Insights into the Electrochemical Oxygen Evolution Reaction with ab

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Insights into the Electrochemical Oxygen Evolution Reaction with Ab Initio Calculations and Microkinetic Modeling: Beyond the Limiting Potential Volcano Colin F. Dickens, Charlotte Kirk, and Jens Kehlet Nørskov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b03830 • Publication Date (Web): 12 Jul 2019 Downloaded from pubs.acs.org on July 18, 2019

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The Journal of Physical Chemistry

Insights into the Electrochemical Oxygen Evolution Reaction with ab initio Calculations and Microkinetic Modeling: Beyond the Limiting Potential Volcano Colin F. Dickens,1,2,† Charlotte Kirk,1,2,† and Jens K. Nørskov1,2,3*

1

SUNCAT Center for Interface Science and Catalysis, Department of Chemical Engineering,

Stanford University, Shriram Center, 443 Via Ortega, Stanford, California, 94305, USA.

2

SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory

2575 Sand Hill Road, Menlo Park, California, 94025, USA

3

Department of Physics, Technical University of Denmark, 2800, Kongens Lyngby, Denmark.

†

Contributed equally to this work

*Correspondence to: [email protected]

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Abstract Density functional theory calculations are potentially useful for both understanding the activity of experimentally tested catalysts and screening for new catalyst materials. For electrochemical oxygen evolution reaction (OER) catalysts, these analyses are usually performed considering only the thermodynamics of the reaction path, which typically consists of adsorbed OH*, O*, and OOH*. Scaling relationships between the stability of these intermediates lead to a limiting potential volcano whose optimum is constrained by the roughly constant offset between the binding energies of OH* and OOH*. In this work, we evaluate OER kinetics at rutile IrO2, RuO2, RhO2, and PtO2 surfaces by computing reaction barriers with an explicit model of the electrochemical interface. We conclude that the kinetics of proton transfer between oxygen atoms at the surface and in the electrolyte are facile and that O-O bond formation is most likely rate determining in all cases. Combining these results with a microkinetic model and a scaling relationship for the OOH* formation barrier, we construct a new activity volcano whose optimum is similar to that of the limiting potential volcano for typical current densities. This kinetic volcano is also shown to agree reasonably well with experimental observations. Based on this analysis, we propose a more precise requirement for improving OER catalysts beyond the state of the art: that the transition state for OOH* formation must be stabilized as opposed to the fully formed OOH* final state as has been previously presumed.

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Introduction The oxygen evolution reaction (OER) plays a critical role in the sustainable production of fuels and chemicals as the anodic reaction that provides protons and electrons from water. When coupled to a source of renewable electricity, these charged species can be used to sustainably produce valuable products such as hydrogen gas, hydrocarbons, oxygenates, and ammonia. While these products are generated at the cathode via reduction of CO2, N2, or protons themselves, the sluggish kinetics of water oxidation at the anode (i.e. the OER) contribute a major source of inefficiency of the overall process. Thus, there is a large effort to design improved OER catalysts that operate at sufficiently high rates with a small overpotential (e.g. 10 mA/cm2 at < 0.3 V) whilst simultaneously maintaining stability under highly oxidizing conditions.1–3 In principle, two design strategies exist to improve the activity of OER catalysts. On one hand, catalyst and support frameworks can be engineered with higher surface area and more exposed active sites, leading to an increase in geometric current density without a significant increase in cost provided that cheap materials are used.4,5 Alternatively, new active sites can be designed that possess intrinsically higher activity. These strategies are of course not mutually exclusive and a combination of both is ideal for practical applications; however, it is likely that further improvement beyond the current state of the art relies primarily on the latter.3 Density functional theory (DFT) is a very popular tool for evaluating the intrinsic activity of electrocatalysts in either an explanatory or predictive manner. One of the most commonly used metrics in these studies is the limiting potential (or the closely related theoretical overpotential),6 the definition of which allows for an efficient activity comparison of different catalytic surfaces and active sites without even a model for the electrochemical interface in its most basic implementation. While it is a useful tool for both catalyst screening and interpretation of experimental observations, it is sometimes taken for granted that the limiting potential (the potential at which all steps of the mechanism are thermodynamically downhill) has no direct correspondence to any experimentally observed quantity in 3 ACS Paragon Plus Environment


general as has been noted previously.6–8 To develop new mechanistic insights and validate conclusions that have arisen from the purely thermodynamic limiting potential model, there have been efforts over the years to construct microkinetic models for electrochemical reactions based on DFT calculations of a single catalyst surface.9–15 The inclusion of linear scaling relationships based on multiple catalyst surfaces allow for the construction of current or overpotential volcanoes, which have direct correspondence to experiment and provide optimal catalyst design criteria. Such analyses have been performed for the hydrogen evolution/oxidation,16 oxygen reduction,17 and CO2 reduction reactions.18,19 For the case of OER, which is the subject of this work, a series of volcanoes were recently constructed for different Co3O4 surface facets with various 3d metal dopants,20,21 and DFT-based microkinetic models have been created individually for RuO2 and IrO2.9,15,22 However, these OER studies have not considered the kinetics of charge transfer reactions, which are methodologically and computationally challenging to calculate as they require some representation of the electrochemical interface and model for including the electrode potential.10,12,14,23,24 In the context of OER, the conventional limiting potential analysis, while possibly over-simplified, has suggested two key insights that have fueled research directions over the past several years. First is that the optimal OER catalyst site which obeys the usual scaling relationships between the binding energies of OER intermediates O*, OH*, and OOH* has a surface hydroxide deprotonation free energy value in the range of 1.5 to 1.7 eV.3,25–29 This material descriptor, often written as ΔGO –ΔGOH, is relevant because it is a measure of the ability of surface oxygen atoms to make and break bonds with hydrogen and oxygen atoms, which tend to be the most thermodynamically difficult reaction steps for OER.25,26 The utility of such a descriptor is that it allows for the rapid screening of active sites with as few as two calculations: one each for O* and OH*. The second insight suggested by these analyses is that a necessary requirement the key to further improving OER catalysts beyond the state-of-the-art is to break the very robust scaling relationship between the binding energies of OH* and OOH* by engineering active sites 4 ACS Paragon Plus Environment

