Vincent Maurice
email: vincent.maurice@chimieparistech.psl.eu

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

Molybdenum effects on the stability of passive films unraveled at the nanometer and atomic scales

Data recently obtained on model FeCrNi(Mo), 316 L stainless steel, and FeCrNiCo(Mo) passivated surfaces by advanced surface analysis and density functional theory modeling are comprehensively discussed to unravel the multiple effects that molybdenum might have at the nanometer and atomic scales to enhance the stability of passive films. The key role played on corrosion protection by the compositional and structural nanoscale defects of the passive film that originate from the pre-passivation mechanisms of the surface is considered. It is shown how Mo, enriched together with Cr in the nanometer-thick passive film, can combine several effects to enhance the resistance to Cl --induced passivity breakdown. Enriched as Mo(VI) species in the outer exchange layer of the passive film, Mo impedes the deep penetration of Cl -ions and limits their access to the inner barrier layer. Dispersed as Mo(IV) at the interface with the inner layer, Mo protects against the entry of Cl -ions into the defect sites of the Cr(III) oxide barrier. Present as Mo(IV + δ) in the Fe-rich compositional nanoscale defects self-generated by the local failure of Cr supply upon initial formation of the barrier layer, Mo enhances the selective dissolution of iron and its replacement by chromium and molybdenum. By impeding the formation of O vacancies, Mo also increases the resistance against chloride entry in the oxide matrix, thereby curing these the Fe-rich weak sites against Cl --induced passivity breakdown.

INTRODUCTION

Passivity is the best of all means of corrosion protection of metals and alloys components in service conditions [START_REF] Macdonald | Passivity-the key to our metals-based civilization[END_REF][START_REF] Schmuki | From Bacon to barriers: a review on the passivity of metals and alloys[END_REF][START_REF] Olsson | Passive films on stainless steels-chemistry, structure and growth[END_REF][START_REF] Strehblow | Passivity of Metals[END_REF][START_REF] Maurice | Passive films at the nanoscale[END_REF][START_REF] Strehblow | Passivity of metals studied by surface analytical methods, a review[END_REF][START_REF] Maurice | Progress in corrosion science at atomic and nanometric scales[END_REF] . However, the presence of aggressive anions such as chloride ions in the environment can trigger the initiation of localized corrosion and thereby limit the durability of the self-protection property of the metallic substrate [START_REF] Frankel | Pitting corrosion of metals a review of the critical factors[END_REF][START_REF] Macdonald | The point defect model for the passive state[END_REF][START_REF] Marcus | Localized corrosion (pitting): a model of passivity breakdown including the role of the oxide layer nanostructure[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][START_REF] Strehblow | Corrosion Mechanisms in Theory and Practice[END_REF][START_REF] Soltis | Passivity breakdown, pit initiation and propagation of pits in metallic materials-review[END_REF][START_REF] Frankel | Localized corrosion: passive film breakdown vs pit growth stability[END_REF] .

On stainless steels (SS) and Ni-based stainless alloys, chromium is the key alloying element added to obtain corrosion protection by passivity. Chromium is enriched in the passive film [START_REF] Olefjord | Surface composition of stainless steels during anodic dissolution and passivation studied by ESCA[END_REF][START_REF] Marcus | Round Robin on combined electrochemical and AES/ ESCA characterization of the passive films on Fe-Cr and Fe-Cr-Mo alloys[END_REF][START_REF] Maurice | XPS and STM study of passive films formed on Fe-22Cr (1 1 0) single-crystal surface[END_REF][START_REF] Maurice | X-ray photoelectron spectroscopy and scanning tunneling microscopy study of passive films formed on (1 0 0) Fe-18Cr-13Ni single-crystal surfaces[END_REF][START_REF] Olsson | Quantifying the metal nickel enrichment on stainless steel[END_REF][START_REF] Keller | XPS investigations of electrochemically formed passive layers on Fe/Cr-alloys in 0.5 M H2SO4[END_REF][START_REF] Maurice | Effects of molybdenum on the composition and nanoscale morphology of passivated austenitic stainless steel surfaces[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][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 | An XPS and ToF-SIMS study of the passive film formed on a model FeCrNiMo stainless steel surface in aqueous media after thermal pre-oxidation at ultra-low oxygen pressure[END_REF] , and it is protective owing to the formation of Cr(III) oxidized species with extremely low dissolution kinetic [START_REF] Strehblow | Passivity of Metals[END_REF] . Compared to air-formed native oxide films, chromium (and molybdenum for Mo-bearing alloys) is further enriched in the passive oxide film when the alloy substrate is immersed in acidic and near-neutral aqueous environments due to the faster dissolution of iron species in the electrolyte [START_REF] Marcus | Round Robin on combined electrochemical and AES/ ESCA characterization of the passive films on Fe-Cr and Fe-Cr-Mo alloys[END_REF][START_REF] Olsson | Quantifying the metal nickel enrichment on stainless steel[END_REF][START_REF] Keller | XPS investigations of electrochemically formed passive layers on Fe/Cr-alloys in 0.5 M H2SO4[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Olsson | An AES and XPS study of the high alloy austenitic stainless steel 254 SMO® tested in a ferric chloride solution[END_REF][START_REF] Zhang | Characterization of film properties on the Ni-Cr-Mo Alloy C-2000[END_REF] . Anodic passivation causes re-growth and dehydrogenation of the oxide film when compared to open circuit (i.e., free) potential conditions [START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF] . The passive oxide films are 1 to 3 nm-thick, depending on alloy composition and passivation conditions [START_REF] Keller | XPS investigations of electrochemically formed passive layers on Fe/Cr-alloys in 0.5 M H2SO4[END_REF][START_REF] Maurice | Effects of molybdenum on the composition and nanoscale morphology of passivated austenitic stainless steel surfaces[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][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] Marcus | The anodic dissolution and passivation of NiCrFe alloys studied by ESCA[END_REF][START_REF] Sugimoto | The role of molybdenum additions to austenitic stainless steels in the inhibition of pitting in acid chloride solutions[END_REF] . Thinning or thickening can also be observed after passivation, depending on the competition between the rates of oxide dissolution and oxide re-growth [START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][START_REF] Wang | XPS and ToF-SIMS investigation of native oxides and passive films formed on nickel alloys containing chromium and molybdenum[END_REF] . The alloy close to the metal/oxide interface exhibits a modified composition, resulting from the preferential oxidation of elements such as chromium [START_REF] Steffen | Oxidation of NiCr and NiCrFe alloys at room temperature[END_REF] with higher affinity to oxygen during oxide film formation and leading to enrichment in metallic nickel under the oxide layer in Fe-based austenitic alloys [START_REF] Maurice | XPS and STM study of passive films formed on Fe-22Cr (1 1 0) single-crystal surface[END_REF][START_REF] Maurice | Effects of molybdenum on the composition and nanoscale morphology of passivated austenitic stainless steel surfaces[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][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] Lim | ESCA studies of Ni-Cr alloys[END_REF] .

In many of the surface analytical studies of passivated stainless steels and alloys, the stratified structure of the nanometer-thick passive film could be characterized. It is described by a bilayer model [START_REF] Maurice | Effects of molybdenum on the composition and nanoscale morphology of passivated austenitic stainless steel surfaces[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][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] Zhang | Characterization of film properties on the Ni-Cr-Mo Alloy C-2000[END_REF][START_REF] Marcus | The anodic dissolution and passivation of NiCrFe alloys studied by ESCA[END_REF] . The inner layer forms a dense oxide barrier highly enriched in Cr(III) oxide, limiting atomic transport across the interface and thus giving the alloy its high corrosion resistance. The outer layer, viewed as the exchange layer with the electrolyte, can contain Cr(III) hydroxide, Fe(II,III) oxide/hydroxide, Ni(II) hydroxide, and Mo(IV,VI) oxide species depending on the alloy composition [START_REF] Maurice | Effects of molybdenum on the composition and nanoscale morphology of passivated austenitic stainless steel surfaces[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][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] Zhang | Characterization of film properties on the Ni-Cr-Mo Alloy C-2000[END_REF][START_REF] Marcus | The anodic dissolution and passivation of NiCrFe alloys studied by ESCA[END_REF][START_REF] Sugimoto | The role of molybdenum additions to austenitic stainless steels in the inhibition of pitting in acid chloride solutions[END_REF] . Since the formation of oxide is usually faster compared to cation diffusion, the possible occurrence of a non-equilibrium solute capture has been suggested [START_REF] Yu | Nonequilibrium solute capture in passivating oxide films[END_REF][START_REF] Cwalina | In operando analysis of passive film growth on Ni-Cr and Ni-Cr-Mo alloys in chloride solutions[END_REF][START_REF] Sherman | Thermodynamics of solute capture during the oxidation of multicomponent metals[END_REF] , defined as a non-stoichiometric Cr(III) oxide matrix in which Ni 2+ , Fe 2+ as well as Mo 4+ cations would be captured.

