Application of Bayesian Inference in Fourier Transformed Alternating

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Application of Bayesian Inference in Fourier Transformed Alternating Current Voltammetry for Electrode Kinetic Mechanism Distinction Jiezhen Li, Gareth F. Kennedy, Luke Gundry, Alan M. Bond, and Jie Zhang Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.9b00129 • Publication Date (Web): 18 Mar 2019 Downloaded from http://pubs.acs.org on March 30, 2019

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Analytical Chemistry

Application of Bayesian Inference in Fourier Transformed Alternating Current Voltammetry for Electrode Kinetic Mechanism Distinction Jiezhen Li,† Gareth F. Kennedy,† Luke Gundry,† Alan M. Bond*,†,‡ and Jie Zhang*,†,‡ † School of Chemistry, Monash University, Victoria 3800, Australia ‡ ARC Centre of Excellence for Electromaterials Science, School of Chemistry, Monash University Corresponding Author *[email protected] *[email protected] ABSTRACT Estimation of parameters of interest in dynamic electrochemical (voltammetric) studies is usually undertaken via heuristic or data optimization comparison of the experimental results with theory based on a model chosen to mimic the experiment. Typically, only single point parameter values are obtained via either of these strategies without error estimates. In this paper, Bayesian inference is introduced to Fourier transformed AC voltammetry (FTACV) data analysis to distinguish electrode kinetic mechanisms (reversible or quasi-reversible, Butler-Volmer or Marcus-Hush models) and quantify the errors. Comparisons between experimental and simulated data were conducted across all harmonics using public domain freeware (MECSim).

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Analytical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

INTRODUCTION In electrochemistry, gaining a detailed understanding of the process that involves the transport of charge across the interface between an ionic conductor (an electrolyte) and electronic conductor (an electrode) is of great interest.1-3 Direct current (DC) cyclic voltammetry is probably the most widely used method to study and parameterize the electrode kinetics and thermodynamics. Historically, transient cyclic voltammetric experiments were undertaken to acquire the electrode kinetic information based on data analysis via the Nicholson method.4 This approach is simple and only requires the measurement of the difference of the reduction and oxidation peak potentials present in a DC cyclic voltammogram as a function of scan rate and reference to look up tables. However, failure to take into account the uncompensated ohmic potential drop in the theory results in an underestimate of the electrode kinetics.5 A more advanced approach to data analysis is to directly compare experimental and simulated voltammograms.6 For a simple one-electron transfer reaction described by equation 1, O + e―⇌ R

(1)

eight parameters including electrode area (A), formal reversible potential (𝐸0f), bulk concentration (c), diffusion coefficient (D), uncompensated resistance (𝑅𝑢), double layer capacitance (𝐶dl), standard electron transfer rate constant (𝑘0) and electron transfer coefficient (𝑎) in Butler-Volmer (BV) or reorganization energy (𝜆) in Marcus-Hush (MH) models of electron transfer are required to generate simulated data. Values of parameters such as c, D, 𝑅𝑢, 𝐶dl and even 𝐸0f sometimes may be known from independent measurements with acceptable accuracy. Under these circumstances, by adjusting the remaining unknown parameters in the model until a good fit is achieved, the kinetic information can be extracted.7-11 2


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In DC cyclic voltammetry, multiple measurements over a wide range of scan rates are required to obtain reliable results for comparison with simulations.12,13 An alternative approach to quantify electron transfer kinetics is Fourier-transformed large-amplitude alternating current voltammetry (FTACV), in which a sinusoidal or another periodic waveform with frequency, f, and amplitude, ΔE, is superimposed onto the dc ramp.14-16 In this technique, a series of higher order harmonics at 2f, 3f, 4f, and 5f etc. are generated, each of which can be resolved by using a FT – band selection and filtering – inverse FT sequence of operations. As a result, data over a broad range of timescales are collected from a single experiment.17 Moreover, the current magnitude is very sensitive to 𝑘0 and 𝑅𝑢, and the charging current contribution is minimal in the higher-order harmonic components.5 Therefore, FTACV provides a very powerful tool for the measurement of electrode kinetics.18,19 Commonly, theory–experiment comparisons used to extract kinetics parameters have been undertaken by a time-consuming heuristic method. In this form of analysis, the outcome depends on the experience and prejudices of the experimenter to determine what combination of parameters provides the best agreement between experimental and simulated data. As computing power has increased, more advanced digital approaches have become available to overcome human-factor limitations by performing theory-experiment comparisons automatically.20-25 In these data optimization procedures, each parameter of interest is assigned a set of values (e.g., 𝑘0 from 0.01 to 1 cm s-1 and 𝑎 from 0.4 to 0.6) and all possible combinations are used for comparison with experiment. A least square (LS) function is then used to gauge the level of agreement between experiment and the simulated data for each parameter set with the best fit having the minimal LS values. However, this strategy only gives useful results when the system is sensitive to all of unknown parameters