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that selectively stabilize the latter.26,30–34 Essentially, this boils down to strengthening surface oxygen’s affinity towards hydroxyl without affecting its affinity towards atomic hydrogen. The primary goals of the present work are to determine whether these conclusions are congruent with the results of a DFT-based microkinetic model and to provide new insights that aid in understanding and designing OER catalysts. We attempt to do so in a simple and transparent manner with the hope that the OER community may more easily inject kinetic principles into their typical analyses. We note that recent works by Plaisance et al. have made significant progress on this front for doped Co3O4 surfaces,20,21 and we will compare important findings of that work to our own, where conductive metal oxides and the kinetics of charge transfer reactions will be considered. Computational Details All DFT calculations were performed using the planewave code, Quantum Espresso via the Atomic Simulation Environment (ASE) python interface.35,36 Exchange and correlation were treated with the RPBE functional,37 and planewave/density cutoffs of 600/6000 eV were used. The importance of using GBRV pseudopotentials to accurately describe RuO2 in particular has been discussed previously,28,38 and in general we use ultrasoft pseudopotentials from this library in all cases,39 with the exception of those involving Ir in which case ultrasoft pseudopotentials from the PSLibrary project were used.40 This is simply for historical reasons, and we have verified that the differences in surface binding energies obtained using either library are negligible for IrO2 calculations. Rutile metal-oxide bulk structures of RuO2, IrO2, RhO2, and PtO2 were optimized by minimizing the total energy with respect to the tetragonal lattice parameters a and c and the reduced oxygen coordinate (often referred to as u). Surfaces were cut along the (110) facet with four complete layers of which the bottom two were fixed to their bulk positions in all subsequent relaxations, and a dipole correction was employed in all cases.41 Spin-polarization was used in all surface calculations except for those involved in determining electrochemical charge transfer barriers, where it was computationally too costly to include in all cases. For those barriers where spin5 ACS Paragon Plus Environment


polarization was included, the effect was a modest 0.1 to 0.2 eV, and thus we have elected to neglect the effect in general. Because all materials considered in this work are metallic, we do not elect to add a Hubbard U to the metal centers.42 All transition states were determined using the climbing image nudged elastic band (NEB) method.43 In light of previous work by Gauthier et al. that demonstrate rather modest solvation effects (c.a. 0.2 eV) for OER intermediates at the IrO2 (110) surface,44 we neglect solvation where possible (i.e. everywhere except electrochemical charge transfer barriers), and we emphasize the primary goal of this work is to construct a general microkinetic model via scaling relations (i.e. a volcano) rather than argue the location of particular surfaces and active sites on the volcano itself. Calculations performed in vacuum correspond to a 1x3 supercell (with 3 × 4 × 1 Monkhorst-Pack k-point sampling),45 which enables the usage of a spectator coadsorbate for steps involving two sites such as the coupling of neighboring CUS oxygen atoms. Calculations performed with explicit solvation (i.e. those involving the search for electrochemical transition states) are in multiples of 1x2 supercells (e.g. 1x2, 1x4, 2x4) using an ice-like water structure previously found by minima hopping at the IrO2 (110) surface.44 These calculations were performed with 4/a × 8/b × 1 Monkhorst-Pack k-point sampling where a and b are the supercell dimensions in the x and y directions, respectively. The absolute electrode potential in these calculations is computed as the potential drop between vacuum above the interface (sampled in a region of zero field before the dipole correction) and the Fermi level and is referenced to the standard hydrogen electrode (SHE) via the experimental constant of 4.44 V.46 Free energy corrections for surface calculations were applied assuming only harmonic degrees of freedom and free energy corrections for gaseous species were determined in the ideal gas limit. Further details can be found in the SI. The free energy of liquid water was determined with DFT by simulating a gas-phase water molecule and applying free energy corrections at its room-temperature vapor pressure using the thermochemistry module of ASE. The free energy of hydrogen gas at standard conditions was 6 ACS Paragon Plus Environment

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determined similarly. The free energy of oxygen gas at standard conditions was determined based on the experimental equilibrium potential for OER, 1.23 V. The electrochemical potential of proton/electron pairs was determined via the computational hydrogen electrode:

H

+

/e-

=

1 0  − eU RHE 2 H2 ( g )

(1)

where URHE is the electrode potential relative the reversible hydrogen electrode (RHE).6

Results and Discussion Mechanism Overview We begin by considering an expanded version of the OER mechanism that is typically used for calculating limiting potentials with DFT: H2O(l) + * → H 2O*

(2)

H 2 O* → OH* + H + /e-

(3)

OH* → O* + H + /e-

(4)

O* + H 2 O(l) → OOH* + H + /e -

(5)

OOH* → O 2 * + H + /e-

(6)

O2 * → * + O2 (g)

(7)

