On the pH dependence of electrochemical proton transfer barriers

Catalysis Today 262 (2016) 36–40

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On the pH dependence of electrochemical proton transfer barriers Jan Rossmeisl a,∗ , Karen Chan b , Egill Skúlason c , Mårten E. Björketun d , Vladimir Tripkovic d a

Department of Chemistry, University of Copenhagen, Universitetsparken 5, København Ø 2100, Denmark SUNCAT Center for Interface Science and Catalysis, Stanford University, Stanford, CA 94305, USA Science Institute and Faculty of Physical Sciences, VR-III, University of Iceland, IS-107 Reykjavik, Iceland d Department of Physics, Technical University of Denmark, Building 311, Lyngby 2800, Denmark b c

a r t i c l e

i n f o

Article history: Received 25 June 2015 Received in revised form 6 August 2015 Accepted 7 August 2015 Available online 7 September 2015 Keywords: Hydrogen evolution/reduction Charge transfer reaction Entropic barrier

a b s t r a c t The pH dependence of rate of the hydrogen evolution/oxidation reaction HER/HOR is investigated. Based on thermodynamic considerations, a possible explanation to the low exchange current for hydrogen reactions in alkaline is put forward. We propose this effect to be a consequence of the change in configurational entropy of the proton as it approaches the surface. As a proton crosses the outer Helmholtz plane, it will lose a fraction of its entropy before it can interact with the electrode surface, which gives rise to an entropic barrier. The size of this barrier will depend on the electrostatic environment in the double layer region. The entropic barrier can be rate determining only when the surface catalysis is fast. Therefore the effect of pH is most pronounced on good catalysts and for fast reactions. This entropic barrier is also in a good agreement with the unusually low prefactor measured in experiments of good catalysts such as Pt. In such catalysts, the enthalpy barrier of 0.1–0.2 eV of the rate-determining step does not come from any of the surface reactions (Volmer, Tafel or Heyrovsky) but instead from the proton transfer into the outer Helmholtz layer. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The electrocatalysis of the water cycle reactions – hydrogen oxidation (HOR), hydrogen evolution (HER), oxygen reduction (ORR) and oxygen evolution (OER) – has been intensively studied the last hundred years. However, some very fundamental questions remain unanswered. In this paper, we address why pH affects the HOR and HER reaction rates. Common to all the water cycle reactions is the exchange of protons between the electrolyte and the reaction intermediates on the surface. Proton transfer is thus a key component of all four reactions, but pH influences the reactions very differently. On Pt, the hydrogen reactions HOR and HER are 2–3 orders of magnitude faster at low pH than at high pH [1,2], whereas ORR and OER are almost unaffected by pH [3,4]. In this study, we present a perspective on the origin of the pH-dependence for HER and HOR. We focus only on coupled proton–electron transfers and neglect reactions involving intermediates with a large dipole moment, where reaction energetics depend on the composition of the elec-

∗ Corresponding author. Tel.: +45 50719584. E-mail address: [email protected] (J. Rossmeisl). http://dx.doi.org/10.1016/j.cattod.2015.08.016 0920-5861/© 2015 Elsevier B.V. All rights reserved.

trolyte. Reactions where only protons or electrons are exchanged depend directly on the absolute electrode potential or pH. The thermodynamic driving force for a combined proton and electron transfer reaction depends only on the reversible hydrogen electrode (RHE) potential, which is given by the sum of the proton and electron chemical potentials: H + = oH + + kT ln aH + = oH + − 2.3kT pH

(1a)

e− = oe− + eUSHE = oe− + eURHE + 2.3kT pH

(1b)

H + + e− = oH + + oe− + eUSHE − 2.3kT pH = oH + + oe− + eURHE (1c) where i is the chemical potential of species i, the superscript o indicates standard conditions, k the Boltzmann constant, T the temperature, and USHE and URHE the potential on an SHE and RHE scale, respectively. Hence, as long as a change in pH is exactly cancelled by a change in the absolute electrode potential, i.e. URHE is constant, the thermochemical driving force will also stay constant. In experiments this correspond to having the hydrogen reference electrode in the same electrolyte as the working electrode. One may argue that under alkaline conditions there are no protons

J. Rossmeisl et al. / Catalysis Today 262 (2016) 36–40

available and that is the reason why proton transfer to the surface is slow or that there is a bigger barrier for proton donation from water compared to hydronium ions. However, water can provide the protons through the reaction. H2 O ↔ HO− + H+ ,

