Integration of Heuristic and Automated Parametrization of Three Unresolved Two‐Electron Surface‐Confined Polyoxometalate Reduction Processes by AC Voltammetry

Abstract

The thermodynamic and electrode kinetic parameters that describe each of the three unresolved proton‐coupled two‐electron transfer processes of surface‐confined Keggin‐type phosphomolybdate, [PMo12O40]3− adsorbed onto glassy carbon electrode in 1.0 M H2SO4 have been elucidated by comparison of experimental and simulated AC voltammmetric data. Modelling of this problem requires the introduction of over 30 parameters, although this may be reduced to about half this number when intelligent forms of data analysis are introduced. Heuristic (i. e., an experimenter based trial and error method) and automated data optimization approaches are integrated in this very extensive parameter estimation exercise. However, obtaining a unique solution remains challenging for reasons that are outlined. In the final analysis and using the automated strategy, estimates of six reversible potentials, lower limits of the six electron transfer rate constants, the double layer capacitance, uncompensated resistance and surface coverage are reported, with others (such as the charge transfer co‐efficient) present in the model being unobtainable for reasons that are provided. The fit to experimental data using parameters obtained by automated data optimisation is excellent and slightly superior to that obtained by heuristic analysis. The parameters obtained by either method account for differences in shapes and current magnitudes of each of the overall two electron processes.

Models and Their Limitations in the Voltammmetric Parameterization of the Six‐Electron Surface‐Confined Reduction of [PMo12O40]3− at Glassy Carbon and Boron‐Doped Diamond Electrodes

“…In the automated parameterization of the 21 variables present in the model, possible values available for k 0 , α R u , C dl , and Γ were constrained to confine the extent of parameter space search and hence computational time needed to complete each data optimization exercise. Sensible range restrictions for these parameters were easily identified based on knowledge gained in the study in 1 M H 2 SO 4 at a GC electrode . Significantly, this earlier study revealed that much tighter and carefully considered restrictions on E rev ranges are needed to avoid physically unrealistic values been obtained via data optimization.…”

“…Each process, despite commonly having a significantly different shape, corresponds to the overall two electron‐two proton processes represented by Eq. 1–3 . Clearly, the background non‐faradaic current also contributes more extensively to the total current (faradaic plus non‐faradaic) in the DC and AC voltammetry derived from the GC electrode than for the BDD one (compare Figures b and 1d and 1c and 1e), and this also is expected to have an impact in the degree of difficulty and indeed level of success encountered in the the modelling. …”

“…Experiment‐theory comparisons are then undertaken at levels ranging from fully heuristic to highly automated data optimization multi‐parameter fitting exercises in attempts to define uniquely the thermodynamic, kinetic, capacitance and resistance and other parameters that are included in the model. In one of the more complex reaction schemes we have quantified, values of thermodynamic, electrode kinetic and other variables present in a model were elucidated for a series of proton‐coupled two‐electron transfer processes associated with the reduction of the polyoxometalate (structure included in Figure a) adsorbed onto a glassy carbon (GC) electrode in 1 M H 2 SO 4 . In this particular example, integration of experimenter‐controlled heuristic and automated data optimization was employed to estimate up to 17 of the parameters employed in the model.…”

“…In this particular example, integration of experimenter‐controlled heuristic and automated data optimization was employed to estimate up to 17 of the parameters employed in the model. spontaneously adsorbs onto many electrode surfaces and has been used widely in this state in many practical applications (see for example). In 1 M HClO 4 , the electrolyte of interest in this study, three well‐defined surface confined reduction processes are found under conditions of DC cyclic voltammetry at glassy carbon (GC) and boron doped diamond (BDD) electrodes (Figures b and 1c respectively) within the potential range employed in these experiments.…”

“…While simulation of the forward multi‐step problem relevant to this study is inherently complex, it can be readily achieved nowadays with the aid of commercially available DigiElch or KISSA or web downloadable software packages such as MECSim. Alternatively, in‐house coding of the algorithms needed to generate the model can be undertaken as in our recent work on the reduction of at a GC electrode in 1 M H 2 SO 4 . The major challenge now is to solve the computationally intense inverse problem in a manner that provides a reliable and physically meaningful understanding of the values and significance of a large number of parameters.…”