Ke Zhang

Linfei Li

Jacek Goniakowski

Claudine Noguera

Hans-Joachim Freund

Shamil Shaikhutdinov

Size effect in two-dimensional oxide-on-metal catalysts of CO oxidation and its connection to oxygen bonding: An experimental and theoretical approach

Keywords: CO oxidation, monolayer catalysts, iron oxide, metal/oxide interface

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Transition metal oxides (TMO) are active players in catalysis, in particular in oxidation reactions. The differentiation of the various oxygen species in activity is important and, therefore, needs speciation. This concept has been propagated prominently by Robert Grasselli [START_REF] Grasselli | Selective Oxidation and Ammoxidation Catalysis: Mechanism and Commercial Practice[END_REF][START_REF] Grasselli | Fundamental Principles of Selective Heterogeneous Oxidation Catalysis[END_REF] with respect to oxidation of hydrocarbons on complex oxides, as well as by Jerzy Haber combining theory and experiment. [START_REF] Bielański | Oxygen in Catalysis on Transition Metal Oxides[END_REF][START_REF] Haber | Oxidation catalysis-electronic theory revisited[END_REF][START_REF] Haber | Catalytic Oxidation -State of The Art and Prospects[END_REF] Even in the case of supported metal catalysts, where the noble metal particles are supported on a TM-oxide, the role of the oxide support often extends beyond that of being solely a template to fix the metal component, in that it actively participates in the process. A notable special case is represented by the so-called "Strong Metal Support Interaction" (SMSI), where the metal oxide may overgrow the supported metal particle and thus affect catalytic performance. [START_REF] Tauster | Strong metal-support interactions[END_REF] A detailed understanding of SMSI, whether it involves structural and/or electronic effects, remains under debate. [START_REF] Haber | Oxidation catalysis-electronic theory revisited[END_REF][START_REF] Solymosi | Importance of the Electric Properties of Supports in the Carrier Effect[END_REF][START_REF] Haller | Metal-Support Interaction: Group VIII Metals and Reducible Oxides[END_REF][START_REF] Bernal | Some recent results on metal/support interaction effects in NM/CeO2 (NM: noble metal) catalysts[END_REF][START_REF]kenshtein, Experiment and the electronic theory of catalysis[END_REF]] SMSI often manifests itself via encapsulation of metal nanoparticles by a very thin oxide film stemming from a TMO support. Therefore, establishing the structure-reactivity relationships for ultrathin TMO films on metals may shed light on the SMSI effects. [START_REF] Shaikhutdinov | Strong Metal-Support Interaction and Reactivity of Ultrathin Oxide Films[END_REF] In particular, the nature of oxygen species in the TMO films and its role in the oxidation reactions need to be better understood.

The interest in such studies increased after a "monolayer" FeO(111) film on Pt(111) was found to be active in low temperature CO oxidation, [START_REF] Sun | The Interplay between Structure and CO Oxidation Catalysis on Metal-Supported Ultrathin Oxide Films[END_REF] and this is the film that encapsulates Pt nanoparticles supported on Fe3O4 as result of SMSI. [START_REF] Qin | Encapsulation of Pt Nanoparticles as a Result of Strong Metal-Support Interaction with Fe3O4(111)[END_REF][START_REF] Willinger | A Case of Strong Metal-Support Interactions: Combining Advanced Microscopy and Model Systems to Elucidate the Atomic Structure of Interfaces[END_REF] However, under reaction conditions at near atmospheric pressures, the FeO(111) film transforms into the "O-rich" film with a compositional stoichiometry close to FeO2 (henceforth referred to as a FeO2-x film). Due to the Moiré-like coincidence structure formed on the initial FeO(111)/Pt(111) surface, [START_REF] Vurens | Growth, structure and chemical properties of FeO overlayers on Pt(100) and Pt(111)[END_REF] the O-rich film has a complex structure. Based on scanning tunneling microscopy (STM) studies [START_REF] Sun | The Interplay between Structure and CO Oxidation Catalysis on Metal-Supported Ultrathin Oxide Films[END_REF] and density functional theory (DFT) calculations [START_REF] Giordano | Oxygen-Induced Transformations of an FeO(111) Film on Pt(111): A Combined DFT and STM Study[END_REF][START_REF] Merte | Identification of O-rich structures on platinum(111)-supported ultrathin iron oxide films[END_REF], the film forms close-packed O-Fe-O trilayer islands, following the Moiré pattern, with an ill-defined structure between the islands. Temperature programmed desorption (TPD) measurements showed that the FeO2-x films contain oxygen species desorbing at much lower temperatures than in FeO. Weakly bound oxygen (WBO) species were attributed to the O atoms in the topmost layer. Accordingly, it is this oxygen that reacts with CO to form CO2. Comparison with other TMO films tested in this reaction suggested that the binding energy of WBO species, determined by TPD, may serve as a descriptor for this reaction. [START_REF] Martynova | CO Oxidation on Metal-Supported Ultrathin Oxide Films: What Makes Them Active?[END_REF] At sub-monolayer oxide coverages, the oxide/metal interface starts to dominate the reactivity. Indeed, the CO2 production rate increases and apparently follows the total island perimeter length, both reaching a maximum at mid-coverages. [START_REF] Pan | Enhanced CO Oxidation on the Oxide/Metal Interface: From Ultra-High Vacuum to Near-Atmospheric Pressures[END_REF][START_REF] Martynova | CO oxidation over ZnO films on Pt(111) at near-atmospheric pressures[END_REF] It was reasonably thought that CO strongly adsorbing on Pt reacts with the O atoms at the oxide island edge, either on the FeO/Pt [START_REF] Sun | Theoretical Study of the Role of a Metal-Cation Ensemble at the Oxide-Metal Boundary on CO Oxidation[END_REF] or FeO2/Pt, [START_REF] Pan | Enhanced CO Oxidation on the Oxide/Metal Interface: From Ultra-High Vacuum to Near-Atmospheric Pressures[END_REF] depending on reaction conditions. However, a more recent study [START_REF] Zhang | Carbon Monoxide Oxidation on Metal-Supported Monolayer Oxide Films: Establishing Which Interface is Active[END_REF] provided evidence for the steady state reaction taking place neither on FeO/Pt nor FeO2/Pt interfaces, but at the boundary between oxidized and reduced phases of the iron oxide film formed under reaction conditions. In this scenario, the metal (Pt) surface solely aids in the initial formation of such boundaries close to the island edges rather than directly participates in the reaction.

In this work, we report substantial size effects of FeO islands on the CO oxidation reaction. Reducing the island size down to 5 nm considerably lowers the light-off temperature for CO2 production. In an attempt to rationalize this effect, we have performed oxygen desorption experiments and have combined them with an extended DFT analysis to provide insight into the bonding of various oxygen species on such systems.

The theoretical results on isolated FeO2 islands indicate a substantially lower stability of the boundary oxygen atoms as compared with the same islands embedded into an FeO layer in qualitative agreement with experiment showing that the smaller the island, the lower the desorption temperature. To the best of our knowledge, the results demonstrate the first example of a size effect in oxidation catalysis on two-dimensional systems.

Methods and Materials

Experimental Section

The experiments were carried out in several UHV chambers (base pressures below 5×10 -10 mbar) all equipped with standard facilities for cleaning the Pt(111) single crystal surface and preparation of the FeO(111) films. In situ STM studies were performed using a variable-temperature scanning tunneling microscope (VT-STM, Omicron). Temperature programmed reaction (TPR) measurements were performed in another chamber using a quadrupole mass-spectrometer (QMS, from Pfeiffer). The prepared samples were characterized with Omicron VT-STM/AFM at room temperature. Oxygen desorption experiments were performed in the third UHV chamber equipped with STM (Omicron) and QMS (Hiden).

In all setups, Fe was deposited using e-beam assisted evaporator (Focus EMT3) from an Fe rod (99.9999%, Sigma Aldrich). FeO(111) islands were prepared by Fe deposition onto a clean Pt(111) substrate in 10 -7 mbar of O2 at room temperature, followed by annealing in 10 -6 mbar O2 at 700 K for 5 min. In order to prepare the smallest FeO(111) islands, Fe was deposited onto a substrate kept at 100 K. The samples were then oxidized in 10 -5 mbar O2 at 573 K to transform FeO(111) into the FeO2-x phase.

Computational Methods

All DFT calculations were performed with the Vienna Ab-initio Simulation Package (VASP), [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium[END_REF] using Projector Augmented Wave (PAW) method [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF][START_REF] Blöchl | Projector augmented-wave method[END_REF] to represent the electron-core interaction, and the Perdew-Wang 91 (PW91) [START_REF] Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF] gradient-corrected exchange-correlation functional. Following our previous studies, [START_REF] Giordano | Interplay between structural, magnetic, and electronic properties in a FeO/Pt(111) ultrathin film[END_REF][START_REF] Prada | Properties of Pt-supported iron oxide ultra-thin films: Similarity of Hubbard-corrected and hybrid density functional theory description[END_REF] iron oxides were treated with the DFT+U approach in the form proposed by Dudarev, [START_REF] Dudarev | Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study[END_REF] with UFe -JFe = 3 eV. FeOX/Pt(111) system was represented by a three-layer-thick Pt(111) slab with the FeOX oxide adsorbed on one side only. The bottom Pt layer was hold fixed while the two surface Pt layers and the oxide film were fully relaxed (threshold on forces equal to 0.01 eV/Å).

The slabs were separated by at least 10 Å of vacuum and the dipole corrections were applied in order to eliminate the residual dipoles in the direction perpendicular to the surface. In order to take into account the effect of lattice mismatch between the Pt(111) substrate and the oxide layer and to mimic the coincidence structures observed experimentally, [START_REF] Ritter | Growth and structure of ultrathin FeO films on Pt(111) studied by STM and LEED[END_REF] a (73 × 73)R5.8°-FeO(111)//(91 × 91)R5.2°-Pt(111) periodic supercell has been used. Its large size makes Γ point sufficient to sample the Brillouin zone.

