3.3.2 - 3.3.2 Student's t-Test for Equality of Mean [Seite 111]
9.3.3.3 - 3.3.3 F-Test for Equality of Variance [Seite 114]
9.3.3.4 - 3.3.4 Chi-Squared Test for Goodness of Fit [Seite 120]
9.3.4 - Problems [Seite 121]
9.4 - 4 Electrical Circuits [Seite 125]
9.4.1 - 4.1 Passive Electrical Circuits [Seite 125]
9.4.1.1 - 4.1.1 Circuit Elements [Seite 125]
9.4.1.1.1 - Response to a Sinusoidal Signal [Seite 127]
9.4.1.1.2 - Impedance Response of Passive Circuit Elements [Seite 129]
9.4.1.2 - 4.1.2 Parallel and Series Combinations [Seite 129]
9.4.2 - 4.2 Fundamental Relationships [Seite 131]
9.4.3 - 4.3 Nested Circuits [Seite 133]
9.4.4 - 4.4 Mathematical Equivalence of Circuits [Seite 134]
9.4.5 - 4.5 Graphical Representation of Circuit Response [Seite 135]
9.4.6 - Problems [Seite 137]
9.5 - 5 Electrochemistry [Seite 139]
9.5.1 - 5.1 Resistors and Electrochemical Cells [Seite 139]
9.5.2 - 5.2 Polarization Behavior for Electrochemical Systems [Seite 142]
9.5.2.1 - 5.2.1 Zero Current [Seite 143]
9.5.2.1.1 - Equilibrium [Seite 144]
9.5.2.1.2 - Nonequilibrium [Seite 144]
9.5.2.2 - 5.2.2 Kinetic Control [Seite 145]
9.5.2.3 - 5.2.3 Mixed-Potential Theory [Seite 147]
9.5.2.4 - 5.2.4 Mass-Transfer Control [Seite 154]
9.5.3 - 5.3 Definitions of Potential [Seite 157]
9.5.4 - 5.4 Rate Expressions [Seite 159]
9.5.5 - 5.5 Transport Processes [Seite 163]
9.5.5.1 - 5.5.1 Primary Current and Potential Distributions [Seite 164]
9.5.5.2 - 5.5.2 Secondary Current and Potential Distributions [Seite 166]
9.5.5.3 - 5.5.3 Tertiary Current and Potential Distributions [Seite 166]
9.5.5.4 - 5.5.4 Mass-Transfer-Controlled Current Distributions [Seite 167]
9.5.6 - 5.6 Potential Contributions [Seite 169]
9.5.6.1 - 5.6.1 Ohmic Potential Drop [Seite 169]
9.5.6.2 - 5.6.2 Surface Overpotential [Seite 169]
9.5.6.3 - 5.6.3 Concentration Overpotential [Seite 170]
9.5.7 - 5.7 Capacitance Contributions [Seite 172]
9.5.7.1 - 5.7.1 Double-Layer Capacitance [Seite 172]
9.5.7.2 - 5.7.2 Dielectric Capacitance [Seite 176]
9.5.8 - 5.8 Further Reading [Seite 176]
9.5.9 - Problems [Seite 177]
9.6 - 6 Electrochemical Instrumentation [Seite 179]
9.6.1 - 6.1 The Ideal Operational Amplifier [Seite 179]
9.6.2 - 6.2 Elements of Electrochemical Instrumentation [Seite 181]
9.6.3 - 6.3 Electrochemical Interface [Seite 183]
9.6.3.1 - 6.3.1 Potentiostat [Seite 184]
9.6.3.2 - 6.3.2 Galvanostat [Seite 185]
9.6.3.3 - 6.3.3 Potentiostat for EIS Measurement [Seite 185]
9.6.4 - Problems [Seite 187]
10 - II Experimental Considerations [Seite 189]
10.1 - 7 Experimental Methods [Seite 191]
10.1.1 - 7.1 Steady-State Polarization Curves [Seite 191]
10.1.2 - 7.2 Transient Response to a Potential Step [Seite 192]
10.1.3 - 7.3 Analysis in Frequency Domain [Seite 193]
10.1.3.1 - 7.3.1 Lissajous Analysis [Seite 194]
10.1.3.2 - 7.3.2 Phase-Sensitive Detection (Lock-in Amplifier) [Seite 201]
