G Ratié

K Zhang

M Iqbal

D Vantelon

F Mahé

C Rivard

M Komárek

M Bouhnik-Le Coz

A Dia

K Hanna

M Davranche

R Marsac
email: remi.marsac@cnrs.fr

Journal Pre-proof

Keywords: Cerium, goethite, adsorption isotherms, redox, XANES Ce L III -edge, speciation modeling

Iron (Fe) oxyhydroxides are major phases that may control the cerium (Ce) behavior in the Earth's Critical Zone. However, understanding Ce behavior with Fe oxyhydroxides remains uncompleted.

Especially, if thermodynamic calculations suggest that Fe(III) is not a sufficiently strong oxidant, several studies reported the presence of Ce(IV) onto Fe oxyhydroxides. In this study, multiple approaches, including modeling and X-ray absorption spectroscopy, deciphered the driving forces of Ce(III) oxidation to Ce(IV) onto goethite. Results showed that oxidized Ce occurred onto goethite with a Ce(III)/Ce(IV) ratio depending on the Ce concentration in the solution ([Ce] tot ). The percentage of Ce(IV) onto goethite ranged from 20 % to 50 %, linearly increasing with [Ce] tot . Comparable observation with a redox-inert Alhydroxide (gibbsite), allowed to rule out the importance of Fe(III) redox reactivity as the main driver of Ce(III) oxidation. Instead, thermodynamic calculations suggested that surface precipitation of Ce(IV)hydroxides, whose formation is favored with increasing [Ce] tot , was an important driving force of the redox reaction. Because the goethite surface seemed to stabilize more strongly Ce(IV) than Ce(III) surface species than does gibbsite, differences in binding mechanisms of Ce(III) and Ce(IV) onto different mineral surfaces have been suggested to play a role on Ce redox speciation.

Introduction

Rare Earth Elements (REE) are considered critical and strategic substances owing to their everincreasing demand for many modern-day technologies [START_REF] Massari | Rare earth elements as critical raw materials: Focus on international markets and future strategies[END_REF]Ramos et al., 2016a).

Their subsequent anthropogenic emissions make REE to be considered as emerging pollutants in the environment (e.g., [START_REF] Brioschi | Transfer of rare earth elements (REE) from natural soil to plant systems: implications for the environmental availability of anthropogenic REE[END_REF][START_REF] Gwenzi | Sources, behaviour, and environmental and human health risks of high-technology rare earth elements as emerging contaminants[END_REF][START_REF] Kulaksız | Rare earth elements in the Rhine River, Germany: first case of anthropogenic lanthanum as a dissolved microcontaminant in the hydrosphere[END_REF]. Although their source and behavior in the Earth's mantle and crust are well characterized [START_REF] Frey | Rare earth element abundances in upper mantle rocks[END_REF][START_REF] Humphris | The mobility of the rare earth elements in the crust[END_REF][START_REF] Mclennan | Rare earth elements in sedimentary rocks: influence of provenance and sedimentary processes[END_REF], many uncertainties persist due to the lack of studies related to the REE environmental mobility in the Earth's Critical Zone [START_REF] Cao | Effects of redox potential and pH value on the release of rare earth elements from soil[END_REF][START_REF] Denys | Evaluation of selectivity of sequential extraction procedure applied to REE speciation in laterite[END_REF][START_REF] Guénet | Experimental evidence of REE size fraction redistribution during redox variation in wetland soil[END_REF][START_REF] Ramos | Rare earth elements in the soil environment[END_REF].

Among REE, cerium (Ce) is the most abundant in the Earth's crust. Under oxic conditions, Ce can be oxidized as tetravalent Ce by adsorption-catalyzed oxidation [START_REF] Bao | Geochemistry of mineralization with exchangeable REY in the weathering crusts of granitic rocks in South China[END_REF][START_REF] Janots | Ce (III) and Ce (IV)(re) distribution and fractionation in a laterite profile from Madagascar: insights from in situ XANES spectroscopy at the Ce LIII-edge[END_REF][START_REF] Takahashi | Direct observation of tetravalent cerium in ferromanganese nodules and crusts by X-ray-absorption near-edge structure (XANES)[END_REF][START_REF] Taylor | The significance of the rare earths in geochemistry and cosmochemistry[END_REF] controlled by oxidizing minerals, e.g., Mn or Fe oxyhydroxides [START_REF] Davranche | Adsorption of REE(III)humate complexes onto MnO2: Experimental evidence for cerium anomaly and lanthanide tetrad effect suppression[END_REF][START_REF] Ohta | REE (III) adsorption onto Mn dioxide (δ-MnO2) and Fe oxyhydroxide: Ce (III) oxidation by δ-MnO2[END_REF][START_REF] Takahashi | Direct observation of tetravalent cerium in ferromanganese nodules and crusts by X-ray-absorption near-edge structure (XANES)[END_REF], from precipitation in the liquid phase as Ce(OH) 4 or CeO 2 [START_REF] Braun | Cerium anomalies in lateritic profiles[END_REF][START_REF] De Baar | Rare earth element distributions in anoxic waters of the Cariaco Trench[END_REF][START_REF] German | Rare earth elements in Saanich Inlet, British Columbia, a seasonally anoxic basin[END_REF][START_REF] Marcus | Forms and distribution of Ce in a ferromanganese nodule[END_REF], and through specific biological processes [START_REF] Kraemer | Negative cerium anomalies in manganese (hydr) oxide precipitates due to cerium oxidation in the presence of dissolved siderophores[END_REF][START_REF] Moffett | Microbially mediated cerium oxidation in sea water[END_REF][START_REF] Tanaka | A specific Ce oxidation process during sorption of rare earth elements on biogenic Mn oxide produced by Acremonium sp. strain KR21-2[END_REF]. The much larger affinity of Ce(IV) for surface, in regards to the other REE(III), makes it highly reactive to particles and colloids that subsequently control its mobility and transport [START_REF] Liu | Rare earth elements sorption to iron oxyhydroxide: Model development and application to groundwater[END_REF][START_REF] Marsac | Effects of Fe competition on REE binding to humic acid: Origin of REE pattern variability in organic waters[END_REF][START_REF] Pédrot | Upper soil horizons control the rare earth element patterns in shallow groundwater[END_REF]Pourret et al., 2007b;[START_REF] Quinn | Sorption of yttrium and rare earth elements by amorphous ferric hydroxide: influence of temperature[END_REF]. However, prediction of the Ce redox behavior is a challenging task because it is affected by several parameters and biogeochemical factors such as pH [START_REF] Bau | Scavenging of dissolved yttrium and rare earths by precipitating iron oxyhydroxide: experimental evidence for Ce oxidation, Y-Ho fractionation, and lanthanide tetrad effect[END_REF], the presence of colloids [START_REF] Davranche | Adsorption of REE(III)humate complexes onto MnO2: Experimental evidence for cerium anomaly and lanthanide tetrad effect suppression[END_REF][START_REF] Pédrot | Upper soil horizons control the rare earth element patterns in shallow groundwater[END_REF]2008), biotic activity [START_REF] Ngwenya | Discrete Site Surface Complexation Constants for Lanthanide Adsorption to Bacteria As Determined by Experiments and Linear Free Energy Relationships[END_REF][START_REF] Tanaka | A specific Ce oxidation process during sorption of rare earth elements on biogenic Mn oxide produced by Acremonium sp. strain KR21-2[END_REF], and the organic matter [START_REF] Gangloff | Characterization and evolution of dissolved organic matter in acidic forest soil and its impact on the mobility of major and trace elements (case of the Strengbach watershed)[END_REF]Pourret et al., 2007a).

The overall view of the REE behavior provides normalized REE patterns, which indicate a signature of their origin and geochemical processes (Henderson, 2013). In response to the behavior of its J o u r n a l P r e -p r o o f trivalent neighbors (La and Pr), a positive or negative anomaly of Ce is observed and often highlights changes in O 2 availability due to the redox-sensitivity of Ce as compared with other REE. Because of this peculiar redox sensitivity, Ce anomalies have, therefore, often been considered as efficient paleo-redox tracers in many environments (Nakada et al., 2013a;[START_REF] Shields | Diagenetic constraints on the use of cerium anomalies as palaeoseawater redox proxies: an isotopic and REE study of Cambrian phosphorites[END_REF][START_REF] Tostevin | Effective use of cerium anomalies as a redox proxy in carbonate-dominated marine settings[END_REF][START_REF] Tostevin | Cerium Anomalies and Paleoredox[END_REF][START_REF] Zhong | Characterization of the zircon Ce anomaly for estimation of oxidation state of magmas: a revised Ce/Ce* method[END_REF]. The positive Ce anomaly associated to Ce(IV) adsorbed to MnO 2 is well-known and documented (e.g., [START_REF] Loges | Negative Ce anomalies in Mn oxides: the role of Ce4+ mobility during water-mineral interaction[END_REF]Nakada et al., 2013b;[START_REF] Takahashi | Chemical and structural control of the partitioning of Co, Ce, and Pb in marine ferromanganese oxides[END_REF].

Thermodynamic data show that the Mn 2+ /Mn(IV)O 2 standard potential (E 0 ) is higher than the E 0 of Ce 3+ /Ce(IV)O 2 [START_REF] Brookins | Aqueous geochemistry of rare earth elements[END_REF]; namely that Mn(IV) is reduced as Mn(III) or Mn(II) by oxidizing Ce(III) to Ce(IV) [START_REF] Ohta | REE (III) adsorption onto Mn dioxide (δ-MnO2) and Fe oxyhydroxide: Ce (III) oxidation by δ-MnO2[END_REF]. The oxidative pathway of Ce by MnO 2 corresponds to the Ce(III) adsorption followed by its oxidation as Ce(IV) and preferential adsorption relative to the REE(III) [START_REF] Ohta | REE (III) adsorption onto Mn dioxide (δ-MnO2) and Fe oxyhydroxide: Ce (III) oxidation by δ-MnO2[END_REF] or as Ce(OH) 4 nanoparticles [START_REF] Nakada | Aqueous speciation is likely to control the stable isotopic fractionation of cerium at varying pH[END_REF]. On the contrary, for Fe oxyhydroxides, the electron transfer should not occur because the E 0 of Fe 2+ /Fe(III)-(oxy)(hydr)oxides is significantly lower than that of Ce 3+ /Ce(IV)O 2 [START_REF] Stumm | The cycling of iron in natural environments: Considerations based on laboratory studies of heterogeneous redox processes[END_REF]. However, contrasted results were obtained, such as (i) low Ce oxidation at acidic pH onto hydrous Fe oxide [START_REF] Bau | Scavenging of dissolved yttrium and rare earths by precipitating iron oxyhydroxide: experimental evidence for Ce oxidation, Y-Ho fractionation, and lanthanide tetrad effect[END_REF], (ii) Ce(IV) adsorption and incorporation to hematite [START_REF] Bolanz | Hematite (α-Fe2O3)-A potential Ce4+ carrier in red mud[END_REF][START_REF] Takahashi | Determination of the oxidation state of cerium in rocks by Ce LIII-edge X-ray absorption near-edge structure spectroscopy[END_REF], (iii) total Ce oxidation onto goethite [START_REF] Yu | Cerium sequestration and accumulation in fractured crystalline bedrock: the role of Mn-Fe (hydr-) oxides and clay minerals[END_REF] or (iv) partial Ce oxidation onto goethite [START_REF] Denys | Evaluation of selectivity of sequential extraction procedure applied to REE speciation in laterite[END_REF][START_REF] Ratié | Cerium anomalies in riverbanks: Highlight into the role of ferric deposits[END_REF]. These results evidence the lack of knowledge about the behavior of Ce oxidation processes in contact with Fe oxyhydroxides. Several mechanisms might be invoked to explain these discrepancies, such as the crystalline structure of the Fe -oxyhydroxides and their synthetic or natural origin, or the redox condition during the interaction of Ce with Fe oxyhydroxides. In addition, thermodynamic aspects at mineral-water interfaces have also been shown for plutonium (Pu), a chemical analogue of Ce. Previous studies revealed that Pu(IV) was more stabilized at clays surfaces than Pu(III). This process highly depended on the Pu-clay binding mechanisms, i.e., surface complexation, cation exchange or surface precipitation [START_REF] Banik | Sorption and Redox Speciation of Plutonium at the Illite Surface[END_REF]Marsac et al., 2015a;2017a).

