Materials Screening by the Descriptor G_max(η): The Free-Energy Span Model in Electrocatalysis

Abstract

To move from fossil-based energy resources to a society based on renewables, electrode materials free of precious noble metals are required to efficiently catalyze electrochemical processes in fuel cells, batteries, or electrolyzers. Materials screening operating at minimal computational cost is a powerful method to assess the performance of potential electrode compositions based on heuristic concepts. While the thermodynamic overpotential in combination with the volcano concept refers to the most popular descriptor-based analysis in the literature, this notion cannot reproduce experimental trends reasonably well. About two years ago, the concept of G_max(η), based on the idea of the free-energy span model, has been proposed as a universal approach for the screening of electrocatalysts. In contrast to other available descriptor-based methods, G_max(η) factors overpotential and kinetic effects by a dedicated evacuation scheme of adsorption free energies into an analysis of trends. In the present perspective, we discuss the application of G_max(η) to different electrocatalytic processes, including the oxygen evolution and reduction reactions, the nitrogen reduction reaction, and the selectivity problem of the competing oxygen evolution and peroxide formation reactions, and we outline the advantages of this screening approach over previous investigations.

1. Introduction

Coal, oil, and natural gas are still the major contributors to global energy production. These conventional, nonrenewable resources are hazardous and environmentally unfriendly because they release toxic elements upon their production and consumption. Besides the environmental consequences, they are being depleted while human demand for energy is ever-increasing.¹⁻³ Hence, energy production from renewables is needed, but most of these sources such as wind, tidal, and solar powers are intermittent. Therefore, energy storage is ultimately called for, implying that overproduced energy can be stored in chemical bonds, such as gaseous hydrogen (H₂), and that in the case of an energy deficit the storage system can feed the grid with the required electricity.⁴⁻⁶ Vital electrochemical devices for energy conversion and storage are fuel cells, electrolyzers, and batteries, which, in the long run, are thought to maintain the necessities of our everyday life, such as mobility, communication, transmission, and industrial manufacturing. At the electrodes in these devices, electrochemical reactions take place. For example, oxygen electrocatalysis comprising the oxygen evolution (OER) and oxygen reduction reactions (ORR) is operative at the cathode of metal–air batteries,⁷⁻⁹ while in electrolyzers the hydrogen evolution (HER) and oxygen evolution reactions (OER) take place at the cathode and anode, respectively.¹⁰⁻¹² Likewise, fuel cells convert chemical energy into electrical energy by the anodic hydrogen oxidation reaction (HOR) and cathodic ORR.¹³⁻¹⁵ To design an electrode material with improved performance, an atomic-scale understanding of these electrochemical reactions is ultimately needed.

With the advancement of computational capacities in the 21st century, a remarkable amount of work has been dedicated to in silico catalyst design for these reactions using electronic structure calculations in the density functional theory (DFT) approximation. The ultimate goal of these studies is to connect the microscopic picture (elementary reaction steps modeled by DFT) with the macroscopic picture in terms of experimental observations. Commonly, free-energy diagrams (FEDs) are used to build a bridge between these two worlds.^16,17 In theory, the elementary steps of an electrochemical process with N electron transfers consist of at least of N – 1 reaction intermediates (RIs) and N transition states (TSs) (cf. Figure 1a). Modeling the entire FED is mathematically exact; yet, it is computationally expensive and not feasible for the investigation of a plethora of different electrode compositions, which, however, is needed to identify suitable candidates within the large phase space of available materials. Therefore, the high-throughput screening of catalysts in a homologous series of materials is based on a heuristic approach in that only the RIs (thermodynamics) in the FED are calculated whereas the TSs (kinetics) are neglected, as shown in Figure 1b.

Figure 1.

(a) Free-energy diagram (FED) of a four-electron process comprising four transition states (TSs), #_j (j = 1, 2, 3, 4), and the associated reaction intermediates (RIs), denoted (i)–(v), under equilibrium conditions (η = 0 V). The RI with the lowest free energy is indicated in yellow. Brown arrows specify the overpotential dependence of each RI and TS, while α_j (j = 1, 2, 3, 4) denotes the transfer coefficient of each TS. (b) Simplification of the same electrochemical reaction by neglecting the TS. Commonly, only the energetics of the RIs are analyzed to assess the electrocatalytic activity in a heuristic fashion by high-throughput approaches.

The first attempt of a heuristic theory for the screening of electrocatalysts applied to energy storage and conversion processes was formulated by Nørskov and co-workers, who introduced the concept of the computational hydrogen electrode (CHE)¹⁸ and the thermodynamic overpotential, η_TD,^18,19 as an activity descriptor. The methodology put forth by Nørskov and co-workers extends the Anderson–Newns model²⁰ by computing the adsorption free energies of the RIs using DFT. While in their initial benchmark work the ORR was studied,¹⁸ in the following, we refer to the OER^21,22 in this perspective. The mononuclear OER mechanism, which consists of the same adsorbates as the suggested ORR mechanism, is given by eqs 1–4:

1

M denotes the catalytically active surface site, and the ΔG_j=1,2,3,4^OER values refer to the free-energy changes of the respective proton–electron (H⁺/e^–) transfer steps in the OER mechanism. The free energies of H⁺ and e^– are referred to half of the free energy of a hydrogen molecule by the CHE approach (cf. eqs 5 and 6), which enables using a reference accessible to gas-phase DFT to study the energetics of adsorbate species at electrified interfaces.

5

6

Note that the use of the CHE approach implies that electrons and protons are transferred in concert rather than in a decoupled fashion, and DFT calculations are carried out in a canonical ensemble (fixed charge) rather than in a grand canonical ensemble with constant electrode potential; the latter refers to the experimental conditions.^23,24 The CHE model can be used to obtain the free-energy change for each electrochemical step according to eq 7:²⁵

7

Equation 7 allows deriving the ΔG_j=1,2,3,4^OER values of the OER mechanism at any applied electrode potential, U, and pH in that only the adsorption energies (ΔE_DFT), zero-point energies (ΔZPE), and entropy changes (ΔS) are evaluated based on DFT calculations or thermodynamic data tables at T = 298.15 K, respectively. It should be noted that, due to the fundamental laws of equilibrium thermodynamics, the free-energy changes associated with the 3 RIs (OH, O, and OOH) sum up to 4.92 eV:

8

Knowledge of the ΔG_j=1,2,3,4^OER values allows classifying materials into active and inactive catalysts by using the Sabatier principle and the Brønsted–Evans–Polanyi relation. In this context, the Sabatier principle states that the RIs should be bound neither too strongly nor too weakly to the catalyst surface.²⁶ Yet, the Sabatier principle does not make any quantitative statement on the RIs’ binding energies.²⁷

The thermodynamic overpotential, η_TD, as an activity descriptor relies on a combination of the Sabatier principle and the Brønsted–Evans–Polanyi (BEP) relation²⁸ by evaluating the highest free-energy change among the set of all free-energy changes at zero overpotential, η = U – 1.23 V:

9

The reasoning of this descriptor is as follows: the smaller the value of η_TD, the higher the activity of a catalyst. η_TD is a very simple descriptor that enables materials screening by the assessment of a single free-energy change. Based on this notion, new electrode compositions could be identified for electrocatalytic processes, such as MoS₂ for the HER²⁹⁻³⁵ or IrO_x/SrIrO₃ for the OER.³⁶ However, both theoreticians and experimentalists have severely criticized the usage of η_TD recently because trends in a homologous series of materials cannot always be captured correctly by this descriptor.³⁷⁻⁴⁰ The elementary step that defines η_TD is denoted as the potential-determining step (PDS). Electrocatalytic activity, however, is governed by the rate-determining step (RDS), which remains unknown in the thermodynamic picture of adsorption energies because TSs are not explicitly investigated. As a fundamental consequence, the concept of η_TD relies on the tacit assumption that the condition PDS = RDS is met;⁴¹⁻⁴³ however, the PDS does not necessarily correspond to the RDS, as exemplarily shown in Figure 2.

Figure 2.