(2)

which is fast and therefore often can be considered in equilibrium. For hydrogen reactions, the change is observed in the exchange current density at equilibrium conditions where the net rate is zero, and where diffusion limitations are not present. Therefore, the lack of protons or slow proton donation in alkaline media does not directly explain why the rate of the hydrogen reactions are affected by a change in pH. Atomic scale simulations of proton transfer reactions across electrochemical interfaces have been performed by a number or research groups, including us [5–11]. Except in a few special cases (proton discharge in the Volmer reaction at low pH and the reaction of protons with adsorbed hydrogen in the Heyrovsky reaction [8]), it has been very difficult to identify any potential energy barrier for the proton transfer in these simulations. It seems almost impossible to obtain a situation where the proton in the outer Helmholtz layer (OHL) is close in energy to the proton adsorbed on the surface. Typically, a simple structural relaxation of the interface automatically triggers the proton transfer. This is normally considered to be due to finite size effects in the simulations [12]. Here, we will hypothesize that it, at least partially, reflects a real physical issue. In this study, we propose an explanation for why the exchange current for hydrogen oxidation and evolution exhibit strong pH dependence, even though the thermochemical driving force is independent of pH. In the first section, we discuss the thermodynamic and electrostatic properties of the electrochemical interface and describe how these are influenced by pH and electrode potential. This discussion draws on our recent work on the pH dependence of the structure of the electrochemical interface [13]. Based on the insights obtained in the first section, we then introduce a qualitative model for the proton transfer from electrolyte to electrode surface. We show that the model captures measured trends in activation barriers and rates for HOR and HER, and we discuss the implications it has for atomistic simulations of proton coupled charge transfer reactions. 2. Thermodynamics and electrostatics of the interface The equilibrium of the hydrogen reaction can be written in both an acidic and an alkaline way: 2H+ (aq) + 2e− ↔ H2 (g)

(3a)

2H2 O(l) + 2e− ↔ H2 (g) + 2OH− (aq)

(3b)

The only difference between Eqs. (3a) and (3b) is that in (3b), reaction (2), which is normally in equilibrium, has been added to reaction (3a). Reactions (3a) and (3b) themselves do not provide any insight into why hydrogen exchange rates are faster in acidic environments. In both cases, the potential at which the two sides are in equilibrium is at 0 V vs. RHE. As mentioned above, on the RHE scale, there is no distinction between the proton and electron chemical potentials, the two terms are lumped together in the potential vs. RHE. In order to understand the pH dependence, the respective contributions of electrons and protons to the RHE potential have to be considered separately [15]. Imagine reactions (3a) and (3b) take place in two different electrolytes with different pH values. The difference in chemical potential between the protons in the two electrolytes is given by the difference in configurational entropy. This is calculated subtracting the chemical potential of the protons (given by Eq. (1a))

37

in two different electrolytes where pH is the difference in pH between the two electrolytes,





 H+ = −TS = kT ln

aH + = −2.3kTpH aoH +

(4)

The proton solvation energy is thus a linear function of pH. The proton solvation energy contains many other contributions than just the configurational entropy, however only the configurational entropy changes with pH. The free energy of H2 (g), however, is not dependent on the pH. Overall, reactions (3a) and (3b) take place through an adsorbed H* intermediate, H+ (aq) + e− ↔ H∗ ↔ 1/2H2 (g),

(5)

For simplicity, we write Eq. (5) in the acid way. We have used the notation H*, denoting hydrogen adsorbed on the electrode, and have neglected a Heyrovsky mechanism for simplicity. The energy difference between H2(g) and H* is the binding energy of hydrogen on the surface. In order to analyze the effect of the pH, we consider the equilibrium case where the three different states of hydrogen (H+ (aq) + e− , 1/2H2 (g) and H*) have the same chemical potential, H + + e− = H • =

