Separating the effects of experimental noise from inherent system variability in voltammetry: the $[$Fe(CN)$_6]^{3-/ 4-}$ process

Sepa rating the effects of exp erimental noise from inherent system va riabilit y in voltammetry: the [ F e(CN) 6 ] 3 − / 4 − p ro cess Ma rtin Robinson, ∗ , † Alexandr N Simonov, ‡ Jie Zhang, ‡ Alan Bond, ∗ , ‡ and David Gavaghan ∗ , † † Dep artment of Computer Scienc e, University of Oxfor d, Wolfson Building, Parks R o ad, Oxfor d, O X1 3QD, Unite d Kingdom. ‡ Scho ol of Chemistry, Monash University, Clayton, Vic. 3800, Austr alia. E-mail: ma rtin.robinson@cs.ox.ac.uk; alan.b o nd@monash.edu.au; david.gavaghan@cs.ox.ac.uk Abstract Recen tly , w e hav e introduced the use of techniques dra wn from Bay esian statistics to recov er kinetic and thermo dynamic parameters from voltammetric data, and w ere able to sho w that the tec hnique of large amplitude ac v oltammetry yielded significan tly more accurate parameter v alues than the equiv alent dc approac h. In this paper we build on this work to show that this approach allows us, for the first time, to separate the effects of random exp erimen tal noise and inheren t system v ariability in v oltammetric exp erimen ts. W e analyse ten rep eated exp erimen tal data sets for the [F e(CN) 6 ] 3 − / 4 − pro cess, again using large-amplitude ac cyclic voltammetry . In eac h of the ten cases w e are able to obtain an extremely go o d fit to the exp erimen tal data and obtain v ery narro w distributions of the reco vered parameters go verning both the faradaic (the re- v ersible formal faradaic potential, E 0 , the standard heterogeneous charge transfer rate 1 constan t k 0 , and the charge transfer co efficient α ) and non-faradaic terms (uncompen- sated resistance, R u , and double la yer capacitance, C dl ). W e then emplo y hierarc hical Ba yesian metho ds to recov er the underlying “h yp erdistribution” of the faradaic and non-faradaic parameters, sho wing that in general the v ariation b et ween the exp erimen- tal data sets is significantly greater than suggested by individual exp eriments, except for α where the inter-experiment v ariation w as relatively minor. Correlations b etw een pairs of parameters are provided, and for example, reveal a weak link b etw een k 0 and C dl (surface activit y of a glassy carb on electro de surface). Finally , we discuss the implications of our findings for voltammetric exp erimen ts more generally . In tro duction In a previous pap er, 1 w e describ ed the use of Bay esian inference for quantitativ e comparison of voltammetric methods for in v estigating electro de kinetics. W e illustrated the utilit y of the approac h b y comparing the information con tent in both dc and ac v oltammetry at a planar electrode for the case of a quasi-rev ersible one electron reaction mec hanism. Using b oth syn thetic and exp erimental data, w e w ere able to demonstrate that realistic lev els of purely random experimental (Gaussian) noise ha ve a relativ ely minor affect on the in verse problem of reco v ering b oth the faradaic (the rev ersible formal p otential, E 0 , the standard heterogeneous charge transfer rate constant k 0 , and the c harge transfer coefficient α ) and non-faradaic (uncomp ensated resistance, R u , and double la yer capacitance, C dl ) parameters that go vern this reaction mechanism. W e also demonstrated the clear adv antages in terms of accuracy of parameter recov ery of the large amplitude ac approach. With this ability of b eing able to reco ver parameter v alues reliably from a single exp erimen tal data set in place, w e are no w in a p osition to go on to inv estigate system level v ariabilit y i.e. if we rep eat the same exp erimen t multiple times and implement our Bay esian parameter reco very pro cedure for each data set indep endently , how repro ducibly do we recov er the go v erning parameters? T o approac h this problem, w e again return to the “pathological” [F e(CN) 6 ] 3 − / 4 − pro cess. 2 It is w ell known that for this system the kinetic parameters rep orted are highly v ariable ev en when using apparen tly identical electro des and conditions (see 2–14 and references cited therein). Using our new approach we are able to sho w that this difficult y is not due to the impact of random exp erimental noise, since for each individual data set w e are able to fit the mathematical mo del to the exp erimen tal data extremely accurately . Ho wev er, we are able to demonstrate that the recov ered v alues from eac h individual data set v ary significantly from one another, i.e. the system itself v aries b et ween exp erimental runs, and so the reco vered parameter v alues are extremely sensitiv e to the precise exp erimental conditions pertaining on that particular run. Since w e hav e already demonstrated its adv antages ov er the dc approac h, w e restrict ourselves to the ac case in this pap er. Metho ds Exp erimen tal metho ds Details of the exp erimen tal data sets used in this pap er ha ve b een giv en previously in. 2,15 In summary , large amplitude ac v oltammetry w as p erformed in a standard three-electrode cell, using a glassy carb on macro disk (diameter 3 mm) w orking electro de. All potentials are rep orted v ersus an Ag/AgCl/KCl(3 M) reference electro de (hereinafter Ag/AgCl). The frequencies, amplitude and scan rate w ere 9.02 Hz, 0.080 V and 0.894 Vs − 1 resp ectiv ely and data were collected o ver the p oten tial range of 0.5 to -0.1 V v ersus Ag/AgCl. The surface area of the electro de was estimated as 0 . 