FED of the OER according to the mechanistic description in eqs 1–4 under equilibrium conditions (η = 0 V). The thermodynamic overpotential, η_TD, evaluates the largest free-energy change (ΔG_j=1,2,3,4^OER) of the elementary reaction steps at zero overpotential to approximate electrocatalytic activity by the concept of the PDS but neglects the kinetics in terms of the TSs #_j=1,2,3,4. In contrast, the RDS is given by the transition (i) → #₃, indicating that the PDS and RDS do not match.

Therefore, the simplicity of η_TD by focusing on the thermodynamic picture while the kinetics is ignored is identified as the main reason activity trends are not always reproduced correctly by this descriptor. More precisely, η_TD is a powerful approach to identify inactive catalysts at the leg of a volcano plot,⁴⁴⁻⁴⁷ but it is a weak descriptor to distinguish between active and highly active materials or to sort catalysts according to their activity.^48,49 Given that our main interest is dedicated to active materials rather than inactive catalysts, this finding calls for the development of advanced concepts that build upon the two major weaknesses of the descriptor η_TD:

• (i)The entire analysis is carried out at the equilibrium potential of the reaction, η_OER = 0 V, whereas experiments quantify production formation only for η_OER > 0 V.
• (ii)The entire analysis relies on adsorption free energies whereas the kinetics is ignored.

The first aspect can be solved by shifting the reference potential in the analysis in that trends are analyzed under OER conditions (e.g., η_OER = 0.3 V) rather than at η_OER = 0 V.^50,51 The second aspect is more difficult, given that an explicit calculation of the kinetics is still computationally too expensive for catalyst screening.⁵²⁻⁵⁵ However, the concept of the free-energy span model,^56,57 as introduced by Kozuch and Shaik in homogeneous catalysis, is a potential opportunity to build a bridge between the thermodynamics in terms of binding energies and the kinetics governing the electrocatalytic activity. Both aspects were taken up by Exner to introduce an advanced descriptor for the high-throughput screening, denoted as G_max(η).^43,58 We emphasize that also other descriptors besides the notion of G_max(η) to capture activity trends for electrocatalytic processes have been postulated, such as the electrochemical-step symmetry index (ESSI),⁵⁹ the electrochemical-step asymmetry index (ESAI),^60,61 and G_max.⁶² The present perspective focuses solely on the approach of G_max(η) and its implications for energy conversion and storage processes. In the further course of this article, we will first introduce the activity descriptor G_max(η) (section 2) and, after that, discuss its application to selected examples including the OER (section 3.1), bifunctional oxygen electrocatalysis (section 3.2), the selectivity challenge of the competing OER and peroxide formation (section 3.3), and the nitrogen reduction reaction (section 3.4).

2. G_max(η): A Universal Activity Descriptor for the Evaluation of Trends

2.1. Mathematical Formalism

The descriptor G_max(η) advances the free-energy span model, suggested by Kozuch and Shaik for homogeneous catalysis,⁵⁶ toward its applications in electrocatalysis, thereby accounting for the applied overpotential and kinetics effects in the analysis.^43,58 Considering the same OER mechanism as discussed in the Introduction (cf. eqs 1–4), the idea of this descriptor, also denoted as G_max(U), relies on a potential-dependent description of the RIs’ free energies (cf. eqs 10–14).

10

11

12

13

14

G_max(U) is then defined as the largest free-energy span between the RIs (cf. eqs 15–25) at a predefined target overpotential, η = U – 1.23 V, with U > 1.23 V vs RHE. In total, ten free-energy spans between the RIs are conceivable:

15

16

17

18

19

20

21

22

23

24

Equations 15–24 and the definition of G_max(η) can be written in a compact format as

25

In eq 25, n represents the total number of plausible free-energy spans, which is strongly dependent on the number of intermediate states. In the case of 3 reaction intermediates, as encountered for the OER, 10 free-energy spans are observed. For even more complicated electrocatalytic processes, such as the nitrogen reduction reaction with five intermediate states, the number of possible free-energy spans amounts to 20.⁶³ We refer to the Supporting Information files of refs (43 and 63) for a detailed discussion of how to determine the respective free-energy spans for electrocatalytic reactions comprising a varying number of proton–electron coupled transfer steps.

A schematic illustration of how to evaluate G_max(η) in a FED is illustrated in Figure 3. G_max(η) approaches the rate-determining transition state (G^#_RDS) and thus mimics overpotential and kinetic effects in the analysis of trends. The term ΔG^#_RDS-max, which is related to the free-energy difference between the TS with the highest free energy and the descriptor G_max(η), is not accounted for in the analysis; however, it has been shown that the term ΔG^#_RDS-max is a constant among different materials with a sensitivity of about 0.2 eV, and thus, G_max(η) allows evaluating trends with an uncertainty of 0.2 eV.^43,58 A detailed description of how to apply the notion of G_max(η) for the assessment of electrocatalytic activity is given in the Supporting Information of ref (43).

Figure 3.

Schematic representation of a FED at a finite overpotential, η > 0 V, for an electrocatalytic process, such as the OER. RIs and transition states are indicated by solid blue lines and #, respectively. The descriptor G_max(η) approaches the rate-determining transition state, #_RDS, by computing the maxima of all plausible free-energy spans between the RIs (cf. eq 25). The meaning of the term ΔG_RDS-max^# is further explained in section 2.3. Reproduced with permission from ref (64).

So far, we have described the mathematical formalism to compute G_max(η) for an electrochemical reaction from the example of the OER. It is evident that the computational cost for the determination of G_max(η) is the same as that of η_TD. Differences between G_max(η) and η_TD, however, refer to the meaningfulness of these descriptors. While η_TD yields information on the PDS alone, the free-energy span model of G_max(η) indispensably approaches the RDS for typical reaction conditions (cf. Figure 3) and thus is much closer to the actual rate-determining states than is η_TD. Furthermore, the notion of G_max(η) can distinguish between electrochemical and chemical steps in a reaction mechanism, whereas chemical steps need to be connected to the subsequent electrochemical step on applying the descriptor η_TD. To extend this discussion, we compare G_max(η) and η_TD in the OER for a reaction mechanism that contains both chemical and electrochemical steps.

2.2. G_max(η) vs η_TD for Chemical and Electrochemical Reaction Steps

The conventional mononuclear mechanism for the OER contains the OH, O, and OOH adsorbates. Alternatively, bifunctional and binuclear OER mechanisms have been proposed in the literature.^44,65 Among these three anticipated mechanisms, the last two both consist of electrochemical and chemical reaction steps. The binuclear mechanism reads:

26

Equation 30 indicates the recombination of two neighboring oxygen surface atoms to form gaseous oxygen, which is a chemical reaction step without charge transfer. At U = 0 V vs RHE, the five free-energy changes of eqs 26–30 sum up to 4.92 eV to meet the requirements of equilibrium thermodynamics:

31

For the application of η_TD, one needs to combine the chemical and electrochemical steps of eqs 29 and 30 so that the precondition of an electrochemical step is met:

32

Hence, η_TD assesses electrocatalytic activity by evaluating the free-energy changes ΔG₁^OER, ΔG₂, ΔG₃^OER, and ΔG_4,rev using eq 9. An example of an arbitrary FED of the binuclear OER mechanism is given in Figure 4. There, it becomes obvious that η_TD is defined by the free-energy change ΔG₁^OER. Yet, the thermodynamic FED of Figure 4 also indicates that ΔG₄ (cf. eq 29) corresponds to the elementary step that is the most uphill in free energy, but this finding cannot be captured by the notion of η_TD because ΔG₄^OER and ΔG_chem are conjoined to ΔG_4,rev^OER (cf. eq 32). Therefore, it is viable to use an activity descriptor that can factor nonelectrochemical steps into the analysis of activity trends. Recalling that G_max(η) relies on analyzing all possible free-energy spans within the reaction mechanism, this concept can also be exerted to chemical steps. For the binuclear OER mechanism, the corresponding free-energy spans are given by eqs 33–47:

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

Figure 4.

FED for the OER according to the binuclear pathway in eqs 26–30 at zero overpotential. The reaction steps (i) → (i′) and (i′) → (ii) are hidden on evaluating electrocatalytic activity by the concept of η_TD (purple). In contrast to this, the descriptor G_max(η) is able to include chemical reaction steps in the evaluation of activity trends, identifying that the transition (i) → (i′) corresponds to the largest free-energy span along the reaction mechanism (green arrow).

