- Home
- Documents
- Kinetics of hydrogen evolution reaction with Frumkin adsorption: re-examination of the Volmer–Heyrovsky and Volmer–Tafel routes

Published on

02-Jul-2016View

213Download

1

Transcript

Kinetics of hydrogen evolution reaction with Frumkinadsorption: re-examination of the VolmerHeyrovsky andVolmerTafel routesM.R. Gennero de Chialvo, A.C. Chialvo *Programa de Electroqumica Aplicada e Ingeniera Electroqumica (PRELINE), Facultad de Ingeniera Qumica, UniversidadNacional del Litoral, Santiago del Estero 2829, 3000 Santa Fe, ArgentinaReceived 3 April 1998AbstractA re-examination of the basic kinetic derivations of the VolmerHeyrovsky and VolmerTafel routes for thehydrogen evolution reaction with a Frumkin adsorption of the intermediate was carried out. Expressions for thedependence of the surface coverage and current density on overpotential were derived for both routes withoutkinetic approximations. On the basis of these dependencies, the kinetic behavior was simulated for dierent valuesof the parameters involved at 298.16 K. Conditions for the existence of Tafelian domains were discussed and theindependence of the Tafel slopes on the type of adsorption of the reaction intermediate was demonstrated. Theresults obtained were critically compared with those derived from approximated expressions customarily used andthe dierences with them were pointed out. # 1998 Elsevier Science Ltd. All rights reserved.Keywords: Hydrogen evolution reaction; Kinetic analysis; Frumkin adsorption; Tafel slopes; Exchange current densities1. IntroductionThe discharge of proton or water through theVolmer step with the formation of adsorbed hydrogenH(a), the electrochemical desorption (Heyrovsky step)and the recombination of the H(a) (Tafel step) are gen-erally accepted as the steps of the kinetic mechanismof the hydrogen evolution reaction (HER). The analy-sis of the kinetic behavior is usually done on the basisof the VolmerHeyrovsky (VH) and the VolmerTafel(VT) routes. Using the approximation of the ratedetermining step (rds), diagnostic criteria were estab-lished and widely used for both routes [17].Furthermore, it has been considered that the adsorp-tion process of the reaction intermediate can bedescribed through the Frumkin isotherm in the domainof surface coverage (y) ranging between 0.2 and 0.8[13, 5, 6]. On this range the relationship y/(1 y) hasbeen approximated to unity and in the resulting ex-pressions of the reaction rate, the dependence of y onoverpotential (Z) in the pre-exponential factor has beenneglected [3, 6]. As a result of these approximations, ithas been concluded that the Tafel slope (b) in the lowoverpotentials region should be influenced for the reac-tion symmetry factor (a) and for the adsorption sym-metry factor (l) as well. Nevertheless, from theanalysis of the results obtained in the study of theHER without kinetic approximations due toEnyo [8, 9], it can be arrived to the conclusion that bdepends only on a. Consequently, a re-examination ofthe basic kinetic concepts is worthwhile in order to elu-cidate this apparent controversy.The present work deals with a kinetic study of theHER under the VolmerHeyrovskyTafel mechanismwith a Frumkin adsorption and without kinetic ap-proximations. The expressions for the variation of thecurrent density and the surface coverage on overpoten-tial will be derived for the VH and VT routes. TheElectrochimica Acta 44 (1998) 8418510013-4686/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.PII: S0013-4686(98 )00233-3PERGAMON* Corresponding author. Fax: +54-42571162; E-mail:achialvo@fiqus.unl.edu.arconditions for the existence of Tafelian domains willbe obtained for both routes and the relation betweenthe amounts resulting from the extrapolation of theselinear regions and the kinetic parameters will be deter-mined.2. Preliminary considerationsThe Tafel slope is one of the experimental kineticparameters often used for the characterization of anelectrochemical reaction, but its determination is inmany cases questionable. It is common