Analytical and Numerical Modeling of Microelectrode Voltammetry in Oblate Spheroidal Coordinates
Authors: Fenton, Jr. Alexis Maguin , Neyhouse Bertrand J. , Tenny Kevin M. , Chiang Yet-Ming , Brushett Fikile R. 
Published: July 2021

Voltammetry is a ubiquitous electroanalytical technique used to examine electroactive compounds for a range of purposes, such as component characterization of an electrochemical device or mechanistic analyses of electrode reactions.1–3 Although most commonly performed on a macroelectrode (radius ~mm), voltammetry conducted on a microelectrode (radius ~μm) enables interrogation of electrolyte compositions inaccessible to macroelectrodes, as the smaller radius minimizes ohmic distortions, reduces charging currents, and enables near-steady-state measurements at slower scan rates (e.g., 10 mV/s).4 These characteristics also mean that microelectrodes hold promise for the examination of concentrated multi-component electrolytes, potentially enabling in situ or operando analysis in practical embodiments.2,4 Methodologies aimed to evaluate electrolyte behavior in electrochemical cells leverage well-established expressions,1–4 and while physically grounded, they do not account for conditions where migration or non-idealities impact the voltammetric response.3 In these instances, more extensive modeling, often framed in cylindrical coordinates,1,5 is needed to estimate the features of interest (e.g., bulk concentrations) with sufficient accuracy. These methods involve partial differential equations with piecewise boundary conditions, most often resulting in expressions of limited adoptability due to either numerical complexity or lack of physicality.5 The mathematical treatment may be strengthened by obviating the piecewise boundary conditions through the use of oblate spheroidal coordinates. Indeed, with this coordinate transform, closed form steady-state and numerical transient voltammograms are obtainable with greater ease.6,7 Although this change has simplified modeling, to the best of our knowledge, existing treatments using both cylindrical and oblate spheroidal coordinates are only applicable to a restricted set of conditions (e.g., equal diffusion coefficients, no bulk product present) rarely encountered in real devices,5–7 limiting their ability to study practical electrolytes.
To extend the applicability of this approach, we use oblate spheroidal coordinates to derive steady-state and transient microelectrode voltammogram models capable of accounting for different electron transfer kinetics, dissimilar diffusion coefficients, and the presence of both reactant and product in the bulk solution (illustrated in the abstract figure). In this presentation, we will discuss the derivation of the closed form steady-state solutions and their evaluation using cases previously described in literature.5,6 We will then describe the development of the finite difference transient models capable of simulating voltammograms over a broader range of conditions (e.g., varying scan rates), which, in turn, are validated using COMSOL®. Finally, we will discuss the application of this procedure to evaluate electrolyte behavior in electrochemical cells, such as examining whether more complex physical phenomena (e.g., migration-induced transport) can be integrated into this framework and whether closed form steady-state solutions can accurately estimate relevant parameters from transient voltammograms.
Acknowledgments: This work was funded by the National Science Foundation under Award Number 1805566. B.J.N. and K.M.T. both gratefully acknowledge the National Science Foundation Graduate Research Fellowship Program under grant no. 1122374. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Figure 1