1 Vol.:(0123456789) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports A simple and effective method for the accurate extraction of kinetic parameters using differential Tafel plots Prashant Khadke1*, Tim Tichter2, Tim Boettcher3, Falk Muench3, Wolfgang Ensinger3 & Christina Roth1 The practice of estimating the transfer coefficient ( α ) and the exchange current ( i 0  ) by arbitrarily placing a straight line on Tafel plots has led to high variance in these parameters between different research groups. Generating Tafel plots by finding kinetic current, i k from the conventional mass transfer correction method does not guarantee an accurate estimation of the α and i 0  . This is because a substantial difference in values of α and i 0 can arise from only minor deviations in the calculated values of i k  . These minor deviations are often not easy to recognise in polarisation curves and Tafel plots. Recalling the IUPAC definition of α , the Tafel plots can be alternatively represented as differential Tafel plots (DTPs) by taking the first order differential of Tafel plots with respect to overpotential. Without further complex processing of the existing raw data, many crucial observations can be made from DTP which is otherwise very difficult to observe from Tafel plots. These for example include a) many perfectly looking experimental linear Tafel plots (R2 > 0.999) can give rise to incorrect kinetic parameters b) substantial differences in values of α and i 0 can arise when the limiting current ( i L  ) is just off by 5% while performing the mass transfer correction c) irrespective of the magnitude of the double layer charging current ( i c  ), the Tafel plots can still get significantly skewed when the ratio of i 0 /i c is small. Hence, in order to determine accurate values of α and i 0  , we show how the DTP approach can be applied to experimental polarisation curves having well defined i L  , poorly defined i L and no i L at all. List of symbol α � Transfer coefficient αA � Anodic transfer coefficient αC � Cathodic transfer coefficient βc � Cathodic symmetry factor βa � Anodic symmetry factor �m � Reorganisation energy per mole η � Overpotential ηTOP � Tafel overpotential ηTA � Useable Tafel overpotential range υ � Viscosity of supporting electrolyte ω � Angular velocity A � Geometric area of a disk electrode c+ � Surface concentration of hydrogen ions cH � Surface concentration of hydrogen C∗ � Bulk reactant concentration D � Diffusion coefficient of reactant in liquid electrolyte EOCV � Open circuit potential ER � Reversible potential OPEN 1Faculty of Engineering Sciences, University of Bayreuth, Universitaetsstr.30, 95440 Bayreuth, Germany. 2Physical and Theoretical Chemistry, Free University of Berlin, Takustr. 3, 14195 Berlin, Germany. 3Department of Materials‑ and Geoscience, Technische Universitaet Darmstadt, Alarich‑Weiss‑Str. 2, 64287  Darmstadt, Germany. *email: prashant.khadke@uni-bayreuth.de http://crossmark.crossref.org/dialog/?doi=10.1038/s41598-021-87951-z&domain=pdf 2 Vol:.(1234567890) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ F � Faraday’s constant i � Measured current ic � Double layer charging current ik � Overall kinetic current ik,A � Anodic kinetic current ik,C � Cathodic kinetic current iL � Limiting current i0 � Exchange current J � Current k2 � Rate constant anodic k3 � Rate constant cathodic np � Number of electrons transferred before the rate determining step nq � Number of electrons involved in rate determining step nr � Number of electrons transferred after the rate determining step n � Overall number of electrons transferred R � Universal gas constant R2 � Coefficient of determination T � Temperature Abbreviations BV � Butler Volmer DTP � Differential Tafel Plot ECP � Exchange current plot GC � Glassy carbon HER � Hydrogen evolution reaction HOR � Hydrogen oxidation reaction OER � Oxygen evolution reaction ORR � Oxygen reduction reaction RDE � Rotating disk electrode rds � Rate determining step rpm � Rotation per minute TOP � Tafel overpotential Tafel plots (TPs) are primarily used to obtain two important catalyst performance indicators namely the transfer coefficient α , and the exchange current i0 . TPs are generated from the well-known Tafel equation, a case of the Butler–Volmer (BV) equation or Erdey-Gruz–Volmer equation1 at high overpotential, η . The BV equation with no mass transfer limitations is given as2 In Eq. (1), ik is the overall kinetic current, αA and αC are the experimentally obtained anodic and cathodic transfer coefficients corresponding to a single or multistep reaction, f = F/RT and η = E − Eeq. The variables F , R, T , E and Eeq are Faraday´s constant, the universal gas constant, the temperature, the operating potential and equilibrium potential, respectively. At larger η one of the exponential terms in Eq. (1) becomes insignificant. Consequently, Eq. (1) reduces to, In Eqs. (2) and (3), the α is either αA or αC depending on the sign of the η . From a plot of ln(ik)vs η , α is obtained from the slope of the curve in the Tafel regime and i0 is obtained from its extrapolation to zero overpotential. Since Erdey-Gruz and Volmer3 introduced the transfer coefficient in 1930s, its physical interpretation has changed significantly. A very detailed discussion on history and interpretation of transfer coefficients has been published elsewhere4. To briefly summarise, after Tafel generalized the current–voltage relation based on his hydrogen evolution reaction (HER) experiments on several different metals5, Erdey-Gruz and Volmer3 were the first to confirm the Tafel equation by applying laws of kinetics and introducing transfer coefficients. This became the theoretical basis of the Tafel equation. They suggested that at any overpotential the current (overall rate) can be expressed as the difference of the rate of forward reaction and backward reaction3. where J is the current, k2 and k3 are the rate constants of neutralization of the hydrogen atom and the ionoziation of the hydrogen respectively, c+ and cH are the surface concentrations of hydrogen ions and hydrogen and ER is the reversible potential1. Equation (4)  at higher overpotential becomes similar to the Tafel equation. Polanyi and Horiuti6 provided for the first time a physical meaning of the transfer coefficient from transition state theory and (1)ik = i0 [ eαAf η − e−αCf η ] (2)ik = i0e αf η (3)ln(ik) = ln(io)+ αf η (4)J = Fk2c+e −α(ER+η)f − Fk3cHe +α(ER+η)f 3 Vol.:(0123456789) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ also