(Keynote) Electrochemistry at Regular and Random Arrays of Nano-Disk Electrodes
Authors: Amatore Christian Andre , Sliusarenko Oleksii , Oleinick Alexander , Svir Irina 
Published: February 2020

Electrochemical microdisk arrays provide great analytical improvement essentially by increasing the signal-to-noise (S/N) ratio of electrochemical experiments. Indeed, in electrochemistry the minimum current noise of electrochemical origin mostly stems from the electrode capacitance so that the S/N value is drastically increased upon minimizing the electrode surface area[1]. In an electrochemical array performing under conditions when diffusion occurs over the whole array surface area A
array the Faradaic current is proportional to A
array while the capacitance of the array is proportional to the surface area of its N
disk disk micro- or nanoelectrodes of radius r
0, so that the S/N value increases upon increasing the area of inactive sections relative to that of the active ones: S/N 
≈ A
array/(N
diskπr
0
2)[1].

However, present theories allowing the precise prediction/rationalization of electrochemical responses of electrode arrays rely on a model we developed ca thirty years ago based on hypothetical arrays consisting of cylindrical unit cells[2]. Indeed, at the time numerical capabilities could not allow developing theory covering realistic arrays, so it was intuitively accepted that the “cylindrical unit cell” model was representing an ideal framework describing the behavior of electrochemical arrays despite that it is impossible to pave regularly a plane with unit cells having a circular section. Doing so involves significant overlap between the unit cells and leaves uncovered a non-negligible fraction of the array surface. This is easily evident from basic geometry for the squared and hexagonal arrays but is even worse for irregular or random arrays.

Since this seminal work was published, many interesting other ones have followed with the aim of establishing effective conversions between real arrays and their “cylindrical equivalents” but none has been able to proceed beyond reasonable intuition to validate the conversion processes proposed[3]. In the present lecture we will establish quantitatively, based on analytical theory and on the outcome of 3-dimension Brownian simulations that “cylindrical equivalents” models can be developed to account for the behavior of regular arrays[4] as well as of irregular ones[5]. Indeed, these two works establish that in all cases, predictions based on “cylindrical equivalents” are correct within a maximum 10% error on chronoamperometric currents when the time-dependent currents are measured on the plateau of a redox wave and in absence of any ohmic drop. This remains true for any electrochemical method (e.g., voltammetry) even in the presence of ohmic drop but only for regular arrays (hexagonal or squared). Indeed in both arrays, the “cylindrical unit cells equivalents” cross-section surface area are identical so as at any time or potential all disk electrodes experience the same current density which ensures that the events in each unit cell are identical.

Conversely, in the case of random arrays each unit cell is specific. This is evident upon considering the outcome of Voronoi construction of unit elements around each disk electrode. Each one has a cross-section surface area which depends on the proximity between the disk electrode it includes and its neighboring ones. As such, when the disk electrodes potential is not extremely rapid (viz., the analyte concentration is not governed by Nernst’s law) and there is possibly ohmic drop the analyte surface concentration depends on the current density in the unit cell, viz., on its cross-section surface area. Conversely, when the disk electrodes potential is poised onto the plateau of the investigated redox wave, the analyte concentration at its surface is nihil which eliminates the influence of the current density. Under such conditions, the dimensionless chronoamperometric response, I
Array(t), of the whole array is predicted within a relative accuracy better than a few percent by a simple weighted summation of the responses, I
cell(t), of each of its independent equivalent cylindrical unit cells.This allows an easy and quick prediction of I
Array(t) since I
cell(t) functions are obtained under simple analytical form as a function of r. to afford the real array current value, i
Array(t). Indeed, I
Array(t) is directly related to this latter one through a simple normalization involving its short-time, i
array
0(t), and long-time current limits, i
array
∞(t), which are achievable experimentally or can be predicted through an analytical law.

References

C. Amatore, in Physical Electrochemistry: Principles, Methods and Applications; (Eds: I. Rubinstein), M. Dekker, New York 1995; Chapter 4.

C. Amatore, J.-M. Savéant and D. Tessier, J. Electroanal. Chem., 147, 39-51 (1983).

For one recent contribution, see e.g.: S.R. Belding and R. G. Compton, J. Phys. Chem. C,
114
, 8309-8319 (2010).

O. Sliusarenko, A. Oleinick, I. Svir and C. Amatore, Electroanalysis, 27, 980-991 (2015).

O. Sliusarenko, A. Oleinick, I. Svir and C. Amatore, ChemElectroChem, in press.