This mechanism is expanded in the sense that we have explicitly included intermediate steps (such as water adsorption and OOH* deprotonation) that are typically left out of the conventional four-step



thermodynamic analysis. This is because the goal of the present work is to perform a kinetic analysis, which is dependent on the rate constants of elementary steps. We note here that all considered electrochemical steps have been written in the “acidic” convention, where each consists of the production of a proton rather than the consumption of a hydroxide ion. Whether the acidic or basic mechanism is considered has no ramifications for the equilibrium constants (thermodynamics) of each electrochemical step as they are only dependent on the potential vs the reversible hydrogen electrode (URHE). However, this is not true for the kinetics of electrochemical steps, as the acidic and basic pathways have their own USHE dependent rate constants. Additionally, the forward rate in basic conditions is proportional to the concentration of hydroxide ions, giving it a mixed USHE/pH dependence as opposed to the purely USHE dependence of the forward rate in acidic conditions. This work will only involve the calculation of electrochemical rate constants for the acidic case, which are unlikely to be relevant in basic conditions as they will be orders of magnitude lower due to the drop in USHE of c.a. 0.8 V when moving from acid to base at constant URHE. Therefore, our kinetic analysis will be performed for pH = 0 where USHE = URHE, and we will primarily compare our results to experiments performed in acidic media. The goal of this work is to utilize microkinetic modeling and DFT calculations to construct current density and overpotential volcanoes for the OER. In principle, this can be achieved by calculating the reaction free energies and barriers (and their potential dependence where applicable) for each step defined above on many surfaces, constructing scaling relationships that map all energetics down to a single descriptor, such as ΔGO –ΔGOH, and plugging those relationships into a steady-state microkinetic model. However, performing such an analysis requires the calculation of many transition-states, which are both methodologically challenging and computationally intensive for electrochemical reactions. Additionally, the results of such a full-blown kinetic analysis can often be difficult to interpret. In this work, we will sequentially introduce and justify a simplifying set of assumptions based on both computational 8 ACS Paragon Plus Environment

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and experimental observations that result in a simple and interpretable microkinetic model for the OER. To this end, we will first narrow down the number of possible rate determining steps without explicit microkinetic modeling, and then we will compare the behavior and feasibility of microkinetic models that assume the remaining possible rate determining steps. The elementary steps considered in the above mechanism comprise steps that are either thermochemical or electrochemical (i.e. involving charge transfer across the electrochemical interface). The two considered thermochemical steps are water adsorption and O2 desorption. The former will be assumed to always be in equilibrium as water binds rather strongly to metal oxide surfaces. This is shown for the surfaces considered in this work in Table S2 and was recently shown for a wider variety of metal oxides and zeolites.47 Additionally, we do not expect there to be any sort of diffusion barrier for water in an aqueous electrolyte. We will not make such an a priori equilibrium assumption about O2 desorption and instead consider it as a possible rate determining step (RDS). This choice is motivated in part by recent crystal truncation rod experiments that indicate coverage of an O-O bond containing intermediate at the RuO2 (110) surface under OER conditions.48 The remaining elementary steps involve the separation of charge across the electrochemical interface and are hence referred to as electrochemical steps. They can be broadly split into two groups: proton transfer between oxygen atoms and OOH* formation, which includes formation of the O-O bond. Previous kinetic OER studies have assumed that proton transfer steps do not have significant barriers and are hence equilibrated.9,15,20,21 In the following, we will present our work to quantify the barriers for both types of charge transfer reactions using DFT.

Quantification of electrochemical barriers



We will first consider the barrier for proton transfer from surface OH* to a water molecule in the water layer to form a hydronium ion (i.e. eqn. (4)). Shown in Figure 1a is the reaction coordinate for such a process at an IrO2 (110) surface in a 1 × 4 supercell with 50% coverage each of OH* and O* at CUS sites and 100% coverage of O* at bridge sites. The surface bound oxygen’s affinity for hydrogen and the electrochemical potential are such that the process has nearly zero driving force (ΔE = 0.02 eV). Thus, we interpret the observed energetic barrier as an intrinsic measure of the kinetic difficulty for proton exchange between surface oxygen and electrolyte. The forward (oxidation) barrier is roughly 0.05 eV, suggesting an essentially instantaneous reaction at room temperature. It is unsurprising that this process is kinetically facile as the proton need only travel a short distance of 0.47 Å from the surface oxygen atom to reach the oxygen atom in the electrolyte and form the hydronium ion (see Figure 1b). For reference, a proton hopping between CUS oxygens must travel more than three times this distance yet may still be considered facile with an energetic barrier of 0.48 eV. Additionally, we note previous works that have identified barriers of less than 0.3 eV for proton transfer from electrolyte to surface oxygen at Pt (111) and TiO2 (110) surfaces.49,50 Ultimately, we conclude that there is no kinetically relevant transition state for proton exchange between surface oxygen and the electrolyte, and we treat eqn (4) as equilibrated in our microkinetic model. We further assume that the analogous steps of proton transfer from either H2O* or OOH* to the electrolyte are also equilibrated. This accounts for all electrochemical steps in our mechanism, with the exception of OOH* formation.


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Figure 1. (a) Reaction coordinate for proton transfer from OH* to solvated H2O at the IrO2 (110) surface illustrating a very small intrinsic barrier of c.a. 0.05 eV. For reference, the electrode potential of the initial, transition, and final states are 1.70, 1.47, and 1.14 V vs SHE, respectively. (b) Image of the corresponding transition state.