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and hence gives a unique solution. Moreover, these approaches usually do not provide a rigorous statistical analysis of parameter uncertainties. The Bayesian inference method has been widely used in many branches of science with great advantage but has had limited application in electrochemistry. Two recent examples in ac voltammetry demonstrated the extraction of noise contributions to error in a single experiment26 and electrode variability with respect to multiple experiments27. In this paper, Bayes’ theorem is used to generate the probability distribution of the electrode kinetic parameters (𝑘0, 𝑎 or 𝜆) of interest. Adopting a Bayesian approach allows the probabilities of the parameters of interest to be updated, given new experimental data. The initial probabilities (priors) are set either by a uniform distribution, where no information is known before conducting the experiment, or by using the final probabilities (posteriors) from a different experiment. The Bayesian concept is ideal for FTACV because each harmonic can be treated as an independent experimental data set thus automatically exploiting the fact that some harmonics are more sensitive to a particular parameter than others. Additional experiments, say with a different frequency, also can be conducted to further improve the precision of the parameter ranges. These Bayesian concepts have been introduced into the freeware software package MECSim21 so that the protocols are now available in the public domain (http://www.garethkennedy.net/MECSim.html). The experimental data employed in this paper include the reversible Ru(NH3)63+/2+ reduction process at a glassy carbon (GC) electrode and the quasi-reversible Fc0/+ (Fc = ferrocene) oxidation process at a boron doped diamond (BDD) electrode using the Butler-Volmer model. The data provided in Ref.21 for the kinetically slow DmFc0/+ (DmFc = decamethylferrocene) oxidation process at a thiol modified gold electrode is also reexamined by the Bayesian approach to quantitatively show how highly preferential the Marcus-Hush (MH) model is in comparison to the Butler–Volmer (BV) model.

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EXPERIMENTAL SECTION Chemicals. 1-butyl-3-methylimidazolium hexafluorophosphate ([BMIM][PF6], Io-li-tec) was dried by nitrogen purging for 48 hours. Acetonitrile (CH3CN, 99.9%, Sigma-Aldrich), was used as supplied by the manufacturer. Tetrabutylammonium hexafluorophosphate ([TBA][PF6], 98%, Wako) was recrystallized twice from ethanol (96%, Merck) before use as the supporting electrolyte in voltammetric experiments undertaken in acetonitrile. Ferrocene (Fc, Sigma-Aldrich, ≥

98 %) was recrystallized from n-pentane (Merck,

EMSURE). Tetrapropylammonium tetrafluoroborate ([TPA][BF4]) was prepared by a metathesis reaction between sodium tetrafluoroborate (Na[BF4], Sigma-Aldrich) and tetrapropylammonium bromide ([TPA]Br, Sigma-Aldrich) in CH3CN and recrystallized from ethanol. Decamethylferrocene (DmFc, 97 %, Sigma-Aldrich), propylene carbonate (≥ 99 %, Sigma-Aldrich), ruthenium hexamine trichloride ([Ru(NH3)6]Cl3) 98% (SigmaAldrich) and potassium chloride (KCl) 99% (Sigma-Aldrich) were used as obtained from the manufacturer. Doubly distilled water was used for the preparation of all solutions. Instrumentation and Procedures. AC voltammetric data related to Case study 2 for the reduction of 0.46 mM Ru(NH3)63+ at a glassy carbon (GC, nominal diameter = 3.0 mm, BAS) electrode in 1.0 M KCl aqueous solution were acquired in three-electrode cells at 23 °C using home built Fourier transform voltammetric instrumentation28. Platinum wire was used as the auxiliary and Ag/AgCl (aqueous 3.0 M KCl) as the reference electrodes. The effective electrode area of the GC electrode was calculated to be 0.066 cm2 from analysis of a plot of DC peak current versus the square root of scan rate for the oxidation of 1.0 mM Fc in CH3CN (0.1 M [Bu4N][PF6]) using the Randles-Sevcik relationship29 and the known diffusion coefficient of 2.24 × 10-5 cm2 s-1 for Fc30 under these conditions. Cdl was quantified from the background current in the fundamental harmonic at potentials where AC 5



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faradaic current is absent. In order to define the potential dependence of Cdl, a nonlinear capacitor model was used, as described elsewhere:28 𝐶dl(𝑡) = 𝑐0 + 𝑐1𝐸(𝑡) + 𝑐2𝐸2(𝑡)

(2)

C0, C1 and C2 are determined to be 1.80 × 10-5 F cm-2, -4.11 × 10-6 F cm-2 V-1 and 3.50 × 106