In addition to chromium, molybdenum is the other key alloying element added in stainless alloys for improving the corrosion resistance. Other elements such as N can also be added [START_REF] Mozhi | The effect of nitrogen on the sensitization of AISI 304 stainless steel[END_REF][START_REF] Marcus | On some fundamental factors in the effect of alloying elements on passivation of alloys[END_REF] . Mo added in stainless steel can enhance significantly the resistance to pitting, crevice and stress corrosion [START_REF] Rockel | Pitting, crevice and stress corrosion resistance of high chromium and molybdenum alloy stainless steels[END_REF] . For 316 L and 304 L austenitic stainless steels of quite similar composition, the 2 at. % Mo added in grade 316 L markedly improve the resistance to chloride-induced pitting corrosion [START_REF] Ilevbare | The role of alloyed molybdenum in the inhibition of pitting corrosion in stainless steels[END_REF] , and it has been shown that 1.5 at. % Mo added in FeCrNi alloys can decrease the active dissolution rate in HCl aqueous solution by a factor of 5 to 10 [START_REF] Newman | The dissolution and passivation kinetics of stainless alloys containing molybdenum-1. Coulometric studies of Fe-Cr and Fe-Cr-Mo alloys[END_REF][START_REF] Newman | The dissolution and passivation kinetics of stainless alloys containing molybdenum-II. Dissolution kinetics in artificial pits[END_REF] . Besides, sodium molybdate is also widely used as a corrosion inhibitor [START_REF] Vukasovich | Molybdate in corrosion inhibition-a review[END_REF][START_REF] El Din | Mechanism of corrosion inhibition by sodium molybdate[END_REF][START_REF] Li | Sodium molybdate as a corrosion inhibitor for aluminium in H3PO4 solution[END_REF] , and it has been argued that adding molybdenum to stainless steels would be like adding a molybdate inhibitor to the solution [START_REF] Kaneko | Effects of molybdenum on the pitting of ferritic-and austenitic-stainless steels in bromide and chloride solutions[END_REF][START_REF] Alshamsi | Corrosion of heat-treated 304SS in the presence of molybdate ions in hydrochloric acid[END_REF] .

Despite many studies, the exact role that molybdenum plays in stabilizing passivity is still debated. Some authors say that it aids in preventing the breakdown of the passive film in the first place [START_REF] Ilevbare | The role of alloyed molybdenum in the inhibition of pitting corrosion in stainless steels[END_REF][START_REF] Bojinov | Influence of molybdenum on the conduction mechanism in passive films on iron-chromium alloys in sulphuric acid solution[END_REF][START_REF] Hashimoto | An X-ray photo-electron spectroscopic study on the role of molybdenum in increasing the corrosion resistance of ferritic stainless steels in HCl[END_REF][START_REF] Horvath | Critical potentials for pitting corrosion of Ni, Cr-Ni, Cr-Fe, and related stainless steels[END_REF] , while for others molybdenum would reduce active dissolution after passivity breakdown and thus promote the selfrepair of the oxide film [START_REF] Newman | The dissolution and passivation kinetics of stainless alloys containing molybdenum-1. Coulometric studies of Fe-Cr and Fe-Cr-Mo alloys[END_REF][START_REF] Tan | The effect of air exposure on the corrosion behavior of amorphous Fe-8Cr-Mo-13P-7C alloys in 1 M HCl[END_REF][START_REF] Lee | Effects of alloying elements, Cr, Mo and N on repassivation characteristics of stainless steels using the abrading electrode technique[END_REF][START_REF] Elbiache | The role of molybdenum in the dissolution and the passivation of stainless steels with adsorbed sulphur[END_REF][START_REF] Pardo | Effect of Mo and Mn additions on the corrosion behaviour of AISI 304 and 316 stainless steels in H2SO4[END_REF][START_REF] Pardo | Pitting corrosion behaviour of austenitic stainless steels-combining effects of Mn and Mo additions[END_REF] . These mechanisms may act solely or concurrently to conduce to the enhancive effect of Mo on corrosion resistance depending on how advanced is the localized corrosion process [START_REF] Frankel | Localized corrosion: passive film breakdown vs pit growth stability[END_REF][START_REF] Sun | Revisiting the effect of molybdenum on pitting resistance of stainless steels[END_REF] .

The view that Mo mitigates passivity breakdown is supported by results showing that Mo species impede the penetration of aggressive ions such as Cl -into the oxide film. Indeed, Mo(IV,VI) and Cr(III) oxide species are found to be concentrated mostly in the outer exchange and inner barrier layers of the duplex passive film, respectively [START_REF] Maurice | Effects of molybdenum on the composition and nanoscale morphology of passivated austenitic stainless steel surfaces[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][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] Zhang | Characterization of film properties on the Ni-Cr-Mo Alloy C-2000[END_REF][START_REF] Marcus | The anodic dissolution and passivation of NiCrFe alloys studied by ESCA[END_REF][START_REF] Sugimoto | The role of molybdenum additions to austenitic stainless steels in the inhibition of pitting in acid chloride solutions[END_REF][START_REF] Yamamoto | Depassivation-repassivation behavior of type-312L stainless steel in NaCl solution investigated by the micro-indentation[END_REF][START_REF] Wang | Chloride-induced alterations of the passive film on 316 L stainless steel and blocking effect of prepassivation[END_REF] , and the penetration of Cl -can be limited to only the outmost part of the oxide film [START_REF] Wang | Chloride-induced alterations of the passive film on 316 L stainless steel and blocking effect of prepassivation[END_REF] . Also, Cl -incorporated islands were reported to form on the surface of 316 L SS with no Cl -penetration underneath after immersion in NaCl aqueous solution, while no islands but pits were formed on 304 L SS immersed in the same solution [START_REF] Vignal | Effect of molybdenum on passivity of stainless steel chloride media using ex situ near field microscopy observations[END_REF] . Besides, the presence of Mo oxides on the surface would prevent the outward diffusion of Cr to the oxide surface [START_REF] Wang | Ion transport mechanisms in the oxide film formed on 316 L stainless steel surfaces studied by ToF-SIMS with 18O2 isotopic tracer[END_REF] , thus preventing the voiding mechanisms proposed as one route leading to passivity breakdown [START_REF] Frankel | Pitting corrosion of metals a review of the critical factors[END_REF][START_REF] Macdonald | The point defect model for the passive state[END_REF][START_REF] Marcus | Localized corrosion (pitting): a model of passivity breakdown including the role of the oxide layer nanostructure[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][START_REF] Strehblow | Corrosion Mechanisms in Theory and Practice[END_REF][START_REF] Soltis | Passivity breakdown, pit initiation and propagation of pits in metallic materials-review[END_REF] . Mo oxides would also suppress the passive film dissolution [START_REF] Lloyd | Mo and W alloying additions in Ni and their effect on passivity[END_REF] , which is another route that can lead to passivity breakdown [START_REF] Frankel | Pitting corrosion of metals a review of the critical factors[END_REF][START_REF] Macdonald | The point defect model for the passive state[END_REF][START_REF] Marcus | Localized corrosion (pitting): a model of passivity breakdown including the role of the oxide layer nanostructure[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][START_REF] Strehblow | Corrosion Mechanisms in Theory and Practice[END_REF][START_REF] Soltis | Passivity breakdown, pit initiation and propagation of pits in metallic materials-review[END_REF] .

In support of the view that Mo promotes self-repair, it has been argued that, although a certain amount of Cr is necessary to ensure the beneficial effect of Mo in increasing the corrosion resistance [START_REF] Sugimoto | The role of molybdenum additions to austenitic stainless steels in the inhibition of pitting in acid chloride solutions[END_REF] , the Mo surface coverage on FeCrMo alloy is small and yet active dissolution is markedly decreased [START_REF] Newman | The dissolution and passivation kinetics of stainless alloys containing molybdenum-1. Coulometric studies of Fe-Cr and Fe-Cr-Mo alloys[END_REF][START_REF] Newman | The dissolution and passivation kinetics of stainless alloys containing molybdenum-II. Dissolution kinetics in artificial pits[END_REF] . On Ni-based alloy surfaces, it was observed that Mo enriches at the surface in the transpassive range, and redissolves when the potential is returned to the passive range, suggesting that Mo dissolution/ precipitation could play an important role in the breakdown/ repassivation process [START_REF] Henderson | Molybdenum surface enrichment and release during transpassive dissolution of Ni-based alloys[END_REF] , possibly involving the formation of molybdates [START_REF] Sun | Revisiting the effect of molybdenum on pitting resistance of stainless steels[END_REF] .

Herein, we comprehensively discuss data recently obtained on FeCrNi(Mo) model alloy, on 316 L stainless steel, and on Cr-and Mo-bearing multi-principal element alloy (MPEA) passivated surfaces using scanning tunneling microscopy (STM), X-Ray photoemission spectroscopy (XPS) and time-of-flight secondary ion mass spectrometry (ToF-SIMS). The discussion highlights the key role played on corrosion protection by the compositional and structural defects of the passive film that originate from the prepassivation mechanisms of the surface. Data obtained from density functional theory (DFT) modeling are also included in the discussion as they contribute to unravel the multiple effects that molybdenum might have at the nanometer and atomic scales to enhance the stability of passive films.

RESULTS AND DISCUSSION

Fe-rich nanoscale defects in Cr oxide barrier layers

The detailed mechanisms by which an oxide film initially forms on a metal substrate until complete surface saturation, i.e., how the surface pre-passivates, can determine the corrosion protection properties of the substrate in its later life. However, for reactive materials such as Fe-or Ni-based stainless alloys, the formation of the native oxide layer is so fast that extremely well-controlled conditions of exposure to the oxidizing environment are required to be able to time-resolve the transition of the surface state from oxide-free to native oxide-covered. On austenitic stainless steel, this was achieved by applying a surface science approach in which a model single crystalline surface, FeCrNi(100), was first cleaned from any residual traces of surface oxide and contamination in ultra-high vacuum (UHV) and then exposed stepwise to gaseous oxygen at ultra-low pressure until saturation of the oxide growth. In situ chemical and structural surface analysis by XPS and STM was applied at each step, including to verify initial surface cleanliness and structure after preparation [START_REF] Ma | Stainless steel surface structure and initial oxidation at nanometric and atomic scales[END_REF][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 work provided direct evidence that the surface oxide film contains nanoscale heterogeneities of composition and structure, even when grown on a substrate of well-controlled crystallographic orientation and free of emergent microstructure defects, owing to the mechanisms of local supply of chromium (Fig. 1).

