Steady-state currents at inlaid and recessed microdisc electrodes for first-order EC′reactions
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The analytical solutions, approximate expressions and a Finite Element simulation of the steady-state currents arising from the diffusion of a regenerating electroactive species towards a disc electrode, either inlaid in an insulator or recessed, are discussed. The results are valid for any reversible charge transfer, regardless of the applied potential (i.e. including limiting currents) and for an equal or unequal diffusion coefficient of the species. For the inlaid disc, derivation of the exact analytical solution, via a reformulation of the diffusion-reaction problem as a dual integral equation that can then be solved using a series of Bessel functions, allows us to assess and review the accuracy of existing approximate expressions. We present three new formulae for the steady-state current under these conditions, among which we highlight one with an accuracy better than 0.27% over the entire range of rate constants and we show that the accuracy of a recently presented two point Padé approximation (L. Rajendran and M.V. Sangaranarayanan, J. Phys. Chem. B 103 (1999) 1518) is better than 0.01%. The analytical solution also allows us to show that the accuracy of the simulation of the same problem using the Finite Element Method is better than 0.4%. For the recessed disc the exact analytical solution is derived, as an extension of the solution of the inlaid disc, by matching the series representing the concentration of the electroactive species and its derivative. Two approximate expressions are suggested, one of which yields at least 2% accuracy. Concentration profiles for the electroactive species provide physical insight into the processes involved.