Importance of the Walden Inversion for the Activity Volcano Plot of Oxygen Evolution

Abstract Since the birth of the computational hydrogen electrode approach, it is considered that activity trends of electrocatalysts in a homologous series can be quantified by the construction of volcano plots. This method aims to steer materials discovery by the identification of catalysts with an improved reaction kinetics, though evaluated by means of thermodynamic descriptors. The conventional approach for the volcano plot of the oxygen evolution reaction (OER) relies on the assumption of the mononuclear mechanism, comprising the *OH, *O, and *OOH intermediates. In the present manuscript, two new mechanistic pathways, comprising the idea of the Walden inversion in that bond‐breaking and bond‐making occurs simultaneously, are factored into a potential‐dependent OER activity volcano plot. Surprisingly, it turns out that the Walden inversion plays an important role since the activity volcano is governed by mechanistic pathways comprising Walden steps rather than by the traditionally assumed reaction mechanisms under typical OER conditions.


Figures S2-S3
illustrate unambiguously that various mechanistic pathways govern the OER activity volcano.The mononuclear description is only a good approximation for the volcano legs, but not for the volcano apex where the highly active materials are situated.Most notably, it becomes evident that highly active OER electrocatalysts reveal a change in the reaction mechanism with increasing electrode potential, a situation that, hitherto, has been completely overlooked in the modeling of OER materials.For further discussion, the interested reader is referred to reference [5].

It is noticeable that the volcano curves in Figures S2-S3
reveal kinks at the volcano legs.This finding is related to the fact that the volcano slope does not only change at the volcano apex, but even is prone to alter at the volcano legs.A change in the volcano slope at the legs is observed if another mechanistic pathway is energetically preferred or if another elementary step becomes the limiting one.For a detailed discussion on this matter and its implication to electrocatalysis, the interested reader is referred to references [13,14].

Thermodynamic analysis of the mononuclear mechanism
The presented modeling approach relies on an in-house methodology that connects the adsorption free energies of the intermediate species in the reaction mechanisms to the electrocatalytic activity by the descriptor Gmax(U) to compile volcano curves for oxygen evolution [14].In the following, this procedure is illustrated on the example of the mononuclear mechanism (cf.equations ( 1) -( 4)).The free energies of the reaction intermediates *, *OH, *O, and *OOH in dependence of the applied electrode potential are given by equations ( 29) -(33): By considering the scaling relations of equations ( 34) and ( 35), the energetics of the intermediate states are: Volcano curves arise by plotting Gmax(U) as a function of G1 at a constant applied electrode potential, U, see Figures S2-S3 as a prototypical example for the entire breadth of OER mechanisms.

Thermodynamic analysis of the mononuclear-Walden mechanism
The mononuclear-Walden mechanism reads: The free energies of the reaction intermediates *OH, *O, *OOH, and *OO in dependence of the applied electrode potential are given by equations ( 29) -(33): Due to G5 = G2 and G6 = G3, the energetics of the mononuclear-Walden mechanism can be related to the scaling relations of equations ( 34) and ( 35).Additionally, it can be shown that G7 = G2, and thus we obtain: Based on the intermediate states' energetics, the free-energy spans for the mononuclear-Walden mechanism can be defined to derive Gmax(U): The energetics of the mononuclear and mononuclear-Walden mechanisms is compiled in a single activity volcano plot based on the above modeling approach to determine the energetically favored pathway in dependence of the descriptor G1.This comprises that, based on the obtained values of Gmax(U) for the mononuclear and mononuclear-Walden mechanisms (cf.equations ( 41) and (56), respectively), the minimum Gmax(U) value is plotted as a function of G1. Figure S4 indicates the raw data for the OER volcano plot at U = 1.23 V vs. RHE, and its analysis culminates into Figure 1a of the main text where only the energetically favored pathway (minimum value of Gmax(U)) is shown.