070 cm 2 . The v alue of the diffusion co efficient, D , of [F e(CN) 6 ] 3 − w as found to b e 7 . 2 × 10 − 6 cm 2 s − 1 , as describ ed in. 15 In this pap er w e mak e use of the ten rep eated data sets for ac voltammetry taken from 15 for the reduction of aqueous 1.0 mM [F e(CN) 6 ] 3 − in 3 M KCl aqueous electrolyte solution. In order to reduce the amount of computation required for the parameter inference, we use a mo ving av erage windo w of length 21 to reduce the n umber of experimental data p oints within eac h data set to ab out 25,000 data p oints. 3 Mathematical Mo delling The details of the mathematical and computational mo delling approac h that we hav e tak en in this pap er were given previously in. 1,15 In summary , our c hosen exp erimen tal system is mo delled as a quasi-reversible reaction A + e − E 0 ,k 0 ,α − − − − * ) − − − − B , (1) where sp ecies A and B are in solution, and E 0 , k 0 , and α are the reversible formal p o- ten tial, standard heterogeneous charge transfer rate constant at E 0 and the c harge trans- fer co efficient, resp ectively . W e assume that the Butler-V olmer formalism applies to the electron transfer pro cess. 16–18 W e also assume that both conv ection and migration can b e neglected since w e are using a macro disk stationary electro de and an excess of supporting electrolyte, resp ectiv ely . Since w e assume equal diffusion co efficients for each sp ecies A and B ( D A = D B = D ) we need to solv e for the concen tration of only one of the species (i.e. the concen trations c A , c B of sp ecies A and B satisfy c A = c ∞ − c B , where c ∞ is the bulk concen tration of sp ecies A ) and w e choose to solv e for sp ecies A . W e can then use Fic k’s second law to mo del the v ariation with time of sp ecies A via ∂ c A ∂ t = D ∂ 2 c A ∂ x 2 , (2) where x is distance from the electro de surface and t is time. The initial and b oundary conditions are 4 c A ( x, 0) = c ∞ c A → c ∞ , as x → ∞ , t > 0 . (3) A t the electro de surface, x = 0, for t > 0, w e ha ve the conserv ation and flux conditions D ∂ c A ∂ x = I f F S , (4) along with the Butler-V olmer condition D ∂ c A ∂ x = k 0  ( c ∞ − c A ) exp  (1 − α ) F RT ( E eff ( t ) − E 0 )  − c A exp  − α F RT ( E eff ( t ) − E 0 )  . (5) Here, I f is the faradaic current, S is the electro de area, and E eff ( t ) is the effe ctive applied p oten tial (defined b elo w). W e complete the mo del b y defining E app ( t ) to b e the applied p otential, then for the case of an ac voltammetry ramp w e hav e E app ( t ) = E start      + v t + ∆ E sin ( ω t ) , 0 ≤ t ≤ t reverse , − v t + 2 v t reverse + ∆ E sin ( ω t ) , t reverse ≤ t ≤ 2 t reverse (6) where v is the sweep rate, E start is the initial p otential, t reverse is the time of switching from the forw ard to the rev erse sw eep in cyclic v oltammetry , ω is the radial frequency and ∆ E is the amplitude of the sine w a ve. The effe ctive applied p oten tial can now b e defined as 5 E eff ( t ) = E app ( t ) − E drop = E app ( t ) − I tot R u where E drop mo dels the effect of uncomp ensated resistance, R u . I tot is the total (measured) curren t, and combines the faradaic current and the background capacitiv e current, I c , which can b e mo delled as I c = C dl dE eff dt , (7) where C dl is the double lay er capacitance (assumed constant in this w ork), and then I tot = I f + I c . (8) Equations 2 – 8 are non-dimensionalised as describ ed previously . 1,15 The resulting non- dimensional system of equations is solved using an implicit finite difference metho d with an exp onen tially expanding grid (again, as describ ed previously 19 ). W e can now see mathemati- cally that the reaction mechanism in Eq. 1 is gov erned b y fiv e parameters ( E 0 , k 0 , α, C dl , R u ), and we will collectively denote these parameters by the vector θ . The inverse problem that w e wish to solv e can b e defined as finding the best p ossible appro ximation to θ given our measured exp erimental output trace of the current I data tot v ersus p oten tial. P arameter Reco v ery Metho ds In a recen t pap er we describ ed in detail how metho ds of Ba yesian inference can b e used to solv e the inv erse problem of parameter reco very 1 from v oltammetric data. W e illustrated ho w these metho ds yield not only a p oint estimate for eac h parameter of in terest, but also 6 a measure of our confidence in that estimate. F ull details of the approach that we adopted, including the algorithms used can b e found in, 1 so that here w e simply give a brief outline of the metho ds. Ba y esian Inference W e denote b y y = ( y 1 , . . . , y T ) the exp erimental data trace, that is, the total measured curren t I data tot, t at eac h time p oin t t . In our previous w ork (see Figure 4 and T able S2 of 15 ) w e demonstrated