In contrast to the case of simple proton transfer between oxygen atoms, the size of the barrier for OOH* formation is substantial, and we will consider it in more detail. Due to long-range electrostatic interactions of the partially charged electrochemical transition states with themselves via periodic images, finite cell size effects that significantly affect the barrier height are unavoidable.16,51,52 It is therefore crucial to “extrapolate” to conditions where periodic images of the reaction coordinate do not interact with each other, i.e. to an infinitely sized unit cell. The extrapolation methodology used in this work is slightly different than those developed previously and is explained in detail in the appendix. Intuitively, the problem with electrochemical barrier calculations in a finite cell is that the electrode potential changes across the reaction coordinate. This is due to the gradual separation of charge across the interface from initial state to final state, i.e. creating solvated protons in the electrolyte and electrons in the electrode at a very high density due to the periodic boundary conditions. This high density results in spurious electrostatic interactions between the charges that would not exist in an infinitely large cell. It is therefore unclear how to relate the calculated barrier from DFT, which has two different



electrode potentials, one associated with the initial state and one associated with the transition state, to the barrier in a macroscopic experimental system where the electrode potential is fixed at a single value. A naïve approach might be to simply average the potential of the initial and transition states of the calculation. It turns out that this approximation is exact given three assumptions: first, that the electrode potential is a linear function of the coverage of charged species (i.e. negligible polarizability); second, that the stability of any charged species along the reaction coordinate is a linear function of the coverage of itself or any other charged species (i.e. assuming linear, mean field interactions), and third, that the interaction energy between two charged species is proportional to the product of their dipole moments. All three of these assumptions are made when treating the reaction coordinate as the continuous charging of a capacitor with constant capacitance, C, as has been done previously.52,53 Ultimately, these assumptions allow us to determine the extrapolated barrier (that corresponds to an infinitely sized unit cell) as a function of potential by plotting the barrier height vs average potential calculated in progressively larger cell sizes. Shown in Figure 2a is an image of the OOH* formation transition state and in 2b the barrier height calculated in three different supercell sizes (1×2, 1×4, 2×4) on IrO2 (110) at the respective average potential of each calculation as well as an image of the transition state in the 2×4 supercell. The initial state in all cases consists of a completely discharged double layer and a surface coverage of 50% each of O* and OH* at CUS sites and 100% O* at bridge sites. The linear fit is very good, with a slope of -0.57, but it is obviously desirable to obtain more than three points for verification. Rather than further increase the cell size, which is computationally intractable, we resort to another means for changing the potential: surface coverage. The work function of the 1×2 IrO2 (110) surface can be modified by more than 1 eV by either adding hydrogen atoms to coadsorbed oxygen to form OH* or H2O* or by creating an oxygen vacancy. Thus, we can further explore the barrier height at different potentials by modifying the surface coverage, 12 ACS Paragon Plus Environment

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which is significantly cheaper computationally compared to increasing the cell size. If we make an additional assumption in our model that the interaction energy between charged species and surface species is also proportional to the product of their dipole moments, then the barrier height is strictly a function of potential regardless of the mechanism (cell size or surface coverage) used to achieve that potential. This is shown to be the case for the reaction energy of a proton transfer between O* and the electrolyte in Figure S3. In Figure 2c, we show five additional data points where the OOH* formation barrier calculated in a 1×2 supercell has been modified by either adding hydrogen to the coadsorbed CUS OH* to form H2O*, adding hydrogen to the coadsorbed bridge or subsurface (referred to as 3c in the figure label) O* to form OH*, or removing the oxygen atom from the bridge or subsurface sites to form oxygen vacancies. In an effort to probe only the indirect, electrostatic impact of the surface coverage on the barrier, these barriers at different coverages have been obtained by fixing the geometry of the original initial and transition states and simply relaxing the local coordinates of the modified adsorbate. We note that it is possible for the structure of the transition state to change due to indirect electrostatic interactions with coadsorbates, but we also note that, by definition, the gradient of the potential energy surface is small near the transition state such that the corresponding energy differences may be neglected. This strategy has the added practical benefit of obviating the need for additional water structure relaxations and NEB calculations. Thus, this method of modifying surface coverage represents an exceptionally cheap way of estimating the potential dependence of an electrochemical process. Ultimately, these calculations suggest a potential dependence of approximately -0.6 for OOH* formation. This corresponds to a capacitance of 12.6 μF/cm2, using a transition state dipole moment of 0.41 eÅ, which is the average of all calculations in Figure 2c (see SI for details).



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Figure 2. (a) Depiction of a transition state for OOH* formation at an IrO2 (110) surface in a 2×4 supercell. (b) Barrier heights for OOH* formation at an IrO2 (110) surface in various supercells plotted against the average electrode potential taken across the initial and transition states. (c) Same as (b) but with additional points (diamonds) corresponding to the 1×2 supercell with modified surface coverages. Raw data can be found in Table S1.

Towards our goal of developing a microkinetic model in continuous material descriptor space (i.e. a volcano relationship), it is necessary to construct a scaling relation for the OOH* formation barrier as a function of both potential and a surface-dependent thermochemical descriptor. The conventional descriptor used for OER, ΔGO –ΔGOH, is a good choice in this instance because it has a very strong scaling with the OOH* formation reaction energy, ΔGOOH –ΔGO. This is because the two generally sum to the roughly constant ΔGOOH –ΔGOH, c.a. 3.2 eV.26 Therefore we assume a linear model of the form

Ga (U SHE , GO − GOH ) =  −  ( GO − GOH ) − U SHE

(8)

Additional OOH* formation barriers were calculated for RuO2, RhO2, and PtO2 (110) surfaces by utilizing the initial state water structures from IrO2 (110). For RhO2 and PtO2, a single NEB calculation was performed in a 1×2 supercell and additional data was generated by modifying the surface coverage. The


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reaction driving force was too uphill to identify a transition state in the 1×2 supercell for RuO2, which is the strongest binding (largest ΔGOOH –ΔGO) surface considered in this work. However, by considering a 1×4 supercell, the potential along the reaction coordinate was increased (and hence the reaction driving force increased) sufficiently such that a transition state could be identified. Figure 3a displays all of the OOH* formation barriers considered in this work as a function of potential and thermochemical descriptor, ΔGO –ΔGOH. The values of ΔGO –ΔGOH used here were computed with a vacuum interface (i.e. without explicit solvation or ions) at the same surface coverage as the corresponding transition state calculation and with spin polarization included. We note the impact of coverage on the descriptor value is quite small (0.1 to 0.2 eV). Also shown in Figure 3a is the plane of best fit to the data according to eqn (8) with coefficients ξ, α, and β of 2.41 eV, 0.48, and 0.58 e respectively. A parity plot for this model is shown in Figure 3b. We note that by performing such a fit, we are implicitly considering yet another way to modify the electrode potential: variations in surface work function due to metal composition. The work functions of oxygen covered RuO2 and RhO2 differ by 0.7 eV.