F cm-2 V-2. Ru was estimated to be 45 Ω experimentally from analysis of the RC time

constant available with the CHI instrument. The value of the diffusion coefficient of Ru(NH3)63+ was estimated as 7.0 × 10-5 cm2 s-1 from the Randles-Sevcik relationship.29 The data related to the Case studies 1 and 3 for the oxidation of 0.20 mM Fc in acetonitrile (0.1 M Bu4NPF6) at a BDD electrode were obtained in a similar manner. In brief, the surface area of the BDD electrode was estimated to be 0.0077 cm2 based on the voltammetric peak current associated with the Fc0/+ process and the application of the Randles-Sevcik relationship. Ru was estimated to be 430 Ω. Cdl was quantified to be 8.0 μF cm−2 and E0 was set at 0.000 V vs. Fc0/+ by definition with f = 9.02 Hz, ΔE=80 mV generated instrumentally. The DmFc0/+ experimental data at a thiol modified gold electrode for Case study 4 were taken from Ref21. The preparation of the electrode can be found in this reference and the parameters used in simulations are summarized here: f = 3.99 Hz, ΔE = 160 mV, Ru = 2020 Ω, Cdl = 3 μF cm-2, A = 8.2 × 10-3 cm2, D = 2.4 × 10-6 cm2 s-1, E0 = 0.000 V vs. DmFc0/+ by definition. Simulation Package. Simulations of the FTACV data were undertaken with MECSim (Monash

Electrochemistry

Simulator)

software

package

(http://www.garethkennedy.net/MECSim.html). Links to the python scripts used for Bayesian analysis are available in the website.

RESULTS AND DISCUSSION Bayesian theory 6


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According to Bayes’ rule, the probability that a given model 𝑀j consisting of a particular set of parameters correctly fits the data 𝑥h compared to another model 𝑀𝑘 is 𝑃(𝑀𝑗│𝑥h) =

𝑃(𝑀𝑗)𝑃(𝑥h│𝑀𝑗)

(3)

𝑁

∑𝑘 = 1𝑃(𝑀𝑘)𝑃(𝑥h│𝑀𝑘)

where N is the number of unique parameter sets. In FTACV, h represents the hth AC harmonic component and can be 1 to 6 in this study. The prior 𝑃(𝑀𝑗) is how likely the parameter set 𝑀𝑗 was thought to be before consideration of the new data 𝑥h. The likelihood 𝑃(𝑥h│𝑀𝑗) can be understood as the probability of the data given a parameter set 𝑀𝑗, which can be derived from the difference between the simulated and the experimental data. The probability of the data 𝑥h given the combination 𝑀𝑗 can be simplistically thought of as given by 𝑃(𝑥h│𝑀𝑗) =

𝐸𝑚𝑖𝑛 𝐸(𝑥h|𝑀𝑗)

(4)

where 𝐸𝑚𝑖𝑛 = 𝑚𝑖𝑛(𝐸(𝑥h│𝑀𝑘)), 𝑘 = 1,2,…, 𝑁. The error function 𝐸(𝑥h|𝑀𝑗) is given by 𝑌

𝐸(𝑥h│𝑀𝑗) =

∑𝑖 = 1(𝑥𝑖 - 𝑓𝑖(𝑀𝑗))2 𝑌

∑𝑖 = 1(𝑥𝑖)2

(5)

where Y is the total number of data points 𝑥𝑖, 𝑥𝑖 is current at time 𝑡𝑖 and 𝑓𝑖(𝑀𝑗) is the current obtained from simulation using parameter set 𝑀𝑗. The error function gives 0 for an exact fit and increasing values for poorer fits. Since 𝐸(𝑥h│𝑀𝑗) is zero for a perfect fit, then a small term (10 -6) is added to ensure the probability 𝑃(𝑥h│𝑀𝑗) does not go to infinity. On this basis, 𝑃(𝑥h│𝑀𝑗) is corrected to 𝑃(𝑥h│𝑀𝑗) =

𝐸𝑚𝑖𝑛 𝐸(𝑥h│𝑀𝑗) + 10 -6

(6)

Protocol for Applying Bayesian Analysis to FTACV data

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The protocol summarized in Scheme 1 can be adopted to obtain the probability of having a particular set of parameters in comparison to other sets. Step 1: The parameters of interest are assigned a range of values. MECSim software is then scripted to cover this range and generates simulated data for each model 𝑀𝑘

(𝑘 = 1,2,…, 𝑁) where N is the total number of parameter sets, which are compared to the experimental data via the error function (eq. 5) for each AC harmonic. 1

Step 2: For the 1st harmonic data, set the prior probability 𝑃(𝑀𝑗) of each parameter set as 𝑁. Using the likelihood given in eq. 6, the probability of the data given a parameter set 𝑃

(𝑥1│𝑀𝑗) is calculated. Step 3: Apply the Bayesian procedure to the 1st harmonic data (eq. 3) to calculate the posterior distribution for each model 𝑃(𝑀𝑗│𝑥1). The obtained posterior probability for each model 𝑀𝑗 is then used as the prior probability 𝑃(𝑀𝑗) of the corresponding 2nd harmonic data. Step 4: Update the posterior probability of the higher harmonic data by setting the prior probability as the posterior probability from the lower harmonic data. Finally, the posterior probabilities of each parameter set given the data from all harmonics are obtained. The protocol is summarised in Scheme 1. Scheme 1. Protocol for updating the confidence of one parameter set compared to other parameter sets using Bayes’ rule for each of the 𝑁𝐻 harmonics with sufficient signal to noise.