In the ultra-early stage of oxidation (which would correspond to a few nanoseconds at atmospheric pressure), local nucleation of chromium oxide was observed at the step edges [START_REF] Ma | Stainless steel surface structure and initial oxidation at nanometric and atomic scales[END_REF][START_REF] Ma | Origin of nanoscale heterogeneity in the surface oxide film protecting stainless steel against corrosion[END_REF] at moderate temperature (250 °C). Chromium is supplied by atomic pumping from the terraces to the step edges for formation of Cr(III)-rich oxide nuclei, via a Cr surface diffusion process involving surface vacancies. With ongoing exposure to oxygen, the formation of Ferich oxide nuclei is favored at locations adjacent to the step edges, where Cr was locally depleted owing to its initial consumption to form chromium oxide nuclei. Subsequently, a Cr(III)-rich layer nearly saturating the surface is formed, involving Cr (and Fe) transport from the metallic planes below the oxide. However, nearby the step edges, three-dimensional Fe-rich oxide islands, initiated owing to local depletion of Cr, continue to grow.

At room temperature [START_REF] Ma | Nanoscale early oxidation mechanisms of model FeCrNi austenitic stainless steel surfaces at room temperature[END_REF] , nucleation of Fe oxide occurs at the defect sites on the terraces while Cr(III) oxide nucleation occurs at the step edges. With ongoing exposure to oxygen, the oxide nuclei grow to grain size of up to 2-3 nm and in density until saturation of the surface by a nanogranular film homogeneously covering the substrate terraces and step edges. Oxide nuclei and nanograins develop no well-ordered structure, unlike at higher temperature (250 °C) where crystalline films are formed. A Cr(III)rich oxide film is formed at saturation with a Cr(III) to Fe(II/III) ratio depending on the local supply of metallic Cr and Fe during nucleation and growth and resulting in compositional heterogeneity at the nanoscale of the granular morphology. At 250 °C, the compositional heterogeneity of the surface oxide is less densified at the nanoscale owing to the higher temperature promoting atomic transport and thus the space separation of the formation of oxide nuclei enriched in either of the elements. Regardless of the temperature, nickel does not enter the composition of the grown oxide films.

To our knowledge, no similar local analysis of the nucleation and growth mechanisms has been conducted with model singlecrystal surfaces of Mo-bearing stainless steel. However, in situ XPS analysis of the nucleation and growth process on (100)-oriented FeCrNiMo single-crystal surfaces using the same approach revealed that initial oxidation of molybdenum occurs after chromium and iron have already begun to oxidize and confirmed that no nickel was found in an oxidized form within the oxide film [START_REF] Lynch | XPS study of oxide nucleation and growth mechanisms on a model FeCrNiMo stainless steel surface[END_REF] . From angular XPS analysis, in which the relative intensities of the species are measured at varying take-off angle of the photoelectrons, thus varying the depth sensitivity of the collected information, Cr(III) and Fe(II) were found enriched closer to the modified alloy-oxide interface with Fe(III) found predominantly at the outer part of the film. Oxidized molybdenum was found as both Mo(IV) and Mo(VI).

Despite the unknown, and yet to be studied, effects of water vapor on the nucleation and growth process, these studies performed on model stainless steel surfaces allow us to better understand the nature and the mechanisms of formation of native surface oxides. They provide us a nanoscale insight that needs to be considered to discuss the impact of the local chemical and structural defects self-generated by the initial oxidation process on the corrosion protection and the initiation of localized corrosion. Indeed, based on this insight, passivity breakdown can be expected in the least Cr(III)-enriched regions of the protective surface oxide, i.e., the Fe-rich defect sites in the passive film, and self-repair can be expected to be ineffective in the metallic surface regions depleted in Cr.

Passivation-induced alterations of native oxide films

Corrosion being observed in water-containing environments, it is a prerequisite to study the alterations of the oxide film initially formed on the surface when exposed to an aqueous electrolyte. This can be done by controlling the initial oxidation conditions and then exposing the native oxide-covered surface to the electrolyte of interest at open circuit potential (OCP) to investigate the naturally formed passive film. Additionally, the interface can be polarized in the passive range to study the alterations brought by anodic polarization. In order to limit the possible artefacts brought by ex situ transfer for surface analysis, a closed system, with direct transfer avoiding exposure to ambient air between UHV analytical platform for surface preparation and analysis and Ar-filled chamber or glove box equipped for electrochemical treatment and measurement can be used [START_REF] Strehblow | Passivity of metals studied by surface analytical methods, a review[END_REF][START_REF] Olefjord | Surface composition of stainless steels during anodic dissolution and passivation studied by ESCA[END_REF][START_REF] Keller | XPS investigations of electrochemically formed passive layers on Fe/Cr-alloys in 0.5 M H2SO4[END_REF][START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF][START_REF] Lynch | An XPS and ToF-SIMS study of the passive film formed on a model FeCrNiMo stainless steel surface in aqueous media after thermal pre-oxidation at ultra-low oxygen pressure[END_REF] . Alternatively, in situ conditions can be studied with specifically-designed electrochemical cells and using the benefits of synchrotronemitted photons for surface analysis, e.g., by near-ambient XPS [START_REF] Yue | Synchrotron-based near ambient-pressure X-ray photoelectron spectroscopy and electrochemical studies of passivation behavior of N-and V-containing martensitic stainless steel[END_REF] .

Using the first approach [START_REF] Wang | Mechanisms of Cr and Mo enrichments in the passive oxide film on 316 L austenitic stainless steel[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF] , it has been shown that on a 316 L SS surface covered by the native oxide spontaneously formed after mechanical polishing and immersed in sulfuric acid solution (selected as model chloride-free corrosive aqueous electrolyte), the passivation mechanism is characterized by faster dissolution of oxidized iron already occurring at free potential (OCP). The thickness of the initial native oxide film decreases; however, its enrichment in Cr(III) and Mo(IV,VI) is promoted, mostly in the exchange outer hydroxide layer of the duplex structure (Fig. 2). Anodic polarization in the passive range causes the steady-state thickness to increase owing to oxide growth less rapidly balanced by oxide dissolution than at OCP. The inner barrier oxide layer of the passive film becomes thicker, reflecting the increase of the enrichment in Cr(III) oxide, while the outer exchange layer is less Cr(III) hydroxide-rich, but maintains the Mo(IV/VI) enrichment obtained at OCP. Metallic Mo concentration is unaltered in the modified alloy region underneath the oxide film compared to the initial surface state.

On a pre-oxidized surface obtained by exposing the oxide-free 316 L SS surface prepared under UHV to low oxygen pressure at room temperature [START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF] , the oxide film grown at saturation is thinner than on the native oxide-covered surface due to minimization of the promoting effect of water on the surface oxidation kinetics (Fig. 2). The formation of the inner layer is limited to less than one equivalent monolayer (i.e., not continuous) and the outer layer is thinner but equally enriched in Mo(IV,VI) than on the native oxidecovered surface. Mo concentration is unaltered in the modified alloy region underneath the oxide film. At OCP, faster dissolution of oxidized iron also promotes the Cr(III) and Mo(IV,VI) enrichment. However, molybdenum enrichment is markedly increased in the exchange outer layer compared to the native oxide-covered surface, which improves the protectiveness against transient active dissolution during the transition from OCP to anodic passive state [START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF] . Like on the native oxide-covered surface, anodic passivation increases the steady-state thickness but with higher Mo-enrichment in the exchange outer layer.

On (100)-oriented FeCrNiMo single-crystal surfaces [START_REF] Lynch | An XPS and ToF-SIMS study of the passive film formed on a model FeCrNiMo stainless steel surface in aqueous media after thermal pre-oxidation at ultra-low oxygen pressure[END_REF] , thermal pre-oxidation at low oxygen pressure and 250 °C was also found to improve the protectiveness of the surface oxide film during the OCP-to-anodic passive state transition when compared with the native oxide formed on the same surface. The thermal oxide film was found to be enriched in iron which later dissolved into the acidic electrolyte after being held at OCP, resulting in the enrichment of chromium and molybdenum in the film like on 316 L SS surfaces. Anodic passivation causes growth of the film and favors dehydrogenation of both chromium and molybdenum. Thermal pre-oxidation at low oxygen pressure promotes the enrichment of molybdenum in the passive film, albeit not as significantly when compared to room temperature pre-oxidation on 316 L SS surfaces. However, it enhances the growth of the chromium inner oxide layer after anodic passivation, which can be Fig. 1 Early oxidation mechanisms at the origin of the nanoscale heterogeneity of the surface oxide film grown on austenitic stainless steel. Cross-section comparative models at room temperature (RT) and 250 °C with height-to-width aspect ratio not to scale.

expected to influence the transport of ions through the passive film.

Owing to their local nature and nanometer dimensions, it is extremely difficult to investigate the fate of the Fe-rich weak sites of protection initially generated in the barrier layer of the surface oxide film upon natural passivation at free potential and/or anodic passivation. However, we can reasonably speculate based on the insight brought by the studies of the global alterations induced by passivation. Iron oxides dissolving faster, it can be inferred that the Fe-rich defect sites of inner barrier layer may dissolve already at OCP, at least partially. Thereby, their content in iron would decrease, as well as their density in the inner layer in case of complete dissolution. Upon re-growth of the oxide film under anodic polarization, iron in these Fe-rich defect sites of the inner barrier layer would be replaced mostly by chromium but possibly also by molybdenum observed to be globally further enriched in the passive films. While the enrichment in Cr(III) oxide is directly reflected by the increase in thickness of the barrier layer (assumed to be 100% Cr(III) oxide in the quantification models), the enrichment in Mo(IV,VI) species is localized in the outer layer of these models. However, detailed analysis of the depth distribution of molybdenum in the protective oxide films shows that it is also present deeper, in the inner barrier layer in the protective oxide film as discussed below.

