Epitaxial Core‐Shell Oxide Nanoparticles: First‐Principles Evidence for Increased Activity and Stability of Rutile Catalysts for Acidic Oxygen Evolution

Abstract Due to their high activity and favorable stability in acidic electrolytes, Ir and Ru oxides are primary catalysts for the oxygen evolution reaction (OER) in proton‐exchange membrane (PEM) electrolyzers. For a future large‐scale application, core‐shell nanoparticles are an appealing route to minimize the demand for these precious oxides. Here, we employ first‐principles density‐functional theory (DFT) and ab initio thermodynamics to assess the feasibility of encapsulating a cheap rutile‐structured TiO2 core with coherent, monolayer‐thin IrO2 or RuO2 films. Resulting from a strong directional dependence of adhesion and strain, a wetting tendency is only obtained for some low‐index facets under typical gas‐phase synthesis conditions. Thermodynamic stability in particular of lattice‐matched RuO2 films is instead indicated for more oxidizing conditions. Intriguingly, the calculations also predict an enhanced activity and stability of such epitaxial RuO2/TiO2 core‐shell particles under OER operation.


Introduction
Core-shell nanoparticle morphologies are a powerful and frequently pursued concept in heterogeneous catalysis to reduce the demand of precious active materials. Protonexchange membrane (PEM) water electrolysis [1] is an eminent application area for this concept. Oxides containing rare Ir and Ru are currently the primary anode electrocatalysts for the oxygen evolution reaction (OER) that exhibit both a reasonably small overpotential and sufficient stability under the harsh acidic PEM operating conditions. [2,3] Despite the already high efficiency of current generation catalysts, significant further reduction of Ir or Ru mass loading is required when considering that for a prospective hydrogen economy gigantic amounts of electrolysis power will be required. [3,4] Within the core-shell concept, large research efforts have therefore been undertaken towards dispersing the precious active oxides on a variety of inexpensive core materials comprising abundant metals, their nitrides, carbides or oxides. [5][6][7][8][9][10] Generally, though, these have been massively loaded composites with incoherent thick Ir or Ru oxide films or small nanoparticles that use the core material more like a high surface-area support. As one example we highlight IrO 2 dispersed on TiO 2 , [11][12][13] as has also already been commercialized in form of the recent Elyst Ir75 0480 catalyst from Umicore. [13][14][15][16][17] Titanium dioxide exhibits a stable rutile modification. This motivates the idea to instead pursue epitaxial core-shell nanoparticles with thin coherent films of the equally rutile-structured IrO 2 or RuO 2 enclosing the cheap core material. In this study we explore this idea with detailed first-principles calculations. Analyzing adhesion, strain and surface energies, we show that prevailing gas-phase synthesis protocols will only be able to stabilize thin films in the few-monolayer regime at some lowindex facets of TiO 2 for both IrO 2 and RuO 2 . This rationalizes in particular the experimentally observed poor wetting behavior of IrO 2 at the prevalent (110) facet of rutile TiO 2 nanoparticles. [18,19] Under more oxidizing synthesis conditions, growth of coherent shell films should instead be feasible. Corresponding epitaxial core-shell particles would obviously minimize the precious metal demand. However, most intriguingly, our ab initio thermodynamics based results additionally indicate an increased stability of such particles under OER operation conditions, as well as an increased activity. At enhanced stability, increased activity and minimized precious metal content, this suggests epitaxial rutile IrO 2 /TiO 2 or RuO 2 / TiO 2 core-shell nanoparticles as a promising target for future synthesis or advanced deposition endeavors. In the case of the (111) facet, there are two possible stoichiometric interfaces denoted as t1 and t2, see Figure S1 in the Supporting Information for a description of all interfacial geometries. In the calculation of g hkl ð Þ interf according to equation (6) we consistently use the optimized TiO 2 bulk lattice constants in the directions parallel to the interface for all three solid-state terms which fixes the value of A hkl ð Þ . This effectively filters out the increasing strain in a coherent MO 2 film of increasing thickness and allows to arrive at a purely interface specific quantity that reflects the intrinsic cost of creating the interface. Compared to other heterostructures [20,21] all