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Examples of the use of the Bayesian analysis To demonstrate the practical utility of Bayesian analysis to experimental data, four cases were chosen. Case 1: Comparison of two sets of parameters The application of Bayesian analysis to FTACV data analysis is demonstrated by comparing two distinct sets of kinetic parameters (𝑘0 and 𝑎) associated with the Fc0/+ process. At a BDD electrode, Fc0/+ was found to be a quasi-reversible one electron oxidation process, with values of 𝑘0 = 0.080 cm s-1 and 𝛼 = 0.50. This parameter set is denoted as the first model 𝑀1 . The second model 𝑀2 is arbitrarily taken as 𝑘0 = 0.070 cm s-1, 𝛼 = 0.55 which slightly deviates from the first model. Input parameters needed in the simulation such as A, 𝑅u, c, D, 𝐶dl and 𝐸0f are summarized in the Experimental Section. Figure 1 shows the comparisons of the experimental and simulated data generated by the two different parameter sets. All the parameters required for simulations can be found in the caption to Figure 1.

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Figure 1. Comparison of simulated (red, blue) and experimental (black) FTAC voltammetric data obtained from oxidation of 0.20 mM Fc in CH3CN (0.10 M [TBA][PF6]) at a BDD electrode with ∆E = 80 mV, f = 9.02 Hz and v = 0.089 V s-1. Panels (a-f) show the 1st to 6th AC harmonic components. The simulation parameters are A = 7.7 × 10−3 cm2, D = 2.3 × 10−5 cm2 s−1, Ru = 374 Ω, E0 = 0 V vs. Fc0/+, Cdl (C0 = 8.0 μF cm−2), T = 295 K, k0 = 0.080 cm s-1 and ɑ = 0.5 for 𝑀1 (red) and k0 = 0.070 cm s-1, 𝛼 = 0.55 for 𝑀2 (blue). The probabilities of 𝑀1 and 𝑀2 being updated after each iteration are also shown in the figure.

The error between the experimental and simulated data 𝐸(𝑥h|𝑀𝑖) for these two parameter sets are calculated according to eq. 5 and summarized in Table 1. Applying eq. 6, the probabilities of the first harmonic component data (when h = 1) given 𝑀1 (𝑘0 = 0.080 cm s-1 and 𝑎 = 0.50) and 𝑀2 (𝑘0 = 0.07 cm s-1 and 𝑎 = 0.55) are 𝑃(𝑥1|𝑀1) =

𝑃(𝑥1|𝑀2) =

𝐸𝑚𝑖𝑛 𝐸(𝑥1|𝑀1) + 10 -6 𝐸𝑚𝑖𝑛

𝐸(𝑥1|𝑀1) + 10 -6

=

=

0.0330 0.0330 + 10 -6

0.0330 0.0467 + 10 -6

= 1.0000 = 100%

= 0.7066 = 70.66%

that these are unscaled probabilities, thus the sum of 𝑃(𝑥1|𝑀𝑗) ≠ 1. 10


(7)

(8)It is noted

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Using the data from the first harmonic component the probability of 𝑀1 is given by 𝑃(𝑀1│𝑥1) =

=

𝑃(𝑀1)𝑃(𝑥1│𝑀1) 𝑃(𝑀1)𝑃(𝑥1│𝑀1) + 𝑃(𝑀2)𝑃(𝑥1│𝑀2)

50% × 1.0000 = 0.5861 = 58.61 % 50% × 1.0000 + 50% × 0.7066

(9)

where the prior for 𝑀1is initially assumed to be 𝑃(𝑀1) = 1 𝑁 = 50% (as there are only 2 parameter sets in this exercise). When the second harmonic component data 𝑥2 is applied then the probability of parameter set 𝑀1 is updated and given by 𝑃(𝑀1│𝑥2) =

58.61% × = 58.61% ×

𝑃(𝑀1)𝑃(𝑥2│𝑀1) 𝑃(𝑀1)𝑃(𝑥2│𝑀1) + 𝑃(𝑀2)𝑃(𝑥2│𝑀2) 0.0197

0.0197 + 10 -6

0.0197 0.0197 + 10 -6

+ 41.39% ×

0.0197

= 0.8715 = 71.04 %

(10)