that, to a very goo d appro ximation, experimental measuremen ts can be assumed to b e sub ject to normally distributed random noise, typically with zero mean and some standard deviation whic h w e will denote b y σ (w e sho wed that typical v alues of the standard deviation of the exp erimental noise are in the range 1 to 2% of the p eak cur- ren t). Our mathematical mo del of the system, Eqs 2 to 8 ab ov e, assumes that the observed exp erimen tal data y is a function of the parameters of interest θ , θ = ( E 0 , k 0 , α, C dl , R u ) . W e no w assume further that the parameters θ are themselves dra wn from a probabil- it y distribution. W e can then frame our in verse problem as trying to find this probability distribution for θ given the observe d values of the data y , and denote this probability dis- tribution as P ( θ | y ) (the v ertical line indicates that the v alues of y are given ). In Ba yesian inference, P ( θ | y ) is termed the p osterior pr ob ability density or p osterior distribution ; this is the distribution that we w an t to appro ximate. W e now mak e use of Bay es’ rule which states P ( θ | y ) = P ( y | θ ) P ( θ ) P ( y ) , (9) where P ( θ ) is called the prior distribution of θ and is chosen to capture any prior kno wledge w e ha ve of θ b efore an y experimental observ ation. The distribution P ( y | θ ) is the probabilit y 7 densit y of the exp erimental data y giv en a mo del parameterised with parameters θ , and is termed the likeliho o d of the data; assuming a kno wn distribution of the error in the data this lik eliho o d can be calculated. P ( y ) is a normalising term (whic h is the in tegral of all p ossible densities P ( y , θ ) = P ( y | θ ) P ( θ ) ov er all v alues of θ ), and ensures that the p osterior densit y P ( θ | y ) integrates to 1. In practice, the calculation of this normalising term (which can be v ery computationally expensive) is a voided by considering ratios of the likelihoo d (see b elo w). Calculating the lik eliho o d W riting the likelihoo d as L ( θ | y ) = P ( y | θ ) , (10) w e can re-arrange Bay es’ rule in Eq. 9 to give P ( θ | y ) ∝ P ( θ ) L ( θ | y ) . (11) Since we assume that the errors are indep enden t at eac h time point, the conditional prob- abilit y density of observing the experimental trace from t = 1 , . . . , T given θ is simply the pro duct of the probabilit y densit y functions at eac h time p oin t, that is, the lik eliho o d is giv en b y L ( θ | y ) = T Y t =1 P ( y t | θ ) = T Y t =1 N ( y t | f t ( θ ) , σ 2 ) (12) = T Y t =0 1 √ 2 π σ 2 exp − ( y t − f t ( θ )) 2 2 σ 2 ! , (13) using the assumption that the exp erimental noise is normally distributed with a mean zero and v ariance of σ 2 . F or notational simplicity w e ha ve set f t ( θ ) = I model tot ,t . 8 Since, in most exp erimental situations in electrochemistry , a priori w e will hav e only a rough idea of what the v alues of the parameters are likely to b e, w e assume an “uninforma- tiv e” prior and use a uniform distribution for eac h parameter across a suitably wide range, that is P ( θ ) =      c, { θ } in some suitably chosen 5-dimensional hypercub e, 0 , otherwise , (14) where c is a non-zero finite normalizing constan t. The b ounds for this h yp ercub e were set to E rev er se + 0 . 1 δ E ≤ E 0 ≤ E start − 0 . 1 δ E , 0 ≤ k 0 ≤ 1 cm s − 1 , 0 . 4 ≤ α ≤ 0 . 6 , 0 ≤ C dl ≤ 200 µ F cm − 2 , 0 ≤ R u ≤ 80 Ω , where δ E = E start − E rev er se . Note that this prior is only used for the single level Mark o v Chain Monte Carlo (MCMC) algorithm; in the hierarc hical MCMC algorithm, this is replaced b y a m ultiv ariate normal, as describ ed b elo w. How ever, these b ounds are still used in the hierarchical MCMC algorithm to prev ent the low er lev el samplers from accepting an y samples that lie outside these bounds. Mark o v Chain Mon te Carlo parameter inference T o obtain a sample from the p osterior distribution P ( θ | y ) we mak e use of the Marko v Chain Mon te Carlo metho d. In outline, this inv olves finding an approximation to the p osterior dis- tribution P ( θ | y ) by drawing a finite (but sufficien tly large to b e accurate) n umber of samples 9 from this distribution. T o do this we sim ulate a Marko v Chain whose limiting distribution is the required posterior distribution using an efficien t implementation of the Metr op olis- Hastings algorithm, 20 within which candidate parameter sets are prop osed from a pr op osal distribution q ( θ cand | θ i ) which dep ends only on the previously accepted parameter set θ i ; w e take q ( θ cand | θ i ) to b e a m ultiv ariate normal distribution. If the prop osed parameter set con tains any parameters outside the range of the prior, then the parameter set is assigned an acceptance probability of 0, i.e. it is rejected, and the previously accepted parameter set is added to the Mark ov chain — that is, θ i +1 = θ i . Otherwise, w e compare θ cand to the current parameter set θ i b y calculating the ratio of the p osteriors of the tw o parameter sets. If the candidate parameter set has a greater p osterior densit y v alue than the existing parameter set then it will b e added to the Marko