Computational Electrochemistry (c) John Alden 1998
DPhil Thesis, Oxford University

1. Introduction

1.1 Why model voltammetry?
1.2 The four components of an electrochemical model
1.3 The mechanism
1.3.1 Heterogeneous kinetics
1.3.2 Homogeneous kinetics
1.4 The electrode geometry
1.4.1 Common geometries
1.4.2 Microelectrodes
1.4.3 Hydrodynamic electrodes
1.4.3.1 Rotating disc electrode (RDE)
1.4.3.2 Wall-jet electrode (WJE)
1.4.3.3 Tubular and channel electrode (ChE)
1.4.3.4 Uniform vs. non-uniform accessibility
1.5 The experimental technique
1.5.1 Obtaining kinetic information
1.5.1.1 Steady-state methods
1.5.1.2 Transient methods
1.5.2 Choice of experimental technique: steady-state vs. transient experiments
1.6 The solution method
1.6.1 Analytical solutions
1.6.1.1 Integral transforms
1.6.2 Numerical simulations
1.6.2.1 Finite difference methods
1.6.2.2 Methods of weighted residuals (MWR)
1.6.2.3 Finite analytic method
1.6.2.4 Network approach
1.7 Outline of the work presented in this thesis

2. Finite difference methods

2.1 Discretisation: explicit vs implicit
2.1.1 Explicit
2.1.2 Implicit
2.1.3 Crank-Nicolson
2.1.4 Richtmyer modification or backward differentiation formulae
2.1.5 Dufort-Frankel
2.1.6 Hopscotch
2.1.7 ADI
2.2 Steady-state
2.3 Space-marching Backwards Implicit (2-D frontal) method
2.4 Boundary conditions
2.5 Simulating kinetics
2.5.1 Heterogeneous kinetics
2.5.2 Homogeneous kinetics
2.6 Solution of linear systems
2.6.1 Direct methods
2.6.1.1 Gaussian elimination / LU decomposition
2.6.1.2 Thomas Algorithm
2.6.1.3 FIFD method
2.6.2 Iterative methods
2.6.2.1 Jacobi
2.6.2.2 Gauss-Seidel
2.6.2.3 Acceleration by Successive Over-Relaxation
2.6.2.4 The Strongly Implicit Procedure
2.6.2.5 Splitting
2.7 Solution of non-linear equations
2.7.1 Newton's method

3. Optimisation of finite difference methods

3.1 Convergence
3.2 Extent of simulation space
3.3 Boundary singularities
3.4 Efficient distribution of nodes
3.4.1 Expanding grids
3.4.2 Conformal mappings
3.4.3 Transformed time
3.5 Reaction layer
3.6 Coupled heterogeneous processes
3.6.1 Back-to-back grid
3.7 Coupled homogeneous processes
3.8 Neumann boundaries
3.9 Current integration

4. New solvers for electrochemical problems

4.1 The multigrid method
4.1.1 MDG1 (NAG Library subroutine D03EDF)
4.1.2 Efficiency
4.1.3 Limitations
4.2 The preconditioned Krylov subspace methods
4.2.1 Basis
4.2.2 Performance
4.2.3 Conclusions
4.3 The best of both worlds?
4.4 Conformal mappings and multigrid methods
4.5 A frontal solver based on the BI method
4.6 Parallel transient BI
4.6.1 Theoretical efficiency of processor usage
4.6.2 Storage
4.6.3 Results

5. A general multidimensional electrochemical simulator

5.1 A general simulation strategy based on a sparse matrix approach
5.1.1 Linear homogeneous kinetics
5.1.2 Non-linear homogeneous kinetics
5.1.3 Heterogeneous electrochemical kinetics
5.1.4 Heterogeneous chemical kinetics
5.1.5 n-point flux calculation
5.1.6 n-point Neumann boundary conditions
5.1.7 Extension to time-dependent simulations
5.2 A multidimensional electrochemical simulator

6. The channel microband electrode

6.1 Multigrid simulations of transport-limited electrolysis
6.1.1 Results: within the Lévêque approximation
6.1.2 Results: without the Lévêque approximation
6.2 Simulating homogenous kinetics
6.2.1 ECE within the Lévêque approximation
6.2.2 EC2E within the Lévêque approximation
6.3 Channel electrodes in a general electrochemical simulator

7. The rotating disc electrode

7.1 Hydrodynamics of the RDE
7.2 Simulation method
7.3 Results
7.3.1 Schmidt number correction of the Levich equation
7.3.2 Working curves and surfaces for kinetic analysis

8. The wall-jet electrode

8.1 Wall-jet simulations in conformal space
8.1.1 Velocity field
8.1.2 Simulation in a general curvilinear space
8.1.3 A conformal space based on the boundary layer
8.1.4 Further refinements to the space transformation for efficient 2-D modelling
8.1.4.1 Sigmoidal expansion of the radial co-ordinate
8.1.4.2 Exponential expansion of the normal co-ordinate
8.1.5 Effects of radial diffusion
8.1.6 Working curves and surfaces for analysis of steady-state voltammetry
8.2 Simulations of the micro-jet electrode
8.2.1 Effects of radial diffusion

9. The microdisc electrode

9.1 Disc-hemisphere equivalence
9.2 Steady-state response
9.2.1 Homogeneous & heterogeneous kinetics at microdisc and spherical electrodes
9.2.2 Simulating the microdisc with a conformal mapping
9.2.3 Simulating a spherical electrode in conformal space
9.2.4 Results and Discussion
9.3 Cyclic voltammetry
9.3.1 Cyclic voltammetry at a hemispherical microelectrode
9.3.2 Cyclic voltammetry at a microdisc electrode
9.4 Conclusions

10. Kinetic analysis based on working surface interpolation

10.1 Introduction
10.2 Dimensionless parameters for common mechanisms & geometries
10.3 Equivalence between hydrodynamic electrodes
10.4 Automated simulation methods
10.4.1 Spherical, microdisc and RDE simulations using PKS methods
10.4.2 Channel microband and wall-jet electrodes
10.4.3 Channel electrode working curves (using the BI method)
10.5 Extrapolation of DQ1/2 to the reaction-layer limit
10.6 A comparison of interpolation methods
10.6.1 Method
10.6.2 Results
10.6.3 Conclusions
10.7 A data analysis service via the World Wide Web

Conclusions


Appendices

A1 Dimensionless variables for the channel electrode
A1.1 FULL TREATMENT
A1.1.1 Steady-state transport-limited current
A1.1.2 Homogeneous kinetics
A1.1.3 Time-dependent behaviour
A1.2 LÉVÊQUE APPROXIMATION
A1.2.1 Steady-state transport-limited current
A1.2.2 Treatment without axial diffusion
A2 Dimensionless variables for the RDE
A3 General wall-jet co-ordinate transformations
A3.1 CURVILINEAR TRANSFORMATION OF Z
A3.2 TRANSFORMATION OF BOTH Z AND R
A3.3 TWO-STEP TRANSFORMATION
A4 Peclet number for the wall-jet electrode
A5 Peclet number for the wall-tube electrode
A6 Dimensional analysis of LSV at microdisc and hemispherical electrodes
A6.1 MICRODISC ELECTRODE
A6.2 HEMISPHERICAL ELECTRODE
A6.3 COMPARISON OF PEAK CURRENTS