Passivity enhancement by Mo (and Cr) enrichment

Controlling the pre-oxidation conditions appears instrumental to promote the surface molybdenum enrichment of the initial surface oxide, as shown on model FeCrNiMo and 316 L SS surfaces, which can be expected to benefit the sealing properties of the exchange outer layer of the passive film against the ingress of aggressive ions. Moreover, the Fe-rich weak sites of the inner barrier layer would be healed by mostly Cr but also Mo replacement. Besides, the re-passivation properties of the modified alloy region after passivity breakdown would not be altered since molybdenum is not depleted in this region. Beneficial effects on passivity breakdown and self-repair were indeed observed on 316 L SS surfaces prepared by a surface finishing pretreatment of vibratory polishing [START_REF] Neupane | Can we enhance passivity with a surface finish? Spectroscopic and electrochemical analysis on 316 L stainless steel[END_REF] . XPS and ToF-SIMS surface analysis confirmed that the vibratory finishing pretreatment increases the Cr and Mo enrichments in the native oxide film. As a result, the bilayer structure was characterized by a thicker Cr(III) oxide inner barrier layer and a more Cr(III) hydroxideand Mo(IV,VI) oxide-enriched outer exchange layer than on the surface prepared by traditional mechanical polishing. Accelerated testing of the resistance to passivity breakdown and the initiation of localized corrosion was performed by potentiostatic polarization in the passive range, first in chloride-free aggressive acidic electrolyte (0.05 M H 2 SO 4 (aq)) until reaching a quasi-steady-state of passivity, and then by sequentially increasing the chloride ions (NaCl) concentration without changing the pH.

Several beneficial effects of the Mo (and Cr) enrichment of the native oxide-covered surface could thus be highlighted (Fig. 3a). Vibratory finishing of the surface was found to enhance passivity, as shown by the lower current density observed in the Cl-free solution at the active/passive transition and in the near steadystate of passivity compared to a surface finish by traditional polishing, which can be assigned to the initial higher Cr enrichment of the surface. Resistance to chloride-induced passivity breakdown was also improved, as shown by a lower frequency of breakdown/self-repair events, i.e., metastable pitting, observed for the more Mo-rich vibratory polished surface. It can be noticed that despite being less frequent the metastable events are of higher amplitude for the vibratory polished alloy implying that the surface re-passivates despite the possible formation of larger metastable pits. Furthermore, a higher chloride concentration was needed to trigger stable pitting/crevice corrosion, and thus to block the self-repair of passivity on the more Mo-rich vibratory polished surface. Thus, with such an accelerated test, it was shown that the initial Mo enrichment of the native oxidecovered surface can act concurrently to enhance the stability of the passive state, by increasing the resistance to Cl-induced breakdown of the passive film, and by mitigating the blocking effect of chloride ions on the re-passivation of the metallic surface exposed by the passive film breakdown.

These enhancing effects of molybdenum on the passivity of stainless steel were confirmed by investigating the corrosion properties of Cr-and Mo-bearing multi-principal element alloys of higher Mo content than 316 L SS [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] . Such alloys (Cr 25 Fe 25 Co 5 -Ni 40 Mo 5 , Cr 15 Fe 10 Co 5 Ni 60 Mo 10 in at. %) were designed, processed and characterized as having a single phase fcc structure that is maintained while varying their relatively high content in Cr and Mo. It was confirmed by ToF-SIMS and XPS surface analysis that the native oxide films had a bilayer structure with the inner barrier layer mostly consisting of Cr(III) oxide, and the outer exchange layer containing Cr(III) hydroxide, Fe(II,III) oxide/hydroxide and Ni(II) hydroxide [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF] . As detailed further on, Mo was enriched as Mo(IV,VI) oxide species, and mostly concentrated in the outer layer, like on stainless steel.

Submitted to the same type of accelerated corrosion test as 316 L SS (Fig. 3b), the beneficial effect of Cr in enhancing passivity was confirmed, as observed by the near steady-state passive currents decreasing in the Cl-free solution with increasing Cr content in the substrate. For an Mo-free MPEA (Cr 35 Fe 20 Co 5 Ni 40 ), Cl-induced metastable pitting was followed by the initiation of localized corrosion despite the high content of the alloy in Cr (35 at. %). In contrast, for both Mo-bearing MPEAs, no passivity breakdown/self-repair events were observed as well as no initiation of localized corrosion, even for a chloride concentration as high as 4.7 M. Like in Cl-free solution, the near steady-state passive currents were even observed to continuously decrease with time, showing no Cl-induced degradation of the passive state.

To understand the absence of effect of chloride ions on passivated Mo-bearing MPEAs, the questions that arise are whether the Cl -ions of the concentrated aggressive environment in these tests interact with the passivated surface or not, and, if yes, how deep they penetrate the protective oxide films. This was answered by applying ToF-SIMS to characterize the depth distribution of Cl from the outer surface of the passive film to the modified alloy region underneath (Fig. 4) [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] . Indeed, the Cl - ions were found to interact with the passivated surface and penetrate the passive film. However, the penetration was limited to the outer layer of the bilayer structure. In the inner layer, no penetration was observed for the Cr 15 4). Besides, pre-passivation in the Cl-free agressive solution also contributes to significantly impede the penetration of the agressive species whereas no compositional changes of the passive film were observed, which was assigned to a less defective nanostructure rather than to compositional enhancement of the passive films [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] . Hence, Mo can be viewed as having -secondary ions. However, the slower intensity decay of the MoO 2 -secondary ions observed afterwards clearly marks the predominance of Mo(IV) oxide species when at the onset of the transition between outer and inner layers. This difference in depth distribution of the Mo(IV) and Mo(VI) oxide species was also confirmed for Cr 15 Fe 10 Co 5 Ni 60 Mo 10 alloy surfaces passivated in the same conditions [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF] .

In addition to Mo(IV) and Mo(VI) oxide species, XPS analysis also revealed an additional Mo species characterized by a 5/2-3/2 doublet component observed in the Mo 3d core level region at a binding energy (231.2-234.3 eV) intermediate between those of the Mo(IV) and Mo(VI) components, at 227.8-231.0 and 232.6-235.7 eV, respectively [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF] . Such an intermediate Mo species has been previously observed and assigned to hydroxylated Mo(IV) [START_REF] Olefjord | Surface composition of stainless steels during anodic dissolution and passivation studied by ESCA[END_REF][START_REF] Lynch | Passivation-induced Cr and Mo enrichments of 316 L stainless steel surfaces and effects of controlled pre-oxidation[END_REF] , Mo(V) oxide [START_REF] Zhang | Characterization of film properties on the Ni-Cr-Mo Alloy C-2000[END_REF][START_REF] Clayton | Electrochemical and XPS evidence of the aqueous formation of Mo 2 O 5[END_REF][START_REF] Eidhagen | Synchrotron XPS and electrochemical study of aging effect on passive film of Ni alloys[END_REF] , or an intrinsic satellite of Mo(IV) oxide due to final state screening effect [START_REF] Gulino | Influence of metal-metal bonds on electron spectra of MoO 2 and WO 2[END_REF][START_REF] Scanlon | Theoretical and experimental study of the electronic structures of MoO3 and MoO2[END_REF] . From angular XPS analysis [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF] , it was observed that the Mo(IV) oxide species were located deeper than the Mo(VI) oxide species, in agreement with the Tof-SIMS data (Fig. 5a), and that the intermediate Mo species were located underneath the Mo(IV) oxide species. The intermediate Mo species observed by XPS could thus be an oxide species present in the Cr(III) oxide matrix of the inner barrier layer, including in the Fe-rich defect sites, and carrying a formal charge higher than +4.

Regarding the concentration of the Mo(IV) oxide species, quantification of the XPS relative intensities was carried out using a bilayer model with Mo(IV) assumed to form and interfacial layer between the outer and inner layers as suggested by the peak of the MoO 2 -secondary ions clearly observed after that of the MoO 3 secondary ions and before that of the CrO -secondary ions characteristic for the Cr(III) oxide species (Fig. 5a). This resulted in equivalent thicknesses of 0.12 nm at the most for the passivated Cr 15 Fe 10 Co 5 Ni 60 Mo 10 alloy [START_REF] Wang | Origin of enhanced passivity of Cr-Fe-Co-Ni-Mo multi-principal element alloy surfaces[END_REF] . This corresponds to only fractions of one oxide monolayer suggesting that, even if accumulated at this interface, the Mo(IV) oxide species would not form a dense monolayer fully covering the inner barrier layer. For the intermediate Mo species and assuming their location in the Cr(III) inner barrier layer, their cationic fraction was calculated to be 6 and 19 at. % for the passivated Cr 25 Fe 25 Co 5 Ni 40 Mo 5 and Cr 15 Fe 10 Co 5 Ni 60 Mo 10 alloys, respectively, which is too high to be consistent with a doping effect of the Cr(III) oxide matrix of the inner layer and rather suggests a mixing effect, including at the Fe-rich oxide defect sites. The ToF-SIMS depth profiles presented in Fig. 5b supports this view of the minor presence of mixed XMoO n (X = Cr, Fe, Ni) oxide species in the inner region of the passive film formed on the Cr 25 Fe 25 Co 5 Ni 40 Mo 5 alloy. Indeed, an intensity maximum is observed near the interface with the substrate for XMoO 2 -and XMoO 3 -secondary ions that would originate from such parent mixed species, with the maxima being more intense for the XMoO 2 -secondary ions. The oxidation state of molybdenum cannot be retrieved from such measurements but this insight is provided by atomistic DFT simulation as discussed below.

Model of the passive oxide film with defect sites cured by Mo Figure 6 presents the model of passivated CrFeCoNiMo alloy surfaces proposed to account for these observations, and considering the localization of molybdenum at the nanometer scale. Like on passivated stainless steel surfaces, a bilayer structure is formed with the outer exchange layer enriched in Cr(III) hydroxide and containing Fe(II,III) oxide/hydroxide and Ni(II) hydroxide. Molybdenum is enriched as Mo(VI) oxide in this outer layer and its major effect is to impede the deep penetration of chloride ions (Fig. 4), as also observed for stainless steel surfaces [START_REF] Wang | Chloride-induced alterations of the passive film on 316 L stainless steel and blocking effect of prepassivation[END_REF] . Thus, it contributes to prevent passivity breakdown even in concentrated NaCl-containing aggressive electrolyte (Fig. 3b).