Up to now, no systematic study of Ce interaction with a single Fe oxyhydroxide has been conducted, which would allow deciphering the reason for the occurrence of Ce(IV) at the Fe J o u r n a l P r e -p r o o f oxyhydroxide surface. Therefore, to quantitively observe the adsorption or precipitation processes responsible for the Ce(IV) bound onto goethite, this study combined isotherm adsorption experiments, numerical modeling, and X-ray absorption spectroscopy focusing on the XANES (X-ray Absorption Near Edge Structure) Ce L III -edge spectrum. To identify the redox control potentially exerted on the Ce adsorption by goethite, a trivalent analog non-redox-sensitive REE, namely Nd, was also studied.

Moreover, to compare the role of goethite surface on the Ce redox behavior, we also performed Ce batch experiments to gibbsite, i.e., inert redox surface. Using this approach, we determined the mechanism behind the Ce(III)/Ce(IV) ratio occurring at the goethite surface, allowing us to explain the different results yet published.

Materials and Methods

All chemicals used were of analytical grade. All experimental solutions were prepared with double-deionized water (Milli-Q system, Millipore). Synthetic Ce and Nd solutions were prepared from nitrate Ce and Nd standard solutions (10 g L -1 , in 5% HNO 3 , AccuTrace Reference Standard).

Polyethylene containers were all soaked in 10% Ultrapure HNO 3 for 48 h at 60°C, then rinsed with deionized water for 24 h at 60°C to remove all contamination sources.

Batch experiments

Preparation of goethite and gibbsite

A goethite suspension was prepared by neutralizing a volume of 500 mL of 0.5 mol L -1 ferric Fe(NO 3 ) 3 ‧9H 2 O solution with a 400 mL 2.5 mol L -1 NaOH solution added at a rate of 1 mL min -1 [START_REF] Hiemstra | Multisite proton adsorption modeling at the solid/solution interface of (hydr)oxides: A new approach: II. Application to various important (hydr)oxides[END_REF]. The solution was stirred vigorously during the NaOH addition in a glove box (N 2 ).

The precipitate was aged for 72 h at 60 °C. The final suspension was dialyzed at 1 kDa (Spectra/Por membrane 2) with Milli-Q water. The water was changed twice a day until the suspension conductivity was close to 0 µS cm -1 . The dialyzed suspensions were stored in polypropylene containers at 4°C. An aliquot of the dialyzed suspension was dried at 60 °C for 24h. X-Ray Diffractogram was recorded on the J o u r n a l P r e -p r o o f dry powders to confirm the synthesis of goethite. The Brunauer-Emmett-Teller (BET) specific surface area of the synthesized goethite was 81 m 2 g -1, and the point of zero charge (PZC) was estimated at 9.1 by potentiometric titration [START_REF] Cheng | Adsorption of Quinolone Antibiotics to Goethite under Seawater Conditions: Application of a Surface Complexation Model[END_REF]. Transmission electron microscopy (TEM, Jeol JEM 1230 microscope) images showed goethite needle particles of around 100 nm long (Figure S1). Gibbsite (Al(OH) 3 ) was synthesized as described in [START_REF] Jodin | Limitations of potentiometric studies to determine the surface charge of gibbsite γ-Al(OH)3 particles[END_REF]. The BET specific surface area of the synthesized gibbsite was 20 m 2 g -1, and the PZC was estimated at 9.8.

Adsorption isotherms

All adsorption isotherm experiments were performed under ambient air, at room temperature, i.e., 20°C  2. Adsorption experiments of Ce and Nd (noted "Ce/Nd" in the following text) by goethite and gibbsite were carried out using 0.5 g L -1 of goethite and 2 g L -1 of gibbsite in 5 mM NaCl solution. The ratio between gibbsite and goethite depends on their BET specific surface. The range of [Ce/Nd] tot varied from 250 to 5000 µg L -1 , noted Ce/Nd250 to Ce/Nd5000. Such concentrations were selected in response to the necessary concentration for XANES investigations. The pH was adjusted to pH 6 (e.g., pH conditions commonly encountered in soils) and monitored over time using 0.1 M NaOH/HCl solution (Hach, sensION+). The Ce/Nd-goethite and gibbsite suspensions were stirred for 24 h, as it is often considered as a sufficient equilibration time in adsorption experiments. Additionally, an aliquot was maintained under stirring for 70 days for Ce-goethite suspension to check potentially slow kinetic processes. The remaining suspension was centrifuged at 3000 g for 30 min. The resulting centrifuge was air dried and stored at room temperature for XANES investigations. The solution was filtered at 0.2µm (PES filter, Sartorius), acidified to 2% HNO 3 for Ce/Nd concentration analysis ([Ce/Nd] aq ), determined by Quadruple ICP-MS using an Agilent Technologies 7700×. Concentrations of Fe and Al were also monitored, and the low values confirmed that the amounts of goethite and gibbsite were negligible in the filtrate, as previously expected or determined in previous comparable studies [START_REF] Cheng | Adsorption of Quinolone Antibiotics to Goethite under Seawater Conditions: Application of a Surface Complexation Model[END_REF][START_REF] Luo | Interactions of Anti-Inflammatory and Antibiotic Drugs at Mineral Surfaces Can Control Environmental Fate and Transport[END_REF][START_REF] Xu | Co-binding of Pharmaceutical Compounds at Mineral Surfaces: Mechanistic Modeling of Binding and Cobinding of Nalidixic Acid and Niflumic Acid at Goethite Surfaces[END_REF].

J o u r n a l P r e -p r o o f

Cerium L III -edge XANES investigation

Using Ce L III -edge XANES spectra allows for identifying the oxidation states +III and +IV of Ce through their different electron transitions during measurement. Overall, the Ce(III) model compounds express a white line corresponding to the 2p 3/2 →(4f 1 )5d transition, while the Ce(IV) model compounds exhibit a white line with two maxima because of the mixed-valence behavior of tetravalent Ce in its final state, due to the interaction between 4f orbitals of the Ce and 2p orbitals of the O ligand in its initial state [START_REF] Malterre | L III-edge spectroscopy on cerium-based intermediate-valent compounds[END_REF]. This corresponds to two electronic transitions: the lower-energy peak is assigned to the 2p →5d transition with 4f 1 L configuration, and the higher-energy peak is assigned to the 2p →5d transition with 4f 0 L configuration [START_REF] Fonda | EXAFS analysis of the L3 edge of Ce in CeO2: effects of multi-electron excitations and final-state mixed valence[END_REF][START_REF] Soldatov | Crystalstructure effects in the Ce ${\mathit{L}}_{3}$-edge x-ray-absorption spectrum of ${\mathrm{CeO}}_{2}$: Multiple-scattering resonances and many-body final states[END_REF].

XANES acquisition

X-ray absorption experiments were performed on the LUCIA beamline (SOLEIL synchrotron) [START_REF] Flank | LUCIA, a microfocus soft XAS beamline[END_REF][START_REF] Vantelon | The LUCIA beamline at SOLEIL[END_REF]. The X-ray beam was monochromatized using a Si (111) double-crystal monochromator. The monochromator was calibrated by setting the first inflection point of a Cr metallic foil XANES spectrum to 5987 eV. Spectra were collected in fluorescence mode (using a Bruker 60 mm 2 mono-element SDD). Cerium L III -edge XANES spectra were collected with a beam size of 2*2 mm² under a secondary vacuum (10 -7 mbar) and at 25 K, using a liquid He cryostat. A set of spectra published by [START_REF] Ratié | Cerium anomalies in riverbanks: Highlight into the role of ferric deposits[END_REF] were used for reference compounds (CePO 4 , Ce(NO 3 ) 3 .6H 2 O, Ce(OH) 4 , Ce(SO 4 ) 2 , Ce-doped δMnO 2 and humic acid).

The Ce-goethite samples were prepared as pellets of finely ground and homogenized powder mixed with cellulose. Measures were performed using the XANES "e-flyscan" continuous mode available on the LUCIA beamline, ranging from 5680 eV to 5800 eV with an energy step of 0.4 eV and a counting time of 300 ms per point. The recorded spectra (up to ten) were superimposable for each sample, confirming that no beam damage occurred during measurements and then merged. XANES spectra were processed using the Athena software [START_REF] Ravel | HEPHAESTUS: data analysis for X-ray absorption spectroscopy using IFEFFIT[END_REF]. Normalized spectra were obtained by J o u r n a l P r e -p r o o f fitting the pre-edge region with a linear function and the post-edge region with a quadratic polynomial function.

Cerium solid speciation

The Ce speciation in the studied sample can be obtained by Linear Combination Fits (LCF) of Ce references XANES spectra. However, for an identical valence, with Ce in different phases, the XANES spectrum does not allow to identify slight different structures [START_REF] J O U R N A L P R E -P R O O F Chalmin | A pre-edge analysis of Mn K-edge XANES spectra to help determine the speciation of manganese in minerals and glasses[END_REF][START_REF] Farges | Ab initio and experimental pre-edge investigations of the Mn K-edge XANES in oxidetype materials[END_REF]. An alternative method exists to focus on the valence state instead of the Ce speciation for allowing the determination of the Ce(III)/Ce(IV) ratio: linear combinations with a database of pure-valence species (the Combo method), already applied for determining Mn valence states [START_REF] Manceau | Determination of Mn valence states in mixed-valent manganates by XANES spectroscopy[END_REF]. For applying to Ce valence states, that method consists in performing LCF with several known Ce valence reference compounds. The attribution to the Ce(III) state is considered as the sum of fractions of each Ce(III) species in the fit and the same for Ce(IV) state for estimating the Ce(III)/Ce(IV) ratio in the goethite samples.