to observeTafel lines drawn over slightly curved but non-linearexperimental dependencies, which we will call pseudo-Tafelian behaviors. Moreover, an exchange currentdensity is obtained by extrapolation of the fittedstraight line. Such experimental kinetic parameters,with arbitrary values, could lead to a wrong interpret-ation of the behavior of the system under study.The explanation for the linear dependence of thelogarithm of the current density ( j) on Z is based ontheoretical dependencies resulting from kinetic andalso mathematical approximations, such as the con-sideration of quasi equilibrium steps, neglect of thevariation of y(Z), etc. Nevertheless, the consistency ofsuch approximations with the rigorous solution of thecorresponding kinetic mechanism is usually not veri-fied. It should be noticed that the usefulness of the ex-perimental kinetic parameters lies on their relationshipwith the kinetic parameters of the elementary steps ofthe reaction mechanism. Consequently, the correct in-terpretation of a Tafelian domain defined as the regionwhere the dependence log j vs. Z is linear and thereforedZ/d log j is constant, depends on the previous clearknowledge of the descriptive capability of a given kin-etic mechanism. Only if this condition is fulfilled, theTafel slope and the exchange current density can bequantitatively related with the corresponding kineticparameters of the elementary steps.3. Theoretical analysisThe expressions of the reaction rate (V) of thehydrogen evolution reaction, corresponding to the fol-lowing stoichiometry:H2O 2e $ H2 g 2OH , 1will be derived in steady state for both VolmerHeyrovsky and VolmerTafel routes. From them andtaking into account the relationship of V with the cur-rent density,j 2FV, 2the dependence of j on the overpotential will be simu-lated by computational calculations at T= 298.16 K.3.1. VolmerHeyrovsky routeThe elementary steps areH2O e $ Ha OH Volmer, 3H2OHa e $ H2 g OH Heyrovsky: 4The equations for the rate of the dierent reactionsteps with a Frumkian behavior of the adsorbedhydrogen, following the treatment given by Enyo [8],can be written asvV veV1 y1 ye sle1a fZ, 5vV veVyyes1leafZ, 6vH veHyyes1le1a fZ, 7vH veH1 y1 ye sleafZ, 8wheres euyye, 9u the interaction parameter between the adsorbedhydrogen atoms, v + i and v i are the forward andbackward reaction rates of step i (i= V, H), respect-ively, v ie is the equilibrium reaction rate of step i, y e isthe equilibrium surface coverage and f= F/RT(38.92039 V1). Furthermore, a and l are the reactionand adsorption symmetry factors, respectively, andthey are considered to be equal for all elementarysteps.On steady state, the rate of Eq. (1) and those ofEqs. (3) and (4) are related by [10]V vV vH 0:5vV vH, vV vV vV,vH vH vH:10Substituting the expressions of the reaction rate of thecorresponding steps (Eqs. (5)(8)) and dividing by vVe ,we obtainM. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841851842VveV 1 y1 ye sle1a fZ yyes1leafZ mHyyes1le1a fZ 1 y1 ye sleafZ 121 y1 ye sle1a fZ yyes1leafZmHyyes1le1a fZ 1 y1 ye sleafZ, 11where mH= vHe /vVe . From Eq. (11), the following im-plicit function y= f(Z, y, mH, ye, u) can be obtained:y yes1mH efZ1 ye1mH efZ yes1mH efZ : 12The amounts y/y e and (1 y)/(1 y e) can be evalu-ated from Eq. (12). Substituting them on Eq. (11) andtaking into account Eq. (2), the general equation ofthe dependence of the current density on overpotentialis obtained,jj0V 2mHsle2a fZ eafZ1 ye1mH efZ yes1mH efZ , 13where jV0 is the exchange current density of the Volmerstep and s= s[y(Z)] is given by Eq. (9).The complete description of the kinetics of the HERwhen the VolmerHeyrovsky route is applicable canbe obtained from the simultaneous resolution ofEqs. (12) and (13).3.1.1. Tafelian domainsThe existence of overpotential domains where thereis a linear dependence of the logarithm of j on Z is notclearly inferred from Eq. (13) and it will depend on thevalues of the parameters. Nevertheless, for certaindomains of mH and ye values, a linear variation can beobtained. The