bayesian data analysis reveals no preference for cardinal
TRANSCRIPT
doi.org/10.26434/chemrxiv.12844715.v1
Bayesian Data Analysis Reveals No Preference for Cardinal Tafel Slopesin CO2 Reduction ElectrocatalysisAditya Limaye, Joy S. Zeng, Adam Willard, Karthish Manthiram
Submitted date: 21/08/2020 • Posted date: 24/08/2020Licence: CC BY-NC-ND 4.0Citation information: Limaye, Aditya; Zeng, Joy S.; Willard, Adam; Manthiram, Karthish (2020): Bayesian DataAnalysis Reveals No Preference for Cardinal Tafel Slopes in CO2 Reduction Electrocatalysis. ChemRxiv.Preprint. https://doi.org/10.26434/chemrxiv.12844715.v1
In this paper, we develop a Bayesian data analysis approach to estimate the Tafel slope fromexperimentally-measured current-voltage data. Our approach obviates the human intervention required bycurrent literature practice for Tafel estimation, and provides robust, distributional uncertainty estimates. Usingsynthetic data, we illustrate how data insufficiency can unknowingly influence current fitting approaches, andhow our approach allays these concerns. We apply our approach to conduct a comprehensive re-analysis ofdata from the CO2 reduction literature. This analysis reveals no systematic preference for Tafel slopes tocluster around certain "cardinal values" (e.g. 60 or 120 mV/decade). We hypothesize several plausiblephysical explanations for this observation, and discuss the implications of our finding for mechanistic analysisin electrochemical kinetic investigations.
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Bayesian data analysis reveals no preference for
cardinal Tafel slopes in CO2 reduction
electrocatalysis
Aditya M. Limaye,† Joy S. Zeng,† Adam P. Willard,∗,‡ and Karthish
Manthiram∗,†
†Department of Chemical Engineering, MIT, Cambridge, MA
‡Department of Chemistry, MIT, Cambridge, MA
E-mail: [email protected]; [email protected]
Abstract1
In this paper, we develop a Bayesian data analysis approach to estimate the Tafel2
slope from experimentally-measured current-voltage data. Our approach obviates the3
human intervention required by current literature practice for Tafel estimation, and4
provides robust, distributional uncertainty estimates. Using synthetic data, we illus-5
trate how data insufficiency can unknowingly influence current fitting approaches, and6
how our approach allays these concerns. We apply our approach to conduct a com-7
prehensive re-analysis of data from the CO2 reduction literature. This analysis reveals8
no systematic preference for Tafel slopes to cluster around certain “cardinal values”9
(e.g. 60 or 120 mV/decade). We hypothesize several plausible physical explanations10
for this observation, and discuss the implications of our finding for mechanistic analysis11
in electrochemical kinetic investigations.12
1
Introduction13
Modern tools in data science provide the capability to reduce or outright eliminate sources14
of human bias in the analysis and interpretation of experimental measurements. Despite15
the wide availability of these tools, many communities continue to rely on more primitive16
and bias-prone methods of data analysis. The calculation of Tafel slopes through linear17
least squares fitting is one prominent example. Here, we present a robust Bayesian approach18
to analyzing electrochemical current-voltage measurements that (1) eliminates the need to19
manually exclude points in limiting-current regimes, and (2) provides a well-defined measure20
of uncertainty in the fitting parameters (e.g. the Tafel slope). By applying this approach21
to a large set of literature data, we identify a systematic but unjustified tendency for the22
assignment of Tafel slopes to specific “cardinal” values. This finding highlights the role that23
modern data science can play in uncovering and eliminating hidden sources of bias that exist24
within various scientific communities.25
Current-voltage measurements are a fundamental characterization tool for electrochem-26
ical systems, as they report on the propensity for system response when pushed out of27
equilibrium by a thermodynamic driving force. In the context of electrochemical catalysis,28
current-voltage behavior is often summarized by the Tafel slope, a parameter that quantifies29
the amount of electrochemical driving force required to produce a logarithmic increase in the30
observed current.1 Nearly all studies that develop a novel electrochemical catalyst report a31
Tafel slope, and it is considered an important figure of merit when comparing catalysts. In32
principle, the Tafel slope contains information about the microscopic mechanism underlying33
the operation of a catalyst. The elementary kinetic steps of idealized reaction mechanisms34
imply a strong tendency for Tafel slopes to exhibit certain “cardinal values”,2,3 and these35
cardinal values are frequently referenced in the kinetic analysis of catalytic materials. How-36
ever, several notable studies have reported Tafel slopes that differ significantly from their37
predicted cardinal values.4,5 Because the kinetic steps associated with these cardinal values38
are ubiquitous, there is a tendency to interpret Tafel slopes with off-cardinal values as if39
2
they represent a truly cardinal value that has been altered by sources of experimental error,40
despite relatively incomplete error quantification.6–12 The flaw in this interpretive strategy41
is that it relies on the validity of idealizing assumptions about the underlying kinetic mech-42
anism, and does not account for the numerous ways in which deviations from ideality can43
influence the Tafel slope.1344
Despite the scientific and engineering relevance of the Tafel slope, current literature ap-45
proaches for estimating this parameter from measured current-voltage data require subjective46
human intervention, and are susceptible to numerous sources of systematic error. Subjective47
considerations in the fitting procedure (namely, the manual demarcation of a linear fit re-48
gion) make it impossible to determine, in a truly unbiased manner, the intrinsic distribution49
of Tafel slopes, and whether they cluster around cardinal values. Additionally, human inter-50
ventions in the fitting procedure enable researcher bias, both inadvertent and intentional, to51
influence a quantitative catalyst benchmark. Such biases are difficult to recognize without52
re-examining primary source data.53
To address these concerns, we advance an alternative Bayesian data analysis method54
that enables unbiased Tafel slope estimation. This method provides robust, distributional55
uncertainty quantification, elucidating the credible range of Tafel slope values consistent56
with the measured data. Because our method eliminates subjectivity in the fitting process,57
it enables us to fairly evaluate the prevalence of cardinal Tafel slopes within re-analyzed58
literature data.59
In this manuscript, we begin by describing common literature practices for assigning Tafel60
slopes from experimental current-voltage data. Subsequently, we develop the mathematical61
formalism behind our Bayesian approach to Tafel slope estimation, and discuss its associated62
benefits compared to existing approaches. Using synthetic data, we illustrate the benefits of63
our approach, and show how it can be combined with iterative data acquisition procedures64
to systematically reduce uncertainty in Tafel slope estimates. Finally, we apply our approach65
to a large set of CO2 reduction catalyst data from the literature, and compare our Tafel slope66
3
estimates to the reported values. We find that clustering of reported Tafel slopes around67
cardinal values is unjustified, and likely reflects systemic bias across the field. We conclude68
by hypothesizing several plausible sources of mechanistic nonideality and estimating their69
ability to modify Tafel slopes from their cardinal values.70
Tafel Slopes in Electrocatalysis71
Electrochemical systems operate by converting the energy stored in chemical bonds into72
electrical work (or vice-versa) by means of electron transport through an external circuit.73
The electron transport process originates at an electrochemical interface, where a portion74
of the external circuit (an electrode) is contacted with a chemical system (a reactive elec-75
trolyte solution) in the presence of a catalyst. Catalyzed interfacial electron transfer serves76
to inextricably link the electrical dynamics of the external circuit to the chemical dynamics77
of the electrolyte. Electrochemical characterization techniques exploit this linkage, using the78
voltage and current measured in the external circuit to report on the thermodynamic driving79
forces and nonequilibrium currents in the chemical system. The simplest possible electro-80
chemical experiment involves setting the applied potential at an electrode and measuring81
the resultant electrical current (or vice-versa), generating a current-voltage trace.82
Several phenomena can influence the shape of current-voltage traces. For example, the83
electronic properties of the electrode, the chemical identity of the catalyst material, the84
transport characteristics of reactive species in the electrolyte, and myriad other factors all85
play an important role in structuring current-voltage behavior.14 When characterizing the86
performance of a catalyst material, we are most interested in the kinetic control regime of a87
current-voltage trace, where the measured current reports directly on the intrinsic rate of a88
chemical reaction at the interface. Under kinetic control, the current is generally expected89
to follow an exponential asymptotic dependence on the overpotential η, which quantifies90
the difference between the applied electrode potential and the equilibrium potential for the91
4
chemical reaction.1 In the high |η| limit (specifically, |η| � kBT/e, the thermal voltage), the92
logarithm of the kinetic current, ikin, should depend linearly on the applied potential. The93
(inverse) slope of this relationship is termed the Tafel slope,94
Tafel Slope ≡ dη
d log10 ikin
∣∣∣∣|η| � kBT/e
, (1)
and is generally reported in units of mV/decade.95
The Tafel slope is an important parameter to judge the performance of a catalyst because96
it quantifies the amount of additional applied potential required to observe a logarithmic in-97
crease in the measured current. Hence, studies that develop a novel electrocatalytic material98
often measure current-voltage data in the kinetic control regime, and then use this data to99
estimate a Tafel slope for their catalytic system. Experimental limitations impose a number100
of practical constraints on the Tafel slope estimation procedure. First, for several electro-101
chemical reactions (CO2 reduction, N2 reduction, organic electrosynthesis, etc.), accurately102
determining a kinetic current for a specific reaction requires product quantification after a103
constant potential/current hold. Due to throughput limitations for quantification techniques,104
the Tafel slope often must be estimated from just a handful of data points (roughly, 3–10).105
Another important practical constraint arises from limiting current phenomena observed in106
many electrocatalytic systems. At sufficiently high overpotentials, the reaction rate exceeds107
the rate of another physical process required for the reaction to proceed. In this regime, the108
system is no longer under kinetic control, and the measured current plateaus to a limiting109
current, becoming independent of the applied voltage.15 Most commonly, limiting currents in110
CO2 reduction systems arise from diffusive transport limitations in the electrolyte, although111
several other physical reasons for plateau currents have been hypothesized and investigated112
in the literature.16–20 Consequently, experimentally measured Tafel data usually starts out113
linear, but curves sub-linearly at sufficiently high overpotentials.114
In the face of these practical limitations, studies in the literature use a relatively standard115
5
Figure 1: Schematic comparison of the traditional approach to Tafel fitting and the newapproach we describe in this manuscript. Starting from raw current-voltage data on the left,the current literature approach begins with manual identification of a linear Tafel region onthe plot. The rest of the data is discarded, and a linear fit to the Tafel region yields a Tafelslope, with an associated uncertainty corresponding to the standard error of the ordinaryleast squares (OLS) estimator. In addition to this quantified uncertainty, an additionalunquantified source of uncertainty arises from manual selection of a Tafel region on the plot.The new approach described here considers all of the data in the context of a nonlinear modelthat smoothly interpolates between the traditional Tafel region and a plateaued region (e.g.,due to mass transport limitations). Our approach uses a Monte Carlo method to sample fromthe Bayes posterior distribution over the parameters of the model, yielding a probabilisticdistribution over Tafel slopes that are consistent with the measured data.