Following our previous studies, soft oxygen pseudopotential (energy cutoff of 280 eV) was used and a row-wise anti-ferromagnetic alignment of Fe spins was systematically imposed. [START_REF] Pan | Enhanced CO Oxidation on the Oxide/Metal Interface: From Ultra-High Vacuum to Near-Atmospheric Pressures[END_REF][START_REF] Giordano | Identification of Active Sites in a Realistic Model of Strong Metal-Support Interaction Catalysts: The Case of Platinum (1 1 1)-Supported Iron Oxide Film[END_REF] Atomic charges were estimated with the partition scheme proposed by Bader, [START_REF] Bader | A quantum theory of molecular structure and its applications[END_REF][START_REF] Henkelman | A fast and robust algorithm for Bader decomposition of charge density[END_REF] and atomic configurations were plotted with VESTA. [START_REF] Momma | VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data[END_REF]

Results and Discussion

Experimental results

We first show how CO interacts with the FeO2-x islands using in situ STM. Figure 1 displays STM images continuously recorded in 10 -6 mbar of CO at 350 K for different exposure time as indicated. Regular protrusions imaged on the island surface reflect O-Fe-O trilayer domains showing the same periodicity (2.5 nm) as the Moiré structure in the original FeO(111)/Pt(111) films. Obviously, the reaction starts at the edge and propagates towards the center of the island until all FeO2 related protrusions disappear due to their reduction back to FeO layer. Although STM tip effects cannot be ruled out, the reduced surface does not look atomically flat because the temperature (350 K) may not be high enough for complete structural relaxations accompanying the oxide reduction.

Interestingly, the time required for CO to reduce FeO2 domains at the perimeter is considerably shorter than that obtained for reduction of the next "spots ring" left (about 20 and 70 min, respectively). This finding suggests that the FeO2 domains close to the island edge are much more reactive to CO than those located in the interior region of the island upon reaction front propagation. Henceforth, we referred to those as the "edge" and the "interior" spots, respectively. Not surprisingly, the observed reaction is a thermally activated process. It proceeds much faster at increasing temperature. In order to get an estimate for the activation barrier, we imaged identically prepared samples all exposed to 10 -6 mbar CO at several different temperatures between 300 and 450 K. In a crude approximation, we assume that the reaction rate is inversely proportional to the time () needed for depleting the "edge" FeO2 spots. The rate vs temperature dependence showed Arrhenius-like behavior (Figure 1), from which we calculated the apparent activation energy of ~40 kJ/mol. The energy barrier can be assigned to the CO + O -> CO2 reaction step, which is considered as the rate-limiting step in the CO oxidation reactions following Mars-van Krevelen type mechanism. This high reactivity of FeO2 spots at the island edge further manifested itself in the temperature programmed reduction (TPR) experiments summarized in Figure 2. Upon heating a sample in 10 -6 mbar of CO, the latter reduces FeO2-x to FeO producing CO2 that has been recorded by a mass-spectrometer. The sample (1) has been prepared in such a way that it primarily exposes islands containing only "edge" spots as shown in the corresponding STM image. The maximum in CO2 production on this sample is observed at 375 K. Then the sample was annealed in UHV at 650 K for 30 min to promote oxide sintering and re-oxidized in O2 to transform islands back into the O-rich, FeO2-x phase (sample 2) before the new TPR run. The procedure was repeated to prepare the samples (3) and ( 4). The morphology of each sample was inspected by STM before and after the TPR experiment. It should be mentioned that the O ad-atoms remaining on the Pt(111) surface after sample preparation (exposure to 5×10 -6 mbar O2 at 573 K for 10 min, then pumping out and cooling to 300 K) readily react with residual CO in the UHV background during STM imaging and hence do not contribute to the CO2 signal measured. Indeed, the four TPR spectra are very different, whereas the integral CO2 signal area remains the same since the total oxide coverage is not changed upon islands sintering, but the island size.

TPR profiles revealed certain correlation with the sample morphology imaged by STM. It is clear that reaction at the lowest temperature occurs on the smallest islands exposing solely "edge" FeO2 spots. As the island size increases, the total number of such spots decreases, and its contribution to overall CO2 production diminishes. Concomitantly, the TPR profile shifts to higher temperatures, which is associated with the reaction propagating into the islands interior, as directly shown by in situ STM (Figure 1). Although the low-temperature peak shifts to higher temperatures (from 375 to 420 K in samples (1)-(4), respectively), it seems to consist of several discrete components (indicated by dashed lines to guide the eye), each associated with the certain island size. Meanwhile a high-temperature signal is represented by one peak, albeit slightly shifting (from 510 to 535 K) at increasing size. Therefore, the combined TPR and STM results point out a strong size effect on CO reaction with O in oxide: The smaller the islands, the lower the light-off temperature and hence the activation energy for CO2 formation.

In an attempt to rationalize exceptional activity of the "edge" FeO2 spots and assuming that the binding energy of WBO species remains the descriptor for CO oxidation on partially covered films as well, we studied oxygen desorption using TPD. Figure 3a shows TPD spectra measured on FeO2-x/Pt(111) films of three different coverages, i.e., 0.25 , 0.5 and 1 ML (the latter corresponds to a dense monolayer (ML) film), determined from STM images shown as insets. Negligible oxygen desorption was detected on the FeOfree Pt(111) sample under the same preparation conditions, most likely due to the reaction with CO in the UHV background. The presented spectra show solely oxygen evolution during thermal reduction of the oxidized FeO2-x phase into FeO, which, in turn, decomposes at temperatures as high as 1200 K. [START_REF] Sun | The Interplay between Structure and CO Oxidation Catalysis on Metal-Supported Ultrathin Oxide Films[END_REF] The spectra clearly show that: (i) the signal intensity is proportional to the oxide coverage; (ii) the entire desorption profile shifts to higher temperatures with increasing coverage and/or island size; (iii) the desorption signals are highly symmetric and almost identical in shape. It is interesting, that such spectra resemble those previously reported for noble metal surfaces oxidized at strongly oxidizing conditions and often referred to as "explosive" desorption. [START_REF] Saliba | Oxidation of Pt(111) by ozone (O3) under UHV conditions[END_REF][START_REF] Weaver | Oxidation of Pt(111) by gas-phase oxygen atoms[END_REF][START_REF] Klötzer | Oxygen-induced surface phase transformation of Pd(111): sticking, adsorption and desorption kinetics[END_REF] In particular on Pt(111), the desorption peak was found to shift from 700 K to 750 on increasing nominal O coverage from 1.5 ML to 2.9 ML, respectively. [START_REF] Weaver | Oxidation of Pt(111) by gas-phase oxygen atoms[END_REF] Other values are reported in Ref. [START_REF] Saliba | Oxidation of Pt(111) by ozone (O3) under UHV conditions[END_REF]: the peak shifts from 680 K at 1.8 ML to 710 K at 2.4 ML. In principle, desorption temperatures observed in Figure 3a fall in the range obtained on oxidized Pt(111) surfaces. The O2 desorption spectra on Pt(111) were originally explained by considering the influence of surface tension on the decomposition rates of PtOx oxide particles presumably covering the surface. [START_REF] Saliba | Oxidation of Pt(111) by ozone (O3) under UHV conditions[END_REF] As particles increase in size, the rate at a given temperature decreases because the surface tension on the particles also decreases. Accordingly, the maximum desorption rate shifts to higher temperatures at increasing coverage. "Explosive" desorption occurs because as the particles shrink during decomposition they become increasingly less stable and hence decompose rapidly during the TPD experiments.

Such TPD behavior was later explained by a kinetic model in which O atoms migrate from oxide domains ("condensed phase") onto the surface regions containing chemisorbed oxygen atoms ("dilute phase"). [START_REF] Weaver | Oxidation of Pt(111) by gas-phase oxygen atoms[END_REF][START_REF] Klötzer | Oxygen-induced surface phase transformation of Pd(111): sticking, adsorption and desorption kinetics[END_REF] Desorption directly from the dense phase was considered to be negligible, and it preferentially occurred from the dilute phase, with the oxide islands and particles acting as a reservoir for O. The migration of species from the dense phase to the dilute phase was assumed to be faster than the desorption from either phase. Under these conditions, the desorption rate is proportional to the fraction of the surface covered by the dilute phase. Since this fraction increases as the dense phase is consumed, desorption is self-accelerating and the rate increases rapidly once desorption is initiated. Basically, explosive desorption occurs because the area of the surface from which species desorb, and hence the total desorption rate, increases until the dense phase is completely decomposed.

The latter mechanism seems hardly plausible for the case of FeO/Pt. First, on noble metal surfaces considered above, surface oxide domains (or particles) are fully reduced to the metal which is incorporated back to the metal surface. In our case, FeO2-x islands only undergo reduction to FeO and do not vanish, thus the area of dilute phase remains constant. Second, the O atoms have to migrate across the FeOx island to reach the perimeter. This step is not included in the above model, and no experimental results exist thus far to get an estimate for diffusion coefficients. Third, the amount of oxygen released during the FeO2-x -> FeO transformation is, at most, 0.8 ML with respect to Pt(111). [START_REF] Sun | The Interplay between Structure and CO Oxidation Catalysis on Metal-Supported Ultrathin Oxide Films[END_REF] Once all these O atoms spread onto the clean Pt(111) surface surrounding oxide islands covering 25% of the surface (the 0.25 ML sample, see Figure 3a), the amount of O adatoms would only be sufficient to form a (2×2)-O ad-layer which normally desorbs at about 750 K, i.e., considerably higher than 640 K observed on this sample. In addition, the kinetic model assumes (2×2)-O as the critical O concentration for the dilute phase, that must be sustained via dissolution of oxide phase.

To gain more insight into oxygen desorption from our films, we studied morphology of the 0.5 ML sample by STM after heating was stopped at 640 K, i.e., shortly after the O2 desorption starts. In the next run, the sample was heated to 690 K, that is the maximum desorption temperature, and again inspected by STM. Desorption traces are shown in Figure 3b together with the one obtained by non-stop heating to 900 K (replotted from Figure 3a) for direct comparison.

At first glance, comparison of STM images before and after heating to 640 K (Figure 3c,d) shows rather random disappearance of the FeO2 spots due to their local thermal reduction to FeO. However, a closer look reveals that all "edge" spots have already disappeared at this point, clearly indicating preferential desorption from the islands edges that dominates in the initial stage of thermal reduction. During the next heating to 690 K, desorption only starts at the end temperature of the previous heating, i.e., 640 K (Figure 3b). The leading edge basically follows the one observed in this temperature range in the "non-stop heated" TPD spectrum as if the desorption process continued from the surface formed in the previous run. After the sample was heated to 690 K, a small amount of FeO2 spots remains on the islands (Figure 3e), which fully disappear after further heating to 900 K (not shown), ultimately recovering the FeO(111) surface. Therefore, STM results show that O2 desorption starts within the FeO2 spots at the island edge and then takes place randomly in the island interior.