OH(U); GM-OOH(U) -GM-OH(U); GM-OO(U) -GM-OH(U); GM-OOH(U) -GM-O(U); GM-OO(U) -GM-O(U); GM-OH+O2(U) -GM-O(U); GM-OO(U) -GM-OOH(U); GM-OH+O2(U) -GM-OOH(U); GM-OH+O2(U)
The following free-energy spans limit the OER volcano plot at U = 1.23 V vs. RHE (cf.Figures S7-S8 confirm that the volcano legs of the OER volcano are governed by the mononuclear-Walden description.This finding coincides with the volcano plot of the main text (cf. Figure 1).At the volcano apex, the mononuclear mechanism is preferred for small overpotentials, U ≤ 1.40 V vs. RHE, whereas for large overpotentials, U ≥ 1.60 V vs. RHE, the two mechanistic descriptions compete.For SRI = 2.80 eV, a competition between the two pathways is already visible at smaller applied electrode potentials (cf.Figures S8), yet competition at the volcano top becomes visible only at U = 1.60 V vs. RHE, in agreement with SRI = 3.00 and 3.20 eV.In summary, the sensitivity analysis reveals that the obtained results relating to the OER volcano curve of the mononuclear and mononuclear-Walden descriptions is robust despite the assumption of the scaling relation between the *OH and *O intermediates.

Thermodynamic analysis of the bifunctional and bifunctional-Walden mechanisms
The bifunctional mechanism reads: Figure S9 illustrates that for low overpotentials, the bifunctional and bifunctional-Walden pathways are favored at the right and left volcano legs including apex, respectively.Even if both mechanisms compete with increasing overpotential on the right-hand side of the volcano, the apex of the OER volcano is governed by the bifunctional-Walden description rather than the bifunctional mechanism.This finding pinpoints the importance of the bifunctional-Walden pathway for the theoretical description of highly active OER catalysts.
The following free-energy spans limit the OER volcano plot of

Precondition of *OH surface groups
In section 2 of the main text, it is discussed that the occurrence of Walden inversion steps relies on the precondition that *OH groups are available under OER conditions.This finding is backed up by equations ( 42) -(45) as as (72) (75) since the catalytic cycle commences from the *OH adsorbate rather than from the unoccupied metal site, M, as in the conventional description (see equations ( 1) -( 4)).
There In reference [24], Calle-Vallejo and coworkers present a detailed analysis on the PDS for a large data set of materials ranging from transition-metal oxides, metal oxides, perovskites, porphyrins, and functionalized graphitic materials by analyzing the largest free-energy change according to equation (6).In contrast to the conventional OER volcano of Figure S1, they report that the of the *OH adsorbate constitutes the PDS in about 12 % of all cases.This is the motivation to inspect the precondition of *OH surface groups under OER conditions in more detail.
While the determination of the actual surface coverage is far beyond the scope of the present manuscript and not possible by simple thermodynamic considerations in terms of analyzing the free-energy changes of intermediate species, Gj, we use the following criterion to inspect whether *OH surface groups are available under OER conditions: if the condition G1 < 1.50 eV (cf.equation ( 1)) is fulfilled, the surface of the electrocatalyst consists of *OH groups, and thus, mechanistic steps including the Walden inversion can occur.We discuss this criterion for several classes of materials based on the data in references [24-27].
Table S1.Analysis of the free-energy change G1 referring to *OH formation for various material classes, and assessment of the electrocatalytic activity in terms of the thermodynamic overpotential (cf.equation ( 6)) in case that G1 > 1.50 eV is met.Table S1 reveals that in more than 70% of all cases, the criterion G1 < 1.50 eV is met, and thus, surface *OH groups are available for the OER catalysis including Walden inversion steps.In less than 30% of all cases, the presence of surface *OH groups is questionable due to the thermodynamic restraint of *OH formation.For this specific situation, it is further analyzed whether these materials are highly active, adopting ηTD < 0.60 eV as a threshold for highly active catalysts [11,25].It turns out that in 5 out of 177 cases, a highly active catalyst (ηTD < 0.60 eV) reveals G1 > 1.50 eV, and thus, the occurrence of Walden inversion steps is debatable.This refers to less than 3 % of the entirely considered material space, and thus, the main conclusions of this manuscript relating to the importance of Walden inversion steps for highly active catalytic materials at the apex of the OER volcano plot does not break down.

Material class
O2(g) + H + + e − G4 Figure S1 reveals that either *OOH formation or *O formation governs the electrocatalytic activity at the left and right volcano legs, respectively, thereby making use of the tacit assumption that the PDS is equal to the rate-determining step [3].Hence, the common approach to enhance the electrocatalytic activity of electrode materials refers to the stabilization of the *OOH or *O adsorbates to lower the value of ηTD, depending on the position of the electrocatalyst in the volcano plot [4].