6
protocol for estimating the Tafel slope, depicted schematically in the upper half of Fig. 1.116
First, a researcher must manually identify an ad hoc cutoff between a linear, kinetically-117
controlled regime (the Tafel regime) and a limiting current (plateau) regime.21 All data points118
in the plateau regime are subsequently discarded, and a Tafel slope is fitted by ordinary least119
squares (OLS) linear regression to the data in the Tafel regime. The OLS procedure offers120
a prescription for extracting the standard error of the Tafel slope; this standard error is121
sometimes used to construct a confidence interval for the Tafel slope estimate.22,23122
We believe the current standard literature practice bears several drawbacks. First and123
foremost, manual identification of a cutoff between the kinetic and limiting-current regimes124
introduces subjectivity into Tafel slope estimation, potentially incorporating undesirable hu-125
man influence into the quantification of an important metric for electrocatalyst performance.126
Second, reporting the error associated with a linear fit to a manually selected set of data127
points systematically under-estimates the actual error associated with estimating a Tafel128
slope from a small number of data points. The OLS slope standard error quantifies the129
uncertainty associated with the linear fit to a given set of data, but there is an additional130
unquantified error associated with selection of the linear regime. Third and finally, ad hoc se-131
lection of a regime cutoff can introduce a systematic bias in the Tafel slope, since the final few132
points of the kinetic regime will suffer at least some effects from limiting current curvature,133
causing the current at these points to deviate slightly from the true kinetic current.134
Bayesian Data Analysis Algorithm135
Our approach for Tafel slope estimation seeks to obviate manual demarcation between the136
linear and plateau regions in current voltage data. To this end, we choose to fit all current-137
voltage data measured in a Tafel experiment to a phenomenological model that smoothly138
7
interpolates between the kinetic control and plateau regimes. The model reads,139
1
i(η)=
1
ilim+
1
i0 exp[m−1T · |η|
] , (2)
where i(η) is the measured current density as a function of overpotential. The unknown140
parameters in the model are ilim, the limiting current density, i0, the exchange current141
density, and m−1T , the inverse Tafel slope. The mathematical structure of Eq. (2) can be142
shown to arise, for example, when the surface concentration of a redox-active species changes143
with applied overpotential due to diffusive transport effects. Alternatively, Eq. (2) could be144
motivated by interpreting the presence of limiting-current phenomena as an additional series145
resistance to current in the equivalent circuit for an electrochemical cell.1 Most generally,146
this model can describe any physical phenomenon that imposes a “speed limit” on the passed147
current.24148
Typically, the parameters in a model like Eq. (2) would be adjusted to achieve optimal149
agreement between the model and experimental data. Despite the nonlinearity inherent to150
the model, numerical optimization schemes for determining the optimal set of parameters are151
mature and well-studied. Here, we employ a different approach based on Bayesian sampling152
that can quantify not only an optimal set of parameters, but also a distribution over the likely153
values of the model parameters given the available experimental data.25 Before carrying out154
any current-voltage measurements, we generally have at least some idea of the reasonable155
values of parameters in Eq. (2). For example, one would be very leery of a Tafel slope156
mT /∈ [101, 103] mV/decade, and one can use tabulated values of a species diffusion coefficient157
and an estimate of the cell boundary layer thickness to compute a ballpark estimate of a158
limiting current, ilim, arising from diffusive transport limitations.1,24 For a general set of159
parameters θ, this knowledge is encoded in a prior distribution over the parameters p(θ).160
Upon observing some data y, we can compute an updated posterior distribution p(θ|y) (the161
8
probability of the parameters given the observed data) using Bayes’ rule,26162
p(θ|y) =p(y|θ)× p(θ)
p(y). (3)
The likelihood p(y|θ) is supplied by a model like Eq. (2), in conjunction with an assumption163
about the distribution over the error at each data point. Here, we assume the error at each164
data point is independently normally distributed with a standard deviation σ = 0.10 log165
units, though our approach is flexible in this respect (see SI for sensitivity analysis with166
respect to the σ parameter).167
Despite the fact that the partition function p(y) is an unknown constant, Eq. (3) yields a168
prescription for sampling from the posterior distribution over parameters p(θ|y). Briefly, one169
can use a Markov chain Monte Carlo sampling scheme27 to glean several parameter samples170
from the posterior distribution, which are expected to concentrate in density around the171
optimal values of the parameters (additional detail on implementation can be found in the172
SI). As depicted in the lower panel of Fig. 1, this Bayesian posterior sampling technique173
reveals both the optimal values of the model parameters and a distributional uncertainty174
estimate over their values, accounting for all sources of uncertainty in the model. If we truly175
want to collapse this distributional information to a single value of the parameter, say for176
quoting a Tafel slope associated with a catalyst, we can compute,177
〈θ〉 ≡∫
dθ · θ · p(θ|y), (4)
where 〈θ〉 is termed the mean a posteriori (MAP) parameter estimate. When the poste-178
rior distribution p(θ|y) is strongly peaked around an optimal set of parameters, the MAP179
estimate will line up with the parameters gleaned from a nonlinear optimization technique.180
When the posterior distribution is broad or multimodal, the MAP estimate and the opti-181
mal parameter values may differ, signaling a high degree of uncertainty associated with the182
optimal parameter estimate.183
9
The Bayesian posterior sampling approach offers several key advantages over the tradi-184
tional approach to Tafel slope estimation. First, it removes subjectivity from the analysis of185
Tafel data: users of our algorithm need only select a model such as Eq. (2) to interpret the186
observed data, which can be justified on the basis of rigorous physical arguments, unlike a187
subjective delineation between linear and plateau regimes. Second, our approach yields ac-188
curate quantification the uncertainty associated with a Tafel slope estimate. Specifically, we189
believe the distributional uncertainty quantification afforded by our algorithm will be useful190
when assessing and discriminating between disparate sets of experimental data. Finally, be-191
cause the model in Eq. (2) analytically extrapolates away curvature-related attenuation of192
the kinetic current, our approach is free of the systematic bias present in current literature193
practice.194
As a caveat, we stress that while the model in Eq. (2) is appropriate for fitting some195
current-voltage data in the CO2 reduction literature, experimental and theoretical stud-196
ies have noted the possibility of multiple kinetic control regimes with distinct Tafel slopes197
before a limiting current plateau regime.3,28,29 This phenomenon can arise due to potential-198
dependent surface coverage effects or a potential-dependent switch in the microscopic reac-199
tion mechanism. When multiple distinct Tafel regimes are present, the experimental data200
must be interpreted under a model that allows for this possibility; once a suitable model is se-201
lected, the Bayesian posterior sampling approach can still be employed (see SI for additional202
discussion on model flexibility).203
Identifying and Addressing Data Insufficiency204
As mentioned previously, practical throughput considerations imposed by product quantifi-205
cation and other experimental requirements limit the amount of data generally used in a206
Tafel analysis to 3–10 points. In a survey of Tafel data reported in CO2 reduction studies207
in the literature, we found that a significant number of papers conduct Tafel analyses on a208
10
set of 3–5 data points within a narrow overpotential window. With such few data points,209
trends often appear linear, and so many studies simply extract the Tafel slope from a linear210
fit to all the data. While seemingly benign, estimating a Tafel slope without accounting211
for the possibility of limiting-current nonlinearity can lead to systematic error arising from212
data insufficiency. This phenomenon is elegantly identified and addressed by our Bayesian213
posterior sampling approach. For the sake of clarity, we illustrate how this systematic error214
can emerge by analyzing a set of synthetic data. The parable narrated by this data is easily215
relatable to a specific set of experimental data, and it emphasizes another distinct advantage216
of the Bayesian posterior sampling approach.217
B
50 75 100 125 150 175 200
Tafel Slope [mV / decade]
0.00
0.01
0.02
0.03
0.04
0.05p(T
afe
l Slo
pe)
Experiment 1
Experiment 1 + 2A
Experiment 1 + 2B
Model I Tafel Slope
Model II Tafel Slope
Linear Fit Tafel Slope
A
log
10
[i pro
d.]
0.05 0.10 0.15 0.20 0.25 0.30 0.35
Overpotential, [V]
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8 Model I
Model II
Experiment 1
Experiment 2A
Experiment 2B
0.06 0.08 0.10 0.12 0.14
2.2
2.0
1.8
1.6
1.4
130 ± 10 mV dec–1
Figure 2: Bimodal posterior distributions signify Tafel slope ambiguity that cannot be clari-fied by the available data. (A, inset) Synthetic current-voltage data collected in hypotheticalExperiment 1 appears linear over a 100 mV overpotential region, with a Tafel slope of 130±10mV/decade. (A) Synthetic current-voltage data over a broader range of overpotentials. Thedashed lines show two models (I, II) with different Tafel slopes and plateau currents thatcould both reasonably fit the Experiment 1 data. Synthetic error bars represent one standarderror of the mean of uncertainty in the synthetic data. Experiment 2A (blue triangles) and2B (red squares) represent two possible outcomes of experiments that probe a broader rangeof overpotentials, which can clearly distinguish between Model I and Model II. (B) Bayesposterior distributions over the Tafel slope determined by our algorithm given various sets ofobserved data. If the algorithm is fed just the Experiment 1 data, the posterior distributionover the Tafel slope is broad and weakly bimodal, indicating that the Experiment 1 datais insufficient to discriminate between Model I and Model II. When fit to the Experiment2A or 2B data in addition to the Experiment 1 data, this bimodality splits cleanly into twoseparate modes centered at the Model I and Model II Tafel slopes. Note that the linear fitto just the Experiment 1 data is distinct from both the Model I and Model II Tafel slopes.