In principle, the observed random desorption favors the mechanism of thermal reduction via O2 desorption from oxide directly into the gas phase rather than via spillover of the O atoms onto the Pt surface from which they desorb associatively. Indeed, the desorption peak obtained for a dense film (i.e., no Pt exposed) has the same shape as the partially covered samples do. In such scenario (desorption into the gas phase), the shift to lower temperatures at decreasing film coverage (Figure 3a) could be interpreted as if oxygen atoms in smaller FeO2-x islands were more weakly bound. In this respect, we note a recent STM study [START_REF] Liu | Enhanced oxidation resistance of active nanostructures via dynamic size effect[END_REF] showing that FeO islands below 3 nm in size are more resistant towards oxidation. This implies that "extra" O species in such small islands are less stable, and, therefore, the islands will more easily release oxygen during heating and thermal reduction. One may argue, that oxide islands studied here (tens of nanometers) are too large for invoking such a strong size effect. If the effect had a purely kinetic origin (larger island needs more time for oxygen desorption, thus causing the peak shifting to higher temperatures), then the TPD spectra would exhibit zero order kinetics with the same leading edge irrespective of the island size, at variance with the experiment. As a hypothesis, we could propose that oxygen desorption is affected by the size-dependent strain in the film as a result of a large (~ 10%) lattice mismatch between the oxide layer and metal surface. Indeed, looking at STM images one may notice that larger islands exhibit roundish and well-ordered FeO2 spots, whereas such spots in the smallest islands as well as at the edge of large islands are considerably distorted.

Theoretical results

To get more insight into the oxygen stability in O-rich islands, we have performed DFT calculations on two models: FeO-embedded islands (Figure 4a) and isolated FeO2 islands (Figure 4b). While the former describes the inner part of relatively large islands, the latter corresponds to the smallest, i.e., single-spot island, but may also mimic the edge spots in larger islands.

The embedded model was obtained from the continuous FeO2 tri-layer film by the removal of the most weakly bound oxygen atoms (see Sec. S1 in the Supporting Information, SI), as to preserve large uniform and well-structured tri-layer FeO2 islands in registry with the Pt substrate, and a continuous FeO(111) lattice in between. Since the precise structure of these islands is yet unknown, we have purposely included both zigzag-and arm-chair-oriented FeO2/FeO boundaries. To allow a direct comparison, the isolated FeO2 islands were constructed with the same stoichiometry, morphology, atomic structure, and position with respect to the substrate as the embedded ones. In both models, the calculated individual oxygen vacancy formation energies (E) tend to be systematically smaller for interface oxygen atoms and are by far the smallest for interfacial oxygen atoms at the island periphery (Secs. S2 and S3 in SI). This points toward a sequential desorption mode, which starts at the periphery of the FeO2 islands and shifts progressively toward their centers. As a consequence, individual desorption energies are not fully representative of the actual energy cost of island reduction. They also neglect the collective effects in desorption of O2 molecules. To better approximate the desorption thermodynamics, we have simulated the desorption process by the stepwise collective removal of interfacial oxygen atoms successively in the three regions (delimited by white lines in Figure 4), starting from the island periphery. After each desorption stage, the size of the FeO2 region decreases, and the FeO/FeO2 boundary shifts toward the island center. Moreover, for the embedded islands, the increase of  when moving towards the island center suggests a slowdown of oxygen desorption, whereas such effect is not obvious for the isolated islands. The systematically lower oxygen stability in the isolated islands and the enhanced stability of inner oxygen atoms in the embedded ones are qualitatively consistent with the measured lower oxygen desorption temperature from smaller islands and to the observed greater facility to desorb oxygen atoms from the edges of large islands. This finding may also explain the in situ STM results (Figure 1), showing considerable increase of the time required for CO to reduce FeO2 domains during the propagation of the reaction towards the island center.

To rationalize the origin of the differences between  values for the isolated and embedded islands, we compared the evolution of their structural characteristics as a function of island size. Synthetic, spatially-resolved information on the oxide structure is provided by the average Fe-Fe distances around cations along the unit cell long diagonal (see Figure 4). Figure 5 shows that, while the initial Fe-Fe distances in the embedded and isolated FeO2 islands display similarities (compare black lines), their behavior upon progressive oxygen desorption differs significantly, thus suggesting that strain release effects play a role in the different oxygen stability in the two types of islands. In the embedded case, the Fe-Fe distances in the interior of the FeO2 islands are nearly size independent and only slightly differ from those in the pristine FeO film in the same region of the Moiré cell. Thus, the embedded O-rich region is not subject to a substantial strain release upon oxygen desorption. However, the structure of the FeO film between the FeO2 islands undergoes important modifications. Indeed, short in-plane lattice parameters of FeO2, consistent with a strong oxide-metal interaction and a large rumpling in the O-Fe-O trilayer film, induce a tensile strain in the inter-island FeO layer.

This strain is particularly strong around the full FeO2 islands (black lines in Figure 5), where it produces the largest expansion of the Fe-Fe distances in the FeO zone with respect to those in the corresponding pristine FeO layer (blue). Unsurprisingly, the effect diminishes upon removal of peripheric O atoms (red) and becomes quasi-negligible in the smallest embedded islands (green). In order to estimate the consequence of such strain release on oxygen desorption, in a separate calculation we have released the strain of the FeO region by removing four oxygen atoms from its center. We found that it results in an increase of at the stage I by as much as 0.1 eV/atom (Sec. S3 in SI). We thus conclude that the strong tensile strain in the embedding FeO regions substantially reduces the binding of the peripheric O atoms and somewhat weakens the binding of the intermediate ones. It, therefore, explains the progressive increase of at the subsequent stages of oxygen desorption from the embedded FeO2 islands (Table 1).

In the isolated island, the Fe-Fe distances in the full FeO2 islands (black), are fairly similar to those in the embedded one, suggesting that dFe-Fe ≈ 3.0 Å represents the inplane lattice parameter in the FeO2/Pt(111) nano-oxide. However, in stark contrast with the embedded case, these distances are much shorter than those in the corresponding fully reduced island (blue). As a consequence, at the successive stages of oxygen desorption, the progressive increase of size of the FeO boundary induces a progressive expansion of the Fe-Fe distances within the remaining FeO2 region. Such behavior indicates the presence of a tensile strain in the FeO2 region, induced by the FeO border.

The effect is clearly the most pronounced at the stage III of desorption, at which the departure of oxygen atoms additionally enables a thorough relaxation of the entire FeO island which is additionally accompanied by a change of its registry with respect to the Pt(111) surface. In direct analogy with the embedded case, the presence of such FeOinduced tensile strain reduces the strength of oxygen binding in the FeO2 region, is responsible for the systematically lower  values in the isolated case, and explains the significant reduction of  for the inner oxygen atoms.

To summarize, calculations reveal the specific character of the FeO2/FeO boundary: a large stability difference between surface and interface oxygen atoms and a substantial binding reduction of the latter. As a consequence, oxygen desorption is likely a sequential process triggered by oxygen binding reduction at the progressing FeO2/FeO boundary.

Moreover, while different ion coordination and Fe-O bond iono-covalency contribute to the stability of boundary oxygen atoms, we show that strain also plays a prominent role.

Indeed, the non-negligible difference of in-plane lattice parameters in the oxygen-rich and oxygen-poor regions affects the oxygen stability at the edges of the embedded islands via a strong tensile strain in the inter-island FeO region. Conversely, the tensile strain induced by the reduced border is the robust cause of oxygen destabilization in the isolated island.

These computational results are consistent with the size effect observed experimentally and clearly show that the thermodynamic stability of oxygen in the FeO2 nano-oxide is sensitive to both its direct environment and to the size of the oxygen-rich phase. However, a translation of the calculated desorption energies  into desorption temperatures using a Redhead analysis as in Ref. [START_REF] Pan | Enhanced CO Oxidation on the Oxide/Metal Interface: From Ultra-High Vacuum to Near-Atmospheric Pressures[END_REF] leads to differences between embedded and isolated islands roughly twice smaller than those measured experimentally. Several effects of different nature may contribute to such an underestimation. On the one hand, regarding the thermodynamics of desorption, the computational models remain somewhat arbitrary. In particular, while the actual state of the embedding inter-island FeO region under desorption conditions is unknown, we have shown that its strain state significantly affects oxygen stability at the FeO/FeO2 boundary.

Interestingly, the roughness of the FeO layer directly after oxygen desorption visible in the STM images (Figure 1) points toward a partial release of the tensile strain in the embedding film. According to our theoretical analysis, this should produce an additional stabilization of oxygen at boundaries of embedded islands, and thus bring the theoretical estimation closer to the measurements. On the other hand, regarding kinetics of oxygen desorption from the boundaries, it may involve activation barriers which do not scale linearly with energy of oxygen desorption, as assumed in Ref. [START_REF] Pan | Enhanced CO Oxidation on the Oxide/Metal Interface: From Ultra-High Vacuum to Near-Atmospheric Pressures[END_REF] An explicit account for energy barriers associated with alternative oxygen recombination and desorption pathways would be necessary to refine the present thermodynamics-driven estimation.

Conclusions

Our experimental results show that the CO oxidation reaction on FeO monolayer islands supported by Pt(111) primarily occurs on FeO/FeO2 interface formed under reaction conditions along the islands edge which can be considered as a dynamically fluctuating "reaction zone". Its width (~ 3 nm) is comparable with the size of a Moiré unit cell in epitaxial FeO(111)/Pt(111) films thus suggesting the peculiar reactivity of the oxide in the "edge" zone, whereas well-ordered interior regions are relatively inert. As a result, reactivity strongly depends on island size. The islands only consisting of the edge reaction zone show considerably lower (by about 200 K) CO2 formation temperature than the extended islands.

In an attempt to rationalize this effect, we performed oxygen desorption experiments and combined them with an extended DFT analysis to provide insight into the bonding of various oxygen species in such systems since O binding energy has been found as a good descriptor for this reaction on the dense monolayer oxide films. The theoretical results on isolated FeO2 islands indicate a substantially lower stability of the boundary oxygen atoms as compared with the same islands embedded into an FeO layer in qualitative agreement with TPD results showing that the smaller the island, the lower the oxygen desorption temperature.

Size effects are well documented in catalysis over highly dispersed metals in structure sensitive reactions, which are commonly rationalized in terms of abundance of under-coordinated surface atoms on smaller particles considered as the most active sites.

Herein, the effect is observed on the two-dimensional system.

It should be mentioned that TMO deposits on metal substrates have been long considered as model "inverse" catalysts [START_REF] Hayek | Surface reactions on inverse model catalysts: CO adsorption and CO hydrogenation on vanadia-and ceria-modified surfaces of rhodium and palladium[END_REF] for studying chemical reactions at the interface between the metal particle and oxide support in "conventional" metal-on-oxide catalysts. Since TMO deposits often form monolayer islands rather than spherical particles, [START_REF] Netzer | Low-dimensional oxide nanostructures on metals: Hybrid systems with novel properties[END_REF] the coverage effects reported for "inverse" model catalysts may be affected by the island size and even cause some discrepancy, if the islands size is not controlled or directly measured.