The free-energy span model purports that G_span #6(U) (cf. eq 38) is reconciled with the descriptor G_max(η). In contrast, when using the concept of η_TD, one would refer to G_span #15(U) (cf. eq 47). As shown in previous works, optimizing a free-energy change that does not correspond to the RDS may cause electrocatalysts to be optimized in the wrong direction toward lower activity.^66,67 This finding can be seen as the main reason the concept of G_max(η) excels over the notion of η_TD for a thorough assessment of activity trends because various pathways, comprising both electrochemical and chemical steps, can be thoroughly evaluated in the mechanistic description to approach the RDS, thereby using simple thermodynamic considerations analyzed by a dedicated evaluation scheme.

2.3. Microkinetic Modeling by the Concept of G_max(η)

In the Introduction, we have outlined two major shortcomings of the concept of η_TD, relating to overpotential and kinetic aspects in the evaluation of trends. While in the previous two subsections we have described how the descriptor G_max(η) accounts for the potential dependence in the analysis, in the following, we refer to the kinetics. Even if G_max(η) evaluates the largest free-energy span from the RI with smallest free energy to the RI with highest free energy, the electrocatalytic activity is governed by the TS with the highest free energy, G_RDS^# (cf. Figure 3). Figure 3 illustrates that G_max(η) does not capture the free-energy difference between the RI with the highest free energy and the the transition state with the highest free energy in the analysis, denoted as ΔG_RDS-max. Henceforth, we are going to outline how G_max(η) can be connected to G_RDS^# so that this activity descriptor can be used for microkinetic modeling.

Starting from the generalized Butler–Volmer equation in the Tafel approximation, the current density as a function of the applied overpotential is given by eq 48:^16,17

48

In eq 48, e, k_B, T, and h denote the elementary charge, the Boltzmann constant, the absolute temperature in Kelvin, and the Planck constant, respectively, while Γ_act represents the number of active sites per unit surface area. β_RDS corresponds to the apparent transfer coefficient and indicates the number of electrons transferred from the starting surface (RI with lowest free energy) to the transition state with the highest free energy.^16,68 For a fixed overpotential, η_target, eq 48 can be translated to eq 49 by making use of the relation in eq 50:

49

50

In previous works by Exner, it was demonstrated that there is an internal BEP relation between G_RDS^#(η) and G_max(η):^43,58

51

The term ΔG_RDS-max^# (cf. Figure 3) is a potential-independent constant with a sensitivity of about 0.2 eV,⁴³ and thus, G_max(η) directly scales with ΔG_RDS(η), irrespective of the applied driving force:

52

Inserting eq 52 into eq 49, it becomes evident that the current density, j(η_target), can be approximated by the descriptor G_max(η) if one is using an educated guess for the constant of eq 52:

53

Previous studies have shown that one may apply a value range between 0.4 and 0.6 eV as lower and upper bounds for the constant, respectively.⁴³

It is noteworthy that the approximation of current densities for electrocatalytic processes as a function of the applied overpotential goes far beyond the notion of η_TD. In the case of η_TD, it has been reported that theoretically calculated values scale with the experimental overpotential to reach 10 mA/cm² in a homologous series of materials.⁶⁹ Yet, it would be incorrect to relate η_TD to any experimental current density because there is no direct connection between the descriptor η_TD and the electrocatalytic rate. Instead, η_TD is an educated guess to comprehend trends by relying on the tacit assumption of PDS = RDS.^41−43,62 The notion of G_max(η) allows an approximation of current densities, but only if the constant in the numerator of the exponential term (cf. eq 53) is accounted for. Therefore, determining current densities by G_max(η) is possible in a qualitative rather than in a quantitative fashion. Yet, this approach can be used to study the upper and lower boundaries of current densities for electrochemical processes dependent on the driving force, which can help to rationalize trends. We use this methodology to discuss the bifunctional oxygen electrocatalysis and the selectivity problem of the oxygen evolution and peroxide formation reactions in sections 3.2 and 3.3, respectively.

2.4. Approaching the Rate-Determining Step by the Concept of G_max(η)

The concept of the thermodynamic overpotential, η_TD, has spurred materials screening by resolving the potential-determining step (PDS) of a mechanistic description.^41−43,62 The PDS refers to the elementary reaction step with the highest free-energy change at zero overpotential (cf. Figure 2). However, as pointed out by Koper for the simple HER, the PDS may differ from the actual RDS governing the rate of the reaction.⁴¹ Exner gave further evidence on the potential-dependent relationship of the PDS and RDS, outlining that for small overpotentials the PDS may differ from the RDS, whereas for large overpotentials the PDS may approach the RDS.⁴³

A peculiarity of the descriptor G_max(η) refers to the fact that this activity measure can resolve the actual RDS in a potential-dependent fashion. This requires including another assumption into the analysis though: namely, the value of the Tafel slope.^16,70,71 Typical Tafel slopes are either 40 or 120 mV/dec, indicating that the second or the first elementary electrochemical step reveals the highest transition-state free energy, respectively. To illustrate this finding on a selected example, the OER over RuO₂(110) is adopted.^43,72Figure 5 depicts the corresponding FED at applied overpotentials of 200 and 300 mV.

Figure 5.

Free-energy diagram of the OER over RuO₂(110) at (a) η_OER = 0.20 V and (b) η_OER = 0.30 V. The descriptor G_max(η) is given by the free-energy span from (a) M–O → M–OOH → M + O₂ or (b) M–OOH → M + O₂. These different free-energy spans have implications on the rate-determining step (RDS) dependent on the Tafel slope.

In Figure 5, the descriptor G_max(η) is highlighted by a green arrow, and the free-energy diagram commences from the intermediate state with lowest free energy.^{17,43,58,61,66} For η_OER = 0.20 V, the free-energy span governing G_max(η) is reconciled with the transition from the O adsorbate to the formation of O₂ (M–O → M–OOH → M + O₂). In contrast, for η_OER = 0.30 V, the transition from the OOH adsorbate to the formation of O₂ (M–OOH → M + O₂) refers to the descriptor G_max(η).

Let us assume that the Tafel slope amounts to 40 mV/dec at an applied overpotential of η_OER = 0.20 V. Hence, the second elementary electrochemical step refers to the decomposition of the OOH adsorbate, and thus, the step M–OOH → M + O₂ reveals the highest transition-state free energy and governs the OER rate. In contrast, if we assume a Tafel slope of 120 mV/dec, the formation of the OOH adsorbate, M–O → M–OOH, is rate determining. Similarly, for a Tafel slope of 120 mV/dec at η_OER = 0.30 V, the decomposition of the OOH adsorbate refers to the RDS. In principle, experimental studies of single-crystalline model electrodes allow extracting the precise potential-dependent value of the Tafel slope,^73,74 which can be merged with the concept of G_max(η) to discuss the RDS. If experimental information on the Tafel slope remains unknown, it is yet possible to discuss the RDS in a qualitative fashion by referring to the standard values of 40 and 120 mV/dec, as outlined above. We use this methodology to discuss the nitrogen reduction reaction over transition-metal oxides in section 3.4 to compare the thermodynamic picture of the PDS with the RDS.

Finally, a short note on the compilation of free-energy diagrams and the discussion of the RDS is needed. As is evident from Figures 4 and 5, the free-energy landscapes commence from the RI with the smallest free energy, and this RI is not necessarily reconciled with the first step of the OER mechanism (cf. eq 1).^16,17 The reason for this finding is that only if the elementary steps commence from the RI with smallest free energy is the free-energy span model of G_max(η) properly linked to the rate-determining states that refer to the RDS.^43,58 It should be noted that the renumbering procedure of the RIs in the free-energy diagram does not affect the numerical value of the descriptor G_max(η), and thus, even without the renumbering procedure activity trends can be assessed in a predictive fashion. We refer to section 2 of the Supporting Information in ref (58), where this aspect is discussed in more detail. The renumbering procedure of the elementary steps is also discussed in several other works of the authors focusing on different electrocatalytic processes.^{16,17,43,58,89,136,157}

2.5. Approximations Made within the Concept of G_max(η)

In sections 2.1–2.4, we have outlined the advantages of the descriptor G_max(η) compared to the commonly applied notion of the thermodynamic overpotential, η_TD. Yet, it is also beneficial to point out the approximations made within the concept of G_max(η).