10.1.3.3 - 7.3.3 Single-Frequency Fourier Analysis [Seite 202]
10.1.3.4 - 7.3.4 Multiple-Frequency Fourier Analysis [Seite 205]
10.1.4 - 7.4 Comparison of Measurement Techniques [Seite 206]
10.1.4.1 - 7.4.1 Lissajous Analysis [Seite 206]
10.1.4.2 - 7.4.2 Phase-Sensitive Detection (Lock-in Amplifier) [Seite 206]
10.1.4.3 - 7.4.3 Single-Frequency Fourier Analysis [Seite 206]
10.1.4.4 - 7.4.4 Multiple-Frequency Fourier Analysis [Seite 206]
10.1.5 - 7.5 Specialized Techniques [Seite 207]
10.1.5.1 - 7.5.1 Transfer-Function Analysis [Seite 207]
10.1.5.2 - 7.5.2 Local Electrochemical Impedance Spectroscopy [Seite 208]
10.1.5.2.1 - Global Impedance [Seite 210]
10.1.5.2.2 - Local Impedance [Seite 210]
10.1.5.2.3 - Local Interfacial Impedance [Seite 210]
10.1.5.2.4 - Local Ohmic Impedance [Seite 211]
10.1.5.2.5 - Global Interfacial Impedance [Seite 211]
10.1.5.2.6 - Global Ohmic Impedance [Seite 211]
10.1.6 - Problems [Seite 212]
10.2 - 8 Experimental Design [Seite 215]
10.2.1 - 8.1 Cell Design [Seite 215]
10.2.1.1 - 8.1.1 Reference Electrodes [Seite 215]
10.2.1.2 - 8.1.2 Flow Configurations [Seite 217]
10.2.1.2.1 - Rotating Disk [Seite 217]
10.2.1.2.2 - Disk under Submerged Impinging Jet [Seite 218]
10.2.1.2.3 - Rotating Cylinders [Seite 218]
10.2.1.2.4 - Rotating Hemispherical Electrode [Seite 218]
10.2.1.3 - 8.1.3 Current Distribution [Seite 219]
10.2.2 - 8.2 Experimental Considerations [Seite 220]
10.2.2.1 - 8.2.1 Frequency Range [Seite 220]
10.2.2.2 - 8.2.2 Linearity [Seite 220]
10.2.2.3 - 8.2.3 Modulation Technique [Seite 231]
10.2.2.4 - 8.2.4 Oscilloscope [Seite 234]
10.2.3 - 8.3 Instrumentation Parameters [Seite 234]
10.2.3.1 - 8.3.1 Improve Signal-to-Noise Ratio [Seite 234]
10.2.3.2 - 8.3.2 Reduce Bias Errors [Seite 236]
10.2.3.2.1 - Nonstationary Effects [Seite 236]
10.2.3.2.2 - Instrument Bias [Seite 236]
10.2.3.3 - 8.3.3 Improve Information Content [Seite 237]
10.2.4 - Problems [Seite 238]
11 - III Process Models [Seite 241]
11.1 - 9 Equivalent Circuit Analogs [Seite 243]
11.1.1 - 9.1 General Approach [Seite 243]
11.1.2 - 9.2 Current Addition [Seite 245]
11.1.2.1 - 9.2.1 Impedance at the Corrosion Potential [Seite 245]
11.1.2.2 - 9.2.2 Partially Blocked Electrode [Seite 246]
11.1.3 - 9.3 Potential Addition [Seite 251]
11.1.3.1 - 9.3.1 Electrode Coated with an Inert Porous Layer [Seite 251]
11.1.3.2 - 9.3.2 Electrode Coated with Two Inert Porous Layers [Seite 252]
11.1.4 - Problems [Seite 255]
11.2 - 10 Kinetic Models [Seite 257]
11.2.1 - 10.1 General Mathematical Framework [Seite 257]
11.2.2 - 10.2 Electrochemical Reactions [Seite 259]
11.2.2.1 - 10.2.1 Potential Dependent [Seite 259]
11.2.2.2 - 10.2.2 Potential and Concentration Dependent [Seite 263]
11.2.2.2.1 - Charge-Transfer Resistance [Seite 266]
11.2.2.2.2 - Diffusion Impedance [Seite 267]
11.2.2.2.3 - Cell Impedance [Seite 270]