Linear CF were performed on the normalized and first derivative of Ce L III -edge spectra from 5700 eV to 5743 eV, to avoid any influence of the first oscillations depending on the coordination sphere Ce-O of each structure of references, and not forcing the weights to sum up to 1, using the Athena software [START_REF] Ravel | HEPHAESTUS: data analysis for X-ray absorption spectroscopy using IFEFFIT[END_REF]. The R-factor and the χ 2 reduced were used to characterize the LCF accuracy. The precision of the LCF-XANES approach, by fitting mechanically-mixed Ce reference compounds, is approximately 10% [START_REF] Janots | Ce (III) and Ce (IV)(re) distribution and fractionation in a laterite profile from Madagascar: insights from in situ XANES spectroscopy at the Ce LIII-edge[END_REF][START_REF] Takahashi | Determination of the oxidation state of cerium in rocks by Ce LIII-edge X-ray absorption near-edge structure spectroscopy[END_REF].

Iterative algorithm

Additionally, if references XANES spectra of Ce(III) and Ce(IV) compounds are not known a priori to be used in LCF, another way to determine the Ce(III)/Ce(IV) is to use an optimization algorithm to minimize the difference between the measurement results and the results of the following decomposition:

𝐴(𝐸, [𝐶𝑒] 𝑡𝑜𝑡 ) = 𝑓 1 ([𝐶𝑒] 𝑡𝑜𝑡 ) × 𝐴 1 (𝐸) + (1 -𝑓 1 ([𝐶𝑒] 𝑡𝑜𝑡 )) × 𝐴 2 (𝐸) (1)
J o u r n a l P r e -p r o o f

Where A values are calculated normalized XANES intensities, f 1 , A 1, and A 2 are the variables to be determined, which correspond to the fraction of Ce(IV), the normalized XANES intensities of pure Ce(IV)

and Ce(III) compounds, respectively. The deviation function to be minimized can be written as:

𝐽(𝑓 1 , 𝐴 1 , 𝐴 2 ) = ∑ ∑ (𝐴(𝐸 𝑖 , [𝐶𝑒] 𝑡𝑜𝑡,𝑖 ) -𝐴_𝑚𝑒𝑎𝑠(𝐸 𝑖 , [𝐶𝑒] 𝑡𝑜𝑡,𝑖 )) 2 𝑚 𝑗=1 𝑛 𝑖=1 (2)
where, n is the number of energies E i and m is the number of concentrations [Ce] tot,i used for the measurements, and A_meas is the measured XANES intensity. To solve this optimization problem, an iterative two-step method is used: A 1 and A 2 are determined assuming that f 1 is fixed, then f 1 is determined assuming that A 1 and A 2 are fixed, and the process is repeated until convergence. f 1 is initialized to values regularly spaced from 0.2 to 0.5, A 1 (E) is initialized to A_meas for [Ce] tot =5000 µg L -1 , and A 2 (E) to A_meas for [Ce] tot =250 µg L -1 . The Sequential Quadratic Programming [START_REF] Nocedal | Numerical optimization[END_REF] method is used to solve each of the internal optimization problems with the Matlab TM software.

Geochemical speciation modeling

Predominance (pH-Eh) diagrams were obtained using PhreePlot [START_REF] Kinniburgh | PhreePlot: Creating graphical output with PHREEQC[END_REF], which contains an embedded version of the speciation code PHREEQC [START_REF] Parkhurst | Description of input and examples for PHREEQC version 3: a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF].

Aqueous Ce(III) and Ce(IV) reactions and equilibrium constants were taken from (Marsac et al., 2017b).

The Davies equation was used to extrapolate thermodynamic constants at various ionic strengths.

Results and discussion

Neodymium and cerium adsorption to goethite and gibbsite

Neodymium adsorption isotherm to goethite reached a plateau at around 19 µg m -2 (Figure 1a). This behavior was fitted using a Langmuir isotherm model:

[𝑁𝑑] 𝑎𝑑𝑠 = 𝑄 𝑚𝑎𝑥,𝑁𝑑 (𝐾 𝑁𝑑 [𝑁𝑑] 𝑎𝑞 ) 1+(𝐾 𝑁𝑑 [𝑁𝑑] 𝑎𝑞 ) (3)
where, Q max,Nd is the maximum amount of adsorbed Nd (µg m -2 ), and K Nd is the Langmuir constant. All parameters were determined by a least-square fit (Table 1). By contrast, the amount of adsorbed Ce J o u r n a l P r e -p r o o f Journal Pre-proof constantly increased to reach 30 µg m -2 at the highest [Ce] tot (5000 µg L -1 ), and no plateau was reached.

The concentration of adsorbed Ce overpassed that of Nd for [Ce/Nd] aq = 1400 µg L -1 (i.e. [Ce/Nd] tot = 2000 µg L -1 ). At low [Ce/Nd] aq , the highest adsorption is expected for Nd than for Ce because Nd is bound more strongly by ligands (including surface -OH groups) than Ce [START_REF] Davranche | Impact of humate complexation on the adsorption of REE onto Fe oxyhydroxide[END_REF][START_REF] Ohta | REE (III) adsorption onto Mn dioxide (δ-MnO2) and Fe oxyhydroxide: Ce (III) oxidation by δ-MnO2[END_REF][START_REF] Quinn | Comparative Scavenging of Yttrium and the Rare Earth Elements in Seawater: Competitive Influences of Solution and Surface Chemistry[END_REF], because of their hydrolysis constants [START_REF] Brown | Hydrolysis of Metal Ions[END_REF][START_REF] Rizkalla | Chapter 127 Lanthanides and actinides hydration and hydrolysis[END_REF]. At high [Ce/Nd] aq , Ce is more adsorbed than Nd in response to the incomplete oxidation of Ce(III) to Ce(IV), which is more adsorbed than Ce(III) by mineral surface [START_REF] Bau | Scavenging of dissolved yttrium and rare earths by precipitating iron oxyhydroxide: experimental evidence for Ce oxidation, Y-Ho fractionation, and lanthanide tetrad effect[END_REF][START_REF] Ohta | REE (III) adsorption onto Mn dioxide (δ-MnO2) and Fe oxyhydroxide: Ce (III) oxidation by δ-MnO2[END_REF][START_REF] Takahashi | Chemical and structural control of the partitioning of Co, Ce, and Pb in marine ferromanganese oxides[END_REF][START_REF] Takahashi | Direct observation of tetravalent cerium in ferromanganese nodules and crusts by X-ray-absorption near-edge structure (XANES)[END_REF]. Interestingly, this oxidation seems fast since Ce adsorption equilibrium was reached at around 24h and did not evolve until 70 days (Figure 1a).

J o u r n a l P r e -p r o o f

J o u r n a l P r e -p r o o f

The difference between Nd and Ce adsorption isotherm suggests different adsorption processes. A unique Langmuir isotherm did not succeed in fitting Ce adsorption by goethite. An additional Freundlich isotherm had to be used to consider the oxidation of Ce(III) as Ce(IV):

[𝐶𝑒] 𝑎𝑑𝑠 = 𝑄 𝑚𝑎𝑥,𝐶𝑒

(𝐾 𝐶𝑒 [𝐶𝑒] 𝑎𝑞 ) 1+(𝐾 𝐶𝑒 [𝐶𝑒] 𝑎𝑞 ) + (𝐾 𝑝𝑜𝑙𝑦 [𝐶𝑒] 𝑎𝑞 ) 𝑛 (4)
Where, Q max,Ce is the maximum amount of adsorbed Ce (µg m -2 ), K Ce is the adsorption constant, K Poly the Freundlich constant, and n is the non-linear parameter. A similar model was previously used for silicate binding and polymerization/precipitation onto Fe oxyhydroxides [START_REF] Kanematsu | Silicate binding and precipitation on iron oxyhydroxides[END_REF]. By assuming that (i) the maximal binding capacity is the same for Nd and Ce(III) (Q max,Ce = Q max,Nd , when expressed in mol m -²) and that (ii) Ce(III) and Nd(III) are adsorbed on the same surface sites, the values of K Ce , K Poly, and n can be determined by a least-square fit. This model allowed us to reproduce the experimental datasets with a RMSE = 0.033. This model will be further used to discuss the Ce redox speciation in combination with XANES results.

Neodynium adsorption isotherm followed the same trend as the gibbsite (Figure 1b). The Langmuir isotherm model allowed us to reproduce the Nd adsorption datasets (Eq. 3) (Table 2). A larger binding capacity but a smaller Langmuir constant were calculated for gibbsite than for goethite. At low

[Ce] tot concentration, the Ce adsorption was smaller than that of Nd, but was higher at the highest [Ce] tot concentration. Cerium adsorption isotherm has also been fitted by a combination of Langmuir and Freundlich isotherms, RMSE = 0.045 (Eq. 4). Figure 1b exhibits the contribution of Ce(III) and Ce(IV) adsorption onto gibbsite, respectively. Cerium(III)/Ce(IV) adsorption onto the mineral depends on the Ce concentration in the solution and the mineral that controls the Ce redox equilibrium.

J o u r n a l P r e -p r o o f

J o u r n a l P r e -p r o o f

Journal Pre-proof

Cerium solid speciation

XANES of Ce references

The Ce(III) references displayed Ce L III -edge XANES spectra with an intense white line at 5726 eV, followed by a broad oscillation centered around 5755-5760 eV (Figure 2a). The Ce(IV) references showed a white line with two maxima, followed by a broad oscillation centered at 5773 eV.

According to the different Ce(IV) references, the first maximum ranged from 5729 eV to 5731 eV, and the second maximum from 5737 eV to 5738 eV. The differences in intensity and position between the first two maxima of the Ce(IV) species are explained by changes in covalent character in the Ce-O bond [START_REF] Kaindl | Pressure-induced changes in L III x-ray-absorption near-edge structure of Ce O 2 and Ce F 4: Relevance to 4 f-electronic structure[END_REF]. Additionally, the first derivative of the spectra was also expressed to reveal the valence state of transition without the effect of coordination and symmetry (Figure 2b). The Ce(IV) references derivative spectra exhibited peaks at 5725 eV and 5735 eV that can be assigned to the +IV electronic state, while the Ce(III) references indicated a peak at 5724 eV that can be assigned to the +III electronic state.