11
The inset of Fig. 2A depicts a set of (synthetic) Tafel data measured over a 100 mV218
overpotential window. To the eye, the data looks entirely linear, and fitting a Tafel slope219
to the entire dataset using OLS regression yields a Tafel slope of 130 mV/decade, with a220
standard error of 10 mV/decade. However, Fig. 2A illustrates the issue with this traditional221
Tafel analysis. Indeed, the original set of green data in the narrow overpotential regime is222
essentially entirely consistent, within experimental error, with two models possessing very223
different Tafel slopes. Model I has a Tafel slope of 80 mV/decade, while Model II has a Tafel224
slope of 120 mV/decade. Despite this wide berth in Tafel slope, the models align in the225
initial overpotential window because of their distinct limiting currents ilim, which differ by226
a modest half order of magnitude. Clearly, the set of data measured in the inset of Fig. 2A227
is insufficient to distinguish between the two models, but this data insufficiency is entirely228
hidden by the traditional analysis approach.229
Unlike the traditional analysis approach, Bayesian posterior sampling correctly identifies230
the Tafel slope ambiguity present in the original data. The green trace in Fig. 2B depicts231
the posterior distribution over the Tafel slope using solely the green data. This distribution232
is broad and markedly multimodal, with concentrations of probability density around the233
Model I and II Tafel slopes. In this manner, Bayesian posterior sampling correctly surmises234
that the original data is insufficient to pin down a value of the Tafel slope with high cer-235
tainty. One possible solution to this issue is to measure additional data over a wider range236
of overpotentials. Depending on the true parameters of the electrochemical system under237
measurement, this experiment could yield either the blue data or the red data in Fig. 2A.238
When the Bayes posterior sampling algorithm is fed a combination of the green data and the239
red data, it correctly predicts a posterior distribution of Tafel slopes concentrated around240
the Model I Tafel slope. Conversely, when fed a combination of the green data and the blue241
data, it correctly predicts a posterior distribution of Tafel slopes concentrated around the242
Model II Tafel slope. In other words, multimodality in the posterior distribution predicted243
by our algorithm is a hallmark of data insufficiency; when the underlying insufficiency is244
12
addressed, the algorithm neatly splits the distributional modes according to the observed245
data.246
We highlight a couple of important conclusions from the synthetic data analysis presented247
in Fig. 2. First, current-voltage data used for Tafel slope estimation should ideally be mea-248
sured until clear curvature is observed. If such a measurement is unnecessarily inconvenient249
or impossible, one should attempt to quantify the limiting current either through back-of-250
the-envelope estimates or through direct experimental control over the limiting current (e.g.251
with a rotating-disk electrode), and ensure that data used to estimate Tafel slopes is collected252
well below the limiting current density. Without information that elucidates the magnitude253
of the limiting current, it is impossible to ascertain the degree of limiting current-induced254
attenuation suffered by the current measured in the Tafel regime. Consequently, Tafel slopes255
estimated on the basis of a linear Tafel plot measured in a small overpotential window are256
likely systematically unreliable, and can harbor significant unquantified uncertainty. Second,257
the synthetic data analysis illustrates how the Bayesian posterior sampling approach can be258
employed iteratively with data acquisition efforts. Since the posterior distributions accu-259
rately quantify the uncertainty associated with a Tafel slope estimated given available data,260
an experimentalist can use this uncertainty information to guide future data acquisition until261
a desirable uncertainty threshold is achieved.262
Evaluating Cardinal Preferences in Literature Data263
In addition to being an important metric in assessing catalyst performance, the Tafel slope264
can be valuable because it may yield insight into the mechanism of a catalyzed electro-265
chemical reaction. The connection between the Tafel slope, a macroscopically-measurable266
quantity, and the microscopic reaction mechanism is derived using microkinetic analysis in-267
voking a whole host of ideality assumptions.2,3 For an electrochemical reaction that proceeds268
through a number of elementary steps, one must assume that a single step determines the269
13
rate, and that all steps prior to the rate-determining step (RDS) are in quasi-equilibrium.270
Each of the quasi-equilibrated elementary steps carries an associated equilibrium constant271
which is possibly dependent on the applied potential. For potential-dependent equilibrium272
constants, one must additionally assume that the potential dependence goes exponentially in273
the (strictly integer) number of electrons transferred in the elementary step. The RDS has274
an associated forward rate constant, which is assumed to have a Butler-Volmer-like depen-275
dence on the applied potential, with a symmetry coefficient α = 1/2.1 Under these restrictive276
assumptions, one can derive (see SI) an equation for the Tafel slope of the entire chemical277
reaction (at T = 298 K),278
Tafel Slope =60 mV/decade
n+ q/2, (5)
where n is the total number of electrons transferred in elementary steps prior to the RDS,279
and q is the number of electrons transferred in the RDS.280
Equation (5) gives rise to so-called “cardinal values” of the Tafel slope, which arise from281
evaluating the Tafel slope for different values of (n, q). Tafel slope values of 120, 60, and282
40 mV/decade are familiar to most electrochemists, and arise from (n, q) = (0, 1), (1, 0)283
and (1, 1), respectively. Researchers routinely appeal to cardinal values to extract micro-284
scopic insight from experimentally-measured Tafel slopes. A common argumentative thrust285
goes: “my catalyst has a Tafel slope of 110 mV/decade, which is reasonably close to 120286
mV/decade, indicating that the reaction proceeds through a rate-limiting first electron trans-287
fer step.” This line of reasoning is only truly valid if the typical catalyst satisfies the ideality288
assumptions involved in deriving Eq. (5), a point that has been emphasized several times in289
the literature.3–5 Comprehensive analysis of literature Tafel data can shed light on whether a290
typical catalyst satisfies these strict assumptions; if this is indeed true, then literature Tafel291
slopes should tend to cluster around the cardinal Tafel values predicted by Eq. (5).292
Our Bayesian posterior sampling algorithm for Tafel slope fitting allows us to carry out293
an unbiased, automated survey of literature Tafel data to quantitatively test whether Tafel294
slope values reported in the literature show any preference for cardinal values. In this study,295
14
we choose to focus on re-analyzing Tafel data from the CO2 reduction literature. We fo-296
cus on this subsection of the literature because CO2 reduction is a burgeoning field with297
diverse catalyst materials and morphologies,30 and because product quantification require-298
ments place Tafel analysis in this field in the low-data regime, as discussed previously. To299
carry out the literature survey, we digitized 344 distinct Tafel datasets from the CO2 reduc-300
tion literature and fed the resultant data to the Bayesian posterior sampling algorithm to301
produce re-analyzed estimates of the Tafel slope. Further information on the data mining302
and analysis procedure can be found in the Methods section.303
Our re-analysis procedure uses Eq. (2) to interpret the literature Tafel datasets. As304
mentioned earlier, this model is only truly appropriate in the case of one kinetic control305
regime associated with a single Tafel slope; it cannot accurately capture current-voltage306
behavior under multiple kinetic regimes, which may be operative in at least some of the307
datasets we have analyzed. However, absent independent experimental confirmation of the308
physical mechanism underlying the observed multiple kinetic regimes (e.g. from spectroscopy309
or surface imaging), it is difficult to rigorously select a single model from the plethora that310
arise from enumerating microkinetic possibilities for intermediates in CO2 reduction. Since311
the papers from the literature that we have re-analyzed fit a single Tafel slope to their data,312
and because they lack the experimental evidence required to pin down a richer physical313
model describing their data, we believe that uniform application of the model in Eq. (2)314
is an appropriate choice for our literature survey study. Our usage of Eq. (2) to interpret315
the data should not be construed as a blanket endorsement of this model in Tafel analysis.316
Indeed, if there is solid experimental evidence motivating the usage of a different kinetic317
model for a specific CO2 reduction catalyst system, it can and should be employed under318
our Bayesian framework.319
Figure 3A depicts a correlation plot of the MAP Tafel slope estimated by the Bayes320
posterior sampling approach versus the literature-reported Tafel slope. A significant fraction321
of the datasets fall within the 20% parity line, a strong sign that our algorithm produces Tafel322
15
PD
F(Tafe
l S
lop
e)
25 50 75 100 125 150 175 200
Tafel Slope [mV / decade]
0.0
0.2
0.4
0.6
0.8
1.0
CD
F(Tafe
l Slo
pe)
Reported
Fitted
Cardinal Values
40
mV
dec–1
60
mV
dec–1
12
0 m
V d
ec–1
0 50 100 150 200 250
Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
0.012 ReportedFittedCardinal Values
60
mV
dec–1
12
0 m
V d
ec–1
40
mV
dec–1A B C
40 60 80 100 120 140 160 180 200
Reported Tafel Slope [mV / decade]
40
60
80
100
120
140
160
180
200
MA
P T
afe
l Slo
pe [
mV
/ d
eca
de]
Figure 3: Unbiased refit of literature data using our Bayesian analysis approach revealslittle preference for cardinal values of the Tafel slope for CO2 reduction catalysts. (A)Correlation plot of reported Tafel slopes from the literature against MAP Tafel slopes fittedby our algorithm on identical data. The solid red line represents perfect agreement, whilethe red filled intervals are lines representing 10% and 20% relative error. (B) Cumulativedistribution function of the Tafel slopes reported in literature data (blue), and those refittedby our algorithm (red). Error intervals correspond to one standard deviation of bootstrappedresamples. (C) Kernel density estimates (KDE) of the empirical probability distributionfunction of Tafel slopes reported in literature data (blue) and MAP Tafel slopes refitted byour algorithm (red). Error intervals correspond to one standard deviation of bootstrappedresamples. Green dashed lines in both (B, C) correspond to cardinal values of the Tafel slopepredicted by Eq. (5).
slopes that are consistent with literature values when seeing identical data. Additionally,323
the MAP Tafel slope does not seem to systematically overestimate or underestimate the324
literature-reported value over a wide range of reported Tafel slopes. We note that complete325
parity between the MAP and literature Tafel slopes should not be expected; as explained326
previously, due to the possibility for subjectivity and systematic error with current literature327
practice for Tafel estimation, the MAP estimates derived by our algorithm are arguably more328
trustworthy than the literature-reported values.329
Figures 3B and 3C depict estimates of the distributional tendencies of the MAP and330
literature-reported Tafel slopes. Figure 3B plots the empirical cumulative distribution func-331
tion (CDF) of the MAP Tafel slope (red trace) and the literature-reported Tafel slopes (blue332
trace). The low-opacity intervals in both Fig. 3B and Fig. 3C span one standard deviation333
of several bootstrapped resamples drawn with replacement (see Methods section for addi-334
tional detail on the bootstrapping procedure), and are useful for examining the sensitivity335
of our distributional results to the specific subsampling of literature data we have chosen to336
16
analyze.31 The CDF value for a given Tafel slope value mT tallies the running fraction of337
datasets that have a Tafel slope value of at most mT. If Tafel slopes truly cluster around338
cardinal values, the running fraction should increase sharply around those preferred values,339
and one would expect to see sigmoidal features in the CDF at the cardinal values. Figure 3340
shows little evidence of such locally sigmoidal behavior; rather, we see something resembling341
a straight line, corresponding to a roughly uniform distribution over the range of Tafel slopes342
considered.343
We can visualize the distributional data in a different way by examining the empirical344
probability distribution function (PDF) of the Tafel slopes. Estimating the PDF of a distri-345
bution given a set of samples is a notoriously difficult problem in statistics, because relatively346
small amounts of sampling noise can result in the presence of spurious peaks in the PDF.347
Several techniques exist for tackling this problem; here, we employ Gaussian kernel density348
estimation, which constructs an estimate of the PDF by summing appropriately-normalized349
Gaussian kernel functions centered at each of the observed data points. Additional details350
on the kernel density estimation procedure are reported in the Methods section. Figure 3C351
shows a kernel density estimate of the PDF of the MAP (red trace) and literature-reported352
(blue trace) Tafel slopes. Based on the shape of the PDFs and their corresponding stan-353
dard errors, we conclude that there is a slight preference to assign Tafel slope values around354
70 and 125 mV/decade in the literature that essentially disappears when the same data is355
re-analyzed with our approach. While a small peak persists around 125 mV/decade in the356
MAP Tafel slopes, the height of the peak is roughly within the error bound. We also note357
that any residual preference for cardinal values in the MAP Tafel slopes can possibly be358
explained by data acquisition biases. While the Bayesian approach we develop removes sub-359
jectivity from data analysis, we cannot remove biases introduced during data collection; it is360
at least plausible that such biases exist given that the literature-reported values significantly361
overestimate the concentration of Tafel slopes around 120 mV/decade compared to the MAP362
values.363
17
The results presented in Fig. 3 combine data from several studies using different catalyst364
materials to assess whether or not a preference for cardinal Tafel slopes exists broadly across365
all CO2 reduction catalysts. In order to confirm that this apparent lack of cardinal preference366
in the entire dataset is not simply an artifact introduced by pooling together separate cata-367
lyst materials which each individually exhibit a cardinal preference, we broke out the PDF368
analysis in Fig. 3 according to catalyst material identity. Upon examining the results from369
the breakout analysis (see SI), we conclude that the apparent lack of cardinality we find in370
Fig. 3 indeed persists when separately examining Tafel slopes from common materials used371
in CO2 reduction catalysts: Ag, Au, Cu, Sn, Zn. Curiously, it appears that Bi-based mate-372
rials do show a preference for Tafel slope values around 120 mV/decade, which may inform373
future mechanistic studies on these catalysts. As a caveat, we note our Bi results comprise374
only 27 distinct Tafel datasets, and are hence subject to a high degree of variability arising375
from the specific set of studies we chose to re-analyze; future work that attempts to weigh376
in on this question should ideally perform new experiments on well-controlled Bi surfaces377
and use the data acquisition and analysis recommendations identified in this work. Taken378
together, the results presented in Fig. 3 and the material breakout analysis in the SI lead us379
to conclude that, when analyzed in an unbiased fashion, experimental data in the literature380
does not support a systematic preference for cardinal values of the Tafel slope among CO2381
reduction catalysts.382
While we are not in a position to identify the cause for deviation from cardinal Tafel slope383
values in each specific Tafel dataset comprising Fig. 3, we advance the hypothesis that these384
deviations could originate from physical non-idealities that violate the assumptions involved385
in deriving Eq. (5). To test this hypothesis, we attempt to ascertain the manner in which386
a select few simple physical non-idealities can adjust the PDF of Tafel slopes that would387
otherwise be concentrated around cardinal values. In this regard, we consider three possible388
physical effects that violate the ideality assumptions used to arrive at Eq. (5). These three389
effects comprise a very small subset of the menagerie of physical non-idealities that could390
18
C
O O
A
0 50 100 150 200 250
Tafel Slope [mV / decade]
Ideal
Variable
Variable
Variable f
Variable ,
Variable , f
Variable , f
Variable , , f
PD
F(Tafe
l Slo
pe)
B
C
Figure 4: Physical hypotheses for the lack of observed cardinality in literature Tafel slopes.(A) Schematic of three selected physical non-idealities which can affect the measured Tafelslope. (B, blue trace) Synthetic kernel density estimate of the probability distribution overTafel slopes for a “random” CO2 reduction catalyst, peaked around the cardinal valuespredicted by Eq. (5). (B, other traces) Several synthetic kernel density estimates of theprobability distributions over the Tafel slope generated from including random values ofdifferent parameters governing physical non-idealities. (C) Schematic illustrating the possi-bility of measuring data across separate kinetic regimes in a Tafel analysis. Due to a switchin mechanism, different overpotential regimes exhibit different Tafel slopes, complicatinginterpretation of a single Tafel slope value fit straddling both regimes.
