Application of Danckwerts’ expression to first-order EC′ reactions. Transient currents at inlaid and recessed microdisc electrodes

MODELING OF NONLINEAR REACTION–DIFFUSION PROCESSES OF AMPEROMETRIC POLYMER-MODIFIED ELECTRODES

A mathematical model of amperometric response for a polymer-modified electrode system has been developed. The model is based on nonstationary diffusion equations containing a nonlinear term related to Michaelis–Menten kinetics of the enzymatic reaction. In particular, the interplay between chemical reaction and substrate diffusion is specifically taken into account. The limiting situations of catalytic site unsaturation and site saturation are considered. The analytical solutions for substrate concentration and transient current for both steady and nonsteady-state are obtained using Danckwerts' relation and variable and separable method. An excellent agreement with the previous analytical results are noted. The combined analytical set of solution of steady-state current in all the nearest sites is also described in a case diagram. A general simple analytical approximate solution for steady-state current for all values of α is also given. A two-point Padé approximation is also derived for the nonsteady-state current for all values of saturation parameter α. Limiting case results (α ≪ 1 and α ≫ 1) are compared with Padé approximation results and are found to be in good agreement.

ANALYTICAL SOLUTION FOR THE STEADY-STATE CHRONOAMPEROMETRIC CURRENT FOR AN EC′ REACTION AT SPHEROIDAL ULTRAMICROELECTRODES

The steady-state chronoamperometric current for an EC′ reactions at spheroidal ultramicroelectrodes is derived from the non-steady-state diffusion limited current. The polynomial expressions pertaining to two extreme limits of reaction rates are combined for all reaction rates. Starting with the result for spheroidal electrode, equations are obtained for the steady-state currents at disc, oblate, hemisphere and prolate electrodes. Tabular compilation of dimensionless current for disc electrodes are reported. A good agreement with previously available simulation results is noticed.

Catalytic mechanism in cyclic voltammetry at disc electrodes: an analytical solution

The theory of cyclic voltammetry at disc electrodes and microelectrodes is developed for a system where the electroactive reactant is regenerated in solution using a catalyst. This catalytic process is of wide importance, not least in chemical sensing, and it can be characterized by the resulting peak current which is always larger than that of a simple electrochemical reaction; in contrast the reverse peak is always relatively diminished in size. From the theoretical point of view, the problem involves a complex physical situation with two-dimensional mass transport and non-uniform surface gradients. Because of this complexity, hitherto the treatment of this problem has been tackled mainly by means of numerical methods and so no analytical expression was available for the transient response of the catalytic mechanism in cyclic voltammetry when disc electrodes, the most popular practical geometry, are used. In this work, this gap is filled by presenting an analytical solution for the application of any sequence of potential pulses and, in particular, for cyclic voltammetry. The induction principle is applied to demonstrate mathematically that the superposition principle applies whatever the geometry of the electrode, which enabled us to obtain an analytical equation valid whatever the electrode size and the kinetics of the catalytic reaction. The theoretical results obtained are applied to the experimental study of the electrocatalytic Fenton reaction, determining the rate constant of the reduction of hydrogen peroxide by iron(II).

Adaptive finite element methods in electrochemistry

In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the edge effect. Our approach to overcoming this problem has involved the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm, allowing calculation of the quantity of interest (the current) to within a prescribed tolerance. We illustrate the generic applicability of the approach by considering a broad range of steady-state applications of the technique.