v c hain, that is θ i +1 = θ cand . Otherwise, (making use of Eq. 11) the prop osed parameter is accepted with probability , r , giv en b y r = min ( P ( θ cand ) L ( θ cand | y ) P ( θ i ) L ( θ i | y ) , 1 ) . (15) If the prop osed parameter set is rejected (with probabilit y 1 − r ), then the previously accepted parameter set is again added to the Marko v c hain — that is, θ i +1 = θ i . Since the likelihoo ds for large samples are extremely small, in practice we work with the natural log of the lik eliho o d which reduces to l ( θ | y ) = − T log ( σ ) − 1 2 σ 2 T X t =1 ( y t − f t ( θ )) 2 , (16) where terms which are constant in θ ha v e b een remo ved (since these will cancel on taking the difference of log-lik eliho o ds in the Metrop olis-Hastings algorithm). More comprehen- siv e descriptions of the theory of MCMC can be found in the statistics literature (see for example 21 ). 10 Practical implementation of the Metrop olis-Hastings Algorithm In practice, w e mak e use of an adaptiv e co v ariance matrix v ersion of the Metropolis-Hastings algorithm whic h helps identify the directions in parameter space whic h hav e the highest like- liho o d v alues. 22 A t each iteration of the algorithm, the cov ariance matrix of the multiv ariate normal distribution is up dated and a scalar v alue is also up dated to define the width of the distribution. In the results presented in this pap er, we run our MCMC c hains for 10,000 samples and discard the first 5,000 samples as ‘burn in’ (see 21 ). T o ensure efficiency in our Mon te-Carlo sampling, w e first find the lo cation of the optim um, that is, the maxim um lik eliho o d estimate of θ using a standard global minimisation algorithm (w e use the cma-es algorithm 23 ), whic h we use as a seed p oint for the MCMC algorithm as describ ed in. 1 In the results section, these samples are sho wn as histograms which illustrate the nature of the p osterior distribution. Hierarc hical Ba y esian Inference In the results section, the MCMC algorithm described ab o ve is used to estimate the pa- rameter v alues and their posterior distributions from each of the ten rep eat runs of the ac v oltammetry experiment for the reduction of aqueous 1 mM [F e(CN) 6 ] 3 − , as describ ed in the Exp erimen tal metho ds section. This will allo w us to sho w the degree of v ariabilit y in the recov ered v alues of the parameters across these ten data sets. This will in turn allow us to p ostulate that on each rep eat of the exp erimen t the parameters θ = { E 0 , k 0 , α, C dl , R u } are themselv es dra wn from a m ultiv ariate normal distribution with mean h yp er-parameters µ = ( ˆ E 0 , ˆ k 0 , ˆ α, ˆ C dl , ˆ R u ) and a co v ariance h yp er-parameter matrix Σ . Our aim no w is to sample from the distributions of µ and Σ to enable us to quan tify ho w θ v aries betw een differen t exp erimen ts. W e first write the p osterior distribution of all of our parameters giv en the data, taking in to accoun t our new hierarchical mo del structure 11 P ( µ , Σ , θ 1 , ..., θ n , | y 1 , ..., y n ) ∝ P ( µ , Σ , θ 1 , ..., θ n ) n Y i =0 P ( y i | θ i ) , (17) = P ( µ , Σ ) n Y i =0 P ( θ i | µ , Σ ) n Y i =0 P ( y i | θ i ) , (18) where θ 1 , ..., θ n are all the b ottom lev el parameters for the n different exp erimen ts, and y 1 , ..., y n are the corresp onding measurements. W e choose a m ultiv ariate normal distribution for the b ottom lev el parameters θ P ( θ i | µ , Σ ) = N ( µ , Σ ) . (19) It no w remains to c ho ose a suitable hyper-prior P ( µ , Σ ), whic h for ease of computation is generally taken to b e a normal-inv erse-Wishart distribution (see for example 24 ) P ( µ , Σ ) = N I W ( µ 0 , κ 0 , ν 0 , Ψ ) . (20) The distributions in Eqs. 19 and 20 are c hosen to b e conjugate, so that the conditional distribution of the hyper-parameters can b e analytically derived as 25 P ( µ , Σ | θ 1 , ..., θ n ) = N I W ( κ 0 µ 0 + n ˆ θ κ 0 + n , κ 0 + n, ν 0 + n, Ψ + C + κ 0 n κ 0 + n ( ˆ θ − µ 0 )( ˆ θ − µ 0 ) T ) , (21) where ˆ θ and C are the sample mean and cov ariance of the b ottom level parameters ˆ θ = 1 n n X i =0 θ i , (22) C = n X i =0 ( θ i − ˆ θ )( θ i − ˆ θ ) T . (23) 12 W e can now use our original adaptive MCMC algorithm to sample from the b ottom level parameters, combined with classical Gibbs sampling and Eq. 21 to sample from the h yp er- parameters (see Algorithm 1 for details). F or the h yp erprior parameters we use κ 0 = 0 and ν 0 = 1. W e set µ 0 to b e the centre p oin t of the 5-dimensional hypercub e in Eq. 14, and Ψ as a diagonal matrix with the standard deviation of each parameter set to half the width of this same hypercub e. T o prev ent the b ottom level MCMC chains from exploring unph ysical parameter regimes, w e automatically reject an y prop osed p oint whic h lies outside of the h yp ercub e. Algorithm 1 Metrop olis within Gibbs s = 0 µ s , Σ s = SampleF rom N I W ( µ 0 , κ 0 , ν 0 , Ψ ) { using Eq. 20 } rep eat for i = 1 to n do θ i = Adaptiv eMCMCStep( µ s , Σ s ) { see Eq. 15 and enclosing section } end for s = s + 1 µ s , Σ s = SampleF rom N I W ( µ 0 , κ 0 , ν 0 , Ψ , θ 1 , ..., θ n ) { using Eq. 21 & 22 } un til finished sampling Generating synthetic data as a test case T o test our inference pro cedure and algorithms we make use of “synthetic” test data i.e. we generate ten syn thetic data sets b y solving Equations 2 to 8 for a randomly c hosen set of v alues of θ = ( E 0 , k 0 , α, C dl , R u ), drawn from a m ultiv ariate normal distribution with mean θ true = (7 . 