suggested that Eq. (4) can be written in a form that a constant term (commonly termed as exchange current) can be multiplied to anodic and cathodic exponential terms resulting in an equation very similar to Eq. (1). Frumkin7–9 proposed a corrected current-overpotential relation arguing that the reactant concentration, c (at low currents) in Eq. (4) may not be the bulk concentration but rather the concentration at the outer Helmholtz plane. While studying the electroreduction of H+ ion, he found that the current measured at the electrode is independent of the H+ concentration. He suggested that the reaction occurs at the outer Helmholtz plane and the local concentration at this site is different than in the bulk. He also suggested that the driving force for the electron transfer must be the difference between the electrode potential and average potential at the outer Helmholtz plane rather than the potential difference between the electrode potential and the potential in the bulk solution. For a multistep reaction (i.e. series of consecutive reaction steps) Parsons10 presented a methodology to determine the rate determining step (rds) from the Tafel slope with the help of a free energy diagram. Based on his work the transfer coefficient of a multistep reaction can be derived as11 where np is the number of electrons transferred before the rds, nq is the number of electrons involved in rds, nr is the number of electrons transferred after the rds, βa and βc are the symmetry factors of the forward and backward reaction, respectively. The nq = 0 when the chemical step becomes rate-determining and is equal to 1, when the electrochemical step is the rds. Marcus12,13, while working on his solvent fluctuation model for electron transfer, showed that the symmetry factor β (or transfer coefficient of a single step single electron reaction) is a function of overpotential as follows where �m is the reorganisation energy per mole. Hence, the modern view of Eq. (4) differs quite a lot from the time, when it was first published by Erdey-Gruz and Volmer3. However, recently Fletcher11 derived the same form of the BV equation (as in Eq. 1) from first principles stating that the outward form of the common BV equation can be still be used for Tafel analysis, but recognizing that the experimental Tafel slope can be a func- tion of overpotential. Referring to Eqs. (5)–(8), it is clear that one may observe (a) a single Tafel slope when the chemical step is rds (b) a double Tafel slope when there are two chemical steps as rds, and each rds exists at a certain overpotential range within the Tafel regime, (c) a single Tafel slope when the electrochemical step is rds and �m>> Fη , (d) curved Tafel slopes when the electrochemical step is rds and �m is small. Adding to these complexities, curved Tafel slopes can also be observed due to the concentration depletion at the surface of the elctrode14 and charging current influence15. Hence, before any mechanistic conclusions are made, it’s absolutely necessary to appropri- ately correct any data for concentration depletion and charging current or at least the distortion caused by these influences in the Tafel regime must be known prior to further analysis. The effect of concentration depletion is generally corrected by an appropriate mass transfer correction (also called the Koutecky–Levich compensation). For example in rotating disk electrode (RDE) experiments using the following equation16, where i is the measured current and iL is the limiting current. The ik determined from Eq. (9) is used in Eq. (3) to generate the TPs. However, for gas evolving reactions such as for oxygen evolution reaction (OER) and hydrogen evolution reaction (HER), the Eq. (9) cannot be used directly because of an unknown iL . In such cases, one resorts to a plot of ln(i)vs η for the extraction of kinetic parameters. When kinetics are sufficiently slow, the extraction of kinetic parameters from a plot of ln(i)vs η can be carried out with reasonable accuracy. The effect of the charging current is corrected by subtracting the polarisation curve without active species (blank measurements) from the polarisation curve with active species. There are two critical issues often overlooked in the above-mentioned correction methods. Firstly, the limit- ing current used for mass transfer correction must be very precise17. In this context, it will be shown later that an offset of only 5% in the limiting current can easily result in curvature of TP leading to incorrect values of α , even though the TPs in certain overpotential ranges may look linear with R2 > 0.99. In other words, a wide range of assumed limiting currents for mass transfer correction can still result in an apparent linear curve with R2 > 0.99. This becomes a critical issue for polarisation curves where the i asymptotically reaches a limiting current. Secondly, the double layer correction by background subtraction is an oversimplification of the actual situation. Considering the current flow through a simple Randles-circuit where the impedance of charge double layer is placed in parallel to the impedance of charge transfer, the current flow through the double layer ( ic ) in absence and presence of a faradaic reaction become significantly different18. Therefore, ic measured in the blank measurements cannot be used to correct for double layer effects in the presence of faradaic reaction. At best, one can detect the negative effects of ic from Tafel plots but measurement of the magnitude of ic is very difficult. In an (5)αa = np + nqβa (6)αc = nr + nqβc (7)βc = 1 2 ( 1− Fη 2�m ) (8)βa = 1 2 ( 1+ Fη 2�m ) (9)ik = iLi (iL − i) 4 Vol:.