calculated interface formation energies are very small (about one order of magnitude smaller than the surface free energies discussed below) and consistently below 30 meV Å À 2 . This shows the expected propensity to form such interfaces between the lattice-matched oxides. [22] Simultaneously, we always obtain positive values. Bonding between the two materials is thus less favorable than the bonding within the pure materials, and there is not energetic driving force for interdiffusion. This is consistent with experiments reporting an (entropically driven) solid solution of IrO 2 in TiO 2 only at temperatures above 900°C. [23] As complementary key quantities, the negative W hkl ð Þ adh in Table 1 indicate the energy it would cost to separate the formed interface. Also here, using the optimized TiO 2 bulk lattice constants in the directions parallel to the interface for all terms entering Equation (7) yields an interface specific quantity that is independent of the thickness of the shell layer. The large negative values obtained for both IrO 2 /TiO 2 and RuO 2 /TiO 2 designate a strong intrinsic adhesion, with the same trend over the five low-index orientations found for both materials: (111) toughest to break and (010)/(100) and (110) also allows to trace the consistently stronger adhesion found for the IrO 2 /TiO 2 interface to the higher respective surface free energies of IrO 2 as compared to RuO 2 (see below). Note that at an almost equal cohesive energy of both oxides (computed as À 16.95 and À 17.16 eV per MO 2 unit at the present DFT Perdew-Burke-Ernzerhof (PBE) functional [63] level, respectively), this difference arises predominantly from a higher polarizability of the larger Ir ion. [24] Consistently offering smaller interface formation energies and stronger adhesion, the analysis up to now would suggest IrO 2 as preferred shell material. However, the focus on the interface specific quantities g hkl ð Þ interf and W hkl ð Þ adh disregards the increasing strain that will build up in pseudomorphic shell layers of increasing thickness. To this end, it is important to realize that the lattice mismatch of RuO 2 À TiO 2 and IrO 2 À TiO 2 is quite different for the two main rutile bulk lattice constants, a and c. Along the longer a axis all three materials exhibit almost identical values. With the employed PBE functional this is a TiO 2 = 4.578 Å and shorter lattice constants by only À 1.1 % for both RuO 2 and IrO 2 . In contrast, along the shorter c axis, this mismatch is larger. Specifically, at PBE level c TiO 2 = 2.955 Å, while RuO 2 and IrO 2 prefer a longer lattice constant by 5.6 % and 7.6 %, respectively. As a consequence of this anisotropy, hardly any strain will build up at the three interface orientations (001), (011)/(101) and (111), whereas much higher strain will build up at the two other orientations (110) and (010)/(100), see the Supporting Information for a detailed account of all values. While the larger strain for IrO 2 might (partially) scotch the intrinsic advantage of this material in terms of the afore discussed interface specific quantities, we note that much more problematic is the fact that precisely those two orientations that predominantly suffer from strain are those two that offer the weakest intrinsic adhesion anyway, cf. Table 1. This already indicates a high directional dependence of potential core-shell concepts.

Surface effects
For the actual realization of epitaxial core-shell nanoparticles, not only the interface matters. While growth itself is kinetics, there will always be a thermodynamic driving force to expose the material with the lower surface free energy at the shell. To this end, Figure 1 summarizes the calculated surface free energies of all three oxides, IrO 2 , RuO 2 and TiO 2 , in an oxygen environment. Equivalent behavior and trends are obtained for the surface free energies in an aqueous environment, [29] cf. the Supporting Information and below, so that the following analysis conceptually also extends to electrodeposition. Explicitly marked in Figure 1 are the synthesis conditions employed by Lee et al. (1 bar, 500°C) [28] to represent typical gas-phase synthesis endeavors. For the corresponding range of O chemical potential, Figure 1 immediately reveals a key bottleneck. The surface free energies of IrO 2 and RuO 2 are generally not smaller than those of TiO 2 , and in particular for the two orientations (010)/(100) and (110) that were already identified as problematic in terms of their interface stability and strain, TiO 2 exhibits even significantly lower respective g hkl ð Þ;s surf than the intended shell oxides.