11.2.3 - 10.3 Multiple Independent Electrochemical Reactions [Seite 272]
11.2.4 - 10.4 Coupled Electrochemical Reactions [Seite 276]
11.2.4.1 - 10.4.1 Potential and Surface Coverage Dependent [Seite 276]
11.2.4.2 - 10.4.2 Potential, Surface Coverage, and Concentration Dependent [Seite 281]
11.2.5 - 10.5 Electrochemical and Heterogeneous Chemical Reactions [Seite 284]
11.2.6 - Problems [Seite 290]
11.3 - 11 Diffusion Impedance [Seite 293]
11.3.1 - 11.1 Uniformly Accessible Electrode [Seite 294]
11.3.2 - 11.2 Porous Film [Seite 295]
11.3.2.1 - 11.2.1 Diffusion with Exchange of Electroactive Species [Seite 295]
11.3.2.2 - 11.2.2 Diffusion without Exchange of Electroactive Species [Seite 301]
11.3.3 - 11.3 Rotating Disk [Seite 306]
11.3.3.1 - 11.3.1 Fluid Flow [Seite 306]
11.3.3.2 - 11.3.2 Steady-State Mass Transfer [Seite 309]
11.3.3.3 - 11.3.3 Convective Diffusion Impedance [Seite 311]
11.3.3.4 - 11.3.4 Analytic and Numerical Solutions [Seite 312]
11.3.3.4.1 - Nernst Hypothesis [Seite 312]
11.3.3.4.2 - Assumption of an Infinite Schmidt Number [Seite 313]
11.3.3.4.3 - Treatment of a Finite Schmidt Number [Seite 314]
11.3.4 - 11.4 Submerged Impinging Jet [Seite 316]
11.3.4.1 - 11.4.1 Fluid Flow [Seite 316]
11.3.4.2 - 11.4.2 Steady-State Mass Transfer [Seite 318]
11.3.4.3 - 11.4.3 Convective Diffusion Impedance [Seite 318]
11.3.5 - 11.5 Rotating Cylinders [Seite 319]
11.3.6 - 11.6 Electrode Coated by a Porous Film [Seite 321]
11.3.6.1 - 11.6.1 Steady-State Solutions [Seite 322]
11.3.6.2 - 11.6.2 Coupled Diffusion Impedance [Seite 327]
11.3.7 - 11.7 Impedance with Homogeneous Chemical Reactions [Seite 328]
11.3.8 - 11.8 Dynamic Surface Films [Seite 341]
11.3.8.1 - 11.8.1 Mass Transfer in the Salt Layer [Seite 342]
11.3.8.2 - 11.8.2 Mass Transfer in the Electrolyte [Seite 344]
11.3.8.3 - 11.8.3 Oscillating Film Thickness [Seite 345]
11.3.8.4 - 11.8.4 Faradaic Impedance [Seite 347]
11.3.9 - Problems [Seite 350]
11.4 - 12 Impedance of Materials [Seite 353]
11.4.1 - 12.1 Electrical Properties of Materials [Seite 353]
11.4.2 - 12.2 Dielectric Response in Homogeneous Media [Seite 354]
11.4.3 - 12.3 Cole-Cole Relaxation [Seite 357]
11.4.4 - 12.4 Geometric Capacitance [Seite 357]
11.4.5 - 12.5 Dielectric Response of Insulating Nonhomogeneous Media [Seite 359]
11.4.6 - 12.6 Mott-Schottky Analysis [Seite 361]
11.4.7 - Problems [Seite 367]
11.5 - 13 Time-Constant Dispersion [Seite 369]
11.5.1 - 13.1 Transmission Line Models [Seite 369]
11.5.1.1 - 13.1.1 Telegrapher's Equations [Seite 371]
11.5.1.2 - 13.1.2 Porous Electrodes [Seite 372]
11.5.1.3 - 13.1.3 Pore-in-Pore Model [Seite 380]
11.5.1.4 - 13.1.4 Thin-Layer Cell [Seite 383]
11.5.2 - 13.2 Geometry-Induced Current and Potential Distributions [Seite 387]
11.5.2.1 - 13.2.1 Mathematical Development [Seite 388]
11.5.2.1.1 - Blocking Electrode [Seite 388]