19
operate in CO2 reduction electrocatalysis; our goal is simply to show that these effects can391
spoil a preference for cardinal values of the Tafel slope, not to single out these particular392
effects as the only non-idealities present in CO2 reduction.393
First, we consider the possibility that the symmetry coefficient α 6= 1/2. Such deviations394
could arise, for example, due to disparate local slopes of the Marcus free energy surfaces at395
their crossing point, or reactant species position fluctuations in the electrochemical double396
layer.32–35 Second, we examine the effect of partial charge transfer or surface dipole formation397
in the adsorption of CO2 to the electrode surface, phenomena that have been hypothesized398
and characterized in prior studies on CO2 reduction.29,36 Mathematically, the formation of399
a surface dipole introduces a potential dependence in the adsorption equilibrium constant400
which mathematically resembles a partial charge transfer parameter γ. Third, we introduce401
a possible Frumkin correction f originating from the protrusion of an electrode-adsorbed402
species into the electrochemical double layer, which attenuates the applied potential due to403
electrostatic screening effects.1 The Frumkin correction is most important at low supporting404
salt concentration (and hence large resultant electrolyte screening length), but it has been405
considered in the context of CO2 reduction electrocatalysis.29 Each of these non-idealities406
depends on the value of a non-ideality parameter. Of course, we have no idea, at least a407
priori, about the distribution of values these non-ideality parameters can take in typical CO2408
reduction catalyst systems. In the absence of information, we make the maximally ignorant409
choice, and assume that the non-ideality parameters are drawn from uniform distributions410
within a reasonable set of bounds. Once we postulate these uniform distributions, we can411
examine how randomly selected non-ideality parameters deform a distribution of Tafel slopes412
that begins concentrated around cardinal values.413
The blue trace in Fig. 4B shows a distribution of Tafel slopes concentrated around the414
cardinal values predicted by Eq. (5). The relative heights of the cardinal value peaks are415
selected by artificially binning the distribution over MAP Tafel slopes into buckets centered416
around the cardinal values. The effects of randomly drawn physical non-idealities on this417
20
distribution can be examined using Monte Carlo simulation. Briefly, we sample a Tafel slope418
from the distribution depicted in the blue trace, sample the values of one or more non-ideality419
parameters, and finally calculate the resultant Tafel slope in the presence of non-idealities420
(see Methods section for additional details). Repeating the sampling procedure several times421
yields distributions over the Tafel slope, depicted as multicolored traces in Fig. 4 for different422
sets of non-idealities. Evidently, even rather mundane pieces of additional physics like the423
ones discussed above can produce stark changes in the distribution of Tafel slopes, spanning424
a range of behavior from moving certain peaks away from cardinal values to smearing out425
the entire distribution. While we cannot prove with certainty that physical non-idealities426
are responsible for the lack of observed Tafel cardinality in the literature, the results in Fig.427
4 demonstrate that this is at least a plausible explanation for the observed behavior.428
Alternatively, the observed lack of cardinality may also be a consequence of interpreting429
current-voltage data measured under several disparate kinetic regimes through the lens of430
Eq. (2), which cannot capture these intricacies. As illustrated schematically in Fig. 4C, the431
kinetic regimes may exhibit different Tafel slopes; in this case, mechanistic interpretation432
of a single Tafel slope extracted by fitting Eq. (2) to the data is inappropriate. Indeed, as433
examined in more detail in the SI, fitting synthetic data generated from a model with multiple434
cardinal Tafel slope regimes using Eq. (2) can produce an off-cardinal Tafel slope value. This435
underscores the need to rigorously characterize the several physical complexities present436
in catalytic systems for CO2 reduction, as they can complicate mechanistic interpretation437
guided solely by the Tafel slope.438
Taken in their entirety, the results presented in Figures 3 and 4 present some compelling439
reasons for the community to rethink the current approach of deducing mechanistic informa-440
tion purely from Tafel slope data. Indeed, the prevalence of Tafel slopes in the CO2 reduction441
literature that do not fall neatly on cardinal values suggests that physical non-idealities, omit-442
ted by Eq. (5), may be commonplace in typical catalytic systems. The familiar approach443
of “rounding” experimentally-measured Tafel slopes to their nearest cardinal value to guide444
21
mechanistic interpretation, then, leaves one prone to interpreting experimental data in an445
overly simplistic manner. Rather than hand-wave away the physical complexities present in446
catalytic systems with strong ideality assumptions, we believe it is important to interpret447
Tafel data alongside several other pieces of experimental data (e.g. more diverse electro-448
chemical kinetic data, surface-sensitive spectroscopy, materials characterization, etc.). In449
this manner, one can take a non-cardinal Tafel slope (and its associated uncertainty based450
on the data) at face value, and build a holistic physical picture that attempts to explain the451
deviation from cardinality in a manner consistent with all other experimental observations.452
Ideally, all these observations can be interpreted in the context of a richer model that allows453
one to determine the true breadth of physical phenomena present across a wide range of454
operating parameters, as has been done in some select studies in the literature.2,28,37 We455
believe our Bayesian data analysis approach will be equally useful for rigorously quantifying456
parametric uncertainties in the suggested new paradigm for kinetic data interpretation.457
Methods458
Data Mining and Re-Analysis459
We built up a dataset of Tafel measurements reported in the literature by manually extract-460
ing figures from published papers and digitizing them using the WebPlotDigitizer tool.38 A461
full accounting of all papers and corresponding figures can be found in the SI. When select-462
ing datasets to analyze, we excluded those that reported continuous current-voltage data,463
because it is difficult to ascertain the underlying data density associated with a continuous464
curve, because our method is meant to address the unique challenges of estimating Tafel465
slopes with a small amount of data, and because continuous current-voltage data may be466
unreliable because product selectivity is not always 100%, especially in CO2 reduction. We467
also excluded datasets that reported current-voltage data but did not report an explicit value468
of the Tafel slope. We assumed that all datasets were collected with appropriate experimen-469
22
tal techniques (IR-correction for solution resistance has already been applied, etc.), and did470
not modify or omit any data from a figure during the digitization process. After digitiza-471
tion, each dataset was tagged with manually entered metadata to facilitate re-analysis. A472
full accounting of the metadata fields, as well as a complete record of all scraped data and473
metadata, is available in the SI.474
Re-analysis of the data was carried out using the Python-based julius package, de-475
veloped in-house to handle data collation and Bayesian posterior sampling workflows. In476
order to determine prior distributions for the parameters, we first find an optimal set of477
parameters θ∗ for the limiting current model using an implementation of the trust-region re-478
flective (TRF) algorithm implemented in the optimization and root finding package included479
in SciPy.39 For each model parameter θi, we select uniform prior distributions supported on480
the interval [0, a× θ∗i ], with a = 10 (note that all parameters we fit in this study are strictly481
non-negative). For all models studied here, we find that the posterior distribution does not482
depend on the value of a, indicating that the data imposes strong preferences on the optimal483
model fit (see SI for detailed sensitivity analysis). For each dataset, we draw N = 4 × 104484
total samples (104 samples from four independent chains, each burning their first 2000 sam-485
ples) from the posterior distribution using the No-U-Turn Hamiltonian Monte Carlo sampler486
(NUTS) implemented in the PyMC3 probabilistic programming package.40487
In total, we re-analyzed 344 distinct Tafel datasets. Figure 3A restricts to both reported488
and MAP Tafel slopes mT ∈ [0, 200], which comprises 300 distinct Tafel datasets. A corre-489
lation plot including all analyzed Tafel datasets is reported in the SI.490
Kernel Density Estimation491
We use kernel density estimation (KDE) to estimate probability distributions given a finite492
set of samples. KDEs are used in the distributional visualizations in Figs. 2, 3, and 4. We493
use the Gaussian KDE function in the statistics package included in SciPy, and use Scott’s494
rule for bandwidth selection in Figs. 2 and 3. Since the estimates in Fig. 4 are meant to495
23
emulate the result of a single simulated experimental observation with some associated error,496
here we use a pre-specified bandwidth of 6 mV/decade.497
Bootstrap Resampling498
We carry out a bootstrap resampling procedure to quantify the degree of variability of the499
results in Fig. 3 associated with our choice of a specific subset of literature data.31 Essentially,500
we posit that the observed distribution over the Tafel slope is a good estimate of the true501
underlying distribution, and then resample several datasets of the same size as the original502
dataset from this distribution with replacement (i.e. samples can show up more than once, or503
not at all). The error intervals presented in Figs. 3A and 3B are gleaned from one standard504
deviation of 20 such bootstrapped resamples.505
Monte Carlo Simulation506
We use Monte Carlo simulation to estimate the distributional changes precipitated in a Tafel507
slope distribution by the physical non-idealities identified in the main text. To carry out508
this procedure, we begin with the distribution presented in Fig. 4A, which is generated509
by artificially bucketing the MAP Tafel slopes from the literature analysis into the bins510
{[0, 50), [50, 90), [90,∞)}. We sample the physical non-ideality parameters according to α ∼511
Unif[0.20, 0.80], γ ∼ Unif[0, 1], f ∼ Unif[0.50, 1.00], where Unif[a, b] signifies a uniform512
distribution supported on the interval [a, b]. For each set of non-idealities, we draw N =513
4 × 104 total posterior samples (104 samples from four independent chains, each burning514
their first 500 samples). The equations governing modifications to the Tafel slope based515
on physical nonidealities are worked out in the SI. A sensitivity analysis of the Tafel slope516
distributions including non-idealities with respect to the bounds of the uniform distributions517
over non-ideality parameters is also reported in the SI. All Monte Carlo simulation is again518
carried out using the PyMC3 probabilistic programming package.40519
24
Acknowledgement520
The authors thank Nathan Corbin, Zachary Schiffer, John Cherian, and Bertrand Neyhouse521
for useful discussions. AML acknowledges a graduate research fellowship from the National522
Science Foundation under Grant No. 1745302. AML and APW additionally acknowledge523
support from the Air Force Office of Scientific Research (AFOSR) under award number524
FA9550–18–1–0420. JSZ acknowledges MathWorks for a MathWorks fellowship.525
Author Contributions526
Conceptualization: AML, APW, and KM; Methodology: AML, JSZ, APW, and KM; Soft-527
ware: AML; Reproduction: JSZ and AML; Investigation: AML, APW, and KM; Writing528
(Original Draft): AML; Writing (Review and Editing): AML, JSZ, APW, and KM; Super-529
vision: APW and KM.530
Competing Interests531
The authors have no competing interests to declare.532
Data and Code Availability533
Code that supports the findings of this study is available under the MIT License (https://534
opensource.org/licenses/MIT) in Zenodo (http://doi.org/10.5281/zenodo.3995021).535
Data that supports the findings of this study is available under CC BY 4.0 (https://536
creativecommons.org/licenses/by/4.0/) in Zenodo (http://doi.org/10.5281/zenodo.537
3995021), with the exception of the excerpted figures from other articles as described in the538
Supporting Information. The excerpted figures are reused under agreement between MIT539
and the publishers of the articles (https://libraries.mit.edu/scholarly/publishing/540
25
using-published-figures/), where the copyright is owned by the publishers.541
26
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Supporting Information:
Bayesian data analysis reveals no preference for
cardinal Tafel slopes in CO2 reduction
electrocatalysis
Aditya M. Limaye,† Joy S. Zeng,† Adam P. Willard,∗,‡ and Karthish
Manthiram∗,†
†Department of Chemical Engineering, MIT, Cambridge, MA
‡Department of Chemistry, MIT, Cambridge, MA
E-mail: [email protected]; [email protected]
Literature Analysis1
Dataset Organization2
In this study, we manually digitized Tafel data included in several different papers from3
the electrochemical CO2 reduction literature. Often, a single paper may report multiple4
sets of Tafel data, possibly with many traces in the same figure. Each distinct paper that5
reports (possibly multiple) Tafel data(sets) receives a unique three-digit identifier, which will6
be referred to as PaperID. Note that PaperIDs are not guaranteed to be consecutive, and7
some numbers are skipped. Within each paper, each dataset receives a separate three-digit8
identifier (unique within the same PaperID), which will be referred to as SetID. A single9
Tafel dataset in our study, then, is uniquely identified by the tuple (PaperID, SetID).10
S-1
Our entire zipped dataset is organized into a hierarchical directory structure, with each11
directory named according to the PaperID identifier. Each individual paper directory con-12
tains several files. First, we include files with names of the form figure*.png, which are13
raw Portable Network Graphics (PNG) images of the paper figures, screenshotted manually14
from PDF copies of the papers. A few paper directories have multiple figure screenshots, and15
they are named according to figure1.png, figure2.png, etc. Directories also include raw16
data files with names of the form dat <SetID>.txt, which contain raw x, y data taken from17
manually digitizing the figure. Associated with each raw data file is a separate metadata18
file with a name of the form metadata <SetID>.txt. The metadata files contain key-value19
pairs stored in the YAML Ain’t Markup Language (YAML) format, and contain several20
pieces of information required for processing the raw x, y data. YAML parsers are avail-21
able in several programming languages, and the official YAML standard is documented at22
https://yaml.org/.23
Metadata Tags24
Tags in a metadata file are associated with YAML keys, and their associated values have25
several types. The types of the values, as well as the information they carry, are documented26
in Table S1. Note that for CO2 reduction catalysis, more negative applied electrochemical27
potentials correspond to more positive overpotentials. Studies in the literature report ei-28
ther the applied electrochemical potential or the overpotential on their voltage axis. The29
v reversed tag contains the information required to properly orient the voltage data in the30
direction of increasing overpotential for each dataset.31
S-2
Table S1: Table describing the types and meanings of the key-value pairs in the metadatafiles associated with each Tafel dataset.
