Stability and mobility of supported Ni n ( n ¼ 1 – 10) clusters on ZrO 2 (111) and YSZ(111) surfaces: a density functional theory study †

The performance of supported metal catalysts, such as nickel nanoparticles decorating yttria-stabilized zirconia (YSZ), depends on their microstructure and the metal – support interface. Here, we have used spin polarized density functional theory (DFT) to evaluate di ﬀ erent Ni cluster geometries and determined the electronic structure of the most stable con ﬁ gurations. We have described the interaction of Ni n ( n ¼ 1 – 10) clusters supported on the cubic ZrO 2 (111) and YSZ(111) surfaces, which show a preference for pyramidal shapes rather than ﬂ at structures wetting the surface. The interfacial interaction is characterized by charge transfer from the cluster to the surface. We also show how yttrium, present in YSZ, a ﬀ ects the Ni – Ni interaction. Through analysing the di ﬀ erence between the cohesive energy and the clustering energy, we show the preference of Ni – Ni bond formation over Ni-surface interaction; this energy di ﬀ erence decreases with the increase of the Ni n cluster size. From the evaluation of the Ni atomic hopping rates on YSZ, we have demonstrated that under di ﬀ erent temperature conditions, Ni atoms aggregate with other atoms and clusters, which a ﬀ ects the cluster size stability. ), the di ﬀ erence between the clustering energy and the cohesive energy ( coh (cid:3) E ) and the interaction energy ( E int ) as a function of the cluster size on top of both ZrO 2 (111) and YSZ(111) surfaces.


Introduction
There is a growing interest in the stabilities and properties of supported clusters and metal-oxide interfaces, since many industrial processes use oxide-supported metal particles as catalysts, i.e. in the water gas shi reaction, 1 steam reforming processes, 2,3 microelectronics, sensors and solid oxide fuel cells (SOFC). [4][5][6] These processes depend, among other things, on the particle sizes and the particlesupport interactions. [7][8][9] For instance, the performance of the electrode in SOFCselectrochemical devices that convert chemical energy into electrical energydepends on the microstructure and the size distribution of the metal particles and the oxide phase in the cermet 10 (ceramic matrix containing metal nanoparticles). A common electrode in SOFCs is metallic nickel, used as an electron conductor, supported on an oxygen conductor, usually an oxide. For instance, Ni supported on zirconia (ZrO 2 ) doped with yttria (Y 2 O 3 ) (Ni/YSZ), is a suitable material for SOFCs as it has a high mixed electronic-ionic conductivity and high mechanical strength. 11 The key reactions in these devices occur at the triple phase boundary (TPB) where the gas phase, Ni particles and the YSZ surface meet. Therefore, tuning and stabilizing the Ni microstructure will affect the SOFC performance and electrode lifetime. 12 For example, the performance decay of the SOFC is associated with the sintering of Ni particles through atomic or cluster diffusion across the oxide support. 13 While atomic-level information on the metal-oxide interface is difficult to obtain experimentally, computational tools facilitate the evaluation of individual factors and thereby can provide crucial insight. 14,15 For example, previous density functional theory (DFT) calculations have been used to study the interaction of certain metals on g-Al 2 O 3 ; [16][17][18][19] Rajesh et al. studied the adsorption of Au n (n ¼ 2-10) on a-Al 2 O 3 (0001), showing favourable metal clustering rather than wetting i.e. spreading across the surface. 20 Di Valentin et al. investigated the stability and growth of Ni clusters on the MgO surface, 21 where small Ni n (n # 9) clusters adsorb weakly on the surface and their diffusion can be stopped by surface defects such as oxygen vacancies. Molina et al. 22 have shown that an Au 20 cluster keeps its tetrahedral geometry upon adsorption on the MgO(100) surface. Here, we have systematically analysed the interaction of Ni n (n ¼ 2-10) clusters on cubic zirconia (c-ZrO 2 )(111) and YSZ(111) surfaces in order to characterize the electronic structure of the interface and the cluster geometry as a function of their size, and to evaluate the stability and sintering rate of Ni clusters on both c-ZrO 2 (111) and YSZ(111) surfaces.

