Chapter
The Blind Men and the Elephant
A Brief Introduction to Impedance Spectroscopy
History of Impedance Spectroscopy
1.1 Why Imaginary Numbers?
1.2.1 The Imaginary Number
1.2.3 Conventions for Notation in Impedance Spectroscopy
1.3 Operations Involving Complex Variables
1.3.1 Multiplication and Division of Complex Numbers
1.3.2 Complex Variables in Polar Coordinates
1.3.3 Properties of Complex Variables
1.4 Elementary Functions of Complex Variables
2.1 Linear First-Order Differential Equations
2.2 Homogeneous Linear Second-Order Differential Equations
2.3 Nonhomogeneous Linear Second-Order Differential Equations
2.4 Chain Rule for Coordinate Transformations
2.5 Partial Differential Equations by Similarity Transformations
2.6 Differential Equations with Complex Variables
3.1.1 Expectation and Mean
3.1.2 Variance, Standard Deviation, and Covariance
3.1.3 Normal Distribution
3.1.5 Central Limit Theorem
3.3.2 Student’s t-Test for Equality of Mean
3.3.3 F-Test for Equality of Variance
3.3.4 Chi-Squared Test for Goodness of Fit
4.1 Passive Electrical Circuits
Response to a Sinusoidal Signal
Impedance Response of Passive Circuit Elements
4.1.2 Parallel and Series Combinations
4.2 Fundamental Relationships
4.4 Mathematical Equivalence of Circuits
4.5 Graphical Representation of Circuit Response
5.1 Resistors and Electrochemical Cells
5.2 Polarization Behavior for Electrochemical Systems
5.2.3 Mixed-Potential Theory
5.2.4 Mass-Transfer Control
5.3 Definitions of Potential
5.5.1 Primary Current and Potential Distributions
5.5.2 Secondary Current and Potential Distributions
5.5.3 Tertiary Current and Potential Distributions
5.5.4 Mass-Transfer-Controlled Current Distributions
5.6 Potential Contributions
5.6.1 Ohmic Potential Drop
5.6.2 Surface Overpotential
5.6.3 Concentration Overpotential
5.7 Capacitance Contributions
5.7.1 Double-Layer Capacitance
5.7.2 Dielectric Capacitance
6 Electrochemical Instrumentation
6.1 The Ideal Operational Amplifier
6.2 Elements of Electrochemical Instrumentation
6.3 Electrochemical Interface
6.3.3 Potentiostat for EIS Measurement
II Experimental Considerations
7.1 Steady-State Polarization Curves
7.2 Transient Response to a Potential Step
7.3 Analysis in Frequency Domain
7.3.2 Phase-Sensitive Detection (Lock-in Amplifier)
7.3.3 Single-Frequency Fourier Analysis
7.3.4 Multiple-Frequency Fourier Analysis
7.4 Comparison of Measurement Techniques
7.4.2 Phase-Sensitive Detection (Lock-in Amplifier)
7.4.3 Single-Frequency Fourier Analysis
7.4.4 Multiple-Frequency Fourier Analysis
7.5 Specialized Techniques
7.5.1 Transfer-Function Analysis
7.5.2 Local Electrochemical Impedance Spectroscopy
Local Interfacial Impedance
Global Interfacial Impedance
8.1.1 Reference Electrodes
8.1.2 Flow Configurations
Disk under Submerged Impinging Jet
Rotating Hemispherical Electrode
8.1.3 Current Distribution
8.2 Experimental Considerations
8.2.3 Modulation Technique
8.3 Instrumentation Parameters
8.3.1 Improve Signal-to-Noise Ratio
8.3.3 Improve Information Content
9 Equivalent Circuit Analogs
9.2.1 Impedance at the Corrosion Potential
9.2.2 Partially Blocked Electrode
9.3.1 Electrode Coated with an Inert Porous Layer
9.3.2 Electrode Coated with Two Inert Porous Layers
10.1 General Mathematical Framework
10.2 Electrochemical Reactions
10.2.1 Potential Dependent
10.2.2 Potential and Concentration Dependent
Charge-Transfer Resistance
10.3 Multiple Independent Electrochemical Reactions
10.4 Coupled Electrochemical Reactions
10.4.1 Potential and Surface Coverage Dependent
10.4.2 Potential, Surface Coverage, and Concentration Dependent