Figure 3. (a) Barriers for OOH* formation at rutile (110) surfaces shown as a function of material descriptor value (ΔGO –ΔGOH) and electrode potential. The color of each point indicates the surface composition (red = RuO 2, blue = IrO2, purple = RhO2, gray



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= PtO2), and the shape of each point takes on the same meaning as in Figure 2. (b) Parity plot (including the mean absolute error) of the 2D linear model, which is represented by the plane in (a).

Overview of microkinetic model The above analysis leaves two possible rate determining steps to consider: OOH* formation and O2 desorption. The turnover frequencies (TOF) for OOH* formation and O2 desorption, respectively, are written as

r1 =  O k1 (1 −  )

(9)

r2 =  O2 k2 (1 −  )

(10)

where θO and θO2 are the coverages of atomic and molecular oxygen, respectively, at active surface sites; k1 and k2 are the molecular rate constants for OOH* formation and O2 desorption, respectively; gamma is the approach to equilibrium of the reaction, which accounts for the backwards reaction rate; kB is Boltzmann’s constant; and T is temperature. In our analyses, we will consider one RDS at a time, and assume that all other steps are quasi-equilibrated, in which case the approach to equilibrium corresponds to the total approach to equilbrium for the OER as a whole, which is defined as 𝑒𝑥𝑝(−4𝜂𝑒/k 𝐵 𝑇) where η is the overpotential. This term can be neglected for overpotentials greater than 30 mV where it has a value of less than 1% at room temperature. To convert from turnover frequency to current density, we use the fact that 4 electrons are generated for each reaction turnover and assume a typical active site density of 1015 cm-2. This results in an almost 1:1 conversion between turnover frequency (molecules O2/site-second) and current density (mA/cm2). Specifically, i = 0.64r where i is the current density and r is the turnover frequency.


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To construct a volcano, these rate expressions (rate constants and coverages) must be specified in terms of electrode potential and thermochemical descriptor (i.e. ΔGO –ΔGOH). Rate constants are calculated via the Eyring equation:

k=

k BT exp ( −Ga / k BT ) h

(11)

and the problem reduces to constructing a linear scaling relationship Ga(U, ΔGO –ΔGOH), as we have done above for OOH* formation. For each RDS, the coverages are computed by assuming all other steps are quasi-equilibrated. For example, assuming OOH* formation to be the RDS results in the following five equilibrium expressions:

K1 =

H O *

(12)

K2 =

 OH H O

(13)

K3 =

O  OH

(14)

K5 =

O  OOH

(15)

* O

(16)

2

2

K6 =

2

2

where the equilibrium constants are computed based on the free energy of the appropriate reactions presented in (2)-(7) at the RHE potential of interest. Combining the above expressions with a statement of site conservation,

 xS

x

= 1 where S = {*, H2O, OH, O, OOH, O2}, gives six independent equations

that can be solved for the six intermediate coverages. Continuing with the example of OOH* formation as



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the RDS, θO is of primary interest for computing reaction kinetics because O* is the precursor to the RDS (equation (9)). Rearranging the statement of site conservation, we arrive at an expression for θO:

O =

1

  H O  OH  OOH  O 1+ * + + + + O O O O O 2

.

(17)

2

where each of the ratios in the denominator can be determined based on the above equilibrium expressions:

O =

1 1 1 1 1 1 1+ + + + + K1 K 2 K 3 K 2 K 3 K 3 K1 K 2 K 3 K 5 K 6 K1 K 2 K 3 K 6

(18)

and related to ΔGO –ΔGOH via linear scaling relationships of adsorbate binding energies. The terms in the denominator can be thought of as effective equilibrium constants for each surface intermediate relative to the species of interest (O* in this case), i.e.

O =

1  K x /O

(19)

xS

where Kx/O is the equilibrium constant for the reaction converting O* into surface species x without traversing the RDS. To demonstrate this point, consider the calculation of K O2 /O . This equilibrium constant should be calculated based on the free energy of the reaction

O* + O 2 ( g ) + 2 H + /e -  O 2 * + H 2O ( l )

K O2 /O =

1 K1 K 2 K 3 K 6

(20)

as opposed to

O* + H 2 O ( l )  O 2 * + 2H + /e -

K O 2 /O = K 4 K 5


(21)

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because the latter has implicitly and incorrectly assumed that the RDS (OOH* formation) is equilibrated. This is a crucial distinction; the equilibrium constant for (20) decreases with potential (reduction) while the equilibrium constant for (21) increases with potential (oxidation). We have demonstrated above that reaction kinetics are necessary in general for determining the surface coverage under reaction conditions, but also that this information can be as minimal as the assumption of a RDS. Having assumed a RDS, the surface species with the highest coverage will be the one with the largest equilibrium constant in equation (19) because all states on either side of the RDS equilibrate with one another via their coverages according to the quasi-equilibrium approximation. This is most easily visualized by creating a FED for the overall reaction where the assumed RDS has been shifted to be the final step such that all intermediates lie on the same side of the RDS. In this representation, the lowest lying (most thermodynamically stable) state on the FED corresponds to the dominant surface intermediate. FEDs of this type will be constructed for each RDS we consider. This analysis raises an important point that is often overlooked in DFT studies of OER and other electrochemical reactions. Namely, that surface coverages under reaction conditions are not only a function of equilibrium constants (thermodynamics) but are dependent on reaction kinetics. Surface Pourbaix diagrams, which are often used to determine the surface coverage as a function of potential,22,54–60 are only strictly valid under conditions of thermodynamic equilibrium (i.e. at the equilibrium potential). For any finite overpotential, the surface coverage is a function of reaction kinetics, and the surface Pourbaix diagram is no longer applicable or even well-defined as at least one atomic species necessarily will have two references of different chemical potentials. For the case of OER, oxygen atoms can either be referenced to liquid water and protons/electrons or ambient pressure O2 gas. This is analogous to the point made by Schmidt and coworkers that oxide materials are thermodynamically unstable under any finite OER overpotential since bulk oxygen atoms also have two possible thermodynamic references.61 It is important to recognize that by taking the reference to be 19 ACS Paragon Plus Environment


