Separating the Effects of Experimental Noise from Inherent System Variability in Voltammetry: The [Fe(CN)₆]^3–/4– Process

Abstract

Recently, we have introduced the use of techniques drawn from Bayesian statistics to recover kinetic and thermodynamic parameters from voltammetric data, and were able to show that the technique of large amplitude ac voltammetry yielded significantly more accurate parameter values than the equivalent dc approach. In this paper we build on this work to show that this approach allows us, for the first time, to separate the effects of random experimental noise and inherent system variability in voltammetric experiments. We analyse ten repeated experimental data sets for the [Fe(CN) 6 ] 3−/4− process, again using large-amplitude ac cyclic voltammetry. In each of the ten cases we are able to obtain an extremely good fit to the experimental data and obtain very narrow distributions of the recovered parameters governing both the faradaic (the reversible formal faradaic potential, E 0 , the standard heterogeneous charge transfer rate 1 arXiv:1810.02187v1 [eess.SP] 18 Sep 2018 constant k 0 , and the charge transfer coefficient α) and non-faradaic terms (uncompensated resistance, R u , and double layer capacitance, C dl ). We then employ hierarchical Bayesian methods to recover the underlying "hyperdistribution" of the faradaic and non-faradaic parameters, showing that in general the variation between the experimental data sets is significantly greater than suggested by individual experiments, except for α where the inter-experiment variation was relatively minor. Correlations between pairs of parameters are provided, and for example, reveal a weak link between k 0 and C dl (surface activity of a glassy carbon electrode surface). Finally, we discuss the implications of our findings for voltammetric experiments more generally.

A perceived paucity of quantitative studies in the modern era of voltammetry: prospects for parameterisation of complex reactions in Bayesian and machine learning frameworks

“…In recent collaborative work with the Oxford University Department of Computer Science, the Monash University Electrochemistry Group [38,39] established a Bayesian framework for quantifying the electrode kinetics of the [Fe(CN) 6 ] 3-/4electron transfer reaction at a GC electrode. Bayesian inference allowed a rigorous assessment of the 'noise', and hence, uncertainty of parameter estimates in a single experiment to be undertaken.…”

“…Additionally, the superiority of AC voltammetric methods over DC ones could be confirmed quantitatively [38]. Analysis of ten replicate AC voltammetric experiment-theory comparisons at a GC electrode was undertaken also to establish the system variability, with parameter estimates and confidence limits of E 0 f , k 0 α, R u and C dl provided [39]. In a Bayesian framework, each experimental data set was used to 'update' the confidence level for reporting parameter values present in the model.…”

“…The [Fe(CN) 6 ] 3-/4is known to be pathologically variable from laboratory to laboratory and even from experiment to experiment in the same laboratory. The Bayesian approach allowed it to be established [39] that while the k 0 electrode kinetic parameter value estimated in each experiment does vary significantly, E 0 f is robustly constant and the fit of the AC voltammetric data to that predicted by the Butler-Volmer relationship always remains excellent. This implies that averaging of the electrode kinetics at microscopically variable regimes of the heterogeneous GC surface does indeed occur on the timescale used via diffusional overlap, to achieve a good approximation of a uniform surface as assumed in the modelling.…”

Liquefied Sunshine: Transforming Renewables into Fertilizers and Energy Carriers with Electromaterials

“…However, the majority of the experiment‐simulation comparisons are undertaken heuristically, i.e., relying on an unavoidably biased experimentalist's decision on what “good” and “bad” levels of agreement are. A fundamental step forward in the field that is desperately needed is the broad implementation of automated fitting methods and extended statistic protocols, which have been used in other fields for many years, to provide a quantitative measure of the validity of a particular electrocatalytic model in relation to the system of interest …”

A Comparison of Bayesian Inference Strategies for Parameterisation of Large Amplitude AC Voltammetry Derived from Total Current and Fourier Transformed Versions

“…[8,[12][13][14] Of particular relevance to the present study, Bayesian inference methods using the Markov Chain Monte Carlo (MCMC) method have been introduced for parameter quantification with uncertainty estimation in DC and AC voltammetry. [15,16] Due to the time series structure of the experimental data and the need to quantify parameters from comparison to simulated data, these and other forms of dynamic voltammetry are naturally suited for implementing data analysis in a Bayesian framework. The Bayesian approach provides a posterior probability distribution for a parameter derived from the experimental data in a format known as a sampled from the posterior distribution.…”