10.5 Electrochemical and Heterogeneous Chemical Reactions
11.1 Uniformly Accessible Electrode
11.2.1 Diffusion with Exchange of Electroactive Species
11.2.2 Diffusion without Exchange of Electroactive Species
11.3.2 Steady-State Mass Transfer
11.3.3 Convective Diffusion Impedance
11.3.4 Analytic and Numerical Solutions
Assumption of an Infinite Schmidt Number
Treatment of a Finite Schmidt Number
11.4 Submerged Impinging Jet
11.4.2 Steady-State Mass Transfer
11.4.3 Convective Diffusion Impedance
11.6 Electrode Coated by a Porous Film
11.6.1 Steady-State Solutions
11.6.2 Coupled Diffusion Impedance
11.7 Impedance with Homogeneous Chemical Reactions
11.8 Dynamic Surface Films
11.8.1 Mass Transfer in the Salt Layer
11.8.2 Mass Transfer in the Electrolyte
11.8.3 Oscillating Film Thickness
11.8.4 Faradaic Impedance
12 Impedance of Materials
12.1 Electrical Properties of Materials
12.2 Dielectric Response in Homogeneous Media
12.3 Cole–Cole Relaxation
12.4 Geometric Capacitance
12.5 Dielectric Response of Insulating Nonhomogeneous Media
12.6 Mott–Schottky Analysis
13 Time-Constant Dispersion
13.1 Transmission Line Models
13.1.1 Telegrapher’s Equations
13.1.3 Pore-in-Pore Model
13.2 Geometry-Induced Current and Potential Distributions
13.2.1 Mathematical Development
Blocking Electrode with CPE Behavior
Electrode with Faradaic Reactions
Electrode with Faradaic Reactions Coupled by Adsorbed Intermediates
13.2.3 Complex Ohmic Impedance at High Frequencies
13.2.4 Complex Ohmic Impedance at High and Low Frequencies
13.3 Electrode Surface Property Distributions
13.3.1 Electrode Roughness
Influence of Roughness on a Disk Electrode
Influence of Surface Roughness on a Recessed Electrode
Capacitance Distribution on Recessed Electrodes
Capacitance Distribution on Disk Electrodes
13.4 Characteristic Dimension for Frequency Dispersion
13.5 Convective Diffusion Impedance at Small Electrodes
13.5.2 Local Convective Diffusion Impedance
13.5.3 Global Convective Diffusion Impedance
13.6 Coupled Charging and Faradaic Currents
13.6.1 Theoretical Development
Mass Transport in Dilute Solutions
Coupled Faradaic and Charging Currents
Decoupled Faradaic and Charging Currents
Steady-State Calculations
13.6.3 Consequence of Coupled Charging and Faradaic Currents
13.7 Exponential Resistivity Distributions
14 Constant-Phase Elements
14.1 Mathematical Formulation for a CPE
14.2 When Is a Time-Constant Distribution a CPE?
14.3 Origin of Distributions Resulting in a CPE
14.4 Approaches for Extracting Physical Properties
14.4.1 Simple Substitution
14.4.2 Characteristic Frequency: Normal Distribution
14.4.3 Characteristic Frequency: Surface Distribution
14.4.4 Power-Law Distribution
14.5 Limitations to the Use of the CPE
15 Generalized Transfer Functions
15.1 Multi-input/Multi-output Systems
15.1.1 Current or Potential Are the Output Quantity
15.1.2 Current or Potential Are the Input Quantity
15.1.3 Experimental Quantities
15.2 Transfer Functions Involving Exclusively Electrical Quantities
15.2.1 Ring–Disk Impedance Measurements
15.2.2 Multifrequency Measurements for Double-Layer Studies
15.3 Transfer Functions Involving Nonelectrical Quantities
15.3.1 Thermoelectrochemical (TEC) Transfer Function
15.3.2 Photoelectrochemical Impedance Measurements
15.3.3 Electrogravimetry Impedance Measurements
16 Electrohydrodynamic Impedance
16.1 Hydrodynamic Transfer Function
16.2 Mass-Transport Transfer Function
16.2.1 Asymptotic Solution for Large Schmidt Numbers
16.2.2 Asymptotic Solution for High Frequencies
16.3 Kinetic Transfer Function for Simple Electrochemical Reactions