11.5.2.1.2 - Blocking Electrode with CPE Behavior [Seite 389]
11.5.2.1.3 - Electrode with Faradaic Reactions [Seite 389]
11.5.2.1.4 - Electrode with Faradaic Reactions Coupled by Adsorbed Intermediates [Seite 391]
11.5.2.2 - 13.2.2 Numerical Method [Seite 392]
11.5.2.3 - 13.2.3 Complex Ohmic Impedance at High Frequencies [Seite 392]
11.5.2.4 - 13.2.4 Complex Ohmic Impedance at High and Low Frequencies [Seite 392]
11.5.3 - 13.3 Electrode Surface Property Distributions [Seite 399]
11.5.3.1 - 13.3.1 Electrode Roughness [Seite 399]
11.5.3.1.1 - Influence of Roughness on a Disk Electrode [Seite 401]
11.5.3.1.2 - Influence of Surface Roughness on a Recessed Electrode [Seite 401]
11.5.3.2 - 13.3.2 Capacitance [Seite 410]
11.5.3.2.1 - Capacitance Distribution on Recessed Electrodes [Seite 411]
11.5.3.2.2 - Capacitance Distribution on Disk Electrodes [Seite 415]
11.5.3.3 - 13.3.3 Reactivity [Seite 419]
11.5.4 - 13.4 Characteristic Dimension for Frequency Dispersion [Seite 419]
11.5.5 - 13.5 Convective Diffusion Impedance at Small Electrodes [Seite 420]
11.5.5.1 - 13.5.1 Analysis [Seite 421]
11.5.5.2 - 13.5.2 Local Convective Diffusion Impedance [Seite 423]
11.5.5.2.1 - Low-Frequency Solution [Seite 424]
11.5.5.2.2 - High-Frequency Solution [Seite 424]
11.5.5.3 - 13.5.3 Global Convective Diffusion Impedance [Seite 424]
11.5.6 - 13.6 Coupled Charging and Faradaic Currents [Seite 427]
11.5.6.1 - 13.6.1 Theoretical Development [Seite 428]
11.5.6.1.1 - Mass Transport in Dilute Solutions [Seite 428]
11.5.6.1.2 - Coupled Faradaic and Charging Currents [Seite 429]
11.5.6.1.3 - Double-Layer Model [Seite 430]
11.5.6.1.4 - Decoupled Faradaic and Charging Currents [Seite 432]
11.5.6.2 - 13.6.2 Numerical Method [Seite 433]
11.5.6.2.1 - Steady-State Calculations [Seite 433]
11.5.6.2.2 - Double-Layer Properties [Seite 433]
11.5.6.2.3 - Impedance Calculations [Seite 434]
11.5.6.3 - 13.6.3 Consequence of Coupled Charging and Faradaic Currents [Seite 434]
11.5.7 - 13.7 Exponential Resistivity Distributions [Seite 439]
11.5.8 - Problems [Seite 442]
11.6 - 14 Constant-Phase Elements [Seite 445]
11.6.1 - 14.1 Mathematical Formulation for a CPE [Seite 445]
11.6.2 - 14.2 When Is a Time-Constant Distribution a CPE? [Seite 446]
11.6.3 - 14.3 Origin of Distributions Resulting in a CPE [Seite 449]
11.6.4 - 14.4 Approaches for Extracting Physical Properties [Seite 451]
11.6.4.1 - 14.4.1 Simple Substitution [Seite 452]
11.6.4.2 - 14.4.2 Characteristic Frequency: Normal Distribution [Seite 452]
11.6.4.3 - 14.4.3 Characteristic Frequency: Surface Distribution [Seite 454]
11.6.4.4 - 14.4.4 Power-Law Distribution [Seite 456]
11.6.4.4.1 - Bounds for Resistivity [Seite 463]
11.6.4.4.2 - Comparative Analysis [Seite 463]
11.6.5 - 14.5 Limitations to the Use of the CPE [Seite 465]
11.6.6 - Problems [Seite 468]
11.7 - 15 Generalized Transfer Functions [Seite 471]