(1234567890) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ even worse scenario, when the ic ≫ io , this may introduce an additional slope (shown later) in the TPs allowing some researchers to incorrectly interpret this slope as the one due to the faradaic reaction. Adding to the problem of the inaccurate mass transfer and charging current correction, there is a signifi- cant uncertainty in the extraction of α from the TPs due to the herein called straight-line-placement-problem. Because there is no clear possibility to define Tafel regime in many experiments, the α is found from the slope of an arbitrarily placed straight line on the TP. Or even sometimes the α is found from the placement of an arbitrary straight line on a curve which is in fact a curved TP. It is not uncommon in literature that for the same reaction- catalyst system different values of α , and i0 are reported. For instance, comparing the literature19–22, the i0 reported for the oxygen reduction reaction (ORR) on Pt in 0.1 M KOH ranges from 10–9 A cm−2 to 10–11 A cm−2. While the reported values of α in these studies are between 0.86 to 1.06 (52–64 mV dec−1) at low η , it is between 0.122 to 0.23 (260–490 mV dec−1) at high η . Similarly, to name few examples, the trend of different Tafel slopes for a similar catalyst-reaction system is also common for OER23–26, HER5,27–29 and methanol oxidation reaction30–32. Such a wide range of values may certainly give rise to quite different mechanistic interpretations for the same reaction-catalyst system. Henceforth the question remains, how can one estimate the relevant α and i0 values? (a) when there is no defined rule for the placement of the straight line on the TP and (b) when there are several possible values of iL and ic that can lead to apparent straight lines in the TP. To some extent the answer lies in what graphs are used for extraction of α and thereby its interpretation. Within the framework of the IUPAC definition, the transfer coefficient is defined as4 Hence the differentiation of TP, herein termed as differential Tafel plot (DTP) gives α . In this article, we dis- cuss the advantages of using DTP (a plot of α vs η ) over standard TP (a plot of ln(ik)vs η ) and how it can lead to accurate estimation of kinetic parameters for the uniformly accessible electrodes under steady state conditions such as for the rotating disk electrodes (RDEs). Due to higher sensitivity of DTP over TP, it becomes easy to spot the negative effects of ic and false mass transfer correction. Although advantageous, this approach is adopted by very few research teams17,33–35. The sen- sitivity of DTP towards different electrode geometry, non-uniformly accessible electrodes and unsteady processes have been extensively studied4,33,36. In this article we focus on two critical ratios, the ratio of exchange current to limiting current ( i0/iL ) and ratio of exchange current to double layer charging current ( i0/ic ). It is shown how the Tafel slope and the Tafel range is sensitive to these ratios. From the shapes of DTP and on the basis of ratio i0/iL and i0/ic , it is shown in which scenarios the TP leads to incorrect estimation of α and i0 . The exact inter- pretation of the value of α in terms of reaction mechanisms is not our main intent or goal and throughout the article our focus has been rather on the accurate extraction of kinetic parameters. In the first part of this paper, we use simulated curves to demonstrate the effect of the iL and the ic , before we continue with experimental data (second part). In all of our experimental data, the α was found to be independent of overpotential for the large portion (up to i ≤ 0.9 iL ) of the polarisation curve. Hence the simulations were carried out considering the BV formalism as given in Eq. (1). DTP is used to analyse the experimental data of ORR, ferricyanide reduction reaction and OER. These reac- tions served as exemplary case for polarisation curves with well-defined iL , ill-defined iL and no iL . With this, we advocate the use of DTP instead of TP for an accurate determination of kinetic parameters and to reduce scattered experimental data for similar reactions and catalyst systems. Results and discussions Tafel overpotential.  The |η| at which the Tafel region starts is the point in the TP where the contribution of the backward reaction is negligible compared to the forward reaction. This potential is herewith defined as Tafel-overpotential (TOP), ηTOP . Mathematically, the contribution of the backward reaction is never zero, but for practical approximation, it can be defined as the η , at which the kinetic current of the backward reaction contributes less than 1% to the overall kinetic current. In other words, ik/ik,A or ik/ik,C is greater than 0.99. Here ik,A and ik,C are the anodic and cathodic kinetic current. Hence, for an electrochemical reaction, a generalised TOP can be calculated by setting ik/ik,A or ik/ik,C = 0.99. Noting ik,A = i0eαAf η and ik = i0[eαAf η − e−αCf η] , then at η = ηTOP , the ratio ik/ik,A becomes, where ∑ α = αA + αC . At 25 °C, the Tafel region starts at ~ 118 mV when ∑ α = 1 whereas it starts at ~ 59 mV when ∑ α = 2. The ηTOP calculated here may not resemble the experimental values where we have observed ηTOP ranging from 0.3 to 0.65 V, largely due to the 200–400 mV of difference between the theoretical equilibrium potential and observed open circuit potential. Despite this, for the simulation purpose, the ηTOP calculated according to Eq. (12) serves as a good reference point to discuss the features of DTP before and after ηTOP. The DTP is generated after differentiating Eq. (1) with respect to η, (10) ∣ ∣ ∣ ∣ 1 f d(ln(ik)) dη ∣ ∣ ∣ ∣ = α (11) i0[e αAf ηTOP − e−αCf ηTOP ] i0eαAf ηTOP = 0.99 (12)|ηTOP | = ln(100) f ∑ α 5 Vol.:(0123456789) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ At |η| > |ηTOP | , one of the exponential terms is negligible yielding the generalized expression as shown in Eq. (10). Hence the DTP, i.e. a plot of 1f d(ln(ik)) dη vs η converges to a constant value beyond |ηTOP | . To determine the evolution of i0 with respect to η , a plot of i0 vs η termed as the exchange current plot (ECP), can be generated from Eq. (2), once α from Eq. (10) is known. The i0 is calculated only for |η| > |ηTOP | , so that Eqs. (2) and (10) are applicable. Figure 1 shows from bottom to top, the TP and its corresponding DTP and ECP. The slope in the TP is equal to αf and at |η| > |ηTOP | , the DTP shows values of α . The shaded region highlights the non-Tafel region and is determined from Eq. (12). The DTP in Fig. 1 is an ideal curve and asymptotically approaches to a constant value beyond |ηTOP | , a char- acteristic necessary for the determination of i0 from ECP. We will show in the subsequent sections how ic and incorrect iL will distort the DTP and ECP behaviour, even when the TP appears to be unchanged. Effect of limiting current.  