The inner barrier layer is mostly constituted of Cr(III) oxide like on passivated stainless steel surfaces. However, compositional and nanostructural defects are present, originating from the initial growth of the oxide film (Fig. 1). Compositional defects are Fe-rich sites, originating from the failure of the local supply of Cr during the pre-passivation process. Structural defects are mostly intergranular boundaries between the nanometer-size oxide grains produced by the nucleation and growth mechanism of the initial oxide film. These defects may be partially cured by Cr and Mo enrichments observed upon passivation in aqueous solution. However, they may subsist even after reaching a steady-state of passivity in Cl-free environment. The major effect of the Mo(IV) species concentrated at the interface between the outer and inner layers region of the passive film would be to form a (discontinuous) layer protecting these local chemical and structural defects. In this model, Fe-rich defect sites are stabilized by partial substitution or insertion of Mo cations at the surface, subsurface or even in the bulk of their oxide matrix. The DFT simulation discussed hereafter provided atomic insight on how substitutional Mo cures the weak sites in the passive film and thus counteracts the destabilizing effect of chlorides. The intergranular boundaries joining the oxide nanograins, expected to be preferential pathways for atomic transport across the protective inner layer 10 , could also be cured by Mo preferentially inserted at these interfaces of the barrier inner layer.

DFT atomistic insights into the role of Mo

Given the complexity of the stratified structure of passivated Crand Mo-bearing alloy surfaces and the relative lack of relevant structural details at the atomic scale, assumptions for atomistic modeling must include what appears as the most relevant feature for corrosion protection. For the systems under consideration here, it is the inner layer of the passive film that must be modeled, even though Mo enriched in the outer layer can impede the deep penetration of chloride ions and thus contribute to stabilize passivity. Indeed, the inner layer forms a dense barrier that slows down the atomic transport across the interface and thus drastically limits the exchange current in the passive state thus ensuring the corrosion protection [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][START_REF] Seyeux | Oxide film growth kinetics on metals and alloys: I. physical model[END_REF] . Ultimately, this barrier must breakdown and fail to self-repair for localized corrosion to initiate [START_REF] Frankel | Pitting corrosion of metals a review of the critical factors[END_REF][START_REF] Macdonald | The point defect model for the passive state[END_REF][START_REF] Marcus | Localized corrosion (pitting): a model of passivity breakdown including the role of the oxide layer nanostructure[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][START_REF] Strehblow | Corrosion Mechanisms in Theory and Practice[END_REF][START_REF] Soltis | Passivity breakdown, pit initiation and propagation of pits in metallic materials-review[END_REF] . On Fe-based stainless alloys, this inner layer of the passive film is mostly constituted of Cr(III) oxide but can contain compositional defects corresponding to Fe-rich sites, in addition to nanostructural defects. Mo, mostly present as Mo(VI) in the outer layer, is also present as Mo(IV) at the interface with the inner layer and possibly as Mo(IV + δ) in the oxide matrix of the inner layer as discussed above.

The strategy adopted for atomistic modeling by DFT of the Mo effects was to build periodic systems of an oxide matrix common to chromium and iron oxides, at the surface of which a Cr or Fe atom was substituted by a Mo atom [START_REF] Huang | Atomistic insights on enhanced passivity: DFT study of substitutional Mo on Cr2O3 and Fe2O3 surfaces[END_REF] . The different possible sites of replacement were investigated as well as the distributions of charges for the most stable substituted systems. Then, the effects of substitutional Mo on the formation energy of point defects, Cr (or Fe) metal vacancies or O vacancies, at the surface of the oxide matrix were calculated. The corundum structure, common to chromite (Cr 2 O 3 ) and hematite (Fe 2 O 3 ) and terminated by its most stable (0001) surface, was used for modeling, thus enabling direct comparison between oxides of the same structure. DFT + U was applied for the calculations and all details have been reported in a recent paper [START_REF] Huang | Atomistic insights on enhanced passivity: DFT study of substitutional Mo on Cr2O3 and Fe2O3 surfaces[END_REF] .

Using this approach, it was shown that, on neutral and vacancyfree surfaces, substitution by Mo is always exothermic and the most stable location is in the third metal plane of the (0001)oriented surface, i.e., in between the first (topmost) and second O planes (Fig. 7). Energetically, this substitution is more favored on Fe 2 O 3 than on Cr 2 O 3 surfaces. Thus, if both Fe-rich and Cr-rich oxide zones coexist in the inner barrier layer the passive film as discussed above, DFT simulation suggests that Mo may be locally more enriched in the Fe-rich oxide zones than in the Cr-rich oxide zones, which would contribute to cure the Fe-rich weak sites present in the passive film.

The calculated distributions of Bader and formal charges showed that, for neutral (uncharged) systems, 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 (Fig. 7a). However, for charged systems, the 1st added positive charge on the Mo-substituted surface is mainly localized on the substitutional Mo which can be considered as in +4 oxidation state. While the 2nd added charge is delocalized, the 3rd added charge is again localized on the substitutional Mo that can reach an intermediate state between +4 and +6 (Fig. 7b). Noteworthy, the localization of the 3rd added charge on the substitutional Mo was only observed on Fe 2 O 3 . Then, the intermediate Mo(IV + δ) species observed experimentally by XPS may be considered as an indicator of the presence of Mo-substituted Fe-rich local zones in the inner barrier layer of the passive film (Fig. 6).

Compared to perfect, unsubstituted surfaces, the energy of formation of metal vacancies (Cr or Fe) was always calculated endothermic. However, the presence of substitutional Mo favors the formation of metal atom vacancies, whereas it disfavors the formation of O vacancies or Cr(Fe)-O coupled vacancies. The Mo effect on vacancy formation is also more pronounced on Fe 2 O 3 than on Cr 2 O 3 surfaces. When a metal atom vacancy is created, the released charge is localized on the substitutional Mo. More charge is localized on Mo in Fe 2 O 3 than in Cr 2 O 3 defective surfaces, with the substitutional Mo reaching an oxidation state between +4 and +6 on Fe 2 O 3 and only +4 on Cr 2 O 3 . Then, the intermediate Mo(IV + δ) species observed experimentally by XPS could also be a marker of substitutional Mo present in defective Fe-rich local zones with cation vacancies.

The calculations for these Mo-substituted defective systems have implications on how molybdenum could mitigate the breakdown of the inner barrier layer of the passive film. Since substitution by Mo decreases the formation energy of Fe vacancies more significantly than that of Cr vacancies, one can expect the selective dissolution of Fe to be accelerated, and thus iron oxide to be replaced by more stable Mo-substituted chromium oxide in the oxide film. This replacement mechanism would lead to Cr and Mo enrichment, so here the effect of molybdenum is an accelerated selective dissolution of iron, assisted by molybdenum, that would contribute to substitute Fe in the weak sites of the inner barrier layer of the passive film by Mo and Cr, thereby improving the resistance to passivity breakdown.

One of the possible routes leading to passivity breakdown is via the Cl -ions penetrating the passive oxide films and destabilizing the inner barrier layer [START_REF] Frankel | Pitting corrosion of metals a review of the critical factors[END_REF][START_REF] Macdonald | The point defect model for the passive state[END_REF][START_REF] Marcus | Localized corrosion (pitting): a model of passivity breakdown including the role of the oxide layer nanostructure[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][START_REF] Strehblow | Corrosion Mechanisms in Theory and Practice[END_REF][START_REF] Soltis | Passivity breakdown, pit initiation and propagation of pits in metallic materials-review[END_REF] . Most of the Cl -ions interacting with the passive film would not access the inner barrier layer thanks to the Mo(VI) oxide species of the outer layer impeding their deep penetration. Still, deep penetration could occur locally leading to interaction of the Cl -ions with the inner barrier layer and their possible penetration in the oxide matrix via O vacancies. However, DFT simulation suggests that substitution by Mo disfavors the formation of O vacancies in both chromium and iron oxide matrices, and more effectively in Fe 2 O 3 . This implies that the effect of substitutional Mo in the Fe-rich defect sites would be to increase their resistance against chloride entry in the oxide matrix and thereby Cl-induced passivity breakdown. This effect of molybdenum would complement the acceleration of the selective dissolution of iron induced by substitutional Mo and its replacement by Mo and Cr oxide species, healing the Fe-rich weak sites.

In summary, this comprehensive review of experimental data and simulation obtained recently on passivated stainless steel and Cr-bearing MPEA surfaces by advanced surface analysis and DFT calculations provides both nanometer-scale and atomic-scale insights into the understanding of the role of molybdenum in enhancing the stability of passive films. It shows how Mo, enriched in the nanometer-thick passive film, can combine several effects to enhance the resistance to Cl --induced passivity breakdown. Enriched as Mo(VI) species in the outer exchange layer of the passive film, Mo acts as a barrier impeding the deep penetration of Cl -ions and limiting their access to the inner barrier layer. Dispersed as Mo(IV) at the surface of the inner layer, it protects against the entry of Cl -ions into the defect sites of the Cr(III) oxide barrier. In the Fe-rich compositional defects self-generated by the failure of Cr supply upon initial formation of the barrier layer, molybdenum enhances the selective dissolution of iron and its replacement by Cr(III) and Mo(IV + δ) species. By impeding the formation of O vacancies, it also increases the resistance against chloride entry in the oxide matrix, thereby curing the Fe-rich weak sites that would otherwise be preferential sites for Cl --induced passivity breakdown.

