Xian Huang

Dominique Costa
email: dominique.costa@chimieparistech.psl.eu

Boubakar Diawara

Vincent Maurice

Philippe Marcus
email: philippe.marcus@chimieparistech.psl.eu

Atomistic insights on enhanced passivity: DFT study of substitutional Mo on Cr2O3 and Fe2O3 surfaces

Keywords: A) stainless steel, (B) DFT, (C) passive films, (C

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The corrosion resistance of many Cr-containing alloys (including Ni-based alloys [START_REF] Wang | XPS and ToF-SIMS investigation of native oxides and passive films formed on nickel alloys containing chromium and molybdenum[END_REF], stainless steels [START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316L stainless steel surfaces and effects of controlled pre-oxidation[END_REF], and high entropy alloys [START_REF] Wang | Enhanced passivity of Cr-Fe-Co-Ni-Mo multi-component singlephase face-centred cubic alloys: design, production and corrosion behaviour[END_REF]) relies on the nanometre-thick native or passive oxide films naturally or electrochemically formed on the surface and protecting the substrate from corrosive attack by the environment. These oxide films are most commonly composed of an inner barrier layer mainly containing chromium oxide and an outer layer containing oxides and hydroxides [START_REF] Wang | XPS and ToF-SIMS investigation of native oxides and passive films formed on nickel alloys containing chromium and molybdenum[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][START_REF] Wang | Enhanced passivity of Cr-Fe-Co-Ni-Mo multi-component singlephase face-centred cubic alloys: design, production and corrosion behaviour[END_REF]. The inner barrier layer may present weak sites of corrosion protection, locally depleted in Cr and enriched in Fe, and originating from the build-up of oxide film in the early stages of oxidation as revealed on stainless steel surfaces [START_REF] Ma | Origin of nanoscale heterogeneity in the surface oxide film protecting stainless steel against corrosion[END_REF][START_REF] Ma | Nanoscale early oxidation mechanisms of model FeCrNi austenitic stainless steel surfaces at room temperature[END_REF]. It is well known that the addition of Mo in these alloys enhances the resistance to breakdown of the passive film and to the initiation of localized corrosion, especially in aggressive Cl-containing environments. In a recent study of the enhanced passivity of Mo-bearing multi-principal element alloys (Cr-Fe-Co-Ni-Mo) [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF][START_REF] Wang | Effects of chloride ions on passive oxide films formed on Cr-Fe-Co-Ni(-Mo) multi-principal element alloy surfaces[END_REF], molybdenum oxide was found enriched mainly as Mo(VI) in the outer layer, and as Mo(IV) concentrated at the inner/outer layer interface including where the Fe-rich weak sites of the inner layer are exposed (Figure 1.a). It was proposed that the addition of Mo may reinforce the protection provided by the inner barrier layer by curing these Fe-rich weak sites of passivity [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF]. However, the mechanisms for this curing effect are still not well understood. In order to understand the detailed mechanisms, more investigations of the Mo effects are needed, including atomistic modelling of the interaction of Mo with chromium and iron oxide surfaces.

Simulations in the frame of quantum mechanics, such as Hartree-Fock (HF) method and density functional theory (DFT), are powerful techniques to study materials at the atomic scale. The energies of low-index surfaces of chromia (α-Cr2O3 phase of corundum structure) have been calculated using HF method [START_REF] Sun | Structure and surface energy of low-index surfaces of stoichiometric α-Al2O3 and α-Cr2O3[END_REF], indicating that the (011 ̅ 2) orientation is the most stable in agreement with high-resolution transmission electron microscopy observation on sintered Cr2O3 [START_REF] Scarano | Interaction of CO with α-Cr2O3 surface: a FTIR and HRTEM study[END_REF]. As for surfaces of hematite (α-Fe2O3 phase also of corundum structure), a DFT+U calculation has suggested the (0001) surface as the most stable orientation [START_REF] Dzade | A density functional theory study of the adsorption of benzene on hematite (α-Fe2O3) surfaces[END_REF]. Since the ultrathin Cr2O3 oxide film grown on Cr(110) [START_REF] Maurice | XPS, LEED and STM study of thin oxide films formed on Cr(110)[END_REF] and the inner barrier layers of the oxide films formed on Cr(110) [START_REF] Maurice | XPS and STM investigation of the passive film formed on Cr(110) single-crystal surfaces[END_REF][START_REF] Zuili | In situ scanning tunneling microscopy study of the structure of the hydroxylated anodic oxide film formed on Cr(110) single-crystal surfaces[END_REF], FeCr(110) [START_REF] Maurice | XPS and STM study of passive films formed on Fe-22Cr(110) single-crystal surfaces[END_REF] and FeCrNi(100) [START_REF] Maurice | X-ray photoelectron spectroscopy and scanning tunneling microscopy study of passive films formed on (100) Fe-18Cr-13Ni single-crystal surfaces[END_REF] are oriented to the [0001] direction according to the results of low-energy electron diffraction (LEED) and scanning tunneling microscopy (STM), the (0001) surface is more of interest here.

Using M for Cr or Fe metal, the (0001) surface can be double-M-terminated (-O3M2), single-M-terminated (-O3M), chromyl-or ferryl-terminated (-O3MO) or O-terminated (-M2O3). The single-M termination is the most often considered in simulations since it possesses null dipole moment for each repeatable stacking unit (M-O3-M) according to the Tasker classification [START_REF] Tasker | The stability of ionic crystal surfaces[END_REF]. The surface phase diagrams for chromia and hematite have been calculated using surface energies obtained from DFT calculations [START_REF] Wang | Surface phase diagram for Cr2O3 (0001) : Ab initio density functional study[END_REF][START_REF] Rohrbach | Ab initio study of the (0001) surfaces of hematite and chromia: Influence of strong electronic correlations[END_REF]. When the oxygen potential increases, the most stable termination changes from single-M-terminated to chromyl-or ferryl-terminated, and then to O-terminated-like (-O3Cr3O3 or -O3Fe3O3) at zero oxygen potential. By using DFT+U method, Rohrbach et al. [START_REF] Rohrbach | Ab initio study of the (0001) surfaces of hematite and chromia: Influence of strong electronic correlations[END_REF] found that the O-terminated-like surfaces are strongly disfavoured and the chromyl-or ferryl-terminated surfaces become the most stable at zero oxygen potential. Using different exchange-correlation functionals or different Hubbard parameters [START_REF] Souvi | Influence of water, dihydrogen and dioxygen on the stability of the Cr2O3 surface: A first-principles investigation[END_REF][START_REF] Souvi | A DFT study of the hematite surface state in the presence of H2, H2O and O2[END_REF][START_REF] Huang | Surface-specific DFT + U approach applied to α-Fe2O3 (0001)[END_REF], different terminations were found favourable at zero oxygen potential, indicating an important influence of the simulation method on the surface phase stability. In the experiments including electron energy loss spectroscopy (EELS) and ion scattering spectroscopy (ISS) [START_REF] Kuhlenbeck | Adsorption and reaction on oxide surfaces: CO and CO2 on Cr2O3(111)[END_REF], synchrotron X-Ray diffraction (SXRD) [START_REF] Th | X-ray structure analysis of the Cr2O3(0001)-(1×1) surface: evidence for Cr interstitial[END_REF] and STM [START_REF] Maurice | XPS, LEED and STM study of thin oxide films formed on Cr(110)[END_REF], single-M termination was observed on the surface of chromia films grown by oxidation on Cr(110) [START_REF] Maurice | XPS, LEED and STM study of thin oxide films formed on Cr(110)[END_REF][START_REF] Kuhlenbeck | Adsorption and reaction on oxide surfaces: CO and CO2 on Cr2O3(111)[END_REF][START_REF] Th | X-ray structure analysis of the Cr2O3(0001)-(1×1) surface: evidence for Cr interstitial[END_REF] and bulk hematite [START_REF] Lübbe | A LEED analysis of the clean surfaces of α-Fe2O3 (0001) and α-Cr2O3 (0001) bulk single crystals[END_REF], while O termination were reported on the surface of hematite film epitaxially grown on Pt(111) [START_REF] Ketteler | Surface structures of α-Fe2O3 (0001) phases determined by LEED crystallography[END_REF].

On a single-M-terminated (0001) surface, the terminating metal atom can locate on three kinds of sites. Structural rearrangement between these sites [START_REF] Bender | Structural rearrangement and surface magnetism on oxide surfaces: a temperature-dependent low-energy electron diffraction-electron energy loss spectroscopy study of Cr2O3(111)/Cr(110)[END_REF] and complicated configurations [START_REF] Th | X-ray structure analysis of the Cr2O3(0001)-(1×1) surface: evidence for Cr interstitial[END_REF] have been observed experimentally on chromia surfaces. To be simple, the terminating metal atom is usually considered on the same site as in the bulk, which was suggested as the most stable site on chromia surfaces by LEED analysis coupled with empirical potential calculations [START_REF] Rohr | Strong relaxations at the Cr2O3(0001) surface as determined via low-energy electron diffraction and molecular dynamics simulations[END_REF]. In this configuration, a strong contraction (compared to bulk) of the interlayer distance between terminating M and first O layer (50 ~ 63% for chromia [START_REF] Rohrbach | Ab initio study of the (0001) surfaces of hematite and chromia: Influence of strong electronic correlations[END_REF][START_REF] Th | X-ray structure analysis of the Cr2O3(0001)-(1×1) surface: evidence for Cr interstitial[END_REF][START_REF] Rohr | Strong relaxations at the Cr2O3(0001) surface as determined via low-energy electron diffraction and molecular dynamics simulations[END_REF][START_REF] Rehbein | Structure of the α-Cr2O3 (0001) surface: An ab initio total-energy study[END_REF] and 27 ~ 79% for hematite [START_REF] Rohrbach | Ab initio study of the (0001) surfaces of hematite and chromia: Influence of strong electronic correlations[END_REF][START_REF] Lübbe | A LEED analysis of the clean surfaces of α-Fe2O3 (0001) and α-Cr2O3 (0001) bulk single crystals[END_REF]) is often reported from both experiments and calculations. The interlayer distance between first O layer and second M layer was found slightly expanded (0 ~ 12%) [START_REF] Rohrbach | Ab initio study of the (0001) surfaces of hematite and chromia: Influence of strong electronic correlations[END_REF][START_REF] Lübbe | A LEED analysis of the clean surfaces of α-Fe2O3 (0001) and α-Cr2O3 (0001) bulk single crystals[END_REF][START_REF] Rohr | Strong relaxations at the Cr2O3(0001) surface as determined via low-energy electron diffraction and molecular dynamics simulations[END_REF][START_REF] Rehbein | Structure of the α-Cr2O3 (0001) surface: An ab initio total-energy study[END_REF] except for chromia surfaces in experiments (contracted 3 ~ 20 % [START_REF] Th | X-ray structure analysis of the Cr2O3(0001)-(1×1) surface: evidence for Cr interstitial[END_REF][START_REF] Rohr | Strong relaxations at the Cr2O3(0001) surface as determined via low-energy electron diffraction and molecular dynamics simulations[END_REF]).