First, the descriptor G_max(η) relies on the assumption of a free-energy span model,^56,57 approximating electrocatalytic activity by the free-energy difference between the RIs with the smallest and highest free energy. The concept of a free-energy span model originates from the field of homogeneous catalysis, where often a single active site is encountered. This situation is fundamentally different from that of solid-state electrodes with a plethora of active sites adjacent to each other, and a critical parameter in this regard corresponds to the coverage of the RIs. It is therefore advisible to determine first the RI coverage such as by the construction of Pourbaix diagrams²⁵ before any claim on activity by means of the descriptor G_max(η) is made. Otherwise, activity trends determined by the free-energy span model are prone to be erroneous.

Second, one may question whether a free-energy span is adequate to describe the activity of electrocatalytic processes occurring on solid-state electrodes. The notion of the free-energy span model relies on the assumption of quasi-equilibrium conditions,⁶⁸ whereas experimental measurements are commonly performed under steady-state conditions. Yet, the steady-state and quasi-equilibrium conditions often reveal identical results in terms of microkinetic modeling,¹⁶ thus rendering the assumption of quasi-equilibrium conditions a viable approximation.

Finally, the descriptor G_max(η) relies on a BEP relationship between the RI and TS with the highest free energy, quantified by the term ΔG_RDS-max^# (cf. Figure 3). This BEP relation was determined by theoretical data from DFT calculations and experimental data from single-crystalline model electrodes for the RIs and TSs, respectively.⁴³ Further computational assessment of this BEP relation is desirable, which is, however, challenging due to the large errors of first-principles investigations on the kinetics in an electrochemical environment.^16,52 Of course, if deviations from the presumption that ΔG_RDS-max is a constant with a sensitivity of 0.2 eV are observed, this would affect the predictive power of the free-energy span model relating to the notion of G_max(η).

3. Application of G_max(η)

3.1. Breaking Scaling Relation in the Oxygen Evolution Reaction

The OER plays a crucial role in proton-exchange-membrane (PEM) electrolyzers and metal–air batteries (MABs). The reaction kinetics of the OER is sluggish, which was related by Koper, Norskøv, Rossmeisl, and others to the binding energies of the RIs OH and OOH.^21,22 Koper argued that each ΔG_j value amounts to 1.23 eV for an ideal OER catalyst, giving rise to a thermoneutral FED at the OER equilibrium potential.^42,75 However, the binding energies of the OH and OOH adsorbates are inherently coupled, and the sum of the free-energy changes ΔG₂^OER and ΔG₃ is always larger than the anticipated theoretical value of 2 × 1.23 eV = 2.46 eV:

54

The precise value of the scaling-relation intercept (SRI) may rely on the individual material class, the exchange-correlation functional employed in the DFT calculations, and the consideration or neglect of solvation.^48,76−79 An offset of 3.2 ± 0.20 eV is the most often reported SRI in the literature,⁶⁹ the value of which is within the functional independent SRI reported by Vegge and co-workers.⁷⁹ Therefore, breaking the SRI of about 3.2 eV to reach the optimum value of 2.46 eV, corresponding to a symmetrical thermodynamics FED, is considered a universal remedy to enhance OER activity.^80,81 Despite its popularity, this strategy is discussed controversially in the literature. According to the work of Calle-Vallejo and co-workers, breaking the scaling relation is a required yet not sufficient measure to ensure improved OER activity.^67,82 Application of the activity descriptor G_max(η) provides further insight into the notion of breaking scaling relations. Figure 6a depicts an arbitrary thermodynamic FED for a scaling-relation-fulfilling catalyst (SRI = 3.2 eV) under typical OER conditions (η_OER = 0.3 V). The electrocatalytic activity of this catalyst can be evaluated by means of the descriptor G_max(η), indicating G_max(η = 0.3 V) = 0.17 eV.

Figure 6.

(a) Scaling-relation-fulfilling catalyst (ΔG₂^OER + ΔG₃ = 3.2 eV) with ΔG₁^OER = 1.00 eV, ΔG₂ = 1.70 eV, ΔG₃^OER = 1.50 eV, and ΔG₄ = 0.72 eV. The activity descriptor G_max(η) amounts to 0.17 eV under a typical OER condition: i.e., η = 0.3 V. (b) The OH vs OOH scaling relation is completely broken (ΔG₂^OER + ΔG₃ = 2.46 eV), and the resulting free-energy penalty of 0.74 eV is completely assigned to the free-energy change ΔG₁^OER. The resulting scaling-relation-free catalyst reveals a lower electrocatalytic activity, quantified by G_max(η) = 0.38 eV at η_OER = 0.3 V.

It is a common practice to break the OH vs OOH scaling relation by reducing the free-energy change ΔG₃^OER until the desired SRI is reached. In this example, the resulting free-energy penalty of 0.74 eV is completely assigned to the free-energy change ΔG₁. We obtain G_max(η = 0.3 V) = 0.38 eV for the scaling-relation-free catalyst, indicating that, in this case, breaking the scaling relation is detrimental to obtaining higher OER activity.

Of course, a comment is needed on why the breaking of the OH vs OOH scaling relation may cause such an unfavorable scenario of lower activity. The main reason for this finding is traced to the fact that breaking the scaling relation relies on the notion that the PDS, as indicated by the descriptor η_TD, is also encountered with the RDS. If the free energy of the PDS is reduced, the electrocatalytic activity is enhanced due to the coupling of the thermodynamics to the kinetics via the BEP relation. As is evident from Figure 6, redistributing the free-energy changes to reach SRI = 2.46 eV has a major influence on the catalytic processes: the active site of the electrocatalyst (RI with the smallest free energy) and the free-energy span governing G_max(η = 0.3 V) are both altered. As such, the presumption PDS = RDS is violated when reducing ΔG₃^OER or any other free-energy change within the breaking of the OH vs OOH scaling relation to meet SRI = 2.46 eV, and this can be seen as the main reason this approach can have a disadvantageous effect on the optimization of electrocatalysts. The unsatisfactory picture relating to the breaking of the OH vs OOH scaling relation may raise the question of whether SRI = 2.46 eV is indeed encountered with the ideal situation in the OER. Recalling that the apex of the volcano plot is prone to change with increasing overpotential for a two-electron process,⁸³⁻⁸⁶ a similar finding may also be encountered for the more complex four-electron OER.^50,87,88 With the support of another activity descriptor, the so-called electrochemical-step asymmetry index (ESAI),^60,66 it was demonstrated that the optimum thermodynamic FED in the OER switches from a symmetrical (thermoneutral) to an asymmetrical free-energy landscape (cf. Figure 7) as soon as the free-energy changes of the mechanistic description are analyzed under the reaction conditions (η_OER > 0 V) rather than the equilibrium conditions. The SRI of the asymmetric thermodynamic free energy landscape, suggested as the thermodynamic ideal under operational conditions (η_OER > 0 V), depends on the amount of overpotential applied. For η_OER = 0.3 V, the asymmetrical thermodynamic FED reveals SRI = 2.76 eV.^60,61 While the notion of the asymmetric thermodynamic FED was introduced by a descriptor-based analysis in a qualitative fashion, a quantitative analysis is required to investigate whether breaking the scaling relation to SRI = 2.76 eV is accompanied by higher electrocatalytic activity than that to SRI = 2.46 eV.

Figure 7.

Comparison of the symmetrical and asymmetrical FED for the OER. In the symmetrical FED, all adsorbates are stabilized thermoneutrally at the OER equilibrium potential. This gives rise to the notion of 2.46 eV as the optimum SRI. On the other hand, for the asymmetric FED, the key adsorbate OOH is stabilized thermoneutrally at the target overpotential. For a target overpotential of 300 mV, the SRI of the asymmetric FED corresponds to 2.76 eV. It is argued that, upon increasing the overpotential, the optimum situation is encountered with the asymmetrical rather than with the symmetrical FED.