27 , 2 . 01 , 0 . 53 , 3 . 70 × 10 − 3 , 1 . 06 × 10 − 2 ) (in non-dimensional units), and standard deviation σ true = (0 . 06 , 0 . 7 , 0 . 005 , 0 . 7 × 10 − 3 , 0 . 3 × 10 − 2 ). W e then add randomly generated Gaussian noise at eac h time p oin t with zero mean and standard deviation of 0.3% of the maxim um curren t (c hosen to match the exp erimental noise level). 13 Results Syn thetic data T o v erify that w e ha ve implemented our hierarchical Bay es algorithms correctly w e briefly describ e the results of using synthetic data to test our inference pro cedure. Figure 1 sho ws the sampled distributions of the hyper-parameters. The dashed lines indicate the exp ected p eak of each distribution, as calculated from the sample mean and v ariance of the ten true v alues of θ true . As can b e seen the hierarchical MCMC algorithm obtains the correct p eak for all the five differen t mean and v ariance hyper-parameters. Exp erimen tal data Figure 2 sho ws an example of the initial fitting process for exp erimental data set 1. Once the b est fit for each data set w as obtained using maximum lik eliho o d estimation and the cma-es optimisation algorithm, each of the b ottom lev el MCMC algorithms w as initialised at these points. Then Algorithm 1 was used to generate 5,000 samples (10,000 total samples, of whic h the first 5,000 w as discarded as burn-in) of the parameters θ i and h yp er-parameters µ and Σ . Figure 3 (left column) shows histograms of the samples of θ i tak en from the b ottom level samplers in Algorithm 1. On the same axis is drawn the p osterior predictiv e distribution for eac h v ariable, calculated b y summing the individual Gaussian distributions describ ed b y eac h sample of the hyper-parameters µ and Σ . The p osterior predictiv e distributions describ es the distribution of eac h parameter that would b e expected from another rep eat of the exp eriment, giv en the results of the 10 already observed exp erimen ts. As can b e seen, this distribution co vers the width of all 10 b ottom lev el samples of θ i , and illustrates the significantly greater v ariation in eac h parameter that is exp ected b etw een subsequen t exp erimen ts. It is in teresting to compare the samples obtained from the hierarc hical mo del to those 14 0.212 0.213 0.214 0.215 0.216 E 0 ( V ) 0 1000 P ( E 0 ) 0 2 4 6 2 E 0 ( V 2 ) 1e 6 0.0 0.5 P ( E 0 ) 1e6 0.0 0.5 1.0 1.5 k 0 ( c m s 1 ) 1e 2 0 200 P ( k 0 ) 0.0 0.2 0.4 0.6 0.8 1.0 2 k 0 ( c m 2 s 2 ) 1e 4 0 25000 P ( k 0 ) 0.520 0.525 0.530 0.535 0 200 P ( ) 0.0 0.5 1.0 1.5 2.0 2 1e 4 0 25000 P ( ) 10 15 20 25 30 35 C d l ( F c m 2 ) 0.0 0.2 P ( C d l ) 0 100 200 300 400 2 C d l ( p F 2 c m 4 ) 0 2 P ( C d l ) 1e 2 2 4 6 8 10 12 R u ( ) 0.00 0.25 P ( R u ) 0 10 20 30 40 2 R u ( 2 ) 0.0 0.1 P ( R u ) Figure 1: Histograms of sampled h yp er-parameter distributions µ (left column) and the v ariances of Σ (righ t column), generated b y using Algorithm 1 on 10 sets of synthesised ac voltammetric data. The dashed lines indicate the sample mean and sample standard deviation of the true parameters used to generate the synthetic datasets, and since this aligns correctly with the maxim um likelihoo d p oint of the histograms w e can b e confiden t that our implemen tation of Algorithm 1 is sampling the correct p osterior distribution for µ and Σ . 15 0 2 4 6 8 10 12 t ( s ) 60 40 20 0 20 40 60 I t o t ( A ) experiment simulation Figure 2: Comparison betw een the simulated current trace and the exp erimental data for the reduction of aqueous 1 mM [F e(CN) 6 ] 3 − (dataset 1 from reference 15 ). Sim ulation parameters w ere obtained b y maxim um likelihoo d estimate, and their v alues in dimensional units w ere E 0 = 0 . 214 V, k 0 = 0 . 010 cm s − 1 , α = 0 . 528, C dl = 16 . 9 µ F cm − 2 , R u = 0 . 00 Ω. 16 tak en using the original non-hierarc hical model (i.e. just running ten indep endent MCMC c hains on the ten differen t data sets), and this is shown in Figure 3 (righ t column). As can b e seen, the result in this case are almost iden tical to the hierarc hical mo del, giving us confidence that we are capturing the distributions of θ i correctly in each case. The c hief b enefit of the hierarc hical mo del is that it allo ws us to quan tify (with the h yp er-parameters) the v ariability of the parameters b etwe en different exp erimen ts. Figure 4 shows the histograms for the hyper-parameters samples of µ (left) and Σ (righ t). Also sho wn as vertical dashed lines are the the sample mean and v ariance of the concatenated ten θ i c hains. The most obvious feature of these plots is that these hyper-parameter distri- butions clearly show a m uc h wider confidence interv al as compared with the individual θ i distributions. While the v alue of each parameter for eac h individual experiment is known with high