A Comparison of Fully Automated Methods of Data Analysis and Computer Assisted Heuristic Methods in an Electrode Kinetic Study of the Pathologically Variable [Fe(CN)₆]^3–/4– Process by AC Voltammetry

Abstract

Fully automated and computer assisted heuristic data analysis approaches have been applied to a series of AC voltammetric experiments undertaken on the [Fe(CN)6](3-/4-) process at a glassy carbon electrode in 3 M KCl aqueous electrolyte. The recovered parameters in all forms of data analysis encompass E(0) (reversible potential), k(0) (heterogeneous charge transfer rate constant at E(0)), α (charge transfer coefficient), Ru (uncompensated resistance), and Cdl (double layer capacitance). The automated method of analysis employed time domain optimization and Bayesian statistics. This and all other methods assumed the Butler-Volmer model applies for electron transfer kinetics, planar diffusion for mass transport, Ohm's Law for Ru, and a potential-independent Cdl model. Heuristic approaches utilize combinations of Fourier Transform filtering, sensitivity analysis, and simplex-based forms of optimization applied to resolved AC harmonics and rely on experimenter experience to assist in experiment-theory comparisons. Remarkable consistency of parameter evaluation was achieved, although the fully automated time domain method provided consistently higher α values than those based on frequency domain data analysis. The origin of this difference is that the implemented fully automated method requires a perfect model for the double layer capacitance. In contrast, the importance of imperfections in the double layer model is minimized when analysis is performed in the frequency domain. Substantial variation in k(0) values was found by analysis of the 10 data sets for this highly surface-sensitive pathologically variable [Fe(CN)6](3-/4-) process, but remarkably, all fit the quasi-reversible model satisfactorily.

Exploiting the Properties of Ti‐Doped CVD‐Grown Diamonds for the Assembling of Electrodes

“…Such values range from 10 −6 to 10 2 cm s −1 going from different carbons to metalbased electrodes. [26,[29][30][31][32] As also found in the case of BDD electrodes, the lower values of app 0 k of the Tidoped diamond electrode with respect to metal systems can be ascribed to semiconductive electronic proper ties of the diamond material that translate into a lower poten tialdependent density of electronic states and carrier density as compared to metals. [31,33] Moreover, the absence of specific functional groups on the diamond surface can be responsible for a further rate decrease since it was evidenced that some of them can be involved in the electrocatalysis process of inner sphere electron transfers, such the one occurring for Fe 2+/3+ couple.…”

Use of Bayesian Inference for Parameter Recovery in DC and AC Voltammetry

“…Significantly,, the resolved higher order AC harmonics are devoid of the background capacitance current that can severely restrict the use of DC and small amplitude AC voltammetry. Alternatively, the total current (DC plus AC) response derived from large amplitude AC voltammetry may be directly used in a comparison of simulated and experimential data without the need to introduce Fourier or any other form of analogue or digital filtering;, this direct approach does however require that the model used to mimic the potential dependence of the double layer capacitance allows a sufficiently accurate representation of the experimental data to be achieved, or that the magnitude of the background double layer capacitance current is small, relative to the faradaic current that governs the electrode kinetics. The frequencies of the applied sine waves we have used in large amplitude AC voltammetry are usually much lower that the upper ones used in EIS and only in the 10 to 1000 Hz range.…”

“…Whilst this optimisation approach gives (at least for “well‐posed” problems) the globally optimal solution for the parameters of interest, it generally gives only point estimates of the parameter values. Whilst these are valuable, no measure of the accuracy of the values is provided, although one can be estimated by making assumptions about the local curvature of the objective function at the optimal value, or by using a bootstrapping technique such as that described previously by Bieniasz and Rabitz and in one of our previous papers . However, the former case relies on an assumption of symmetry of the distribution of the parameters (and typically requires complex symbolic manipulation when dealing with differential equations), and the latter case (bootstrapping) is computationally very expensive (since it involves repeated solution of the full inverse problem) unless combined with approximation techniques that obviate the need to solve the underlying system of differential equations repeatedly …”

“…We also show that the AC approach can therefore yield accurate estimates across a wider range of experimental parameters, and in particular can distinguish higher values of the rate constant k 0 . The results of using a MCMC approach to solving the inverse problem are given for both synthetic data (i. e. simulated data for which the true parameter values are known and to which realistic levels of simulated experimental noise have been added), and for an illustrative case of experimental data for the one‐electron reduction of [Fe(CN) 6 ] 3− to [Fe(CN) 6 ] 4− . By using synthetic data, we are also able to quantify (for the first time) the improvements in accuracy in parameter estimation resulting from using the technique of AC compared to DC voltammetry in the presence of purely random (in this case, Gaussian) experimental noise.…”

Planar diffusion to macro disc electrodes—what electrode size is required for the Cottrell and Randles-Sevcik equations to apply quantitatively?

“…Given the renaissance of macro electrode measurements, it is interesting to consider under which conditions Eqs. (1) and (2) can be applied quantitatively not least since it has recently been asserted that macro electrode experiments may lead to a great diversity of results when applied to the same system, but by different groups [16][17][18]. Moreover, this variation has been such as to apparently stimulate the introduction of elaborate new techniques such as fast Fourier transformation (FFT) voltammetry in an attempt to improve the reproducibility [19,20].…”