We employed a data-driven methodology in combination with the descriptor G_max(η) to account for the statistical significance of the symmetrical and asymmetrical thermodynamic FEDs.⁸⁹ For this purpose, we describe the available phase space of materials by a basis set in that we predefine the free-energy changes ΔG₁^OER and ΔG₂ as follows: ΔG₁^OER = ΔG₂ = [start = 0.00, stop = 2.50; step = 0.01] eV (the descriptions “start”, “stop”, and “step” are skipped throughout the rest of this perspective). This gives rise to about 63 000 cases in total. For all these cases, we start from the scaling-relation-fulfilling catalyst (SRI = 3.2 eV) and analyze the energetics at different electrode potentials, U = [1.33, 1.43, 1.53] V vs RHE, corresponding to OER overpotentials of η_OER = (100, 200, 300) mV, respectively. For all data points, G_max(η) is calculated by eq 25. In the next step, we alter the SRI to 2.46 or 2.76 eV corresponding to the thermodynamic symmetrical or asymmetrical FED, respectively. This is achieved by reducing the value of ΔG₃^OER to meet an SRI of 2.46 or 2.76 eV, and the corresponding energy penalty is reassigned to the free-energy change ΔG₄ so that the OER equilibrium condition is not violated. For the symmetrical and asymmetrical free energy landscape, the electrocatalytic activity is quantified by the descriptor G_max(η), and the obtained G_max(η) values dependent on the applied electrode potential are compared to the scaling-relation-fulfilling catalysts (SRI = 3.2 eV). We evaluated the percentage of cases for which the electrocatalytic activity of the asymmetrical or symmetrical FED is equal to or higher than that of the scaling-relation-fulfilling catalysts. Consequently, the conditions of eq 55 or 56 need to be fulfilled:

55

56

The results of our statistical analyses are summarized in Table 1.

Table 1. Breaking the SRI from 3.2 eV to 2.76 (Asymmetrical FED) or 2.46 eV (Symmetrical FED) at Different Overpotentials^a.

η/V	breaking scaling relation	equal or higher activity (%)
0.1	SRI = 3.2 eV → SRI = 2.46 eV	90.6
0.1	SRI = 3.2 eV → SRI = 2.76 eV	100
0.2	SRI = 3.2 eV → SRI = 2.46 eV	89.1
0.2	SRI = 3.2 eV → SRI = 2.76 eV	96.0
0.3	SRI = 3.2 eV → SRI = 2.46 eV	88.0
0.3	SRI = 3.2 eV → SRI = 2.76 eV	95.0

^aThe last column indicates the percentage of electrocatalysts based on our chosen basis set that reveal equal or higher activity in the approximation of G_max(η) (cf. eqs 55 and 56).

It is evident from Table 1 that breaking the scaling relation from 3.2 to 2.76 eV (asymmetrical FED) yields higher or equal activity in more cases than breaking the scaling relation from SRI = 3.2 eV to SRI = 2.46 eV (symmetrical free energy landscape), independent of the amount of overpotential applied. Surprisingly, the impact of breaking the scaling relation is potential dependent and turns out to be beneficial at small rather than large overpotentials. One may argue that this finding may be related to the fact that formation of the OOH adsorbate, corresponding to the free-energy change ΔG₃^OER, is rate determining for small overpotentials.^{44,59,82,90,91} However, for overpotentials exceeding 300 mV, also decomposition of the OOH adsorbate can be reconciled with the RDS, and this step as the RDS does not have a positive impact on the breaking of the OH vs OOH scaling relation.^17,92−94

In summary, the activity descriptor G_max(η) contributed to extend our understanding of breaking the scaling relation in the OER. Our analysis reveals that SRI = 2.76 eV is statistically more significant than SRI = 2.46 eV to reach higher electrocatalytic activity, suggesting a change in the mindset when thermodynamic concepts are applied for catalyst optimization.

3.2. Bifunctional Oxygen Electrocatalysis

The data-science approach presented in section 3.1 for the OER was also exerted to investigate the bifunctional oxygen electrocatalysis at the cathode of MABs. This type of battery is discussed as a potential replacement for lithium-ion batteries (LIBs) due to its higher energy density than that of LIBs.^7−9,95,96 However, the performance of MABs is restrained by the four-electron ORR and OER taking place during discharge and charge of the device, respectively. While the mechanistic description of the OER has been discussed beforehand (cf. eqs 1–4), the elementary steps of the ORR are given by eqs 57–60:⁹⁷⁻¹⁰⁰

57

These four energy changes sum up to −4.92 eV at U = 0 V vs RHE:

61

Similar to eq 54, the inherent coupling of the OH and OOH adsorbates yields

62

For optimum performance, a highly active bifunctional electrocatalyst that catalyzes both the OER and ORR at low overpotentials with a reasonable current density is required.^101−103 Experimentally, the performance of bifunctional materials is assessed by the so-called bifunctional index (BI) (cf. Figure 8), defined as¹⁰³

63

Figure 8.

Current density, j, vs applied electrode potential, U, for the oxygen electrocatalysis, consisting of the OER (red) and ORR (blue), as obtained from cyclic voltammetry experiments. The bifunctional performance of electrocatalysts in the oxygen electrocatalysis can be assessed by the difference in electrode potential between j_OER = 10 mA/cm² and j_ORR = −1 mA/cm², denoted BI. Reproduced with permission from ref (64).

Please note that up to three different definitions for the reduction process (ORR) can be found in the literature. Besides the electrode potential to reach j_ORR = −1 mA/cm², also j_ORR = −3 mA/cm² or the half-wave potential (HWP) is reconciled with performance measures to characterize electrode materials in the ORR.^104,105 In the remainder of this discussion, j_ORR = −1 mA/cm² is used. While the BI can be determined by experimental cyclic voltammetry, we emphasize that in previous theoretical studies the BI was approximated by the sum of the thermodynamic overpotentials, η_TD, for the OER and ORR:^67,106

64

While in a homologous series of materials η_TD may scale with the current density to reach 10 mA/cm², it is evident that the cathodic ORR process cannot be captured by this simple descriptor (cf. eq 63). Consequently, application of the formula in eq 64 would result in BI values that always exceed the experimentally measured values. Given that the application of the descriptor G_max(η) allows an approximation of current densities by microkinetic modeling (see section 2.3), we make use of this feature to put forth a method of how to determine the BI from a dedicated analysis of adsorption free energies in conjunction with data-science techniques. Our procedure is similar to that reported in the previous section in that we set up a basis set that may describe the phase space of available materials. Identically with the investigation of breaking the scaling relation,⁸⁹ our basis set is defined by ΔG₁^OER = ΔG₂ = [0.00, 2.50; 0.01] eV and ΔG₄^ORR = ΔG₃ = [−2.50, 0.00; 0.01] eV. Knowledge of ΔG₂^OER/ΔG₃ enables determination of ΔG₃^OER/ΔG₂ by referring to the SRI (cf. eqs 54 and 62), for which we choose an array of [2.90, 3.20; 0.10] eV for the OER and ORR. These values are selected based on the reported SRIs in the literature.^{48,50,69,76,107−109} Consequently, ΔG₄^OER/ΔG₁ can be directly obtained from the equilibrium condition (cf. eqs 8 and 61 for the OER and ORR, respectively), and G_max(η) can be determined for all cases by eq 25.

In the next step, G_max(η) is used to determine the current density as a function of the applied electrode potential (cf. eq 53). This requires the definition of additional input parameters: namely, the density of active surface sites, Γ_act, and the constant that connects the descriptor G_max(η) to the rate-determining transition state. We choose Γ_act = [10¹⁴, 5 × 10¹⁴, 10¹⁵] cm^–2 and const = [0.4, 0.5, 0.6] eV in the analysis. For a rough screening of the parameter space, we use applied electrode potentials of U = [1.13, 1.03, 0.93] V vs RHE and [1.33, 1.43, 1.53 V] vs RHE for the ORR and OER, respectively, corresponding to applied overpotentials of η = (±100, ±200, ±300) mV, to calculate j_OER and j_ORR. All of these values are stored in a multidimensional database consisting of more than 13.6 million data points (cf. Figure 9).

Figure 9.