cases in which two Tafelian domainscan be distinguished will be analyzed first,(a) For y ecorresponding to a Langmuir adsorption [8, 11].Taking into account that at high vZv values v i3v + i(i= V, H), when y(Z)= y * Eq. (11) written on termsof current density is reduced tojj0V 2mH y*ye s*1l e1a fZ 2 1 y*1 ye s*le1a fZ, 17which shows the existence of a Tafelian domain athigh overpotential values with a slope bh=2.3026RT/(1 a)F. The exchange current density at high vZvobtained by extrapolation ( j hext) isjexth 2mHj0Vy*yes*1l 2mHj0Vs*1lye 1 yemHs* : 18This behavior is clearly illustrated in Figs. 1 and 2. Inthe dependence ln( j/jV0 ) vs. Z, the values of ln( j hext/jV0 )= ln 2mHs*(1 l)y */y e are shown as open circles,evaluated on each case by Eq. (18). Furthermore, inthe y vs. Z relationship, it can be clearly distinguishedthe overpotentials range where y(Z)= y *.(d) The existence of a unique linear dependence ofln j in the whole range of overpotentials is possible forcertain values of the parameters y e and mH. It isnecessary that the condition y(Z)3y * be fulfilled at anvZv value suciently low such that the first Tafeliandomain cannot be developed. In order to determinethe y e and mH values that obey such condition, anoverpotential Z # such that y(Z #) y *=102 wasdefined and calculated from Eqs. (12) and (16), withu= 5. Such vZ #v, which denotes the beginning of thelinear domain corresponding to the high overpoten-tials, should be less than 0.2 V. In this case the corre-sponding Tafelian dependence follows Eq. (17) withs *31,j jexth e1a fZ, jexth 2mHj0Vye 1 yemH : 19Fig. 4 shows the dependence Z # vs. log[y e/(1 y e)] fordierent mH. It can be easily established that fory e3 103 or for y e>0.5 andmHa= l= 0.5, u= 10 and mH=104. In the range 0.05VR vZvR 0.20 V, where 0.2083R y(Z)R 0.5976, a slightlycurved dependence is observed. The linear regression insuch range gives a slope equal to 57.6 mV dec1 and anorigin ordinate equal to 8.2218. These results will beanalyzed in detail in Section 4.3.1.3. Interpretation of measurable quantitiesFrom the experimental determination of the depen-dence of current density on a wide range of overpoten-tial, the existence of Tafelian domains can beestablished and the values j lext, j hext, bl and bh, or someof them, can be calculated.The relationship between the extrapolated currentdensities and the kinetic parameters y e, mH, u and lcan be evaluated through Eqs. (14), (15) and (18). Itcan be observed that for j lext there are two alternatives(Eq. (14) for y evT veT1 y21 ye2 s2l, 25where v +T and v T are the forward and backwardreaction rates of the Tafel step and v Te is the equili-brium reaction rate of such step. On steady state, therate of Eq. (1) is given by [10]V 0:5vV vT vV vT, vV vV vV,vT vT vT:26Substituting Eqs. (5), (6), (24) and (25) in Eq. (26)givesVveV 0:51 y1 ye sl e1a fZ yyes1l eafZ mTy2ye2s21l 1 y21 ye2 s2l1 y1 ye sl e1a fZ yyes1l eafZmTy2ye2s21l 1 y21 ye2 s2l, 27where mT= v Te /vVe . Reordering the last two terms ofEq. (27), the following implicit function of y can bedefined:ayy2 by, Zy cy, Z 0, 28whereay 2mTs21lye2 s2l1 ye2, 28aby, Z 4mTs2l1 ye2 eafZefZsl1 ye s1lye, 28bcy, Z 2mTs2l1 ye2 e1a fZsl1 ye , 28cand s= s(y(Z)) is given by Eq. (9).Eqs. (27), (28) and (28a)(c), together with Eq. (2),describe completely the dependence of the current den-sity on overpotential for the VolmerTafel route with-out kinetic approximations.3.2.1. Tafelian domainsAs in the previous case, the existence of overpoten-tial domains where a linear dependence is verified isnot straightforward. However, for certain ranges of y eand mT values, Tafelian domains can be found.