Tag Name Value Type Value Meaningxdat Enum[‘current’, ‘voltage’] Indicates which data is reported on the x axis.
xunit String String describing units of measurement for the xaxis data, for example mV or mA/cm2.
xlog Bool Indicates whether the x axis data is reported on alogarithmic scale.
ydat Enum[‘current’, ‘voltage’] Indicates which data is reported on the y axis.
yunit String String describing units of measurement for the yaxis data, for example mV or mA/cm2.
ylog Bool Indicates whether the y axis data is reported on alogarithmic scale.
rval Float Reported Tafel slope value.
rerr Union[Float, None] Error in reported Tafel slope value, if reported. Ifnot reported, then None.
runit String String describing units of measurement for the xaxis data, for example mV/dec.
v reversed Bool If false, then the voltage data increases in valuewith increasing direction of the axis. If true, thenthe voltage data decreases in value with decreasingdirection of the axis.
i reversed Bool If false, then the current data increases in valuewith increasing direction of the axis. If true, thenthe current data decreases in value with decreasingdirection of the axis.
cat tags List[String] A list of tags describing the catalyst material. Forexample, a CuO catalyst would get the tags [‘Cu’,‘O’].
S-3
Analyzed Papers32
Table S2 documents the literature sources of all Tafel datasets analyzed in this study.33
Table S2: Provenance of all Tafel datasets analyzed in this study. The PaperID defines theunique identifier assigned to the paper in the zipped dataset.
PaperID Data Location Document Object Identifier (DOI)
000 Figure 5 10.1021/ja5065284S1
002 Figure 4 10.1021/ja309317uS2
004 Figure 4 10.1246/bcsj.68.1889S3
005 Figure 4 10.1002/anie.201604654S4
009 Figure 5B 10.1021/acsnano.5b01079S5
010 Figure 4B 10.1002/anie.201713003S6
011 Figure 1C 10.1021/jacs.7b09074S7
012 SI Figure 8 10.1002/anie.201900499S8
014 Figure 10 10.1002/smll.201701809S9
015 Figure 6D 10.1088/1361-6528/aa8f6fS10
016 Figure 7A 10.1021/jacs.5b02975S11
017 SI Figure 6 10.1002/smll.201602158S12
021 Figure 5C 10.1002/celc.201700517S13
022 Figure 3A, 3C 10.1002/cssc.201600202S14
023 Figure 4A 10.1021/ja501923gS15
024 Figure 4B 10.1021/acsenergylett.8b00472S16
025 SI Figure 12 10.1021/ja4113885S17
026 Figure 2D 10.1016/j.elecom.2016.05.003S18
027 Figure 3A, 3C 10.1021/ja2108799S19
028 Figure 7A 10.1021/acs.jpcc.7b01586S20
029 SI Figure 13 10.1021/acscatal.7b00707S21
031 Figure 7 10.1016/j.electacta.2016.03.182S22
S-4
032 Figure 8 10.1016/j.jcou.2017.05.024S23
033 Figure 2D 10.1002/cssc.201902859S24
035 Figure 4 10.1021/acsami.8b03461S25
036 SI Figure 8 10.1021/acsami.7b10421S26
039 Figure 3A 10.1002/asia.201800946S27
040 Figure 7 10.1002/cssc.201802409S28
042 SI Figure 4 10.1021/acsaem.8b00356S29
043 SI Fiigure 14 10.1021/acs.jpcc.8b06234S30
045 SI Figure 13 10.1021/acsaem.8b02048S31
048 Figure 6C 10.1002/celc.201801132S32
049 Figure 4C 10.1002/adma.201706194S33
050 Figure 3 10.1038/ncomms4242S34
051 Figure 4 10.1021/acscatal.5b01235S35
052 Figure 4E 10.1002/aenm.201701456S36
053 Figure 3F 10.1016/j.chempr.2017.08.002S37
054 Figure 4C 10.1002/anie.201608279S38
055 Figure 7A 10.1016/j.apcatb.2018.01.001S39
056 Figure 4 10.1021/ja3010978S40
057 Figure 2C 10.1016/j.apcatb.2018.09.025S41
058 Figure 5D 10.1021/jacs.6b10435S42
059 Figure 6 10.1021/jacs.6b12217S43
060 Figure 2F 10.1002/ange.201805696S44
061 Figure 6 10.20964/2017.03.72S45
062 Figure 3B 10.1002/cssc.201702229S46
063 Figure 3D 10.1002/aenm.201801536S47
064 Figure 4A 10.1016/j.jcis.2018.09.036S48
065 Figure 7 10.1016/j.nanoen.2018.03.023S49
S-5
066 Figure 3D 10.1021/jacs.7b12506S50
067 Figure 2D 10.1021/acscatal.5b00922S51
068 Figure 3F 10.1021/acsnano.7b03029S52
069 Figure 4D 10.1021/acscatal.7b03449S53
070 Figure 8 10.1016/j.apsusc.2016.10.017S54
071 Figure 3 10.1002/anie.201802055S55
072 Figure 6A 10.1016/j.electacta.2018.04.047S56
074 Figure 2G 10.1021/jacs.5b08212S57
076 Figure 3A 10.1002/anie.201809255S58
077 Figure 4A 10.1002/anie.201711255S59
078 Figure 3G 10.1016/j.nanoen.2018.09.053S60
080 Figure 2H 10.1002/anie.201800367S61
081 Figure 7 10.1016/j.cej.2016.02.084S62
083 Figure 2B 10.1016/j.apcatb.2017.06.032S63
084 Figure 2E 10.1016/j.jcat.2018.05.005S64
085 Figure 3E 10.1002/anie.201803873S65
086 Figure 3C 10.1002/anie.201806043S66
087 Figure 4D 10.1016/j.electacta.2018.09.080S67
088 Figure 5 10.1023/B:JACH.0000003866.85015.b6S68
089 Figure 6 10.1021/jp9822945S69
090 Figure 5C 10.1002/smll.201704049S70
091 Figure 5D 10.1016/j.nanoen.2016.11.004S71
092 Figure 5B 10.1002/anie.201701104S72
093 Figure 3A 10.1021/jacs.6b12103S73
094 Figure 4A 10.1002/anie.201903613S74
095 SI Figure 8C 10.1002/anie.201900499S8
096 Figure 2B 10.1002/adfm.201800499S75
S-6
097 SI Figure 6 10.1016/j.cattod.2015.05.017S76
098 Figure 4C 10.1021/acsami.7b16164S77
099 Figure 3 10.1002/smll.201703314S78
100 Figure 4A 10.1016/j.apcatb.2018.08.075S79
101 Figure 4D 10.1073/pnas.1711493114S80
102 Figure 5B 10.1002/cssc.201800925S81
103 Figure 1C 10.1021/acscatal.5b02424S82
104 Figure 3F 10.1021/acsenergylett.8b00519S83
105 Figure 4B 10.1002/anie.201612194S84
106 Figure 3C 10.1002/anie.201703720S85
108 SI Figure 4 10.1021/acs.nanolett.5b04123S86
109 Figure 3F 10.1002/anie.201901575S87
110 Figure 3D 10.1016/j.electacta.2018.12.116S88
111 Figure 5A 10.1021/acscatal.8b01022S89
112 Figure 3 10.1002/cssc.201701673S90
113 Figure 2D 10.1002/aenm.201900276S91
114 Fgiure 2D 10.1021/acsenergylett.8b01286S92
115 Figure 2D 10.1002/anie.201712221S93
116 Figure 2D 10.1002/anie.201807571S94
117 Figure 3A 10.1021/acsenergylett.7b01096S95
118 Figure 5 10.1021/acsaem.8b01692S96
119 Figure 6 10.1039/C5CP03559GS97
120 SI Figure 6 10.1126/science.aaw7515S98
121 Figure 10 10.1002/celc.201801036S99
122 Figure 5A 10.1002/adma.201705872S100
123 Figure 4C 10.1021/acscatal.8b04852S101
124 Figure 2F 10.1016/j.jcou.2019.05.026S102
S-7
125 Figure 3D 10.1002/ange.201810538S103
126 Figure 3C 10.1016/j.joule.2018.11.008S104
127 Figure 1D 10.1002/aenm.201700759S105
128 Figure 5B 10.1002/celc.201700935S106
129 Figure 4B 10.1016/j.apcatb.2019.03.047S107
130 SI Figure 5B 10.1002/anie.201911995S108
131 SI Figure 12 10.1002/aenm.201702524S109
132 Figure 4C 10.1002/ange.201805256S110
133 SI Figure 10 10.1002/aenm.201601103S111
134 Figure 2A 10.1038/s41467-018-07970-9S112
135 Figure 2B 10.1016/j.nanoen.2019.05.003S113
136 Figure 4 10.1039/C6TA04325AS114
137 Figure 2D 10.1002/celc.201800806S115
138 Figure 3B 10.1002/anie.201805871S116
139 Figure 4 10.1002/chem.201603359S117
140 Figure 3D 10.1002/aenm.201903068S118
141 Figure 3 10.1002/cssc.201501637S119
142 Figure 3F 10.1002/anie.201907399S120
143 Figure 3E 10.1016/j.electacta.2018.08.002S121
144 Figure 4A 10.1021/jp509967mS122
145 SI Figure 8 10.1021/acs.est.5b00066S123
146 SI Figure 10 10.1021/acscatal.6b00543S124
147 SI Figure 39 10.1016/j.joule.2019.05.010S125
148 Figure 3B 10.1016/j.jcou.2019.02.007S126
149 Figure 4D 10.1002/cssc.201903117S127
150 Figure 3C 10.1002/chem.201803615S128
151 Figure 5D 10.1002/celc.201800104S129
S-8
152 Figure 4A 10.1039/C9EE00018FS130
153 SI Figure 10 10.1016/j.joule.2018.10.015S131
154 Figure 5 10.1002/celc.201900725S132
155 Figure 4B 10.1038/ncomms14503S133
156 Figure 2D 10.1016/j.elecom.2018.10.014S134
157 Figure 2C 10.1002/aenm.201803151S135
158 Figure 4E 10.1016/j.elecom.2019.03.017S136
159 Figure 4B, 4C 10.1002/cssc.201802725S137
160 Figure 12 10.1016/j.matchemphys.2017.02.016S138
161 Figure 4 10.1002/adfm.201802339S139
162 Figure 4D 10.1016/j.nanoen.2016.06.043S140
163 Figure 4C 10.1038/ncomms12697S141
164 Figure 4F 10.1021/acssuschemeng.9b03502S142
165 Figure 5B 10.1039/C8TA03328ES143
166 Figure 6C 10.1039/C7TA03005CS144
167 Figure 3A 10.1002/anie.201908735S145
168 Figure 6D 10.1021/acsaem.9b02324S146
169 Figure 2C 10.1021/acs.jpcb.9b09730S147
170 Figure 3A 10.1073/pnas.1602984113S148
171 Figure 2 10.1021/acscatal.8b02181S149
Data Licensing34
Code that supports the findings of this study is available under the MIT License in Zenodo35
(http://doi.org/10.5281/zenodo.3995021). Data that supports the findings of this study36
is available under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) in Zen-37
odo (http://doi.org/10.5281/zenodo.3995021), with the exception of the excerpted fig-38
S-9
ures from other articles as described in the Supporting Information. The excerpted fig-39
ures are reused under agreement between MIT and the publishers of the articles (https:40
//libraries.mit.edu/scholarly/publishing/using-published-figures/), where the41
copyright is owned by the publishers.42
Residuals vs. Literature-Reported Values43
Residual analysis can help sniff out systematic or correlated errors in the results presented44
in Fig. 3A of the main text. We can define residuals between the literature-reported values45
and the MAP values in several different ways. Here, we will consider two definitions of the46
residuals, normalized either to the literature-reported Tafel slopes,47
Residual Normalized to Reported =Treported − TMAP
Treported, (1)
or to the MAP Tafel slopes,48
Residual Normalized to MAP =Treported − TMAP
TMAP
. (2)
Figure S1 depicts plots of the residuals versus the literature-reported and MAP Tafel slopes49
(A and C, respectively), as well as kernel density estimates of the distribution over the50
residuals normalized to the literature-reported and MAP Tafel slopes (B and D, respectively).51
To the eye, the residuals appear roughly unbiased around zero, and there appear to be no52
spurious correlations between the residuals.53
Correlation Plot over Full Range54
Figure S2 depicts a correlation plot of the MAP Tafel slope versus the literature-reported55
Tafel slope, including all datasets considered in this study.56
S-10
25 50 75 100 125 150 175 200Reported Tafel Slope [mV / decade]
1.25
1.00
0.75
0.50
0.25
0.00
0.25
0.50
Rela
tive R
esi
dual to
Report
ed
A
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00Relative Residual to Reported
0.0
0.5
1.0
1.5
2.0
2.5
3.0
KD
E(R
ela
tive R
esi
dual to
Report
ed)
B
40 60 80 100 120 140 160 180 200MAP Tafel Slope [mV / decade]
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rela
tive R
esi
dual to
MA
P
C
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00Relative Residual to Fitted
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
KD
E(R
ela
tive R
esi
dual to
Fit
ted)
D
Figure S1: Residual Analysis. (A) Plot of the relative residual to the literature-reported Tafelslope, defined in Eq. (1), versus the literature-reported Tafel slope, including only reportedTafel slopes less than 200 mV/decade. (B) Kernel density estimate of the distribution overthe relative residual to the literature-reported Tafel slope. (C) Plot of the relative residual tothe MAP Tafel slope, defined in Eq. (1), versus the MAP Tafel slope, including only reportedTafel slopes less than 200 mV/decade. (D) Kernel density estimate of the distribution overthe relative residual to the MAP Tafel slope.