liquid water and protons/electrons, as is generally done when constructing a surface Pourbaix diagram, one has implicitly assumed that the last step of the mechanism is rate-determining, e.g. O2 desorption in the case of OER. Having established our methodology for calculating coverages and rate constants, we will now investigate and compare two possible rate determining steps, OOH* formation and O2 desorption, with a linear scaling analysis.

Rate determining step comparison: OOH* formation and O2 desorption The rate constant for OOH* formation can be determined as a function of potential and descriptor value utilizing the activation energy scaling relationship established in eqn (8) with fitted ξ, α, and β of 2.41 eV, 0.48, and 0.58 e respectively. To help gain some intuition for the surface coverage under reaction conditions, free energy diagrams have been constructed from the scaling relations in Figure S6 for different values of ΔGO –ΔGOH and URHE and are shown in Figure 4. Because the FEDs have been constructed such that the assumed RDS is the final step, we can immediately recognize the dominant surface species to be either OH* or O* because they are the lowest lying states for all cases considered. These four FEDs are simply points in a 2D plane of possible ΔGO –ΔGOH and potential values, and also shown in Figure 4 is the dominant surface coverage across a large region of the 2-D plane. Because OH* and O* coverages are dominant across the relevant part of the plane, we can reduce the sum in equation (19) to have a single term and write the coverage of oxygen very simply as 1/(1+KOH/O). This is especially convenient because it means that the coverage of oxygen can be calculated from our thermochemical descriptor alone without the need for any scaling relations. Note however that the use of scaling relations provided the evidence to make this simplification. Ultimately, we may express the rate of OOH* formation as


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r1 =

k BT 1 exp − ( −  ( GO − GOH ) − U SHE ) / k BT (1 −  ) 1 + K OH/O h

(

)

(22)

where the barrier for OOH* formation has been explicitly written out in terms of the 2D linear scaling parameters.

Figure 4. (a) Surface coverage as a function of ΔGO –ΔGOH and URHE assuming that OOH* formation is rate determining. (b)-(e) Free energy diagrams corresponding to surfaces with particular descriptor values (1.2 eV or 1.8 eV) at different potentials (1.4 V or 1.8 V) as indicated by the labeled points in (a). Error bars represent 95% confidence intervals based on the linear scaling relations in Figure S6.

We now turn towards computing the rate of OER assuming O2 desorption to be rate determining. We will not attempt to determine the rate constant for O2 desorption with DFT for two reasons. Primarily, there exists a significant error associated with GGA-level DFT’s description of O2 along its reaction coordinate between surface and gas phase. Secondly, because harmonic transition state theory is not applicable to desorption processes, sampling of the system’s dynamics is required to estimate the entropy along the reaction coordinate, 62 which is computationally intensive, prohibitively so in the presence of an 21 ACS Paragon Plus Environment


electrochemical interface. Therefore, we resort to experiment as a source for the O2 desorption rate constant. Specifically, we leverage temperature programmed desorption (TPD) results obtained for RuO2 (110) by Ertl and coworkers where a sharp peak at 130 K was attributed to molecular O2 desorption with O* coadsorbates.63 Based on this peak temperature, we estimate a room temperature free energy barrier for desorption of 0.28 eV using a Redhead type analysis (see SI for details).64 To extend this analysis to the other surfaces considered in this study, we assume Brønsted–Evans– Polanyi (BEP) scaling such that the desorption barrier is linearly related to the binding energy of O2.65 As shown in Figure 5, the O2 binding energy does not scale with ΔGO –ΔGOH and spans a range of approximately 0.7 eV bounded by O*-covered RuO2 on the weak binding side and OH*-covered IrO2 on the strong binding side. We will analyze these extremes by evaluating two separate models using either an O2 desorption free energy barrier of 0.28 eV or 0.98 eV, i.e. assuming a BEP scaling slope of unity.

Figure 5. O2 desorption energy at various rutile (110) surfaces with various surface coverages as a function of ΔGO –ΔGOH. Coverage labels correspond to coadsorbates at CUS sites and coadsorbates at bridge sites, in that order.


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Analogously to Figure 4 for the assumed RDS of OOH* formation, Figure 6 summarizes the results of the coverage analysis for the assumed RDS of O2* desorption. Note that in this case, the FEDs are constructed such that O2 desorption is the final step because we are assuming it to be rate-determining. This means that all surfaces will eventually saturate to O2* coverage given a large enough overpotential. The error bars in the FEDs represent 95% confidence intervals of the scaling relationships (shown in Figure S7), and we point out that they are quite large (~1 eV) for the vacancy and adsorbed water states, suggesting that the scaling between O2* and these states is poor. However, these states are so unstable relative to the others (even considering the confidence intervals) that they need not be considered when determining the coverage. We note that if these states were stable enough to be competitive, then a single-descriptor scaling analysis would not be applicable to determine the surface coverage. Ultimately, because the dominant species in Figure 6 are OH*, O*, and O2*, we can express the O2* coverage in terms of two equilibrium constants:

O = 2

1 1 + K OH/O2 + K O/O2

.