Very few studies have addressed Mo effects on chromia and hematite surfaces [START_REF] Niu | Boosting electrocatalytic ammonia synthesis of bio-inspired porous Mo-doped hematite via nitrogen activation[END_REF][START_REF] Ng | DFT study of oxygen reduction reaction on chromia and hematite: insights into corrosion inhibition[END_REF]. Mo doping has been simulated on the Fe2O3 surface for studying its effect on the nitrogen reduction reaction [START_REF] Niu | Boosting electrocatalytic ammonia synthesis of bio-inspired porous Mo-doped hematite via nitrogen activation[END_REF]. Regarding corrosion resistance, Ng et al. [START_REF] Ng | DFT study of oxygen reduction reaction on chromia and hematite: insights into corrosion inhibition[END_REF] have investigated Mo effects on hydrous (0001) surfaces of chromia and hematite with an OH termination using DFT+U. It was found that on Mo-free surfaces, the energies needed to activate the oxygen reduction reaction are higher on the chromia than on the hematite surface, suggesting a better resistance on the chromia surface for a corrosion reaction under cathodic control. After substitution of Cr or Fe by Mo, the activation energies on both surfaces increase, suggesting that substitutional Mo may reinforce the corrosion resistance by slowing down the counterpart cathodic activity.

The formation of intrinsic point defects has been well studied in bulk chromia and hematite and recent studies have summarized the calculations performed using different techniques [START_REF] Huang | A DFT study of defects in paramagnetic Cr2O3[END_REF][START_REF] Banerjee | Critical assessment of the thermodynamics of vacancy formation in Fe2O3 using hybrid density functional theory[END_REF]. On surfaces, intrinsic point defects are also involved [START_REF] Nguyen | Defective α-Fe2O3 (0001): an ab initio study[END_REF][START_REF] Pang | The effect of surface vacancies on the interactions of Cl with a α-Fe2O3 (0001) surface and the role of Cl in depassivation[END_REF][START_REF] Feng | Characteristics of dopant distribution and surface oxygen vacancy formation for modified Fe2O3 in chemical looping combustion[END_REF][START_REF] Sarfo | Density functional theory study of the initial stages of Cl-induced degradation of α-Cr2O3 passive film[END_REF][START_REF] Yin | Cl-induced passivity breakdown in α-Fe2O3 (0001), α-Cr2O3 (0001), and their interface: A DFT study[END_REF]. Nguyen et al. [START_REF] Nguyen | Defective α-Fe2O3 (0001): an ab initio study[END_REF] have investigated vacancies on hematite surfaces, but only in the outermost and near outermost surface atom layers. Pang et al. [START_REF] Pang | The effect of surface vacancies on the interactions of Cl with a α-Fe2O3 (0001) surface and the role of Cl in depassivation[END_REF] have reported a surface reconstruction resulting from the creation of O single and Fe-O pair vacancies on hematite surfaces, where the outermost iron layer previously above the outermost oxygen layer was displaced into the subsurface by structural relaxation. It was also reported that increasing the U parameter in DFT+U study increases the formation energies of Fe and O vacancies [START_REF] Pang | The effect of surface vacancies on the interactions of Cl with a α-Fe2O3 (0001) surface and the role of Cl in depassivation[END_REF]. Regarding Mo effects on surfaces containing intrinsic point defects, it has not yet been studied whilst the chromia-enriched inner layer protective against corrosion is often considered to grow via the diffusion of intrinsic defects [START_REF] Young | High Temperature Oxidation and Corrosion of Metals[END_REF].

Here we report on substitutional Mo effects on (0001)-oriented surfaces of chromia and hematite (α-Cr2O3 and α-Fe2O3 phases hereafter denoted as Cr2O3 and Fe2O3, respectively) simulated by DFT+U calculations. The thermodynamic framework in which our study was carried out is first explained, then the study of bulk materials and perfect surfaces is presented before investigating substitution by Mo on neutral and charged surfaces. This is followed by reporting the study of vacancy formation on Mo-free and Mo-substituted surfaces. The work provides atomistic insights into the mechanisms by which Mo may enhance the stability of the inner barrier layers of passive films formed on the surfaces of stainless steel and other Crcontaining alloys.

Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543

Method

Thermodynamic model

The native or passive oxide films formed on Cr-containing alloys under air or aqueous solutions, respectively, [START_REF] Wang | XPS and ToF-SIMS investigation of native oxides and passive films formed on nickel alloys containing chromium and molybdenum[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][START_REF] Wang | Enhanced passivity of Cr-Fe-Co-Ni-Mo multi-component singlephase face-centred cubic alloys: design, production and corrosion behaviour[END_REF] contain generally an inner barrier layer markedly enriched in chromium oxide and an outer layer formed by oxides and hydroxides of chromium and other alloying elements. Figure 1.a illustrates such a chemical duplex structure reported for the passive film formed on a Cr-Fe-Co-Ni-Mo multi-principal element alloy [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF]. To simplify the calculation of defect formation energies, gaseous environment containing O2 is considered through an atomistic thermodynamic approach in the present study. Cr2O3 (or Fe2O3) in contact with the environment is firstly considered before investigating more complex bilayer structures in a further study. Figure 1.b shows the simplified model used in this study and to be compared with the experimental one. To explain how defect formation energy is calculated in the present work, we use Cr2O3 as an example. A thick Cr2O3 lattice is in contact with gas that contains O2 on one side (surface) and with a Cr-containing substrate alloy on the other side (interface). The atomic planes of the oxide near surface and interface and those of the alloy near interface may deform due to surface relaxation and crystallographic mismatch.

The effect of pressure on solid state is omitted, and the assumption of perfect gas is used so that the free energy at different pressures can be estimated by that at the standard pressure. The formation entropy is considered negligible with respect to formation enthalpy, and the enthalpies at the standard pressure are approximated by energies at 0 K. With these assumptions, defect formation energy as function of temperature and pressure can be calculated based on DFT energies from ground state calculations. At the alloy/oxide interface, we consider the partial dissociation of the oxide by escape of a Cr atom into the alloy lattice for creation of a Cr vacancy in the Cr2O3 lattice. The defect formation reaction and its energy ∆ f 𝐸 def for Cr vacancies in the oxide at the interface can be written as:

Cr 2𝑛 O 3𝑛 (int) → Cr 2𝑛-1 O 3𝑛 (int) + Cr(int) (1)
∆ f 𝐸 def,int = 𝐸 Cr 2𝑛-1 O 3𝑛 (int) -𝐸 Cr 2𝑛 O 3𝑛 (int) + 𝐸 Cr(int) ( 2
)
where 𝐸 is the energy (in eV) obtained by DFT calculations, 𝑛 any integer, and "int" stands for the interface parts of oxide or alloy.

If the difference of energies between interface, surface and bulk are negligible compared to the defect formation energy, then the energies of the oxide and alloy parts near the interface can be estimated by those of the oxide part near surface and of the bulk alloy, respectively. If the heat of mixing of the alloy can be omitted, the energy of Cr in the alloy can be estimated by that of pure Cr metal. With these terms easier to be calculated, Equation ( 2) is rewritten as:

∆ f 𝐸 def,int = ∆ f 𝐸 def = 𝐸 Cr 2𝑛-1 O 3𝑛 (surf) -𝐸 Cr 2𝑛 O 3𝑛 (surf) + 𝐸 Cr(bulk) (3)
where "surf" refers to the surface part of the oxide.

On the oxide surface, the formation reaction of Cr2O3 between oxide lattice and O2 in the gas can create 3/2 new O lattice sites for each vacant Cr lattice site in the oxide [START_REF] Young | High Temperature Oxidation and Corrosion of Metals[END_REF]. Cr vacancies are thus created on the oxide surface according to the following reaction and related defect formation energy:

Cr 2(𝑛- 1 2 ) O 3(𝑛- 1 2 ) (surf) + 3 4 O 2 (gas) → Cr 2𝑛-1 O 3𝑛 (surf) (4)
∆ f 𝐸 def,surf = 𝐸 Cr 2𝑛-1 O 3𝑛 (surf) -𝐸 Cr 2(𝑛-1 2 ) O 3(𝑛-1 2 ) (surf) - 3 4 𝐸 O 2 - 3 4 𝑘 B 𝑇 ln ( 𝑃 O 2 𝑃 0 ) (5)
where 𝑘 B is the Boltzmann constant (8.617×10 -5 eV.K -1 ), 𝑇 the temperature in K, 𝑃 O 2 the O2 partial pressure in atm, and 𝑃 0 the standard pressure that equals to 1 atm.

We consider the formation reaction of Cr2O3 at the standard pressure:

2Cr(bulk) + 3 2 O 2 (gas) → Cr 2 O 3 (bulk) (6)
∆ f 𝐻 Cr 2 O 3 0 = 𝐸 Cr 2 O 3 (bulk) -2𝐸 Cr(bulk) - 3 2 𝐸 O 2 (7)
where ∆ f 𝐻 0 is the formation enthalpy in eV.

If the simulated surface is thick enough to write the term 𝐸 Cr 2(𝑛-1

2 ) O 3(𝑛-1 2 ) (surf) + 1 2 𝐸 Cr 2 O 3 (bulk)
as 𝐸 Cr 2𝑛 O 3𝑛 (surf) , then Equation ( 5) can be rewritten as:

∆ f 𝐸 def,surf = ∆ f 𝐸 def + 1 2 ∆ f 𝐻 Cr 2 O 3 0 - 3 4 𝑅𝑇 ln ( 𝑃 O 2 𝑃 0 ) . (8)
The same treatment of oxide dissociation or formation can be applied for the creation of O vacancies. At the alloy/oxide interface, the reaction between oxide and alloy lattice and the related defect formation energy for O vacancies can be written as:

Cr 2(𝑛- 1 3 ) O 3(𝑛- 1 3 ) (int) + 2 3 Cr(int) → Cr 2𝑛 O 3𝑛-1 (int) (9)
∆ f 𝐸 def,int = ∆ f 𝐸 def + 1 3 ∆ f 𝐻 Cr 2 O 3 0 . ( 10
)
where

∆ f 𝐸 def = 𝐸 Cr 2𝑛 O 3𝑛-1 (surf) -𝐸 Cr 2𝑛 O 3𝑛 (surf) + 1 2 𝐸 O 2 . (11)
On the oxide surface, an O atom of the Cr2O3 lattice could escape into the gas, leaving an O vacancy on the surface:

Cr 2𝑛 O 3𝑛 (surf) → Cr 2𝑛 O 3𝑛-1 (surf) + 1 2 O 2 (gas) (12)
∆ f 𝐸 def,surf = ∆ f 𝐸 def + 1 2 𝑘 B 𝑇 ln ( 𝑃 O 2 𝑃 0 ) . (13)
If a Cr vacancy and a O vacancy are formed together, according to the above equations, the formation energies of the coupled vacancies are

∆ f 𝐸 def,int = ∆ f 𝐸 def + 1 3 ∆ f 𝐻 Cr 2 O 3 0 (14)
∆ f 𝐸 def,surf = ∆ f 𝐸 def + 1 2 ∆ f 𝐻 Cr 2 O 3 0 - 1 4 𝑅𝑇 ln ( 𝑃 O 2 𝑃 0 ) (15)
where

∆ f 𝐸 def = 𝐸 Cr 2𝑛-1 O 3𝑛-1 (surf) -𝐸 Cr 2𝑛 O 3𝑛 (surf) + 𝐸 Cr(bulk) + 1 2 𝐸 O 2 . ( 16
)
If a thermodynamic equilibrium is established at alloy/oxide interface (i.e., no energy is released or absorbed for the formation or dissociation of Cr2O3), the equivalent O2 partial pressure at the equilibrium 𝑃 O 2 ,eq (in atm) can be defined [START_REF] Young | High Temperature Oxidation and Corrosion of Metals[END_REF] by writing the formation enthalpy ∆ f 𝐻 (in eV) for the reaction (6) as:

∆ f 𝐻 Cr 2 O 3 = ∆ f 𝐻 Cr 2 O 3 0 - 3 2 𝑘 B 𝑇 ln ( 𝑃 O 2 ,eq 𝑃 0 ) = 0 . ( 17
)
When the O2 partial pressure in the gas is as low as the equivalent value at the equilibrium, the defect formation energies on the surface and at the interface are equivalent (∆ f 𝐸 def,surf = ∆ f 𝐸 def,int ).