1 H2 2

(6)

where the pH and potential dependence of the H + + e− term is given by Eq. (1c). Under this equilibrium condition, a given change in pH (i.e. a change in proton solvation energy in solution) corresponds to the same change in the electrode potential, eUSHE (Eq. (1c)), which is equal to the electron work function at the metal|solution interface, set up by the interface dipole. In other words, the interface dipole changes with pH to match the solvation energy of protons in bulk solution, the equilibrium structure of the interface changes as the pH and electrode potential change, even if the thermodynamic driving force, URHE , remains constant. The adsorption energy, coverage and entropy of adsorbed hydrogen (H*(interface)) in Eq. (6) are expected to be almost independent of the thermochemistry of protons and electrons, a conclusion that is supported by cyclic voltammograms, the underpotential hydrogen deposition region of e.g. Pt(1 1 1) at different pH is identical when plotted vs. RHE [14,15]. From this it follows that the binding energy of H* relative to 1/2H2 (g) is virtually unaffected by electrode potential, and is at most only a weak function of the field, which has also been seen by DFT simulations [16]. A shift in pH (at fixed URHE ) corresponds to a change in the dipole and corresponding field at the interface, meaning that the binding energy is close to independent of pH [17]. Thus any pH dependency for the proton transfer has to arise somewhere between the proton solvated in the bulk of the electrolyte and the adsorbed H* on the electrode surface. 3. Proton transfer reaction The chemical potential of a proton in the bulk of the electrolyte varies linearly with pH. During discharge, the proton is transferred to a surface where it adsorbs. On the surface, as argued, its binding energy and entropy are almost independent of pH. We will now discuss what consequences this has for the kinetics of the proton transfer. Let us again examine a proton transfer that takes place under equilibrium conditions: H+ (aq) + e− ↔ H∗

(7)

We are studying a single proton moving from the bulk of the electrolyte toward the surface of the electrode. The pH and potential, and thereby also the dipole of the interface, are constant. We

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J. Rossmeisl et al. / Catalysis Today 262 (2016) 36–40

The simple picture outlined above explains why proton transfer becomes impeded as the pH is increased. Since the entropic stabilization of H+ (aq) is larger for high pH, the protons simply lose more entropy when they approach the OHL. At the same time the enthalpy gained by reacting the proton from the OHL to the surface is larger (the difference between the high and low pH situations can be seen by comparing the two plots in Fig. 2. An implication of this is that the barriers for hydrogen reactions in alkaline environments are mainly entropic, and the transition state is located in the OHL or further into the bulk solution. This is to some extent similar to the picture for ion dissolution, where a barrier for dissolution arises as a result of energy transition between solvation in the electrolyte and interaction with the surface, as reported in Ref. [18]. 4. Comparison with experiments

Fig. 1. Proton transfer from the bulk of the electrolyte to the electrode surface. (a) Schematic representation of the interface and the proton transfer process at high pH (a) and low pH (b). The arrows indicate the direction, from negative to positive, of the water dipoles in the outer Helmholtz layer (OHL).

consider only the entropy change and the electrostatics. Due to the imposed equilibrium, G of reaction (7) is zero, which means that H(bulk) − TS(bulk) = H(surface) − TS(surface),

(8a)

H = H(bulk) − H(surface) = TS(bulk) − TS(surface) = TS.

(8b)

Generally, the proton loses the configurational entropy (given by −2.3kTpH in the bulk) when it approaches the surface. It follows from Eq. (8b) that this loss has to be exactly counteracted by an opposite change in enthalpy (given by the work function eUSHE ) at the surface. This cancellation holds for the initial relative to the final state, but not necessarily at other points along the reaction path. In particular, there could be situations where the loss of entropy occurs before the gain in enthalpy, and that would result in a free energy barrier for transferring the proton to the surface. Consider an intermediate state, the proton in the Outer Helmholtz Layer, (OHL). The OHL is located at the water layer closest to the surface. With the inclusion of the H+ (OHL) , intermediate state, reaction (7) can be written as (cf. Fig. 1) +

H

(aq)

−

+e → H

+

(OHL)

−

+e → H ∗ .

(9)

Since the water in OHL is in between liquid water and a solid surface, it is reasonable to expect that the entropy of H+ (OHL) is more or less independent of the pH in the bulk, just as it is for H*. The atomic vibrations in the OHL are almost constant no matter the pH, and so is the configurational entropy. The proton has thus lost all, or a part of the entropy when reaching the OHL, and the −TS term has gone up. At the same time, the proton has not yet sensed a change in electrostatic potential when it reaches the OHL, as most of the potential difference is found in the narrow region between the first water layer and the electrode surface. This means that H+ (OHL) has still not gained much enthalpy from the interaction with the electrostatic environment. Summing the entropy and enthalpy terms, we obtain the free energy of the proton in the OHL, which is higher than the energies in the bulk electrolyte and the surface of the electrode (cf. Figs. 1 and 2). The proton free energy in the OHL can thus be viewed as an estimate of the transition state for proton transfer, and the barrier is equal to the reaction energy of the proton as it relaxes toward the surface.