Goethite-Ce samples

The Ce L III -edge XANES spectra of the goethite samples exhibited a singular white line at 5726 eV, matching the Ce(III) references (Figure 3). The intensity of the white line decreased from sample Ce250 to sample Ce5000. This decrease is coupled with the increase of a shoulder at 5737 eV, matching with the second maximum of the Ce(IV) references spectra, which intensity increased from sample Ce250 to sample Ce5000. The XANES spectra exhibited three isosbestic points indicating that Ce speciation can be described using solely two Ce phases or a set of Ce phases whose proportions do not vary (Figure 3).

J o u r n a l P r e -p r o o f [Ce] TOT corresponds to the initial concentration of Ce in the solution.

Ce(III)/Ce(IV) ratios

The normalized and first derivative Ce L III -edge XANES were deconvoluted using the Combo method (Figure S2). As previously described, first derivative spectra allow focusing on the oxidation state instead of the crystalline structure. Both results were compared, and the numerical results of LCF are reported in Table S1. The reduced χ² used to characterize the quality of the fit varies from 1 10 -3 to 5 10 -4 for normalized spectra, allowing us to discuss the Ce(III)/Ce(IV) ratios in the goethite samples. J o u r n a l P r e -p r o o f

The contribution of Ce(III) resulted from the contribution of CePO 4 , and Ce(III) adsorbed to humic acid, while the Ce(IV) contribution resulted from the contribution of Ce(SO 4 ) 2 and Ce(IV) adsorbed to δMnO 2 (Table S1). The Ce(III)/Ce(IV) ratio oxidation state progressively increased relative to the initial Ce concentration in the solution (Table S1). From Ce250 to Ce5000, Ce(IV) ranged from 21% to 49% for normalized spectra and from 20% to 48% for first derivative spectra. The homogeneous results between normalized spectra and first derivative spectra LCF highlighted the negligible influence of the variation of the normalized spectra of Ce(IV) references for LCF, considering the analytical error of LCF, estimated to ± 10%.

As a complementary approach using an iterative algorithm based only on Ce L III -edge XANES absorption normalized spectra determined the normalized intensities of components Ce(III) and Ce(IV) (Figure S3). This allowed estimating the Ce(IV) contribution. From Ce250 to Ce5000, Ce(IV) ranged from 19% to 53% (Table S1).

Both LCF and the iterative algorithm provided consistent results showing that the Ce(IV) amount increased proportionally with the Ce concentration in the solution (Figure 4). At the lowest Ce concentration ([Ce] tot = 250 µg L -1 ), the Ce(IV) proportion represented ≈ 20 % of the total adsorbed Ce, whereas at the highest Ce concentration ([Ce] tot = 5000 µg L -1 ), the Ce(IV) proportion reached ≈ 50%.

These results are in good agreement with the modeling approach assuming that the Langmuir (left member of eq.2) and Freundlich (right member of eq.2) isotherms correspond to the uptake of Ce(III) and Ce(IV) by goethite, respectively (Figure 4).

J o u r n a l P r e -p r o o f

Thermodynamic aspects

Interpretation of Nd and Ce adsorption isotherms onto goethite evidenced the incomplete oxidation of Ce(III) to Ce(IV), which was confirmed by XANES results on goethite. That result suggests, at first glance, the possible role of Fe(III) for Ce oxidation to goethite surface, as very often observed in the presence of MnO 2 . However, by contrast with the Mn(II)/MnO 2 redox couple, the Fe(II)/goethite redox potential is significantly lower than that of Ce(III)/Ce(IV) [START_REF] Stumm | The cycling of iron in natural environments: Considerations based on laboratory studies of heterogeneous redox processes[END_REF]. In A linear relationship exists between [Ce(IV)] goe and the total concentration of Ce in the goethite suspension (Figure 4). Cerium(IV) contribution is, therefore, rather related to the solution saturation than to a redox surface effect. The redox stability range of Ce(IV) increases with the increasing total Ce concentration, as can be observed on the pH-E H predominance diagram (Figure 5a) in response to the Ce(IV)-mineral formation. Amorphous Ce(OH) 4(am) is rather expected than crystalline CeO 2(s) (Marsac et al., 2017b;[START_REF] Neck | Solubility of crystalline thorium dioxide[END_REF][START_REF] Neck | Solubility and hydrolysis of tetravalent actinides[END_REF], according to the following half-reaction: [START_REF] Ratié | Cerium anomalies in riverbanks: Highlight into the role of ferric deposits[END_REF][START_REF] Yu | Cerium sequestration and accumulation in fractured crystalline bedrock: the role of Mn-Fe (hydr-) oxides and clay minerals[END_REF].

𝐶𝑒 3+ + 4𝐻 2 𝑂 ⇌ 𝐶𝑒(𝑂𝐻) 4(𝑠) + 4𝐻 + + 𝑒 - (5
If Ce(OH) 4(am) precipitation explains the increasing thermodynamic stability of Ce(IV) over Journal Pre-proof distinct oxidation states (e.g., Pu(III) and Pu(IV)) [START_REF] Banik | Neptunium sorption and redox speciation at the illite surface under highly saline conditions[END_REF][START_REF] Banik | Sorption and Redox Speciation of Plutonium at the Illite Surface[END_REF]Marsac et al., 2017aMarsac et al., , 2015aMarsac et al., , 2015b)). Accordingly, goethite stabilizes more strongly Ce(IV) than gibbsite. J o u r n a l P r e -p r o o f

Conclusions

This study provided adsorption datasets of Nd and Ce sorption onto goethite and gibbsite. Fitting the datasets of both goethite (oxidative surface) and gibbsite (non-redox sensitive surface) with a combined Langmuir and Freundlich model, we conclude that Fe(III) redox reactivity is not the main driver of the Ce(III) oxidation as Ce(IV) onto goethite. The XANES results showed that 20 % to 50 % of the total Ce bound to goethite was as Ce(IV) and that this linearly increased with the increasing Ce concentration in the solution.

This study obviously assessed an intimate link between both the Ce concentration in solution and the Ce(III) oxidation to Ce(IV) onto goethite. This finding explains the opposite results reported in the literature at high and low [Ce] tot [START_REF] Denys | Evaluation of selectivity of sequential extraction procedure applied to REE speciation in laterite[END_REF][START_REF] Ratié | Cerium anomalies in riverbanks: Highlight into the role of ferric deposits[END_REF][START_REF] Yu | Cerium sequestration and accumulation in fractured crystalline bedrock: the role of Mn-Fe (hydr-) oxides and clay minerals[END_REF]. The key result highlighted by the study is, therefore, that the concentration of dissolved Ce strongly constraints the Ce redox speciation in solution, i.e., Ce(III) and Ce(IV) aqueous form, leading to spontaneous uptake of Ce(IV) by goethite and more generally by minerals. The Goethite surface seems to stabilize more strongly Ce(IV) than the gibbsite. Therefore, further studies are required for quantifying the effect of the mineral on Ce redox speciation from various Fe oxyhydroxides and other minerals. Moreover, the present findings might apply to other redox sensitive metals exhibiting a poorly soluble oxidation state as compared to other oxidation states, e.g., chromium (Cr(III) vs. Cr(VI)) or tetravalent uranium (U(IV) vs. U(V) or U(VI)), and whose redox speciation might be affected by surface precipitation processes.

Regarding the use of the Ce anomaly as an environmental tracer, except for chemical controlling parameters such as pH [START_REF] Bau | Scavenging of dissolved yttrium and rare earths by precipitating iron oxyhydroxide: experimental evidence for Ce oxidation, Y-Ho fractionation, and lanthanide tetrad effect[END_REF], other biogeochemical factors can influence the Ce redox behavior: the colloidal fraction [START_REF] Davranche | Adsorption of REE(III)humate complexes onto MnO2: Experimental evidence for cerium anomaly and lanthanide tetrad effect suppression[END_REF][START_REF] Pédrot | Upper soil horizons control the rare earth element patterns in shallow groundwater[END_REF]2008), the biological parameters [START_REF] Ngwenya | Discrete Site Surface Complexation Constants for Lanthanide Adsorption to Bacteria As Determined by Experiments and Linear Free Energy Relationships[END_REF][START_REF] Tanaka | A specific Ce oxidation process during sorption of rare earth elements on biogenic Mn oxide produced by Acremonium sp. strain KR21-2[END_REF], and the organic matter [START_REF] Gangloff | Characterization and evolution of dissolved organic matter in acidic forest soil and its impact on the mobility of major and trace elements (case of the Strengbach watershed)[END_REF]Pourret et al., 2007a).

Additionally, to these well-known biogeochemical factors inferring in the use of Ce anomaly, such as an environmental tracer, we, therefore, highlight the role of the Ce soluble concentration and speciation on the amplitude of the Ce anomaly development.

J o u r n a l P r e -p r o o f

Figure 1 :

 1 Figure 1: Nd and Ce adsorption isotherms onto (a) goethite (0.5 g L -1 ) and (b) gibbsite (2 g L -1 ), at pH 6 in 5 mM NaCl. Experimental data (exp.) are represented by circles, whereas modeled data are displayed as lines. The green line represents the adsorbed Nd(III) using the Langmuir model, and the black line corresponds to the sum of adsorbed Ce(III) (black dashed line) calculated with the Langmuir model and the adsorbed Ce(IV) and/or Ce(IV) surface precipitation (black dotted line) calculated with a Freundlich model.

Figure 3 :

 3 Figure 3: Ce L III -edge XANES absorption normalized spectra of the goethite samples. The arrows represent the Ce oxidation state, i.e., lower Ce(III) and higher Ce(IV) contributions, as a function of the increasing Ce concentration in the solution. The dashed grey circles highlight the isosbestic points.

Figure 4 :

 4 Figure 4: Comparison between the [Ce(IV)] ads determined from XANES spectra (LCF and iterative algorithm) and by the model (adsorption isotherm) as a function of the initial Ce concentration in solution. The dotted dash line represents the linear regression of the [Ce(IV)] ads (LCF XANES) as a function of the [Ce] tot in solution. The error bar for LCF is considered as ± 10%.

  r n a l P r e -p r o o f addition, the formation of Ce(IV) in the presence of gibbsite, which is redox-inert, shows that the mineral oxidizing capacity is not the only driving process in the oxidation of Ce(III) to Ce(IV).