Models and computational methods
We have employed calculations based on density functional theory (DFT) 23 as implemented in the Vienna Ab initio Simulation Package (VASP), [24][25][26][27] where the generalized gradient approximation (GGA) 28 with the Perdew-Burke-Ernzerhof (PBE) density functional was used to approximate the exchange-correlation functional. Long-range dispersion interactions were described by using the semiempirical method of Grimme 29 and we have employed the projected augmented wave method (PAW) 30 to describe the interaction between the valence and the core electrons, where O (2s, 2p), Ni (3d, 4s), Zr (4d, 5s) and Y (4s, 4p, 4d, 5s) atomic orbitals have been treated as valence electrons. All the calculations were spin polarized. The unit cell of c-ZrO 2 has the uorite structure with space group Fm3m. We have converged the bulk energy for different k-points (from 7 Â 7 Â 7 to 11 Â 11 Â 11) and cutoff energies (from 350 eV to 550 eV) with a convergence criterion of 0.001 eV per atom. The bulk convergence was reached for a kinetic energy cutoff of 500 eV and a Monkhorst-Pack grid of 9 Â 9 Â 9 mesh of k-points. For the 1 Â 1 slab calculations, we have tried four sets of k-points (from 6 Â 6 Â 1 to 9 Â 9 Â 1) with the same convergence criterion and convergence was reached for a 7 Â 7 Â 1 mesh of k-points. For the 2 Â 2 slab calculations, three sets of kpoints were tried (3 Â 3 Â 1, 4 Â 4 Â 1, and 5 Â 5 Â 1) with convergence reached for the 3 Â 3 Â 1 mesh. The tetrahedron method with Blöchl corrections was used to dene the Brillouin zone.
The ZrO 2 (111) surface was obtained with the METADISE code, 31 which takes into account the atomic charges and the periodicity of the plane leading to a stacking of atomic layers with zero dipole moment perpendicular to the surface plane.
We have modelled the most stable zirconia surface, ZrO 2 (111) 32-35 ( Fig. 1(a) and (b)), considering a slab with nine periodic atomic layers (three O-Zr-O trilayers). The ve top atomic layers were relaxed during the optimization, while the four bottom layers were kept xed at their bulk equilibrium position to represent the rest of the crystal. The vacuum size between slabs was set to 15Å in order to minimise interactions with the perpendicular images.
In our previous investigation, 35 we determined the band gap to be 3.14 for the ZrO 2 (111) surface. Although the calculated band gap is in agreement with previous computational studies, 36,37 it is lower than the experimentally measured bulk values (5-6 eV), 38,39 as DFT calculations tend to underestimate the band gap.
We substituted two Zr with two Y atoms and removed one oxygen atom in order to obtain the yttria-stabilized zirconia (YSZ) surface with a dopant concentration of 9.09%, which is in the range of an optimized oxygen transport YSZ electrode (8- 10%). We have previously evaluated the energy of different non-equivalent congurations of the YSZ(111) surface and found that the most stable structure had the two yttrium atoms located in the top and sub O-Zr-O trilayers. 35 The oxygen vacancy is positioned in the lower oxygen layer of the top O-Zr-O trilayer and at the next nearest neighbour site (NNN) to the two Y atoms ( Fig. 1(c) and (d)).
There are many possible shapes and adsorption sites for Ni n clusters. We have chosen the Ni n geometries based on the stability of fcc Ni surfaces [40][41][42][43] and small particles 44 to form triangular based structures in agreement with previous studies. We have therefore grown clusters exposing the Ni(111)-facet 45,46 on the ZrO 2 (111) and YSZ(111) surfaces, where the initial atomic spacing between the Ni atoms is 2.2 A, similar to the interatomic distance in the Ni(111) surface. We have based the initial positions of our clusters on our previous ndings. 35 On the ZrO 2 (111) surface, the clusters interact with the oxygen atoms located in the top oxygen layer, O u ( Fig. 1(a) and (b)), which act as a trap for the Ni. On the YSZ(111) surface, Ni atoms adsorb preferentially on top of the oxygen vacancy, away from the Y atoms, 35 and hence, we have grown the Ni n (n ¼ 2-10) clusters away from the Y atoms.