For comparison with experiments we now focus on the implication of the entropic barrier on hydrogen evolution to H2 . The experimental literature contains many records of barriers and prefactors of electrochemical proton transfer reactions, measured under different conditions. It has proven very demanding to get a good estimate of the kinetics in acid environments, as the proton transfer is extremely fast. Around two-three orders of magnitude difference in exchange rates between acid and alkaline media has been reported in experiments where the rate was not limited by mass transport [19–21]. As the rate of hydrogen reaction is so fast on Pt in acid media [3] and since there is no straightforward way to account for Ohmic losses [22], it is very hard to get an accurate estimate of the activation energy. In contrast, in alkaline media due to slower kinetics the barrier was measured to be 0.30 eV [3]. We believe that the activation barriers in the two environments are the same and that the difference in exchange rate originates mostly from different prefactors in the Arrhenius analysis. A difference in prefactors is consistent with the notion of an entropic barrier, as can be seen from a simple rate expression. G† , H† , S† refers the free energy, enthalpy and entropy difference between the initial and transition state of a reaction respectively.



r ∝ exp

−G† kT



 



= exp − H † − TS † kT

 = exp

−S † k



 exp



−H † kT

 (10)

The barrier measured in an Arrhenius plot (the slope of ln(r) vs. 1/T) is only the enthalpy barrier H† , whereas the prefactor will contain the entropy term S† . In this simple picture, the prefactor will change an order of magnitude per pH unit, i.e. a much larger change than the experimentally observed difference in exchange rates of 2–3 orders of magnitude over 14 pH units. In that case the entropic barrier is the activity of the protons in solution. This is the upper limit for the effect of pH as it corresponds to all configurational entropy loss, no interactions with the electrostatic potential for H+ (OHL) and a perfect catalyst with no barriers related to the surface catalysis. A good catalyst for the hydrogen reaction is characterized by having a high exchange rate. The highest exchange rate can be obtained if the binding energy of hydrogen on the surface is zero and the barriers for the surface reactions are small. However, for non-perfect catalysts, the exchange rate can never be higher than what is defined by the binding energy of hydrogen. This means that the increase in exchange rate in acidic compared to alkaline environments is dependent on how good the catalyst is. For the perfect catalyst (e.g., like Pt) the effect of pH would be larger than on a bad catalyst e.g. Au, for which the exchange rate probably will be limited by the weak binding of hydrogen no matter the pH.

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Fig. 2. Schematic drawings illustrating entropy and enthalpy changes (left) and the entropic barrier changes (right) along the reaction path at low and high pH.

In a very recent study, Durst et al. [2] find similar differences in exchange rates on good metal catalysts, Ir, Pd and Pt. However, it is interesting to note that the effect of pH is largest on Pt [2], which is also the surface with the highest exchange rate at acid conditions and the material with the binding strength of hydrogen closest to zero [8]. This suggests that the better the catalyst, the bigger the effect of pH, in agreement with the hypothesis here. At acidic conditions, the exchange rate is dominated by the hydrogen binding at the surface, as the entropic barrier is small. At alkaline conditions, all good catalysts become limited by the entropic barrier. However, the exchange rates in alkaline environments is not exactly the same on Pt, Pd and Ir as one would expect if the exchange rate was totally dominated by the entropic barrier. We expect that even in alkaline environment on good catalysts the energy of protons in the OHL is slightly dependent on the nature of the catalyst surface. This effect does not have to very large to explain the 10-fold difference between Pd and Pt in an alkaline environment, which corresponds to an energy difference of 2.3kT for H+ (OHL). Overall, the part of the reaction that is responsible for the barrier depends on the reversible hydrogen potential. The model can be summarized as: For −TS < e − eURHE , the barrier is dominated by e − eURHE For −TS > e − eURHE , the barrier is dominated by −TS.where  is the overpotential for hydrogen evolution on the specific material and is related to the binding of hydrogen on the surface. In the latter situation, −TS > e − eURHE , the barrier becomes a constant defined by the pH. This is the situation for a good catalyst where e is small, or at high pH where −TS is large. The former situation, −TS < e − eURHE , corresponds to low pH and/or poor catalysts and

hence a small overall driving force. There, the rate is defined by the surface catalysis. The two situations are schematically drawn in Fig. 3.