The value of iL used for the calculation of ik in Eq. (9), is either observed from the polarisation curve or calculated from the Levich equation after experimentally determining the value of diffusion coefficient, viscosity and bulk concentration. According to the Levich equation, iL = 0.62nFAD2/3υ−1/2ω1/2C∗ , where n is the total number of electrons, A is the geometric area, D is the diffusion coefficient, υ is the viscosity, ω is the angular velocity and C∗ is the bulk concentration. Experimentally, small deviations in reading the iL value from the respective polarisation curve can occur, when the iL is not well-defined or when the i asymptotically reaches a limiting current. Small deviations in the calculated value of iL from Levich equation can occur, when there is a small measurement error in the determined value of either diffusion coefficient, viscosity, or bulk con- centration. The characteristics of TP, DTP and ECP for the two cases, when iL is either over- or underestimated by 5% are shown in Fig. 2a. To make simulation results comparable to experiments, the sequence as shown in Fig. 2b has been applied to calculate ik. Between the η range of − 0.13 V to − 0.25 V, all calculated TPs are fitted with a straight-line equation and the fitting parameters are summarized in Table 1. Undoubtedly, the curve appears to be linear between − 0.13 to − 0.25 V in TPs because the R2 for all three TP is greater than 0.999. Nonetheless a different slope and hence different α is obtained for each iL value even though they are separated by only 5%, when using the over- or underestimated values. This imparts an error in estimation of α and eventually in the prognosis of the reaction mechanisms. Even though these errors cannot be easily inferred from TPs, the DTP is sensitive to them and shows significant differences, where there is a clear distinction between the three cases. Only when the iL is correct, the DTP converges to a constant α value beyond ηTOP , which is important for the calculation of i0 . From these plots we also see that when the value of iL is over- or underestimated, the α values in DTP and consequently i0 in ECP are never approaching constant values with respect to η . Consequently, the same close to perfect linear stretch (13) 1 f d(ln(ik)) dη = αAe αAf η + αCe −αCf η eαAf η − e−αCf η Figure 1.   Theoretical TP (bottom), DTP (middle) and ECP (top). Assumed values: i0 = 20 µA, iL = 1 mA, αA = 0.6, αC = 0.4, T = 21 °C, f=39.5 V−1.  6 Vol:.(1234567890) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ of the TP appears to be far from its ideal behavior in the DTP and ECP. The variance of α with respect to η has been previously reported in the literature37–39 and this is interpreted in terms of reaction mechanism. However, referring to Fig. 2a, we would like to point out, that such a variation of α can also arise from the use of over- or underestimated values of iL in the calculation of ik. Tafel range with mass transfer correction.  When the polarisation curve is mass transfer corrected, a substantial Tafel range can be easily obtained for slow reactions since the limiting current appears only at large overpoten- tials. However, for fast reactions, the Tafel region is experimentally observed only when iL in the polarisation curve appears at |η|>|ηTOP | . In this case, the usable range of η for Tafel analysis, |ηTA| can be found by consider- ing the i0/iL ratio and |ηTOP | . We can set i = 0.95iL when it is assumed that Tafel analysis can be performed on polarisation curve until the current reaches 95% of the iL.Then substituting i = 0.95iL and ik = i0e αf η , in Eq. (9) we get (full derivation in supplemental information), This is the η at which the measured current is 95% of iL . Considering Eqs. (12) and (14), the usable range of η for Tafel analysis is, In Fig. 3, the usable range of η for Tafel analysis is shown for various scenarios. When i0/iL = 5, few mVs are available only for low α values, hence Tafel analysis is not recommended. When the α > 0.5, also a i0/iL ratio of 2 is not suitable for Tafel analysis. Only when the i0/iL< 0.05, Tafel analysis can be performed for all values of (14)η = ln(19 iL/i0) αf (15)|ηTA| = ∣ ∣ ∣ ∣ ln(19 iL/i0) αf ∣ ∣ ∣ ∣ − |ηTOP | Figure 2.   (a)Theoretically generated TP (bottom), DTP (middle) and ECP (top) for iL=1.05 mA (red), iL=1 mA (black) and iL = 0.95 mA (blue). Assumed values: i0 = 4 µA, αA = 0.35, αC = 0.65, T = 21 °C, f = 39.5 V−1. (b) Sequence for the calculation of ik when iL used for mass transfer correction changes by ± 5%. Table 1.   Kinetic and fitting parameters corresponding to the TP of Fig. 2a. The η range for fitting is from − 0.13 to − 0.25 V. iL (mA) R2 i0 (µA) α Slope Intercept 1.05 0.99983 4.46 0.63 24.86 12.32 1.0 1.00000 4.00 0.65 25.61 12.44 0.95 0.99978 3.44 0.675 26.61 12.58 7 Vol.:(0123456789) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ α . Hence Tafel analysis for very fast reactions such as the hydrogen oxidation reaction (HOR) on Pt in acidic media might be difficult. The HOR on Pt in acidic media is so fast that the diffusion limited current in an RDE set up is observed already at small overpotential40,41. According to HOR studies on microelectrodes42, the i0 for the HOR is between 27 and 80 mA·cm−2. In HOR studies on an RDE, the ratio of i0/iL would be between 9–27 considering iL = 3 mA·cm−241. Hence, unless i0/iL is reduced to less than 2 (assuming α = 0.541,43) by experimental design, there will be no visible range of η for Tafel analysis. Thus, irrespective of the α value, the foremost criteria for the polarisation curves with mass transfer correction is that the ratio i0/iL should be less than 5 for any Tafel analysis to be conducted. This ratio becomes more and more restrictive as the α value increases. Tafel range without mass transfer correction.  For some experiments like HER and OER, the limiting currents are not easily observed and hence using Eq. (9) mass transport correction cannot be performed. In these cases, one resorts to extraction of kinetic parameters from a plot of ln(i)vs η instead of ln(ik)vs η . This is, in general, a risky approach, however, in some cases where the i0/iL is below a critical value, α and i0 can be still extracted with reasonable accuracy. Rearranging Eq. (9), substituting ik = i0e αf η and taking the natural logarithm, the Eq. (9) reduces to (full derivation given in supplemental information), Equation (16) is only valid for |η| > |ηTOP | . A plot of ln (i) vs η is mathematically never linear by default, but approximates to linear curve when iL >> i0e αf η . When iL >> i0e αf η , Eq. (16) in principle reduces to the Tafel equation. Setting iL to be at least 100 times greater than i0eαf η , the suitable range of i0/iL for the Tafel analysis can be found. For example, when iL = 100 · i0e αf η , then (full derivation given in Supplementary Information), Subsequently, the usable range of η for Tafel analysis is, Similar to Fig. 3, in Fig. 4, the usable range of η for Tafel analysis is shown for various scenarios, except here the polarization curves are not mass transfer corrected. At a given α , the ηTA reduces as i0/iL is increased. This graph shows that when i0/iL > 0.003, the plot of ln (i) vs η is not suitable for extraction of kinetic parameters for any value of α . At α = 0.5, the iL should be at least 1000 times larger than i0 to observe any potential region for Tafel analysis. Only when i0/iL < 10–4, the kinetic parameters can be determined with reasonable accuracy from a plot of ln (i) vs η for all α values. Here the conditions for extracting kinetic parameters are much more restrictive compared to Fig. 3. In this sense, the plot of ln (i) vs η is quite limited for the extraction of kinetic parameters and care must be practiced when it is assumed that i ∼ ik at |η| > |ηTOP |. Effect of double layer charging current.  