All aspects, surface, strain and interface, can be combined in a rough estimate of the wetting tendency. As illustrated in Figure 2, this estimate takes the surface free energy g hkl ð Þ;s surf 1 ML MO 2 =TiO 2 ; 100 % ð Þ, cf. Equation (9), of a model where a pseudomorphic one monolayer (ML) film of the shell material completely covers the TiO 2 (hkl) surface, and compares it to the surface free energy estimates  [25][26][27] The vertical gray dotted lines indicate the thermodynamic bulk oxide stability (O-poor limit) as computed from the bulk heat of formation (for TiO 2 this limit is at À 3.76 eV outside of the shown range). In the top x axis, the dependence on Δμ O is translated into a temperature scale at different oxygen pressures. The black vertical dotted line in the phase diagram for IrO 2 and RuO 2 represent the synthesis conditions as employed by Lee et al. [28]  In all cases, the most stable surface terminations σ at the experimental gas-phase synthesis conditions of Lee et al. [28] were assumed, in accordance with the surface phase diagrams presented in Figure 1.  (2) and (3) thus evaluate is the cost, when the same total amount of shell material is not homogeneously dispersed on the entire surface as in g hkl ð Þ;s surf 1 ML MO 2 =TiO 2 ; 100 % ð Þ, but instead forms 2 or 3 ML high islands, respectively, with the remainder of the TiO 2 surface area uncovered. The simple linear superposition in Equations (2) and (3) Þ of the fully dispersed monolayer to be lower than the surface energies of the two competing island models. Figure 2 compiles the corresponding data for both IrO 2 and RuO 2 as shell materials and using the most stable surface terminations under the synthesis conditions by Lee et al., [28] cf. Figure 1. Not surprisingly, a wetting tendency is only obtained for the (001) and (111) orientation. These are the orientations with strongest adhesion, minimum strain penalty and comparable or lower MO 2 surface energies than TiO 2 . The latter lower surface energies result in fact from a qualitatively different behavior of the intended shell oxides that provides an important lead to future synthesis endeavors. Over the entire range of oxygen chemical potential shown in Figure 1, the stoichiometric termination is the most stable termination for all TiO 2 facets and their surface free energies correspondingly remain constant. In contrast, both shell MO 2 are able to stabilize O-rich terminations so that their surface free energies decrease with increasing Δμ O . The (001) and (111) facets are able to stabilize such terminations already at lowest oxygen chemical potentials, which is why their surface free energies are already quite low for the synthesis conditions of Lee et al., cf. Figure 1. However, all other facets will eventually also stabilize such terminations, which is why more favorable wetting will generally result for increasingly O-rich conditions where the MO 2 surface free energies will continuously decrease.
Unfortunately, for gas-phase synthesis such conditions are harder to obtain. Most straightforwardly, they would be realized by lowering the temperature, cf. the temperature scales in Figure 1, but then kinetic limitations will increase. Correspondingly, growth has typically been attempted for even less O-rich conditions than the ones by Lee et al. [28] For instance, in Ref. [19], Abb et al. even used a low oxygen partial pressure of 10 À 7 bar at 700 K. In full agreement with the understanding derived from Figure 1, they found IrO 2 (110) thin films at TiO 2 (110) not to be stable under such conditions. Elevated pressures might instead be a route to achieve more favorable O-rich conditions in gas-phase synthesis. However, in light of our results we believe electrodeposition or advanced atomic layer deposition to be more promising routes with easier access to oxidizing conditions. [30,31] We illustrate this in Figure 3 with data for the wetting model as in Figure 2, but now computed in an aqueous environment and using the most stable surface terminations in Equation (8) that result at an applied potential corresponding to the OER equilibrium potential (see below). Under these more oxidizing conditions, a much more favorable wetting tendency is obtained. In particular for RuO 2 /TiO 2 , all but the (110) orientation now exhibit a clear preference for wetting. While the (110) orientation is thus certainly the most difficult, we stress that the simple estimates for the 2 or 3 ML island models in Figure 3 disregard any additional costs from the island edges. The corresponding surface free energies should thus be seen as lower bounds to the true surface free energies, and a wetting tendency is almost obtained already when comparing against these lower bounds (see the essentially flat orange line for the (110) facet in Figure 3). In this respect, we tentatively conclude from the present thermodynamic data that growth of epitaxial core-shell particles should be feasible at sufficiently O-rich conditions. Importantly, no band gap opening is found in our calculations even for only 1 ML thick films and for all five low-index facets. This suggests that corresponding particles would also exhibit sufficient electronic conductivity as required for electrocatalytic performance.