  ) Similarly, the apparent (i.e., concentration dependent) redox potential of the prevailing Ce(IV)/Ce(III) species (E' Ce(IV)/Ce(III) ) can be calculated both in the aqueous and solid phases of the suspension, that correspond to the Ce(IV)-Ce(III) borderline of the Pourbaix diagram at pH = 6 (Figure 5b). Values of E' Ce(IV)/Ce(III) remained constant at low [Ce] tot because no precipitation occurred, hence equal to the apparent redox potential of the Ce(OH) 3 + /Ce 3+ at pH = 6 in 5 × 10 -3 M NaCl. When [Ce] tot increased, E' Ce(IV)/Ce(III) decreased because of the formation of Ce(OH) 4(am) . If we assume that the experimental redox potential (E H ) is independent of the [Ce] tot , since controlled by atmospheric O 2 , we can conclude that Ce(III) oxidation to Ce(IV) is promoted when [Ce] tot increases. This explains the present and the opposite results in the literature at high and low [Ce] tot

  Ce(III) for increasing[Ce] tot , it does not fully explain all the present datasets. Cerium(IV) was observed onto goethite at the lowest [Ce] tot , whereas Ce(IV) was expected to occur in a significant amount onto gibbsite for the highest [Ce] tot . The binding mechanisms of Ce(III) and Ce(IV) onto different minerals might thus control the Ce redox speciation at equilibrium. Previous studies revealed that the redox speciation of actinides (Np and Pu) was affected by clay surfaces presenting different affinities for two J o u r n a l P r e -p r o o f

Figure 5 :

 5 Figure 5: (a) Eh-pH diagram for Ce in 5 × 10 -3 M NaCl solution accounting or not for the precipitation of

Table 1 :

 1 Experimental results of Nd and Ce sorption onto goethite and parameters used for modeling Nd and Ce sorption onto goethite surfaces, as well as the Ce(III)/Ce(IV) adsorbed ratio.

					Journal Pre-proof
	Goethite		Experiment results	Modeling parameters	Modeling results
	Sample name	[ ] total µg L -1 [ ] aq µg L -1 [ ] ads µg m -2		Nd	Ce	[ ] ads µmol m -2 [Ce(III)] ads µg m -2 [Ce(IV)] ads µg m -2
			Nd (1 day)		Qmax (µg m -2 ) 19.87 19.31	Nd (1 day)
	Nd250		61	4.8	logKads	-2.40 -3.01	3.9
	Nd500 Nd750 Nd1000 Nd2000 Nd3000 Nd5000 Ce250 Ce500 Ce750 Ce1000 Ce2000 Ce3000 Ce5000 Ce250 Ce500 Ce750 Ce1000 Ce2000	1049 2070 3006 5219 2075 3223 5116 2075	177 312 532 1376 2269 4479 Ce (1 day) 79 231 451 591 1398 2202 3913 Ce (70 days) 77 227 445 613 1396 J o u r n a l P r e -p r o o f 8.2 m 1.00 1.00 8.2 10.9 logKpoly --0.04 11.0 12.9 n -0.87 13.5 17.4 16.8 18.4 17.9 18.5 18.8 4.3 1.9 6.6 4.8 7.5 8.2 9.9 9.9 16.9 17.2 25.5 22.2 30.1 30.1 4.358 6.726 7.600 9.363 16.967	Ce (1 day) 1.4 3.5 5.9 7.1 11.1 13.2 15.3	0.5 1.3 2.3 2.9 6.1 9.0 14.8
	Ce3000	3223	2160	26.558
	Ce5000	5116	3865	31.267

Table 2 :

 2 Experimental results of Nd and Ce sorption onto gibbsite and parameters used for modeling Nd and Ce sorption onto gibbsite surfaces, as well as the Ce(III)/Ce(IV) adsorbed ratio.

	Gibbsite		Experiment results		Modeling parameters		Modeling results
	Sample name	[ ] total µg L -1 [ ] aq µg L -1 [ ] ads µg m -2		Nd	Ce	[ ] ads µmol m -2 [Ce(III)] ads µg m -2 [Ce(IV)] ads µg m -2
			Nd (1 day)		Qmax (µg m -2 ) 58.70 57.02		Nd (1 day)
	Nd250	206	75	3.0	logKads	-3.53 -3.81	1.3
	Nd500	465	282	4.2	m	1.00 1.00	4.5
	Nd750	702	479	5.5	logKpoly	-	-2.91	7.3
	Nd1000	942	628	7.2	n	-	3.82	9.2
	Nd2000	1886	1188	16.0				15.3
	Nd3000	2875	1899	23.9				21.2
	Nd5000	4887	3601	31.2				30.3
			Ce (1 day)						Ce (1 day)
	Ce250	229	90	3.3				0.8	0.8	0.0
	Ce500	459	185	6.3				1.6	1.6	0.0
	Ce750	678	465	5.3				3.8	3.8	0.0
	Ce1000	926	635	6.9				5.1	5.1	0.0
	Ce2000	1841	1491	8.7				11.2	10.7	0.5
	Ce3000	2748	2087	15.7				15.7	13.9	1.8
	Ce5000	4865	3518	33.1				33.1	20.1	13.0

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J o u r n a l P r e -p r o o f

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement N° 857989, by the French ANR project "C-FACTOR" (number ANR-18-CE01-0008) and French CNRS NEEDS project "PREDIRPLUCE". G.

Ratié acknowledges the Faculty of Environmental Sciences, Czech University of Life Sciences, Prague (Grant n° IGA FŽP 2020B0025) and the Labex VOLTAIRE, France (10-LABX-0100). Through the support of the GeOHeLiS analytical platform of Rennes University, this publication is also supported by the European Union through the European Regional Development Fund (FEDER), the French Ministry of Higher Education and Research, the French Region of Brittany, and Rennes Metropole. The authors acknowledge the SOLEIL synchrotron for beamtime allocation at the LUCIA beamline (proposal 20200442). We also thank Prof. Karen H. Johannesson for handling our paper.

 B,C,D,and E correspond to 5726 eV,5729 eV,5731 eV,5737 eV,and 5738 eV,respectively. Peaks F,G,and H correspond to 5724 eV,5725 eV,and 5735 eV,respectively. Ce_HAJ o u r n a l P r e -p r o o f


domain: phys spi

text:
F Uncu

- B Franc ¸ois

- R Barrier

-N Buffaz

-S Le Guyader

RANS PREDICTION OF THE AERODYNAMIC LOSSES IN A LINEAR TURBINE CASCADE WITH AN UPSTREAM CAVITY AND A PURGE FLOW

Keywords: Linear Cascade, Low-Pressure Turbine, Cavity Flow, RANS Turbulence Modelling 1 Subscript

In a turbomachine, gaps separate stator and rotor disks. For turbines, often a purge flow is blown through these gaps in order to seal and avoid hot air to penetrate deep in the turbine disk components and over-solicit them. The interaction of this purge flow with the main flow creates and amplifies vortex structures, responsible for aerodynamic losses. Accurately predicting these losses remain a challenge with RANS approach for engine manufacturers that wish to design an efficient cavity geometry minimising the losses while ensuring the sealing. The Boussinesq approximation fails in such flows where a high turbulence level and anisotropy are present. Therefore, in RANS, the use of two-equation linear eddy-viscosity turbulence models is questionable. The present work assesses a second order Reynolds Stress Model (RSM), as well as two eddy-viscosity models, in a linear turbine cascade with an upstream cavity from which a purge flow emanates. The specificity of this cascade is the high external turbulence intensity (6%), which is all the more challenging in a RANS approach. Different purge mass flows and three configurations are evaluated: two with different cavities and one without any. The RANS simulations are compared to experimental data. All the RANS models show the same trends: the presence of a cavity induces additional pressure losses; increasing the purge mass flow helps to seal the cavity entry but nourishes the passage vortex and thus leads to additional losses. The RSM simulation shows the best agreement with the measurements and is the only one that predicts the loss evolution when the cavity chute geometry changes.

Static pressure coefficient (P i,1 -Ps(x)) /(P i,1 -P s,2 ) ζ Stagnation pressure loss coefficient (P i,1 -P i,2 ) /(P i,2 -P s,2 ) θ Normalised fluid temperature (T i -T i,leakage ) /(T i,∞ -T i,leakage ) S / Ω Shear / Vorticity tensor INTRODUCTION A 50% drop in the purge mass flow could lead to a 0.5% increase in the efficiency of a two stage high-pressure turbine and to a 0.9% decrease in the specific fuel consumption [START_REF] Johnson | Turbine rim seal gas path flow ingestion mechanisms[END_REF]. Although less used in low-pressure turbines, it is essential to reduce purge flow and to improve cavity sealing to minimise interaction with mainstream flow. However, increasing the combustion chamber exit temperature in modern engines requires more purge flow. The interaction between cool purge flow and main flow creates a shear layer causing viscous losses. In addition, the flow angle is modified at the leading edge of the downstream blade, which disrupts the turbine work. Furthermore, main flow ingress can also occur; this is mainly because of disk pumping, vane/blade periodic pressure field and cavity chute seal geometry [START_REF] Johnson | Turbine rim seal gas path flow ingestion mechanisms[END_REF]. The first two mechanisms are also functions of the latter. Bailey and Owen [START_REF] Bayley | The Fluid Dynamics of a Shrouded Disk System With a Radial Outflow of Coolant[END_REF] showed that disk pumping depends directly on the spacing between the two walls of the chute. A spacing smaller than the boundary layer thickness helps in reducing purge flow necessary for sealing. Complex geometries with overlaps between the two walls of the cavity help reduce the flow exchange between the cavity and the main channel, resulting in significant reductions in aerodynamic losses [3; 4].

The prediction of cavity flows and their impact on the pressure losses of the turbine must be taken into account in RANS simulations during the design stage. However, in regions of complex geometry that present multiple separations, reattachments and mixing process, like in the cavity sealing flows, RANS modelling may not predict accurate results [START_REF] Kluge | Sensitivity analysis, design, instrumentation, and experimental validation of a novel labyrinth seal rig[END_REF]. The high turbulence production and important turbulence anisotropy in such non-equilibrium regions are still challenging for the classical eddy-viscosity turbulence models.

The main studies on cavity flows use the k -ω SST model with a Kato-Launder limiter ( [6; 7]). Horwood et al. [8; 9] showed that RANS simulations well-predict the velocity field and sealing efficiency in the cavity. However, when the purge flow increases, the simulation under-predicts the ingress. RANS modelling produces too much eddy viscosity which stabilises vortex structures leading to a misprediction of the sealing. The level of ingress and egress also modifies the size of the passage vortex which is the source of pressure losses. These losses are mispredicted with RANS turbulence models in a simple turbine stage.

Izenson et al. [START_REF] Izenson | Turbulent Flow Computations for Turbine Disk Cavity Flows[END_REF] compare measurements with k -ε and an RSM model based on Rotta's distribution term. The components of the Reynolds stress tensor calculated by the RSM model are anisotropic. The directional dependence of the velocity fluctuations is absent in a linear eddy-viscosity model. In highly rotational stress flow, where streamline curvature is high, the anisotropy is important, and in such an environment the RSM model provides predictions closer to the experiment. Furthermore, the measurements of Porreca et al. [START_REF] Porreca | Turbulence measurements and analysis in a multistage axial turbine[END_REF] at a stator/rotor interface show a difference of 45% between radial and streamwise component of fluctuations.