S2. Oxygen stability in the FeO-embedded FeO 2 islands.

In the following we focus on the properties of large well-structured FeO 2 islands which form in regions in which interface anions are located on top of surface Pt atoms.

As shown in Sec. S1, the oxygen-rich oxide film displays in these regions a regular and Table S2 shows a relatively strong site dependence of the calculated vacancy formation energies E (0.9 eV < E < 3.3 eV). Surface and interface oxygen atoms in the very island center (sites s, t, u, v) display fairly similar large stabilities (E = 1.5 -1.7 eV), which are close to those found for the same sites in the continuous FeO 2 film, Sec S1.

However, when moving from island center toward its boundaries, surface oxygen atoms become more strongly bound (E = 1.8 -2.1 eV) than those at the interface (E = 1.4 - Table S2. Calculated vacancy formation energies E (eV) for individual surface and interfacial oxygen atoms in FeO-embedded FeO 2 islands on the Pt(111) surface. Site labels a-v refer to those marked in Fig. S3.

eV). The difference between the two types of atoms becomes paramount at the very

These results reveal the specific character of the very FeO 2 /FeO boundary for the oxygen stability. On the one hand, since the interfacial atoms at the boundary become by far the most prompt to desorption, the remaining strongly bound surface atoms will assure the continuity of the FeO(111) lattice, thus facilitating the reversibility of oxygen uptake and release by the embedded FeO 2 islands. More importantly, the particularly small E values of interfacial boundary sites (E = 0.9 eV, compared to 1.6 eV in the island center) show that any temperature programmed oxygen desorption will start there. Interestingly, we find that since the removal of these weakly bound oxygen atoms (sites a and c and all equivalent by symmetry) shifts the FeO Table S3. Calculated average vacancy formation energies  (eV/atom) for the simultaneous removal of interfacial oxygen atoms from the FeO 2 /FeO boundary in FeOembedded FeO 2 islands on the Pt(111) surface. The three stages correspond to three successive locations of the boundary in the island. The labels of the removed oxygen atoms (a total of 18 atoms at each stage) refer to those marked in Fig. S3.

Table S3 shows that the collective desorption energies of boundary interfacial oxygen atoms  are indeed systematically smaller than the corresponding averages of individual desorption energies E, Tab. S2, and fully confirm the specific role of As to highlight the differences between the isolated and embedded cases, in the following we analyze the electronic (electron exchange with the Pt substrate and oxygen charges) and structural (elastic strain) characteristics of the FeO 2 islands, with a special focus on the characteristics of oxygen sites at the island boundaries, and on their evolution as a function of island size. The considered embedded and isolated islands at the three stages of oxygen desorption are depicted in Fig. S6.

We recall that the oxygen-rich FeO 2 nano-oxide is mainly stabilized by an electron transfer from the Pt substrate. Indeed, Fig. S7 (top panels) shows that a substantial positive charging of the Pt atoms underlying the FeO 2 islands takes place, in contrast to a smaller negative one below the FeO film. Our results clearly show that the effect of positive Pt charging is fairly independent of the island type and size. We note that the Pt charge value in the FeO regions between the full FeO 2 embedded islands (black symbols, vicinity of site E, Fig. S7, top left panel) substantially departs from those in the FeO layer.

This effect is much smaller and becomes quasi-negligible in smaller embedded islands. In the embedded case, the Fe-Fe distances in the interior of the FeO 2 islands are nearly size independent and differ only very little from those in the pristine FeO film in the same region of the Moiré cell. Thus, the embedded oxygen-rich region is not subject to a substantial strain release upon oxygen desorption. However, Fig. S8 shows that the structure of the FeO film between the FeO 2 islands undergoes important modifications.

Indeed, short in-plane lattice parameters of FeO 2 , consistent with a strong oxide-metal interaction and a large rumpling in the trilayer film, induce a tensile strain in the interisland FeO layer. This strain is particularly strong around the full FeO 2 islands (black),

where it produces the largest expansion of the Fe-Fe distances in the FeO zone with respect to those in the corresponding pristine FeO layer (blue). Unsurprisingly, this effect diminishes upon removal of peripheric oxygen atoms (red) and becomes quasinegligible in the smallest embedded islands (green). In order to estimate the consequence of such strain release on oxygen desorption, in a separate calculation we have partially released the strain of the embedding FeO film by removing one and four oxygen atoms from its center, Fig. S9. We found that such removal results in an enhancement of E value of the site a by as much as 0.3 eV (single oxygen atoms removed), and in an increase of at the first stage by as much as 0.1 eV/atom (four atoms removed). We thus conclude that the strong tensile strain in the embedding FeO regions substantially reduces the binding of the peripheric oxygen atoms and somewhat weakens the binding of the intermediate ones. It thus explains the calculated progressive increase of at the subsequent stages of oxygen desorption from the embedded FeO 2 islands.

Figure 1 .

 1 Figure 1. In situ STM images of an individual FeO2-x island recorded in 10 -6 mbar of CO at 350 K. Exposure time is indicated. (The image size is 24 nm × 20 nm; tunneling parameters: -2 V, 0.1 nA.) The panel shows the Arrhenius plot for the initial rate of reduction (measured by STM ex situ) as a function of temperature (300 -450 K).

Figure 2 .

 2 Figure 2. Temperature programmed reduction profiles measured on FeO2-x/Pt(111) samples (1-4) shown in STM images (the scale bars are 10 nm) taken prior to the TPR run. The spectra are obtained by recording the CO2 (44 amu) mass-spec signal in 10 -6 mbar of CO upon linear heating with the rate 2 K/s.

Figure 3 .

 3 Figure 3. (a) O2 (32 amu) TPD spectra recorded on FeO2-x/Pt(111) at three different film coverages as indicated. STM images of the "as prepared" 0.25 and 0.5 ML samples (presented in the differentiated contrast) are shown in inset. The scale bar is 20 nm. (b) Black line reproduces the TPD spectrum for the 0.5 ML sample (1) from panel a). The newly prepared 0.5 ML sample (2) was first heated to 640 K (red line) and imaged with STM at 300 K. Then the sample was heated to 690 K (green line) and inspected by STM again. Dashed lines show (inertial) desorption traces after heating was stopped. The heating rate in all spectra is 2 K/s. (c-e) STM images of the sample "as prepared" (c) and after heating to 640 K (d) and 690 K (e), respectively.

Figure 4 .

 4 Figure 4. Models of FeO-embedded (a) and isolated (b) FeO2 islands on the Pt(111) surface. Pt, Fe, and O atoms are represented by gray, blue, and red (light for interface, dark for surface) spheres, respectively. The peripheric, intermediate, and center zones of the FeO2 islands are delimited (white lines) and the periodic (√73×√73)R5.8°-FeO(111)//(√91×√91)R5.2°-Pt(111) unit cell is indicated (black lines).

Table 1

 1 reports the average desorption energies ()at each stage. The lower values of compared to the average of individual values E for the atoms upon consideration (Secs. S2 and S3 in SI) highlight the importance of collective effects in the desorption process. Moreover, contrary to an average of all E values, the average of the three  values reproduces very well the total energy cost (per atom) of a full island reduction (simultaneous removal of all 54 interfacial oxygen atoms in the model), equal to 1.08 eV and 1.01 eV per atom for the embedded and isolated islands, respectively. This difference corresponds to a nearly 4 eV larger energy cost of the full reduction of the embedded island.

Figure 5 .

 5 Figure 5. Local average values of Fe-Fe distances (Å) for cations along the cell long diagonal for the embedded (left) and isolated (right) islands: initial FeO2 (black); with peripheric O atoms desorbed (red); with peripheric and intermediate O atoms desorbed (green); and fully reduced FeO (blue) islands. (For atomic representations of islands at the various stages of reduction see Figure S6 in SI.)

Figure S2 :

 S2 Figure S2: Key structural (distance from the Pt substrate) and electronic (Bader charges) characteristics of anions (full circles) and cations (empty circles) in the FeO 2 (red) and FeO (black) films. Surface and interface oxygen atoms are located at a height larger and smaller than c.a. 2.5 Å from the Pt(111) surface, respectively.

  uniform O-Fe-O trilayer structure and is characterized by the strongest binding of both its surface and interfacial anions (the largest desorption energies E of individual oxygen atoms, Tab. S1). To this goal we have removed the most weakly bound interfacial oxygen atoms from the continuous FeO 2 film, as to preserve the large and well-structured trilayer FeO 2 islands in the O-top regions and a continuousFeO(111) lattice in the inter-island regions, Fig.S3. Since the precise morphology and atomic structure of such embedded oxygen-rich islands are unknown, we have purposely included both zig-zag-and arm-chair-oriented FeO 2 /FeO boundaries. All inequivalent oxygen sites in the irreducible part of the FeO 2 island are labeled a-v.

Figure S3 .

 S3 Figure S3. Model of a FeO 2 island embedded in the FeO(111) layer on the Pt(111) surface. Inequivalent oxygen sites are labeled a-v. Pt, Fe, and O atoms are represented by gray, blue, and red (light for interface, dark for surface) spheres, respectively. The periodic unit cell is indicated.

FeO 2 /

 2 FeO boundary where surface sites with E of about 3 eV (sites b, d), characteristic of a pristine 1ML-FeO(111)/Pt(111), coexist with interface sites with E smaller than 1 eV (sites a and c), characteristic of some less well-structured inter-island regions of the continuous FeO 2 film, Sec. S1.

FeO 2 /

 2 FeO boundary for the ease of oxygen desorption. Unsurprisingly, the most important difference concerns the island center (stage III), where  = 1.15 eV is by as much as 0.4-0.5 eV/atom smaller than individual values E = 1.5 -1.6 eV. The effect is also quite substantial (-0.15 eV) for the energetics of the initial island edge (stage I), as already hinted above by the results on co-operative removal of two oxygens from neighboring sites a. It is worth stressing that, contrary to an average of E values, the average of the three  values reproduces very well the energy cost (per atom) of the full reduction of the embedded island (simultaneous desorption of all 54 interfacial atoms) equal to 1.08 eV/atom. Finally, the present results also reveal an increase of oxygen binding as the FeO 2 /FeO boundary progresses towards the island center. Indeed, it is the weakest at the boundary of the initial islands (stage I), but increases progressively as the islands become smaller (stage II and III).

S3.