5. Other reactions Oxygen reduction is known not to be affected the same way by pH as the hydrogen reactions, even though it involves coupled proton-electron transfers to the surface. Let us consider the oxygen/water equilibrium: 4(H+ + e− ) + O2 ↔ 2H2 O Even on the best catalyst surfaces, this reaction is related to a huge barrier at equilibrium conditions. The exchange current is very small. It is probably difficult to find reaction conditions where the entropic barrier will limit the reaction rate, as the surface reaction barriers at small overpotentials are huge. This argument is similar to the discussion above for poor hydrogen catalysts. In the case of ORR, there exists no very good catalyst where the effect of pH can be observed. Thus the effect of pH via the entropic barrier is not relevant for reactions that require a high overpotential. We expect reactions that on the best catalyst proceed at low overpotentials (e.g. Cl2 evolution) could be limited by the entropic barrier.

Fig. 3. A Schematic drawing of the entropic barrier for (a) −TS > e − eURHE and (b) −TS < e − eURHE .

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6. Consequences for simulations of proton transfer reactions The picture above also explains why no or only very small barriers for proton-coupled charge transfer from the OHL to the surface have been observed in atomic-scale simulations. Only at very low pH has a small barrier been identified, this is the situation where one would expect the entropy barrier to be small and it is a good assumption to expect equilibrium between protons in the OHL and the electrolyte, as the biggest barrier is between H+ (OHL) and H* (Heyrovsky reaction). However, simulations of a high pH interface do not give any potential energy barrier, but only a large decrease in energy when the structure is relaxed due to spontaneous proton transfer. In this situation, it is no longer valid to consider protons in the OHL to be in equilibrium with protons in the bulk, as there is a large entropic barrier associated with bringing the proton from the bulk to the OHL. This is not an effect of a too small simulation cell. Rather, it is because the proton in the OHL represents the transition state and not the initial state for a proton transfer to the surface. The initial state that can be in equilibrium with H* is the proton in bulk of the electrolyte. 7. Summary In this paper, we have seen that the dipole of the electrochemical interface depends on pH and electrode potential, the more acidic, or the higher the RHE potential, the smaller the dipole of the interface. We have further seen that a proton moving from bulk electrolyte toward the electrode surface must convert entropy into enthalpy. The higher the pH, the larger the amount of entropy converted. This picture gives an intuitive explanation of the effect of pH on electrochemical barriers. It becomes a matter of balance between the entropy of the proton in the bulk electrolyte and the enthalpy of the proton in the outer Helmholtz layer. For fast surface reactions and good catalysts the entropy is lost before or in the OHL and the barriers are directly proportional to pH. For slow surface reactions or poor catalysts, this entropic barrier is probably never rate limiting. This suggests that the effect of pH should be larger the better the surface catalysis. This is supported by experiments where it is observed that the exchange current for the hydrogen reactions on Pt is affected more by pH than Pd and Ir. For slow reactions such as the oxygen reaction, the surface catalysis is limiting even for the best catalyst, where the effect of pH is not observed. Apart from providing the general understanding of the pH effect in hydrogen evolution and oxidation, this picture also directly explains why no or only very small proton transfer barriers have been observed in simulations, as the loss of entropy in bringing the proton to the interface has not been included. The results presented here provide an insight into the activity of the best HER/HOR catalysts such as Pt. In these catalysts, the rate is determined not by the elementary steps (Volmer, Tafel or Heyrovsky), but by the transfer of protons into the OHL. Our analysis

