Validation of a Central Approximation in Theories of Regular and Random Electrochemical Electrode Arrays. Extraction of Statistical Information from Amperometric Response of Random Electrode Arrays

Abstract

Regular electrode arrays generally possess a symmetry which (under certain conditions) allows to reduce their mathematical description to consideration of a single electrode of this array within the laterally bounded domain (known as ‘unit cell’ or ‘diffusion domain’) [1, 2]. In this case by multiplying the simulated current of the single cell by the number of electrodes in the array one would obtain an electrochemical response of the whole array. The electrodes in arrays are generally ordered in a hexagonal or square patterns, which result in hexagonal or square cross-sections of the unit cells, respectively. A straightforward approach to the mass-transport problem in such cells would require complex and time-consuming 3D simulations. Therefore almost always an approximation is made in simulations by replacing the original unit cell by the one with circular cross-section of the same surface area. At one hand, this greatly facilitates the simulations due to the possibility to formulate the mass-transport problem in two dimensions (2D). On the other hand, although this approach was proven to be effective in practice, the error introduced by this approximation, to the best of our knowledge, was never been studied or even estimated. The error associated with the approximation is based on the fact that it is impossible to pave the plane with non-intersecting circles contrary to the perfect plane paving with hexagons or squares as well as due to the different possible diffusion pathways within the single cells with different cross-sections. In order to assess quantitatively this approximation we performed 3D Brownian motion simulations for the single cells of square, hexagon and circular cross-sections equipped with disk electrode. Moreover the situations with inlaid, recessed and protruding electrodes were also considered for each cross-section shape. Notwithstanding the completely different diffusion patterns observed in these systems, the obtained results showed that the approximation gives rise to an experimentally indistinguishable error (less than 5%) for all considered systems [2]. The only exception (i.e. when the error is larger than 5%) was the case when the electrode radius was extremely close to the characteristic size of the cross-section, i.e. the situation when the neighbouring electrodes in the array almost touch each other. Such systems are of low experimental interest and, in addition, exhibit the same behavior as a planar electrode with the surface area of the whole array (except of the very initial times of the experiment). With the same idea in mind, to check the validity of the circular diffusion domain approximation, the random arrays were considered. The arrays with randomly distributed electrodes of the same size were partitioned by means of Voronoi tessellation, which creates a unit cell with polygonal cross-section around each electrode. These polygonal unit cells were replaced then by the unit cells with circular cross-section having the same surface area. Taking into account distribution of the unit cell sizes one can compute the response of the whole array. By comparison the outcome of this procedure with the full scale 3D Brownian motion simulations of the whole array we showed that employment of the circular unit cells in simulations of the responses of the random electrode arrays is validated [3]. These results allow us to develop an easy to evaluate analytical approximation for the electrochemical current of the random array requiring only a distribution of sizes of the unit cells as an input. The analytical approximation in turn makes possible retrieval of the unit cell size distribution from the amperometric response of the random array.