We have evaluated the stability of the growing Ni cluster by using the clustering energy per atom (E clus ), eqn (1): where E Ni n /surf , E surf and E Ni are the energies of the slab with the Ni n cluster, the clean oxide surface and the Ni metal atom in the bulk, respectively, and n is the number of adsorbed Ni atoms. A positive clustering energy means that the Ni cluster growth is favourable. The evaluation of the cohesive energy (E coh ) (eqn (2)) allowed us to understand the interactions between the Ni atoms in the clusters: where E Ni n is the energy of the Ni n cluster with the same adsorbed geometry in vacuum (single point calculation). Finally, we have studied the perpendicular interaction between the surface and the Ni clusters by determining the interaction energy (E int ) (eqn (3)): where E 0 surf is the energy of the geometry of the clean oxide surface taken from the optimized Ni n /surf structure and n C is the number of Ni atoms in contact with the surface. The charge transfer between the surface and the molecules was analysed using Bader analysis as implemented in the Henkelman algorithm. 47 We also implemented the DFT results in kinetic Monte Carlo simulations to study the coalescence of Ni atoms and clusters on YSZ(111), thus providing information regarding coverage and time at specic temperatures.

Results and discussion
3.1. Structural analysis 3.1.1. Ni n /ZrO 2 (111). In our previous investigation, 48 we showed that Ni nanoclusters adopt a hexagonal arrangement similar to the shape of the (111)- facet of face-centred cubic metals. The clusters sit on the surfaces, optimizing the interactions with O u atoms and transferring some electrons to the surface. Based on these preliminary ndings, we have positioned up to Ni 10 by maximizing the interactions with O u surface atoms, where we have considered at and pyramidal morphologies. The Ni n (n ¼ 1-5) atoms prefer a 3D structure rather than wetting the surface with planar sheets. 48 For example, we have tried four initial congurations for the Ni 6 cluster on ZrO 2 (111) (Fig. S1 in the ESI †): two at and two in a three-dimensional shape. The most stable shape has ve Ni atoms at the base of the structure, see Fig. 2, maximising their interaction with the O u atoms and having an E clus of 1.51 eV. The average Ni-Ni distance in this cluster is 2.4Å, which is 0.2Å shorter than those in the Ni(111) surface. The optimized structure is different from the initial guess, where before relaxation the Ni atoms were positioned in a at distribution in contact with the surface. Aer relaxation, the atoms adopted a Ni(111)-facetted tetrahedron shape on the zirconia surface. Similar studies 20,49 have shown the preference of aggregation of metal clusters on oxide surfaces. For example, Zhang et al. 49 have shown that for n $ 4 Ru n adopts a 3-D geometry on TiO 2 (101), whereas Rajesh et al. 20 have shown that the aggregation of Au atoms is more favourable than wetting the Al 2 O 3 surface. The analysis of the interfacial distances in the Ni 6 /ZrO 2 (111) system shows that each Ni atom is at $1.9Å from its nearest O u neighbour. In addition, we note that these O u atoms are shied from their initial position: the O u -O u distance is 3.6Å in the clean ZrO 2 (111) surface, whereas it is approximately 4.1Å upon Ni 6 adsorption. The O u atoms are pushed apart in order to accommodate the cluster and minimise the forces on it. The adsorption of this cluster also affects the electronic structure of ZrO 2 . We have evaluated the atomic charges (Table 1) and observed a total charge transfer of 0.5 e from the metal cluster to the surface. The charge transfer is slightly higher for the Ni 9 and Ni 10 clusters: +0.6 and +0.7 e, respectively. This charge transfer causes an electronic rearrangement of the surface atoms interacting with the cluster: the negative charge of the O u atoms decreases from À1.2 to À1.1 e, while the charge of the Zr atoms (near the cluster) decreases from 2.3 to 2.1 e. We have also noted that only the ve Ni atoms directly bonded to the surface transfer charge to ZrO 2 (111), since each of those Ni atoms are +0.1 e while the Ni atom at the top of the cluster remains uncharged. Generally, Ni atoms at the vertex are either negatively charged (À0.1 e) or neutral, which indicates that the atoms at the base of the cluster transfer charge to the surface, while the low coordinated metal atoms, i.e. vertex and corners, accumulate electron density. 