(a) Considering the case in which mTConsequently, the VolmerTafel route at overpoten-tials suciently high always defines, as it is wellknown, a limiting kinetic current density independentlyof the behavior in the low overpotentials region. Opensquares in Fig. 6 illustrate the values of ln( jTlim/jV0 ) cor-responding to lines a and b.3.2.2. Pseudo-Tafelian dependenceThe existence of Tafel lines with slopes near2.3026RT/F has been also proposed for the VolmerTafel route [3, 6], that cannot be justified from Eqs. (2)and (28), as in the previous case. Nevertheless, pseudo-Tafelian behaviors can be observed at low Z, as it is il-lustrated in Fig. 7, where the following parametersvalues were used: mT=105, y e=101, l= a= 0.5and u= 10. In the range 0.05 VR vZvR 0.20 V, where0.2084R y(Z)R 0.6077, a slightly curved dependence isobserved. The least squares linear regression in suchrange, also shown in Fig. 7, gave a slope equal to54.74 mV dec1 and an origin ordinate equal to10.3427. These results will also be analyzed in detailin Section 4.Furthermore, the simulations shown in Fig. 6allowed finding another overpotential range where aslight curvature of ln j vs. Z dependence is observed.They can also be considered as a pseudo-Tafeliandomain. On these cases, Eq. (30) cannot be reduced toa linear expression and the extrapolation of the fittedlines at Z= 0 does not give an amount related to theparameter y e, mH, etc. For example, for line a inFig. 6, a pseudo-linear domain can be found in therange 0.35 VR vZvR 0.65 V, with a Tafel slope equalto 168.6 mV dec1. The same for line b in the range0.30 VR vZvR 0.65 V, where the Tafel slope is equal to136.7 mV dec1. It should be noticed that as |Z|increases, the limiting current will be achieved, accord-ing to Eq. (35).3.2.3. Interpretation of measurable quantitiesThe experimental determination of the dependenceof current density on a wide range of overpotentialsallows in this case the evaluation of j lext, jTlim and bl.Two dierent Tafelian behaviors can be obtained atlow overpotentials. One of them is characterized bybl=2.3026RT/2F and a j lext value evaluated byEq. (31). The other has bl=2.3026RT/(1 a)F andj lext follows Eq. (33). On the other hand, the determi-nation of j Tlim allows infer that the HER is taking placethrough the VT route. This limiting current densitycontributes also to the calculation of the kinetic par-ameters. However, its determination is not alwayspossible and rather unusual [13, 14].Finally, a non-linear regression of all the experimen-tal points should be done when a clear Tafeliandomain at low Z is not observed.Fig. 6. Dependence of ln( j/jV0 ) and y on Z for the VT route.y e=105; a= l= 0.5; u= 5; mT=(a) 107, (b) 105, (c)103, (d) 101, (e) 10. ln( j lext/jV0 ): (*) Eq. (31), (w) Eq. (33);(q) ln( jTlim/jV0 ).Fig. 7. Dependence of ln( j/jV0 ) and y on Z for the VT route.y e=101; a= l= 0.5; u= 10; mT=105. ln(j lext/jV0 ): (W)linear fitting; (r) Eq. (41).M. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841851 8474. DiscussionThe dependencies of current density and surface cov-erage on overpotential for the HER, when it takesplace through the VolmerHeyrovsky or the VolmerTafel routes and the adsorbed intermediate is modelledby the Frumkin isotherm, have been simulated throughthe resolution of implicit equations and the conditionsfor the existence of Tafelian domains were determined.Such behaviors can be used for the interpretation ofthe experimental results when the double layer eectsare considered virtually eliminated by the presence ofan excess of supporting electrolytes [15].At first, the pseudo-Tafelian domains at low overpo-tentials values will be analyzed. For the VolmerHeyrovsky route, slightly curved dependencies can beobserved. When they are fitted by linear regression,pseudo-Tafel slopes with a value near 2.3026RT/F areobtained, as well as the corresponding origin ordinate( j ext). It has been also found that the surface coveragevaries strongly with overpotential in such domains.This behavior is usually interpreted with an approxi-mated kinetic analysis, which assumes that theHeyrovsky step is the rds [3, 6] and on this context thecurrent density becomes equal toj 2FvH 2FveHyyes1l e1a fZ: 36On this approximated analysis, the dependence of yon Z is derived applying the quasi-equilibrium con-dition to the Volmer step, resulting from Eqs. (5) and(6),y1 y ye1 ye s1 efZ: 37In the range 0.2R yR 0.8, another