11.7.1 - 15.1 Multi-input/Multi-output Systems [Seite 471]
11.7.1.1 - 15.1.1 Current or Potential Are the Output Quantity [Seite 475]
11.7.1.2 - 15.1.2 Current or Potential Are the Input Quantity [Seite 477]
11.7.1.3 - 15.1.3 Experimental Quantities [Seite 479]
11.7.2 - 15.2 Transfer Functions Involving Exclusively Electrical Quantities [Seite 479]
11.7.2.1 - 15.2.1 Ring-Disk Impedance Measurements [Seite 479]
11.7.2.2 - 15.2.2 Multifrequency Measurements for Double-Layer Studies [Seite 479]
11.7.3 - 15.3 Transfer Functions Involving Nonelectrical Quantities [Seite 484]
11.7.3.1 - 15.3.1 Thermoelectrochemical (TEC) Transfer Function [Seite 484]
11.7.3.2 - 15.3.2 Photoelectrochemical Impedance Measurements [Seite 488]
11.7.3.3 - 15.3.3 Electrogravimetry Impedance Measurements [Seite 489]
11.7.4 - Problems [Seite 491]
11.8 - 16 Electrohydrodynamic Impedance [Seite 493]
11.8.1 - 16.1 Hydrodynamic Transfer Function [Seite 495]
11.8.2 - 16.2 Mass-Transport Transfer Function [Seite 498]
11.8.2.1 - 16.2.1 Asymptotic Solution for Large Schmidt Numbers [Seite 501]
11.8.2.2 - 16.2.2 Asymptotic Solution for High Frequencies [Seite 501]
11.8.3 - 16.3 Kinetic Transfer Function for Simple Electrochemical Reactions [Seite 503]
11.8.4 - 16.4 Interface with a 2-D or 3-D Insulating Phase [Seite 504]
11.8.4.1 - 16.4.1 Partially Blocked Electrode [Seite 505]
11.8.4.2 - 16.4.2 Rotating Disk Electrode Coated by a Porous Film [Seite 507]
11.8.4.2.1 - Steady-State Solutions [Seite 508]
11.8.4.2.2 - AC and EHD Impedances [Seite 509]
11.8.5 - Problems [Seite 514]
12 - IV Interpretation Strategies [Seite 515]
12.1 - 17 Methods for Representing Impedance [Seite 517]
12.1.1 - 17.1 Impedance Format [Seite 517]
12.1.1.1 - 17.1.1 Complex-Impedance-Plane Representation [Seite 520]
12.1.1.2 - 17.1.2 Bode Representation [Seite 523]
12.1.1.3 - 17.1.3 Ohmic-Resistance-Corrected Bode Representation [Seite 525]
12.1.1.4 - 17.1.4 Impedance Representation [Seite 526]
12.1.2 - 17.2 Admittance Format [Seite 528]
12.1.2.1 - 17.2.1 Admittance-Plane Representation [Seite 530]
12.1.2.2 - 17.2.2 Admittance Representation [Seite 531]
12.1.2.3 - 17.2.3 Ohmic-Resistance-Corrected Representation [Seite 533]
12.1.3 - 17.3 Complex-Capacitance Format [Seite 533]
12.1.4 - 17.4 Effective Capacitance [Seite 538]
12.1.5 - Problems [Seite 541]
12.2 - 18 Graphical Methods [Seite 543]
12.2.1 - 18.1 Based on Nyquist Plots [Seite 544]
12.2.1.1 - 18.1.1 Characteristic Frequency [Seite 544]
12.2.1.2 - 18.1.2 Superposition [Seite 549]
12.2.1.2.1 - Mass Transfer [Seite 549]
12.2.1.2.2 - Evolution of Active Area [Seite 551]
12.2.2 - 18.2 Based on Bode Plots [Seite 551]
12.2.2.1 - 18.2.1 Ohmic-Resistance-Corrected Phase [Seite 554]
12.2.2.2 - 18.2.2 Ohmic-Resistance-Corrected Magnitude [Seite 555]
12.2.3 - 18.3 Based on Imaginary Part of the Impedance [Seite 555]