The present work proposes to assess RANS turbulence models in a turbine cascade with a high level of external turbulence. A hub cavity from which a purge flow emanates is present upstream of the blade leading edge. Different cavity geometries and purge mass flows are simulated. The aim is to assess the ability of RANS simulations in accurately predicting the pressure losses related to the cavity and understand the importance of turbulence modelling in such flows. A second order Reynolds Stress Model (SSG/LRR-ω [START_REF] Cécora | Differential reynolds-stress modeling for aeronautics[END_REF]) as well as two eddyviscosity models (k -ω SST [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] and k -l [START_REF] Smith | A near wall model for the k-l two equation turbulence model[END_REF]) are evaluated.

Firstly, the paper introduces the three simulated configurations. Then, the numerical setup and the meshes are presented. Finally, results with different turbulence models are discussed and compared to the measurements.

CONFIGURATIONS Instrumentation

The simulated configuration is a linear cascade from the EU project MAGPI1 , which aimed to improve turbine cavity designs. It is composed of five stator blades. The hub and shroud are flat, and a cavity is present just before the leading edge at the hub. A turbulence grid located at seven axial chords (C x ) upstream imposes the turbulence intensity on the incoming flow. The rig test and measurement instrumentation are detailed by Schuler et al. [START_REF] Schuler | Investigation of the Influence of Different Rim Seal Geometries in a Low-Pressure Turbine[END_REF] and summarised in Fig. 1.

A five-hole probe located at 1 C x upstream of the leading edge of the blade (LE) measures the angle, stagnation temperature (with a thermocouple) and stagnation pressure. Measurements are also done at 1/4 C x downstream of the trailing edge of the blade (TE). The static pressure is measured on the central blade at 6% channel height. 2. They are located at the same axial position (0.044 C x upstream of the LE). The seal of the first one is orthogonal to the hub of the channel and has sharp edges; the latter is tangential and has a more complex geometry. The compound cavity outlet is not straight; the flow has to cross the fillet obstacle and bend for reaching the main channel. The chute of the cavity is the only geometrical difference between both configurations. The terms used in the following sections are defined in Fig. 3.

NUMERICAL SIMULATIONS

The computational domain has been reduced to one blade considering that the flow is laterally periodic. The domain starts at 1 C x upstream of the LE and ends at 1 C x downstream of the TE. The experimental stagnation pressure P i,1 , stagnation temperature T i,1 and flow angle are imposed at the inlet, and the exit pressure corresponds to the experimental wall static pressure. The purge mass flow is directly imposed at the cavity inlet so that it represents 0% (no purge flow), 0.5% and 1.0% of the channel mass flow. As in the experiment, the purge flow is injected with an angle of 45°in the stream-normal plane (Fig. 3) and is 30 K cooler than Finite volume elsA solver [START_REF] Cambier | The Onera elsA CFD Software: Input from Research and Feedback from Industry[END_REF] has been used for the steady RANS simulations. The Roe upwind scheme [START_REF] Roe | Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes[END_REF] with a second order van Albada flux limiter [START_REF] Van Albada | A Comparative Study of Computational Methods in Cosmic Gas Dynamics[END_REF] have been used. The time integration scheme is the implicit Euler method. The assessed turbulence models are the k -l Smith, the k -ω SST Menter and the RSM SSG/LRR-ω.

The channel has been meshed following an "O-6H" structured topology (Fig. 4) created with Numeca AutoGrid software using hexahedral elements. The O-block allows to maximise the cells' orthogonality. The H-block upstream of the LE is very small. It has been limited by the presence of the upstream hub cavity. The minimum angle between two contiguous face of the cells (orthogonality) is 30°. For the same configuration, Fiore [START_REF] Fiore | Influence des Ecoulements de Cavité Inter-disque sur l'Aérodynamique d'une Turbine[END_REF] has worked with a mesh of 30°minimum orthogonality. The fully hexahedra mesh constitutes the reference configuration without cavity.

For the configurations with a cavity, the same channel mesh is used, and to account for the cavity, tetrahedral elements were used in it, in the chute and nearby in the channel area. The use of tetrahedral elements offers more flexibility and allows local refinement without modifying the channel mesh. Figure 5 shows the grids in which the two parts are visible: the channel and The experimental configuration has a high external turbulence level which needs be modelled in the numerical simulations. The turbulence intensity measured upstream of the LE is 6%. A turbulent dissipation length (8 mm) consistent with the experimental turbulence grid is used in the simulations. For eddy-viscosity models, this leads to a large value of eddy viscosity (µ t /µ=1150) at the inlet of the computational domain. Yet, such high values of eddy viscosity for Boussinesq models result in excessive turbulent diffusion increasing the mixing. Figure 6 shows that the k -ω SST model (Tu=6%) reduces the gradients of stagnation pressure losses (ζ) and overpredicts the losses by a factor of 2, while the SSG/LRR-ω model predicts losses close to the measurements. When the external turbulence rate is reduced (Tu=1%), the effect of the turbulent diffusion for the k -ω SST model is also decreased. Indeed the loss peaks predicted by the k -ω SST model (Tu=1%) at 30% and 70% channel height are less spread with reduced losses over the entire channel height. Only the SSG/LRR-ω model is able to accurately account for the high external turbulence level without altering the behaviour of the mean flow. Unlike Boussinesq models, which model a turbulent viscosity (µ t ) increasing the diffusion term in the RANS momentum equations, RSM models do not incorporate turbulent viscosity and instead model each Reynolds stress component separately.

In a Boussinesq model, the decay of turbulence quantities is given by [START_REF] Spalart | Effective Inflow Conditions for Turbulence Models in Aerodynamic Calculations[END_REF]:

k k Freestream = 1 + (C ε2 -1) ε k Freestream x U -1 C ε2 -1 (1a) ν t ν t,Freestream = 1 + (C ε2 -1) ε k Freestream x U C ε2 -2 C ε2 -1 (1b)
with a constant U velocity, ε represents the turbulence dissipation, ν t = µ t /ρ, C ε2 = 1.92. The decay of k is faster than that of ν t (exponent in Eq.1a is -1.1 vs. -0.09 in Eq.1b). The high level of turbulent viscosity injected at the inlet of the domain persists throughout the computational domain due to this slow decay. It actively participates in the diffusion of the secondary structures, leading to increased aerodynamic losses as shown in Fig. 6. Therefore, the external turbulence will be ignored in simulations with eddy-viscosity models (k -l and k -ω SST) in the present work. 2 This simplification is not expected to change the major conclusions concerning the comparison between the models. The purge flow has an essential role in sealing the cavity from the hot air. It also feeds vortex structures in the inter-blade region. Figure 7 shows the amplification of the near-hub passage vortex compared to near-shroud regions with isovalue surfaces of Q-criterion. The near-hub structure is larger than the one at the top of the blade which is far from the purge flow. The passage vortex is created at the LE and migrates towards the suction side of the neighbouring blade under the effect of the pressure gradient. The recirculation visible after the inner-cavity fin reduces the vena contracta and thus helps in sealing the cavity.

Flow topology

Purge flow influence

The influence of the purge flow has been investigated through the variation of its mass flow in the axial configuration. The pressure losses at the exit of the cascade, the cavity sealing and the influence of the purge on the flow around the blade have been scrutinised. The stagnation pressure maps at 1/4 C x downstream of the TE are gathered in Fig. 8. The losses appear in green and blue in areas that correlate with the wake of the blade and different vortex structures that develop in the channel inter-blade zone. Three distinct structure footprints are visible in the measurements, which are attributed to the wall (A), passage (B), and corner (C) vortices [START_REF] Schuler | Investigation of the Influence of Different Rim Seal Geometries in a Low-Pressure Turbine[END_REF]. In the reference purgeless case, the passage vortex is the biggest among these structures. The Purge = 0% Purge = 1% Exp.

k -l k -ω SST SSG/LRR-ω Figure 8: Stagnation pressure (P i -P i,amb ) map at 1/4 C x downstream of the TE (axial chute) Purge = 0.5% Purge = 1% k -l k -ω SST SSG/LRR-ω Figure 9: Iso-surface of stagnation temperature θ below mainstream temperature (θ = 0.85) colored by (P i,inlet -P i ) (axial chute) presence of the cavity amplifies this vortex, causing the footprint of the pressure losses in the lower part of the channel to be greater than the upper part. When a purge flow is injected (purge mass flow = 1%), this vortex is further amplified. Apart from the passage vortex, the rest of the flow seems to be undisturbed by the purge flow whose influence on the pressure field is limited to the lower 40% of the channel. The simulations with the three turbulence models show the same trend but with overestimating losses in the core of the vortices. The same behaviour is also shown by Franc ¸ois and Gand [START_REF] Franc | Hybrid RANS/LES Simulations of Secondary Flows in a Linear Cascade of Turbine Blades with External Turbulence[END_REF] with a Spalart-Allmaras turbulence model. They also show that a hybrid RANS/LES simulation reduces significantly the overestimations.

It is worth noting that while the k -l model predicts a slight increase in the passage vortex related loss, the two other models predict a footprint which is in better agreement with the measurements (Fig. 8). The purge flow feeds the passage vortex and due to its radial migration the losses are not confined near the hub. Figure 9 shows how the flow exiting the cavity contributes to the amplification of the passage vortex. An iso-surface of temperature θ = (T i -T i,leakage )/(T i,∞ -T i,leakage ) reveals how far the penetration of the cooling purge flow disturbance in the channel goes (θ = 1: channel flow temperature, and θ = 0: cavity temperature). Due to the potential effect of the blade LE, the flow exits the chute periodically in the channel inter-blade area. Once out, the purge flow is mixed with the near-wall low velocity passage vortex which amplifies with the increasing purge. This structure is then pushed towards the suction side of the blade by the pressure differential. As opposed to the k -ω SST and SSG/LRR-ω models, the k -l model predicts a contribution of the purge flow on the development of the passage vortex that is only slightly modified when the mass flow increases. The k -ω SST model predicts a thinner temperature isovalue surface but with a more intense pressure loss. This analysis must be verified by the level of mixing in the cavity. Indeed, the iso-surfaces presented in Fig. 9 are highly dependant on the cavity and chute temperature mixing.

