Scanning Electrochemical Microscopy: Diffusion on a Microdisk Electrode
Scanning Electrochemical Microscopy (SECM) uses a microelectrode tip to image the microscopic domain. Because of the very small size of the microelectrode (of the order of micrometers) the mass transport in a disk-shaped microelectrode can be considered to occur in a semi-infinite region. This Demonstration shows the concentration profile of electroactive space during diffusion in and out of the surface of the microdisk electrode.
Semi-infinite mass transport occurs to a disk-shaped electrode in which the following electrochemical reaction takes place:
The mass transport to the electrode can be modelled by the cylindrical diffusion equation
, where is the concentration of oxidant () and , represent the spatial coordinates.
In this Demonstration, the radius of the microelectrode is taken as on the axis. The boundary conditions are
at the surface of microelectrode,
and above the surface of the microelectrode, where ,
The exact solution was given in  and can also be calculated with Mathematica's built-in function NDSolve.
 J. Crank and R. M. Furzeland, "The Treatment of Boundary Singularities in Axially Symmetric Problems Containing Discs," Journal of the Institute of Mathematics and Its Applications, 20(3), 1977 pp. 355–370.