S-11
0 100 200 300 400 500 600Reported Tafel Slope [mV / decade]
0
100
200
300
400
500
600
MAP
Taf
el S
lope
[mV
/ dec
ade]
Figure S2: Correlation plot of reported Tafel slopes from the literature against Tafel slopesfitted by our algorithm on identical data. The solid red line represents perfect agreement,while the red filled intervals are lines representing 10% and 20% relative error.
S-12
PDF Plot Using all Posterior Samples57
For every single dataset considered in the study, we draw N = 4 × 104 samples from the58
posterior distribution over the Tafel slope. In Fig. 3C in the main text, we depict a kernel59
density estimate of the distribution over the MAP Tafel slope from each of these posterior60
distributions. We choose to display this data because the MAP Tafel slope is a straightfor-61
ward point estimate of the Tafel slope given a posterior distribution over the parameter, and62
hence is likely the quantity one would quote if asked what the Tafel slope of a catalyst is.63
However, the averaging operation involved in computing the MAP Tafel slope can collapse64
broad features in the posterior distribution down to a single value; this is especially true65
in the case of bimodal posterior distributions arising from insufficient datasets. Figure S366
depicts a kernel density estimate of the distribution over the Tafel slope using all samples67
collected from the posterior distribution for each dataset. The essential conclusions reported68
in the main text are upheld when examining the data in Fig. S3. A very small preference69
for Tafel slopes around 45 mV/decade emerges in this analysis. However, given the small70
amount of total distributional mass under this peak and the high degree of sampling vari-71
ability (as evinced by the bootstrap standard deviations), we do not believe it should be72
interpreted strongly.73
Catalyst Breakout Results74
As described in the main text, we split out our results on the distributions over the Tafel75
slope by catalyst material identity in order to confirm that our conclusion of a lack of Tafel76
cardinality in CO2 reduction catalysis is not an artifact of pooling together data from several77
catalyst materials, each of which individually exhibit cardinality. Figures S4–S9 depict these78
results for catalysts containing Cu, Ag, Au, Sn, and Bi. Each figure has three panels: the79
left-most panel plots a kernel density estimate of the distribution over Tafel slopes using all80
samples from the posterior distribution, akin to Fig. S3. The center panel plots a kernel81
density estimate of the distribution over the MAP Tafel slope from each dataset, akin to main82
S-13
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
KD
E(T
afe
l Slo
pe)
Reported
Fitted
Cardinal Values6
0 m
V d
ec–1
12
0 m
V d
ec–1
40
mV
dec–1
Figure S3: Kernel density estimates (KDE) of the empirical probability distribution functionof Tafel slopes reported in literature data (blue) and those refitted by our algorithm (red).Error intervals correspond to one standard deviation of bootstrapped resamples. Greendashed lines correspond to cardinal values of the Tafel slope.
S-14
text Fig. 3C. The right-most panel plots a CDF of the distributional breadth Bi for each83
dataset considered. The distributional breadth is parametrized by the threshold parameter84
t, and is defined as,85
Bi(t) = CDF−1i (1− t)− CDF−1
i (t), (3)
where CDFi(mT) represents the CDF of the Tafel slope for the i’th dataset. Intuitively,86
a high distributional breadth implies that the experimental data measured in the dataset87
does not provide information to pin down the value of the Tafel slope with high confidence.88
Correspondingly, if the CDF of the distributional breadth increases quickly, then most of89
the datasets in the catalyst material subset predict tight distributions over the Tafel slope.90
Conversely, if the CDF of the distributional breadth climbs slowly, then several datasets in91
the catalyst material subset do not determine a Tafel slope value with high confidence.92
For some catalysts, the KDE constructed from all Tafel slope samples (left panel) and the93
KDE constructed from the MAP Tafel slope samples (center panel) show different behavior.94
The former plot looks more visually noisy than the latter plot; this is to be expected,95
since taking the mean of the posterior distribution over the parameters is a “smoothing”96
operation. The two ways of visualizing the data convey slightly distinct information, since the97
KDE constructed from all samples preserves the uncertainty information retained in a single98
fit, while the KDE constructed from the MAP Tafel slopes is the most direct comparison99
to the distribution of literature values (which do not carry an associated uncertainty, in100
most cases). In certain cases (Ag, Sn), the KDE comprising all samples appears to have101
more defined peaks than the KDE comprising MAP samples. First, we note that in these102
cases, the bootstrap standard errors for these peaks are much greater than in other areas103
of the distribution, suggesting that this peaking behavior is controlled by a small number104
of samples, and hence more variable with respect to a change in the specific datasets re-105
analyzed in this study. Second, peaking behavior in the KDE comprising all samples that106
does not appear in the KDE comprising MAP samples is a sign of some underlying data107
insufficiency issues highlighted by Fig. 2 in the main text; these peaks can easily disappear108
S-15
or be shifted upon measuring additional data. To declare confidently that a certain dataset109
espouses a cardinal Tafel slope preference, we contend that both ways of visualizing the110
distribution of Tafel slopes for a certain catalyst should show peaking around a cardinal value.111
This standard, while stringent, enforces that the available experimental data confidently112
determines a cardinal value of the Tafel slope, free of latent data insufficiency issues. While113
we observe this for the Bi breakout results in Fig. S9, the remainder of the catalyst breakout114
datasets do not meet this standard. Hence, though we do not foreclose the possibility115
that additional future data collection on these catalysts may reveal a cardinal Tafel slope116
preference, we argue that broadly, when considering the data extant in the literature, the117
typical CO2 reduction catalyst does not exhibit a strong preference for cardinal values of the118
Tafel slope.119
10 20 30 40 50 60 70 80Distributional Breadth
0.0
0.2
0.4
0.6
0.8
1.0
CD
F(D
istr
ibuti
onal B
readth
)
t= 0.10
t= 0.15
t= 0.20
0 50 100 150 200 250Tafel Slope [mV / decade]
0.0000
0.0025
0.0050
0.0075
0.0100
0.0125
0.0150
0.0175
KD
E(T
afe
l Slo
pe)
Reported
Fitted
Cardinal Values
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
KD
E(T
afe
l Slo
pe)
Reported
Fitted
Cardinal Values
Figure S4: Tafel Slope statistics for catalysts containing Ag (Ndatasets = 38). (Left) Kerneldensity estimates (KDE) of the empirical probability distribution function of Tafel slopesreported in literature data (blue) and those refitted by our algorithm (red). Error intervalscorrespond to one standard deviation of bootstrapped resamples. Green dashed lines corre-spond to cardinal values of the Tafel slope. (Center) Same as (Left), but KDEs are computedusing only the MAP Tafel Slope values for each dataset. (Right) CDF of the distributionalbreadth, as defined by Eq. (3).
S-16
10 20 30 40 50 60 70 80Distributional Breadth
0.0
0.2
0.4
0.6
0.8
1.0
CD
F(D
istr
ibuti
onal B
read
th)
t= 0.10
t= 0.15
t= 0.20
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
Figure S5: Tafel Slope statistics for catalysts containing Au (Ndatasets = 50). (Left) Kerneldensity estimates (KDE) of the empirical probability distribution function of Tafel slopesreported in literature data (blue) and those refitted by our algorithm (red). Error intervalscorrespond to one standard deviation of bootstrapped resamples. Green dashed lines corre-spond to cardinal values of the Tafel slope. (Center) Same as (Left), but KDEs are computedusing only the MAP Tafel Slope values for each dataset. (Right) CDF of the distributionalbreadth, as defined by Eq. (3).
10 20 30 40 50 60 70 80Distributional Breadth
0.0
0.2
0.4
0.6
0.8
1.0
CD
F(D
istr
ibuti
onal B
read
th)
t= 0.10
t= 0.15
t= 0.20
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
Figure S6: Tafel Slope statistics for catalysts containing Ag or Au (Ndatasets = 88). (Left)Kernel density estimates (KDE) of the empirical probability distribution function of Tafelslopes reported in literature data (blue) and those refitted by our algorithm (red). Errorintervals correspond to one standard deviation of bootstrapped resamples. Green dashedlines correspond to cardinal values of the Tafel slope. (Center) Same as (Left), but KDEsare computed using only the MAP Tafel Slope values for each dataset. (Right) CDF of thedistributional breadth, as defined by Eq. (3).
S-17
0 50 100 150 200Distributional Breadth
0.0
0.2
0.4
0.6
0.8
1.0
CD
F(D
istr
ibuti
onal B
read
th)
t= 0.10
t= 0.15
t= 0.20
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
Figure S7: Tafel Slope statistics for catalysts containing Cu (Ndatasets = 54). (Left) Kerneldensity estimates (KDE) of the empirical probability distribution function of Tafel slopesreported in literature data (blue) and those refitted by our algorithm (red). Error intervalscorrespond to one standard deviation of bootstrapped resamples. Green dashed lines corre-spond to cardinal values of the Tafel slope. (Center) Same as (Left), but KDEs are computedusing only the MAP Tafel Slope values for each dataset. (Right) CDF of the distributionalbreadth, as defined by Eq. (3).