For substitution at the interface of a metal cation by a foreign atom (e.g., Mo), the same treatment as used to obtain Equation (3) from Equation ( 2) can be applied to write the substitution reaction and energy as:

Cr 2𝑛 O 3𝑛 (int) + Mo(int) → Cr 2𝑛-1 MoO 3𝑛 (int) + Cr(int) (18)
∆ f 𝐸 def,int = 𝐸 Cr 2𝑛-1 MoO 3𝑛 (surf) -𝐸 Cr 2𝑛 O 3𝑛 (surf) + 𝐸 Cr(bulk) -𝐸 Mo(bulk) . (19)
The above treatment was also used for vacancy formation on surfaces containing substitutional atoms and for substitution on surfaces containing vacancies. The same was done for Fe2O3 surfaces as for Cr2O3 surfaces.

Calculation details

According to the above formalism, the energies of oxide surfaces with and without Mo and those of pure metals (Cr, Fe and Mo) and oxygen gas are needed to estimate the defect formation energies. The calculated oxide formation enthalpies which could be deduced from the DFT energies of bulk oxides, metals and oxygen gas were used rather than experimental values in order to reduce the systematic errors.

In this study, ground state calculations were performed within the frame of DFT using 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]. Several versions of VASP from 5.3.3 to 5.4.4 were tested and all of them gave nearly the same energy (difference smaller than 10 -5 eV) when simulating bulk Cr2O3 (30 atoms) with experimental structure [START_REF] Newnham | Refinement of the α Al2O3, Ti2O3, V2O3 and Cr2O3 structures[END_REF]. The Perdew-Burke-Ernzerhof (PBE) functional using generalized gradient approximation (GGA) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] was selected as the exchange-correlation functional. The projector-augmented wave (PAW) method [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] was used for treating the pseudopotentials. The projection was done in real space with fully automatic optimization of projection operators unless for calculations of gaseous phase, since it is recommended for large system. For O, the suggested standard version of pseudopotential was chosen. For Cr, Fe and Mo, the 3p semi-core states were treated as valence states in the pseudopotential to obtain higher precision. The overall cutoff energy was set at 520 eV, which corresponds to 1.3 time the highest cutoff energy in the pseudopotentials. Electronic loops were stopped when the difference of energies between two successive iterations was lower than 10 -6 eV. The default settings were kept for the other parameters which are not involved here.

The DFT+U method using the approach of Dudarev et al. [START_REF] Dudarev | Electron-energyloss spectra and the structural stability of nickel oxide: An LSDA+U study[END_REF] was implemented to take into account the strong electronic correlation when calculating localized electronic state such as transition metal oxides. For calculations of metals with delocalized electronic states, the DFT+U method was also used to avoid the systematic error occurring when subtracting a DFT (without U) energy from a DFT+U energy [START_REF] Wang | Oxidation energies of transition metal oxides within the GGA+U framework[END_REF]. A discussion of this part is given in the appendix. The parameter J for the effective on-site exchange interactions was always 1 eV. For Cr-and Fe-d electrons, the parameter U for the effective on-site Coulomb interactions was 5 eV as suggested by Rohrbach et al. [START_REF] Rohrbach | Ab initio study of the (0001) surfaces of hematite and chromia: Influence of strong electronic correlations[END_REF]. For Mo-d electrons, the parameter U was chosen to be 6 eV so as to obtain good predictions of the electronic properties according to Ding et al. [START_REF] Ding | Computational investigation of electron small polarons in α-MoO3[END_REF]. The one-centre PAW charge densities were passed through the charge density mixer up to a lquantum number of 4, as recommended for DFT+U calculations treating d-electrons in order to converge to the ground state more rapidly.

Calculations were performed with collinear spin-polarized configurations for magnetic states.

For oxides, no initial magnetic moment was distributed for O since no magnetic moment is measured on O in Cr2O3 and Fe2O3 [START_REF] Hill | Crystallographic and magnetic studies of mesoporous eskolaite, Cr2O3[END_REF][START_REF] Hill | Neutron diffraction study of mesoporous and bulk hematite, α-Fe2O3[END_REF]. For oxygen gas, the initial magnetic moment of O was set to 2 µB. For oxides and metals, the initial magnetic moments of 4 and 5 µB were distributed for Cr and Fe atoms, respectively, which are slightly higher than their final magnetizations in Cr2O3 and Fe2O3.

For structure optimizations, the recommended conjugate gradient algorithm was used. During bulk optimizations, the atom positions, cell shape and cell dimensions were relaxed keeping the symmetry of structure fixed, while during surface optimizations, only the atom positions were relaxed. Ionic loops were stopped when the norm of all forces was lower than 10 -2 eV.Å -1 , in which case the difference of energies between two successive iterations was generally lower than 10 -4 eV. The orbital partial occupancies were treated with the recommended Gaussian smearing method. The width of smearing was set to 0.1 eV since lower smearing width does not bring significant changes to the energy (< 10 -6 eV) according to a test using bulk Cr2O3.

According to test results, 3×3×1 and 11×11×11 k-point grids were used for calculations of bulk oxides and metals, respectively. To calculate gases, a cell size of 10 Å with 1 k-point was sufficient.

In order to generate (0001)-oriented surface cell with single-Cr (or -Fe) termination where the terminating metal atoms are in the bulk site position, oxide slabs were created by inserting vacuum with a thickness at least 25 Å in [0001] direction at equidistance of two O layers. A larger vacuum thickness does not significantly change the energy (about 10 -3 eV) according to a preliminary test. Symmetric (containing 40 and 160 atoms for 1×1 and 2×2 cells, respectively) and asymmetric slab structures (containing 30 and 120 atoms for 1×1 and 2×2 cells, respectively) were both used in this study. The dipole correction [START_REF] Neugebauer | Adsorbate-substrate and adsorbate-adsorbate interactions of Na and K adlayers on Al(111)[END_REF] was used for calculations of asymmetric structures in which the bottom surfaces were frozen and the top surfaces were relaxed. According to a test performed with 1×1 Cr2O3(0001) surface cell, the dipole correction brings a considerable change of energies of about 0.01 eV for asymmetric structures, but a negligible influence for symmetric structures (~10 -6 eV). The formation of defects (vacancies and substitutional atoms) was investigated on both surfaces of symmetric structures but only on the top relaxed surface of asymmetric structures. For calculations of bulk and 1×1 (0001) surface cells of Cr2O3 and Fe2O3, a 3×3×1 k-point grid was used and for calculations of 2×2 (0001) surface cells, a 2×2×1 k-point grid was enough.

Bader charge analysis [START_REF] Tang | A grid-based Bader analysis algorithm without lattice bias[END_REF] requires high quality charge density. Compared with the above parameters used for structure optimizations, a denser grid for fast Fourier transformations, a denser k-point grid and another smearing method (tetrahedron method with Blöchl corrections) were tested but yielded the same Bader charges for bulk Cr2O3. In consequence, the charge density generated during structure optimizations was used for Bader charge analysis.

Results and discussion

Bulk materials

Both Cr2O3 and Fe2O3 have the corundum structure in which the O atoms form a hexagonal close packed lattice with 2/3 of the O octahedral sites occupied by Cr or Fe atoms [START_REF] Hill | Crystallographic and magnetic studies of mesoporous eskolaite, Cr2O3[END_REF][START_REF] Hill | Neutron diffraction study of mesoporous and bulk hematite, α-Fe2O3[END_REF]. The hexagonal unit cells simulated in this study contained 30 atoms. Figure 2 shows the optimized bulk structures of Cr2O3 and Fe2O3. The calculated lattice parameters and oxide formation energies were in good agreement with experimental values as shown by Table S1 and Table S2 in the supporting information (SI).

Cr2O3 is antiferromagnetic with spins on Cr atoms with commensurate magnetic moment and ↑↓↑↓ order along the [0001] direction [START_REF] Corliss | Magnetic structure of Cr2O3[END_REF] if the temperature is below 318 K [START_REF] Volger | Anomalous specific heat of chromium oxide (Cr2O3) at the antiferromagnetic Curie temperature[END_REF] (Néel temperature); otherwise, it is paramagnetic. The Néel temperature of Fe2O3 is about 966 K [START_REF] Białek | Antiferromagnetic resonance in α-Fe2O3 up to its Néel temperature[END_REF], below which the spins on Fe atoms are perpendicular to the [0001] direction. This configuration could be decomposed into two ferromagnetic sublattices coupled antiferromagnetically, resulting in a small net moment. When the temperature is cooled down to about 260 K [START_REF] Levinson | Temperature dependence of the weak ferromagnetic moment of hematite[END_REF] (Morin temperature), a reorientation of spins along the In bulk Cr2O3 and Fe2O3, both metal atoms (Cr and Fe) have a Bader charge of about +1.8 e and O atoms have a Bader charge of about -1.2 e where e is the elementary charge. The formal charges in these oxides are supposed to be +3 e for the metal atoms and -2 e for O. Thus, a ratio about 0.6 between Bader and formal charges is found for these corundum structures, close to the ratio of 0.56 found in another DFT study for Cr2O3 [START_REF] Kumar | Effects of surface orientations of Cr2O3 on CO2 adsorption: A DFT approach[END_REF]. The conversion ratio of 0.6 is systematically used in the following.