44 Thus, the Ni located at the vertex of the cluster could be a source of electrons for an eventual reaction with electron receptor molecules approaching the cluster from the gas phase. 50 Analysing the electron density difference between the cluster and the surface (Fig. 3), we have conrmed a charge accumulation between the Ni atoms directly bonded to the surface and the surface atoms; there is no charge relocation between the top Ni atom which remains fully metallic. We also note the well-localised accumulation of electron density between the O u and the Ni atoms, indicating the formation of a Ni-O bond. A previous report has also shown this electron rearrangement between Pt clusters and the zirconia surface. 51 Next, we have analysed the interaction of Ni 7 , Ni 8 , Ni 9 and Ni 10 with the ZrO 2 (111) surface where we have tried, respectively, ve, six, three and four nonequivalent initial congurations (Fig. S2 to S5 in the ESI †). The most stable shapes are shown in Fig. 2 and the calculated clustering energies are 1.47, 1.42, 1.39 and 1.31 eV, respectively ( Table 1). The four congurations have a similar pyramid shape and the only difference is the number of the Ni atoms in the top Ni layer of each cluster (Fig. 2).
3.1.2. Ni n /YSZ(111). In the YSZ system, we have again built several clusters with non-equivalent initial adsorption congurations, both three-dimensional and at, and adsorbed them on top of the YSZ(111) surface. A single Ni atom on top of the YSZ(111) surface sits on top of the oxygen vacancy, as far away as possible from the yttrium atom, with a clustering energy of 2.35 eV. 35 Table 1 Calculated clustering (E clus ), cohesive (E coh ) and perpendicular interaction (E int ) energies (in eV), and Ni n charges (e) of the most stable configuration of both Ni n /ZrO 2 (111) and Ni n /YSZ(111) systems. Ni n /ZrO 2 (111) (n ¼ 2-5) values are taken from ref. 48  We have optimised several initial congurations for the at Ni 2 /YSZ(111) and Ni 3 /YSZ(111) clusters ( Fig. S6 and S7 in the ESI †) and the most stable ones are shown in Fig. 4, with an average clustering energy of 2.21 eV (Table 1).
For Ni n (n ¼ 4-7), the average E clus is 1.76 eV and the clusters adopt a Ni(111) facet shape, in agreement with a previous study by Li et al. 14 who showed that the Ni 4 cluster prefers to adopt a similar pyramid shape on top of g-Al 2 O 3 . Similarly,  Carrasco et al. 52 have also found that the pyramid Ni 4 geometry is preferred over the planar one, when the cluster is adsorbed on the isostructural CeO 2 (111) surface. In the Ni n /YSZ(111) (n ¼ 2-7) congurations, the Ni-surface atomic spacing is 2.3Å, i.e. slightly larger than in the pure ZrO 2 (111) systems. In addition, each O u is pushed towards the neighbouring vacancy, since we observe an average decrease of 0.1Å of the O u -vacancy distance. This movement explains the preference of the Ni clusters to adsorb on this site: it is easier to drive the O u towards a vacancy where it tends to ll this defect. Moreover, the most stable adsorption site is the one involving two O u atoms with one neighbouring oxygen vacancy.
Charge analysis (Table 1) shows a slight charge transfer from the cluster to the surface of $+0.2 e. As was observed for the Ni n /ZrO 2 (111) systems, this charge transfer affects the electronic structure of the surface: the charges of Zr and Y atoms (near the metal cluster) decrease from 2.2 to 2.1 e and the negative charge of O u decreases from À1.2 to À1.1 e. We also observe charge accumulation on the Ni atom at the vertex of the pyramid of the Ni n /YSZ(111) (n ¼ 4-7) congurations, since this latter atom has a À0.1 e charge. The electron density difference plot (Fig. 5) shows this charge accumulation between the cluster and the surface and shows the localized orbitals of O u and Y atoms interacting with the cluster.