0.0341 + 10 -6

where the updated prior 𝑃(𝑀1) is given by the posterior probability from eq. 9. The Bayesian analysis is repeated up to 6th harmonic component to obtain the final 𝑃(𝑀1│𝑥6), similarly for the other parameter set 𝑀2. The posterior probability values of 𝑃(𝑀1│𝑥𝑖) and 𝑃(𝑀2│𝑥𝑖) (i = 1,2...6) calculated using the analysis procedure described above are summarised in Table 1. With each iteration, the values of 𝑃(𝑀1│𝑥𝑖) and 𝑃(𝑀2│𝑥𝑖) become larger and smaller, respectively. In this exercise, a much larger 𝑃(𝑀1│𝑥6) of 99.72% is obtained compared to 𝑃(𝑀2│𝑥6) of 0.28%, which means the probability of 𝑀1 being correct given the experimental data is much higher than 𝑀2. Table 1 Summary of the Bayesian analysis protocol applied to two parameter sets 𝑀1 and 𝑀2.

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Harmonic

Error

Prior Probability

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Posterior Probability

E(x|𝑀1)

E(x|𝑀2)

P(𝑀1)

P(𝑀2)

P(𝑀1│xi)

P(𝑀2│xi)

1st

0.0330

0.0467

0.5000

0.5000

0.5861

0.4139

2nd

0.0197

0.0341

0.5861

0.4139

0.7104

0.2896

3rd

0.0241

0.0556

0.7104

0.2896

0.8495

0.1505

4th

0.0241

0.0872

0.8495

0.1505

0.9534

0.0466

5th

0.0236

0.1241

0.9534

0.0466

0.9908

0.0092

6th

0.0479

0.1588

0.9908

0.0092

0.9972

0.0028

The data analysis protocol described above can be easily extended to consider more parameter sets as demonstrated below. Case 2: N parameter sets for a fast process The second example considered is the reduction of 0.46 mM Ru(NH3)63+ at a GC electrode in 1.0 M KCl aqueous solution. Ru(NH3)63+ is known to undergo very fast one electron transfer under these conditions.31 In this case, Butler–Volmer electrode kinetics, with mass transport governed by linear diffusion were assumed. Input parameters such as A, 𝑅u, c, D, 𝐶dl were as described in the Experimental Section with 𝛼 assumed to be 0.50. Thus, 𝐸0f and 𝑘0are the parameters to be derived by Bayesian analysis following the procedure described in Scheme 1. A parameter sweep was performed using 41 by 50 evaluations equally spaced through the search space in the range - 0.210 ≤ 𝐸0f ≤ - 0.170 V and 0.1 ≤ 𝑘0 ≤ 5.0 cm s -1. The posterior probability for each of the parameter sets are presented in the form of a contour map (Figure 2) as a function of the two variables. The region in parameter space giving the optimal fit to the experimental data is indicated in red. It should be noted that the probabilities are relatively high since the probability is distributed across 2050 parameter sets, thus each prior had a probability of 1/2050. Examination of the contour plot in Figure 2 shows that the high probability region for 𝐸0f is tightly constrained around -0.194 V while 𝑘0 has a lower bound but no upper limit is found. Contour plots have been shown to be 12


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extremely valuable in understanding the fidelity of the parameterization in previous work.2225

These methods are all based on calculating the difference between experimental and

simulated data using least squares with an equation the same as or similar to eq. 5. The Bayesian approach has the advantage of obtaining meaningful values close to the optimal fit, which is ideal for determining the errors for a given parameter.

Figure 2. Posterior probabilities of each parameter sets obtained by Bayesian analysis for the reduction of 0.46 mM Ru(NH3)63+ at a GC electrode in 1.0 M KCl aqueous solution, as a function of two parameters, 𝐸0f and 𝑘0.

To better understand the data, the contour map is transferred into two two-dimensional figures (Figure 3), where the abscissa is the parameter 𝐸0f or 𝑘0 and the ordinate is the sum of the probability referring to a specific 𝐸0f or 𝑘0 value. Figure 3a (black line) shows the probability of a given simulated data being equal to the experimental data as a function of its corresponding 𝐸0f values. It is straightforward to see that the largest probability is located where 𝐸0f = -0.193 mV, giving P = 36%. Moreover, in order to obtain the range of parameters sets with a certain confidence, a Gaussian distribution is applied to Figure 3a

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using the inbuilt function available in Origin. The Gaussian function is given by following expression: -2 A y = y0 + e ω π/2