Perfect surfaces

Here we are interested in the (0001) surfaces of Cr2O3 and Fe2O3 with single-M termination, where the terminating metal atoms are in their bulk site positions. Two kinds of slab structures were used for simulation: (i) symmetric slabs with both top and bottom surfaces relaxed, which requires more calculation resources, and (ii) asymmetric slabs with unrelaxed bottom surface and relaxed top surface, which requires dipole correction [START_REF] Neugebauer | Adsorbate-substrate and adsorbate-adsorbate interactions of Na and K adlayers on Al(111)[END_REF] not available for calculations of charged slabs. Both slab structures gave the same surface energy and relaxation if the slab model was thick enough. As shown in Figure 2, the bulk hexagonal cell could be considered as a structure piling up 6 M-O3-M unit layers. For symmetric slabs, a thickness of 8 unit layers (40 atoms in total for a 1×1 slab) was enough for surface relaxation. All unit layers were relaxed here and even the 2 unit layers nearest to the symmetric centre (one above and one below) could be frozen without affecting the surface relaxation. For asymmetric slabs, 4 unit layers on top were relaxed and 2 unit layers at the bottom were frozen (30 atoms in total for a 1×1 slab). Both complete slab structures are shown in Figure S1 while Figure 3 Except for the terminating layers of Cr and Fe (Cr1 and Fe1, respectively), "up" and "dn" are used to indicate the metal atoms closer to surface and bulk, respectively, in the same metal atom layer. To reduce the main text, surface energy (Table S3) and relaxation (Table S4) are presented in SI. Figure 4 shows the change of Bader and formal charge for the atoms involved by the surface formation. Compared to bulk, only the outermost atomic layers are affected by surface effects for both Cr2O3 and Fe2O3. The formal charges decrease about 0.2 e and 0.25 e for "Cr1" and "Fe1" atoms, respectively, and increase about 0.1 e for "O1" atoms. The evolution of formal charge on the other atoms is not significant (< 0.1 e). In experiments, Cr 2+ was reported on the surface of (111)-oriented Cr2O3 films grown on (110) Cr by EELS and X-Ray photoelectron spectroscopy (XPS) [START_REF] Xu | Adsorption and reaction on oxide surfaces: NO, NO2 on Cr2O3(111)/Cr(110)[END_REF]. However, it was re-assigned to Cr 3+ in a later article [START_REF] Bender | Structural rearrangement and surface magnetism on oxide surfaces: a temperature-dependent low-energy electron diffraction-electron energy loss spectroscopy study of Cr2O3(111)/Cr(110)[END_REF] based on the results of ab initio calculations. In an ab initio embedded cluster calculation [START_REF] Mejias | Electronic states of the Cr2O3 (0001) surface from ab initio embedded cluster calculations[END_REF], Mulliken charges of Cr and O were found to be +3 e and -2 e in bulk Cr2O3 and +2 e and -1.67 e in the topmost layers, respectively, resulting in a variation of -1 e and +0.33 e. But in another calculation using HF method [START_REF] Rehbein | Structure of the α-Cr2O3 (0001) surface: An ab initio total-energy study[END_REF], the variation of Mulliken charge on terminating Cr atoms surface was found to be -0.188 e, close to our calculations. The partial reduction of formal charge of about -0.2 e calculated in the present work for terminating metal atoms is then considered reasonable.

Substitutional Mo effects

In order to introduce Mo, a metal atom M (representing Cr or Fe) was replaced by a Mo atom.

One site is substituted on each relaxed surface cells on 1×1 and 2×2 slabs for Cr2O3 and Fe2O3. Simulations are first performed on neutral surfaces to study the substitution energy and then on surfaces positively charged in order to investigate the oxidation state of substitutional Mo. The latter correspond to electrochemical passivated interfaces under anodic polarization.

Neutral surfaces

Different substitutional sites from the "M1" site in the terminating layer to the "M3dn" site in the third metal layer (Figure 3) were tested on neutral surfaces. The substitutional Mo could have the same or inverse spin orientation as the substituted metal atom. According to a test with the 1×1 Cr2O3(0001) surface cell, the substitution of Cr by a Mo atom with the same spin orientation was found to be energetically more favourable than substitution with opposite spin orientation, whatever the substitution site. The calculations were then performed with the "same spin" configuration. Mo substitution did not visibly influence the surface structures after relaxation.

Figure 5 shows the Mo substitution energies obtained on the different tested sites, whose positions are illustrated in Figure 3. No significant difference of energies was found between 1×1 and 2×2 surface cells for Cr2O3 and Fe2O3. Mo substitution was found to be always exothermic which appears consistent with experiments that report Mo enrichment in native and passive films formed on Mo-containing substrate alloys, including in the Cr-rich inner oxide layer [START_REF] Wang | XPS and ToF-SIMS investigation of native oxides and passive films formed on nickel alloys containing chromium and molybdenum[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][START_REF] Wang | Enhanced passivity of Cr-Fe-Co-Ni-Mo multi-component singlephase face-centred cubic alloys: design, production and corrosion behaviour[END_REF]. Noteworthy is the result that Mo substitution is always more energetically favoured on the Fe2O3 surface than that on the Cr2O3 surface. This supports the hypothesis that Mo could stabilize the Fe-rich (Cr depleted) weak sites of protection present in the inner layer by

Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543 preferentially substituting Fe as previously proposed [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF].

When the substitution site changes from "M1" to "M2up", a decrease of energy is observed for Fe2O3 but not for Cr2O3. The most favoured site for Mo substitution is the "M2dn" site on both Cr2O3 and Fe2O3 surfaces, which corresponds to the lower metal atom on the 2 nd metal layer from the surface. This result is consistent with a study of dopants in Fe2O3 [START_REF] Feng | Characteristics of dopant distribution and surface oxygen vacancy formation for modified Fe2O3 in chemical looping combustion[END_REF] that conclude that small dopants (< 1.7 A) with valences higher than Fe, which is the case for Mo, tend to be located on the subsurface rather than on the topmost surface. For unrelaxed substitution on the relaxed 1×1 Cr2O3 surface (i.e. same coordinates between substitutional Mo and substituted Cr), "Cr2dn" is still the most favoured site. However, for unrelaxed substitution on the unrelaxed 1×1 Cr2O3 surface (i.e. same coordinates between the atoms on the surface and those in the bulk), the most favoured site becomes the "Cr1" site, which corresponds to the terminating metal atom. This comparison suggests that the surface relaxation, that decreases the interlayer spacing between terminating metal layer and 1 st oxygen layer (see SI), stabilizes the substituted terminating metal layer. Substitution by Mo on the most favourable "M2dn" site was also tested with different surface sizes from 1×1 to 3×3 in order to simulate the effect of concentration, but as shown by Figure S2 in SI, the concentration effect is negligible (difference of energies about 0.05 eV) compared to the effect of substitution site.

Figure 5. Mo substitution energy from terminating "M1" site in the topmost layer to "M3dn" site in the 3 rd metal layer calculated with 1×1 and 2×2 Cr2O3 and Fe2O3 (0001) surface cells.

Figure 6 shows the changes of Bader and formal charges caused by Mo substitution on the most favourable site "M2dn" calculated with 2×2 surface cells. The variation of charge is mainly located on the substitutional Mo atom and is negligible for the other atoms. Compared with the charges of the substituted metal atoms, the charges on substitutional Mo are slightly higher (about +0.25 e in formal charge). Since the increase of formal charge is much smaller than 1, the substitutional Mo atoms on the neutral Cr2O3 and Fe2O3 surfaces can still be considered as in +3 oxidation state, i.e., carrying approximately the same charge as the substituted metal atoms.

Charged surfaces

In this part, we consider Mo effects on surfaces positively charged by removal of electrons (addition of positive charge) from the simulated systems, corresponding to electrochemical passivated interfaces under anodic polarization. This allows us to investigate if the substitutional Mo could be in an oxidation state higher than +3 found on neutral surfaces. Symmetric slabs were used to avoid using dipole corrections. On the symmetric slab with 2×2 surface cell, there are two inequivalent configurations for substitution by one Mo on each surface. Both yielded the same energy according to a test on the neutral Fe2O3 slab. Therefore, they are not distinguished in the following discussion.

The most favourable sites for substitutional Mo were expected to be the same as on neutral surfaces, which was verified by a test with the 1×1 Cr2O3 symmetric slab with 2 added charges.

One added charge was localized on each surface and mainly on the substitutional Mo no matter where it substitutes.

Calculations of charged slabs under periodic conditions include the electrostatic interaction between the charged surfaces and their periodic images [START_REF] Komsa | Finite-size supercell correction for charged defects at surfaces and interfaces[END_REF] and their energies cannot be compared directly if the charge distributions are not identical. Thus, we only compare here the Bader charges between the simulated surfaces with different added charges.

Figure 7 shows the changes of Bader and formal charges on Mo-substituted surfaces induced by the addition of positive charge. Added charges are mostly localized on the substitutional Mo atom. In order to estimate the oxidation state, the charges of the substitutional Mo and substituted Cr (or Fe) on perfect surfaces are compiled in Table 1. On the Mo-substituted Cr2O3

Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543 surface, the 1 st added charge is mainly localized on the substitutional Mo (about +0.6 e in formal charge), resulting a total formal charge of +3.8 e. The 2 nd and 3 rd added charges are no more localized on the substitutional Mo, but mainly delocalized on the near-surface Cr and O atoms. Mo can then be considered in near +4 oxidation state on all charged surfaces. On the Mosubstituted Fe2O3 surface, the behaviour of the first two added charges is like on the Cr2O3 surface, while the 3 rd added charge is again partially localized on the substitutional Mo (about +0.7 e in formal charge), indicating a higher oxidation state (near +5) of Mo in this case. The comparison between the two surfaces shows that substitutional Mo can localize more charge on Fe2O3 than on Cr2O3 surfaces. To summarize this section, substitution of Cr or Fe by Mo is exothermic on both perfect surfaces and more favoured on Fe2O3 than on Cr2O3. The most favoured location of Mo is the "M2dn" site for both. Mo takes the same oxidation degree as the parent metal cation, i.e., +3, unless the system is positively charged. In this case, the 1 st positive added charge is mainly localized on the Mo atom, yielding an oxidation state approaching +4. With increasingly positive charging, the Mo oxidation state still increases to between +4 and +6 on Fe2O3, but not on Cr2O3.

These results suggest that, in the inner oxide layer of a native or passive film in which Cr-rich and Fe-rich oxide zones co-exist: i) Mo location in a Fe-rich zone is more favoured thermodynamically, ii) Mo is +3 oxidation state in a non-charged environment and approaches +4 in an electron-depleted environment (corresponding to an anodically polarized interface),

iii) Mo can be further oxidized into an intermediate state between +4 and +6 in the Ferich oxide zones of anodically-formed passive film, but not in the Cr-rich oxide zones.

Vacancy formation

In this section, we discuss first the formation of vacancies on neutral Mo-free surfaces. Three types of vacancies were investigated in the present work: Cr(Fe) single vacancies, O single vacancies and Cr(Fe)-O pair vacancies. The vacancies were created on each relaxed surface of 1×1 and 2×2 slabs for Cr2O3 and Fe2O3. Secondly, the formation of vacancies was studied on Mo-substituted surfaces, which allows us to investigate the effects of Mo on vacancy formation in the oxide. In the simulations, the substitutional Mo was always placed in the most preferable "M2dn" site according to the previous section.

Mo-free surfaces

For metal atom vacancies, different vacancy sites from terminal "M1" site to sub-surface "M3dn" site of the Cr2O3 and Fe2O3 surfaces were tested. Figure 8 shows the calculated vacancy formation energies for the different tested sites.

Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543

On the Mo-free Cr2O3 surface (Figure 8.a), the most favourable sites to create a Cr vacancy are the "Cr2up" sites followed by the "Cr1" sites according to the test with the 2×2 surface cell.

The test with the 1×1 surface cell gives a different result: the most favourable site being the "Cr1" site followed by the "Cr2up" site. This disagreement may be due to the remarkable difference of defect concentrations between 1×1 and 2×2 surface cells since one defect was created per surface cell. Indeed, the creation of a Cr vacancy on the "Cr1" site of the 1×1 surface cell completely removes the terminating Cr layer forming an O-terminated surface, which is not the case with the 2×2 surface cell. Another effect comes from structural relaxation. In some cases, reconstruction of the surface structure was observed. Figure 9 shows the first four terminating layers ("Cr1", "O1", "Cr2up" and "Cr2dn") of the relaxed Mo-free 2×2 Cr2O3 surface cell with a Cr vacancy on "Cr2up" site. Some "Cr1" atoms (the three "Cr1" atoms nearest to the created vacancy and marked in yellow in Figure 9) are displaced through the "O1" layer into the "Cr2" layer, occupying the octahedral sites of the oxygen sub-lattice initially unoccupied by Cr as seen in Figure 3. Such local rearrangement the atomic stacking along the [0001] direction (from -Cr-O3-Cr to -Cr-Cr-O3) was also observed when creating "Cr2dn" vacancies in the 2×2 surface cell, as shown by Figure S3 in SI, but not for the other sites of Cr vacancies and neither with the 1×1 surface cell. This may be a hint to explain the difference of energy profiles obtained with 1×1 and 2×2 surface cells (Figure 8.a).

On the Mo-free Fe2O3 surface (Figure 8.a), both tests performed with 1×1 and 2×2 surface cells suggest that the most favourable sites to create Fe vacancies are the "Fe3up" sites followed by the "Fe1" sites. Compared to Mo-free Cr2O3 surfaces where atomic rearrangement occurred only when creating "M2up" and "M2dn" vacancies, atomic rearrangement of the Mo-free Fe2O3 surfaces was observed not only for vacancies on these sites, but also for vacancies created in the deeper sites of the 3 rd metal layer as shown in SI (Figure S4). This rearrangement phenomenon may also contribute to decrease the vacancy formation energies for these deep sites, resulting in "Fe3up" site becoming more favoured than the "Fe2up" site for vacancy formation in contrast with Mo-free Cr2O3 surfaces. Figure 10 shows the O vacancy formation energies obtained on different sites from 1 st O layer ("O1" sites) to sub-surface O layers ("O2" and "O3" sites). 1×1 and 2×2 surface cells were tested but little effect of cell size was found, unlike for metal atom vacancies. As a result, only the results obtained with 2×2 surface cells are presented here.

The "O1" sites are the most favorable sites on Mo-free Cr2O3 and Fe2O3 surfaces. The surface reconstruction leading local rearrangement of atomic stacking, in which the "Fe1" atom is displaced into the "Fe2" layers to stay in the initially unoccupied Fe sites, is observed when creating "O1" or "O3" vacancies on Fe2O3 surfaces, as shown by Figure S5 in SI, but no surface reconstruction occurs on Cr2O3 surfaces.

Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543

Figure 10. Defect formation energy ∆ f 𝐸 def of oxygen vacancies on different sites of Cr2O3 and Fe2O3 (0001) surfaces without and with substitutional Mo as calculated with 2×2 surface cells and using Equation [START_REF] Maurice | XPS, LEED and STM study of thin oxide films formed on Cr(110)[END_REF].

The formation of a metal atom vacancy and an O vacancy nearby (called coupled vacancies here) was also investigated. Since the "Cr1" ("Fe1") sites were found to be nearly always the most favorable, a Cr (Fe) vacancy was created on these sites of the 2×2 Cr2O3 (Fe2O3) surfaces and an O vacancy was created nearby as coupled vacancy on the "O1" sites, which were also found to be nearly always the most favorable sites for O vacancy formation on both Cr2O3 and Fe2O3 surfaces. With the existing metal atom vacancy, there are 4 inequivalent "O1" sites to create the O vacancy whose distances to the metal atom vacancy are different. The site closest to the metal atom vacancy was selected here to investigate the formation of couple vacancies. Indeed, according to tests performed on Mo-free Cr2O3 and Fe2O3 surfaces, the "O1" site closest to the metal atom vacancy on the "Cr1" or "Fe1" site was found the most favorable site to create the coupled vacancies. The defect formation energies are shown by Table S5 in SI and will be further discussed on Mo-substituted surfaces.

Mo-substituted surfaces

Vacancy sites at the same depth should be equivalent for the Mo-free 2×2 surface cell, but the substitutional Mo breaks this symmetry. In this study, only the site the closest to the substitutional Mo was investigated among the vacancy sites at the same depth since the effect of Mo is of interest here.

After substitution by Mo on Cr2O3 surfaces on the most favoured "Cr2dn" sites, the most favourable sites for the formation of Cr vacancies become the "Cr1" sites rather than the "Cr2up" sites on the Mo-free surface, indicating that substitutional Mo destabilizes the "Cr1" atoms (Figure 8(b)). A slight repulsion of the "Cr1" atom occurred if the surface was Mo-substituted, which also shows destabilization by the substitutional Mo. Surface reconstruction was observed only for the defective surface with vacancies on "Cr2" atomic layer, like on Mo-free surfaces.

The energy profiles for 1×1 and 2×2 surface cells show the same trend for the Mo-substituted Cr2O3 surfaces (Figure 8(b)). However, for the Mo-substituted Fe2O3 surfaces, the most favourable sites to create Fe vacancies were found different with 1×1 and 2×2 surface cells still due to the local rearrangement of the atomic stacking. With 2×2 surface cells, the rearrangement was observed except for the defective surface with "Fe1" vacancies, like on Mo-free surfaces.

The most favourable sites are also the "Fe3up" sites like on Mo-free surfaces. However, with 1×1 surface cells, the rearrangement was not observed when creating the "Fe3up" vacancies, resulting in vacancy formation energy increasing continuously with sub-surface depth.

For O vacancies, the substitutional Mo on Cr2O3 surfaces does not influence significantly the energy profile (Figure 10). On Mo-substituted Fe2O3 surfaces, the "O1" and "O2" vacancies, which are closer to the substitutional Mo atoms than the "O3" vacancies, become less stable than those on Mo-free surfaces. As a result, the most favorable sites to create O vacancies become the "O3" sites closely followed by the "O1" sites. No new surface reconstruction is observed on Mo-substituted surfaces compared to Mo-free surfaces.

The effect of defect concentration was investigated with slab cells of different sizes and containing one vacancy per relaxed surface. Figure 11 compares the defect formation energies for single and coupled vacancies calculated with 1×1, 2×2 and 3×3 Cr2O3 surface cells. Energies do not vary when surface size increases beyond 2×2. Considering that larger surface cells consume more calculation resources, the 2×2 surface cell was considered large enough to simulate dilute defects. Substitutional Mo shows a significant effect on the Cr isolated vacancies but not on O isolated vacancies nor Cr-O coupled vacancies. 3), ( 11) and ( 16), respectively.

Figure 12 shows the vacancy formation energies ∆ f 𝐸 def,surf as function of oxygen partial pressure at room temperature (25 °C). The calculated oxide formation energies shown in Table S5 were used to calculate the defect formation energies ∆ f 𝐸 def,surf and the equivalent oxygen partial pressures at the equilibrium of formation/dissociation of the oxides. At the thermodynamic equilibrium, O vacancies are expected to be more favourable at low oxygen partial pressure and metal atom vacancies at high oxygen partial pressure [START_REF] Latu-Romain | The role of oxygen partial pressure on the nature of the oxide scale on a NiCr model alloy[END_REF]. This trend is confirmed in Figure 12 and the coupled metal atom and O vacancies are always less favourable whatever the oxygen partial pressure.

Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543

With substitutional Mo, the formation energy of metal atom vacancies becomes lower (-0.57 eV change for Cr vacancies on "Cr1" sites and -1.36 eV for Fe vacancies on "Fe1" sites), indicating that the presence of Mo on the oxide surface is in favour of the formation of metal atom vacancies. The trend is opposite for O vacancies (+0.07 eV on Cr2O3 and +0.43 eV on Fe2O3 on "O1" sites) and for coupled vacancies (+0.16 eV on Cr2O3 and +0.63 eV on Fe2O3) but with smaller amplitude. The effect of Mo is more pronounced on the Fe2O3 than on the Cr2O3 surface. The above content shows the energetical influence of substitutional Mo on vacancy formation. Reversely, Table 2 shows the same trend for the variation of energy for substitution by Mo on "M2dn" sites resulting from the presence of vacancies. Compared to the values for perfect surfaces, the presence of metal atom vacancies favours substitution by Mo on "M2dn" sites. In contrast, O vacancies and coupled vacancies disfavour basically substitution by Mo. The effect is more pronounced on Fe2O3 than on Cr2O3 surfaces. The formation of neutral vacancies by removing atoms releases charges that can re-localize on the neighbour atoms. Figure 13 shows the changes of charges on Mo-substituted surfaces resulting from vacancy formation. For metal atom vacancies, the charges released by the removal of cation are localized not only on the substitutional Mo but also on the Cr (Fe) and O atoms and the values are comparable. In contrast, the localization of added charges on the Cr (Fe) and O atoms was tiny relative to that on substitutional Mo on the vacancy-free surfaces (Figure 7). The formation of oxygen and coupled vacancies affects basically the charges of Cr (Fe) and O atoms, but not substitutional Mo.

In order to estimate the oxidation state of the substitutional Mo atom, Table 3 compiles the charges of the substitutional Mo and the substituted Cr on perfect surfaces. According to the charge analysis of neutral perfect surfaces, substitutional Mo is supposed to have a charge state (Mo 3+ ) similar to that of the substituted metal atom. Three positive charges will be released if a Cr (Fe) atom is missing, which may form a Mo 6+ state. However, the charges localized on the substitutional Mo on Cr2O3 surface (+3.7 e in formal charge) suggest that substitutional Mo on surfaces containing Cr vacancies is near a Mo 4+ state. This is consistent with the experimental observation in which Mo observed in the surface oxide film near the inner Cr-rich layer is present as Mo 4+ rather than Mo +6 [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF].

When a single metal atom vacancy is created, leaving three positive charges, more released charges are re-localized on Mo on Fe2O3 (+1.0 e in formal charge) than on Cr2O3 (+0.5 e in formal charge) surfaces. A similar result was also obtained on vacancy-free charged surfaces (+0. Formal charges are converted from the Bader charges using the ratio of 0.6.