The at shapes of the Ni n /YSZ(111) (n ¼ 2-7) structures, shown in the ESI, † are less stable by $0.11 eV (which corresponds to a thermal energy of 638 K), i.e. the Ni atoms prefer to aggregate rather than spread over the surface, as was also observed for the Ni n /ZrO 2 (111) systems.
It is worth noting that for the Ni 8 /YSZ(111) cluster, the at and pyramid shapes are in equilibrium since their clustering energies differ by only 0.01 eV (58 K) ( Table 1). These two congurations are also the lowest energy ones from the six congurations considered (Fig. S16 and S17 in the ESI †). In the at conguration  ( Fig. S16 in the ESI †) the Ni 8 cluster interacts with six O u atoms at an average Ni-O u distance of 2.0Å. In addition, the oxygen vacancy is lled by one displaced O u which contributes to the stability of the system. Indeed, the movement of O u towards the vacancy allows the cluster to optimise its interaction with the other six O u surface atoms. This could be a drawback, for example in SOFCs, since the role of the oxygen vacancies is to enhance the oxygen transport. As for the threedimensional shape (Fig. 4), the cluster interacts with ve O u atoms but none of the oxygen vacancies are lled by an O u atom. This shape is similar to the one found for Ni 7 /YSZ(111), although the next Ni atom avoids interaction with Y from the surface and forms a (111)-facetted shape with the rest of the cluster. Bader charge analysis (Table 1) shows that the charge transfer from the cluster to the surface is slightly higher in the at conguration (+0.3 e) than in the pyramid shape (+0.2 e) owing to the former's interaction with an extra O u atom. This comparison shows the importance of the two parameters responsible for the stability of the cluster on top of the surface: the number of O u atoms interacting with the cluster and the shape of the cluster. The last cluster studied in this work is Ni 10 /YSZ(111), where three congurations have been considered ( Fig. S20 and S21 in the ESI †), and the most stable shape found is the pyramid one (Fig. 5). The calculated clustering energy is 1.58 eV (Table 1) which is similar to the value found for Ni 9 /YSZ(111). This is predictable since both Ni 9 /YSZ(111) and Ni 10 /YSZ(111) have exactly the same shape and the only difference is the tenth Ni atom located at the apex of the pyramid in Ni 10 /YSZ(111). Therefore, the modication of the surface geometry upon adsorption and the number of O u atoms involved in the interaction are similar to Ni 9 /YSZ(111). As to the Bader charge (Table 1), the cluster is +0.3 e charged and we note that the Ni atom located at the apex of the cluster is À0.2 e charged, while all the other Ni atoms are positively charged (average of +0.1 e). We also note the same variation of the charge of the surface atoms interacting with Ni 10 as was observed for Ni 9 /YSZ(111). Thus, the Ni atoms close to the surface transfer charge to the surface atoms and to the Ni atom located at the top of the cluster, which can be seen from the electron density difference plot (Fig. 5), where we have a gain of charge between the cluster and the surface.
In general, in Ni n /YSZ(111) (n ¼ 2-10), the Ni n clusters interact with O u atoms with an average distance of 1.9Å. The metal clusters transfer charge to the surface, depending on the size, ranging from +0.1 to +0.3 e. The threedimensional cluster shape is more favourable due to the repulsive interaction between Y and Ni.

Ni sintering on top of ZrO 2 (111) and YSZ(111)
We have shown that Ni n clusters on both ZrO 2 (111) and YSZ(111) surfaces prefer to adopt a three-dimensional structure rather than at shapes for clusters containing at least 4 atoms. The same conclusion was drawn for a similar system, CeO 2 -supported Au nanoparticles, 53 where it was shown that planar Au 13 on top of the CeO 2 surface is unstable compared to three-dimensional Au 13 . Pan et al. 54 have also shown that a Ni 4 cluster prefers to adopt a 3-D pyramid shape on top of the g-Al 2 O 3 (110) surface, with a large clustering energy. Giordano et al. 55 have also demonstrated that the Ni 4 cluster prefers to adopt a tetrahedron shape on top of the MgO(001) surface.