Figure S1 .
Figure S1.Generalized volcano plot for the oxygen evolution reaction based on the mononuclear mechanism under equilibrium conditions (U = 1.23 V vs. RHE).*OOH and *O formations are reconciled with the potential-dependent steps (PDS) at the left and right volcano legs, respectively.Figure reproduced with permission from reference [5].
)) 2 e -water oxidation: 2 H2O → H2O2 + 2 H + + 2 e - U 0 = 1.76 V vs. RHE (26) M + H2O → M-OH + H + + e − Gt (27) M-OH + H2O → M + H2O2 + H + + e − Gu (28) The free-energy changes Gα (α = a, …, u) of the various mechanistic pathways are related by a rigorous thermodynamic treatment [10] to the free energies of the reaction intermediates, thereby making use of the scaling relations between the *OH and *O as well as the *OH and *OOH adsorbates [2].Knowledge of the reaction intermediates' free energies enables determining the activity descriptor Gmax(U) [11], an advanced activity measure based on the idea of the freeenergy span model [12].In contrast to ηTD, Gmax(U) offers a potential-dependent description of the energetics, and thus, enables approximation of the electrocatalytic activity not only under equilibrium conditions (U = 1.23 V vs. RHE) but equally under OER conditions encountered during experimental measurements.Figures S2-S3 show potential-dependent volcano plots for the OER, thereby taking all the above pathways into consideration.

Figure S2 .
Figure S2.Potential-dependent volcano plots for various pathways of the oxygen evolution reaction at a) U = 1.23 V vs. RHE and b) U = 1.40 V vs. RHE.The energetically favored mechanisms in the approximation of G max (U) as a potential-dependent activity measure are indicated in dependence of the adsorption free energy of the *OH intermediate, G1.To derive the volcano curves, the following scaling relations are considered: G2 + G3 = 3.20 eV and G2 = 2 × G1. Figure reproduced with permission from reference [5].

Figure S3 .
Figure S3.Potential-dependent volcano plots for various pathways of the oxygen evolution reaction at a) U = 1.60 V vs. RHE and b) U = 1.80 V vs. RHE.The energetically favored mechanisms in the approximation of Gmax(U) as a potential-dependent activity measure are indicated in dependence of the adsorption free energy of the *OH intermediate, G1.The black volcano curve in panel b) refers to the two-electron water oxidation with H2O2 as the main product.To derive the volcano curves, the following scaling relations are considered: G2 + G3 = 3.20 eV and G2 = 2 × G1. Figure reproduced with permission from reference [5].

)
For the scaling-relation intercept (SRI), we adopt a benchmark value of SRI = 3.20 eV and consider SRI = 3.00 eV and 2.80 eV in the realm of sensitivity analyses[14].For the scaling relation between the *OH and *O adsorbates (cf.equation (35)), different correlations are considered for sensitivity analyses, namely G2 = 2.3 x G1 and G2 = 1.5 x G1.The freeenergy change G1 serves as the descriptor on the x axis in the activity volcano, and it is varied within the free-energy regime of G1 = [-0.50,2.50] eV with a step size of 0.01 eV.Based on the energetics of the intermediate states, the descriptor Gmax(U) is evaluated by considering all possible free-energy spans between the reaction intermediates: Figure S4.Potential-dependent volcano plot for the mononuclear (blue) and mononuclear-Walden (red) mechanisms of oxygen evolution at U = 1.23 V vs. RHE.To derive the volcano lines, the activity measure G max (U) (cf.equations (41) and (56)) is evaluated for the intermediate states of the mechanistic pathways (cf.equations (36) -(40) and (51) -(55)) in dependence of the descriptor G1, which is varied between -0.5 eV and +2.5 eV in steps of 0.01 eV.In the analysis, the following scaling relations are considered: G2 + G3 = 3.20 eV and G2 = 2 × G1.
Figure S5.Potential-dependent volcano plots for the mononuclear and the mononuclear-Walden pathways of the oxygen evolution reaction at a) U = 1.23 V vs. RHE, b) U = 1.40 V vs. RHE, c) U = 1.60 V vs. RHE, and d) U = 1.80 V vs. RHE.The energetically favored mechanisms in the approximation of Gmax(U) as a potential-dependent activity measure are indicated in dependence of the adsorption free energy of the *OH intermediate, G1.To derive the volcano curves, the following scaling relations are considered: G2 + G3 = 3.00 eV and G2 = 2 × G1.