accuracy , once the v ariability b etw een exp eriments is tak en in to account we see that the p ossible parameter range is significantly broadened. Discussion The Bay esian data analysis approach in tro duced in this pap er pro vides access to funda- men tally new kno wledge that assists in elucidating n uances that hav e contributed to the highly non-reproducible electro de kinetic data published for the “pathologically v ariable” [F e(CN) 6 ] 3 − / 4 − pro cess. T raditional heuristic and data analysis optimisation metho ds pro- duce only single p oin t v alues for a limited set of parameters and do not quan tify the sys- tem v ariability , which is crucial information. Using a statistically based Bay esian inference approac h, w e are no w able to sho w that the difficult y in ac hieving repro ducibilit y in the v oltammetry is not asso ciated with the impact of random noise, since for each data set we are able to fit the exp erimental data extremely accurately using a mo del derived from use of Butler-V olmer electron transfer kinetics, mass transp ort by planar diffusion, uncomp ensated resistance and double lay er capacitance. Thus, while substantial v ariation in k 0 from ab out 17 0.214 0.216 0.218 E 0 ( V ) 0 20000 P i ( E 0 ) 1.0 1.5 k 0 ( c m s 1 ) 1e 2 0 20000 P i ( k 0 ) 0.515 0.520 0.525 0.530 0 500 P i ( ) 16 18 20 22 24 C d l ( F c m 2 ) 0 20 P i ( C d l ) 0 20 40 R u ( ) 0.0 0.2 P i ( R u ) 0.9 1.0 1.1 ( A ) 0 100 P i ( ) 100 200 P ( E 0 ) 25 50 75 P ( k 0 ) 25 50 75 P ( ) 0.05 0.10 P ( C d l ) 1 2 P ( R u ) 1e 2 0.213 0.214 0.215 0.216 0.217 0.218 E 0 ( V ) 0 10000 P i ( E 0 ) 0.75 1.00 1.25 1.50 1.75 k 0 ( c m s 1 ) 1e 2 0 20000 P i ( k 0 ) 0.515 0.520 0.525 0.530 0 500 P i ( ) 16 18 20 22 24 C d l ( F c m 2 ) 0 10 P i ( C d l ) 0 20 40 60 R u ( ) 0.0 0.2 P i ( R u ) 0.90 0.95 1.00 1.05 1.10 ( A ) 0 100 P i ( ) Figure 3: Analysis of ten indep enden t ac voltammetric exp erimen ts for the reduction of aqueous 1 mM [F e(CN) 6 ] 3 − using the hierarc hical MCMC algorithm 1. The samples obtained from the 10 low er level adaptive MCMC samplers (i.e. θ i ) are sho wn as histograms with the axis lab el P i ( · ), and the differen t chains from i = 1 , .., 10 are shown with differen t colours. The left column plots sho w the samples obtained using the hierarchical model, and these also sho w the p osterior predictiv e distribution P ( · ) for eac h v ariable, calculated by summing the Gaussian distributions describ ed b y the samples of the hyper-parameters µ and Σ (see Figure 4 for histograms of these hyper-parameter samples). F or comparison, the right column plots sho w the samples of θ i with no hierarc hical mo del (and thus no h yp er-parameter samples). This is identical to the analysis done in our previous pap er. 1 18 0.210 0.212 0.214 0.216 0.218 E 0 ( V ) 0 500 P ( E 0 ) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 2 E 0 ( V 2 ) 1e 5 0 2 P ( E 0 ) 1e5 0.0 0.5 1.0 1.5 2.0 k 0 ( c m s 1 ) 1e 2 0 200 P ( k 0 ) 0.2 0.4 0.6 0.8 2 k 0 ( c m 2 s 2 ) 1e 4 0.0 0.5 P ( k 0 ) 1e5 0.510 0.515 0.520 0.525 0.530 0 200 P ( ) 0.0 0.5 1.0 1.5 2 1e 4 0 20000 P ( ) 12.5 15.0 17.5 20.0 22.5 25.0 27.5 C d l ( F c m 2 ) 0.0 0.2 P ( C d l ) 0 10 20 30 40 50 60 2 C d l ( p F 2 c m 4 ) 0.00 0.05 P ( C d l ) 20 0 20 40 R u ( ) 0.00 0.05 P ( R u ) 0 500 1000 1500 2000 2 R u ( 2 ) 0 2 P ( R u ) 1e 3 Figure 4: Histograms of h yp er-parameter samples of µ (left) and Σ (right) obtained b y ap- plying the hierarc hical MCMC algorithm 1 to the ten experimental ac voltammetric datasets for the reduction of aqueous 1 mM [F e(CN) 6 ] 3 − . Note that only the diagonal elements of Σ (i.e. the v ariances σ 2 ) are shown. F or comparison, the dashed lines show the sample mean and v ariance taken across all ten low er level MCMC samples of θ i . 19 0.002 to 0.018 cm s − 1 is evident in 10 individual exp eriments at a nominally identical elec- tro de surface (Figure 3), conformance to the quasi-reversible model is exceptionally go o d for eac h individual exp eriment. The v ariation in performance of the now very widely used glassy carb on electro de w as iden tified as a p oin t of concern so on after the material was introduced into electroanalytical c hemistry. 26 In essence, the exact nature of the glassy carb on, and indeed other carb on based surfaces, has their origin in the metho d (e.g. temp erature) of man ufacture, nature of pre-treatment and its history as an electro de (see for example 13,14,26–32 and references cited therein). Accordingly , it seems likely that the widely v ariable electro de kinetic data asso ciated with published studies on the [F e(CN) 6 ] 3 − / 4 − pro cess (summarised in Eq. 24) F e(CN) 3 − 6 + e − − − * ) − − F e(CN) 4 − 6 ( E 0 , k 0 , α, C dl , R u ) , (24) mimics the v ariability of the surface state used in the different publications. In practice, carb on electro des are highly heterogeneous with surface defects and organic functional groups in abundance when used in aqueous electrolyte media. 