0 25 50 75 100 125 150 175 200Distributional Breadth
0.0
0.2
0.4
0.6
0.8
1.0
CD
F(D
istr
ibuti
onal B
readth
)
t= 0.10
t= 0.15
t= 0.20
0 50 100 150 200 250Tafel Slope [mV / decade]
0.0000
0.0025
0.0050
0.0075
0.0100
0.0125
0.0150
0.0175
KD
E(T
afe
l Slo
pe)
Reported
Fitted
Cardinal Values
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
KD
E(T
afe
l Slo
pe)
Reported
Fitted
Cardinal Values
Figure S8: Tafel Slope statistics for catalysts containing Sn (Ndatasets = 37). (Left) Kerneldensity estimates (KDE) of the empirical probability distribution function of Tafel slopesreported in literature data (blue) and those refitted by our algorithm (red). Error intervalscorrespond to one standard deviation of bootstrapped resamples. Green dashed lines corre-spond to cardinal values of the Tafel slope. (Center) Same as (Left), but KDEs are computedusing only the MAP Tafel Slope values for each dataset. (Right) CDF of the distributionalbreadth, as defined by Eq. (3).
S-18
10 20 30 40 50 60Distributional Breadth
0.0
0.2
0.4
0.6
0.8
1.0
CD
F(D
istr
ibuti
onal B
read
th)
t= 0.10
t= 0.15
t= 0.20
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
0 50 100 150 200 250Tafel Slope [mV / decade]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
KD
E(T
afe
l S
lop
e)
Reported
Fitted
Cardinal Values
Figure S9: Tafel Slope statistics for catalysts containing Bi (Ndatasets = 27). (Left) Kerneldensity estimates (KDE) of the empirical probability distribution function of Tafel slopesreported in literature data (blue) and those refitted by our algorithm (red). Error intervalscorrespond to one standard deviation of bootstrapped resamples. Green dashed lines corre-spond to cardinal values of the Tafel slope. (Center) Same as (Left), but KDEs are computedusing only the MAP Tafel Slope values for each dataset. (Right) CDF of the distributionalbreadth, as defined by Eq. (3).
S-19
Limiting Current Statistics120
Figure S10 depicts kernel density estimates of the distribution of fitted limiting currents ilim121
from a re-analysis of literature data. Roughly, the distribution appears to be in agreement122
with the value of the transport-limited current density for CO2 reduction at an aqueous123
flooded electrode.S150
4 2 0 2 4 6
log10[ilim. ] [mA / cm2]
0.0
0.1
0.2
0.3
0.4
0.5
KD
E(l
og
10[i
lim.]
)
0 2 4 6 8
log10[ilim. ] [mA / cm2]
0.0
0.1
0.2
0.3
0.4
0.5
KD
E(l
og
10[i
lim.]
)
Figure S10: Limiting current statistics for all datasets analyzed in the study. (Left) Kerneldensity estimates (KDE) of the empirical probability distribution of the limiting current, ilimfitted when interpreting the data through the model in main text Eq. (2). (Right) Same as(Left), but KDEs are computed using only the MAP limiting current values for each dataset.
124
Bayesian Fitting125
Mathematical Detail126
As quoted in the main text, Bayes’ rule reads,127
p(θ|y) =p(y|θ)× p(θ)
p(y). (4)
In the context of this work, y represents the measured current data at a set of voltage points.128
We will use the subscript notation yk to denote a single current data point, where the index129
k = 1, . . . , Npts. The parameters of a model for interpreting current-voltage data are denoted130
S-20
by θ; in the context of this work, the relevant parameters for the limiting current model are131
ilim, the limiting current density, i0, the exchange current density, and m−1T , the inverse Tafel132
slope. We will denote the model’s predictions at each voltage point by the subscript notation133
Mk(θ).134
To successfully apply Bayes’ rule to glean p(θ|y), the posterior distribution over the135
model parameters given measured data, we need to identify mathematical forms for the136
prior distribution p(θ) and the likelihood function p(y|θ). In all fits conducted in this study,137
we employ a uniform prior distribution (also known as an “uninformative” prior distribution)138
over a certain parameter range. Since the prior is uniform in the selected parameter range, as139
long as the range includes the values of the parameters for which p(θ|y) has high probability140
mass, the choice of prior is unimportant (see further on for a numerical confirmation of this141
fact). Given this fact, the choice for the range of our uniform prior is determined by first142
using a standard nonlinear least-squares optimization algorithm (TRF) to determine a point143
estimate of the optimal set of parameters, θ∗. Formally,144
θ∗ = arg minθ
{Npts∑k=1
[yk −Mk(θ)]2
}. (5)
With the optimal parameters θ∗ in hand, we select a uniform prior p(θ) that is supported in145
the range [0, a× θ∗i ] for each parameter i = 1, . . . , Nparams. We choose the very conservative146
value a = 10 to ensure that the prior distribution has support over a very broad range147
around the optimal parameters. In principle, this choice of a results in a wide parameter148
space, which may affect the computational efficiency of a posterior sampling algorithm. In149
practice, we find very little computational disadvantage for choosing a = 10 as compared to150
a = 2 when using the No-U-Turn Sampler implemented in PyMC3.151
The likelihood function for the data given the parameters, p(y|θ), is determined by assum-152
ing that the experimental measurement represents a ground truth measurement described153
S-21
by the model, polluted by unavoidable experimental error,154
yk =Mk(θ) + εk. (6)
We assume that errors at different data points are uncorrelated, and further assume that the155
error εi at any single data point is drawn from a Gaussian distribution with zero mean and156
variance σ2,157
p(εk) =1√2πσ
exp
[− ε2k
2σ2
]. (7)
Because the errors at each point are uncorrelated, the likelihood now factorizes over all the158
data points,159
p(y|θ) =
Npts∏k=1
1√2πσ
exp
[−(yk −Mk(θ))
2
2σ2
]. (8)
With a likelihood function p(y|θ) and a prior distribution p(θ) in hand, we can plug this160
information into a Monte Carlo sampler of our choice to draw samples from the posterior161
distribution p(θ|y). Equation (8) also makes apparent how one can generalize the Bayesian162
posterior sampling approach to more general models M(θ). After we write down a suitable163
model, we simply evaluate the model predictions with different parameters whenever we164
need to compute the likelihood function for a given set of parameters during the sampling165
procedure.166
Gaussian Error Estimates from Nonlinear Optimization167
The Bayesian posterior sampling approach advanced in this work provides a way to glean168
distributional uncertainty information about the estimated values of model parameters given169
observed data. A similar set of information can also be obtained by analyzing the Hessian170
matrix determined by a nonlinear optimization algorithm seeking an optimal point estimate171
S-22
of the model parameters. Specifically, if the optimizer seeks to minimize a loss function,172
L(θ) =1
2σ2
Npts∑k=1
[yk −Mk(θ)]2 , (9)
then it often also produces an estimate of the Hessian,173
Hij =∂L
∂θi∂θj
∣∣∣∣θ∗, (10)
evaluated at the optimal value of the parameters θ∗. If we assume that the experimental174
data is generated by the random process described by Eq. (6), and further assume that the175
errors at different data points are uncorrelated and drawn from Gaussian distributions with176
mean zero and variance σ2, then we can form a Gaussian approximation to the posterior177
distribution around θ∗,178
p(θ|y) ≈ 1
(2π)d/2 (detH)1/2exp
[−1
2(θ − θ∗)TH (θ − θ∗)
], (11)
where d is the number of parameters being estimated, and H is guaranteed to be positive179
definite by virtue of being evaluated at the optimal point θ∗.180
We stress that the expression provided by Eq. (11) is an approximation to the true pos-181
terior distribution; due to its Gaussian form, this expression can never accurately represent182
bimodality in the posterior distribution. In this sense, the Bayesian sampling approach is183
superior, although it comes at significant additional computational expense as d increases.184
In this work, d = 3, and this additional expense is essentially negligible given the compu-185
tational power available on a typical laptop or desktop computer. Hence, we suggest that186
posterior distributions over the Tafel slope fitted using the model in Eq. (2) of the main187
text should always be computed using the Bayesian posterior sampling algorithm described188
in the previous section. We have simply included mention of the Gaussian approximation189
for the sake of completeness, and to guide possible future work that attempts to fit models190
S-23
with significantly more parameters.191
Sensitivity to Error Distribution Width192
One important parameter of the Bayesian fitting approach is the width of the normal distri-193
bution governing the probability of deviations from the model, which arises when evaluating194
the quantity p(y|θ) in Bayes’ rule. Figure S11 studies the sensitivity of the posterior dis-195
tribution over the Tafel slope to the parameter σ, the standard deviation of the normal196
distribution governing the statistics of the model error. As expected, lowering the value of
0.00 0.05 0.10 0.15 0.20 0.25 0.30Overpotential, [V]
2.2
2.0
1.8
1.6
1.4
log
10
[i pro
d.]
True Model
Sampled Data
20 40 60 80 100 120 140Tafel Slope [mV / decade]
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
p(T
afe
l S
lop
e)
0.02
0.04
0.06
0.08
0.10
A B
Figure S11: Sensitivity to the width of the error distribution at each point for a simple fitto synthetic data. (A) Synthetic data sampled from a model with an underlying Tafel slopeof 80 mV/decade. (B) Several traces of the posterior distribution computed using differentvalues of σ, the standard deviation of the normal error distribution at each point.
197
σ causes the algorithm to become more confident in its estimate. At σ = 0.01 logarithmic198
units, the model essentially nails the true Tafel slope of 80 mV/decade. For larger values of σ,199
a clear distributional drift to lower values of the Tafel slope is observed. This occurs because200
most of the data in Fig. S11A lies in the plateau region, and the model faces less penalty201
for down-weighting these points as the value of σ is increasing. Hence, the model drifts to202
larger slopes on the Fig. S11A plot, which corresponds to lower values of the Tafel slope.203
Note that the distributional widening is significantly greater than the drift in the mean,204
suggesting that we should not put much stock into the mean drift. The upshot: for high205
S-24
values of the σ parameter, this particular set of data does not contain enough information206
to pin down the value of the Tafel slope accurately.207
Sensitivity to Prior Distribution208
As explained in the main text, our Bayesian approach requires specification of a prior dis-209
tribution p(θ) over the parameters. Since we know very little about the true distribution of210
the Tafel slope at the outset, we choose an uninformative uniform box prior over the interval211
[0, a × θ∗i ] for each parameter θi, where θ∗i is the optimal value of the parameter gleaned212
from the TRF algorithm described in the Methods section. Here, we conduct a sensitivity213
analysis on the a parameter. Given that we are using an uninformative prior, we should214
expect that the prior width should not influence the posterior distribution as long as the215
data expresses some opinion about the ideal value of the parameters. Figure S12 depicts216
the results of the sensitivity analysis on a set of synthetic data. Indeed, as expected, the
0.00 0.05 0.10 0.15 0.20 0.25 0.30Overpotential, [V]
2.2
2.0
1.8
1.6
1.4
log
10
[i pro
d.]
True Model
Sampled Data
20 40 60 80 100 120 140Tafel Slope [mV / decade]
0.00
0.01
0.02
0.03
0.04
p(T
afe
l Slo
pe)
2
4
6
8
10
12
14
16
18
20
a
A B
Figure S12: Sensitivity to the width of the uniform prior distribution bounds. (A) Syntheticdata sampled from a model with an underlying Tafel slope of 80 mV/decade. (B) Severaltraces of the posterior distribution computed using different values of a, a parameter influ-encing the width of the prior distribution fed to the Bayesian posterior sampling algorithm.All posterior distributions are computed with σ = 0.05.