Enhanced stability and activity
Ir and Ru oxides are the primary current OER electrocatalysts. However, even they are known to degrade in the harsh acidic PEM operation conditions, involving a hitherto only incompletely characterized transformation to some amorphous hydrous state. [2,16,[32][33][34][35][36][37][38][39] In previous work for IrO 2 , [29] we had established a simple thermodynamic descriptor for this degradation. At applied potentials in the OER regime the surface free energies were found to turn negative, indicating a potential thermodynamic instability of the rutile crystal lattice. We here obtain fully analogous results also for RuO 2 and summarize these findings in Figure 4. As detailed in the Supporting Information, more than 100 terminations σ for all five low-index facets have been computed to systematically consider the different possibilities to adsorb O, H, OH, OH 2 , OOH, and OO species in (1 × 1) and (1 × 2) surface unit-cells. Figure 4 shows the surface free energies of the resulting most stable terminations as a function of the applied potential from open-circuit (0 V vs. RHE) to conditions relevant for technological PEM operation U > 1.3 V. The corresponding values at the OER equilibrium potential U = 1.23 V were also used in the wetting model in Figure 3 above. As in the analogous Figure 1 for gasphase conditions, at each facet different terminations become most stable with increasing potential, reflected in Figure 4 by a changing slope of the g hkl ð Þ;s surf line. Generally, we find the expected sequence from fully hydrated or hydroxylated surfaces at open-circuit conditions to gradually deprotonated terminations at increasing potentials until pure O-terminations and eventually terminations with higher oxidized superoxo species become most stable at OER-relevant potentials. A detailed account of these findings and their very good consistency with existing theoretical and experimental data in particular for the best characterized (110) facet is provided in Figure S3 in the Supporting Information.
More relevant for the present context is the pronounced decrease of the surface free energies with increasing potentials. As such, it is in particular g 111 ð Þ;s surf of the (111) facet that first turns negative already at a critical potential U RuO 2 111 ð Þ critical = 1.27 V, that is, just at the OER onset. This reflects a potential thermodynamic instability, as the particle could gain energy by decaying and generating more such surfaces. With the current computational settings and pursuing the exact same approach, for IrO 2 this critical potential where g 111 ð Þ;s surf turns negative is computed as U IrO 2 111 ð Þ critical = 1.39 V. The concomitantly indicated thermodynamic stability up to higher applied potentials is fully consistent with the well-established better corrosion resistance of this material [28,[40][41][42][43][44] and further confirms U critical as one of the useful thermodynamic descriptors for evaluating the catalyst stability. Intriguingly, when we compute exactly the same set of surface free energies for a pseudomorphic 2 ML film of RuO 2 on TiO 2 , we find the decreasing surface free energies of all facets to be shifted toward higher potentials, and thereby obtain higher U critical where the surface energies turn negative, cf. Figure 4. In particular, for the (111) facet which is still the one turning negative first, the critical potential is now increased to U 2ML RuO 2 =TiO 2 111 ð Þ critical = 1.36 V, that is, 0.1 V higher than for the pure RuO 2 particle and close to the corresponding facet of pure IrO 2 . For other facets, this relative increase is up to 0.2 V as detailed in Table S4 in the Supporting Information. According to this thermodynamic descriptor the core-shell RuO 2 /TiO 2 particle thus exhibits trends of an increased stability, which is for all facets now essentially the same or better than for a pure IrO 2 particle.
This striking finding begs the question for its physical origin. Numerically, the shifted U critical values result from increased surface free energies of the core-shell particle compared to those of the native shell oxide, cf. Figure 4. These higher values arise in turn from the additional interface formation energy and strain. Obviously, the shell oxide is not in its optimum state, and if it were for the shell oxide alone, it would be thermodynamically preferable to form a relaxed MO 2 particle. However, for the entire system a corresponding dewetting or other strain relieving alternatives like extended defects are not competitive as they would necessarily involve the formation of more surface area or expose TiO 2 with its higher surface free energy at these OER operation conditions. For thicker films this will be changed by the accumulating strain. However, for few ML epitaxial films the increased stability of once formed core-shell particles results simply out of lack of alternatives and this is how the thermodynamic descriptor should be read.
This cautious reading extends also to another interesting aspect reflected by the calculated surface free energies in Figure 4. The strong decrease in particular of g 111 ð Þ;s surf with increasing potential as compared to the surface free energies of the other facets shows that it becomes increasingly favorable to form (111) facets. Indeed, combining the fully analogous results for native IrO 2 within a Wulff construction, we had shown in previous work that the thermodynamically preferred particle shape at OER operation conditions would be one that exclusively exhibits (111) facets. [29] Here, the data summarized in Figure 4 indicates exactly the same thermodynamic driving force to reshape RuO 2 particles and epitaxial RuO 2 /TiO 2 particles away from the familiar rutile form with its predominant (110) facets. [29] However, in particular for the core-shell particle it is uncertain in how much such thermodynamics really applies. To one end, the mechanical hardness of the TiO 2 core may lead to a very slow kinetics. Additionally, as long as the shell is intact, the TiO 2 core does not contact the electrolyte and will at least be partially screened from the applied potential. It could thus well be that epitaxial core-shell particles show an increased resistance to the driving force to reshape -and concomitantly against degradation, either in terms of general mass loss or the fraction of exposed (111) facets with their particularly low U critical , cf. Figure 4.