Implications for passivity

The results of the present work provide insights into the possible mechanism of the enhanced resistance to passivity breakdown and initiation of localized corrosion experimentally observed with Mo-containing stainless steels and other alloys [START_REF] Wang | XPS and ToF-SIMS investigation of native oxides and passive films formed on nickel alloys containing chromium and molybdenum[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][START_REF] Wang | Enhanced passivity of Cr-Fe-Co-Ni-Mo multi-component singlephase face-centred cubic alloys: design, production and corrosion behaviour[END_REF]. If both Fe-rich and Cr-rich oxide zones coexist on passivated alloys as a result of the local inhomogeneity of the Cr supply in the very first stages of formation of the protective oxide film [START_REF] Ma | Origin of nanoscale heterogeneity in the surface oxide film protecting stainless steel against corrosion[END_REF][START_REF] Ma | Nanoscale early oxidation mechanisms of model FeCrNi austenitic stainless steel surfaces at room temperature[END_REF], the present work, showing that the energy of substitution by Mo is lower on Fe2O3 than on Cr2O3 surfaces, suggests that Mo may be locally more enriched in the Fe-rich oxide zones than in the Cr-rich oxide zones. This has implications for the Fe-rich oxide zones, which are the weak sites of protection against corrosive attack, including in the presence of chloride ions.

Obviously, the systems considered here are model systems (corundum structures) and Mo was considered in substitution of Cr and Fe in their crystalline position, whereas other structures and Mo locations could be possible (O sites, unoccupied sites). Mixed oxides could as well as combination of cation and anion vacancies could also be considered. Nonetheless, despite the (over) simplifications of the present models, our results support the hypothesis that Mo heals the Fe rich zones.

Our results suggest that Mo has a negative effect on oxygen vacancy formation and a positive effect on metal atom vacancy formation. The decrease of donor density (which include oxygen vacancies) was indeed reported in the past by Jang and Kwon [START_REF] Jang | In situ study on the effects of Ni and Mo on the passive film formed on Fe-20Cr alloys by photoelectrochemical and Mott-Schottky techniques[END_REF] on passive films on Fe-Cr-Ni and Fe-Cr-Ni-Mo alloys by means of Mott Schottky plots. Our results also partially agree with those of Montemor et al. [START_REF] Montemor | The role of Mo in the chemical composition and semiconductive behaviour of oxide films formed on stainless steels[END_REF] on heat-formed oxide film on 304 and 316 SS, who found that the presence of Mo decreases the density of acceptors and donors in p-type Cr-and n-type Fe-rich oxide layers, respectively. If intrinsic vacancies are the main defects, the acceptors and donors in p-type Cr-and n-type Fe-rich oxide layers will be Cr vacancies and oxygen vacancies, respectively. These experimental results seem to be coherent with the negative effect of Mo on oxygen vacancy formation and incoherent with the positive effect on metal atom vacancy formation as concluded in this work. However, the Mott-Schottky plots analyze the entire passive films often containing several kinds of impure oxides and hydroxides in complex multilayer structures. Comparisons with Mott Schottky plots which give a global measurement on the passive films must be taken with caution.

According to the "point defect model" [START_REF] Macdonald | The point defect model for the passive state[END_REF][START_REF] Macdonald | The history of the Point Defect Model for the passive state: A brief review of film growth aspects[END_REF] and the generalized model [START_REF] Seyeux | Oxide film growth kinetics on metals and alloys: I. physical model[END_REF] which also considers vacancies, metal cations dissolve in the solution at oxide/solution interface, leaving metal atom vacancies in the oxide. Since substitution by Mo decreases the formation energy of Fe vacancies more significantly than that of Cr vacancies as shown in the present work, one can expect the selective dissolution of Fe to be accelerated, and thus the replacement of iron oxide by more stable Mo-substituted chromium oxide to be promoted in the oxide film. This replacement mechanism of accelerated selective dissolution would thus contribute to curing the Fe-rich weak sites of passivity.

Cl -can penetrate in passive oxide films and induce breakdown [START_REF] Marcus | Localized corrosion (pitting): A model of passivity breakdown including the role of the oxide layer nanostructure[END_REF]. It is reported that the presence of O vacancies in Cr2O3 favours the penetration of Cl -from the surface to the subsurface [START_REF] Sarfo | Density functional theory study of the initial stages of Cl-induced degradation of α-Cr2O3 passive film[END_REF]. The present study suggests that substitution by Mo disfavours the formation of O vacancies in Cr2O3 and more effectively in Fe2O3. This implies that substitutional Mo in passive films could prevent passivity breakdown in chloride-rich environment by reinforcing the Fe-rich weak sites of protection against chloride penetration.

Conclusion

Substitution of a metal cation by Mo and its effects on vacancy formation were investigated on (0001)-oriented chromia and hematite surfaces by quantum modelling in order to gain atomistic insight on the enhanced passivity, experimentally observed on Mo-bearing stainless steels and other alloys. The results obtained with DFT+U calculations show that, on neutral and vacancyfree surfaces, substitution by Mo is always exothermic and more favoured energetically on Fe2O3 than on Cr2O3 surfaces. The charge on the substitutional Mo is only a little higher than that on the substituted metal atom so that the substitutional Mo can be considered as in +3 oxidation state. The 1 st added charge on the Mo-substituted surface is mainly localized on the substitutional Mo which can be considered as in +4 oxidation state. The 2 nd added charge is delocalized but the 3 rd added charge is again localized on the substitutional Mo reaching an intermediate state between +4 and +6. The localization of the 3 rd added charge on the substitutional Mo is only observed on Fe2O3.

The presence of substitutional Mo favours the formation of metal atom vacancies, but disfavours that of O vacancies or Cr(Fe)-O coupled vacancies. The Mo effect on vacancy formation is more pronounced on Fe2O3 than on Cr2O3 surfaces. When a metal atom vacancy is created, the released charges localize on the substitutional Mo. More charge is localized on Mo on Fe2O3 than on Cr2O3 defective surfaces, with the substitutional Mo reaching an oxidation state between +4 and +6 on Fe2O3 and only +4 on Cr2O3.

The energy of substitution by Mo, lower on Fe2O3 than on Cr2O3 surfaces, suggests that Mo may be locally more enriched in the Fe-rich than in the Cr-rich oxide zones of the barrier layer of the passive film, and thus could cure the Fe-rich weak sites of the passive film, thus enhancing the corrosion resistance. Since substitution by Mo decreases the formation energy of Fe vacancies more significantly than that of Cr vacancies, the selective dissolution of Fe may be accelerated, resulting in the replacement of iron oxide by more stable Mo-substituted

Appendix: Energies of chemical references

The energy of the metal is needed as the reference value when creating a defect in its oxide. While the oxide is in localized state that should be calculated in DFT+U method, the metal is in delocalized state that should be calculated without the U effect and systematic error occurs if we subtract the DFT (without U) energy of metal from the DFT+U energy of oxide [START_REF] Wang | Oxidation energies of transition metal oxides within the GGA+U framework[END_REF]. The following lists several ways to avoid this situation and their corresponding limits:

1. We can calculate the energy of oxygen gas in DFT and deduce the energy of the metal from the energy of the oxide and the experimental formation energy of the oxide. Oxygen in triplet state is simulated in this case, which is found more stable than in singlet state for O2 as predicted by experiments [START_REF] Linstrom | NIST Chemistry WebBook, NIST Standard Reference Database Number 69[END_REF]. However, the bond energy of dioxygen obtained by DFT calculations is not correct (1.5 eV different from the experimental value [START_REF] Johnson | NIST Computational Chemistry Comparison and Benchmark Database[END_REF]) due to the incorrect prediction of the self-interaction effect [START_REF] Klüpfel | The effect of the Perdew-Zunger self-interaction correction to density functionals on the energetics of small molecules[END_REF], indicating that the energy of oxygen is not so reliable.

2. We can calculate the energy of metal in DFT+U calculations using the same U parameter as for its oxide. This is the chosen method in the present work. Cr, Fe and Mo metals all have the bcc structure which cubic unit cell contains only 2 atoms. Cr metal is reported in spin density wave (SDW) state but here it is approximated by the antiferromagnetic (AFM) state observed in the literature [START_REF] Fawcett | Spin-density-wave antiferromagnetism in chromium[END_REF]. Fe and Mo metals are in ferromagnetic (FM) and paramagnetic (approximated by nonmagnetic, NM) states, respectively. All these magnetic states are confirmed by the DFT calculations as the most stable states. However, the atomization energies of metal (shown in Table S6) become incorrect if the metals are calculated in DFT+U.

3. We can use a fitted value for the energy of metal. In this method, several oxides of one metal will be simulated. By using the experimental oxide formation energy, the energy of metal can be fitted with or without fixing the energy of dioxygen. Cr, Fe and Mo have been investigated with this method [START_REF] Jain | Formation enthalpies by mixing GGA and GGA+U calculations[END_REF] using different U parameters. New fitting was carried out for this study. Cr2O3 and CrO2, FeO and Fe2O3, and MoO2 and MoO3 were used for fitting the energies of Cr, Fe and Mo, respectively. Simulation of oxides was globally successful, except that the calculated DOS type of FeO was found metallic while FeO should be a semi-conductor [START_REF] Bowen | Electrical and optical properties of FeO[END_REF]. Table S7 shows the calculation details of the oxides. With or without using a fixed dioxygen energy with a correction of 1.36 eV [START_REF] Wang | Oxidation energies of transition metal oxides within the GGA+U framework[END_REF], the difference of fitted energies of metal could attain 1.8 eV (Table S8).

Figure 1 .

 1 Figure 1. (a) Model of protective oxide films proposed from experiments on Cr-Fe-Co-Ni-Mo multi-principal element alloys [6] and (b) model used in present simulations.

  [0001] direction with a deviation of about 10° occurs, with a loss of net magnetic moment. In the present work, Fe2O3 bulk structures were tested with different spin orientations ([0001], [1000] and [1200]) in collinear spin-polarized calculations, resulting in a difference of energy of only 10 -8 eV. Consequently, the direction of spins for Fe2O3 was set to [0001] direction like for Cr2O3 in order to simplify the calculations. The different spin configurations of optimized bulk antiferromagnetic Cr2O3 and Fe2O3 are shown in Figure2. The magnetic moments of Cr and Fe in such structures are 3.0 and 4.2 µB, respectively, close to the experimental values in the literature (2.7[START_REF] Hill | Crystallographic and magnetic studies of mesoporous eskolaite, Cr2O3[END_REF] and 4.2[START_REF] Hill | Neutron diffraction study of mesoporous and bulk hematite, α-Fe2O3[END_REF] µB, respectively).

Figure 2 .

 2 Figure 2. Optimized bulk structures of Cr2O3 and Fe2O3. O atoms are in red, spin up Cr and Fe atoms in blue, and spin down Cr and Fe atoms in light grey or white. Cell contour is marked by a black line.

  enlarges the relaxed Cr2O3 and Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543 Fe2O3 (0001)-oriented surface cells with the denomination of atomic layers used hereafter.

Figure 3 .

 3 Figure 3. Optimized (0001) surface cells of Cr2O3 and Fe2O3. O atoms are in red, spin up Cr and Fe atoms in blue, and spin down Cr and Fe atoms in light grey or white. Cell contour is marked by black line.

Figure 4 .