  Fig.S5. The individual vacancy formation energies E of the two types of islands show several quantitative and qualitative similarities. E values tend to be systematically smaller for interface oxygen atoms and the least stable (weakly bound) oxygen atoms are systematically found at the very island periphery (sites a and c). This shows that the scenario of oxygen desorption triggered by E reduction at the progressing FeO 2 /FeO boundary is essentially the same in the isolated and embedded islands. However, Fig.S5alsodemonstrates that the stability of several surface and interface sites is substantially lower in the isolated case. This difference principally concerns atoms at and close to the isolated island boundary and is particularly well pronounced for surface sites b and d, which display E values more than 1 eV lower than in the embedded case.Similarly, binding of surface atoms j, f, l and h and of the interface atoms e, k, and m is by 0.1-0.4 eV smaller. Conversely, in the island interior (sites o-v), there is a slight average increase of E with respect to the embedded case. Interestingly, this increase also concerns interface sites a and c at the very island periphery. As far as  values are concerned, Fig.S5evidences that isolated islands systematically require less energy to be reduced. The difference in  is marginal at the first stage (0.95 vs 0.95 eV/atom for isolated and embedded islands, respectively), increases at the second one (1.06 vs 1.13 eV/atom), and becomes paramount at stage III (0.99 vs 1.15 eV/atom). This translates into a 4 eV lower energy cost of the full reduction of the isolated island. Moreover, in the embedded case, the progressive increase of  when moving towards the island center suggests a slowdown of oxygen desorption during island reduction, whereas such effect is absent in the isolated object. The systematically lower average oxygen stability in the isolated islands corresponds well to the measured lower oxygen desorption temperature from smaller islands and to the observed greater facility to desorb oxygen atoms from the edges of large islands.

Figure S6 .

 S6 Figure S6. Atomic structure of the embedded (top panels) and isolated (bottom panels) FeO 2 islands at the three stages of oxygen desorption. From left to right: full, with peripheric oxygen atoms desorbed, with peripheric and intermediate oxygen atoms desorbed, and fully reduced islands. Pt, Fe, and O atoms are represented by gray, blue, and red (light for interface, dark for surface) spheres, respectively.

Figure S7 .

 S7 Figure S7. Bader charges of surface Pt (top) and oxygen (bottom) atoms, taken along the segments indicated in Fig. S6, for the embedded (left) and isolated (right) islands: full (black), with peripheric oxygen atoms desorbed (red), with peripheric and intermediate oxygen atoms desorbed (green), and fully reduced (blue). Labels E, H, K, and N refer to sites in Fig. S1. Labels a, i, t, v, and m refer to Figs. S3 and S4.

Figure S8 .

 S8 Figure S8. Local average values of Fe-Fe distances (Å) for cations along the cell long diagonal for the embedded (left) and isolated (right) islands: full (black), with peripheric oxygen atoms desorbed (red), with peripheric and intermediate oxygen atoms desorbed (green), and fully reduced (blue) islands.

Figure S9 .

 S9 Figure S9. Atomic structure of the full embedded FeO 2 islands with one (left) and four (right) atoms removed from the strained embedding FeO layer. Large outward relaxation of the surrounding lattice witnesses a substantial strain relief.

Table 1 .

 1 Average vacancy formation energies  (eV/atom) for the stepwise removal of oxygen atoms in embedded and isolated FeO2 islands. Each successive stage corresponds to the desorption of 18 interfacial oxygen atoms from the peripheric, intermediate, and inner regions, respectively, as shown in Figure4.

	 (eV/atom)	Stage I	Stage II	Stage III
		(Peripheric)	(Intermediate)	(Inner)
	Embedded	0.95	1.13	1.15
	island
	Isolated island	0.95	1.06	0.99

Calculated  values show that oxygen stabilities in the two types of island differ quite considerably. While the difference in  is marginal at the first stage I, it becomes significant at the stage II (0.07 eV/atom), and substantial at stage III (0.16 eV/atom).

Table S1 .

 S1 Desorption energies E (eV) of individual oxygen atoms from different regions of the continuous FeO 2 film, Fig. S1. Larger oxygen stability is associated to larger E values.

  2 /FeO boundary towards the island interior, it simultaneously lowers the binding of interfacial oxygen atoms at the newly exposed FeO 2 /FeO boundary (sites g, e, and i) compared to the corresponding E values in Tab. S2 (by -0.3 eV, -0.2 eV, and -0.1 eV, respectively). These finding points toward a sequential mode of oxygen desorption from the embedded FeO 2 islands, which involves a removal of interfacial oxygen at the FeO 2 /FeO boundary, accompanied by a progressive boundary shift towards the island center. Such sequential oxygen desorption triggered by E reduction at the progressing FeO 2 /FeO boundary makes the individual vacancy formation energies, Tab. S2, not fully representative of the actual energy cost of island reduction. Indeed, an average of E from TableS1(1.34 eV/atom) is noticeably larger than that calculated for a simultaneous desorption of all interfacial oxygen atoms (1.08 eV/atom). Moreover, since E values clearly depend on the local environment of the oxygen atoms, a cooperative desorption of anions (e.g., of two neighboring anions, as to form an O 2 molecule) differs from a sum of individual vacancy formation energies. For exemple, we find that a simultaneous removal of two atoms from neighboring sites a requires 0.75 eV/atom, much less than 0.9 eV/atom found for the individual desorption, Tab. S2.As to better account for these collective and co-operative effects on the thermodynamics of oxygen desorption in the embedded FeO 2 islands, we have mimicked the actual desorption process by the removal of interfacial boundary oxygen atoms in three stages, Tab S3. The first stage consists of a simultaneous removal of all interfacial oxygen atoms at the initial FeO 2 /FeO boundary (sites a, c, e, and all equivalent by symmetry) and is characterized by the corresponding average oxygen vacancy formation energy . Further on, the subsequent interfacial atoms are simultaneously removed from the newly formed boundary (sites g, k, I, m, and all equivalent by symmetry) at the second stage. Finally, the remaining interfacial atoms (sites o, r, t, v, and all equivalent by symmetry) are simultaneously removed at stage III.

	Desorption stage	Stage I	Stage II	Stage III
	Site labels	a, c, e	g, k, I, m	o, r, t, v
	 (eV/atom)	0.95	1.13	1.15

Acknowledgements

The work was supported by Fonds der Chemischen Industrie. K.Z. acknowledges financial support from the NFFA-Europe Project (Grant 654360).

SUPPORTING INFORMATION for:

Size effect in two-dimensional oxide-on-metal catalysts of CO oxidation and its connection to oxygen bonding: An experimental and theoretical approach Ke Zhang, Linfei Li, Jacek Goniakowski,* Claudine Noguera Hans-Joachim Freund, Shamil Shaikhutdinov*

The supporting information provides a detailed description of the results obtained from the DFT simulations of a continuous FeO 2 film (Section S1), of FeO-embedded FeO 2 islands (Section S2), and of isolated FeO 2 islands (Section S3).

S1. Structural, electronic, and stability characteristics of the continuous

Pt(111)-supported FeO 2 film.

To


Lucien Fournier

Longitude by Wireless

Michael Kershaw

During the second half of the 19 th century, a world-wide web of longitude measurements was constructed using cable telegraphy. The telegraph had allowed, for the first time, the precise establishment of simultaneity at two far-distant observatories, so that their local times could be compared and their difference in longitude deduced. Nonetheless, because of its piecemeal construction over decades using all sorts of different apparatus, this web of longitude was known to be imperfectly accurate.

In the early years of the 20th century the innovation of wireless offered a new and potentially better method of establishing simultaneity, by receipt of a single wireless signal at different locations. These new wireless techniques came to dominate precision longitude measurement, not being superseded until the use of satellites in the 1980s. The Bureau des longitudes continued to play an important role. The leading French figure in the use of wireless for military and scientific purposes during the first three decades of the 20 th century was Général Gustave Ferrié (1868-1932, and a correspondent member of the Bureau from 1911). It was he who had in 1899 been the French observer of first cross-channel transmissions by Guglielmo Marconi (1874Marconi ( -1937)), and he who identified the then-redundant Eiffel Tower as the ideal support for the long aerials needed for wireless transmission (Figure 1). With the participation of the Bureau des longitudes, successful experiments to measure longitude by wireless were made from 1910 to 1914: from Paris to Brest, Bizerte, Brussels, and even as far as Washington. But Ferrié's further plans were brought to a halt by the outbreak of war.

He was able to improve and revive those plans soon after the Armistice, and a typewritten memorandum entitled Détermination d'un Réseau Mondial de Longitudes was presented at the séance of the Bureau des longitudes on 5 th March, 1919 (Figure 2). His proposal was that the relative longitude of three observatories in the northern hemisphere, forming a triangle to which other locations could later be linked, be determined with the utmost precision by wireless. The three points, with meridians roughly 8 hours in time apart, were Paris, Shanghai (a French settlement) and San Francisco. A further station at the antipode of France, in New Zealand, and a connection to the Greenwich meridian were also suggested. The crucial test of the operation would be the error of closure of the primary triangle, whose total difference of longitude should be exactly 24 hours. A critical part of these measurements was that sufficiently powerful transmitters should be available for their signals to be received clearly and simultaneously at two observatories many thousands of kilometres apart. It was the rapid progress in wireless techniques during the war that had made this possible, most clearly evidenced in France. Following the American entry into the war a new transmitter was built at Bordeaux, with a great deal of American help (Figure 3). This huge transmitter with a power of 1 megawatt, needing access to local hydroelectric generation and with an aerial composed of eight 250 metre towers, was the most powerful in the world. It was built to communicate reliably across the Atlantic in conjunction with another transmitter in Maryland. Transmissions could now reach half way across the globe. Although the technology was ready, Ferrié still had to wait a while. That was because of the complex architecture of post-war science: two newly-established bodies, the International Astronomical Union and the International Union of Geodesy and Geophysics were both interested in the project, and established a joint commission chaired by Ferrié. There followed lengthy technical debates, and much refinement of the project, so it took some years to obtain final approval. But it was finally executed in 1926, and was considered a great success: the closing error of the primary triangle was two-thousandths of a second in time (or less than one metre, expressed in distance). And this 'First World Longitude Operation' was followed by two more, in 1933 and 1958. Whilst the later operations became more international and less French in character, the fundamental techniques were unchanged. Such continuity emphasises the historical importance of Ferrié's memorandum to the Bureau des longitudes in 1919: with it, he defined the techniques of precision longitude determination for the entire wireless era.