(23)

In this case, the coverage cannot be tautologically related to the thermochemical descriptor, ΔGO –ΔGOH, and instead we must rely on two scaling relationships, one for ΔGOH –ΔGO2 and one for ΔGO –ΔGO2, to determine the coverage as a function of ΔGO –ΔGOH. We then arrive at the following expression for the net rate assuming that O2 desorption is rate determining:

r2 =

1 1 + K OH/O2 + K O/O2

k BT exp −Ga ,O2 / k BT (1 −  ) h

(

where Ga,O2 is the free energy barrier for O2 desorption.


)

(24)


Figure 6. Analogous to Figure 4 with the assumption that O2 desorption is rate determining.

Having established expressions for the rate constants and coverages for cases of either OOH* formation or O2 desorption being rate determining, we can now evaluate the rates of each as a function of applied potential and thermochemical descriptor (ΔGO –ΔGOH). We take the step that yields the lower rate to be the true rate determining step. This classification is indicated across potential-descriptor space in Figure 7 for two different values of the O2 desorption free energy barrier (0.28 eV and 0.98 eV) in accordance with the data in Figure 5. Clearly, the results are highly sensitive to the value of the O2 desorption barrier chosen with O2 desorption becoming the more likely RDS as its barrier is increased. When considering the higher O2 desorption barrier, which DFT calculations indicated should be relevant for IrO2 (110) based on its O2 binding energy at any coverage, O2 desorption is rate determining for the majority of the 2D plane. Such a barrier results in a maximum current density of c.a. 0.1 μA/cm2 and a loss of potential dependence of the rate once the surface becomes saturated with O2 (most of the 2D plane in Figure 6) as shown in Figure S8. This is not in line with experimental observation and potentially calls into


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question our description of adsorbed O2 at IrO2 (110). Therefore, experimental TPD measurements similar to those performed for RuO2 (110) would be highly valuable.63 We note that O2 desorption plays no role when considering the experimentally derived O2 desorption barrier of 0.28 eV. Ultimately, we rely on the results obtained using this barrier value in which case OOH* formation dominates and disregard O2 desorption for the remainder of this work.

Figure 7. Comparison of possible rate determining steps, OOH* formation and O2 desorption, for two different values of the O2 desorption free energy barrier (0.28 eV at left and 0.98 eV at right).

We note recent experimental results obtained by Shao-Horn and coworkers in which crystal truncation rod experiments suggest a high coverage of an O-O bond containing species at the RuO2 (110) surface.48 Our analysis does not recover this behavior as the barrier for O2 desorption from O*-covered RuO2 (110), which is based solely on experimental observation, is surmountable at room temperature. It is possible that coadsorbed OH* species could stabilize O2* via hydrogen bonding, increasing the desorption barrier. However, our DFT calculations agree with theirs in that we do not predict OH* species 25 ACS Paragon Plus Environment


Page 26 of 60

at the RuO2 (110) surface under reaction conditions as shown in Figure 6. Even if such coadsorbed species were stable, we find their stabilization effect on O2* to be rather insignificant for coadsorbed CUS OH* (0.04 eV) as shown in Figure 5. It is also possible that solvation, which is neglected here, plays a significant role in stabilizing O2*; however, we note a previous work that computed the binding free energy of O2 on IrO2 (110) with O* coadsorbates and explicit solvation to be 4.20 eV, which compares very well to our value of 4.17 eV. 44

Rate-determining-step comparison: OOH* formation and O-O coupling In the following, we will investigate an alternative path to O-O bond formation, namely the coupling of two surface oxygen atoms. This type of Langmuir-Hinshelwood mechanism has been considered for many decades in the literature.9,66–71 The rate of the process we will consider here is written:

r3 =  O2 k3 .

(25)

Because we are still considering that an O-O bond formation step is rate determining, the coverage of oxygen is given by the equilibrium between OH* and O* alone regardless of whether the Eely-Rideal type OOH* formation or Langmuir-Hinshelwood type O-O coupling process is being considered. Regarding the rate constant for O-O coupling, we have constructed a scaling relation between the thermochemical descriptor, ΔGO –ΔGOH and the activation energy for O-O coupling as shown in Figure 8.


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Figure 8. Scaling relationship for O-O coupling barrier for various rutile (110) surfaces at various surface coverages as a function of ΔGO –ΔGOH. Coverage labels correspond to coadsorbates at CUS sites and coadsorbates at bridge sites, in that order.

Similar to Figure 7 where we compared the rates of OOH* formation and O2 desorption, in Figure 9 we compare OOH* formation to O-O coupling, although in this case we choose the true RDS as the step with the higher rate because the processes run in parallel rather than in series. Our results suggest that OOH* formation and O-O coupling are competitive with one another at the rutile (110) surface, especially for intermediate values of ΔGO –ΔGOH. However, we note that surfaces that are dominated by O-O coupling exhibit somewhat unusual Tafel behavior as shown in Figure S9. Specifically, they have a low potential Tafel slope of 30 mV/decade (when the surface is OH* covered) and become singular (i.e. they lose potential dependence) at the potential where O* coverage becomes favorable. Further increases in potential facilitate OOH* formation and eventually linear Tafel behavior resumes once the OOH* formation pathway begins to dominate. Such singularities in Tafel behavior are not typically observed experimentally, which provides some reason to disregard O-O coupling. It is also important to keep in mind that the O-O bond strength is significantly overestimated with GGA-level DFT.72 It is unclear how much of that error translates to the partial O-O bond formed in the transition state for O-O coupling, but 27 ACS Paragon Plus Environment


considering even a modest +0.2 eV correction causes O-O coupling to be slower than OOH* formation over the entire range of conditions shown in Figure 9.