 4 Figure 4. Bader and formal charges of Cr2O3 and Fe2O3 (0001) surfaces relative to bulk values. Each point represents an atomic layer whose nature is indicated by the background image of the perfect Cr2O3 surface according to the [0001] coordinate. O atoms are in red and metal atoms in grey.

Figure 6 .

 6 Figure 6. Bader and formal charges of Mo-subtituted Cr2O3 and Fe2O3 (0001) surfaces relative to the Mo-free surfaces calculated with 2×2 surface cells. Each point represents an atomic layer whose nature is indicated by the background image of the perfect Cr2O3 surface according to the [0001] coordinate. O atoms are in red and metal atoms in grey. Formal charges are converted from the Bader charges using the ratio of 0.6.

Figure 7 .

 7 Figure 7. Bader and formal charges of Mo-substituted (a) Cr2O3 and (b) Fe2O3 (0001) charged surfaces relative to the Mo-substituted neutral surfaces. Each point represents an atomic layer whose nature is indicated by the background image of the perfect Cr2O3 surface according to the [0001] coordinate. O atoms are in red and metal atoms in grey. Formal charges are converted from the Bader charges using the ratio of 0.6.

Figure 8 .

 8 Figure 8. Defect formation energies ∆ f 𝐸 def of metal atom vacancies on different sites of Cr2O3 and Fe2O3 (0001) surfaces (a) without and (b) with substitutional Mo as calculated with 1×1 and 2×2 surface cells and using Equation (3).

Figure 9 .

 9 Figure 9. Top and side views of the local rearrangement of the Mo-free Cr2O3 (0001) surface structure with a "Cr2up" vacancy observed with a 2×2 surface cell.

Figure 11 .

 11 Figure 11. Defect formation energies ∆ f 𝐸 def of isolated Cr or O and coupled Cr-O vacancies calculated on Mo-free and Mo-substituted Cr2O3 (0001) surfaces of different cell sizes. Cr vacancies are on "Cr1" sites and O vacancies on "O1" sites. Formation energies of Cr, O and coupled vacancies were calculated using Equations (3), (11) and (16), respectively.

Figure 12 .

 12 Figure 12. Defect formation energy ∆ f 𝐸 def,surf of vacancies as a function of O2 partial pressure (O chemical potential) for (a) Cr2O3 and (b) Fe2O3 (0001) surfaces without and with substitutional Mo. Vacancies are on their most stable sites.

Figure 13 .

 13 Figure 13. Bader and formal charges of Mo-substituted (a) Cr2O3 and (b) Fe2O3 (0001) defective surfaces relative to the Mo-substituted vacancy-free surfaces. Each point represents an atomic layer whose nature is indicated by the background image of the perfect Cr2O3 surface according to the [0001] coordinate. O atoms are in red and metal atoms in grey.Formal charges are converted from the Bader charges using the ratio of 0.6.

Table 1

 1

	Added charge	Cr2O3	Fe2O3

. Charges (in e) on the substituted Cr or Fe atom and substitutional Mo on neutral and charged Cr2O3 and Fe2O3 (0001) surfaces.

Table 2 .

 2 Energy of substitution by Mo on "M2dn" sites on defective surfaces relative to perfect surfaces. Values on Cr2O3 and Fe2O3 perfect surfaces are -0.80 and -1.78 eV, respectively.

		Site Cr2O3 Fe2O3
	Metal atom vacancy M1	-0.56 -1.37
		M2up -0.06 -1.78
		M3up -0.72 -1.19
		M3dn -0.63 -1.21
	Oxygen vacancy	O1	0.08	0.43

  6 e on Mo on Cr2O3 and +1.4 e on Fe2O3 in formal charge), indicating that it is easier to transfer the valence electron of Mo to Fe in Fe2O3 than to Cr in Cr2O3. This is consistent with Allen electronegativity (in Pauling units) defined according to valence electron energies, showing that Fe (1.80) is more electronegative than Cr (1.65) and Mo (1.47)[START_REF] Allen | Electronegativity is the average one-electron energy of the valence-shell electrons in ground-state free atoms[END_REF][START_REF] Mann | Configuration energies of the d-block elements[END_REF]. The formal charge on Mo is +0.5 e higher on Fe2O3 than on Cr2O3 for defective surfaces. Since Mo on Cr2O3 defective surfaces can be considered as in Mo 4+ state, one can expect that Mo on Fe2O3 surfaces is in intermediate state between Mo 4+ and Mo 6+ , even if the formal charge (+4.2 e) on Mo on Fe2O3 surface is not much higher than +4. Since the oxidation state of Mo can be expected higher than +4 on both vacancy-free charged and vacancy-containing neutral surfaces Fe2O3, the intermediate oxidation state between Mo 4+ and Mo 6+ observed experimentally[START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF] may be considered as an indicator of the presence of Mo-substituted Fe-rich local zones in the inner barrier layer of the oxide film (which mostly consists of Cr oxide).In the study of Larsson et al. for passive films on Ni-Cr-Mo alloys[START_REF] Larsson | The Oxygen Evolution Reaction Drives Passivity Breakdown for Ni-Cr-Mo Alloys[END_REF], Mo 4+ and Mo 5+ detected at relatively low potential could be oxidized into Mo 6+ at more anodic polarization condition. However, it is also reported in the article that Mo 6+ species may be present as MoO3, mixed Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543with Cr(OH)3 and Ni(OH)2 in an amorphous hydrous network, which is not the crystal structure simulated in our DFT calculations. This may explain why the Mo 6+ state is not found in the present work.

Table 3 .

 3 Bader and formal charges (in e) on substituted Cr (Fe) atom and substitutional Mo atom on vacancy-free and vacancy-containing surfaces. Vacancies are on their most stable sites.

		Vacancy site	Cr2O3	Fe2O3
			Bader	Formal	Bader	Formal
	Substituted Cr (Fe) atom	Vacancy-free	+1.8	+3.0	+1.8	+3.0
	Substitutional Mo	Vacancy-free	+1.9	+3.2	+1.9	+3.2
		Cr1/Fe3up	+2.2	+3.7	+2.5	+4.2

  Author preprint published in Corrosion Science https://doi.org/10.1016/j.corsci.2023.111543 programme Investissements d'Avenir supervised by the Agence Nationale pour la Recherche.

Acknowledgement

This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No. 741123).

The authors acknowledge GENCI for high performance calculations in the national CEA centre under the agreement A0040802217 and the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the

Data Availability

Data will be made available on request.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Author preprint published in Corrosion


Vlad Nicolicin-Georgescu

Vincent Benatier

Rémi Lehn

Henri Briand
email: henri.briand@univ-nantes.fr

Remi Lehn

An Ontology-Based Autonomic System for Improving Data Warehouse Performances

Keywords: Business Intelligence, Decision Support Systems, Autonomic Computing, Data Warehouse, Ontology, Cache, Business Rule

published or not. The documents may come

Introduction

As the 21 st century is well on its way, in a civilized and modern world, we realize that the most important asset needed in order to keep the pace with this new rhythm is knowledge. Knowledge is the source of power and truly the new edge of the power shift [START_REF] Toffler | Powershift: Knowledge, Wealth and Power at the edge of the 21 st Century[END_REF]. As technology comes greatly to our help, we find it normal to research, discover and improve ways of gathering, processing and using all information available. Our purpose is to develop a system aimed at helping enterprises analyze and improve their decision making process by providing a unified representation of certain aspects involving the knowledge available for this process (from software support documents to human expert experience) and an autonomic system that makes use of this knowledge and acts upon it. Simpler and often referred to as DSSs, Decision Support Systems are defined as computerized systems whose main goal is to analyze a series of facts and give various propositions for actions regarding the facts involved [START_REF] Druzdel | Decision Support Systems, Encyclopedia of library and information science[END_REF]. This is why the process of decision making, based on such systems and the elements involved, is known as business intelligence (BI) process.

The applicative area of this paper is cache memory allocation for the Oracle Hyperion Essbase BI 1 cubes. This is a common configuration problem that BI experts are faced with. The cubes represent the data warehouse whose performances are to be improved. The system we propose makes use of knowledge based on system information (from architecture to cube cache parameters) and on sets of rules representing constraints and advice for the cache allocations (taken from the Essbase documents and from our human experts). The purpose is to provide two main functionalities. First, to compute a system's degree of improvement based on cache allocations and performance indicators. Second, to propose an improved cache configuration, that gives (if possible) the optimal performances. Two main aspects have been taken into consideration along with this approach.

The first aspect is knowledge representation. The knowledge regroups several sources: describing software and hardware architectures, system performance measurement, system analysis and improvement practices (described as sets of ECA (event condition action) rules [START_REF] Huebscher | A Survey on Autonomic Computing -Degrees, Models and Applications[END_REF]). Knowledge representation describes how this information is unified into knowledge bases. If for the system architecture, models are being developed and even adopted as w3c standards 2 , then for the data representation of system report performances and the rules of system analysis, we are obliged to turn to specific representations (using ontologies [START_REF] Gruber | What is an ontology?[END_REF] and ontology based rules).

Second, the improvement process itself, meaning having a fast response (from the moment a demand for improvement is made) and having a good (if not the best) response for any type of decision request (in our case a new cache allocation). In order to achieve this, IBM has proposed a solution to help automate various processes. The solution is called Autonomic Computing [START_REF]et Autonomic Computing. Powering your business for success[END_REF], [START_REF] Huebscher | A Survey on Autonomic Computing -Degrees, Models and Applications[END_REF] and its applications extend way beyond the business intelligence sector. We propose the usage of autonomic computing with the cache allocation improvement process.

Section 2 gives an insight of how we manage the knowledge in our system in order to drive the data warehouse. First we present how the knowledge base is organized for managing data warehouses and then how autonomic computing is used in the decision making process. Section 3 focuses on the description of our model and how this approach is used to perform analysis and improvement. A schema of a DSS together with the description of the data warehouses and an example of associated rules are presented. Section 4 provides a view of the experimentation and the results obtained. Finally, we sum up the work presented and take a glance at the future directions.

Data Warehouse Management

Knowledge Management

In brief, Knowledge Management is the process through which organizations generate value from their intellectual and knowledge-based assets, disseminating this knowledge and sharing it in an effort to get competitive advantage [START_REF] Oliveira | Managing knowledge about resources for autonomic computing, 1 st latin american autonomic computing symposium[END_REF]. Data warehouse (DW) management is a key element in the decision making process. A data warehouse is a repository of an organization's electronically stored data and is designed to facilitate reporting and analysis [START_REF] Inmon | Tech topic: what is a data warehouse?[END_REF]. Managing a data warehouse includes the process of analyzing, extracting, transforming and loading data and metadata. Our interest in knowledge management comes from the types of data involved in the decision making process.

The knowledge management into our work is based on system analysis and functioning. These refer to a complex set of rules that describes the functioning and non trivial interdependencies between the elements of the system. The main objective is to describe the rules for the analysis and improvement processes. Representing data under the form of rules [START_REF] Stojanovic | A framework for knowledge management on the semantic web[END_REF] gives a completely different approach to knowledge management. Practically, we create a business rule knowledge base that serves for the process of analysis and improvement. This process is supported both by the human expert and by the autonomic system. We propose to divide these rules into two main components: constraints and advice.