Figure 2 -

 2 Figure 2 -The title page of Ferrié's memorandum of 5th March, 1919

Figure 3 -

 3 Figure 3 -The Bordeaux transmitter building and six of its aerial towers. It was named Lafayette, after the French Marquis who fought alongside America in the revolutionary war. Construction having started in 1918, it was inaugurated in 1920 and destroyed during the Second World War (Source: Bibliothèque nationale de France)


Harinadha Gidituri

Alois Würger

Kevin Stratford

Juho S Lintuvuori

Keywords: Colloidal particle, contact angle, wetting, Lattice-Boltzmann Method (LBM), drag force, torque

We study the dynamics of a spherical colloidal particle pulled along fluid-fluid interface using lattice Boltzmann (LB) simulations. We consider an interface with a finite width and include both the effects of the thermodynamics of the interface and the particle wetting, characterised by the contact angle θ between the particle surface and the interface, in addition to the viscosity ratio λ between the two fluids. We characterise the particle dynamics by applying a constant pulling force along the interface and measure both the translational and the rotational dynamics as a function of the contact angle and the viscosity ratio. We observe that the hydrodynamic drag is reduced and the particle rotation is increased when the particle resides more in the low viscosity fluid, in agreement with previous hydrodynamic theories. We also present a case where the particle rotation is suppressed. Then, the drag is observed to increase as compared to the situation where rotation is allowed, while the general behaviour of reducing drag while the particle thermodynamically prefers the lower viscosity fluid, is retained.

I. Introduction

Colloidal particles trapped at fluid-fluid interface can be seen in plethora of systems, such as stabilization of emulsions [START_REF] Dinsmore | Colloidosomes: Selectively permeable capsules composed of colloidal particles[END_REF][START_REF] Aveyard | Emulsions stabilised solely by colloidal particles[END_REF][START_REF] Clegg | Colloid-stabilized emulsions: behaviour as the interfacial tension is reduced[END_REF] , foams [START_REF] Horozov | Foams and foam films stabilised by solid particles[END_REF][START_REF] Lam | Pickering stabilization of foams and emulsions with particles of biological origin[END_REF] , bijels [START_REF] Cates | Bijels: a new class of soft materials[END_REF][START_REF] Stratford | Colloidal jamming at interfaces: A route to fluidbicontinuous gels[END_REF][START_REF] Aland | A continuum model of colloid-stabilized interfaces[END_REF][START_REF] Arabjamaloei | Controlling structure of materials derived from spinodally decomposing liquids[END_REF] and liquid marbles [START_REF] Bhosale | Mechanically robust nanoparticle stabilized transparent liquid marbles[END_REF][START_REF] Janardan | Liquid marbles: Physics and applications[END_REF] . Colloidal particles are spontaneously adsorbed on to the fluid-fluid interface and energetics govern the state of the system to reach a minimum energy as first observed by Ramsden [START_REF] Ramsden | Separation of solids in the surface-layers of solutions and 'suspensions' (observations on surface-membranes, bubbles, emulsions, and mechanical coagulation[END_REF] and Pickering [START_REF] Pickering | Cxcvi.-emulsions[END_REF] . The dynamics of a spherical particle trapped at an interface between two fluids or fluid and gas is a widely studied problem both experimentally [START_REF] Radoev | Brownian motion at liquid-gas interfaces. 1. diffusion coefficients of macroparticles at pure interfaces[END_REF][START_REF] Petkov | Measurement of the drag coefficient of spherical particles attached to fluid interfaces[END_REF][START_REF] Walder | Phospholipid diffusion at the oilwater interface[END_REF][START_REF] Wang | Probing diffusion of single nanoparticles at water-oil interfaces[END_REF][START_REF] Sriram | Stokes-einstein and desorption-mediated diffusion of protein molecules at the oil-water interface[END_REF] and theoretically [START_REF] Danov | Influence of the surface viscosity on the hydrodynamic resistance and surface diffusivity of a large brownian particle[END_REF][START_REF] Pozrikidis | Particle motion near and inside an interface[END_REF][START_REF] Dörr | Drag and diffusion coefficients of a spherical particle attached to a fluid-fluid interface[END_REF][START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF][START_REF] Loudet | Drag force on a particle straddling a fluid interface: Influence of interfacial deformations[END_REF][START_REF] Lecrivain | Direct numerical simulation of a particle attachment to an immersed bubble[END_REF][START_REF] Gaddam | Slippage on a particle-laden liquid-gas interface in textured microchannels[END_REF] , due to its importance in various chemical, oil and mineral flotation industries. The hydrodynamic drag of the colloid along the interface depends crucially on the viscosity difference between the two fluids and the particle wetting. In several experiments it has been observed that the hydrodynamic drag decreases when a larger portion of the particle is immersed in the low viscosity fluid or gas [START_REF] Radoev | Brownian motion at liquid-gas interfaces. 1. diffusion coefficients of macroparticles at pure interfaces[END_REF][START_REF] Petkov | Measurement of the drag coefficient of spherical particles attached to fluid interfaces[END_REF][START_REF] Walder | Phospholipid diffusion at the oilwater interface[END_REF][START_REF] Wang | Probing diffusion of single nanoparticles at water-oil interfaces[END_REF][START_REF] Sriram | Stokes-einstein and desorption-mediated diffusion of protein molecules at the oil-water interface[END_REF] . This behavior has been reproduced in theoretical studies, where the drag force is calculated using Stokes hydrodynamics with various boundary conditions, including the presence of shear and dilational surface viscosities [START_REF] Danov | Influence of the surface viscosity on the hydrodynamic resistance and surface diffusivity of a large brownian particle[END_REF] , a finite ratio of bulk viscosities [START_REF] Pozrikidis | Particle motion near and inside an interface[END_REF] , or a solid or membrane boundary on the opposite surface of a thin film [START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF] .

In most previous theoretical approaches [START_REF] Danov | Influence of the surface viscosity on the hydrodynamic resistance and surface diffusivity of a large brownian particle[END_REF][START_REF] Pozrikidis | Particle motion near and inside an interface[END_REF][START_REF] Dörr | Drag and diffusion coefficients of a spherical particle attached to a fluid-fluid interface[END_REF][START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF] , the particle dynamics was studied from a purely hydrodynamic perspective, with the boundary conditions implemented through an infinitely thin interface. The thermodynamics of the interface was reduced to the definition of a contact angle. A very recent study concerning the effects of buoyancy-driven interface deformations on the effective a) Electronic mail: alois.wurger@u-bordeaux.fr b) Electronic mail: juho.lintuvuori@u-bordeaux.fr drag coefficient [START_REF] Loudet | Drag force on a particle straddling a fluid interface: Influence of interfacial deformations[END_REF] , accounted for the thermodynamics of the interface between the two fluids in 2-dimensions.

In this paper we consider the hydrodynamics of an interface particle, and take into account both the thermodynamics of a finite-thickness interface and particle wetting. Using lattice Boltzmann simulations, we explicitly solve the coupled Navier-Stokes and Cahn-Hilliard equations for the binary fluid with surface wettable colloidal particles [START_REF] Kendon | Inertial effects in three dimensional spinodal decomposition of a symmetric binary fluid mixture: A lattice boltzmann study[END_REF] in 3-dimensions. The hydrodynamic boundary conditions at the interface arise from the thermodynamic stress. To simulate the particle dynamics, we set the particle wetting properties, the interfacial tension and the viscosities of the two fluids. The particle is pulled along the interface by a constant force F simulating an active microrheology experiment. The translational and rotational dynamics of the particle are governed by the thermo-hydrodynamic stress integrated over the particle surface as a response to the applied force. We consider two situations: In the first one, the particle rotates due to the torque resulting from the unlike traction exerted on the upper and lower hemispheres, which occurs for a free interface and a perfectly spherical particle. In the second case, particle rotation is inhibited, which in physical terms may occur for surfactant laden interfaces or for contact-line pinning on the particle surface.

II. Methods and simulation details

A. Binary fluid model

To model the thermodynamics of the interface, we consider a phase separating binary fluid characterised by a composition φ(r). The phase behavior is described by a Ginzburg-Landau free energy functional [START_REF] Kendon | Inertial effects in three dimensional spinodal decomposition of a symmetric binary fluid mixture: A lattice boltzmann study[END_REF] ,

G[φ] = dV - A 2 φ 2 + B 4 φ 4 + κ 2 |∇φ| 2 , (1)
where A, B and κ are constants. The first two terms describe the bulk properties; the choice of A = B > 0 corresponds to a phase separated state with equilibrium compositions φ * = ±A/B = ±1. The third term represents the energy cost of interfaces. The interfacial thickness χ and tension σ are related to the above parameters,

χ = 2κ |A| , σ = 8κ|A| 3 9B 2 . ( 2
)
At the equilibrium, the composition profile is given by

φ/φ * = tanh(z/χ), ( 3
)
where z is the coordinate normal to the interface, and the location of an idealized interface can be identified by φ = 0. The temporal evolution of phase field variable φ is governed by a Cahn-Hilliard advection-diffusion equation,

∂ t φ + v • ∇φ = M ∇ 2 µ, (4)
where v is the fluid velocity, M is the mobility and µ is the chemical potential derived from the free energy functional in Eq. 1 by µ = δG δφ = Aφ + Bφ 3 -κ|∇ 2 φ|. The fluid velocity is obtained by solving incompressible Navier-Stokes equation,

ρ∂ t v + v • ∇v = η∇ 2 v -∇P -φ∇µ, (5)
where the gradient of the hydrodynamic pressure P is supplemented by the thermodynamic force φ∇µ.

Binary fluids with different wetting preferences towards a solid surface can be modeled by including an additional surface free energy G s (φ s ) to Eq. 1, where φ s is the order parameter in contact with the surface. The total free energy of the combined system (binary fluid + colloid) is given by 27

G total = G[φ] + G s (φ s )ds, (6)
with a surface contribution

G s = C 2 φ 2 s + Hφ s . (7)
The equilibrium order parameter profile φ s and the boundary condition at the solid surface

dG s dφ s = κ|∇φ • n| ( 8
)
where n is the particle surface normal, can be obtained from the functional minimization of the total free energy (Eq. 6) [START_REF] Desplat | Ludwig: A parallel lattice-boltzmann code for complex fluids[END_REF] . Different wetting behaviors can modeled by tuning of C and H values in Eq. 7. A finite value of C should be considered for systems with a wetting phase transition. In the current study, we considered C = 0 in all cases. The two fluid-solid interfacial tensions can be readily be modified by changing the parameter H. For H = 0 the surface will have an equal affinity towards both fluids, corresponding to a neutrally wetting colloid (θ = 90 • ). A non-zero value of H causes an asymmetry which results in a deviation of the contact angle from 90 • .