Schematic illustration of our multidimensional database, consisting of more than 13 million data points, to determine the BI in the oxygen electrocatalysis for any hypothetical electrode material from theory. ΔG_j^OER/ORR, SRI, and U denote the adsorption free energies, scaling-relation intercept, and applied electrode potential, respectively. Behind these variables, their respective number of cases is given in parentheses. Reproduced with permission from ref (64).

The database of Figure 9 is the starting point to derive the BI for our basis set and to compare the obtained BI values with experiments. While the concrete statistical treatment is described in detail in ref (64), we want to focus on the main conclusions that can be rendered based on the application of G_max(η). Our study aims to identify highly active bifunctional catalysts, and we have extracted those from our database and compared them to experiments (cf. Figure 10). While previous theoretical studies on the BI in the approximation of the thermodynamic overpotential reported that theoretical BI values are in good agreement with the experimental values,¹⁰⁶ our statistical approach clearly demonstrates that there is still plenty of room for improvement for the performance of bifunctional materials in the oxygen electrocatalysis. This statement holds true despite scaling relations, and even for the most common SRI of 3.2 eV an amelioration of at least 0.1 V relating to the BI appears feasible. Yet, a comment is needed how the deviation from our theoretical predictions to the experimental BI values is interpreted.¹¹⁰

Figure 10.

Average BI (blue dots) accompanied by their standard deviation dependent on the SRI, as obtained from our data-driven model for highly active bifunctional OER/ORR catalysts. The shaded region indicates the experimentally reported BIs of highly active bifunctional materials. Reproduced with permission from ref (64).

In our model, we assume that the adsorption free energies in the potential range of the OER (U > 1.23 V vs RHE) as well as the ORR (U < 1.23 V vs RHE) are the same. Thus, it is tacitly presumed that the active sites in the OER and ORR are identical, a supposition that may not always be fulfilled. Consequently, for experimentally synthesized materials the adsorption free energies in the OER regime may differ from the adsorption free energies in the ORR because the morphology of the electrode surface is prone to change when the applied electrode potential is altered. This finding suggests that, for experimentally synthesized materials, the set of adsorption free energies is optimized for only one process within the oxygen electrocatalysis, whereas the set of adsorption free energies is not fully optimized for the reverse reaction. It would therefore be required to design bifunctional electrocatalysts with two different active sites in that the first active site reveals an optimized set of ΔG_j=1,2,3,4^OER values in the potential range of U > 1.23 V vs RHE, while the second active site should have an optimized set of ΔG_j=1,2,3,4 values in the potential range of U < 1.23 V vs RHE. For an example of a bifunctional catalyst with two different active sites, the reader is referred to ref (110).

We conclude that the application of G_max(η) in conjunction with data-science techniques may contribute to bridging the gap between ab initio theory and experiments by providing advanced insight into the performance of bifunctional electrocatalysts for the oxygen electrocatalysis beyond the approximation of the thermodynamic overpotential. Further theoretical studies are needed to comprehend implications for efficient bifunctional oxygen electrocatalysis on the atomic scale and to determine how the prediction of BI values by G_max(η) is related to the proposed limit of BI = 0.74 V based on scaling relations.¹⁰⁶

3.3. Oxygen Evolution vs Peroxide Formation: Application of G_max(η) to a Selectivity Problem

H₂O₂ (hydrogen peroxide) is an important industrial chemical: e.g., for water disinfection. While so far H₂O₂ is mainly produced by the anthraquinone process containing of hazardous materials, the electrochemical oxidation of water is an alternate, sustainable pathway to generate H₂O₂. Yet, the two-electron oxidation of water to H₂O₂, also denoted as the peroxide formation reaction (PFR), is thermodynamically restrained because of its equilibrium potential of 1.76 V vs RHE, which is about 500 mV higher than that of the four-electron OER. Despite this, the PFR is kinetically preferred because it requires fewer electron transfers than the OER. Ultimately, the production of H₂O₂ by electrolysis requires overcoming the selectivity challenge of the competing OER and PFR. Consequently, an electrode material with high affinity toward the PFR, which at the same time efficiently suppresses the OER, is needed. Previous theoretical studies in the DFT approximation used the descriptor η_TD to compile volcano curves at the equilibrium potential of the PFR.^111−114 Quite in contrast, the descriptor G_max(η) allows studying the selectivity problem of the competing reaction channels in a potential-dependent fashion, as outlined in the following.

The reaction mechanism of the PFR is given by eqs 65 and 66:

65

66

In comparison to the OER, the PFR shares a common intermediate (*OH) with this pathway, but in the second proton–electron transfer step, the OH adsorbate is directly converted into H₂O₂. Consequently, an active PFR catalyst may not be prone to oxidize and thus may not bind the O adsorbate strongly because this may indicate a tendency to catalyze the OER rather than the PFR. In the literature, carbon-based materials, such as boron-doped diamond (BDD), have been discussed as anode materials for the electrocatalytic production of H₂O₂.^115−117

We modeled BDD in a first-order approximation as a single-atom-thin layer of graphite in the [001] direction because BDD can be reconciled with a partially graphitized diamond surface.^118,119 Various slab models with different adsorbate densities were optimized. Altogether, eight coverage densities were studied, and the models containing adsorbate densities of 3–12.5% and 17–25% correspond to the low- and high-coverage compositions, respectively. Models are labeled GN, where N is the number of carbon atoms per supercell, while the coverage density is equal to 1/N. Two of these structures are schematically shown in Figure 11. The G6 and G16 models correspond to coverage densities of 17% and 6%, respectively, thus representing high and low adsorbate densities.

Figure 11.

Example of the optimized geometry for the G6 model with high coverage (top) and the G16 model with low coverage (bottom) for the OH, O, and OOH adsorbates (from the left to the right). Unit cells are shown by solid lines. Both coverage concentrations are presented in two different projections. Reproduced with permission from ref (120).

For the low- and high-coverage compositions, we compiled FEDs of our BDD model for the OER and PFR at the PFR equilibrium potential (cf. Figure 12). The free-energy diagrams allow determining the descriptor G_max(η = 0 V) for the OER and PFR, using the free-energy span model from the RI with the lowest free energy to the RI with highest free energy, as introduced in section 2. For an accurate determination of the PFR free-energy landscape, gas-phase error corrections are called for, as discussed by Calle-Vallejo and co-workers.¹²¹ In the presented approach, we make use of the experimental equilibrium potential rather than calculate the H₂O₂ molecule because the latter may result in an inaccurate equilibrium potential for the PFR.

Figure 12.

FEDs of 2e^– PFR (green), and 4e^– OER (black) at the PFR equilibrium (U = 1.76 V vs RHE, pH = 0), generalized for graphene-like surfaces by averaging the values of ΔG for models with high and low coverage. The descriptors G_max^PFR(η = 0) and G_max(η = 0), defining the thermodynamic penalty to move from the RI with the lowest free energy to the RI with the highest free energy for the PFR and OER, respectively, are indicated by blue arrows. Figure adopted with changes from ref (120).

Our analysis shows that, for the 2e^– PFR, the free-energy span associated with G_max(η) refers to the formation of the OH adsorbate with G_max^PFR in a range of 0.90–1.08 eV. Moreover, for the 4e^– OER, the transition from the O adsorbate to the OH intermediate is thermodynamically the most uphill in the free energy landscape with G_max in a range of 1.34–1.52 eV. This finding demonstrates that at U = 1.76 V vs RHE the selectivity should be in favor of peroxide formation due to the lower thermodynamic restraints, resulting in superior kinetics according to the BEP relation.^28,122−124

While the free-energy span associated with the descriptor G_max^PFR is related to G_M–OH – G_M, the most stable surface termination corresponds to M–O or M–OOH for high and low coverages, respectively (cf. Figure 12). Accordingly, one may claim that the M–O or M–OOH surface compositions first need to be converted into M so that the PFR can occur. The transition of M–O or M–OOH to M may add a free-energy penalty to the descriptor G_max, which affects the selectivity analyses. Given that the coverage density does not exceed 25% (cf. Figure 11), there are M–O, M–OOH, and M surface sites available, and the first two surface sites facilitate the OER whereas the latter is the starting point for the PFR. However, we emphasize that, in the case of higher coverages, it is important to consider the poisoning effects of adsorbates that can be incorporated by means of a free-energy penalty into the free-energy span model of G_max(η).

