The Ultramicrodisk Electrode, UMDE

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 [425]). One can regard this as a disk (or any shape) at the bottom of a deep well, 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 [365] 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/a2. General solutions were - and are - needed. There have been no analytical solutions holding for all times, but some limiting expressions, and an approximate one, have been derived. 

Consider Fig. 12.1, depicting the UMDE in a cylindrical 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. 

The classic study of Saito (1968) [490] is often cited. Saito derived the steady state current iss at a UMDE, setting the left-hand side of (12.2) to zero, and arrived at 

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