217
posterior distributions are insensitive to the choice of prior. Note that in all fits considered218
in the main text, we use a value of a = 10, as mentioned in the Methods section.219
S-25
Derivations220
Cardinal Tafel Slopes Equation221
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,
A(n+q)+ + e− −−→←−− A(n+q−1)+
...
A(q+1)+ + e− −−→←−− Aq+
Aq+ + q e− −−→ A.
The overall current is determined by the rate of the RDS. Assuming Butler-Volmer kinetics222
for the forward rate constant of the RDS, we have,223
rate = k0 {aAq+ exp [−βqe(1− α) · (φ− φeq.)]− aA exp [+βqeα · (φ− φeq.)]} , (12)
where k0 is the rate prefactor (sometimes called the Arrhenius prefactor), ai is the activity224
of species i, β ≡ (kBT )−1 is the inverse thermodynamic temperature, e is the fundamental225
charge, α is the symmetry coefficient, φ is the applied potential, and φeq. is the equilibrium226
potential for the RDS. At sufficiently high reductive overpotentials φ − φeq. � 0, only the227
first term survives,228
rate ≈ k0aAq+ exp [−βqe(1− α) · (φ− φeq.)] . (13)
To make further progress, we have to solve for the activity of the intermediate species Aq+229
in terms of the activity of the reactant species for the overall reaction, A(n+q)+. If we assume230
S-26
that all steps prior to the RDS are fast and equilibriated, we can extract this activity231
by analyzing the thermodynamics of the steps prior to the RDS. The free energy change232
associated with the r’th reaction reads,233
∆Fr = −β−1 log aA(n+q−(r+1))+ + β−1 log aA(n+q−r)+ + eφ. (14)
Then, the equilibrium constant for reaction r goes as,
Kr ≡ exp [−β∆Fr] (15)
Kr =aA(n+q−(r+1))+
aA(n+q−r)+
× exp [−βeφ] (16)
Kr = Kr × exp [−βeφ] , (17)
where Kr 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,
aAq+ =
[n∏r=1
Kr
]exp [−nβeφ]× aA(n+q)+ (18)
aAq+ =
[n∏r=1
Kr
]exp [−nβe (φ− φeq.)] exp [−nβeφeq.]× aA(n+q)+ . (19)
Plugging back into Eq. (13),
rate ≈ k0
([n∏r=1
Kr
]exp [−nβe (φ− φeq.)] exp [−nβeφeq.]× aA(n+q)+
)exp [−βqe(1− α) · (φ− φeq.)]
(20)
rate = k0
([n∏r=1
Kr
]exp [−nβeφeq.]× aA(n+q)+
)exp [−βe (φ− φeq.) (n+ q · (1− α))] . (21)
This is a mess, but we only care about the potential-dependent terms when extracting the234
Tafel slope, which means we only have to consider the last factor on the RHS. Taking the235
S-27
logarithm yields,236
log [rate] = −βe (φ− φeq.) (n+ q · (1− α)) + C, (22)
where C is a constant independent of potential. For a reduction reaction, the Tafel slope is237
defined as,238
Tafel Slope =
[∂ log10 [rate]
∂ [− (φ− φeq.)]
]−1
. (23)
Hence, we have,239
Tafel Slope = [log10 (exp(1)) · βe]−1 · 1
n+ q · (1− α). (24)
Appropriate unit scalings yield,240
Tafel Slope =60 mV/decade
n+ q · (1− α), (25)
which reduces to the equation quoted in the main text when α = 1/2.241
Physical Non-Idealities242
Tafel Slopes with Physical Non-Idealities243
Eq. (25) already accounts for the non-ideality effects introduced by α 6= 1/2. If we assume244
that the CO2 adsorption step has partial charge transfer character quantified by γ, then245
despite the fact that the adsorption step is purely chemical, we assume that its equilibrium246
constant carries a non-integer order dependence on the applied potential. This can also be247
motivated by considering the formation of a permanent dipole on the surface species, which248
can access additional thermodynamic stabilization due to a dipole Stark shift from electric249
fields present at theinterface. The manner in which surface dipole formation augments the250
Tafel slope depends on whether or not the CO2 adsorption step is the rate determining251
step. For the case (n, q) = (0, 1), the adsorption step is the RDS. The Frumkin correction252
S-28
attenuates the applied potential for the rate-determining step by a factor f , which simply253
multiplies the latter term in the denominator of Eq. (25). Hence, the Tafel slope for (n, q) =254
(0, 1) is,255
Tafel Slope =60 mV/decade
γ · f · (1− α). (26)
For the case (n, q) = (1, 0), the adsorption step is the RDS, and rather than contributing256
n = 1 to the order, it instead contributes according to the γ parameter,257
Tafel Slope =60 mV/decade
γ. (27)
Finally, for the case (n, q) = (1, 1), the adsorption step occurs before the RDS, and the Tafel258
slope reads,259
Tafel Slope =60 mV/decade
γ + f · (1− α). (28)
Sensitivities to Parameter Bounds260
Figures S13 and S14 study the sensitivity of the distributional results presented in Fig. 4261
of the main text to the bounds of the uniform distributions over non-ideality parameters.262
Broadly, our claim is supported by the sensitivity analysis; within reasonable parameter263
bounds, we still see that we can get essentially arbitrary distributional shapes depending on264
the nonidealities included in the model.265
Multiple Kinetic Regimes266
Here, we examine the consequences of fitting current-voltage data from a system exhibiting267
multiple kinetic regimes to a model that only allows a single Tafel slope (as in Eq. (2) in the268
main text) by analyzing synthetic data. The synthetic data is generated from the model,269
1
i(E)=
1
ilim+
1 + exp(−e [kBT ]−1 × [E − Eeq,1]
)exp
(e [kBT ]−1 × α [E − Eeq,2]
) , (29)
S-29
0 50 100 150 200 250Tafel Slope [mV / decade]
0
2
4
6
8
10
KD
E(T
afe
l S
lop
e)
Ideal
Variable
Variable
Variable f
Variable ,
Variable , f
Variable , f
Variable , , f
0 50 100 150 200 250Tafel Slope [mV / decade]
0
2
4
6
8
10
KD
E(T
afe
l S
lop
e)
Ideal
Variable
Variable
Variable f
Variable ,
Variable , f
Variable , f
Variable , , f
0 50 100 150 200 250Tafel Slope [mV / decade]
0
2
4
6
8
10
KD
E(T
afe
l S
lop
e)
Ideal
Variable
Variable
Variable f
Variable ,
Variable , f
Variable , f
Variable , , f
Figure S13: Several synthetic kernel density estimates of the probability distributions overthe Tafel slope generated from including random values of different parameters governingphysical non-idealities. Different panels use different uniform distributions over the symme-try coefficient parameter, α.
0 50 100 150 200 250Tafel Slope [mV / decade]
0
2
4
6
8
10
KD
E(T
afe
l S
lop
e)
Ideal
Variable
Variable
Variable f
Variable ,
Variable , f
Variable , f
Variable , , f
0 50 100 150 200 250Tafel Slope [mV / decade]
0
2
4
6
8
10
KD
E(T
afe
l S
lop
e)
Ideal
Variable
Variable
Variable f
Variable ,
Variable , f
Variable , f
Variable , , f
0 50 100 150 200 250Tafel Slope [mV / decade]
0
2
4
6
8
10
KD
E(T
afe
l S
lop
e)
Ideal
Variable
Variable
Variable f
Variable ,
Variable , f
Variable , f
Variable , , f
Figure S14: Several synthetic kernel density estimates of the probability distributions overthe Tafel slope generated from including random values of different parameters governingphysical non-idealities. Different panels use different uniform distributions over the Frumkincorrection parameter, f .
S-30
where E is the applied potential, and the model has free fitting parameters α, ilim, Eeq,1,270
and Eeq,2. This model can be shown to arise when a reaction proceeds through a rate-271
limiting surface reaction involving a surface intermediate generated through a one-electron272
transfer, and present at non-negligible surface coverages. The specifics of how this model273
arises are less relevant to this analysis than the fact that the model exhibits two different274
Tafel regimes. When α = 1/2, the first regime has Tafel slope mT = 40 mV decade−1 for275
Eeq,1 < E < Eeq,2, and the second regime has a Tafel slope mT = 120 mV decade−1 for276
E > Eeq,2, before topping out at the limiting current ilim.277
0.0 0.1 0.2 0.3 0.4 0.5 0.6Overpotential [V]
3
2
1
0
1
2
log(C
urr
ent)
Underlying Model
Noised Sample
0.0 0.1 0.2 0.3 0.4 0.5 0.6Applied Potential [V]
3
2
1
0
1
2
log(C
urr
ent)
Two-Regime Model
Series Resistances Model
42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0Tafel Slope [mV/decade]
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
p(T
afe
l Slo
pe)
A B C
Figure S15: (A) Current-voltage trace generated from Eq. (29) using the parameters α =1/2, ilim = 100 mA cm−2, Eeq,1 = 0.05 V, and Eeq,2 = 0.15 V (red trace), along withartificially noised data sampled from the model (black points). (B) Current-voltage tracesevaluated from the MAP fit parameters for Eq. (2) in the main text and Eq. (29), alongwith the noised data used for the fits (black points). (C) Posterior distribution over the Tafelslope for the model described by Eq. (2) in the main text.
Figure S15A shows a trace of the current-voltage behavior predicted by Eq. (29), with278
parameters α = 1/2, ilim = 100 mA cm−2, Eeq,1 = 0.05 V, and Eeq,2 = 0.15 V, as well279
as artificially noised data sampled from this model at a sparsely sampled set of voltage280
points. Figure S15B shows current-voltage traces from both Eq. (29) (red) and Eq. (2)281
from the main text (blue), each evaluated under the MAP parameters determined from their282
respective Bayesian fits. Finally, Fig. S15C shows the Bayes posterior distribution for the283
Tafel slope fitted using Eq. (2) from the main text. As expected, when using the single284
Tafel regime model to fit data generated from multiple Tafel regimes, the MAP value of the285
Tafel slope does not coincide with the Tafel slope from either kinetic regime in Eq. (29).286
S-31
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14E0, 1 [V]
0
2
4
6
8
10
p(E
0,1
)
Mean Value
True Value
0.08 0.10 0.12 0.14 0.16 0.18E0, 2 [V]
0
5
10
15
20
25
p(E
0,2
)
Mean Value
True Value
0.3 0.4 0.5 0.6 0.7 0.80
1
2
3
4
5
6
p(
)
Mean Value
True Value
60 80 100 120 140 160 180 200
ilim [mA cm 2]
0.000
0.005
0.010
0.015
0.020
0.025
p(i
lim)
Mean Value
True ValueA B
DC
Figure S16: Posterior distributions for α, ilim, Eeq,1, and Eeq,2 (A, B, C, D, respectively)gleaned from Bayesian posterior sampling on the artificially noised data in Fig. S15A.
S-32
However, as illustrated in Fig. S16, when the original data is fit to Eq. (29), the posterior287
distributions are peaked around the true values of the parameters.288
This analysis yields two important takeaways. First, it lends credence to the idea that289
fitting a single Tafel slope to data collected under multiple Tafel regimes, each individu-290
ally exhibiting a cardinal Tafel slope, can produce an off-cardinal value of the Tafel slope,291
providing an alternative possible explanation for the lack of observed cardinality in the lit-292
erature analysis in the main text. Second, and perhaps more importantly, it shows that293
the Bayesian framework presented here can successfully estimate the parameters of more294
complicated physical models that incorporate the effects of multiple different Tafel regimes,295
or some of the physical nonidealities discussed in the main text. Practically, we furnish the296
following recommendation: if one knows in advance that multiple kinetic regimes are at play297
in a set of current-voltage data, and the nature of these regimes can be encoded into a ki-298
netic model like Eq. (29), then one should carry out Bayesian fitting to such a model. In the299
absence of sufficient independent evidence to pin down a kinetic model that resolves multiple300
kinetic regimes, Eq. (2) from the main text is a viable alternative, but one should be very301
cautious about over-interpreting the mechanistic implications of a Tafel slope determined in302
this manner.303
S-33
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