A general method for electrochemical simulations .2. Application to the simulation of steady-state currents at microdisk electrodes: Homogeneous and heterogeneous kinetics - Alden, J; Compton, R (1997)
ILU preconditioned Krylov subspace methods are used with conformal mappings to simulate the steady-state response of microdisk and hemispherical electrodes with the influence of homogeneous and heterogeneous kinetics. For the microdisk electrode, the conformal mapping of Amatore and Fosset (J. Electroanal. Chem. 1992, 328, 21) is shown to be superior to that of Verbrugge and Baker (J. Phys. Chem. 1992, 96, 4572), both in its efficiency for simple electron-transfer problems and in terms of the conditioning of the matrix it produces. The efficiency improvement arising from the use of multipoint Taylor series expressions for boundary conditions is investigated and is found to be highly significant for these systems where edge singularities are removed by the conformal mapping. Convergence at high rate constants is also addressed. The simulations were used to generate working curves at microdisk and spherical/hemispherical electrodes for ECE, DISP1, EC2E, DISP2, and EC′ mechanisms and a working surface for quasi-reversible heterogeneous kinetics. These allow quantitative mechanistic analysis for these mechanisms without the need for any further simulation. A suite of programs is available to perform this analysis via the World Wide Web (http://physchem.ox.ac.uk:8000/ wwwda/). Approximate analytical expressions, where available, are compared with simulated results. The approximate 'equivalence' between microdisk and spherical/hemispherical electrodes is assessed in the presence of heterogeneous and homogeneous kinetics. The results show that while there is no formal physical basis for an equivalence relation, an approximation for many common mechanisms can be generated by treating a microdisk electrode of radius r as if it were a spherical electrode of radius 2r/π. Slightly better results are obtained if the rate constant, k, for a coupled homogeneous reaction at the microdisk electrode is treated as a rate constant of πk/4 at a spherical electrode of radius π/4. Caution is advised, however, since the quality of the approximation is mechanism dependent: the error is reasonably uniform across the ECE, EC2E, and DISP reactions but increases significantly with substrate concentration in the EC′ reaction.