It might be interesting to indicate that the descriptor G_max^PFR, evaluated at zero overpotential, is identical with the thermodynamic overpotential because the PFR is a two-electron process consisting of only one intermediate. The difference between the free-energy span approach and η_TD manifests as soon as the analysis is conducted in a potential-dependent fashion, as discussed in the following.

The evaluation of G_max^PFR and G_max as a function of applied PFR overpotential is indicated by G_max^PFR(η) and G_max(η), respectively. The difference between these two descriptors, G_sel(η), at a fixed electrode potential is a benchmark for PFR selectivity dependent on the driving force:

67

Following the selectivity model of Exner for the competing chlorine and oxygen evolution reactions,^122,125 an expression for the potential-dependent PFR selectivity can be derived taking the Butler–Volmer formalism into account:

68

In eq 68, k_B denotes the Boltzmann constant and T amounts to 298.15 K. PFR sel (η) can adopt values between 0 and 1, denoting 100% selectivity for the OER or 100% selectivity for the PFR, respectively. Evaluating G_max^PFR(η) and G_max(η) to calculate G_sel(η) and PFR sel (η) has the following advantages: (i) PFR selectivity can be discussed as a function of the applied electrode potential because PFR selectivity may change upon modifying the driving force and (ii) current densities for the PFR can be approximated using the formalism introduced in section 2.3. Knowledge of current densities enables assessing, at least in a qualitative fashion, whether the produced amount of H₂O₂ per time is sufficient for experimental applications. These results are summarized in Figure 13. For the high-coverage regime (17–25%), the BDD model maintains a large PFR selectivity up to 100% up to U = 2 V vs RHE, whereas for U > 2 V vs RHE, PFR selectivity deteriorates. In contrast, for the low-coverage regime (3–12.5%), PFR is the dominant reaction for U < 2.11 V vs RHE, but for higher electrode potentials the OER is superior to the PFR. Reference (126) highlighted an inverse correlation between the current density and the selectivity toward H₂O₂; hence, a condition for low working overpotentials inherently implies low current densities. Our investigation underpins this inverse relation where the highest current density of about 10^–5 mA/cm² is achieved for the lowest adsorbate concentrations (3% and 4%) @U = (2.05–2.10) V vs SHE (@pH = 0). The optimum adsorbate density is below 8% coverage because these surface compositions can achieve current densities above 10^–6 mA/cm² while maintaining reasonable PFR selectivity.

Figure 13.

PFR selectivity and theoretically calculated PFR current densities as a function of the applied bias (@U vs SHE @pH = 0) for all graphene models (G4, 25%; G6, 17%; G8, 12.5%; G12, 8.3%; G16, 6%; G24, 4%; G32, 3%), in which the coverage densities are indicated in percentages. Current densities are calculated by referring to the formalism introduced in section 2.3. Reproduced with permission from ref (120).

Applying the concept of G_max(η) to the selectivity problem of the OER and PFR enabled us also to gain further insight compared to the notion of η_TD. The analysis of the FEDs in Figure 12 illustrates that the relation η_TD^OER = η_TD is met at U = 1.76 V vs RHE, resulting in the conclusion that the PFR selectivity is 50% irrespective of the coverage. Since the thermodynamic overpotential does not account for the effect of the applied overpotential on the energetics, PFR selectivity in the approximation of η_TD is also independent of the driving force and remains 50% when increasing the applied overpotential is increased. Additionally, approximating electrocatalytic activity and selectivity by η_TD does not allow predicting current densities in contrast to the approach of G_max(η). Given that the PFR selectivity is a strong function of coverage and the applied overpotential according to experimental investigations,¹¹⁷ we conclude that the application of G_max(η) to a selectivity problem is a significant improvement compared to the use of η_TD, thus opening new doors in the understanding of competing reaction channels.

3.4. Kinetic Volcano Plots for the Nitrogen Reduction Reaction over Transition-Metal Oxides

Ammonia (NH₃), an important precursor to produce fertilizers, dyes, and medicaments, is mainly manufactured by the traditional Haber–Bosch process.^127−129 Unfortunately, this process is environmentally unfriendly because it releases 450 million tons of CO₂ per annum, corresponding to more than 1% of the global carbon dioxide emissions.¹³⁰ A promising sustainable route refers to electrochemical ammonia synthesis. However, the major hurdle in this regard is the competing HER, which takes place in the same potential window as the nitrogen reduction reaction (NRR).^131,132 Besides, the NRR requires larger overpotentials than the HER because six electrons are transferred in the overall reaction to produce one molecule of ammonia (cf. eq 69), whereas the HER is a facile two-electron process (cf. eq 69):

69

Previous theoretical studies on the NRR have focused on the concept of η_TD in combination with the construction of volcano plots to determine the PDS that may limit the overall reaction in the thermodynamic picture. Volcano curves can be constructed for the proposed mechanistic descriptions of the NRR that range from a dissociative pathway to an associative mechanism (cf. Figure 14). Given that breaking the N–N triple bond is energetically demanding, most electrode materials may follow the associative pathway, and in the following we are focusing on the volcano plot for this mechanistic description. While the associative mechanism can be differentiated into alternating and distal pathways, the volcano plots of Skúlason and co-workers for metals, transition-metal oxides, and other material classes indicate that the PDS is encountered with either the first reaction step (formation of the NNH adsorbate) or the last reaction step (formation of NH₃ out of the NH₂ adsorbate).^55,133−136 Both the alternating and distal mechanisms have these steps in common, so that the precise nature of the electrochemical protonation steps between the first and last reaction steps were thought not to be of the highest importance for materials development. A schematic overview of the thermodynamic volcano, where the formation of ammonia or the formation of the NNH adsorbate is limiting at the left or right volcano legs, respectively, is given in Figure 15.

Figure 14.

Schematic representation of the associative and dissociative NRR mechanisms. Reproduced with permission from ref (136).

Figure 15.

Schematic illustration of the volcano plot for the NRR over transition-metal oxides. The formation of ammonia (blue) or the formation of the NNH adsorbate (red) is reconciled with the limiting reaction steps in the thermodynamic picture at the left and right volcano legs, respectively. Figure adopted with changes from ref (134).

In the following, we apply the descriptor G_max(η) to factor overpotential and kinetic effects into the analysis of trends in the NRR, thereby aiming to resolve the RDS as introduced in section 2.4. In this context, we refer to the material class of transition-metal oxides which follows the conventional scheme of NNH or NH₃ formation as the PDS at the right and left volcano legs.^134,135 While we want to refrain from a long mathematical derivation, the respective details of how to apply G_max(η) to the NRR can be found in a recent article.¹³⁶ Also, we want to point out the importance of gas-phase error corrections for the energetic description of the NRR, because without these mandatory corrections, inaccurate or even erroneous equilibrium potentials, free-energy changes, and activity predictions are observed.¹³⁷ Starting with the associative distal mechanism, which comprises the *NNH, *NNH₂, *N, *NH, and *NH₂ adsorbates (cf. Figure 14), we extract the largest free-energy span corresponding to the descriptor G_max(η) at an applied electrode potential of U = −0.2 V vs RHE by varying the descriptor ΔG₁, which corresponds to the adsorption free energy of the NNH adsorbate, between −2.5 and +2.5 eV. The electrode potential of U = −0.2 V was chosen because it may refer to the conditions in experimental studies.¹³⁸ The result of this investigation is shown in Figure 16, indicating that different free-energy spans may limit the rate dependent on ΔG₁. Most notably, it becomes evident that a combination of several elementary reaction steps rather than a single reaction step is energetically most uphill in free energy at the right volcano leg (cf. Figure 16a), a feature than cannot be resolved by means of η_TD.

Figure 16.

(a) Volcano plot for the NRR over transition-metal oxides according to the associative distal mechanism. (b) RDS derived at U = −0.2 V vs RHE based on the concept of G_max(η), which goes beyond the common volcano interpretation in terms of the PDS (cf. Figure 14). Reproduced with permission from ref (136).