Advice represents business rules (BR) and best practices for the DSS giving the measure of a system improvement level in these terms. This means how 'close' a configuration is to satisfy sets of advice and therefore is able to generate an advice scoring. This scoring is built upon a point allocation system that grades the level of implementation of an advice set which we call a BR improvement points system, and which we describe in Section 3.

Constraints represent limitations imposed (i.e. the index cache cannot be under 1 Mb) and a violation of such constraints leads to an error in the system analysis.

To the division above, an entire set of rules is added, (from initial fact deduction to planning and action rules). These sets of rules are considered as state specific rules and are modeled for each state of the autonomic computing manager (presented in the next section).

Autonomic Computing in Data Warehouse Management

Most of the IT organizations spend a lot of time reacting to problems that occur at the IT infrastructure component level. This prevents them from focusing on monitoring their systems and from being able to predict and prevent problems before end users are impacted [START_REF] Manoel | Problem determination using self-managing autonomic technology[END_REF]. Autonomic computing (AC) is the ability for an IT infrastructure to adapt and change in accordance with business policies and objectives. Quite simply, it is about freeing IT professionals to focus on higher-value tasks by making technology work smarter, with business rules guiding systems to be self-configuring, self-healing, self-optimizing and self-protecting [START_REF]et Autonomic Computing. Powering your business for success[END_REF]. This subject is of great interest to enterprises and has already been put into practice for improving database performance by IBM [START_REF] Markl | LEO : An Autonomic Optimizer for DB2[END_REF], [START_REF] Lightstone | Toward autonomic computing with DB2 universal database[END_REF] and Microsoft [START_REF] Mateen | Autonomic Computing in SQL Server[END_REF]. There is great interest of development into applications of autonomic computing on managing data warehouses, as experts can no longer face the quantity of information available IBM specifications link autonomic computing with the notion of autonomic manager as the entity that coordinates the activity of the autonomic process. Four separate phases are distinguished for the manager: monitoring, analyzing, planning and executing [START_REF]et Autonomic Computing. Powering your business for success[END_REF], [START_REF] Parshar | Autonomic Computing: Concepts, Infrastructure and Applications[END_REF]. We propose an implementation of the autonomic manager connected to our knowledge base and based on the data warehouse performance. As it is rule based knowledge, we differentiate the sets of rules for each of these phases. The illustration of this process is shown in Section 3. Similar alternatives to autonomic computing were made in real BI [START_REF] Nguyen | Sense & Response Service Architecture (SARESA)[END_REF] but the idea is the same: to be able to analyze and improve (in our case) a given system through a closed loop that differentiates a series of states.

The loop formed by the four states mentioned is regularly run. By regularly we mean once per night during batch operations, when statistics on the data warehouse usage for that day are gathered. Each loop, according to the new query times, modifies the values of the caches, and this is repeated until the desired times are achieved. This process is based on both the feedback from the previous response times and on the sets of advice mentioned earlier. Consequently, what our solution proposes is to include business rule in the loop so that the modifications to the cache values are more substantial and relevant, and thus the time needed to reach the desired performance level is greatly reduced.

Knowledge Base

We have seen the information we need to formalize and we have found the ontology [START_REF] Gruber | What is an ontology?[END_REF] as a model of knowledge representation. The ontology representation suits our needs as it provides the solution for two main problems: knowledge unification and knowledge interchange. Works in this area have already been done by [START_REF] Stojanovic | The role of ontologies in autonomic computing systems[END_REF] and we found this model fully applicable to our system as it covers both knowledge formalization and rule usage. We propose a division of the knowledge aspect into two main categories: static and dynamic.

Static knowledge base

The static aspect of knowledge contains all the knowledge representation under the form of ontology concepts: classes, individuals and the properties linking them. Our implementation uses OWL3 as ontology description language and Protégé 4 4 as software support for ontology manipulation. The ontology contains over 150 concepts and over 250 axioms. We propose two main data types for the static knowledge base:

System information and architecture refers to all data concerning the software and hardware specifications of the DSS system (i.e. the quantity of RAM memory installed on a server). This subject has been approached [START_REF] Ackerman | Using ontologies to support development of software architectures[END_REF] and detailed by a certain number of editors [START_REF]Understanding system definition model (SDM) and its practical application in 2006 to[END_REF]. Fig. 1 shows how we model the DSS architecture hierarchically, with the use of a UML 5 diagram. Several entities are distinguished, starting from the top of the hierarchical tree (the DSS) and, at each level one or more sub-components can be identified with the specific parameters. System report and performance contains aspects regarding the DSS, in particular the data warehouse, parameters and performance indicators. There are many indicators to take into consideration: memory cache values, query response times, report editing times, aggregation operation times, etc. This paper considers three types of the Essbase cube cache: Index Cache, Data File Cache and Data Cache and the query response time as performance indicator. The implication is that by a good configuration of these three caches according to a given situation (i.e. at night some bases are not used, some cache allocations are useless, etc.) we gain substantial processing time. Although the problematic of performance improvement in data warehouses throughout caches is debated [START_REF] Malik | Automated Physical Design in Database Caching[END_REF], [START_REF] Saharia | Enhancing Data Warehouse Performance through Query Caching[END_REF] the issue is always addressed either through the physical design or the design of algorithms to determine which information is likely to be stored in cache memories. Cache allocation improvement is an important aspect of data warehouse tuning and there are little or no works on cache improvement in the context of data warehouses.

Dynamic knowledge base

The dynamic aspect is the challenge in our work and provides the main innovating approaches. It formalizes the functioning system and the analysis aspect of data warehouse management presented in 2.1. It contains all the rules that are part of the AC loop and all the rules that determine property interdependence and individual inference in the ontology. For reasons of simplicity and efficiency we have chosen to use Jena Rules via the Jena Java API 6 for ontology development. The rules are divided according to their area of activity in conformity with the AC loop phases.

For the business rule illustration, we give below an example of an analysis business rule that informally states: The closer the Index Cache value is to the Index File Size for a base, the better. We formalize this rule and obtain a discrete point allocation for the different proportions:

[rule1: (?base rdf:type cp:c_Base) (?base cp:dp_hasIndexFileSize ?ifs) (?base cp:dp_hasIndexCache ?ic) quotient(?ic, ?ifs, ?rap) ge(?rap, "0.95"^^xsd:double) -> (?base cp:dp_hasPoints_Advice_IndexCacheAllocation "1000"^^xsd:int)] [rule2: (?base rdf:type cp:c_Base) (?base cp:dp_hasIndexFileSize ?ifs) (?base cp:dp_hasIndexCache ?ic) quotient(?ic, ?ifs, ?rap) lessThan(?rap, "0.95"^^xsd:double) ge(?rap, "0.85"^^xsd:double) -> (?base cp:dp_hasPoints_Advice_IndexCacheAllocation "900"^^xsd:int)] …

The formalization process of business rules such as this one a very important aspect, and always requires an expert hand. By applying all the analysis rules we obtain an overall scoring and we can calculate a performance level of all the data warehouses from the point of view of business rules.

For the autonomic computing rule illustration we propose a simple example of the passage through the 4 states of the autonomic manager. We show the index cache is modified in a cycle.

Monitor: retrieval of the response time and the current cache values for a DW. These are stored in the knowledge base via the java program.

Analyse: we compare the average response time of the DW with its desired response time. If it is greater, the caches must be increased:

(?base cp:dp_hasAvgResponseTime ?avgt) (?base cp:dp_hasDesiredResponseTime ?dt) ge(?avgt, ?dt) -> (?base cp:hasState cp:IncreaseCache) Plan: we try to see if the increased cache state can be applied for the index cache. If the new cache value (increased with 10% of its current value) is not greater than the allocated memory, then plan the change to the index cache:

(?base cp:hasState cp:IncreaseCache) (?base cp:dp_hasIndexCache ?ic) (?base cp:dp_hasAllocatedMemory ?am) product(?ic, '1.1'^^xsd;double, ?newic) le(?newic, ?am) -> (?base cpdp_:dp_hasIndexCache ?newic) Execute: Execute a modification script with the new proposed value of the index cache. This is done via the java program.

Experimentation and Results

For our experiments we have considered a test suite that simulates a real environment with the associated parameters. On an existing server we have chosen an Essbase cube as the data warehouse whose performances were improved. For the pertinence of the tests, the cube was created starting from the "Sample" base provided by Essbase. The cube contains in average 11 principal axes and 27 level 2 axes and the data file has an average size of 300MB.

With the respective cube we carried out several tests corresponding to several configurations. We had to simulate the night/day loop passage faster so we made a series of 5 queries (from very fast to very slow as time of response) and applied them to each configurations. This process was iterated 10 times therefore simulating a day/night cycle. At the end of each cycle we fetch the average response times for each of the bases and pass through the autonomic computing loop so that we could optimize the cache allocation where it was necessary. The evolution of cache allocation with the response times can be observed in Fig. 2. We can see the difference between the minimum cache allocations in configuration C1 which is almost 6 times slower than the maximum possible allocation in configuration C6. As a maximum allocation is not always possible due to the quantity of memory available, we have to try our best to improve the performances. Applying our approach, configuration C3 maximizes the BR improvement points, and with a passage through the AC loop several times we get a configuration which has a smaller response time C4. The main improvement is that C4 is using only 174MB for its caches whereas C3 requires 240MB. In addition, the average response times in comparison with a normal or random configuration (as in C2 and C3) are improved at least 2 times.

Conclusions

This article presented how autonomic computing and ontologies can be used for helping DSS experts improve the cache memory allocations for data warehouses. It is not the first approach that tries to combine the two elements together [START_REF] Stojanovic | The role of ontologies in autonomic computing systems[END_REF], [START_REF] Gonzales | Self-adapted service offering for residential environments[END_REF] but the premiere is its application in the field of business intelligence and data warehouse improvement using business rules.

There are many positive aspects to this approach of decision support system management: the simple and intuitive yet powerful ontology representation, the facilities of web semantics and rule support brought to the domain of BI and, last but not least, the process of autonomic computing to manage all these elements.

Our future directions are to expand the Data Warehouses described above so that our small prototype can prove its efficiency on more than a 'few simple rules'. Our purpose is to integrate the prototype presented here with more than one aspect (data warehouse cache allocations based on response times) of decision support systems.

As the domain is relatively new and not much has been written on the subject yet, we try to bring as much support as possible for future development in the direction of autonomic decision support systems. We follow these changes and hope that our work will equally add something to this expanding environment.

Fig. 1

 1 Fig. 1 DSS hierarchical architecture

Fig. 2 .

 2 Fig. 2. Query average response times evolution with the cache configurations

http://download.oracle.com/docs/cd/E10530_01/doc/epm.931/html_esb_dbag/frameset.htm?ds tcache.htm

http://www.w3.org/TR/2008/CR-sml-20081125/

http://www.w3.org/2007/OWL/wiki/OWL_Working_Group

http://protege.stanford.edu/download/registered.html

http://www.uml.org/

http://jena.sourceforge.net/inference/#rules