Can one hear the shape of an electrode? I. Numerical study of the active zone in Laplacian transfer

The concept of active zone in the Laplacian transport to and across irregular interfaces is rigorously introduced. It applies to arbitrary geometries and uses the coarse-graining method proposed by Sapoval to compute the flux across an irregular interface from its geometry without solving the general Laplace problem. Such transport play a dominant role in electrochemistry, heterogeneous catalysis and physiological diffusion processes. In the held of electrochemistry, the method permits one to predict the impedance of an electrode of arbitrary geometry for any value of the frequency. It shows that, for systems with aspect ratios of the order of a few times unity or less, impedance spectroscopy yields in principle a reliable approximate measure of the length of the chord corresponding to a perimeter length inversely proportional to the interface capacitance and frequency. For these cases, impedance spectroscopy can determine the shape of an electrode to the extent that the knowledge of the average chord length as a function of the perimeter determines the shape. For systems of arbitrary geometry, it is shown that impedance spectroscopy permits a measure of the size of the active zone. These results can be transposed to several problems related to Laplacian transfer, such as etching of irregular solids and catalysis in the Eley-Rideal regime.