approximation isapplied to Eq. (37), which consists in considering y/(1 y)31. On these conditions, obtaining s fromEq. (37), substituting it in Eq. (36) and taking logar-ithm, the following expression is obtained:ln j ln2FveHyel1 yel1 ln y 2 a lfZ: 38This expression does not define a Tafelian domain, asthe slope depends on Z. Nevertheless, trying to justifya domain such that the variation of ln y on Z is neg-lected. On this basis, an slope equal to 2.3026RT/(2 a l)F is obtained [3, 6]. If this argumentationwould be correct, then the origin ordinate would begiven byln jextl ln2jeVmHyel1 yel1: 39Consequently, the linear fitting of the rigorouslysimulated results should give an origin ordinate coinci-dent with that of Eq. (39). Nevertheless, the simu-lations lead to the conclusion that there is noagreement between the value obtained by extrapolationand that calculated by Eq. (39). For example, thevalues corresponding to the case illustrated in Fig. 5are ln( j lext/jV0 )= 8.2218 (extrapolation) and ln( j lext/jV0 )= 6.1093 (Eq. (39)). Besides, the slope obtainedfrom the linear regression (57.6 mV dec1) is slightlyless than that resulting of considering l= a= 0.5 in2.3026RT/(2 a l)F (59.16 mV dec1).Consequently, the approximation consisting of neglect-ing the term ln y in Eq. (38) is incorrect.A similar situation can be found for the VolmerTafel route at low vZv if the Tafel step is considered asthe rds and the Volmer step at equilibrium. The ex-pression of the current density is in this caseln j ln2jeVmTye2l1 ye2l1 2 ln y 21 l fZ, 40where a Tafelian domain cannot be defined and, as inthe previous case, the variation of ln y on Z was neg-lected in order to justify a Tafel slope equal to2.3026RT/2(1 l)F [3, 6]. The corresponding originordinate of this linear approximation isln jextl ln2jeVmTye2l1 ye2l1: 41As in the previous case, the amount resulting fromthe application of Eq. (41) is far from the origin ordi-nate obtained from the extrapolation of the linear re-gression of the pseudo-Tafelian domain, as isillustrated in Fig. 7. These values are ln( j lext/jV0 )= 10.0641 and ln( j lext/jV0 )= 8.4118 for theextrapolation and Eq. (41), respectively. Besides, theslope obtained from the linear regression (54.74 mVdec1) diers from that resulting of consideringl= 0.5 in 2.3026RT/2(1 l)F (59.16 mV dec1).The results described above demonstrate that theregions usually considered linear with slopes equal to2.3026RT/(2 a l)F and 2.3026RT/2(1 l)F arepseudo-Tafelian domains. Notwithstanding, thesedomains are useful because they are only possiblewhen ur 5 and therefore they are a clear indication ofa Frumkian behavior, in spite of that the kinetic par-ameters of the elementary reaction steps cannot beobtained, as in the case of real Tafelian domains.The results obtained in the present work allow toconclude that Tafelian domains at low vZv can be foundonly if y is constant or else if it has a small variationon overpotentials. In this case, the Tafel slope can takethe value 2.3026RT/(2 a)F or 2.3026RT/(1 a)F forthe VolmerHeyrovsky route and the values 2.3026RT/2F or 2.3026RT/(1 a)F for the VolmerTafel route.Therefore, the values that can take the Tafel slope areindependent of the adsorptive characteristics of theM. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841851848substrate, due to the constancy of the surface coveragein such domain. A similar behavior was found for thehigh vZv region, where y= y* and b= 2.3026RT/(1 a)F for the VolmerHeyrovsky route and y= 1and b= 1 for the VolmerTafel route. Besides, a re-lationship between the extrapolated current density atZ= 0 and the kinetic parameters of the elementarysteps can be established for each case. It should benoticed that the same Tafel slope value for substrateswith dierent electrosorption characteristics does notimply that the corresponding kinetic behaviors are alsoequal. For example, the extrapolated current density(Eq. (18)), the limiting values y* (Eq. (16)) and j lim(Eq. (35)), etc. demonstrate the existence of very dier-ent behaviors, in spite