Understanding Voltammetry: Simulation Of Electrode Processes

Description

This is the first textbook in the field of electrochemistry that will teach experimental electrochemists how to carry out simulation of electrode processes. Processes at both macro- and micro-electrodes are examined and the simulation of both diffusion-only and diffusionconvection processes are addressed. The simulation of processes with coupled homogeneous kinetics and at microelectrode arrays are further discussed. Over the course of the book the reader's understanding is developed to the point where they will be able to undertake and solve research-level problems. The book leads the reader through from a basic understanding of the principles underlying electrochemical simulation to the development of computer programs which describe the complex processes found in voltammetry. This second edition has been revised throughout, and contains new material relating to random walks in electrochemistry, as well as expanded materials on the checking and validation of simulations, pulse techniques, and square wave voltammetry. Contents Preface to the Second Edition Preface to the First Edition 1. Introduction 1.1. Electrochemical Systems 1.1.1. Mass transport 1.1.2. Boundary conditions 1.1.3. Reaction mechanisms 1.2. Voltammetric Techniques 1.3. Finite Difference Methods 1.4. Voltammetry: A Selected Bibliography References 2. Mathematical Model of an Electrochemical System 2.1. Cyclic Voltammetry 2.2. Diffusion: Fick’s Second Law 2.2.1. Equal diffusion coefficients 2.3. Boundary Conditions 2.4. Current 2.5. Dimensionless Parameters 2.5.1. Electrode surface boundary 2.5.2. Scan rate 2.5.3. Current 2.6. Summary References 3. Numerical Solution of the Model System 3.1. Finite Differences 3.1.1. Central differencing 3.1.2. Second derivative 3.2. Time Evolution: Discretising Fick’s Second Law 3.2.1. The explicit method 3.2.2. The implicit method 3.3. The Thomas Algorithm 3.4. Simulation Procedure 3.4.1. Example program 3.5. Testing Simulations 3.5.1. General tests 3.5.1.1. Self-convergence analysis 3.5.1.2. Unchanging stoichiometry-weighted total number of particles 3.5.1.3. Unchanging stoichiometry-weighted total concentration 3.5.2. Specific tests: Analytical solutions and reference values 3.5.2.1. Any electrode kinetics 3.5.2.2. Fully reversible electron transfers 3.5.3. Comprehensive simulation testing 3.6. Performance and Runtime Analysis 3.6.1. Memory requirements References 4. Diffusion-Only Electrochemical Problems in One-Dimensional Systems 4.1. Unequally Spaced Grids 4.1.1. Expanding spatial grid 4.1.2. Expanding time grid: Chronoamperometry and pulse techniques 4.1.3. Example program 4.2. Finite Electrode Kinetics 4.2.1. The Butler–Volmer model 4.2.2. Testing Simulations: Specific tests 4.2.3. The Marcus–Hush model 4.3. Unequal Diffusion Coefficients 4.4. Other One-Dimensional Electrode Geometries 4.4.1. Testing Simulations: Specific tests 4.4.1.1. Small Tmax limit: Macroelectrodes 4.4.1.2. Intermediate Tmax regime 4.4.1.3. Large Tmax limit: Steady state References 5. First-Order Chemical Kinetic Mechanisms 5.1. First-Order ECirre Mechanism: Basic Concepts 5.1.1. Reaction layer 5.2. First-Order Catalytic Mechanism: Coupled Equation Systems 5.3. LU Decomposition and Extended Thomas Algorithm 5.4. First-Order ECrev Mechanism: Including a Third Species 5.4.1. Testing Simulations: Specific tests 5.5. Multiple-Electron Transfer Processes 5.5.1. Testing Simulations: Specific tests 5.6. Heterogeneous Chemical Processes References 6. Second-Order Chemical Kinetic Mechanisms 6.1. Second-Order Catalytic Mechanism: The Newton–Raphson Method 6.2. Multiple-Electron Transfers: Adaptive Spatial Grids 6.3. Adsorption References 7. Electrochemical Simulation in Weakly Supported Media 7.1. The Nernst–Planck–Poisson Problem 7.2. Weakly Supported Cyclic Voltammetry and Chronoamperometry References 8. Hydrodynamic Voltammetry 8.1. Rotating Disc Electrodes 8.1.1. Testing Simulations: Specific tests 8.2. Channel Electrodes References 9. Two-Dimensional Systems: Microdisc Electrodes 9.1. Microdisc Electrodes: The Model 9.1.1. Diffusion 9.1.2. Boundary conditions 9.1.3. Current 9.1.4. Normalisation 9.2. Numerical Solution 9.2.1. Finite differences in two dimensions 9.2.2. The ADI method 9.2.2.1. Z-sweep discretisation 9.2.2.2. R-sweep discretisation 9.2.3. Boundary conditions 9.2.3.1. Implicit terms 9.2.3.2. Explicit terms 9.2.4. Flux 9.3. Implementation 9.3.1. Multithreading 9.3.2. Testing Simulations: Specific tests 9.4. Microband Electrodes 9.4.1. Numerical solution 9.4.2. Flux References 10. Heterogeneous Surfaces 10.1. Arrays of Microdisc Electrodes 10.1.1. Diffusion domain approximation 10.1.2. Implementation 10.1.2.1. Simulation procedure 10.1.3. Diffusional modes 10.1.4. Random arrays 10.1.4.1. Partially blocked electrodes 10.2. Microband Arrays 10.2.1. Highly ordered pyrolytic graphite 10.2.2. Interdigitated microbands 10.3. Porous Electrodes 10.4. Conclusions References 11. Stochastic Electrochemistry 11.1. Stochastic Electrochemistry and Partial Differential Equations 11.1.1. Example: Chronoamperometry at Low Concentrations 11.1.1.1. Implementation in C++ 11.2. Stochastic Electrochemistry and Stochastic Differential Equations 11.2.1. Theoretical Background 11.2.1.1. Friction 11.2.1.2. The Equipartition Theorem 11.2.1.3. The Langevin Equation 11.2.1.4. A General Solution to the Diffusion Equation 11.2.1.5. Einstein-Smoluchowski Relation and Stokes-Einstein Equation 11.2.1.6. Binomial Distributions and the De Moivre-Laplace Theorem 11.2.1.7. Construction of the Random Walk 11.2.1.8. Random Walks in Two and Three Dimensions 11.2.2. Example: Reversible Voltammetry at a Macroelectrode 11.2.2.1. Implementation in C++ References Appendix A: Review of C++ A.1. Hello World A.2. Variables, Types and Operators A.2.1. Arrays, vectors, and strings A.3. Logic and Program Control A.3.1. The if statement A.3.2. The for and while loops A.4. Functions References Appendix B: Microdisc Program References Index