28–30 This seems to translate in to an electron transfer pro cess o ccurring at a surface consisting of microscopically small and distinctly differen t regions that must b e sufficiently close so that complete o v erlap of diffusion la yers occurs on the measuremen t timescale, 33 presumably allo wing the entire surface to b e successfully mo delled by appro ximating the mass transfer by planar diffusion. Th us, even though highly heterogeneous with v ariable k 0 v alues at the microscopic level, the surface b ehav es as though it is fully homogeneous, hence giving rise to measuremen t of an apparen tly single ”av eraged” k 0 v alue. Electron transfer reaction mechanisms are often classified in to inner and outer sphere categories in b oth homogeneous chemical redo x reactions that o ccur in the solution phase and in heterogeneous reactions in electro c hemistry that take place across an electrode-solution in terface. 16,33 In a chemical redo x reaction, the homogeneous outer sphere class mechanism 20 is characterised by w eak in teractions of the reactant and pro duct, as in Eq. 25, ∗ F e(CN) 3 − 6 (aq) + F e(CN) 4 − 6 (aq) − − * ) − − ∗ F e(CN) 4 − 6 (aq) + F e(CN) 3 − 6 (aq) , (25) whic h is the homogeneous analogue of the electro chemical one of interest in this study , while in the inner class, ligands in v olving bridging to a common metal cen tre ma y b e in v olved. In an outer sphere electrochemical process, the plane of closest approach to the Outer Helmholtz Plane do es not allo w p enetration of the lay er of non-sp ecifically adsorb ed or co- ordinated solv en t adhered to the electro de surface by reactan ts. In contrast, inner sphere electron transfer pro cesses inv olve sp ecifically adsorb ed reactants, 16,33 and therefore are an- ticipated to exhibit electro de kinetics that are strongly dep enden t on the c hemical nature of the electro de surface. Th us, the [F e(CN) 6 ] 3 − / 4 − pro cess can b e designated as outer sphere under homogeneous chemical redox reaction conditions but inner sphere under electro chem- ical conditions at a glassy carb on electro de. Presumably , electron transfer at such electro des is accompanied b y in teraction with surface functional groups, facilitated by the high negative c harges asso ciated with the [F e(CN) 6 ] 3 − reactan t and [F e(CN) 6 ] 4 − pro duct whic h allow sp e- cific binding or electrostatic attractive and repulsive interactions to accompany the electron transfer pro cess. On this basis, the [F e(CN) 6 ] 3 − / 4 − electro de kinetics are strongly dep enden t on the treatment and origin of the glassy carb on. Before attempting to iden tify the factors that may b e most relev ant to the v ariation encoun tered in the electro de kinetics, it is worth while reviewing the electro de pre-treatment regime. A nominally 3 mm diameter GC working electro de was purc hased from BAS. Prior to eac h set of measurements the surface of the GC electro de was thoroughly p olished with 0.3 µ m alumina p owder in the form of an aqueous slurry on a w et p olishing cloth (BAS). After p olishing, the electro de was rep eatedly washed with high purity w ater and sub jected to sonication for 10 to 20 s. T o pro vide effectiv e remo v al of any residual alumina p owder, after the initial sonication, the electro de was carefully wip ed with a clean wet p olishing 21 cloth, again w ashed with w ater and sonicated for 10 to 20 s in high purit y w ater. Finally , the w orking electro de w as dried under a nitrogen stream. As far as p ossible, each electro de preparation was undertaken under identical conditions; nev ertheless, the electro de kinetics differ substan tially from exp erimen t to exp eriment. Ho wev er, of course the history of the electro de could b e imp ortant since the electro de used for exp erimen t ten will hav e a more extensiv ely p olished electro de than that used in exp erimen t one. The narro w distribution of eac h P i ( · ) in Figure 3 enables us to conclude that the con- tribution to parameter v ariability from noise asso ciated with each particular exp eriment is minimal for all fiv e parameters estimated. F rom p erusal of Figures 3 and 4, we can conclude that while the degree of experiment-to-experiment v ariation changes with resp ect to eac h parameter, it is in general significan tly greater than the parameter v ariability from noise alone (i.e. the width of each P i ( · )). F rom the h yp er-parameter samples w e can calculate the distribution P ( · ) of each parameter that would b e exp ected from another rep eat exp er- imen t (sho wn in Figure 3, left column). Quan titativ ely , the parameter mean v alues (with one standard deviation v alues in paren thesis) as deduced from this analysis are as follows: E 0 = 0 . 215 V vs Ag/AgCl (0.002), k 0 = 0 . 010 cm s − 1 (0.005) α = 0 . 521 (0.006), C dl = 20 . 1 µF cm − 2 (4.5), R u = 15 . 