Figure 9. Comparison of two different mechanisms for O-O bond formation: OOH* formation via water from the electrolyte (electrochemical) and thermal coupling of two CUS oxygen atoms. The dominant mechanism is shown as a function of potential and descriptor value, ΔGO –ΔGOH.

We also acknowledge that the O-O coupling barrier is likely quite structure dependent (e.g. dependent on the distance between oxygen sites). CUS oxygen at rutile (110) sites are bound to edge-sharing octahedra, which results in a significantly shorter distance compared to corner sharing octahedra at e.g. perovskite (100) surfaces. Thus, it is likely that such a coupling mechanism between singly-coordinated oxygens is inaccessible for other surfaces. We have not considered the coupling of bridge and CUS oxygen at the rutile (110) surface in this analysis in large part because we find that O2 is significantly more strongly bound at bridge sites compared to CUS, preventing its desorption (see Table S3). With the goal of making a more generalizable model and in light of potential errors in describing the transition state for O-O coupling, we will continue our analysis considering only OOH* formation as the RDS, but we acknowledge


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a more detailed study that focuses on OER kinetics via O-O coupling at various oxide surfaces would be of great interest.

Characteristics of a kinetic model limited by OOH* formation In the following, we will take a closer look at the implications of considering OOH* formation as the primary RDS for OER. The kinetic model represented by eqn. (22) consists of three continuous variables: rate (or current density), potential, and thermochemical descriptor (ΔGO –ΔGOH) and three parameters: α, β, and ξ, from the transition state scaling for OOH* formation. First, we will look at potential as a function of ΔGO –ΔGOH at different current densities (i.e. a volcano plot), and next we will look at potential as a function of current density for different values of ΔGO –ΔGOH (i.e. a tafel plot). Our goals are to highlight important features of the model that can be compared with experimental observations and to simply explain the sensitivity of the results to the three model parameters in light of the uncertainty associated with the OOH* barrier calculations. We emphasize that the goal is not to compare our rates of reaction directly to experiment for the handful of rutile (110) surfaces that we have performed calculations of and for which reliable experiments with well-defined active sites do not exist. We believe it is reasonably justified to extend our analysis beyond rutile (110) surfaces as the considered mechanism proceeds through singly-coordinated oxygen atoms and thus is not expected to exhibit a significant structural dependence.

Volcano analysis Shown in Figure 10 are volcano curves that relate potential to ΔGO –ΔGOH for various current densities. These curves must be generated through numerical solution of eqn. (22) because it is not



Page 30 of 60

possible to analytically solve for U. Although we have only a single rate determining step, a typical volcano shape emerges due to the tradeoff between thermodynamics (coverage) and kinetics (rate constant). Over the range of descriptor values considered, increasing the descriptor value, ΔGO –ΔGOH, always results in an improved rate constant for OOH* formation. However, increasing ΔGO –ΔGOH also shifts the surface equilibrium towards OH*, which may have a negative impact on the overall OER rate through the coverage term. For surfaces with KO/OH >> 1 (equivalently, ΔGO –ΔGOH/e < URHE), the oxygen coverage is approximately unity and an increase in descriptor value has a positive impact on the rate by only influencing the rate constant. Increasing the descriptor value too far, however, will eventually result in KO/OH 0 is not actually of any practical interest as the transition state is a transient species. Therefore, in the event that the equality in eqn (30) does not actually hold, the correct capacitance to use is the one based on κTS/FS. However, the existence of eqn (30) significantly reduces the number of NEB calculations required to estimate the capacitance. In the language of cell extrapolation, estimating κTS/TS requires only a single cell extrapolation, whereas estimating κTS/FS requires multiple cell extrapolations at different values of θFS. *

†

For example, E(0,0), E(1,0), E(½, 0), which require NEB calculations in two different sized cells.


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Thus, we see that the non-differential forward barrier (finite cell) is equivalent to the differential forward barrier (infinite cell, eqn (28)) at the average transition state coverage,

EaPBC (TS , FS ) = Ea (TS , FS ) for  TS =  TS

(34)

or equivalently with the assumption of eqn (30), at the average potential of the chosen initial and transition states

EaPBC (U ) = Ea (U ) for U = U

(35)

This allows us to directly fit the slope and intercept of eqn (31) using non-differential barriers from DFT calculations by treating them as differential barriers at the average potential of the initial and transition states of the NEB. Fitting this linear function requires at minimum two barrier calculations in different cell sizes. The aforementioned charge extrapolation scheme yields an estimate for the capacitance, C, (and hence the slope of eqn (31)) from the Bader charges of the initial and transition states in a single cell size, thus requiring only a single barrier calculation to fully specify eqn (31). We do not employ that scheme in this work because we find the assignment of atomic Bader charges in the transition state to belong to either the surface or electrolyte is ambiguous in this case and has a significant impact on the resulting potential dependence of the barrier (see Figure S4). The above model can be extended include interactions arising from coadsorbed species, which will be generally referred to as θsurf. Thus, the system energy, E(θTS, θFS, θsurf), and potential, U(θTS, θFS, θsurf), are now considered a function of the surface coverage as well. The equality in eqn (30) can be analogously extended to include the “capacitance” due to the interaction between transition states and surface coverage



C

 TS/TS  U     TS 

2

=

 TS/FS  TS/surf = .  U  U   U  U          TS   FS    TS  surf 

Page 46 of 60

(36)

Once again, this is the necessary and sufficient condition for expressing the differential forward barrier strictly as a linear function of potential, i.e. eqn (31). In other words, if differential barriers calculated for different θTS and different θsurf fall on the same line when plotted against potential, the fulfillment of eqn. (36) is implied, as shown in Figures Figure 3 and S3.

Supporting Information •

Document containing supporting figures and raw data tables as referenced in the main text

•

Zip file containing all relevant atomic structures in either ASE-trajectory or cube formats

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