12.2.3.1 - 18.3.1 Evaluation of Slopes [Seite 555]
12.2.3.2 - 18.3.2 Calculation of Derivatives [Seite 556]
12.2.4 - 18.4 Based on Dimensionless Frequency [Seite 556]
12.2.4.1 - 18.4.1 Mass Transport [Seite 557]
12.2.4.2 - 18.4.2 Geometric Contribution [Seite 557]
12.2.5 - 18.5 System-Specific Applications [Seite 562]
12.2.5.1 - 18.5.1 Effective CPE Coefficient [Seite 562]
12.2.5.2 - 18.5.2 Asymptotic Behavior for Low-Frequency Mass Transport [Seite 562]
12.2.5.3 - 18.5.3 Arrhenius Superposition [Seite 568]
12.2.5.4 - 18.5.4 Mott-Schottky Plots [Seite 571]
12.2.5.5 - 18.5.5 High-Frequency Cole-Cole Plots [Seite 572]
12.2.6 - 18.6 Overview [Seite 573]
12.2.7 - Problems [Seite 575]
12.3 - 19 Complex Nonlinear Regression [Seite 577]
12.3.1 - 19.1 Concept [Seite 577]
12.3.2 - 19.2 Objective Functions [Seite 579]
12.3.3 - 19.3 Formalism of Regression Strategies [Seite 580]
12.3.3.1 - 19.3.1 Linear Regression [Seite 580]
12.3.3.2 - 19.3.2 Nonlinear Regression [Seite 582]
12.3.4 - 19.4 Regression Strategies for Nonlinear Problems [Seite 584]
12.3.4.1 - 19.4.1 Gauss-Newton Method [Seite 584]
12.3.4.2 - 19.4.2 Method of Steepest Descent [Seite 584]
12.3.4.3 - 19.4.3 Levenberg-Marquardt Method [Seite 584]
12.3.4.4 - 19.4.4 Downhill Simplex Strategies [Seite 585]
12.3.5 - 19.5 Influence of Data Quality on Regression [Seite 586]
12.3.5.1 - 19.5.1 Presence of Stochastic Errors in Data [Seite 587]
12.3.5.2 - 19.5.2 Ill-Conditioned Regression Caused by Stochastic Noise [Seite 587]
12.3.5.3 - 19.5.3 Ill-Conditioned Regression Caused by Insufficient Range [Seite 587]
12.3.6 - 19.6 Initial Estimates for Regression [Seite 591]
12.3.7 - 19.7 Regression Statistics [Seite 592]
12.3.7.1 - 19.7.1 Confidence Intervals for Parameter Estimates [Seite 592]
12.3.7.2 - 19.7.2 Statistical Measure of the Regression Quality [Seite 593]
12.3.8 - Problems [Seite 594]
12.4 - 20 Assessing Regression Quality [Seite 595]
12.4.1 - 20.1 Methods to Assess Regression Quality [Seite 595]
12.4.1.1 - 20.1.1 Quantitative Methods [Seite 595]
12.4.1.2 - 20.1.2 Qualitative Methods [Seite 596]
12.4.2 - 20.2 Application of Regression Concepts [Seite 596]
12.4.2.1 - 20.2.1 Finite-Diffusion-Length Model [Seite 598]
12.4.2.1.1 - Quantitative Assessment [Seite 599]
12.4.2.1.2 - Visual Inspection [Seite 599]
12.4.2.2 - 20.2.2 Measurement Model [Seite 602]
12.4.2.2.1 - Quantitative Assessment [Seite 603]
12.4.2.2.2 - Visual Inspection [Seite 603]
12.4.2.3 - 20.2.3 Convective-Diffusion-Length Model [Seite 605]
12.4.2.3.1 - Quantitative Assessment [Seite 607]
12.4.2.3.2 - Visual Inspection [Seite 608]
12.4.3 - Problems [Seite 613]
13 - V Statistical Analysis [Seite 615]
13.1 - 21 Error Structure of Impedance Measurements [Seite 617]
13.1.1 - 21.1 Error Contributions [Seite 617]
13.1.2 - 21.2 Stochastic Errors in Impedance Measurements [Seite 618]
13.1.2.1 - 21.2.1 Stochastic Errors in Time-Domain Signals [Seite 619]