(x - xc)2 ω2

(11)

where y0 is the offset, xc is the position of the center of the peak, ω is the width factor which controls the width of the peak and A is the peak area. In this case, the standard deviation 𝜎 = ω/2. Figure 3a shows a distribution close to the ideal Gaussian curve (r2 = 0.97) (red line) along with the probability curve (black line), giving an estimated distribution G (-0.193, 0.0000932), with A = 8.1 × 10 -4, y0 = 0.0023. Note that in many existing statistical test approaches32,33, the 95% confidence interval is the value widely adopted for statistical significance, which lies within two standard deviation of the mean. Therefore, instead of giving a best fit through LS analysis as in Ref21, the result now can be presented as a mean value with a two standard deviation error. In this example, it gives a value of 0.1930±0.0002 V for𝐸0f at the 95% confidence interval. Figure 3b shows the sum of probabilities as a function of their corresponding 𝑘0 values. Unlike figure 3a obeying a Gaussian distribution, figure 3b behaves like a sigmoid function. The probability increases sharply when 𝑘0 changes from 0.1 to 1 cm s-1. Beyond, 1 cm s-1, the probability becomes insensitive to the 𝑘0 value. This implies that the process has approached the reversible limit and that a single value determination of 𝑘0 is not available. Using Bayesian analysis we can constrain the rate constant to be 𝑘0 > 0.7 cm s-1, where the sum of probabilities in this region is 95%. However, it is noted that the 95% confidence interval is conditional and depends on the choice of the maximum 𝑘0 value (5.5 cm s-1 in this example) which is set before preforming the Bayesian analysis. Figure S1 shows the experimental data alongside simulations using the mean values of E0 = -0.193 V and k0 > 0.7 cm s-1.

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Figure 3. (a) Summed posterior probability distribution of 𝐸0f values (black) for the reduction of 0.46 mM Ru(NH3)63+ at a GC electrode in 1.0 M KCl aqueous solution. The red line represents a fit to the posterior probability distribution with a Gaussian function, as defined by eq. 11. (b) Summed posterior probability distribution of 𝑘0 values. The red line represents a region where the sum of probabilities in this region is 95% given the imposed constraint of 0 ≤ 𝑘0 ≤ 5.5 cm s-1. Case 3: N parameter sets for a quasi-reversible process In this third example, Bayesian analysis is applied to the determination of kinetics parameters (𝛼 and 𝑘0) for the oxidation of Fc at a BDD electrode in CH3CN (0.10 M [TBA][PF6]), a process which is quasi-reversible under FTACV conditions. The values for parameters of 𝐸0f, A, 𝑅u, c, D and 𝐶dl are considered as known, leaving 𝛼 and 𝑘0 as the only variable parameters to be assigned. Figure 4 shows a contour map of the posterior probabilities as a function of the two variables. As the process is quasi-reversible, the probability converges to a small region best described as 𝛼 = 0.50 ± 0.04 and 𝑘0 = 0.082 ± 0.006 cm s-1 with a 95% level of confidence as shown in Figure 5a and b, respectively. Figure S2 shows the experimental data against simulated data for the best fit values ɑ = 0.5 and k0 = 0.082 cm s-1.

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Figure 4. Posterior probabilities obtained by Bayesian analysis for the oxidation of 0.20 mM Fc at a BDD electrode in CH3CN (0.10 M [TBA][PF6]) as a function of 𝛼 and k0.

Figure 5. (a) Summed posterior probability distribution of 𝛼 values (panel a) and 𝑘0 values (panel b) for the oxidation of 0.20 mM Fc at a BDD electrode in CH3CN (0.10 M [TBA][PF6]. The red lines represent a fit to the posterior probability distribution with a Gaussian function, as defined by eq. 11. Case 4: Comparison of Butler-Volmer (BV) and Marcus-Hush (MH) models. The BV and MH models are the most commonly employed ones for studying electrode kinetics.29,34 The former is an empirical model and mathematically simple while the latter offers physical insights into the nature of the electron transfer process. It has been proven that differences between BV and MH models can be revealed for kinetically controlled currents at sufficiently large overpotentials.35,36 We have previously reported the use of

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FTACV to assess the applicability of the BV and MH models by studying the electrode kinetics of the DmFc0/+ process in propylene carbonate at an octanethiol modified Au electrode21. Based on the LS analysis, the MH model (LS = 94.3%) was shown to provide better agreement with the experimental data than the BV model (LS = 76.5 %). However, comparison of LS values does not provide information on exactly how preferential the MH model is relative to the BV one. In this study, Bayesian analysis now is applied to find out how superior the MH model is compared to the BV model for the same experimental data set. Similarly to Case 3, the kinetic parameters (𝑘0 and λ for MH model and 𝑘0 and 𝛼 for BV model) are varied while 𝐸0f, A, 𝑅u, c, D, 𝐶dl are kept fixed and take the same values as in Ref21. A parameter sweep was performed using 21 by 61 evaluations equally spaced in the range 0.001 ≤ 𝑘0 ≤ 0.003 cm s -1 and 0.2 ≤ 𝜆 ≤ 0.8 cm s -1 for MH model, and 18 by 21 in the range 0.0005 ≤ 𝑘0 ≤ 0.0022 cm s -1 and 0.4 ≤ 𝛼 ≤ 0.6 cm s -1 for BV model. Posterior probabilities are shown in Figures S3 and S4 for MH and BV models respectively and the associated probability distributions are shown in Figures S5 and S6. The 95% level of confidence around the best fits are 𝑘0 = 0.00174 ± 0.00016 cm s-1 and 𝜆 = 0.42 ± 0.17 eV and 𝑘0 = 0.00150 ± 0.00026 cm s-1 and 𝛼 = 0.50 ± 0.06 for the MH and BV models, respectively. The results obtained here are consistent with the values reported in reference21 which are k0 = 0.0017 cm s−1 and 𝜆 = 0.40 for the MH model and k0 = 0.0010 cm s−1 and 𝛼 = 0.50 for the BV model. The experimental data along with simulated data for the BV and MH models using mean values as representative are shown in Figures S7 and S8 for the first 6 harmonics. To quantify how much better one model is than the other we take the best fit from each and apply the Bayesian analysis in a similar manner to case one. Firstly we set the priors for BV or MH being correct to be 50%, i.e. begin without any bias. Then the experimental data is compared to the simulated data, one harmonic at a time. For example comparison between