Figure S6 .
Figure S6.Potential-dependent volcano plots for the mononuclear and the mononuclear-Walden pathways of the oxygen evolution reaction at a) U = 1.23 V vs. RHE, b) U = 1.40 V vs. RHE, c) U = 1.60 V vs. RHE, and d) U = 1.80 V vs. RHE.The energetically favored mechanisms in the approximation of Gmax(U) as a potential-dependent activity measure are indicated in dependence of the adsorption free energy of the *OH intermediate, G1.To derive the volcano curves, the following scaling relations are considered: G2 + G3 = 2.80 eV and G2 = 2 × G1.

Figure S7 .
Figure S7.Potential-dependent volcano plots for the mononuclear and the mononuclear-Walden pathways of the oxygen evolution reaction at a) U = 1.23 V vs. RHE, b) U = 1.40 V vs. RHE, c) U = 1.60 V vs. RHE, and d) U = 1.80 V vs. RHE.The energetically favored mechanisms in the approximation of Gmax(U) as a potential-dependent activity measure are indicated in dependence of the adsorption free energy of the *OH intermediate, G1.To derive the volcano curves, the following scaling relations are considered: G2 + G3 = 3.20 eV and G2 = 2.3 × G1.

Figure S8 .
Figure S8.Potential-dependent volcano plots for the mononuclear and the mononuclear-Walden pathways of the oxygen evolution reaction at a) U = 1.23 V vs. RHE, b) U = 1.40 V vs. RHE, c) U = 1.60 V vs. RHE, and d) U = 1.80 V vs. RHE.The energetically favored mechanisms in the approximation of Gmax(U) as a potential-dependent activity measure are indicated in dependence of the adsorption free energy of the *OH intermediate, G1.To derive the volcano curves, the following scaling relations are considered: G2 + G3 = 3.20 eV and G2 = 1.5 × G1.

Figure
S9 at U = 1.23 V vs. RHE: a) -0.5 eV < G1 < 0.62 eV (bifunctional-Walden mechanism): GM-OH+O2(U) -GM-OO+*O(U) b) 0.62 eV < G1 < 0.74 eV (bifunctional-Walden mechanism): GM-OH+O2 (U) -GM-OO(U) c) 0.74 eV < G1 < 0.81 eV (bifunctional-Walden mechanism): GM-OO+*O(U) -GM-OO(U) d) 0.81 eV < G1 < 1.23 eV (bifunctional mechanism): GM-O(U) -GM-OH(U) e) 1.23 eV < G1 < 2.50 eV (bifunctional mechanism): GM-OO(U) -GM(U) [23]22]ferent levels of sophistication to determine whether *OH surface groups are available for the OER catalysis.Coverage of intermediate species under steady-state reaction conditions is determined by the reaction kinetics rather than by the thermodynamics[18].Therefore, it is a viable measure to identify whether the formation of the *OH adsorbate refers to the rate-determining step (RDS) under OER conditions.Following reference[19]based on a generalized microkinetic model, *OH formation is never met with the RDS under typical OER conditions.Therefore, it can be concluded that the precondition of having *OH adsorbates on the surface is likely not violated.The main issue of discussing the RDS refers to the fact that most theoretical studies in the density functional theory (DFT) approximation discuss the potential-determining step (PDS)[20]based on the reaction energetics of the elementary steps (thermodynamics) rather than the RDS since the calculation of transition states by conventional canonical (constant-charge) approaches is still in its infancy and not fully mature yet[21,22].Therefore, for a dedicated discussion of the coverage of intermediate species for catalysts within a class of materials, it is rather needed to inspect the reaction energetics in terms of free energies, G, following the popular computational hydrogen electrode approach[23].Therefore, one needs to consider this evaluation scheme with a grain of salt.Following the conventional OER volcano (cf.FigureS1) based on linear scaling relationships between the *OH, *O, and *OOH intermediates [2], *OH formation does not refer to the PDS.Hence, it can be concluded that the formation of the *OH adsorbate is not a limiting factor, and thus, the precondition of *OH surface groups under OER conditions is likely met.
[12]20]ghtforward evaluation scheme refers to inspect whether *OH formation is reconciled with the PDS (cf.equation (6)) since if the generation of *OH surface groups is potential determining, this may cause a kinetic limitation when making use of the tacit assumption that PDS = RDS is met[11,20].It should be noted though that the presumption of identical potential-and ratedetermining steps can be violated, particularly for low overpotentials[12].