When measuring the polarisation curve via linear sweep vol- tammetry, always a non-faradaic current flows along with the faradaic current. This non-faradaic current is due to charging or discharging of the double layer. Although the ic is relatively small at scan rates below 20 mV s−1, it can still have a significant effect on the progression of the DTP. In experiments, the measured linear sweep (16)ln(i) = ln ( ioiL iL + i0eαf η ) + αf η (17)η = ln(iL/100 i0) αf (18)|ηTA| = ln(iL/100 i0) αf − |ηTOP | Figure 3.   The usable potential range for Tafel analysis for the mass transfer corrected polarisation curves, when i0/iL = 0.05 (black), i0/iL = 1.0 (red), i0/iL = 2 (blue) and i0/iL = 5 (green). Example: at α = 0.5, the Tafel behaviour is seen for a potential range of 185 mV and 30 mV when i0/iL is equal to 0.05 and 1.0, respectively. 8 Vol:.(1234567890) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ voltammetry current, itot is approximately equal to the sum of i and ic . Similarly, the experimentally measured limiting current, iL,tot = iL + ic . Extraction of kinetic parameters is performed by considering itot and iL,tot. Hence to mimic this scenario, in our simulations the α and i0 has been determined from the modified kinetic current ik,tot . According to Eq. (9) this is given by ik,tot = iL,tot itot/(iL,tot − itot) . To generate ik,tot , the sequence as shown in Fig. 5b is adopted. Figure 5a presents three scenarios with different ratios of i0/ic . Here, the iL is intentionally chosen high so that in all three cases iL >  > i0 and iL >  > ic . When i0 is significantly larger than ic , the DTP behaves normal, i.e. Figure 4.   The usable potential range for Tafel analysis for the polarisation curves without mass transfer correction, when i0/iL = 10–5 (black), i0/iL = 10–4 (red), i0/iL = 10–3 (blue) and i0/iL = 3. 10–3 (green). Example: at α = 0.5, the Tafel behaviour is seen for a potential range of 115 mV when i0/iL is equal to 10–4. Subsequently no usable potential range at α = 0.5 is observed for Tafel analysis when i0/iL ≥ 10–3. Figure 5.   (a) Effect of double layer charging current on the TP (bottom), DTP (middle) and ECP (top) for i0/ic = 10 (black), i0/ic = 1 (red), i0/ic = 0.1 (blue). For all curves iL = 10 mA, i0 = 4 µA, αA = 0.35, αC = 0.65, T = 21 °C and f  = 39.5 V−1. (b) Sequence for calculating ik under the influence of double layer charging current. 9 Vol.:(0123456789) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ α and i0 are constant with respect to η at |η|> |ηTOP | . When i0 is significantly smaller than ic , the DTP is severely distorted making the estimation of α and i0 difficult. The lower the i0 value with respect to ic , the larger becomes the distortion. Hence, it would be misleading to assume that ic has only a minor influence (for example at low scan rates) on the polarisation curves by comparing the relative magnitudes of i (or iL ) to ic . The most crucial ratio here is i0/ic . If ic is significantly larger than i0 , then irrespective of the magnitude of iL , the DTP and ECP will get distorted in the Tafel region and, α , i0 cannot be estimated accurately. The negative effect of ic on TPs has been reported previously15,44, and from our TPs, some differences between the curves can also be seen, however they are not as pronounced as in DTP. This result is crucial for catalysts with low catalytic activity and a comparably large double layer capacity, such as metal oxide catalysts, which are either supported or mixed with carbon material. As these materials often have low electron conductivities, the carbon powder is added to the catalyst layer to make it conductive45–47. However, while carbon does not increase the catalytic activity, it increases the double layer capacity. Therefore, when the carbon additive content with respect to the active material is increased, then i0 would remain the same, while ic increases. In such cases, the ic may be significantly larger than i0 and there is a good chance that the values of α and i0 estimated from TPs would be incorrect. In such cases, DTP will be a valuable tool to detect the distortion in the TPs due to ic and identify the true Tafel range. Even though we have made simpler assumption that the ic to be independent of overpotential here, the benefits of DTP in detecting the distortion in TPs due to ic will be still applicable. We have seen that by combining the results of Figs. 3 and 4, at any given α , it is possible to predict the whole range of i0/iL , where (a) Tafel analysis is not feasible (b) Tafel analysis is feasible, but only with mass transfer correction and (c) Tafel analysis is feasible with or without mass transfer correction. In addition, from Fig. 5a, we can identify the Tafel regime distorted by double layer charging current. This gives us some guidelines on how to process the experimental data. A recommended scheme is shown in Fig. 6. Experiments.  Two difficulties in representation of DTP and ECP for the experimental data are worth mentioning here. First due to the small fluctuations of the measured polarization data and second due to the unknown value of ηTOP . In a DTP representation, the small fluctuations in polarization curves are amplified and hence the DTPs and corresponding ECPs are not smooth compared to their TPs and polarization curves. Therefore, fluctuations in DTP and ECP representation should not be necessarily considered as an indicator of unreliable data. For our measurements, without compromising the original shape and trend of curves, the data in DTP has been smoothened by moving average method. As discussed previously, the value of i0 extracted from DTPs are valid only for |η| > |ηTOP | , i.e. when the α in the DTP approaches a constant value. In our experiments, Figure 6.   