We note in passing that CHE is a simplified theoretical approach and that a negative surface energy resulting from such a treatment is not a sound indicator of what detailed physiochemical processes ensue, but merely one readily available stability descriptor which can be used to gauge trends between similar systems. This descriptor neither accounts for the chemical properties of the decomposition products nor for mass transport effects as has been pointed out recently. [39] Such influences can be treated approximately by simple models based on the Poisson-Nernst-Planck equations which, to lowest order, will induce shifts in the phase diagrams. [45] For the highly similar systems considered here, we assume that these shifts are approximately constant over the range of systems studied and that our current approach delivers an acceptable semi-quantitative estimate of which structural modifications might improve the relative stability of the catalyst within its class.
According to our calculations, epitaxial RuO 2 À TiO 2 core-shell particles would not only exhibit an improved stability, but also catalytic activity. Figure 5 demonstrates this by comparing the computed reaction energetics for the prevalent (110) facet for pristine RuO 2 and for 2 ML RuO 2 À TiO 2 , as well as for RuO 2 that is equally strained as in the epitaxial core-shell system. Specifically, this is the reaction energetics along the classic OER peroxide pathway suggested by Rossmeisl et al.: [42] Here, we note that alternative mechanistic pathways, such as the lattice oxygen mechanism, have recently been investigated. [46,47] However, for a simple estimate of the relative activity of closely related catalyst systems, the energetic comparison of reaction intermediates within the conventional OER mechanism on mechanically (un)strained materials can still serve as a suitable descriptor. Fully consistent with previous work on RuO 2 (110), [48,49] these energetics reveal the fourth, oxygen evolution step as the potential-determining one, requiring the largest minimum thermodynamic overpotential η TD = 0.44 V to make all reaction steps exergonic. In contrast, we compute a much lower overpotential of only η TD = 0.26 V for the coherent 2 ML film of RuO 2 on TiO 2 . This reduction results exclusively from the strain imposed on RuO 2 in the core-shell system, as demonstrated by the almost identical overpotential obtained for an equally strained RuO 2 (110) surface, cf. Figure 5. This strain effectively weakens the O binding that is too strong for the native RuO 2 (110) and that is correspondingly brought closer to the optimum value at the top of the volcano in the epitaxial core-shell system. [42] Concomitant with this weakening, we also obtain a shift in the potential-determining step to the final O 2 evolution, which would then also affect the Tafel slope for such particles. To demonstrate this, we compute the Tafel slope b for pristine RuO 2 (110) and the related core-shell system. It is governed by the apparent transfer coefficient β, which specifies the number of electrons transferred counting from the catalyst resting state to the transition state in the free energy diagram [49,50] (see the Supporting Information for details): As a result, the computed Tafel slope b of RuO 2 (110) is 39.3 mV dec À 1 at U = 1.23 V. However, by increasing U beyond 1.57 V, the resting state changes from the surface oxo (*O) to the hydroperoxo (*OOH) intermediate, corresponding to a shift of the transition state from the second to the nearest neighbor position in the free energy profile, resulting in a Tafel slope of 118 mV dec À 1 (see Figure S4). This fully coincides with experimental data under OER conditions. [51] For the core-shell system, the resting state is given by the surface hydroxo (*OH) intermediate at the OER equilibrium potential (η = 0). The transition state is the third electron transfer step following the (*OH) state. The Tafel slope thus changes twice with increasing U, from b = 23.6 to 39.3 and finally to 118 mV dec À 1 by consecutively moving the transition and resting states closer to each other (see Figure S5). Intriguingly, the transition bias between the two Tafel regimes (39.3 and 118 mV dec À 1 ) has been lowered for the core-shell system by more than 0.2 V from U = 1.57 to 1.35 V. This phenomenon of lowering the transition bias may provide a validation and a guiding principle for the successful design of such core-shell systems: the lower the transition bias, the better one achieves the desired core-shell structure. As a further activity descriptor, we adopt the electrochemical-step symmetry index (ESSI), which was proposed as a measure to quantitatively assess how close a catalyst approaches the catalytically ideal free energy profile: [52][53][54] Here, DG þ i are the reaction energies of those steps in the OER peroxide pathway that are larger than 1.23 eV and E 0 = 1.23 V is the OER equilibrium potential. For pristine RuO 2 the relevant DG þ i are the third and fourth steps in Figure 5 (left), that is, SOOH formation and O 2 evolution, and the corresponding ESSI descriptor becomes 0.391 V. For the core-shell system, the ESSI descriptor yields 0.153 V which is much smaller and indicates that it is significantly closer to the ideal catalyst than pristine RuO 2 . The reduction is, as discussed above, due to the strain induced destabilization of the O binding in the *O state. The equally strained RuO 2 has an intermediate ESSI value of 0.226 V. In spite of the nearly identical overpotentials η TD of the core-shell and the strained RuO 2 system, this large ESSI change can simply be attributed to a change in the number of terms n contributing to the average in Equation (5). The second reaction step at the strained RuO 2 is slightly below (by À 0.02 eV) the threshold of 1.23 eV while it is slightly above (+ 0.08 eV) for the core-shell system. Such large deviations in the ESSI descriptor by a threshold induced discretization error for the average should be taken with a grain of salt based on typical DFT errors. The simple η TD estimate is more robust for such cases. As shown in Figure 5 (right), we can generally conclude though that the RuO 2 /TiO 2 core-shell system, especially for the most relevant (011)/(101), (110) and (111) surfaces, could provide a promising RuO 2 -based OER electrode as the free-energy change of the second reaction step ranges between 1.03 and 1.43 eV and it exhibits ESSI values below 0.45 V. [53] Conclusions Systematically analyzing the interfacial stability, strain and surface free energies, our first-principles calculations predict a general thermodynamic feasibility of epitaxial core-shell particles, in which an ultrathin film of RuO 2 coherently encapsulates a rutile-structured TiO 2 core. A high directional dependence of adhesion and lattice mismatch disfavors in particular the (110) orientation, which suffers from maximum strain and offers the smallest density of interfacial bonds. While the same trend with regards to direction is obtained for IrO 2 , its intrinsically higher surface energies render this oxide a less suitable material for such core-shell concepts.
A key aspect for the practical realization is the ability of both studied MO 2 to stabilize O-rich surface terminations at more oxidizing conditions. This increasingly lowers their surface free energies relative to TiO 2 , for which only stoichiometric terminations are found as most stable up to high oxygen chemical potentials. Generally favoring wetting, this points towards growth protocols operating at oxidizing conditions, as for example achievable by electrodeposition. Apart from reducing the precious oxide content, our calculations further indicate two additional benefits of corresponding pseudomorphic core-shell particles. On the one end, an increased stability at OER operation conditions results as the implied higher surface area and/or exposure of TiO 2 penalizes a dewetting or strain-relieving formation of extended defects. On the other end, the additional strain in particular of the prevalent (110) facet leads to a lowered thermodynamic overpotential for few monolayer, coherent RuO 2 films at rutile TiO 2 . We hope that these insights will stimulate advanced growth endeavors to overcome the presently realized massively loaded composites with their incoherent thick, precious oxide films towards tailored epitaxial core-shell particles with a predicted increased stability and activity at simultaneously minimized mass loading.

Methods
In order to assess the interfacial stability we use ab initio thermodynamics [55] to calculate two general key quantities, the interface formation energy g hkl ð Þ interf and the work of adhesion W hkl ð Þ adh of a stoichiometric interface with crystallographic orientation (hkl). The prior indicates the cost of creating the interface from the respective bulk materials and is defined as Here, A (hkl) is the area of the interface model employed in the calculations, G hkl ð Þ interf its Gibbs free energy, n hkl ð Þ Ti its number of Ti atoms, and n hkl ð Þ M its number of M=Ir or Ru atoms. G TiO 2 ,bulk is the Gibbs free energy per TiO 2 formula unit of a rutile bulk unitcell and G MO 2 ,bulk correspondingly the Gibbs free energy per MO 2 formula unit of a rutile bulk unit-cell. The work of adhesion is the reversible work required to separate the interface into two free surfaces in vacuum where G hkl ð Þ TiO 2 ;surf and G hkl ð Þ MO 2 ;surf are the Gibbs free energies of surface models of TiO 2 and MO 2 , respectively, exhibiting the (hkl) facet and the same stoichiometric surface termination as the interface model.
In analogy to g hkl accurate electronic structure of the metal-insulator interface system, Hubbard-corrected DFT has been applied to the Ti 3d states using an effective on-site parameter of 4.5 eV. [67] Double counting in this DFT + U approach has been treated in the fully localized limit. [67] A detailed account of the computational settings employed in the DFT calculations is provided in the Supporting Information.