13.1.2.2 - 21.2.2 Transformation from Time Domain to Frequency Domain [Seite 619]
13.1.2.3 - 21.2.3 Stochastic Errors in Frequency Domain [Seite 621]
13.1.3 - 21.3 Bias Errors [Seite 625]
13.1.3.1 - 21.3.1 Instrument Artifacts [Seite 625]
13.1.3.2 - 21.3.2 Ancillary Parts of the System under Study [Seite 626]
13.1.3.3 - 21.3.3 Nonstationary Behavior [Seite 626]
13.1.3.4 - 21.3.4 Time Scales in Impedance Spectroscopy Measurements [Seite 626]
13.1.4 - 21.4 Incorporation of Error Structure [Seite 629]
13.1.5 - 21.5 Measurement Models for Error Identification [Seite 631]
13.1.5.1 - 21.5.1 Stochastic Errors [Seite 633]
13.1.5.2 - 21.5.2 Bias Errors [Seite 636]
13.1.6 - Problems [Seite 643]
13.2 - 22 The Kramers-Kronig Relations [Seite 645]
13.2.1 - 22.1 Methods for Application [Seite 645]
13.2.1.1 - 22.1.1 Direct Integration of the Kramers-Kronig Relations [Seite 647]
13.2.1.2 - 22.1.2 Experimental Assessment of Consistency [Seite 648]
13.2.1.3 - 22.1.3 Regression of Process Models [Seite 648]
13.2.1.4 - 22.1.4 Regression of Measurement Models [Seite 649]
13.2.2 - 22.2 Mathematical Origin [Seite 650]
13.2.2.1 - 22.2.1 Background [Seite 650]
13.2.2.2 - 22.2.2 Application of Cauchy's Theorem [Seite 654]
13.2.2.3 - 22.2.3 Transformation from Real to Imaginary [Seite 655]
13.2.2.4 - 22.2.4 Transformation from Imaginary to Real [Seite 657]
13.2.2.5 - 22.2.5 Application of the Kramers-Kronig Relations [Seite 659]
13.2.3 - 22.3 The Kramers-Kronig Relations in an Expectation Sense [Seite 660]
13.2.3.1 - 22.3.1 Transformation from Real to Imaginary [Seite 661]
13.2.3.2 - 22.3.2 Transformation from Imaginary to Real [Seite 662]
13.2.4 - Problems [Seite 664]
14 - VI Overview [Seite 665]
14.1 - 23 An Integrated Approach to Impedance Spectroscopy [Seite 667]
14.1.1 - 23.1 Flowcharts for Regression Analysis [Seite 667]
14.1.2 - 23.2 Integration of Measurements, Error Analysis, and Model [Seite 668]
14.1.2.1 - 23.2.1 Impedance Measurements Integrated with Error Analysis [Seite 668]
14.1.2.2 - 23.2.2 Process Models Developed Using Other Observations [Seite 670]
14.1.2.3 - 23.2.3 Regression Analysis in Context of Error Structure [Seite 671]
14.1.3 - 23.3 Application [Seite 671]
14.1.4 - Problems [Seite 677]
15 - VII Reference Material [Seite 679]
15.1 - A Complex Integrals [Seite 681]
15.1.1 - A.1 Definition of Terms [Seite 681]
15.1.2 - A.2 Cauchy-Riemann Conditions [Seite 684]
15.1.3 - A.3 Complex Integration [Seite 685]
15.1.3.1 - A.3.1 Cauchy's Theorem [Seite 685]
15.1.3.2 - A.3.2 Improper Integrals of Rational Functions [Seite 689]
15.1.4 - Problems [Seite 691]
15.2 - B Tables of Reference Material [Seite 693]
15.3 - C List of Examples [Seite 695]
16 - List of Symbols [Seite 701]
17 - References [Seite 713]
18 - Author Index [Seite 741]
19 - Subject Index [Seite 747]
20 - EULA [Seite 766]