These combined effects of molybdenum enriched in the chromium oxide-based passive film are essential for preventing passivity breakdown in Cl -solutions.

Fig. 2

 2 Fig. 2 Equivalent thickness (nm) and relative concentration (at. %) of the surface oxide films and modified alloy regions of 316 L SS surfaces as calculated from XPS data using a bilayer model of photoelectron intensity attenuation. a native oxide-covered surface; b surface pre-oxidized at ultra-low pressure (ULP) of oxygen sequentially treated at open circuit potential (OCP) and under anodic polarization (AP) in 0.05 M H 2 SO 4 . Reproduced from ref. 23 . Copyright owned by authors.

  Fe 10 Co 5 Ni 60 Mo 10 MPEA and only trace penetration was observed for the Cr 25 Fe 25 Co 5 Ni 40 Mo 5 MPEA in the most severe conditions, i.e., without pre-passivation of the surface in Cl-free solution. On both alloys, penetration did not reach the metal substrate. This absence of Cl -penetration until the interface with the substrate in the most severe tested conditions, whereas traces access the barrier inner layer, reflects the curing effect of Mo on the Fe-rich weak sites of the inner barrier layer. Comparison of the two alloys shows that increasing the Mo content in the alloy and its enrichment in the outer layer of the passive film (about 30 and 17 Mo at. % for Cr 15 Fe 10 Co 5 Ni 60 Mo 10 and Cr 25 Fe 25 Co 5 Ni 40 Mo 5 , respectively, as quantified by XPS analysis) decreases the amount of penetrated chorides (Fig.

Fig. 4

 4 Fig. 4 In-depth distribution of chloride ions on passivated Cr 15 Fe 10 Co 5 Ni 60 Mo 10 (MPEA-15Cr10Mo) and Cr 25 Fe 25 Co 5 Ni 40 Mo 5 (MPEA-25Cr5Mo) surfaces as measured by ToF-SIMS. Samples were passivated in 0.05 M H 2 SO 4 + 4.7 M NaCl at 0.64 V/SHE for 1 h with (broken line) or without (plain line) pre-passivation in the NaClfree electrolyte. The margin of error for the maximum and interface Cl -signal intensities is smaller than 5%. Adapted from ref. 70 . Copyright owned by authors.

Fig. 3

 3 Fig. 3 Accelerated testing of resistance to passivity breakdown and initiation of localized corrosion. a 316 L SS samples prepared by traditional polishing (TP) and vibratory polishing (VP); b Cr 15 Fe 10 Co 5 Ni 60 Mo 10 (MPEA-15Cr10Mo), Cr 25 Fe 25 Co 5 Ni 40 Mo 5 (MPEA-25Cr5Mo) and Cr 35 Fe 20 Co 5 Ni 40 (MPEA-35Cr) samples prepared by traditional polishing. Passivation current transients were recorded in 0.05 M H 2 SO 4 (aq) after stabilization at OCP. Samples were polarized at (a) +0.3 V/SHE and (b) +0.64 V/SHE in the Cl-free base electrolyte before stepwise increasing the NaCl concentration as marked. Adapted from references (a) 67 and (b) 69 . Copyright owned by authors.

Fig. 5

 5 Fig. 5 ToF-SIMS elemental in-depth distribution for passivated Cr 25 Fe 25 Co 5 Ni 40 Mo 5 surfaces. Passivation was carried out in NaCl-free 0.05 M H 2 SO 4 + solution at 0.64 V/SHE for 1 h after stabilization at OCP. a Main oxide and hydroxide species constituting the passive films. Adapted from ref. 69 . Copyright owned by authors. b Minor mixed XMoO n (X = Cr, Fe, Ni) oxide species concentrated in the inner layer near the interface with the MPEA substrate.

Fig. 6

 6 Fig. 6 Model of the duplex passive oxide film formed on CrFeCoNiMo MPEA surfaces. Fe-rich and defect sites of the inner barrier layer are protected against corrosive attack by the local enrichment in Mo(IV + δ) and Mo(IV) oxide species. Adapted from ref. 69 . Copyright owned by authors.

Fig. 7

 7 Fig. 7 DFT optimized (0001)-oriented Cr 2 O 3 and Fe 2 O 3 surfaces with Cr(Fe) substituted by Mo in sub-surface and calculated Bader and formal charges. a Neutral surfaces relative to the Mo-free surfaces. b Charged surfaces relative to neutral surfaces. Each point represents an atomic layer whose nature is indicated by the background image of the perfect surface according to the [0001] coordinate. O atoms are in red and metal atoms in gray. Adapted from ref. 76 with permission from Elsevier.

npj Materials Degradation (2024) 3Published in partnership with CSCP and USTB 1234567890():,;

Published in partnership with CSCP and USTB npj Materials Degradation (2024) 3

npj Materials Degradation (2024) 3Published in partnership with CSCP and USTB

ACKNOWLEDGEMENTS

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: Corrosion Initiation Mechanisms at the Nanoscale and Atomic Scale). Dimitri Mercier, Dominique Costa, Frédéric Wiame, Sandrine Zanna, Antoine Seyeux, Lorena H. Klein, Boubakar Diawara, Li Ma, Zuocheng Wang, Ben Lynch, Xueying Wang, Xian Huang, Eirini-Maria Paschalidou, Shova Neupane are acknowledged with thanks for their contributions to the production of the original data previously published and further discussed herein.

DATA AVAILABILITY

The relevant data previously unpublished and discussed herein are available from the corresponding authors.

AUTHOR CONTRIBUTIONS

This review was conceived by V.M. and P.M. V.M. wrote the first draft, reviewed by P.M. V.M. contributed to funding acquisition and P.M. both funding acquisition and project management. Both authors approved the final version.

COMPETING INTERESTS

The authors declare no competing interests.

ADDITIONAL INFORMATION

Correspondence and requests for materials should be addressed to Vincent Maurice or Philippe Marcus.

Reprints and permission information is available at http://www.nature.com/ reprints Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Junior Momo Ziazet

Brigitte Jaumard

Adel Larabi

Nicolas Huin

Placement of Logical Functionalities in 5G/B5G Networks

Keywords: 5G provisioning, logical functionalities, UPF placement, DU and CU placement, latency requirements

5G technology has brought tremendous growth in connectivity, mobile traffic capacity, and enhanced performance with greater throughput, lower latency, ultra-high reliability, higher connectivity, and an expanded range of mobility. We present here a unified E2E logical functionality placement with the joint placement of distributed units (DU), centralized Units (CU) and user plane functions (UPF).

The problem is modeled as a large-scale integer linear program, solved using decomposition techniques. It includes all key network and IT resources and optimizes DU, CU and UPF placements and their numbers.

Numerical results are presented on an open Montreal traffic dataset. Sensitivity analysis is performed, investigating the impact of the number of locations hosting DU, CU and UPF functionalities on the delays of the request flows.

I. INTRODUCTION

As 5G continues to evolve and we begin planning for 6G, difficult questions remain related to logical functionality, considering cost, energy and latency requirements. On the one hand, it is good to host the logical functionalities as close as possible to the radio access network, in order to satisfy the latency requirements, and in particular the latency requirements of the Ultra-Reliable Low Latency Communications (uRLCC) 5G traffic class. Conversely, sharing computing resources is easier when moving away from the radio access network. The question then arises as to how much computing resources we should schedule close to the RAN, and how much we can move further away to be able to meet user needs at any time, taking into account traffic fluctuations.

Moreover, considering the increase in quality of service (QoS) of 5G applications and the increase in overall 5G traffic, we have to deal with the huge power consumption of data centers involved in the different clouds hosting the growing software part of 5G networks. Again energy consumption needs to be kept in mind when taking care of the logical functionalities as to favor the sharing of their compute resources.

We therefore study here the placement of logical functionalities, DU, CU and UPF from an E2E network point of view. We propose a comprehensive mathematical model which integrates the key constraints and solve it using large-scale optimization techniques. We next use that model to explore the compromise between delays and number of locations for hosting the logical functionalities.

Several studies have been devoted to the placement of Virtual Network Functions and Service Function Chains, e.g., [START_REF] Huin | Optimal network service chain provisioning[END_REF], [START_REF] Hyodo | Virtual network function placement for service chaining by relaxing visit order and non-loop constraints[END_REF], and much less on the placement of 5G logical functionalities. On the one hand, there are studies on the placement of DU and CU ( [START_REF] Liu | Provisioning optimization for determining and embedding 5G end-to-end information centric network slice[END_REF], [START_REF] Klinkowski | Latency-aware DU/CU placement in convergent packet-based 5G fronthaul transport networks[END_REF], [START_REF] Yu | DU/CU placement for C-RAN over optical metro-aggregation networks[END_REF]), then on UPF [START_REF] Leyva-Pupo | Optimal placement of user plane functions in 5G networks[END_REF] on the other hand, but none on all the logical functionalities of 5G.

II. MATHEMATICAL MODEL: HOW TO OPTIMIZE OPTICAL NODE PLACEMENT

A. Notation and Problem Statement

We consider a 5G Software-Defined Network (SDN) that is represented by a graph G = (V, L) where V represents the set of nodes and L the set of physical links. We assume that only a subset of nodes V △ ⊆ V, △ ∈ {DU, CU, UPF} have computational resources available to host 5G logical functionalities, respectively for the DU, CU and UPF functionalities [START_REF] Gpp | System architecture for the 5G system (5GS)[END_REF]. Following ITU-R, 5G traffic covers three traffic categories: Enhanced Mobile Broadband (eMBB), Ultra-Reliable Low Latency Communications (URLLC), and Massive Machine Type Communications (mMTC). On the user plane, each traffic flow request must follow a logical functionality chain made of an ordered set of logical functionalities as described in Figure 1, where we only present the dominant part of the traffic. Demand is then defined by a set of flow requests, Each E2E logical functionality chain (LFC) c is defined as an ordered sequence of 5G logical functionalities. We denote by f i the ith logical functionality in any given logical functionality chain, where i ∈ I CoS k = {1, . . . , |c CoS k |}. Each logical functionality chain is mapped to a request depending on its service category. Let C be the set of all logical functionality chains.