Scheme for extracting accurate kinetic parameters from the polarisation data. 10 Vol:.(1234567890) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ since ηTOP is unknown, an approach of α towards constancy has been taken as an indicator to define the data representation in ECP. Polarisation curve with well‑defined limiting current.  Figure  7a shows the ORR curve in alkaline media for glassy carbon (GC) disk coated with Pt/C particles. The loading of Pt is kept at 20 µgPt·cm−2. With such high loadings, the Pt particles are uniformly distributed throughout the surface of the GC and hence the limiting cur- rent is well defined. The corresponding TP, DTP and ECP are presented in Fig. 7b. These plots are generated by assuming the iL value shown as grey dot in Fig. 7a. Following the conventional approach for ORR on Pt in alkaline media, the TP can be interpreted as a curve consisting of two slopes19–22 or even three slopes if one considers the data between − 0.175 to − 0.275 V. Only from the corresponding DTP it becomes clear that the first slope (− 0.175 to − 0.275 V) is due to ic , second Tafel slope (− 0.31 to − 0.45 V) is indicated by constant α and evidence for the presence of the third slope (− 0.5 to − 0.6 V) is rather weak. Although the shape of DTP is distorted due to the effects of ic at low overpotentials, its effects subdues at high overpotentials and it does not have significant impact on the estimation of α and i0 . The kinetic values extracted from the third slope are not of any practical use as the potentials are very close to limiting current region where the electrochemical device is unlikely to operate. Hence, relying on TPs alone can easily lead to unwarranted slope values. Here we would like to stress again that BV formalism is a reasonable model to explain our experimental results because a clear distinction can be made between cases in a DTP, (a) when α appears to change with η as a result of incorrect mass transfer correction, or (b) when α appears to change linearly with η according to Eqs. (5)–(8)11. In the former case, the DTP will show a constant α at one particular value of iL(used in the calculation of ik ) and at all other values of iL the α appears to change non-linearly with η, as already seen in Fig. 2a. In the latter case, the DTP will show a linear dependency of α on η at one particular value of iL , and at all other values of iL the α appears to change non-linearly with η . For our experimental data, after appropriate mass transfer correction we always observed the former case. Therefore, and for simplicity, only the BV formalism is considered for the analysis. Polarization curves of slow reaction with ill‑defined limiting current.  The DTP approach is especially useful for the interpretation of polarization curves having a poorly defined iL . One such very prominent example for ill-defined limiting currents is the ORR in alkaline media at GC coated with low loading of Pt/C particles. Fig- ure 8a compares the ORR curve of a GC coated with low loading of Pt/C with bare GC disk. The loading of Pt is 5 µgPt·cm-2 and it is kept intentionally low to ensure some GC area remains uncovered. During polarization, Pt catalyzes ORR at low η and at high η , additionally the underlying GC also catalyzes ORR48. This is due to the substantial activity of bare GC towards ORR at high η. Therefore, due to the background current of GC at high η , the observed limiting current for partially covered GC is not well defined. Had there been no catalytic activity from GC, the current would have reached a defined limiting current as commonly seen in acidic media49,50. The magnitude of the measured iL is proportional to the projected geometric area of the catalyst layer. For partially covered GC like the one described above, the projected geometric area of the catalyst layer is unknown and the iL with respect to the Pt nanoparticles is masked within the polarization curve. The platinum’s limiting current, Figure 7.   (a) ORR curves for GC disk coated with 20 µgPt cm−2. The ORR curves were measured at 10 mV s−1, 1600 RPM, 60 °C and 0.1 M KOH. The grey dot shows a point where the value of limiting current is defined. (b) The corresponding TP (bottom), DTP (middle) and ECP (top). 11 Vol.:(0123456789) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ however, is the value needed to find the kinetic parameters of Pt. It is shown in Fig. 8b how the features of the DTP can be exploited to obtain the kinetic parameters in this and similar cases. It shows from bottom to top TP, DTP and ECP of GC disk coated with 5 µgPt·cm−2. In order to find the correct value of iL for the extraction of kinetic parameters, three arbitrary iL are chosen. This is shown as point 1, 2, 3 in Fig. 8a,b. These three iL values are used to calculate ik and generate the TP, DTP and ECP. Unquestionably, there is visibly no difference for point 1, 2 and 3 in the respective TP. In fact, between − 0.4 and − 0.5 V, all the TPs presented in Fig. 8b have a slope with R2 > 0.999. Consequently, this feature makes it dangerous to extract kinetic parameters from TP and proceed further with mechanistic conclusions. In contrast to the TP, DTP and ECP are distinctly different and very sensitive to the three different cases. Only at point 2, where iL = − 293 µA, α becomes constant and hence this is the value of α , that should be used for the determination of i0 . Please notice, that even though points 1 and 3 are quite close to point 2, there is a significant difference in α value in the DTP. This is in very good agreement with what has been shown previously in Table 1. Furthermore, in TP of Fig. 8b, depending on the overpotential range chosen for extraction of α , the value of i0 may differ by two orders of magnitude. Perhaps this is one of the causes for large variations in i0 values in literature. Yet another possibility in large variation in i0 is due to the calculation of overpotential. It can be calculated as η = E − Eeq or η = E − EOCV . For ORR on Pt electrodes, this difference is between 200 and 300 mV. This directly translates into two to three orders of difference in the extracted value of i0 from the TP. Compared to GC coated with 20 µgPt·cm−2, here the ic distorts DTP even at higher η . The higher influence of ic in DTP for GC disk coated with 5 µgPt·cm−2 is attributed to the effective decrease in the ratio i0/ic because of the additional double layer capacity of uncovered GC. Polarization curves of fast reaction with ill‑defined limiting current.  