2 Ω (21.0). In triguingly , experiment-to-experiment v ariation in α is very small. That is, its v alue lies in a very narrow range of ab out 0.515 to 0.525 with a standard deviation of just 0.006. It is also notable that while the v alue of R u co v ers a range from zero to ab out 35 Ω, it is alwa ys small. This means that the more imp ortan t ohmic drop term that can distort voltammograms also is small and hence not highly significan t. The large standard deviation of 21.0 Ω with a mean v alue of 15.2 Ω allows us to conclude the R u parameter only has a minor impact on the ac v oltammetry (highly conducting 3 M K Cl aqueous electrolyte) and hence on E 0 , k 0 , or α v ariability . Although the exp eriment- to-exp erimen t v ariability of E 0 is m uch greater than predicted b y P i ( E 0 ), it is still fitted to a tight regime (ab out 2 mV) when compared with the total voltage scan range. Indeed, in the absence of adsorption, E 0 is theoretically predicted to b e completely indep endent of 22 electro de material or state, suggesting that the in ter-exp eriment v ariability of E 0 is due to comp ensating factors from other parameters. C dl displa ys some v ariability with v alues lying within the range of ab out 16 to 23 µF cm − 2 whic h confirms that the electro de is not in an iden tical surface state, and hence do es hav e a v ariable level of activity for eac h exp erimen t. k 0 as noted ab ov e encompasses a wide range of ab out 0.002 to 0.018 cm s − 1 (Figure 3) with a mean v alue of 0.010 cm s − 1 and a standard deviation of 0.005 cm s − 1 . The question that arises is whether there is a correlation b etw een C dl or surface activity and k 0 . Figure 5 sho ws quan titatively the correlation betw een pairs of parameters using scat- terplots of the mean h yp er-parameter µ samples. Clearly , R u and α are not correlated at all with eac h other or an y other parameter (i.e. they giv e “shot gun” correlation plots). Ho w ever, there app ears to b e a w eak correlation of k 0 and C dl with larger k 0 v alues b eing asso ciated with data sets having higher C dl v alues. There also app ears to b e a weak corre- lation betw een k 0 and E 0 , with the more positive E 0 v alues coinciding with the larger C dl v alues. One needs to b e careful not to ov er interpret the significance of weak correlations, as other non-quantified parameters also can b e op erative. Ho wev er, a larger capacitance cur- ren t caused b y a larger activ ation of the glassy carb on surface (more surface functionality) ma y b e exp ected to increase the rate of an inner sphere pro cess, like the [F e(CN) 6 ] 3 − / 4 − one prob ed in this work. The origin of the w eak correlation of k 0 and E 0 is more problematical. The presence of extremely weak adsorption not accommo dated in the mo del is one p ossible explanation, but a small drift in reference electro de p otential cannot b e ruled out. Conclusion In summary , the Bay esian inference-inspired strategy introduced in this pap er for data ev al- uation represen ts a significant adv ance in understanding the contribution of differen t param- eters to v oltammetric data and their significance in exp eriment-to-experiment v ariabilit y at a heterogeneous electro de in a manner that has not b een p ossible in earlier studies. Perhaps 23 Figure 5: P airwise correlation plots for the mean hyper-parameter samples µ = ( ˆ E 0 , ˆ k 0 , ˆ α, ˆ C dl , ˆ R u ) (see Algorithm 1) obtained for 10 independent ac voltammetric exper- imen ts for the reduction of aqueous 1 mM [F e(CN) 6 ] 3 − . Eac h dot in the scatter plots sho ws a sample dra wn from the p osterior distribution P ( µ , Σ ) (Eq. 20), showing the correlation b et ween parameter pairs. The diagonal plots sho w histograms of eac h individual mean h yp er-parameter. 24 remark ably , eac h data set in the pathologically v ariable [F e(CN) 6 ] 3 − / 4 − pro cess conforms exceptionally w ell with sim ulated data deriv ed from the Butler-V olmer mo del of electron transfer and mass transp ort by planar diffusion even though the v ariation in k 0 is quite substan tial. In triguingly , α , unlik e k 0 , do es not rev eal significant exp eriment-to-experiment v ariation in this data analysis exercise. This ma y b e exp ected if conformance to the Butler- V olmer mo del is strong. In early electro de kinetic studies, the ideal and very homogenous mercury electro de was used. It is no w eviden t that electro de design is b ecoming very sophisticated, particularly when adv ances in materials science are used to generate highly heterogeneous electro de materials. 34 Th us, there is a tendency in electro chemistry no wada ys to use far more complex electro des than the historically imp ortant liquid mercury and pure solid metal surfaces. The new breed of highly heterogeneous electro des imply that data analysis strategies in the future will also need to b e more sophisticated. That is, use of models significan tly more complex than used in this study with Butler-V olmer theory for electron transfer and mass transp ort b y planar diffusion, as well as Bay esian forms of data analysis, will b ecome increasingly essen tial if informative exp erimental v ersus sim ulation comparisons are to b e rep orted. Ac kno wledgemen ts AMB, DJG and JZ would lik e to ackno wledge the supp ort Australian Research Council through the aw ard of a Discov ery Gran t DP170101535. 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The solid line shows the p redictive p osterior fo r E 0 , which describ es the distribu- tion of each parameter that would b e exp ected from another rep eat of the exp eriment. 30