The total energies of four individual Ni atoms (4Ni) compared to the Ni 4 cluster adsorbed on the surface are 3.38 eV and 2.20 eV less stable on the ZrO 2 (111) and YSZ(111) surfaces, respectively, thus showing that aggregation of the Ni atoms is clearly preferred energetically over dispersion. We have also compared the total energy of two Ni 4 clusters, separated by approximately 6.0Å, with the Ni 8 /surface system and here we also found that (Ni 4 + Ni 4 ) is less stable than Ni 8 , now by 1.67 eV and 1.11 eV on ZrO 2 (111) and YSZ(111), respectively.
Furthermore, in Fig. 6(a) we have plotted the clustering energy (E clus ) as a function of the cluster size, for both ZrO 2 (111) and YSZ(111) surfaces. This graph shows a probable aggregation of Ni on both surfaces owing to a thermodynamic driving force as the cluster size increases. The trend in the clustering energy shows that for the same cluster size, the clustering energy is lower on ZrO 2 (111), in agreement with the two aggregation examples calculated for Ni 4 and Ni 8 clusters. The Y atoms affect the geometry of the surface and anion rearrangement, making the interaction between the clusters and the O u surface atoms less favourable, thus enhancing the preference for the formation of Ni clusters.
We have also evaluated the difference between the cohesive energy (E coh ) and the clustering energy (E clus ) as a function of the cluster size. The difference between those energies expresses the trend to form a Ni-Ni bond instead of a Nisurface bond. Fig. 6(b) shows that for both surfaces, this energy difference decreases with increasing cluster size due to the preference of the Ni-Ni interaction over Ni-surface interactions. The E coh À E clus graph also shows that for the same cluster size, the energy difference is lower for YSZ(111) than for ZrO 2 (111), Fig. 6 (a), (b) and (c) represent, respectively, the plot of the clustering energy (E clus ), the difference between the clustering energy and the cohesive energy (E coh À E clus ) and the interaction energy (E int ) as a function of the cluster size on top of both ZrO 2 (111) and YSZ(111) surfaces. indicating that the degree of interaction between Ni atoms is greater for Ni n clusters on top of YSZ(111). The interaction energy (E int ) calculated as a function of the cluster size ( Fig. 6(c)) conrms the affinity of the Ni n clusters for the ZrO 2 (111) surface over YSZ(111). Note that the Ni cluster interaction energy is more favourable for zirconia than yttria-stabilized zirconia, whose preference is even more striking for larger clusters. This demonstrates that cluster aggregation is more favourable on the YSZ(111) surface than on ZrO 2 (111), in good agreement with, for instance, the difference in energy between two Ni 4 clusters and a Ni 8 cluster, which is larger on ZrO 2 (111) than YSZ(111).
