Recent work [1-3] has shown that electrodes, often made of platinum or basal plane graphite, can be usefully modified with random arrays of microdroplets of insoluble oils such that oxidation or reduction of redox groups within the oil molecules is accompanied by counter ion insertion into the droplet to maintain electroneutrality. Thus, for example, the reduction of 4-nitrophenyl nonyl ether droplets was shown [2] to induce proton and alkali metal insertion from the aqueous electrolyte bathing the droplet array. Similarly [3], anion insertion is observed in the case of oxidation of droplets of tetra-alkylphenylene diamines. A variety of evidences [1-3] has been presented to suggest that these charge transfer reactions take place exclusively at the three phase boundary formed between the droplet, the electrode surface and the electrolyte solution. Accordingly, it is of interest to examine the voltammetric response of a single droplet under such conditions. Early calculations using the dual reciprocity finite element method showed that charge insertion at the three phase boundary could be distinguished voltammetrically from other mechanisms of charge insertion [4] but these calculations were not readily extended beyond the case of simple charge transfer, notably for the case of coupled chemical reaction kinetics inside the droplet. Nor was the coupling of transport processes outside the droplet. Accordingly, we propose in this paper a new numerical approach based on the use of the conformal mapping technique. For the simulation of 2D electrochemical problems at microelectrodes this approach represents the most efficient and easiest way of obtaining accurate solution of diffusion problems and allows transformation back to real space where the flux lines may be extremely curved from the transformed space in which the flux lines are almost parallel [5]. Although we consider here only the simulation of the droplet interior problem (ω_in area), the suggested conformal mapping also allows to solve the droplet problem considering additionally the exterior electrolyte (ω_out area) as well.

References
1. J. D. Wadhawan, R. G. Compton et al., J Amer. Chem. Soc., (2003) in press.
2. A. J. Wain, N. S. Lawrence, P. R. Greene, J. D. Wadhawan, R. G. Compton, PhysChemChemPhys., 5 (2003) 1867.
3. J. D. Wadhawan, R. G. Evans, C. E. Banks, S. J. Wilkins, R. R. France, N. J. Oldham, A. J. Fairbanks, B. Wood, D. J. Walton, U. Schroeder, R. G. Compton, J. Phys. Chem. B, 106 (2002) 9619.
4. Q. Fulian, J. C. Ball, F. Marken, R. G. Compton, A. C. Fisher, Electroanalysis, 12 (2000) 1012.
5. C. Amatore, in I. Rubinstein (Ed.), Physical Electrochemistry: Principles, Methods and Applications, Marcel Dekker, New York. 1995. (Ch. 4).

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1. Simulation of the Double Hemicylinder Electrode System Through Conformal Mapping. Application to Steady-state and Transient Electrogenerated Chemiluminescence