B. Simulation details

The simulations were carried out using a hybrid method, where the evolution of the order parameter was solved using finite difference and the flow field v(r), arising from the continuity and Navier-Stokes equations, using lattice Boltzmann 28 . The colloids were modelled as spherical particles with a radius R lattice sites. The no-slip boundary condition at the particle surface was realised by a standard method bounce-back on links method. [START_REF] Ladd | Numerical simulations of particulate suspensions via a discretized boltzmann equation. part 1. theoretical founda-tion[END_REF][START_REF] Ladd | Numerical simulations of particulate suspensions via a discretized boltzmann equation. part 2. numerical results[END_REF] , and the boundary condition for φ was solved numerically [START_REF] Desplat | Ludwig: A parallel lattice-boltzmann code for complex fluids[END_REF] .

We use a 3-dimensional simulation domain consisting of 128 × 128 × 128 lattice sites with periodic boundary conditions in all directions.

The simulation parameters below are given in lattice units, where the lattice spacing, time step and density are fixed to unity: ∆x = 1, ∆t = 1 and ρ = 1, as customary in LB simulations. The size of the colloid is fixed at R = 12∆x. The validation of the domain and colloid sizes are presented in the appendix.

The viscosity of the φ = +1 fluid was fixed η 1 = 0.625 throughout the paper. The viscosity ratio (λ = η 2 /η 1 ) is varied by changing the viscosity η 2 of the φ = -1 fluid. To have a continuous viscosity across the interface, a composition dependant viscosity η(φ) following an Arrhenius relation [START_REF] Langaas | Lattice boltzmann simulation of a binary fluid with different phase viscosities and its application to fingering in two dimensions[END_REF] ,

η(r) = η 1-φ 2 2 η 1+φ 2 1 (9)
was used.

The densities of both of the fluids and the colloids were fixed for a constant value of 1.0. For the binary fluid A = B = 0.00258, surface penalty κ = 0.004 and mobility M = 4.0 were used. Using the parameters above and Eq. 2, we get an interfacial width χ ≈ 1.76 and interfacial tension σ ≈ 0.003 in lattice units.

We consider both freely rotating particles, and particles without rotational motion. To inhibit the rotation, a director ŝ is assigned on the particle and strong magnetic field B 0 is applied to align the particle director ŝ along the applied force F. The resulting torque T = B 0 × ŝ inhibits the rotational motion of the particle. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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C. Particle contact angle

We study the dynamics of a colloidal particle at a fluidfluid interface. A spherical colloidal particle of radius R is placed at an interface between two fluids with different viscosities η 1 ≥ η 2 (Fig. 1). Depending on the surface wetting, the particle adopts a contact angle θ in equilibrium which can be measured using a relation

cosθ = ∆l R ( 10
)
where ∆l is the difference between interface position and center of the colloid (Fig. 1).

In the simulations, the contact angle can be tuned by varying the free energy parameters and follows theoretically

27 cosθ = 1 2 [-(1 -h) 3 2 + (1 + h) 3 2 ] ( 11
)
where the wetting parameter h is given by,

h = H √ κB . (12)

D. Non-dimensional numbers

The hydrodynamic non-dimensional numbers describing the interface deformation and flow behavior are the capillary and Reynolds numbers, respectively

Ca = η 1 V σ (13) Re = ρRV η 1 , ( 14
)
where V is the steady state particle velocity, and the viscosity η 1 of the fluid φ = +1 was used in the calculation of both Re and Ca. Using the simulations parameters, we recover Re ∼ O(10 -3 -10 -4 ) and capillary number Ca ∼ O(10 -2 ). Due to low value of the Ca, we are in the regime where deformation of the interface can be neglected.

We consider a density matched system, thus the gravitational Bond number

Bo = (ρ p -ρ 1 )gR 2 /σ = 0. ( 15
)
An additional non-dimensional measure associated with interface thickness is given by Cahn number,

Cn = χ R (16)
In the current study, Cn is fixed at a constant value of 0.15.

III. Results and Discussion

A. Contact angle validation

Depending the wetting of the particle surface, the colloid is displaced vertically with respect to the interface (Fig. 2). This allows the measurement of the contact angle θ using equation 10. In Fig. 2(a) the influence of wetting parameter h on the contact angle observed from the simulations is compared with the theoretical values. The simulations are in good agreement with the theoretical estimation for θ ∈ [60 • , 120 • ]. However, a deviation from the theoretical prediction is observed towards the two extremes h = ±0.4 (Fig. 2(a)). This deviation probably arises due to the discretisation of the particle surface on the lattice. We have thus limited our range between θ ∼ 108 • and θ ∼ 72 • respectively as upper and lower limit for θ in the simulations of the particle dynamics.

B. Drag on a neutrally wetting colloid

To validate our simulation method, we conducted a numerical experiment with a neutrally wetting colloidal particle (θ = 90 • ) dragged along the fluid-fluid interface with a viscosity ratio λ = 1 (Fig. 2c).

Under Stokes flow, the drag force on a translating spherical particle in the bulk fluid is given by,

F = 6πηRV ( 17
)
where η is the viscosity of the fluid, R is the radius of the particle and V is its velocity. By varying F , we find a linear force-velocity relation as expected from Stokes's law (Fig. 3). Since our domain size is finite and we used periodic boundary conditions our drag coefficient would be af-This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. fected by finite size effects. To correct for these, we use a correction by Hasimoto 32 derived for the Stokes' drag of a periodic array of spherical particles. For the case of simple cubic arrangement, corresponding a single colloid in a cubic box with periodic boundary conditions, the FIG. 3. The force-velocity characteristics for the colloidal particle of radius R = 12.0 at the interface with contact angle θ = 90 • and viscosity ratio λ = 1.0. The simulation data is fitted with Stokes drag (Eq. 17) of a spherical particle and includes Hashimoto 32 correction parameter Q Eq. 18 (see the text for more details).

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correction is given by 32

Q = 1 -1.7601 3 C τ + C τ -1.5593C 2 τ (18) C τ = 4π 3 R 3 L 3 (19)
where C τ is the volume fraction, L is the box length.

Using the correction, we recover an excellent agreement between the simulations and the Stokes' result (Eq. 17) for a neutrally wetting colloid at an interface with λ = 1 (Fig. 3).

C. Translation dynamics: the effect of the viscosity ratio and a contact angle

The main aim of this work is to study the hydrodynamic drag of a spherical colloidal particle trapped at an interface between two fluids. The drag coefficient is determined as the function of the viscosity contrast λ = η 2 /η 1 and the contact angle θ. To investigate the translational dynamics, the particle was pulled along the interface by a constant force F = 0.01125. After initial transient, the particle velocity was averaged over the last 1 × 10 5 steps in the steady state.

For a contact angle different from 90 • or for a viscosity ratio different from 1, the drag on the upper and lower hemispheres differs, resulting in a torque around an axis perpendicular on the interface normal and the direction of the translational velocity V . The resulting translationrotation coupling leads to a superposition of gliding and rolling motion with a finite angular velocity Ω, as discussed in more detail in the section III D. Further, we consider the case where the rotation is inhibited and find a velocity V which in general differs from V .

In Fig. 4(a) we plot both V and V as a function of ǫ = cos θ, for various values of the viscosity ratio λ. The This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. velocity shows a roughly linear variation with ǫ and decreases with increasing viscosity ratio λ. Both dependencies are qualitatively understood when noting that cos θ provides a measure of the fraction of the particle volume immersed in the less viscous fluid, and that 1+λ accounts for the mean viscosity of the two fluids in the units of η 1 .

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Regarding the effect of the rolling motion, we find that the velocity of rotating particles is in general larger, V > V , and that this effect increases with ǫ yet decreases with λ. These dependencies are readily understood from the data in Fig. 4: The overall reduction of the viscous stress due to the rotation is most efficient for a large volume fraction immersed in the low-viscosity fluid (large ǫ) and for a large viscosity contrast (small λ).

The drag coefficient ζ = F/V of a freely rotating particle is given by the ratio of the applied force F and the velocity V ; we also define a reduced drag coefficient

ζ r = ζ(λ) ζ(1) . ( 20
)
Similarly we define ζ and ζr when the particle rotation is inhibited. The drag ratio ζ r is plotted against contact angle for different values of the viscosity ratio λ in Fig. 4(b). We find that at constant ǫ, the drag increases with increasing viscosity ratio λ. At constant λ < 0, the drag increases with the immersion parameter ǫ, that is, with the fraction of the particle volume immersed in the high-viscosity phase, in agreement with the previous studies [START_REF] Danov | Influence of the surface viscosity on the hydrodynamic resistance and surface diffusivity of a large brownian particle[END_REF][START_REF] Pozrikidis | Particle motion near and inside an interface[END_REF][START_REF] Dörr | Drag and diffusion coefficients of a spherical particle attached to a fluid-fluid interface[END_REF][START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF][START_REF] Fischer | The viscous drag of spheres and filaments moving in membranes or monolayers[END_REF] .

V 𝜆 = 1, ϵ < 0 Ω < 0 V 𝜆 = 1, ϵ > 0 Ω > 0 V 𝜆 < 1, ϵ = 0 Ω > 0 a) c) b)

D. Rotational dynamics

The hydrodynamic drag forces on the upper and lower hemispheres differ in general, thus exerting a torque which results in a steady-state rotation of the particle, as shown schematically in Fig. 5. We calculated the angular velocity Ω of the colloid using the same averaging procedure as the calculation of V .

In Fig. 6 we plot Ω as a function of the immersion parameter ǫ = cos θ for different values of the viscosity ratio λ. For identical viscosities, λ = 1, the angular velocity is small and roughly linear in ǫ. The sign of Ω is readily understood when noting that the interface position with respect to the particle center reads Rǫ. Close to the contact line the flow field is strongly perturbed by the presence of the interface, thus engendering additional viscous stress and enhancing the backward drag force on the lower hemisphere in Fig. 5a and on the upper hemisphere in Fig. 5b. As a consequence, the particle rotates clockwise in (a) and counter-clockwise in (b).

Much larger values of Ω occur for a finite viscosity contrast between the two fluids. In our simulations λ < 1 indicates higher viscosity in the upper halfspace, such that the backward drag force is larger on the upper hemisphere and the particle rotates counterclockwise (Fig. 5c). In the case λ < 1 and ǫ < 0, the mechanisms (a) and (c) partly cancel each other. The data in Fig. 6 show that the viscosity ratio has a stronger effect; for λ < 0.8 the angular velocity is positive, even for negative ǫ.