To better understand the differences in the development of the passage vortex, an investigation of the flow near the chute exit is necessary. For that purpose, Fig. 10 induces the ingress in the LE plane which is predicted by all of the models. However, the level of ingress seems to be higher for k -ω SST and SSG/LRR-ω where the hot air penetrates deeper and the temperature mixing occurs in the entire cavity, while this happens mainly in the nearchute area with the k -l model. Therefore, the temperature isovalue surface in Fig. 9 is thicker for the k -l model: the temperature mixing continues out of the cavity. One explanation of this phenomena lies in the recirculation that is present in the outlet of the chute (Fig. 10). This vortex structure limits the flow crossing the chute from the channel to the cavity. The creation of this vortex structure can be attributed to the production of turbulent kinetic energy in the shear layer at the intersection of the two flows. Uncu et al. [START_REF] Uncu | Assessment of RANS Turbulence Models on Simplified Geometries Representative of Turbine Blade Tip Shroud Flow[END_REF] found a similar behaviour of overproduction of TKE for the k -l model in a jet in crossflow. It is worth mentioning the inlet turbulence decay for the RSM model in the channel which is not visible for the two other models according to the modelling choice. The purge flow also changes the mean angle of the flow coming to the blade. This angle modification disrupts the velocity triangle and the pressure distribution. The predictions of the pressure field, C p = (P i,1 -Ps(x)) /(P i,1 -P s,2 ), around the blade in the near-hub region (6% channel height) are presented in Fig. 11. At the pressure side, with no purge flow, all the models predict a pressure distribution close to the experiment. At the suction side, they predict a decrease of pressure higher than the measurements from the mid-chord. The distribution of pressure is all the more discriminatory when purge flow is injected (Fig. 11-b). The ingress and egress topology are different between the k -l model and the other simulations. As a consequence, the work on the blade is modified and different pressure distributions are noticeable from the LE on both sides. More precisely, a pressure decrease at the pressure side between x/C x = 0.2 and x/C x = 0.5 is visible with this model, and the pressure is also underestimated at the suction side from x/C x = 0.3 which corresponds to the location where the purge-amplified passage vortex migrates towards the blade suction side. The SSG/LRR-ω model is the one that fits the measurements best, both on suction and pressure sides.

Cavity geometry influence

An increased purge flow improves the chute sealing at the expense of additional aerodynamic losses. A better sealing is possible by changing the chute geometry and maintaining the same purge mass flow. The simulation results from the two geometries presented in Fig. 2 are now compared. Figure 12 shows the ingress and egress on the LE and inter-blade planes. The temperature field shows that the level of ingress is very low for the compound geometry. Even at the LE plane, where the pressure potential field pushes mainstream flow into the cavity for the axial configuration, the recirculations created at the compound chute exit blocks the passage. The same phenomenon (recirculation inside the seal) deeper in the compound chute occurs in the inter-blade plane. The predictions of the structures for the compound case Axial Compound k -l k -ω SST SSG/LRR-ω Figure 12: Streamlines and temperature, θ, in the cavity exit region (purge flow = 1%) are very similar for the three models. The axial configuration, which does not have an inner fin, is exposed to the mainstream flow. Levels of ingress and egress are stronger. Another difference lies in the distance of radial penetration of the purge jet in the channel. Figure 13 shows that the smooth angle between the compound chute and the hub helps realign the streamlines in the direction of the mainstream flow, while for the axial case the penetration is deeper. The k -ω SST and SSG/LRR-ω models predict a recirculation downstream of the platform edge in the inter-blade plane of the axial cavity (Fig. 12). The ingress/egress map presented in Fig. 14 shows that the amplitude of flow entering the cavity is quite high in the axial case. The redirection at the compound chute exit decreases the effect of the potential field. Except the k -l model, which shows less egress in the upstream part of the compound chute, the Axial Compound models predict the same quantities and regions where the flow is entering and exiting the compound chute. The main differences occur for the axial geometry as discussed in the precedent section. For the axial geometry, the k -l model predicts an egress which is inferior to the two other models (Fig. 14). This is due to the lower ingress in the LE plane (Fig. 10) and the conservation of mass.

Axial

Compound k -l k -ω SST SSG/LRR-ω Figure 14: Cavity chute ingress (blue) / egress (red) (normalised normal velocity) upstream of the LE (purge mass flow = 1%)

The difference in the topology of the jet exiting the chute also modifies the secondary structures created in the inter-blade area. An increased size of the passage vortex is expected for the axial geometry according to the egress map. Consequently, the aerodynamic losses that are shown in Fig. 15, and defined as ζ = (P i,1 -P i,2 ) /(P i,2 -P s,2 ), are higher for the axial case. The peak at 30% height corresponds to the passage vortex (zone B in Fig. 8) and the peak at 40% to the wall vortex (zone A). These two vortices seems to be amplified when a cavity is present and when a purge flow is injected (Fig. 8). However, the difference between the two cavities in terms of aerodynamic losses lies only on the size of the passage vortex. The peak corresponding to the passage vortex radial location is bigger for the axial cavity. The cavity generates additional aerodynamic losses until 60% of channel height. Above, the channel remains undisturbed by the presence of the cavity. The k -l model predicts aerodynamic losses that are very similar for both of the cavity shapes. Its sensitivity towards the chute geometry is absent. Indeed, the k -l model does not predict an amplification of the passage vortex when an axial chute is

CONCLUSIONS

Two cavity chute geometries and three purge mass flows have been simulated with three different RANS turbulence models: the SSG/LRR-ω, k -l and k -ω SST. The high level of external turbulence in the configuration discriminates the two eddy-viscosity models which predict too much turbulent diffusion causing an overestimation of pressure losses. The external turbulence rate is accounted for only with the SSG/LRR-ω model. All the utilised models predict the increase in the passage vortex size when a cavity with purge is present. The main differences between the models appear at the exit of the axial chute where an extra vortex structure is predicted by the k -l model, which modifies the ingress/egress topology. The pressure distribution around the blade predicted by the same model does not agree well with the measurements. Only the SSG/LRR-ω model is able to capture the difference of losses between an axial and compound cavity. While all the models show that the ingress is eliminated with the compound chute, the two eddy-viscosity models are not able to predict the benefit of this geometry in terms of aerodynamic losses. The non-linear RSM SSG/LRR-ω model seems to be better-suited for simulating purge/mainstream flow interactions. In the future, it would be worth assessing the SSG/LRR-ω model in a real turbine configuration where the flow is compressible and rotational stress is created by the rotating cavity chute walls.

Figure 1 Figure 3 :

 13 Figure 1: Rig instrumentation Figure 2: Simulated cavities

Figure 4 :

 4 Figure 4: Channel grid with hexahedra elements

Figure 5 :Figure 6 :

 56 Figure 5: Hybrid mesh of the linear cascade with (a) axial and (b) compound cavity

Figure 7 :

 7 Figure 7: Q-criterion (SSG/LRR-ω)

Figure 10 :

 10 Figure 10: θ and normalised TKE in the LE plane of the cavity (axial chute, purge flow = 1%)

Figure 11 :

 11 Figure 11: C p around the blade at 6% channel height for a purge mass flow of (a) 0% and (b) 1% (axial chute)

Figure 13 :

 13 Figure 13: Streamlines at chute exit interblade area (purge flow = 1%, SSG/LRR-ω)

Figure 15 :

 15 Figure 15: Radial profiles of ζ with the (a) k -l, (b) k -ω SST and (c) SSG/LRR-ω model for the configurations w/o, axial and compound cavities (purge flow = 1%)

Main Annulus Gas Path Interactions : https://cordis.europa.eu/project/id/30874

The authors wish to indicate that slight anomalies were observed locally at the inlet and outlet (unexplained rise of ∼ 10 Pa) of the computational domain with the RSM model for large turbulence rate. Besides these numerical artefacts, that remain to be understood, the authors are confident with the numerical results with this model.

ACKNOWLEDGEMENTS

The authors wish to thank Georges (Sorbonne University), and Jacques Demolis (Safran Helicopter Engines) for their help in the analysis of turbulence modelling impacts.


Juan Calvo
email: juancalvo@ugr.es

Erwan Hingant
email: erwan.hingant@u-picardie.fr

AND Romain Yvinec
email: romain.yvinec@inrae.fr.

SOME REMARKS ABOUT THE WELL-POSEDNESS OF LIFSHITZ-SLYOZOV'S EQUATIONS WITH NUCLEATION KINETICS

The Lifshitz-Slyozov model is a nonlocal transport equation that can describe certain types of phase transitions in terms of the temporal evolution of a mixture of monomers and aggregates. Most applications of this model so far do not require boundary conditions. However, there is a recent interest in situations where a boundary condition might be needed -e.g. in the context of protein polymerization phenomena. Actually the boundary condition may change dynamically in time, depending on an activation threshold for the monomer concentration. This new setting poses a number of mathematical difficulties for which the existing literature is scarce. In this contribution we construct examples of solutions for which the boundary condition becomes activated (resp. deactivated) dynamically in time; we also discuss how to approach the well-posedness problem for such situations.

Introduction

The Lifshitz-Slyozov model describes the temporal evolution of a mixture of monomers and aggregates. This is done in terms of the monomer concentration u(t) and the number density for aggregates, f (t, x), where x is a size variable. The model reads ∂f (t, x) ∂t

+ ∂ ∂x [(a(x)u(t) -b(x))f (t, x)] = 0 , t > 0 , x ∈ (0, ∞) , (1.1)
u(t) + ∞ 0 xf (t, x) dx = ρ , t > 0 (1.2)
for some given ρ > 0, subject to the initial condition

f (0, x) = f in (x) , x ∈ (0, ∞). (1.3)
Here the transport term in (1.1) describes the changes in size aggregates according to the following interactions: (i) a monomer attaches to a pre-existing aggregate of size x, with a rate a(x), (ii) a monomer detaches from a pre-existing aggregate of size x, with a rate b(x). We interpret ρ > 0 as the total mass of the system, hence (1.2) encodes mass conservation during evolution.

The former model was initially introduced to describe phase transitions in supersaturated solid solutions [START_REF] Lifshitz | The kinetics of precipitation from supersaturated solid solutions[END_REF]. In this context we think of x as a volume variable and we take a(x)

x 1/3 , b(x) 1. More general kinetic rates can be considered and in this way we get a general framework that can be applied to describe various problems in the science of materials (concerning e.g. crystal precipitation or metal alloys, see for instance [START_REF] Friedman | A model of crystal precipitation[END_REF][START_REF] Tavare | Simulation of Ostwald ripening in a reactive batch crystallizer[END_REF]). For those descriptions x usually accounts for aggregate volume and the kinetic rates a, b are such that small aggregates tend to diminish in size. Actually, the standard dynamics in those scenarios resembles the so-called "Ostwald ripening" phenomena, where the growth of large aggregates is promoted thanks to the monomers gained from the shrinkage of small aggregates.

On mathematical grounds, those works analyzing Ostwald ripening phenomena in the literature assume from the start kinetic rates such that the transport term in (1.1) always points outwards at size x = 0 [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF][START_REF] Laurencot | Weak solutions to the Lifshitz-Slyozov-Wagner Equation[END_REF][START_REF] Niethammer | On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening[END_REF][START_REF] Niethammer | Well-posedness for measure transport in a family of nonlocal domain coarsening models[END_REF]. Therefore no boundary condition is needed to render the problem (1.1)-(1.3) well-posed. However, this need not be always the case. We discuss below -Proposition 2.6-a family of kinetic rates such that the transport term becomes characteristic at x = 0 during the evolution. This shows that in general we must be prepared for the possibility of having to prescribe a (dynamic) boundary condition in order to have a well-posed problem. The interesting point here is to relate the eventual boundary condition with relevant mechanisms of the process we are trying to describe.