gives the good exchange rate if one assumes a low enthalpy barrier of around 0.1–0.2 eV, a similar barrier is observed for proton transfer in bulk water or ice [23,24], and a large entropic barrier, which gives an unusually low prefactor on the order of 103 site−1 s−1 . Acknowledgments The Catalysis for Sustainable Energy initiative is funded by the Danish Ministry of Science, Technology and Innovation. Support from the Icelandic Research Foundation and the Danish Center for Scientific Computing is gratefully acknowledged. References [1] R. Subbaraman, D. Tripkovic, D. Strmcnik, K.-C. Chang, M. Uchimura, A.P. Paulikas, V. Stamenkovic, N.M. Markovic, Science 334 (2011) 1256–1260. [2] J. Durst, A. Siebel, C. Simon, F. Hasche, J. Herranz, H.A. Gasteiger, Energy Environ. Sci. 7 (2014) 2255–2260. [3] W. Sheng, H.A. Gasteiger, Y. Shao-Horn, J. Electrochem. Soc. 157 (2010) B1529–B1536. [4] N.M. Markovic, H.A. Gasteiger, P.N. Ross, J. Phys. Chem. 100 (1996) 6715–6721. [5] J.S. Filhol, M. Neurock, Angew. Chem. Int. Ed. 45 (2006) 402–406. [6] R. Jinnouchi, A.B. Anderson, J. Phys. Chem. C 112 (2008) 8747–8750. [7] C.D. Taylor, S.A. Wasileski, J.-S. Filhol, M. Neurock, Phys. Rev. B: Condens. Matter 73 (2006) 165402. [8] E. Skúlason, V. Tripkovic, M. Björketun,.E. Gudmundsdóttir, S. Karlberg, G.S. Rossmeisl, J. Bligaard, T. Jónsson, H.J.K. Nørskov, J. Phys. Chem. C 114 (2010) 18182–18197. [9] E. Skúlason, G.S. Karlberg, J. Rossmeisl, T. Bligaard, J. Greeley, H. Jónsson, J.K. Nørskov, Phys. Chem. Chem. Phys. 9 (2007) 3241–3250. [10] M.J. Janik, C.D. Taylor, M. Neurock, J. Electrochem. Soc. 156 (2009) B126–B135. [11] M. Otani, I. Hamada, O. Sugino, Y. Morikawa, Y. Okamoto, T. Ikeshoji, J. Phys. Soc. Jpn. 77 (2008) 024802. [12] J. Rossmeisl, E. Skúlason, M.E. Björketun, V. Tripkovic, J.K. Nørskov, Chem. Phys. Lett. 466 (2008) 68–71. [13] J. Rossmeisl, K. Chan, R. Ahmed, V. Tripkovic, M.E. Björketun, Phys. Chem. Chem. Phys. 15 (2013) 10321–10325. [14] M.J.T.C. van der Niet, N. Garcia-Araez, J. Hernández, J.M. Feliu, M.T.M. Koper, Catal. Today 202 (2013) 105–113. [15] This is not the Case for Pt(533) and Pt(553), Probably due to Strong Adsorption of H2 O, OH or O on the Under-coordinated Pt Atoms. The Hydrogen Region is Therefore Affected as the Water from These Sites has to be Removed Before Hydrogen can Adsorb, Which in Turn Means that the Potential Region Normally Associated with Hydrogen is Changed. The Binding of Water may Depend on the pH as Water has a Large Dipole Moment that can Interact with the Field Across the Double Layer Region. [16] J. Rossmeisl, J.K. Nørskov, C.D. Taylor, M.J. Janik, M. Neurock, J. Phys. Chem. B 110 (2006) 21833–21839. [17] We note that this is not Necessarily the Case for Adsorbates with a High Dipole Moment, Which Would be Expected to Interact Strongly with the Interface Field and Thereby Show a pH-dependence. [18] M.C.L. Pinto, E. Spohr, P. Quaino, E. Santos, W. Schmickler, Angew. Chem. Int. Ed. 52 (2013) 7883–7885. [19] K.C. Neyerlin, W. Gu, J. Jorne, H.A. Gasteiger, J. Electrochem. Soc. 154 (2007) B631–B635. [20] X. Wang, R. Ahluwalia, A. Steinbach, J. Electrochem. Soc. 160 (2013) F251–F261. [21] S. Chen, A. Kucernak, J. Phys. Chem. B 108 (2004) 13984–13994. [22] D. van der Vliet, D.S. Strmcnik, C. Wang, V.R. Stamenkovic, N.M. Markovic, M.T.M. Koper, J. Electroanal. Chem. 647 (2010) 29–34. [23] N. Agmon, Chem. Phys. Lett. 244 (1995) 456–462. [24] O. Pecina, W. Schmickler, Chem. Phys. 228 (1998) 265–277.