The ferricyanide reduction reaction on a gold electrode is a good example of a fast reaction. Figure 9a shows the polarisation curve of a ferricyanide reduction reaction measured on the Au disk electrode. In order to increase the mass transfer rate and thereby reduce the ratio of i0/iL , the polarisation curve has been measured at higher rotation rate of 4900 rpm. The iL is however, still not well defined and asymptotically reaches a constant value. It can be also seen that the so-called kinetic region indicated by the sharp exponential drop in potential at low current, is here absent. The shape of polarisation curve appears rather like a concentration-overpotential curve51 suggesting that kinetics in this example are indeed fast. Similar to Fig. 8, we have chosen here three arbitrary points (− 43.0 µA, − 43.20 µA and − 43.52 µA) in the limiting current region for the calculation of ik and thereafter generate TP, DTP and ECP. This is presented in Fig. 9b. The α and i0 were found to be nearly constant for a small range of η , when iL = − 43.2 µA was used for the calculation of ik . Even though the chosen values of limiting current vary by less than 1%, we see a sharp deviation in α and i0 in DTP and ECP. This implies that for fast reactions, the kinetic parameters are extremely sensitive to selected value of the limiting current and due to large margin in the error, accurate estima- tion of α and i0 is rather unlikely. In this respect, the DTP offers a robust methodology, wherein the α and i0 can Figure 8.   (a) ORR curves for GC disk (solid line) and GC disk coated with 5 µgPt·cm−2 (dotted dashed line). The ORR curves were measured in 0.1 M KOH, 10 mV·s−1, 1600 RPM at 60 °C. The locations 1, 2, 3 are the positions at which iL is − 273µA, − 293µA and − 313µA respectively. (b) TP (bottom), DTP (middle) and ECP (top) for GC disk coated with 5 µgPt cm−2 assuming iL = − 273 µA (blue), iL = − 293µA (black) and iL = − 313 µA (red). 12 Vol:.(1234567890) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ be estimated accurately by fine adjustment of limiting current. The obtained values of α and i0 are about 0.4 and − 40 µA, respectively, when iL is chosen as − 43.2 µA. Thus, the ratio of i0/iL ~ 1.0. From Eq. (15), the resulting ηTA is circa 70 mV. This is roughly what can also be seen from the DTP of Fig. 9b confirming the predictions of Eq. (15). Hence, even when the exchange current is nearly equal to limiting current, a Tafel range is plausible and α and i0 can be estimated accurately when the precise mass transfer correction has been carried out. Notice that due to the high value of i0 , the ratio i0/ic is quite large and hence effects of ic are not visible in DTP in accord with the results presented in Fig. 5a. Polarization curves without experimentally observed limiting current.  The OER on a Pt disk is a classic example for an electrochemical reaction, in which no limiting current is observed for a wide range of η (see Fig. 10a). In this case, one resorts to estimation of kinetic parameters from a plot of ln (i)vs η . Observing the evolution of α with respect to η in the DTP, it can be found, whether the estimation of kinetic parameters from a plot of ln (i)vs η is reasonable. In Fig. 10b the TP, DTP and ECP are generated without correcting the OER polariza- tion data for mass transfer effects. Observing the trend in the TP, it might be tempting to place two straight lines between 0.60 to 0.75 V and 0.3 to 0.45 V and thereby conclude that two α values exist for OER. However, from DTP we can see that this is a false conclusion and leads to unwarranted mechanistic conclusions. The α in DTP and i0 in ECP are nearly constant only at η > 0.65 V. The presence of constant α and i0 indicates that kinetic parameters from OER polarisation curves can be extracted without mass transfer correction. Hence with DTP we can confirm the presence of multiple α and determine if the polarisation curves require mass transfer correc- tion before the extraction of kinetic parameters. Considerations for the design of the experimental set up.  In the previous sections we have dis- cussed the importance of the ratio i0/iL and i0/ic . Depending on the α , there is a maximum permissible value of i0/iL in order to obtain a sufficiently large Tafel range. Consequently, certain measures can be considered when planning for the experiment. The experiments can be designed to achieve a reasonable range of η for Tafel analy- sis. For a given catalyst, lowering the i0 and increasing the mass transport will increase the accessible Tafel region. The i0 can be lowered by reducing the ratio of microscopic area to geometric area, i.e. the catalyst layer should be as thin as possible. Among other possible methods, the mass transport can be increased by either increasing the rotation rate in an RDE set up or by utilizing a microelectrode. Although the mass transfer rate can also be increased by increasing the bulk concentration of reactant, this also proportionately increases the i02. Therefore, increasing the reactant concentration in some cases can essentially keep the ratio i0/iL unchanged and may not extend the Tafel range for the analysis. Another possibility of increasing Tafel range is to reduce ic when feasible. If the increase in ic is caused by additional carbon/support material, the content of this carbon/support material may be reduced to keep the ratio i0/ic high. Figure 9.   (a) Polarisation curve for potassium ferricyanide reduction on Au disk electrode. The locations 1, 2, 3 are the positions at which iL is − 43.0 µA, − 43.20 µA and − 43.52 µA respectively. The polarization curve was measured at a scan rate of 10 mV s−1, 4900 RPM and 21 °C in a solution consisting of 0.001 M K3Fe(CN)6, 0.001 M K4Fe(CN)6 and 0.1 M KCl. (b) TP (bottom), DTP (middle) and ECP (top) for potassium ferricyanide reduction on Au disk assuming iL = − 43.0 µA (blue), iL = − 43.2 µA (black) and iL = − 43.52 µA (red). 