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experimental and simulated data for the 1st harmonic with priors of 0.5 gives a posterior of 66.95% for the MH one being correct. Bayes’ theorem is then applied for each successive harmonic in turn and the results are summarized in Table 2. It is concluded that the MH model provides a much superior prediction (99.78%) to FTACV experimental data than the BV model (0.22%) for the oxidation of dissolved DmFc at an octanethiol modified Au electrode. Table 2 Summary of the Bayesian analysis protocol applied to MH and BV models for oxidation of DmFc in propylene carbonate at a gold modified electrode.21 Harmonic

Error

Prior Probability

Posterior Probability

E(MH)

E(BV)

P(MH)

P(BV)

P(MH|H_i)a

P(BV|H_i)

1st

0.0171

0.0346

0.5000

0.5000

0.6695

0.3305

2nd

0.0332

0.0479

0.6695

0.3305

0.7448

0.2552

3rd

0.0479

0.0924

0.7448

0.2552

0.8491

0.1509

4th

0.0561

0.2255

0.8491

0.1509

0.9576

0.0424

5th

0.0757

0.3566

0.9576

0.0424

0.9907

0.0093

6th

0.1037

0.4469

0.9907

0.0093

0.9978

0.0022

a

H_i is the experimental data from harmonic i

CONCLUSION Bayesian inference has been used to compare experimental FTACV data to simulated data generated from public domain freeware MECSim software in order to quantify electrode kinetic parameters and distinguish between different classes of mechanism. A Bayesian approach has the advantage over single point parameter value estimates usually reported in that errors and confidence regions can be established. This feature is especially useful in distinguishing reversible from quasi-reversible near the reversible limit and Butler-Volmer from Marcus-Hush models.

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In this paper, the Bayesian approach is introduced in the context of a simple comparison between two models in Case 1 (Fc0/+ at a BDD electrode) where its use correctly confirmed the literature values relative to a set of deliberately deviated electrode kinetic parameters. The key advantage of having easy to interpret 95% confidence ranges in Case 2 (Ru(NH3)63+/2+ at a GC electrode) enables 𝐸0f to be accurately identified while the 𝑘0 value was constrained to the range 𝑘0 > 0.7 cm s-1, consistent with a reversible process. Case 3 (same system as Case 1) provides an example where both 𝛼 and 𝑘0 were found to be well constrained for this quasi-reversible process. In Case 4 (DmFc0/+ process in propylene carbonate at an octanethiol modified Au electrode), Bayesian analysis was used to quantitatively show the superiority of the MH to BV model. For the experiment data considered here the MH model was found to be the preferred model by 99.78% versus only 0.22% for the BV model. The work presented in this study provides a protocol for applying Bayesian inference to parameter estimation in dynamic electrochemistry by updating the probabilities of each parameter set using different sets of experimental data. In the case of FTACV, each harmonic can be treated as an independent experiment. The same protocol is equally applicable to all branches of analytical chemistry in that Bayesian inference allows the final (posterior) probabilities from one experiment to be used as the initial (prior) probabilities for another experiment. At presented in this paper, the proposed protocol requires a pre-set range for a parameter sweep which is decided by the experimenter undertaking the data analysis and hence based on their experience. Global search optimization methods, such as Markov Chain Monte Carlo methods26,27 can be considered to avoid the possibility of human introduced bias.

Acknowledgements This research was supported by the Australian Research Council through the Discovery 19



Project (grant no. DP170101535).

Supporting Information FTACV theory−experiment comparisons for reduction of Ru(NH3)63+, oxidation of Fc and oxidation of DmFc. Contour maps of posterior probabilities and summed posterior probability distribution of the values for each kinetic parameter associated with the DmFc0/+ process.