In the next step, we use the largest free-energy span based on the descriptor G_max(η) to determine the RDS. In this context, we assume a Tafel slope of 120 mV/dec, which assigns the first elementary reaction step to be rate limiting. This assumption is corroborated by the fact that the cathodic potential regime of U = −0.2 V vs RHE corresponds to an applied overpotential of about 350 mV when referring to the standard equilibrium potential, and for these overpotential regimes typically the first step may be the limiting one. On the left side of the volcano, the elementary step *NH₂ → NH₃ is reconciled with the largest free-energy span governing G_max(U = −0.20 V vs RHE). Consequently, the RDS is the transition from *NH₂ to NH₃, since this is the first elementary step in the reaction sequence. A different situation, however, is encountered close to or at the volcano apex. There, the sequences of steps *NNH → *NNH₂ or *NNH → *NNH₂ → *N → *NH are related to the G_max(U = −0.20 V vs RHE) span. Since b = 120 mV/dec purports the first elementary step to be rate limiting, for both regions *NNH → *NNH₂ is identified as the RDS. In the same fashion, N₂ → *NNH is obtained as the RDS at the right leg of the volcano for the G_max(U = −0.20 V vs RHE) spans of N₂ → *NNH → *NNH₂ → *N → *NH and N₂ → *NNH → *NNH₂ → *N → *NH → *NH₂. The kinetic volcano plot indicating the RDS for the associative distal mechanism is depicted in Figure 16b.

In the same fashion as for the associative alternating mechanism, we apply the concept of G_max(η) to the associative distal pathway, which consists of the *NNH, *NHNH, *NHNH₂, *NH₂NH₂, and *NH₂ adsorbates (cf. Figure 14). Using the same reasoning and approximations, we resolve the RDS for this pathway (cf. Figure 17a). Similar to the associative alternating mechanism, it turns out that the correlation RDS = PDS is fulfilled at the volcano legs for the associative distal pathway because formation of either NH₃ or the *NNH adsorbate limits the rate at the left and right volcano legs, respectively. However, at the volcano apex, a surface reaction, namely the step *NHNH₂ → *NH₂NH₂, governs the reaction kinetics at U = −0.2 V vs. RHE.

Figure 17.

(a) Volcano plot for the NRR over transition-metal oxides according to the associative alternating mechanism, following the same procedure as for the associative distal mechanism in Figure 15. The rate-determining step (RDS) is determined for a Tafel slope of b = 120 mV/dec at U = −0.2 V vs RHE based on the concept of G_max(η), which goes beyond the thermodynamic information in terms of the potential-determining step (cf. Figure 15). (b) Comparison of the associative distal (blue) and associative alternating (red) mechanisms to elucidate the preferred mechanistic description and limiting reaction steps. Reproduced with permission from ref (136).

The volcano plots for the associative alternating and distal mechanisms are combined to identify which mechanism is preferred, dependent on the descriptor ΔG₁. Figure 17b indicates that the associative distal and associative alternating mechanisms are preferred for strong and weak binding of the intermediate *NNH, respectively. Close to the apex of the volcano plot, surface reactions in terms of *NNH → M–NNH₂ or *NHNH₂ → *NH₂NH₂ govern the rate of the reaction rather than the formation of NH₃ or of the *NNH adsorbate, the steps of which are kinetically limiting at the volcano legs in agreement with the thermodynamic picture of the PDS.

In conclusion, the concept of G_max(η) has deepened our understanding of the NRR over transition-metal oxides by outlining a change in the preferred mechanistic description and the limiting reaction steps at the volcano apex. These insights cannot be resolved by the notion of η_TD and go beyond thermodynamic considerations at the equilibrium potential, thus illustrating the advantages of the application of G_max(η) for an understanding of trends and limiting steps in a class of materials.

4. Conclusions and Outlook

Computational electrochemistry^139−148 addressing electrified solid/liquid interfaces can be broadly classified into three categories: (a) first-principles or continuum model studies of the electric double layer at the interface,^149−152 (b) method development to move from a constant-charge description, as encountered with the computational hydrogen electrode approach, to a constant-potential formalism,^153−155 and (c) computational electrocatalysis by focusing on the energetics of the elementary reaction steps, thereby applying heuristic tools to quantify electrocatalytic activity.^84,156 The present perspective primarily addresses the last category while not explicitly focusing on the other two, yet highly important, research fields. We have aimed to provide an overview on the descriptor G_max(η) for the heuristic materials screening of electrode compositions for electrochemical processes with applications to energy conversion and storage. For almost two decades, the theoretical community in electrocatalysis has made extensive use of the thermodynamic overpotential, η_TD, as an activity measure by discussing thermodynamically limiting reaction steps, denoted as the potential-determining step (PDS). To the best of our knowledge, the concept of G_max(η) is the first attempt to go beyond a pure thermodynamic treatment of the elementary reaction steps when using heuristic analyses. We have introduced the mathematical formalism of G_max(η) (cf. section 2.1) and illustrated the advantages of G_max(η) compared to η_TD relating to the assessment of chemical reaction steps in the analysis of mechanistic pathways (cf. section 2.2). The notion of G_max(η) can be further used within a microkinetic approach to approximate current densities (cf. section 2.3) or to resolve the rate-determining reaction step (RDS) of a reaction mechanism if the Tafel slope is predetermined (cf. section 2.4), thereby considering the approximations made within the free-energy span model (cf. section 2.5). We have discussed these aspects on four selected examples, namely breaking the scaling relation in the oxygen evolution reaction (OER) (cf. section 3.1), introducing a new method to determine the bifunctional index (BI) for the reversible oxygen electrocatalysis from theory (cf. section 3.2), comprehending a selectivity problem by the descriptor G_max(η) (cf. section 3.3) and resolving the RDS for the nitrogen reduction reaction (NRR) over transition-metal oxides by G_max(η) (cf. section 3.4).

As is evident from the above summary, so far, the application of the descriptor G_max(η) has focused on comprehending the activity or selectivity of electrode compositions.^{61,64,66,89,120,157} The stability aspect of electrocatalysts is often overlooked from the viewpoint of theory, which is mainly related to the fact that powerful concepts to understand degradation processes are yet missing, albeit urgently needed.^158−160 Hitherto, mainly bulk-related properties have been used to comprehend catalyst stability on the atomic scale.^161,162 This picture is, of course, rather simplistic because chemical information on the decomposition products would be required to gain a profound insight into catalyst degradation. A major step forward in this regard represents a recent study by Andreussi and co-workers, who modeled the oxygen evolution and reduction reactions over two-dimensional transition-metal dichalcogenides in conjunction with different decomposition pathways for highly active OER materials.¹⁶³ An overview of the suggested degradation steps is given in Figure 18.

Figure 18.

Schematic representation of the decomposition pathway for the formation of aqueous products (X–O and X–O₂) in the presence of an applied electrode potential. The initial steps denoted by orange arrows are of electrochemical nature, and the steps denoted by black arrows correspond to chemical processes. Transition metal and chalcogen (X = S, Se, or Te) atoms are visualized in gray and yellow, respectively. Figure adopted with permission from ref (163).

As is evident from Figure 18, the formation of decomposition products such as X–O or X–O₂ (X = chalcogen) may comprise both electrochemical and chemical steps. This is where the descriptor G_max(η) may come into play by determining the energetics of the degradation pathways dependent on the driving force, which ultimately can be compared to the free-energy span governing G_max(η) for the elementary steps of product formation. The competition between decomposition and product formation can be seen as two different reaction channels, thus giving rise to a selectivity-type analysis as discussed in section 3.3. More precisely, this would allow extending current activity–selectivity investigations of electrocatalytic processes to more refined activity–selectivity–stability studies and allows the inclusion of stability investigations of electrode materials in the high-throughput screening of electrocatalysts at moderate computational costs. Comprehending the stability of electrocatalysts is of pivotal importance in theoretical studies and may advance screening activities in electrocatalysis,⁵¹ particularly with respect to the OER, where catalyst stability is the major bottleneck in acidic electrolytes,¹⁶⁴ to the next level. We conclude that the concept of G_max(η), based on the free-energy span model, may contribute to extend our understanding of catalyst stability for electrocatalytic processes at electrified solid/liquid interfaces in the next few years, and the gained insight may spur the design of equally active and stable electrode coatings for energy conversion and storage processes.

The authors declare no competing financial interest.