Let F be the set of 5G logical functionalities, indexed by f . Each functionality f requires some compute resources, i.e., CPU, RAM and storage for its processing. Let CPU f , RAM f , STO f be the amount of CPU, RAM and storage required by f for each unit of bandwidth (it is not necessarily a linear function). Therefore, assuming f is one occurrence of the LFC functions in request k, VM □ f k represents the fraction of □ ∈ {RAM, CPU, STO} of virtual machine (VM) (we assume that all VMs have the same configuration) that it needs for each of its occurrences. Each logical functionality has a limited number of replicas, denoted by REPLICA f . Note that each VM can serve several demands or functionalities, with each demand/functionality only using a fraction of the compute resources of each VM. Note that the complexity of a functionality may not always scale linearly with the unit of bandwidth of the functionality. The Placement of 5G/B5G Logical Functionalities problem is critical for efficiently provisioning 5G requests and making sure they can go through their required logical functionality chain. We formally state it as follows. For a given set of request flows, where each flow k is characterized by a 5-tuple,

(s k , d k , c CoS k , b k , δ CoS k )
, identify the best function locations in order to provision the set of LFCs and requests, while minimizing the overall network load subject to the transport and compute capacities (i.e., CAP ℓ for the links and by CAP □ for □ ∈ {CPU, RAM, STO} for the nodes), and optionally to a limit on the number of logical functionality replicas.

When provisioning a request k, its required logical functionalities are encountered in the same order as in c CoS k , with some functions possibly located at the same node.

B. Layered Graph

In the context of Virtual/Container Network Function (VNF/CNF) placement, many authors have used the concept of layered graphs, see [START_REF] Dwaraki | Adaptive service-chain routing for virtual network functions in software-defined networks[END_REF] for one of the early references, and then, e.g., [START_REF] Huin | Optimal network service chain provisioning[END_REF], [START_REF] Hyodo | Virtual network function placement for service chaining by relaxing visit order and non-loop constraints[END_REF]. Here, we propose a revisited layered graph adapted to the logical functionalities of 5G, and in particular, to certain location restrictions of logical nodes, i.e., DU and CU in the access network, while UPF is either in transport or in the core network, depending on latency requirements.

A layered graph G L is defined for each request flow k and its associated E2E logical functionality chain as follows. For a complete E2E chain RU → DU → CU → UPF → CU → DU → RU, the layered graph has 5 layers, and them 3 layers if the request is an uplink/downlink one, see Figure 3 for an illustration of the layered graph for an E2E request.

The initial network graph G is transformed into a layered graph G L (not counting the RU SRC and RU DST nodes), where each layer is either associated with the transport network graph (or a subgraph of it) or the combination of the transport and core networks.

For every node v ∈ V , let v i be the corresponding node in the ith layer i ∈

I CoS k . Every (i -1, i) layer pair is connected by cross-layer links from v i-1 to v i , if the function f c
i-1 can be installed and run on node v, see Figure 4 for an illustration. Therein, request k first goes through its RU SRC following the base station to which it connects, and then to the access network in order to go through a DU and then a CU. Next, it needs to identify a UPF, either in the transport or in the core network, and then again a CU and then a DU, before reaching a RU and then the destination. Note that we do not need to consider the overall transport network, e.g., in Figure 2, we only need to consider TN SRC k for the DU-UL/CU-UL layers, and similarly only TN DST k for the CU-DL/DU-DL layers. Finding a path and a chain placement for a request (s k , d k , c CoS k , b k , δ CoS k ) consists in finding a path on the layered graph G L from node RU SRC to node RU DST . Note that each layer represents the progression of the chain, e.g., being on the second layer means that the placement of the first function of the chain is already decided. The placement of the logical functionality associated with layer i is given by the cross-layer link used to switch between layers i and i + 1.

C. Mathematical Model

We propose a model, called LFP CG model, which relies on the concept of configurations. Therein, a configuration is defined by a potential path provisioning, called service path in the sequel, which is next described formally below for a given request.

A Service Path p for request flow k is a path, i.e., an ordered set of nodes from the source to the destination node of the request, going through node hosting the logical functionalities The objective is to minimize the network workload. For a given request provisioning and its associated routing (path), the network workload can be calculated by multiplying its hop count by the request's bandwidth requirement.

The set of constraints can then be expressed as follows.

Constraints

One path per demand

p∈P k y k p = 1 k ∈ K (2)
Link capacity

k∈K p∈P k b k NUM p ℓ y k p ≤ CAP ℓ ℓ ∈ L (3)
Compute node resource capacity

k∈K i∈ICoS k p∈P k VM □ fik b k a p iv y k p ≤ CAP □ v □ ∈ {CPU, RAM, STO}, v ∈ V △ (4)
Limited # of logical functionality occurrences

v∈V △ z vf ≤ REPLICA f f ∈ F (5)
Checking the existence of a service path, i.e., all its functions are hosted on a node along it

p∈P k a p iv y k p ≤ z vfi v ∈ V △ , i ∈ I CoS k , k ∈ K (6)
Logical functionality placement

z vf ≤ PLACE vf f ∈ F, v ∈ V △ . ( 7
)
Limited number of UPF nodes

v∈V △ z UPF v ≤ UB f ∈ F UPF ( 8
)
z vf ≤ z UPF v f ∈ F UPF , v ∈ V △ . ( 9
)
For a given RU, outgoing and incoming flows must go through the same DU and CU

k∈K UL RU p∈P k a p △,UL,v y k p + k∈K DL RU p∈P k a p △,DL,v y k p ≤ |K RU |x △ RU,v v ∈ V △ , RU ∈ V RU , △ ∈ {DU, CU} ( 10
) v∈V △ x △ RU,v = 1 △ ∈ {DU, CU}, RU ∈ V RU . ( 11
)

D. Solution of the Mathematical Model

The mathematical model of the previous section has an exponential number of variables and therefore requires a decomposition solution scheme using column generation techniques [START_REF] Lübbecke | Selected topics in column generation[END_REF] in order to scale. The latter scheme reuses the mathematical model of the previous section as a so-called Master Problem (MP) and the Restricted Master Problem (RMP), i.e., MP with a very small subset of configurations/columns, and the so-called Pricing Problem (PP), i.e., a configuration generator. Consequently, the Restricted Master Problem corresponds to (1) -( 11) with a very limited number of variables. Its role is to select the best provisioning, one for each request, while the pricing problem generates improving configurations, i.e., configurations such that, if added to the current RMP, improve the value of its linear relaxation. Once the linear relaxation of MP is solved, an integer solution is sought. Reader who is not familiar with column generation and how to seek an integer solution is referred to, e.g., [START_REF]Linear Programming[END_REF] and [START_REF] Barnhart | Branch-and-price: Column generation for solving huge integer programs[END_REF]. Figure 5 provides an illustration of the flowchart of the solution when combining column generation and integer linear programming techniques.

We now describe the pricing problem whose role is to generate a valid Service Path for a given request. Once again, the formulation relies on the layer graph (G L ) introduced in Section II-B. Its objective is defined by the so-called reduced cost (see [START_REF]Linear Programming[END_REF] if not familiar with linear programming concepts).

• u (j) represents the vector of dual variables of constraints (j) in the RMP.

Variables:

• α iv ∈ {0, 1}, where

α iv = 1 if f cCoS k i
is installed on node v, 0 otherwise.

• φ i ℓ ∈ {0, 1}
, where φ i ℓ = 1 if the flow is forwarded on link ℓ on layer i, i.e., links in each layer in graph G L , 0 otherwise.

• β vf ∈ {0, 1}, where β vf = 1 if function f is installed in node v. Note that chains can contain multiple occurrences of the same function.

• α vf cCoS k i ∈ {0, 1}, where α vf cCoS k i = 1 if function f cCoS k i
is installed in node v, allowing a function to intervene several times (e.g., firewall application) in a service chain, and to be run in different locations, i.e., potentially a different one for each occurrence. The service path generator (so-called pricing problem in mathematical programming and decomposition models) is written for each request k. We describe it below for the case of an uplink request, and it can be written in a similar way for the two other cases, i.e., for a downlink or a bidirectional request. Its objective function, so-called reduced cost, can be expressed as follows.

min ℓ∈L |cCoS k | i=1 φ i ℓ × b k -b k u (3) ℓ -u (2) k + □∈{CPU,RAM,STO} b k v∈V △ u (4) v,□ |cCoS k | i=1 VM □ f cCoS k i α vf cCoS k i + v∈V △ |cCoS k | i=1 α vf cCoS k i u (6) k,v,i + COST UL + COST DL , ( 12
)
where (10) RU U L k ,△,v for uplink and E2E requests and 0 for downlink requests.

• COST UL = △∈{DU,CU} v∈V △ α △,UL,v u
• COST DL = △∈{DU,CU} v∈V △ α △,DL,v u (10) RU DL k ,△,v for downlink and E2E requests and 0 for uplink requests. Constraints are written as follows. Flow conservation: they correspond to flow constraints (i.e., route) from one logical functionality to the next one in the 5G E2E logical functionality chain for the SRC k ⇝ DST k request for which the pricing problem is solved (constraints ( 13)), and then flow constraints from the source node to the location of the DU function of the service chain (constraints ( 14)), and similarly from the location of the last function, i.e., UPF for an uplink request, of the service chain to the destination node (constraints (15)). Note that a i v = 0 for all nodes that do not host any logical node/functionality and that we take care of the possibility that several logical functionalities can be located on the same node, including on the source or destination nodes. Let k ∈ K UL . Denote by TN k the transport network associated with k as in Figure 2.