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(16) Li, J.; Bond, A. M.; Zhang, Probing electrolyte cation effects on the electron transfer kinetics of the [α-SiW12O40]4−/5− and [α-SiW12O40]5−/6− processes using a boron-doped diamond electrode. J. Electrochim. Acta 2015, 178, 631-637. (17) Guo, S.-X.; Bond, A. M.; Zhang, J. Fourier transformed large amplitude alternating current voltammetry: principles and applications. J. Rev. Polarography 2015, 61, 21-32. (18) Bentley, C. L.; Li, J.; Bond, A. M.; Zhang, J. Mass-transport and heterogeneous electron-transfer kinetics associated with the ferrocene/ferrocenium process in ionic liquids. J. Phys. Chem. C 2016, 120, 16516-16525. (19) Li, J.; Bentley, C. L.; Bond, A. M.; Zhang, J. Dual-frequency alternating current designer waveform for reliable voltammetric determination of electrode kinetics approaching the reversible limit. Anal. Chem. 2016, 88, 2367-2374. (20) Kennedy, G. F.; Bond, A. M.; Simonov, A. N. Modelling ac voltammetry with MECSim: facilitating simulation–experiment comparisons. Curr. Opin. Electrochem. 2017, 1, 140-147. (21) Li, J.; Kennedy, G. F.; Bond, A. M.; Zhang, J. Demonstration of superiority of the Marcus–Hush eectrode kinetic model in the electrochemistry of dissolved decamethylferrocene at a gold-modified electrode by Fourier-transformed alternating current voltammetry. J. Phys. Chem. C 2018, 122, 9009-9014. (22) Peachey, T.; Mashkina, E.; Lee, C.-Y.; Enticott, C.; Abramson, D.; Bond, A. M.; Elton, D.; Gavaghan, D. J.; Stevenson, G. P.; Kennedy, G. F. Leveraging e-Science infrastructure for electrochemical research. Phil. Trans. R. Soc. A 2011, 369, 3336-3352. (23) Mashkina, E.; Peachey, T.; Lee, C.-Y.; Bond, A. M.; Kennedy, G. F.; Enticott, C.; Abramson, D.; Elton, D. Estimation of electrode kinetic and uncompensated resistance parameters and insights into their significance using Fourier transformed ac voltammetry and e-science software tools. J. Electroanal. Chem. 2013, 690, 104-110. (24) Simonov, A. N.; Morris, G. P.; Mashkina, E. A.; Bethwaite, B.; Gillow, K.; Baker, R. E.; Gavaghan, D. J.; Bond, A. M. Inappropriate use of the quasi-reversible electrode kinetic model in simulation-experiment comparisons of voltammetric processes that approach the reversible limit. Anal. Chem. 2014, 86, 8408-8417. (25) Bond, A. M.; Mashkina, E. A.; Simonov, A. N. Developments in Electrochemistry: Science Inspired by Martin Fleischmann; Pletcher, D., Tian, Z-Q. & Williams, D. E. (eds.). UK: John Wiley & Sons, p. 21 - 47. (26) Gavaghan, D. J.; Cooper, J.; Daly, A. C.; Gill, C.; Gillow, K.; Robinson, M.; Simonov, A. N.; Zhang, J.; Bond, A. M. Use of Bayesian inference for parameter recovery in DC and AC voltammetry. ChemElectroChem. 2018, 5, 917-935. (27) Robinson, M.; Simonov, A. N.; Zhang, J.; Bond, A. M.; Gavaghan, D. Separating the effects of experimental noise from inherent system variability in voltammetry: The [Fe(CN) 3–/4– process. Anal. Chem. 2019, 91, 1944-1953. 6] (28) Bond, A. M.; Duffy, N. W.; Guo, S.-X.; Zhang, J.; Elton, D. Changing the look of voltammetry.. Anal. Chem. 2005, 186A-195A. (29) Bard, A. J.; Faulker, L. R. Electrochemical Methods, 2nd ed.; Wiley: New York, 2001. (30) Wang, Y.; Rogers, E. I.; Compton, R. G. The measurement of the diffusion coefficients of ferrocene and ferrocenium and their temperature dependence in acetonitrile using double potential step microdisk electrode chronoamperometry. J. Electroanal. Chem. 2010, 648, 15-19. (31) Bano, K.; Zhang, J.; Bond, A. M.; Unwin, P. R.; Macpherson, J. V. Diminished electron transfer kinetics for [Ru(NH3)6]3+/2+, [α-SiW12O40]4–/5– and [α-SiW12O40]5–/6– processes at boron-doped diamond electrodes. J. Phys. Chem. C 2015, 119, 12464-12472. (32) Altman, D. G.; Bland, J. M. Interaction revisited: the difference between two estimates. BMJ 2003, 326, 219. 22


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Application of Bayesian Inference in Fourier Transformed Alternating

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