13 Vol.:(0123456789) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ Conclusion The DTP approach is proposed as a simple and more precise alternative to TP for the accurate estimation of α and i0 . The presence of linear curve segment (even with R2 > 0.999) in TP cannot be considered as sufficient criteria for accurate estimation of α and i0 . For instance, we found rather weak evidence for the presence of second Tafel slope for ORR and OER on Pt and DTP revealed the presence of only one Tafel slope. The range, extent of linear- ity and presence of multiple Tafel slopes should be confirmed by DTP which ideally shows no to little variation of α with respect to η for a given Tafel slope. This provides a clear quality criterion for assessing both measured and processed data. The role of iL in defining the Tafel slope has been often overlooked in literature. When the incorrect iL is chosen for mass transfer correction, the resulting α appears to be η dependent. This behaviour is not obvious from either TP or polarisation curves. The DTP shows that α values are also severely distorted at small η due to the charging of the double layer. When the ratio of i0/ic < 0.1, the distortion extends into the Tafel region making it difficult to estimate α and i0 accurately. The effect due to double layer charging current can be detected very well and the effect of mass transfer limitations can be corrected with the DTP method. The DTP method can be applied to polarisation curves having poorly defined limiting current (ex.: ORR on low loading Pt/C), no limiting current (ex.: OER on Pt disk) and well-defined limiting current (ex.: ORR on high loading Pt/C). Using these examples, we experimentally confirmed the capability of the DTP method to screen different catalyst materials in diverse electrochemical reactions, even under conditions which challenge conventional TP analysis. In summary, the DTP method shown in this article represents an accurate tool for interpreting polarisa- tion data, allowing one to circumvent unwarranted mechanistic conclusions. Experimental section The ORR experiments were carried out on in-house prepared GC disk (5 mm diameter) coated with Pt/C (60 wt% Pt, 40 wt% carbon) (Alfa Aesar) particles. A thin film of anion exchange ionomer was coated on Pt/C catalyst layer prior to recording ORR curves. The measurements were conducted in O2 saturated 0.1 M KOH (anhydrous, ≥ 99.95%, Sigma-Aldrich, ultrapure water Millipore 18.2 M � cm) at 60 °C. The OER experiments were carried out on Pt disk (3 mm diameter) in 0.1 M KOH at 21 °C. The Potassium Ferricyanide (III) (99%, Sigma-Aldrich) reduction was carried out on Au disk (3 mm diameter) in a deaerated solution consisting of 0.001 M K3(Fe(CN)6), 0.001 M K4(Fe(CN)6) and 0.1 M KCl (≥ 99.0%, Sigma-Aldrich)at 21 °C. The Pt/C catalyst ink was prepared by adding 50 mg of 60 wt% Pt/C to 40 mL of deionized (DI) water (Mil- lipore, 18.2 MΩ cm), which was sonicated for 20 min in an ice bath and was further diluted so as to obtain 0.1 μgPt in 1 μL of catalyst suspension. This suspension was sonicated each time in an ice bath for 10 min before the desired volume was dropped on the GC surface. The catalyst ink was dried under ambient conditions. Ionomer films of 0.5 μm thickness covering the catalyst coated GC disks were obtained by dropping 10 μL of ionomer solution, diluted accordingly from commercially obtained AS-4 ionomer solution (5 wt% solids in Isopropyl alcohol) supplied by Tokuyama, Japan. The density of recast ionomer film was assumed to be the same as A201 membrane (ρ = 1.06 g·cm−3) supplied by Tokuyama, Japan. Figure 10.   (a) Polarisation curve for OER on Pt disk electrode at 10 mV·s−1, 1600 RPM, 21 °C in 0.1 M KOH solution. (b) Respective TP (bottom), DTP (middle) and ECP (top). 14 Vol:.(1234567890) Scientific Reports | (2021) 11:8974 | https://doi.org/10.1038/s41598-021-87951-z www.nature.com/scientificreports/ A standard three electrode RDE setup (Metrohm Autolab) was used for all electrochemical measurements. Mercury-mercurous oxide (0.165 V and 0.171 V vs SHE at 21 °C and 60 °C, respectively) (ALS Co. Ltd) in alkaline media and in-house prepared saturated Silver-Silver Chloride (0.199 V vs SHE at 21 °C) in 0.1 M KCl was used as reference electrode. A Pt coil was used as counter electrode for all measurements. The working electrodes were GC and GC coated with Pt/C for ORR, Pt disk for OER and Au disk for Ferricyanide reduction reaction. 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Electrochem. Soc. https://​doi.​org/​10.​1149/2.​01719​08jes (2019). Acknowledgements The financial support by the German Research Foundation (DFG RO 2454/17-1) is gratefully acknowledged. The financial support for open access is provided by the University of Bayreuth. Author contributions Conceptualization: P.K. Experiments: P.K., T.B. Methodology: P.K., T.T. Formal Analysis: P.K., T.T. Resources— C.R., W.E. Writing original draft: P.K. Writing-editing and review: All authors supervision—C.R., F.M., W.E. Project Administration—C.R., F.M., W.E. Funding acquisition—C.R., F.M., W.E. Funding Open Access funding enabled and organized by Projekt DEAL. Competing interests  The authors declare no competing interests. Additional information Supplementary Information The online version contains supplementary material available at https://​doi.​org/​ 10.​1038/​s41598-​021-​87951-z. Correspondence and requests for materials should be addressed to P.K. 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To view a copy of this licence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/. © The Author(s) 2021 https://doi.org/10.1016/j.jelechem.2020.114185 https://doi.org/10.1039/c0cp02471f https://doi.org/10.1002/celc.202001107 https://doi.org/10.1021/j100558a009 https://doi.org/10.1021/j100538a012 https://doi.org/10.1149/2.0501514jes https://doi.org/10.1021/jp048641u https://doi.org/10.1149/2.0981501jes https://doi.org/10.1007/s10008-011-1337-4 https://doi.org/10.1007/s10008-011-1337-4 https://doi.org/10.1149/1.3456630 https://doi.org/10.1016/j.est.2016.06.007 https://doi.org/10.1016/j.jpowsour.2007.08.007 https://doi.org/10.1016/j.jelechem.2004.11.022 https://doi.org/10.1016/j.egypro.2017.10.155 https://doi.org/10.1016/j.egypro.2017.10.155 https://doi.org/10.1149/2.0171908jes https://doi.org/10.1038/s41598-021-87951-z https://doi.org/10.1038/s41598-021-87951-z www.nature.com/reprints http://creativecommons.org/licenses/by/4.0/ A simple and effective method for the accurate extraction of kinetic parameters using differential Tafel plots Results and discussions Tafel overpotential. Effect of limiting current. Tafel range with mass transfer correction. Tafel range without mass transfer correction. Effect of double layer charging current. Experiments. Polarisation curve with well-defined limiting current. Polarization curves of slow reaction with ill-defined limiting current. Polarization curves of fast reaction with ill-defined limiting current. Polarization curves without experimentally observed limiting current. Considerations for the design of the experimental set up. Conclusion Experimental section References Acknowledgements