FEM Modelling of Diffusional Electrochemical Impedance Spectroscopy at a Channel Electrode
Although the full equation has not been solved analytically, two sets of assumptions have been used previously, and the resulting equations were solved analytically:
i) The axial diffusion is neglected and the Poiseuille flow is simplified as a Lévêque flow1,2.
ii) Convection is neglected and it is solved as a simple diffusion process3.
However, although the assumptions intuitively allow for solving the problem in a large parameter space, parameters for which these approximations are valid have not been determined.
We demonstrate by using FEM software (Comsol Multiphysics™), that the full problem can be solved numerically. The solution obtained by FEM software is compared to experimental results and the two analytical solutions. The method shows good correspondence with the experimental results and a large flexibility, making it possible to solve more complex problems such as different diffusion coefficients4 and unconventional geometries. An example of a comparison between experimental results, numerical results using FEM software, and an analytical solution is shown in Fig. 1. The parameter space of the analytical solutions is investigated using two non-dimensional parameters, A and B, and a validity area is found where the analytical solutions are well-suited. The parameter A is a version of the Péclet number, A = 6vavh/D, and B is the ratio of electrode width and channel height, B = w/h.
 B. A. Coles, R. G. Compton, J. Electroanal. Chem. 127,37 (1981).
 R. G. Compton, G. R. Sealy, J. Electroanal. Chem. 145,35 (1983).
 R. G. Compton, J. Winkler, J. Phys. Chem. 99,5029 (1995).
 Y. Wang, J. G. Limon-Petersen, R. G. Compton J. Electroanal. Chem. 652,13 (2011).
Fig 1: Nyquist plot of impedance spectroscopy at a channel electrode showing experimental values (blue boxes), values modelled in Comsol (blue line), values calculated analytically by neglecting axial diffusion and simplifying the flow (upward pointing black triangles), and values calculated by neglecting convection (downward pointing black triangles). Impedance values are dimensionless.