From the energy proles in Fig. 7, we note that the Ni 5 /YSZ(111) conguration is more stable than (Ni + Ni 4 )/YSZ(111), indicating that Ni atoms prefer to aggregate to form larger clusters rather than wetting the surface. This is in good agreement with the graph in Fig. 6, where we have shown a decrease of the E coh À E clus difference, indicating the energetic favourability of aggregation of Ni atoms over their dispersion on both surfaces. From Fig. 7, we also observe that the activation energy, i.e. the energy difference between the (Ni + Ni 4 )/YSZ(111) structure and the transition state, is DE Ni 5 /YSZ ¼ 0.46 eV, which is lower than the same Ni collection adsorbed on ZrO 2 (111) (DE Ni 5 /ZrO 2 ¼ 0.72 eV). 48 This difference in activation energies for the addition of a Ni atom to a larger cluster implies that single Ni atoms can more easily join a bigger cluster when those metal atoms are adsorbed on the YSZ(111) surface rather than ZrO 2 (111). It indicates that the aggregation of Ni atoms to form larger clusters is facilitated when the zirconia surface is doped with yttria, which is in good agreement with our E clus and E coh À E clus plots in Fig. 6. From the calculated activation energies, we have evaluated the hopping rate of one Ni atom from state (Ni + Ni n )/YSZ(111) to state Ni (n+1) /YSZ(111): kÃ B ¼ n exp(ÀDE/k B T) where A is either state (Ni + Ni 4 )/YSZ(111) or (Ni + Ni 10 )/YSZ(111) and B is either state Ni 5 /YSZ(111) or Ni 11 /YSZ(111) (Fig. 8). The Boltzmann constant is k B ¼ 8.6 Â 10 À5 eV K À1 and the vibrational frequency, n, is accepted as 10 12 s À1 . We have therefore calculated k A/B for a range of temperatures corresponding to the working temperature of a SOFC (T ¼ 500-900 C). Fig. 8 shows the variation of k A/B (for ve Ni atoms on the surface) from 1.05 Â 10 8 to 1.09 Â 10 10 s À1 , which is higher than the values found for ZrO 2 (111) in our previous investigations (k A/B varies from 1.87 Â 10 7 to 7.66 Â 10 8 s À1 ), 48 indicating that the diffusion of a Ni atom towards a cluster is more favourable on the YSZ(111) surface. This hopping rate is even higher when the cluster diffuses towards a larger cluster, since for eleven Ni atoms k A/B varies from 586.99 Â 10 7 to 703.93 Â 10 8 s À1 (Fig. 8). Thus, this evaluation of the hopping rate further strengthens the conclusions drawn from Fig. 6: the aggregation of Ni atoms is preferred over dispersion on the surface and this aggregation is more favourable on the YSZ(111) surface.

Microkinetics for Ni n /YSZ(111)
We have used kinetic Monte Carlo simulations in order to understand how time (t) and Ni coverage (q) inuence the coalescence of Ni clusters on top of the YSZ(111) surface. To achieve our kinetic Monte Carlo study, we have used the same equations as the ones described in the ESI of the investigation by Tafreshi  et al. 56 Additionally, quantum-mechanical tunnelling 57 has been taken into account. Coverage has been dened in eqn (4): where q * (t) is the coverage of free sites and q i (t) is the coverage of the Ni n clusters. The reactions considered are: Ni n + Ni 1 4 Ni (n+1) . We considered as a free site a 2 Â 2 clean YSZ(111) surface, while the occupied site is a 2 Â 2 YSZ(111) surface with Ni n clusters adsorbed on top of it.
The rst system considered is a YSZ(111) surface with a coverage of 50% of Ni 1 (q Ni 1 ¼ 50%) as a starting point. In Fig. 9 we show a schematic representation of the initial conditions where one Ni atom covers 50% of the system, i.e., at t ¼ 0.0 seconds, half of the system is considered as a clean 2 Â 2 YSZ(111) surface and the other half is Ni 1 supported on YSZ(111).
The coverage (q i (t)) of the Ni n clusters as a function of time for a xed temperature of 500 K is represented in Fig. 10(a).
During the rst 50 seconds the concentration of Ni 1 drops to 0.0%. In the meantime, q Ni 4 increases to reach a plateau of 12%. Thus, at T ¼ 500 K, if 50% of the YSZ(111) surface is covered by single Ni atoms, the metal atoms will aggregate spontaneously to generate Ni 4 clusters. This quick coalescence process is in good agreement with the previous section: for high temperatures, the Ni atoms tend to form clusters on top of YSZ(111) rather than wet the surface.
In Fig. 10(b) we plot the coverage as a function of time, for T ¼ 500 K with an initial q Ni 1 of 100%. The result is similar to the previous initial condition (q Ni 1 ¼ 50%): aer 50 seconds, all of the single Ni atoms aggregate to form Ni 4 pyramids. Indeed, from t ¼ 50 s, q Ni 4 ¼ 24% and q * ¼ 76%. Furthermore, whatever the initial coverage of single Ni atoms is, the aggregation speed is the same as for both initial coverages (q Ni 1 ¼ 50% and q Ni 1 ¼ 100%), within 50 seconds Ni 4 pyramid clusters are generated. We have also calculated the reaction rates and noted that the one corresponding to Ni 4 + Ni / Ni 5 has the lowest value. Thus, Ni atoms will aggregate quickly to generate Ni 4 but the formation of larger clusters is negligible.