Das et. al. [START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF] considered spherical neutrally wetting particles straddling on a liquid film on a solid surface. Our value ΩR/V = 0.206 for λ = 0.1 and ǫ = 0, differs This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. by less than 2% from ΩR/V = 0.21 calculated by Das et. al. [START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF] for λ = 0 and ǫ = 0 at large film depth. At present it is not clear whether this tiny discrepancy arises from the difference in λ or to the fact that Das et al. consider an ideal interface (see below).

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E. Comparison with theory

Our simulations for non-rotating particles are largely motivated by the existence of theoretical predictions for particles moving along an ideally thin and stiff interface. Regarding the dependence on the contact angle, we compare our findings for the velocity V with the perturbative expression for the drag force at a gas-liquid interface (λ = 0), first derived by Brenner [START_REF] Brenner | The stokes resistance of a slightly deformed sphere[END_REF] and reinterpreted by Dörr and Hardt [START_REF] Dörr | Driven particles at fluid interfaces acting as capillary dipoles[END_REF] ,

F = 3πη 1 R V 1 - 9 16 cos θ + ... . (21)
This formula is exact to linear order in ǫ; for ǫ = 0 it reduces to half of the Stokes drag in a bulk liquid. Solving for V and taking the derivative with respect to ǫ we find

1 V d V dǫ ǫ=0 = 9 16 (λ = 0). (22)
In Fig. 7 we show this derivative obtained from our simulation as a function of λ. As expected the derivative vanishes at λ = 1 and increases for smaller values of λ, that is, for increasing viscosity contrast. The simulation data at small λ agree almost quantitatively with the theoretical result (22), indicated as an open star. Still, extrapolating our data to λ = 0 would suggest a derivative slightly larger than the value 9 16 expected for an ideally thin and stiff interface. Now we turn to the dependence of the drag on a neutrally wetting particle as a function of the viscosity ratio, which has been derived by Pozrikidis 20 ,

F = 3πη 1 R V (1 + λ), (ǫ = 0). (23)
The corresponding drag ratio reads as

ζr = 1 + λ 2 , (ǫ = 0). (24)
For λ = 1 this corresponds to a bulk fluid, whereas for λ = 0 it represents a gas-liquid interface, where the drag force is half of the bulk value. In Fig. 8 we compare the drag ratio obtained from our simulation data for ǫ = 0 with Pozrikidis's result. For viscosity ratios λ > 1 2 we find a good agreement. For smaller values, however, our simulation shows a lower drag ratio than expected from theory for interfaces of zero thickness and infinite capillary number. In the limit λ → 0 the simulation data for ζr (open circles) clearly do not converge toward 1 2 but toward a significantly smaller value below 0.4. This effect occurs for both rotating and non-rotating particles, though it is slightly larger for the former. Furthermore, we also present the simulation data of Das et al. [START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF] (open and closed stars) for an ideal gas-liquid interface (λ = 0). Their data for a non-rotating particle (open star) is in excellent agreement with Pozrikidis's data, whereas the drag ratio for a rotating particle (closed star) is reduced by about ten percent. This reduction in the drag ratio for rotating particles compares well with the findings at small λ of the present work.

To rationalize the deviation of our data for the drag ratio ζ r from theory, we remind that our simulations are done for a Cahn-Hilliard diffuse interface of a finite width and a finite surface tension, whereas the theory results, eqs. ( 22) and (24), and the data of Das et al. are obtained for an an ideal interface of infinite tension and zero width. In the view of the rather small capillary number used in our simulations, Ca = 0.01, the strong This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. discrepancy shown visible in Fig. 8 can hardly be related to deformation of the interface. This suggest that the observed deviations may stem from the finite interfacial width, and could be related to a decoupling of the thermodynamic and hydrodynamic interfaces, defined by zero composition variable φ = 0 and zero normal velocity v ⊥ = 0, respectively. In the steady-state and for small Reynolds number, the Navier-Stokes equation ( 5) simplifies to

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η∇ 2 v -∇P -φ∇µ = 0. ( 25
)
At the thermodynamic definition for the interface, φ = 0, the last term vanishes. Yet there does not seem to be an obvious argument why the vertical velocity should vanish at the surface defined by φ = 0. In Fig. 9 we plot the phase φ and the viscosity η across the interface. Both vary smoothly at the scale of the interface width χ, and so does the stress term in (25). Intuitively one would expect that, in order to minimize the overall dissipation, the flow field becomes smoother in the more viscous fluid above the interface, and that the sharp velocity changes required at the contact line rather occur in the less vis-cous phase. This would imply an asymmetric flow profile in the vicinity of the surface φ = 0, and an overall reduction of dissipation with respect to the ideal interface of zero thickness. This could explain our observation of the reduced drag coefficient shown in Fig. 8. The spatial resolution of the interface in our simulations was not sufficient to provide unambiguous data on the flow profile within and close to the surface φ = 0. In future studies, this could improved by increasing the interfacial width χ.

IV. Conclusions

In summary, we studied the dynamics of a spherical colloidal particle at fluid-fluid interface using Lattice-Boltzmann Method (LBM). Our simulations include the effects of interfacial thermodynamics and particle wetting, in addition to the hydrodynamics. The translational and angular velocities of the particle were analysed both as the function of the viscosity ratio λ between the two fluids, and the contact angle θ between the fluids and the colloidal surface.

Our results are in overall good agreement with previous hydrodynamic theories. The drag is reduced when the particle prefers thermodynamically a position where its surface is more in the low viscosity fluid. Further, the viscosity contrast leads to a rotational motion of the particle. The effect of the rotation was analysed by applying a strong aligning field, which stops the rotation and slighlty increases the hydrodynamic drag. Comparison with theoretical findings and previous simulations for ideally thin and stiff interfaces, reveals a discrepancy in the drag coefficient, which could be related to the finite width of the interface considered here. racy of the numerical results is sensitive to choice of both size of the simulation box (L) and particle radius (R). Table I shows the domain independence study on drag force calculations with particle radius R = 12, viscosity ratio λ = 1.0 and contact angle θ = 90 • using three values of domain sizes 108, 128 and 160. It is evident from the table that, increasing the domain size from 128 to 160, there is a negligible change (< 1%) in the value of drag force value. Since the error is under an acceptable limit, we have used domain size of 128 × 128 × 128 in all our simulations to reduce the computational cost. After fixing the domain size at a satisfactory value, it is also important to find satisfactory size of the particle. Fig. 10, shows the deviation from the theoretical drag force calculation for different particle sizes with viscosity ratio λ = 1.0 and contact angle θ = 90 • using two values of interfacial widths χ = 1.13, 1.76. The error percentage is small (< 2%) for the particle sizes 10, 12 and 14 with χ = 1.76. We have chosen the colloid radius R = 12 and interfacial width χ = 1.76 in the present work.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

FIG. 1 .

 1 FIG. 1. Schematic representation of a colloid particle with radius R at fluid-fluid interface with two different viscosities η1 and η2 respectively. The contact angle (θ) is defined as the angle measured between the tangent at the colloid surface and fluid-fluid interface.

  FIG. 2. (a) Contact angle versus wetting parameter defined as h = H 1 κB for the case of viscosity ratio λ = 1.0. Blue circles are the values obtained from the simulation using Eq. 10 and red line is from theoretical calculation using Eq. 11. Different values of colloidal contact angles measured at the interface. (b-d) Snapshots from the simulations for different contact angles: (b) θ ≈ 72 • , (c) θ ≈ 90 • and (d) θ ≈ 108 • . Red colour corresponds to φ = +1 and and blue to φ = -1 fluids with viscosities η1 and η2, respectively.

  FIG. 4. (a) Velocity versus ǫ = cos θ for different values of the viscosity ratio λ. Closed and open symbols correspond to observed velocities with V and without rotation V , respectively. (b) The effective drag coefficient ratio ζr and ζr, as defined in Eq. (20).

FIG. 5 .

 5 FIG. 5. Schematic view of translation-rotation coupling of a particle moving at velocity V to the right. (a) The viscosity is the same above and below the interface, λ = 1, wheres the large contact angle θ > 90 • implies ǫ < 0. Because of the additional viscous stress at the contact line, the drag force on the lower hemisphere is larger than on the upper one, as indicated by double arrows, resulting in a clockwise rotation of the particle. (b) For a small contact angle θ < 90 • , the contact line is on upper hemisphere, and the particle rotates counter-clockwise. (c) The viscosity in the upper halfspace is larger than in the lower, λ < 1. The larger viscous drag on the upper hemisphere favors counter-clockwise rotation.

FIG. 6 .

 6 FIG. 6. The angular velocity of the colloid Ω versus cosine of contact angle ǫ = cos θ for different values of the viscosity ratio λ.

FIG. 7 .

 7 FIG. 7. Velocity derivative taken with respective to ǫ in the close neighbourhood around ǫ = 0, where linear approximation is valid (slopes from Fig. 4(a) normalized with colloid velocity at ǫ = 0 at corresponding λ value. Open star corresponds to theory (Eq. 21); closed squares correspond to case with rotation and open squares correspond to without rotation of the colloid.

FIG. 8 .

 8 FIG.8. Comparison of non-dimensional drag ratio (ζr) given in Eq. 20 for the colloidal particle at fluid-fluid interface with contact angle ǫ = 0 for different values of viscosity ratio (λ) with Refs.[START_REF] Pozrikidis | Particle motion near and inside an interface[END_REF][START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF] . Here circles are from the simulations, closed and open stars correspond to Das et. al.[START_REF] Das | The translational and rotational dynamics of a colloid moving along the air-liquid interface of a thin film[END_REF] with and without particle rotation. The solid line is obtained from a theoretical estimation given by Eq. 23.

1 𝜙FIG. 9 .

 19 FIG.9. Phase and viscosity variation through the interface, with the length scale χ defined in (2).

  FIG.10. Error percentage between drag force obtained from simulation and theory (Eq. 23) for different values particle radius (R) and interfacial thickness (χ). Simulations were carried out using a periodic simulation domain 128×128×128.

TABLE I .

 I Relative error in drag force measurement from simulations and theory (Eq.17

This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0047530

Acknowledgements HG would like to thank Dr. Zaiyi Shen and Sotiris Samatas for engaging and helpful discussions. HG and JSL acknowledge IdEx (Initiative d'Excellence) Bordeaux for funding, Curta cluster for computational time. JSL acknowledges support by the French National Research Agency (ANR) project GASPP through Contract No. ANR-19-CE06-0012-01. AW acknowledges support through contract No. ANR-19-CE30-0012-01

Appendix Domain and grid independence

The simulations were initialised with a fully separated binary fluid mixture with a well-defined interface between the fluids following an equilibrium profile (Eq. 3). A colloidal particle is placed at the interface, and a constant pulling force F is applied along the interface. The accu-This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0047530