Comparison of flux approximations in electrochemical digital simulation. Part 2: Complications due to homogeneous chemical reactions, charge estimation and application to the ultramicrodisk electrode

In numerical simulation of electrochemical systems the estimation of current flow or passed charge from simulated concentrations plays a crucial role and various approaches within the framework of finite differences are examined in this paper. Both on an exponentially expanding grid in X-space, and the corresponding equidistant grid in transformed Y-space, fluxes were expressed as either gradient approximations using a variable number of points, or as the spatial integral of rates of changes of the concentrations with time over the whole diffusion space, in the presence of homogeneous chemical reactions. Also, the passed charge was computed for these cases. The cases of chronoamperometric boundary conditions at a shrouded plane and an ultramicrodisk electrode (UMDE) were simulated. Results show that in all cases, fluxes computed on the basis of the spatial concentration gradient at the electrode are the most accurate provided that the Rudolph modification of the two-point flux in Y-space is used; it also works very well in the presence of homogeneous chemical reactions. The flux approximation on the basis of spatial integration of the time derivative of concentration comes a close second in Y-space in the case of one-dimensional simulations when K is not too large, and also works rather well in the case of the UMDE in transformed space. Charge computed as the time integral of flux can be quite accurate if two separate simulation runs are carried out, combining the results by extrapolation. Charge computed as the spatial integration of concentration loss in the diffusion space (not practical in the presence of a homogeneous chemical reaction) can also be quite accurate, both in the one- and two-dimensional case.