16.4 Interface with a 2-D or 3-D Insulating Phase
16.4.1 Partially Blocked Electrode
16.4.2 Rotating Disk Electrode Coated by a Porous Film
IV Interpretation Strategies
17 Methods for Representing Impedance
17.1.1 Complex-Impedance-Plane Representation
17.1.2 Bode Representation
17.1.3 Ohmic-Resistance-Corrected Bode Representation
17.1.4 Impedance Representation
17.2.1 Admittance-Plane Representation
17.2.2 Admittance Representation
17.2.3 Ohmic-Resistance-Corrected Representation
17.3 Complex-Capacitance Format
17.4 Effective Capacitance
18.1 Based on Nyquist Plots
18.1.1 Characteristic Frequency
18.2.1 Ohmic-Resistance-Corrected Phase
18.2.2 Ohmic-Resistance-Corrected Magnitude
18.3 Based on Imaginary Part of the Impedance
18.3.1 Evaluation of Slopes
18.3.2 Calculation of Derivatives
18.4 Based on Dimensionless Frequency
18.4.2 Geometric Contribution
18.5 System-Specific Applications
18.5.1 Effective CPE Coefficient
18.5.2 Asymptotic Behavior for Low-Frequency Mass Transport
18.5.3 Arrhenius Superposition
18.5.4 Mott–Schottky Plots
18.5.5 High-Frequency Cole–Cole Plots
19 Complex Nonlinear Regression
19.3 Formalism of Regression Strategies
19.3.2 Nonlinear Regression
19.4 Regression Strategies for Nonlinear Problems
19.4.1 Gauss–Newton Method
19.4.2 Method of Steepest Descent
19.4.3 Levenberg–Marquardt Method
19.4.4 Downhill Simplex Strategies
19.5 Influence of Data Quality on Regression
19.5.1 Presence of Stochastic Errors in Data
19.5.2 Ill-Conditioned Regression Caused by Stochastic Noise
19.5.3 Ill-Conditioned Regression Caused by Insufficient Range
19.6 Initial Estimates for Regression
19.7 Regression Statistics
19.7.1 Confidence Intervals for Parameter Estimates
19.7.2 Statistical Measure of the Regression Quality
20 Assessing Regression Quality
20.1 Methods to Assess Regression Quality
20.1.1 Quantitative Methods
20.1.2 Qualitative Methods
20.2 Application of Regression Concepts
20.2.1 Finite-Diffusion-Length Model
20.2.3 Convective-Diffusion-Length Model
21 Error Structure of Impedance Measurements
21.2 Stochastic Errors in Impedance Measurements
21.2.1 Stochastic Errors in Time-Domain Signals
21.2.2 Transformation from Time Domain to Frequency Domain
21.2.3 Stochastic Errors in Frequency Domain
21.3.1 Instrument Artifacts
21.3.2 Ancillary Parts of the System under Study
21.3.3 Nonstationary Behavior
21.3.4 Time Scales in Impedance Spectroscopy Measurements
21.4 Incorporation of Error Structure
21.5 Measurement Models for Error Identification
22 The Kramers-Kronig Relations
22.1 Methods for Application
22.1.1 Direct Integration of the Kramers–Kronig Relations
22.1.2 Experimental Assessment of Consistency
22.1.3 Regression of Process Models
22.1.4 Regression of Measurement Models
22.2.2 Application of Cauchy’s Theorem
22.2.3 Transformation from Real to Imaginary
22.2.4 Transformation from Imaginary to Real
22.2.5 Application of the Kramers–Kronig Relations
22.3 The Kramers–Kronig Relations in an Expectation Sense
22.3.1 Transformation from Real to Imaginary
22.3.2 Transformation from Imaginary to Real
23 An Integrated Approach to Impedance Spectroscopy
23.1 Flowcharts for Regression Analysis
23.2 Integration of Measurements, Error Analysis, and Model
23.2.1 Impedance Measurements Integrated with Error Analysis
23.2.2 Process Models Developed Using Other Observations
23.2.3 Regression Analysis in Context of Error Structure
A.2 Cauchy–Riemann Conditions
A.3.2 Improper Integrals of Rational Functions
B Tables of Reference Material
Electrochemical Impedance Spectroscopy
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