Bayesian data analysis reveals no preference for cardinal Tafel slopes in CO2 reduction electrocatalysis

The Tafel slope is a key parameter often quoted to characterize the efficacy of an electrochemical catalyst. In this paper, we develop a Bayesian data analysis approach to estimate the Tafel slope from experimentally-measured current-voltage data. Our approach obviates the human intervention required by current literature practice for Tafel estimation, and provides robust, distributional uncertainty estimates. Using synthetic data, we illustrate how data insufficiency can unknowingly influence current fitting approaches, and how our approach allays these concerns. We apply our approach to conduct a comprehensive re-analysis of data from the CO2 reduction literature. This analysis reveals no systematic preference for Tafel slopes to cluster around certain "cardinal values” (e.g. 60 or 120 mV/decade). We hypothesize several plausible physical explanations for this observation, and discuss the implications of our finding for mechanistic analysis in electrochemical kinetic investigations.

able in several programming languages, and the official YAML standard is documented at     Table 2: Provenance of all Tafel datasets analyzed in this study. The PaperID defines the unique identifier assigned to the paper in the zipped dataset.
PaperID Data Location Document Object Identifier (DOI) where CDF i (m T ) represents the CDF of the Tafel         As quoted in the main text, Bayes' rule reads, In the context of this work, y represents the measured current data at a set of voltage points. 129 We will use the subscript notation y k to denote a single current data point, where the index slope. We will denote the model's predictions at each voltage point by the subscript notation 134 M k (θ).

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To successfully apply Bayes' rule to glean p(θ|y), the posterior distribution over the 136 model parameters given measured data, we need to identify mathematical forms for the 137 prior distribution p(θ) and the likelihood function p(y|θ). In all fits conducted in this study, 138 we employ a uniform prior distribution (also known as an "uninformative" prior distribution) 139 over a certain parameter range. Since the prior is uniform in the selected parameter range, as With the optimal parameters θ * in hand, we select a uniform prior p(θ) that is supported in

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The likelihood function for the data given the parameters, p(y|θ), is determined by assum-153 ing that the experimental measurement represents a ground truth measurement described 154 by the model, polluted by unavoidable experimental error, We assume that errors at different data points are uncorrelated, and further assume that the 156 error i at any single data point is drawn from a Gaussian distribution with zero mean and 157 variance σ 2 , Because the errors at each point are uncorrelated, the likelihood now factorizes over all the 159 data points, With a likelihood function p(y|θ) and a prior distribution p(θ) in hand, we can plug this then it often also produces an estimate of the Hessian, evaluated at the optimal value of the parameters θ * . If we assume that the experimental where d is the number of parameters being estimated, and H is guaranteed to be positive 180 definite by virtue of being evaluated at the optimal point θ * .

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We stress that the expression provided by Eq. (11) is an approximation to the true pos-182 terior distribution; due to its Gaussian form, this expression can never accurately represent 183 bimodality in the posterior distribution. In this sense, the Bayesian sampling approach is 184 superior, although it comes at significant additional computational expense as d increases.

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In this work, d = 3, and this additional expense is essentially negligible given the compu- we use a value of a = 10, as mentioned in the Methods section. 220 We will work with a generic reaction scheme, assuming that we begin with a starting species A (n+q)+ which undergoes n electron transfers prior to the rate-determining step, and then q electron transfers at the RDS. In practice n will be an integer, and q will be either zero or one. Here we assume the reaction taking place is reductive; however, the derivation is entirely analogous for an equivalent oxidation reaction. Schematically, the reactions read, The overall current is determined by the rate of the RDS. Assuming Butler-Volmer kinetics 223 for the forward rate constant of the RDS, we have, where k 0 is the rate prefactor (sometimes called the Arrhenius prefactor), a i is the activity 225 of species i, β ≡ (k B T ) −1 is the inverse thermodynamic temperature, e is the fundamental 226 charge, α is the symmetry coefficient, φ is the applied potential, and φ eq. is the equilibrium 227 potential for the RDS. At sufficiently high reductive overpotentials φ − φ eq. 0, only the 228 first term survives, To make further progress, we have to solve for the activity of the intermediate species A q+ 230 in terms of the activity of the reactant species for the overall reaction, A (n+q)+ . If we assume 231 that all steps prior to the RDS are fast and equilibriated, we can extract this activity 232 by analyzing the thermodynamics of the steps prior to the RDS. The free energy change 233 associated with the r'th reaction reads, Then, the equilibrium constant for reaction r goes as, whereK r is defined by Eq. (17), and is independent of potential. Since Eq. (17) holds for all reactions r before the RDS, we can easily solve for the activity of the intermediate, Plugging back into Eq. (13), This is a mess, but we only care about the potential-dependent terms when extracting the 235 Tafel slope, which means we only have to consider the last factor on the RHS. Taking the 236 logarithm yields, 237 log [rate] = −βe (φ − φ eq. ) (n + q · (1 − α)) + C, where C is a constant independent of potential. For a reduction reaction, the Tafel slope is 238 defined as, Hence, we have,
Appropriate unit scalings yield,

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Tafel Slope = 60 mV/decade which reduces to the equation quoted in the main text when α = 1/2.
For the case (n, q) = (1, 0), the adsorption step is the RDS, and rather than contributing Here, we examine the consequences of fitting current-voltage data from a system exhibiting 268 multiple kinetic regimes to a model that only allows a single Tafel slope (as in Eq. (2) in the 269 main text) by analyzing synthetic data. The synthetic data is generated from the model,  Tafel regimes. When α = 1/2, the first regime has Tafel slope m T = 40 mV decade −1 for 276 E eq,1 < E < E eq,2 , and the second regime has a Tafel slope m T = 120 mV decade −1 for 277 E > E eq,2 , before topping out at the limiting current i lim .            Gold catalyst reactivity for CO2 electro-reduction: From nano particle to layer. Catal.