In Fig. 10(c) we describe the evolution of a Ni 10 cluster on YSZ(111) with a starting coverage of 5%. Within 50 seconds Ni 10 coverage decreases (3.7%) in order to generate Ni 9 (0.65%) and Ni 11 (0.65%). The formation of the Ni 9 clusters is the consequence of the generation of Ni 11 : one Ni atom, belonging to Ni 10 , detaches to aggregate with a neighbouring Ni 10 cluster giving Ni 9 and Ni 11 .
In Fig. 10(d), we show q Ni 10 with an initial coverage of 10%, and the evolution of the cluster size is similar to an initial q Ni 10 of 5% because the interaction between metal moieties is neglected in the model. Fig. 9 Schematic representation of the coverage of the free sites (q * ) and occupied sites (q i ) by Ni atoms on the YSZ(111) surface. Here, q i ¼ q * ¼ 50%. Finally, this kinetic Monte Carlo study shows that Ni n always tend to form larger clusters; single Ni atoms and Ni 10 clusters generate Ni 4 and Ni 11 , respectively. However, under experimental conditions no specic cluster size will be dominant, as we saw, for instance, that Ni 10 can generate Ni 9 and Ni 11 .

Conclusions
We have used spin polarized DFT and kinetic Monte Carlo calculations to study the interaction of Ni n (n ¼ 1-10) clusters with both ZrO 2 (111) and YSZ(111) surfaces. The general trend observed for Ni n clusters (n ¼ 2-10) adsorbed on the ZrO 2 (111) and YSZ(111) surfaces shows the importance of the interaction of Ni with the O u atoms, as well as the shapes of the adsorbed Ni n clusters. We have seen that the O u atoms facilitate the adsorption of the clusters and that these atoms are shied from their initial position upon adsorption. In Ni n /YSZ(111) systems, the clusters pushed the O u atoms towards lling neighbouring vacancies which could be a drawback in, for instance, SOFCs since these vacancies play an important role in oxygen transport. Bader charge analysis of the clusters revealed that there is charge transfer from the cluster to the surface, in particular from the metal atoms at the interface. The distribution of the charge within the cluster is similar for all clusters with a pyramid shape (Ni n , n ¼ 4-10): the Ni atoms bound Fig. 10 Plot of the coverage, q i , of the Ni n clusters as a function of time. q i has been calculated from t ¼ 0.0 s to t ¼ 3600 s but here we plot q i up to t ¼ 100 s, as from t ¼ 50 s, the values of q i are unchanged. For Ni 8 , both pyramid and flat shapes have been considered. (a) and (b) show q Ni 1 for an initial coverage of 50% and 100%, respectively. (c) and (d) show q Ni 10 for an initial coverage of 5% and 10%, respectively. to the surface are positively charged and those at the top of the pyramid are either charge-neutral or even some negative charge is accumulated at the apex. In some instances, the Ni atoms located at the top of the clusters have a non-negligible amount of charge, which could play a role in the adsorption of molecules: for example, an electrophilic molecule would adsorb on top of the Ni cluster, rather than at the meeting point between the cluster and the surface. Finally, from calculation of the clustering and cohesive energies and evaluation of the diffusion barriers and hopping rates, we conclude that, on both ZrO 2 (111) and YSZ(111) surfaces, the aggregation of the Ni atoms takes place spontaneously, especially on the YSZ(111) surface. Kinetic Monte Carlo simulations showed that Ni atoms aggregate on the YSZ(111) surface once they are exposed to medium temperature: single Ni atoms tend to form Ni 4 clusters, while Ni 10 clusters generate Ni 9 and Ni 11 clusters. We also noted that the sintering speed does not depend on the initial coverage.