of the same Tafel slope value.A second aspect to be analyzed is related to the in-fluence of the interaction parameter u on the behaviorof the ln j vs. Z relationship. It should be particularlyinteresting to analyze the influence on the low vZvregion. Fig. 8 shows the simulation corresponding tothe VolmerHeyrovsky route with the parametersy e=103, l= a= 0.5, mH=107 and 0R uR 10. Asmall range of overpotentials can be observed wherey(Z)3y e, which constitutes a Tafelian domain with aslope equal to 2.3026RT/(2 a)F and a j lext valuegiven by Eq. (14), shown as a solid circle in Fig. 8. Inthe following Z range, a slight curvature can beobserved, directly related to the sharp increase of y(Z),generating a pseudo-Tafelian domain, which widelyincreases as u increases. The influence of u on the highvZv region is also illustrated in Fig. 8, where the opencircles indicate the values of ln( j hext/jV0 ) for the dierentlines. A similar behavior can be found in the case ofthe VolmerTafel route. The influence of the inter-action parameter on the ln( j/jV0 ) and y vs. Z dependen-cies can be appreciated in Fig. 9, where the followingvalues of the parameters were used: y e=103,mT=106, l= a= 0.5 and 0R uR 10. At low vZvthere is a small Tafelian domain with a slope equal to2.3026RT/2F and a j lext value given by Eq. (31),shown as a solid circle in Fig. 9. In the following Zrange there is a pseudo-Tafelian domain, whichwidely increases as u increases. Finally, the influenceof u on the high vZv region is illustrated in Fig. 9,where the open squares indicate the values ofln( jTlim/jV0 ) for the dierent lines calculated byEq. (35). These examples demonstrate the origin ofthe pseudo-Tafelian behaviors at low overpotentialspreviously discussed.Another aspect that should be useful to analyzeis the distinction between the Frumkian andLangmuirian behavior for the VolmerHeyrovskyroute. On this sense, for certain values of the par-ameters of the elementary steps, the HER becomesFig. 8. Dependence of ln( j/jV0 ) and y on Z for the VH route.y e=103; a= l= 0.5; mH=107; u= (a) 0, (b) 2.5, (c) 5,(d) 10. (.) ln( j lext/jV0 ); (w) ln( j hext/jV0 ).Fig. 9. Dependence of ln( j/jV0 ) and y on Z for the VT route.y e=103; a= l= 0.5; mT=106; u= (a) 0, (b) 2.5, (c) 5,(d) 10. (.) ln( j lext/jV0 ); (q) ln( jTlim/jV0 ).M. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841851 849independent of the behavior of the adsorbed inter-mediate. It should be taken into account that thedierence between the expressions of the reactionrate of the elementary steps when the Frumkin orLangmuir adsorption are considered is the factor s,which at suciently high vZv reaches the value s *.Therefore, if s *31 for certain values of the par-ameters involved, the behavior will be similar tothat corresponding to a Langmuirian adsorption, inspite of the dierent adsorption properties (u$ 0).This case corresponds to vy *y ev[10] M.R. Gennero de Chialvo, A.C. Chialvo, J. Electroanal.Chem. 388 (1995) 215.[11] M.R. Gennero de Chialvo, A.C. Chialvo, J. Electroanal.Chem. 415 (1996) 97.[12] J.OM. Bockris, J.M. Carbajal, B.R. Scharifker, K.Chandrasekaran, J. Electrochem. Soc. 134 (1987) 1957.[13] G. Kreysa, B. Hakansson, P. Ekdunge, Electrochim.Acta 33 (1988) 1351.[14] O. Nomura, H. Kita, J. Res. Inst. Catalysis, HokkaidoUniv. 15 (1967) 35, 49.[15] L.J. Krishtalik, in: B.E. Conway, J.OM. Bockris, E.Yeager, S.U. Khan, R.E. White (Eds.), ComprehensiveTreatise of Electrochemistry, Vol. 7. Plenum Press, NewYork, 1983, p. 87.[16] B.V. Tilak, B.E. Conway, Electrochim. Acta 21 (1976)745.[17] B.V. Tilak, C.G. Rader, B.E. Conway, Electrochim. Acta22 (1977) 1167.M. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841851 851

Recommended

Beyond the Butler-Volmer equation. Curved Tafel slopes ... University Institutional Repository Beyond the Butler-Volmer equation. Curved Tafel slopes from steady-state current voltage curves [keynote lecture]Authors: Stephen Fletcher Thomas S VarleyAffiliation: Loughborough UniversityAbout: Electrochemistry Electron transferDocuments