Adaptive Finite Element Methods in Electrochemistry ² David J. Gavaghan,* Kathryn Gillow, and Endre Su ¨li Oxford UniVersity Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, U.K. ReceiVed April 28, 2006. In Final Form: August 7, 2006 In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the edge effect. Our approach to overcoming this problem has involved the derivation of an a posteriori bound on the error in the numerical approx- imation for the current that can be used to drive an adaptive mesh-generation algorithm, allowing calculation of the quantity of interest (the current) to within a prescribed tolerance. We illustrate the generic applicability of the approach by considering a broad range of steady-state applications of the technique. 1. Introduction Microelectrodes are widely used for a variety of electroanalysis and electrochemical measurement techniques. Because in general there is no direct means of relating the measured quantity (current, potential, etc.) to the underlying chemical system of interest, all such techniques are underpinned by mathematical models that are used to relate these output measurements to the input and chemical parameters of interest (applied potential, concentrations, partial pressures, reaction rates, etc.). The mathematical models take the form of reaction-convection-diffusion equations, coupled with boundary conditions describing the particular electrochemi- cal control technique. Modern microdevices allow the analysis of very fast reactions and extend clinical use to in-vivo measurements. For these devices, mathematical models are analytically intractable. As we outline below, numerical solution is complicated by boundary singularities, the complexity of the chemical processes, and, for clinical devices, by semipermeable membranes used to prevent surface spoiling (e.g., by protein deposition). As a result, previous numerical approaches (see ref 1 for a comprehensive review) have had difficulty demonstrating accuracy for general problems. The long-term goal of our research has therefore been to develop a general, and preferably completely automated, approach that will overcome all of these difficulties and generate approximations for the quantity of interestsin our case, generally the current flowing in an electrochemical cellsto a user-prescribed tolerance. In this article, we describe our work to date on the development of adaptive finite element methods (using both continuous and discontinuous approaches, to be described later) for amperometric techniques at a variety of microelectrode geometries and configurations. All of this work uses the technique of deriving an a posteriori error bound to drive the adaptivity, which is somewhat technical and involved from a mathematical viewpoint. In this review of our work to date, we therefore describe in detail only our work on steady-state problems, giving only a brief overview of how we have extended this work to time- dependent problems. Technical mathematical details are given only for the simplest model problem in the Appendix, with readers referred to earlier work for details of the underpinning math- ematics for the more complex examples. We begin this work with a brief review of previous approaches to the mathematical modeling and numerical simulation of electrochemical processes at microelectrodes, together with a brief overview of the use of adaptive numerical techniques in electrochemistry by other authors. 2. Previous Work Diffusion processes at microelectrodes are typically modeled in two dimensions by utilizing an inherent symmetry in the electrode geometry. However, the small size of the electrode (typically in the range of 10-100 µm) and the typical time span of the experiments (the range from the transient out to the steady state) mean that an account must be taken both of the concentration gradient normal to the electrode surface (as for a large planar electrode) and the concentration gradient parallel to the electrode surface. This parallel component results in a nonuniform current distribution over the electrode surface that is usually termed the edge effect, and it is this effect that is usually quoted as making the microdisk problem “challenging” (Michael et al. 2 ). In practice, the edge effect means that the standard finite difference approaches on uniform spatial meshes that have been used for 1D electrochemistry simulation problems since they were introduced by Feldberg in a seminal paper in 1964 3 have very slow rates of convergence and thus a prohibitively large number of unknowns is required to compute an accurate solution. In general, previous approaches that have attempted to overcome these problems fall into five categories: 1. (approximate) analytical solutions; 2. special integration technique at the electrode edge to allow for increased current; 3. matching a locally valid series solution near the electrode edge to the far-field numerical solution; 4. conformal maps that either increase the number of points near the singularity or effectively remove the singularity; and 5. mesh refinement to put more points near the singularity. ² Part of the Electrochemistry special issue. * Corresponding author. (1) Alden, J. A. D. Phil. Thesis, University of Oxford, Oxford, U.K., 1998. (2) Michael, A. C.; Wightman, R. M.; Amatore, C. A. J. Electroanal. Chem. 1989, 267, 33. (3) Feldberg, S. W. Anal. Chem. 1964, 36, 505. 10666 Langmuir 2006, 22, 10666-10682 10.1021/la061158l CCC: $33.50 © 2006 American Chemical Society Published on Web 09/27/2006