Selecting the CU placement

ℓ∈ω + (v) φ TN CU k ℓ - ℓ∈ω -(v) φ TN CU k ℓ + α CU,UL,v -α DU,UL,v = 0 v ∈ V TN CU k (13)
Selecting the DU placement

ℓ∈ω + (v) φ TN DU k ℓ - ℓ∈ω -(v) φ TN DU k ℓ + α DU,UL,v = 1 if v = SRC k 0 else v ∈ TN DU k (14)
Selecting the UPF placement

ℓ∈ω + (v) φ TN UPF k ∪ CN ℓ - ℓ∈ω -(v) φ TN UPF k ∪ CN ℓ -α UPF,v = -1 if v = DST k 0 else v ∈ TN UPF k ∪ CN. ( 15
)
Link capacity.

b k |cCoS k | i=0 φ i ℓ ≤ CAP ℓ . ℓ ∈ L. (16)
Compute node capacity. For v ∈ V LN ,

|cCoS k |-1 i=0 (VM □ f cCoS k i b k ) × α iv ≤ CAP VM □ v VM □ ∈ {CPU, RAM, STO}. ( 17
)
Delay constraint for request k: it includes a link delay (propagation and average queuing delays: it corresponds to the contribution of k to the overall queuing delay on link ℓ, with the assumption that queuing delay depends linearly on the bandwidth , and a node delay (function processing delay).

ℓ∈L

|cCoS k | i=0 δ ℓ,k φ i ℓ + v∈V LN |cCoS k |-1 i=0 δ v,f cCoS k i α iv ≤ δ k (18)

III. NUMERICAL RESULTS

We now report on the numerical results. First, we describe the data set we used and present the placement produced by the proposed LFP CG model as well as the function's logical connectivity. Next, we examine the trade-off between the number of logical functionality locations and the request's path delays. Finally, we study the impact of reducing the number of logical functionality locations on the overall capacity utilization. The proposed LFP CG model was implemented on an AMD Ryzen Threadripper 2990WX-32-Core processor @ 2.95 GHz with 128 GB of RAM in a Python environment with the GUROBI library.

To assess the performance of the proposed LFP CG model, we considered four scenarios with different numbers of CU, DU, and UPF in the network. The details of each scenario are represented in Table III. In scenarios 1 and 2, the numbers of DU and CU are the same, respectively, 18 and 9, as well as in scenarios 3 and 4, respectively, 12 and 6. In scenarios 1 and 3, UPF AR , UPF MIOT , and UPF I4.0 can be placed in the core network, and two replicas are allowed per UPF type, while in scenarios 2 and 4, they cannot be placed in the core, and only one replica is allowed. As UPFs of different services can be collocated, the parameter UB indicates the overall number of admitted UPFs locations.

A. Dataset

The data set used in this work is an extended version of the data set proposed by Ziazet et al. [START_REF] Ziazet | A dynamic traffic generator for elastic 5G network slicing[END_REF]. It contains E2E traffic generated by refactoring the urban data of the city of Montreal. The 5G cell locations are inspired by the real cell locations of a mobile operator in the city of Montreal. 5G transport and core nodes are mapped by aggregating cells in the same neighbourhood, and network connectivity is done as in Figure 6. The algorithm will select nodes where we can place processing points for logical functionalities. We considered six types of services, as illustrated in Table I with different bandwidths and delay requirements. Table II provides the compute resource modeling that we used with the required amount of CPU (in terms of percentage of CPU per user), RAM, and storage for each logical node. These values do not come from real use cases (due to lack of access to real data) and only provide a certain order of magnitude. More details about the data set can be found in [START_REF] Ziazet | A dynamic traffic generator for elastic 5G network slicing[END_REF].

B. Placement analysis

We provide placement results in Figure 7 along with logical functionality connectivity for Scenario 1, for UPF placement Fig. 6: 5G network where the placement will be done: radio, transport and core infrastructures and their hierarchy in the core network. As expected, UPFs of services that we associated with URLLC, i.e., MIoT, AR, and I4.0 in our use case, are placed in the MEC by the proposed approach. This is in line with what is done in real life, as these services are latency-sensitive and their UPFs have to be located near the end user. Some processing points co-locate DU, CU, and UPF and are located in areas with dense populations (downtown Montreal). This is surely to accommodate the peak of traffic that can be encountered in such areas.

From Figure 6d, while in general RU is connected to the closest DU, we can see that the model does not necessarily connect the RU with its closest DU. This might be because of the capacity constraints of that processing point. Another reason might be that the traffic pattern in that area is such that the selected connectivity is the one aligned with the goal of minimizing network workload while maintaining acceptable delay. It is counter-intuitive, as we might think RU should be connected to the closest DU. However, the benefit of our approach is that it will perform the connectivity that benefits the goal and that sometimes is not trivial to find by heuristics or other rule-based approaches. The same interpretation can be made for the connectivity of DU and CU. The proposed LFP CG model concentrates the backhaul network in the downtown part of the city, where we have more traffic, and it positions the UPFs in the area. Back-haul logical links show that a CU might be logically connected to multiple UPFs. This is because, based on the type of requested service and the network status, physical paths connecting a CU and any UPFs might be created on the core network.

C. Delay vs. number functions

To examine the trade-off between the number of logical functionality locations and the path delay, we considered the four scenarios previously described and compared their spare delays. Figure 8 presents the resulted spare delay for each scenario for the augmented reality service. We can see that for all the cases, the LFP CG model can find solutions so that the overall sparse delay is more than 50% of the requirement. This is beneficial as it guarantees delay QoS is met, and this spare delay converted into energy will represent an important reduction in energy consumption.

As we reduce the number of replicas of logical functionalities (i.e., DU from 18 to 12, CU from 9 to 6, and UPF from 2 to 1), we observe an increase in the path delay. This is because, with fewer available logical functionalities, requests would need longer paths to access the available ones and reach the destination. This increase is not too important based on the results indicating the capacity of the model to find an optimized placement given the conditions.

Similar results are obtained for the five other services, i.e., cloud gaming, video streaming, VoIP, MIoT, and Industry 4.0.

D. Capacity utilization

Figure 9 shows the link capacity utilized in all four scenarios. Overall, the obtained results are in line with those of the delay comparison in the previous section. Indeed, for the same reasons as the delay, reducing the number of replicas of logical functionalities will result in an increase in link capacity utilization. Comparing scenarios 1 and 3 which differ from the number of DU and CU replicas and scenarios 1 and 2 which differ from the number of UPFs, we see that in our use case, reducing the number of UPFs has less negative impact on the link capacity utilization than reducing the number of DU and CU. While being influenced by the characteristics of the traffic generated, the placement of DUs and CUs is also very dependent on the dimensioning of the transport network and the availability of computing resources.

IV. CONCLUSIONS

This paper formulates the DU, CU, and UPF placement problem as a large-scale integer linear program and solves it using decomposition techniques. The proposed solution does not only find the placement of logical functionalities but also the paths to serve the requests, which are later used to generate the logical connectivity of the 5G functions. Experiments with Montreal traffic data showed that the placement is done so as to reduce link capacity utilization, and the requests are provisioned with relatively small delays. The next step would be to quantify the gain, i.e., spare capacity and spare delay, in terms of energy for a more energy-efficient network.

  Streaming (downlink from video server) Example: Massive IoT (data collection) Example: VoIP

Fig. 1 :

 1 Fig. 1: 5G E2E logical functionality chain (user plane)

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig. 2: Overview of the various networks

Fig. 5 :

 5 Fig. 5: CG ILP flowchart

ACKNOWLEDGMENT

  First author was supported by a NSERC/INNOVEE internship in collaboration with Ericsson, GAIA Montreal. A special thanks to Jennie Diem Vo for clarifying some concepts of the current 5G technology.

Fig. 7 :Fig. 8 :Fig. 9 :

 789 Fig. 7: Placement of logical functionalities in radio (a), transport (b) and core (c) network. Obtained logical connectivity associated to each logical functionality (d)

TABLE I

 I

	: Logical functionality chains (bandwidth and latency
	values adapted from [8])
	Services	Class	Logical node sequences	Bandwidth per user or IoT system	E2E latency	Request bundles
	Cloud gaming (CG)	eMBB	UPFCG -CU -DU -RU	4 Mbps	80 ms	[40-55]
	Augmented reality (AR)	eMBB	UPFAR -CU -DU -RU	100 Mbps	10 ms	[1-4]
	VoIP	eMBB RU -DU -CU -UPFVOIP-CU -DU -RU	64 Kbps	100 ms [100-200]
	Video streaming (VS)	eMBB	UPFVS -CU -DU -RU	4 Mbps	100 ms [50-100]
	Massive IoT (MIoT)	mMTC	RU -DU -CU -UPFMIOT	[1-50] Mbps	5 ms	[10-15]
	Industry 4.0 (I4.0) uRLLC RU -DU -CU -UPFI4.0 -CU -DU -RU	70 Mbps	8 ms	[1-4]

TABLE II :

 II CPU/RAM/Storage core usage for logical nodes

	Logical vCPU RAM (Gb) Storage (Gb) Logical node processing time
	nodes		per 100 Mbps		per 0.01 msec unit
	RU	1	4	7	12
	DU	9	5	1	6
	CU	11	15	2	22
	UPF CG	13	15	7	14
	UPF AR	5	2	5	16
	UPF VOIP	5	2	5	2
	UPF CG	5	11	10	4
	UPF MIOT	5	3	20	4
	UPF I4.0	5	4	11	4

TABLE III :

 III Input configurations per scenario: blue, green and yellow colors indicate that UPF can be located only in MEC, only in core and in both MEC and Core, respectively

				REPLICAf
	Scenario	RU	DU CU UPFCG	UPFAR	UPFVOIP UPFVS	UPFMIoT UPFI4.0	UB
	1	100	18