The Ultramicrodisk Electrode

Initially, efforts were made to find expressions for the deviation of currents at the UMDE from that at a so-called planar electrode, which is the unidimensional case (called a shrouded plane by Oldham [23]). One can regard this as a disk (or any shape) at the bottom of an insulating deep well of the same cross-sectional shape as the electrode, so that the system can be reduced to one dimension. Here the Cottrell equation defines the current for a potential step, as in Chap. 2 (2.37) and (2.44). 

At a flush UMDE, the current deviates from the Cottrell value very soon after the potential jump. Lingane [24] suggested that to a good approximation and for a range of small values of time t, the current Iumde at a UMDE could be expressed as 

These attempts to express deviations of the current at a UMDE from the Cottrell current are somewhat fruitless because the expressions do not hold for other than rather small t values or rather, dimensionless values of the normalised time, Dt/a. General solutions were—and are— needed. There have been no analytical solutions holding for all times, but some limiting expressions, and a rather accurate approximate one, have been derived. 

Consider Fig. 12.1, depicting the UMDE in a cyUndrical coordinate system. The electrode of radius a is flush with an infinite insulating plane. The pde that governs diffusion around the UMDE is then 

For other kinds of experiments, the second, Cottrell condition, would be replaced with another. 

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Blitz D, 0sterby O, Strutwolf J (2008) Comparison of flux approximations in electrochemical digital simulation. Part 2 complications due to homogeneous chemical reactions, chaige estimation and application to the ultramicrodisk electrode. J Electroanal Chem 622 51-58... 

The above applies largely to one-dimensional problems, but two-dimensional simulations are more and more the norm. All the methods described can be applied to these, but there are some sensible choices. For the ultramicrodisk electrode case, the present authors have recently done a detailed study and concluded [79] that one of the more useful transformations such as the Verbrugge/Baker [80] (see Chap. 12 for this and others) is best, combined with multipoint approximations (optimally using five or seven points), and either BI with extrapolation or BDF with the BI start. 

For 2D simulations testing new algorithms, and where there are no proper analytical solutions it is convenient to have reliable reference flux values to compare with. We have provided some of these UMDE REF FLUX for the ultramicrodisk electrode, BAND REF FLUX for the band, and CYL REF FLUX for the cylinder electrode. 

Ferrigno et al. [239] describe the use of FEM for steady state simulations of recessed, flush and protruding ultramicrodisk electrodes, giving a good description of FEM. The method was made adaptive by Nann (and Heinze) [407,408], and Harriman et al. later published an extensive series of papers on adaptive FEM [287,288,289,290,291]. 

Che G, Dong S (1992) Application of ultramicroelectrodes in studies of homogeneous catalytic reactions- Part 111. The condition for quasi-first and second order homogeneous catalytic reactions at ultramicrodisk electrodes in the steady state. Electrochim Acta 37 2701-2705... 

Three-dimensional simulations of single electrodes have been done by explicit FD [96, 276], ADI [277] and FEM (COMSOL Multiphysics ) [278]. A three dimensional dual ultramicrodisk system was simulated by FuUan and Fisher using the boundary element method (BEM) [279]. BEM[87, 176, 177, 280] and FEM (using COMSOL Multiphysics ) [281] were employed to simulate complex three dimensional tip-substrate geometries in SECM. Arrays with a small number of electrodes were simulated using ADI [282] and commercial FEM packages Fluxexpert [283, 284] and COMSOL Multiphysics [285, 286]. 

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