One specific context where these ideas sit in naturally is that of protein polymerization phenomena, where we think of aggregates as linear chains of length x. To trigger the linear growth of a certain aggregate by monomer addition at both ends, a minimal stable structure is needed first to build upon it. We can represent the formation of such seeds (zero-size aggregates) via a nucleation step involving a few monomers, e.g. [START_REF] Oosawa | Thermodynamics of the Polymerization of Protein[END_REF][START_REF] Knowles | An Analytical Solution to the Kinetics of Breakable Filament Assembly[END_REF][START_REF] Szavits-Nossan | Inherent Variability in the Kinetics of Autocatalytic Protein Self-Assembly[END_REF]. Therefore, in this context the following boundary condition seems reasonable:

(a(0)u(t) -b(0))f (t, 0)χ {a(0)u(t)-b(0)≥0} = n(u(t))χ {a(0)u(t)-b(0)≥0} .
(1.4)

Here we think of n as a mass action kinetics, i.e. n(u) u n ; however, more general choices for n can be considered, which opens the room for additional applications of this description apart from those to protein polymerization.

To date, the literature on (1.1)-(1.3) with an inflow boundary condition like (1.4) is scarce. Some instances of this boundary condition have been deduced in [START_REF] Collet | The Becker-Döring system and its Lifshitz-Slyozov limit[END_REF][START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF]. The well-posedness for (1.1)-(1.4) has been treated in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF], where it is shown that only local-in-time solutions can be expected in the general case. This again is due to the fact that the transport field may change its character at x = 0 in a dynamic fashion, in such a way that a boundary condition may be needed at earlier times but not at later times.

Therefore, a global analysis of (1.1)-(1.4) seems necessary but is still lacking. The aim of this contribution is to give some partial results about its global wellposedness. Some of these results were announced without proof in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF].

Kinetic rates and local-in-time solutions

It is handy to discuss the properties of (1.1)- (1.3) in terms of the ratio between polymerization and depolymerization rates, which we now introduce. Definition 2.1. Assume that a(x) > 0 for a.e. x ∈ R + . We introduce

Φ : R + → R + , Φ(x) := b(x) a(x) .
Let Φ sup := sup x≥0 Φ(x) and Φ inf := inf x≥0 Φ(x). We also define

Φ 0 := lim x→0 Φ(x) ≥ 0.
We have that 0 ≤ Φ inf ≤ Φ 0 ≤ Φ sup ; note that Φ 0 , Φ sup need not be finite.

Well-posedness for (1.1)-(1.3) can be expected to be true if the boundary condition is always "on" (by which we mean a(0)u(t) -b(0) > 0 for every time) or always "off". However, temporal dynamics for which u(t) switches between these two regimes imply a change in the properties of the system. On modeling grounds, the physical meaning of this need not be clear a priori, depending on the specific features of the case under study. Besides, this switching may induce purely mathematical effects whose impact on the well-posedness of the model is not clean cut at first sight. Our investigation here tries to shed some light on this problem. In that regard, the quantity Φ 0 will constitute an important threshold as it will separate two different dynamic regimes.

To state our results we should make clear what do we mean by solutions to (1.1)- (1.3). We need to consider several notions of (weak) solutions. Those are readapted from [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF]. Definition 2.2 (Inflow and outflow solutions). Let T ∈ (0, ∞]. We say that a function f is an inflow solution with nucleation kinetics (resp. outflow solution) to the Lifshitz-Slyozov equation (1.1)-(1.2) on [0, T ) with mass ρ > 0, kinetic rates {a, b, n} (resp. {a, b}) and initial value f in ∈ L 1 (0, ∞) + if the following statements are satisfied:

(1) The function f belongs to C([0, T ), w-L 1 ((0, ∞), (1+x) dx)), is nonnegative and for each T * < T , it also belongs to L ∞ ((0, T * ), L 1 ((0, ∞),

(1 + x) dx)). (2) For every t ∈ [0, T ), u(t) := ρ - ∞ 0 xf (t, x) dx > Φ 0 .
(2.1)

(Resp. with < Φ 0 .) (3) For all ϕ ∈ C 1 c ([0, T ) × [0, ∞)), there holds that T 0 ∞ 0 (∂ t ϕ(t, x) + (a(x)u(t) -b(x))∂ x ϕ(t, x)) f (t, x) dx dt + T 0 ϕ(t, 0)n(u(t)) dt + ∞ 0 ϕ(0, x)f in (x) dx = 0 . (2.2) (Resp. for all ϕ ∈ C 1 c ([0, T ) × (0, ∞)), in which case the term in n is not present.)
One of our aims is to generalize these notions to the case in which an admissible solution can be constructed by merging outflow solutions with inflow solutions. We introduce the following concept. Definition 2.3 (Matching solutions). We say that a pair (u, f ) is a matching solution of (1.1)-(1.3) in [0, t * ) with nucleation boundary condition (1.4) and initial data (u in , f in ) if the following conditions hold:

(1) The set S := {t ∈ [0, t * )/u(t) = Φ 0 } is denumerable. Let (0, t * )\S = ∪ n∈N (t n , t n+1 ). (2) If u(t) > Φ 0 in (t n , t n+1 ) then f is an inflow solution of (1.1)-(1.4) in [t n , t n+1 ) with initial data (f (t - n ), u(t - n ))
-that is, we take f in when n = 0, otherwise we take f (t - n ) as the limit when t t n in the weak topology of 4) u(t) is continuous at each point of S and f (t) is also continuous at each point of S, with respect to the weak topology of L 1 ((0, ∞), (1 + x) dx).

L 1 ((0, ∞), (1 + x) dx). (3) If u(t) < Φ 0 in (t n , t n+1 ) then f is an outflow solution of (1.1)-(1.3) in [t n , t n+1 ) with initial data (f (t - n ), u(t - n )). (
Next we state our hypotheses on the initial datum and the kinetic rates. We will not be aiming at full generality in this contribution. Therefore, we shall adopt the following running assumptions:

(1) f in ∈ L 1 ((0, ∞), (1 + x)dx) + and u in := ρ - ∞ 0 x f in (x) dx ∈ (0, ρ],
(2) The function Φ(x) := a(x)/b(x) is defined for a.e. x > 0 and has a limit Φ 0 ∈ [0, ∞] as x → 0 + . (3) Attachment and detachment rates are globally Lipschitz and positive for x > 0. As a consequence, there holds that

a(x) ≤ A(1 + x) for some A > 0. (2.3) (4) n ∈ L ∞ ([Φ 0 , ρ]) + .
A number of additional assumptions will be required at several points of the document in order to construct specific examples:

There exist x L , C > 0 such that a(x)(ρ -Φ(x)) ≤ C for x ≥ x L , (2.4)
a(x) ≥ a > 0 for every x ≥ 0.

(2.5)

Here (2.4) is a way to state that for large sizes attachment does not overwhelm detachment, whereas (2.5) guarantees that we have a net attachment rate even for small aggregates.

A suitable adaptation of the results and proofs in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF][START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF] provides us the following local existence result. Proposition 2.4. Under our running assumptions, the following statements hold true:

(1) Let u in > Φ 0 . Then, there exists some t * > 0 such that there is a unique inflow solution with nucleation kinetics of (1.1)-(1.3) on [0, t * ).

(2) Let u in < Φ 0 (or u in = Φ 0 with (u in ) < 0). Then, there exists some t * > 0 such that there is a unique outflow solution of (1.1)-(1.3) on [0, t * ).

Furthermore, the following statements are fulfilled in both cases:

(1) u ∈ C 1 ([0, t * )). Actually, the time derivative is given by the following expression:

du dt = ∞ 0 b(x)f (t, x) dx -u(t) ∞ 0 a(x)f (t, x) dx .
(2) Either t * = +∞ or t * < ∞ and lim t t * u(t) = Φ 0 .

(3) Provided that t * < ∞, then u (t) has a limit as t t * and f (t) has a limit as t t * in the weak topology of L 1 ((0, ∞), (1 + x) dx).

To investigate prolongability and matching solutions we shall first rule out situations where inflow solutions will surely not take place. The following provides a partial generalization of similar results in [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF][START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF]: Lemma 2.5. Assume that the initial pair (u in , f in ) is such that ρ < Φ inf . Then any outflow solution of the Lifshitz-Slyozov model is global in time. Furthermore, under any of the following conditions the outflow solution verifies that u(t) ρ as t → ∞:

(1) a(x) ≥ a 0 x for some a 0 > 0 and for every x ∈ [0, ∞).

(2) f in is compactly supported and (2.5) holds.

Proof. Solutions are global in time as u(t) cannot reach Φ 0 . Let us discuss now the long time behavior. For the first case we use that

u = ∞ 0 a(x)(Φ(x) -u(t))f (t, x) dx ≥ (Φ inf -ρ) ∞ 0 a(x)f (t, x) dx .
This provides both the time integrability of the first moment and the monotonicity of u. Thanks to the time integrability we can find a divergent sequence of times along which the first moment converges to zero. Therefore u converges to ρ as t → ∞ thanks to (1.2).

For the second case we use the representation of outflow solutions in terms of characteristics, see for instance [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF]. Under our hypotheses, the time derivative of the characteristic curve with data (t, x) can be estimated from above as d ds X(s; t, x) = a(X(s; t, x))(u(s) -Φ(X(s; t, x)))

≤ (ρ -Φ inf )a(X(s; t, x)) < a(ρ -Φ inf ) < 0 .
Then for a given datum (t, x) the associated characteristic curve leaves [0, +∞) at some finite instant t > t. Therefore the aggregate distribution f vanishes, actually in finite time.

For more general sets of kinetic rates, we cannot presently rule out concentration behavior at zero size when ρ < Φ inf -think for instance in a(x) = x α with α ∈ (0, 1). See [START_REF] Calvo | Long-time asymptotics of the Lyfshitz-Slyozov equations with nucleation[END_REF] for a more detailed investigation of this pathological phenomenon. At any rate, we will impose the condition Φ 0 < ρ in the sequel, in order to allow for inflow solutions with nucleation kinetics.

Next we state and prove a result concerning local solutions of the Lifshitz-Slyozov model in the outflow case. We provide a family of kinetic rates for which such local solutions cannot be extended to a global outflow solution. Thus, not every initial configuration that starts to evolve as an outflow solution will undergo an Ostwald ripening dynamics right away. Even in those cases in which that turns out to be true, a transient with inflow dynamics might take place before that. A restricted instance of this result was announced in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF]. Proposition 2.6. Assume that lim inf x→∞ Φ(x) > Φ 0 = Φ inf > 0 and that Φ inf is only attained at x = 0. Assume also that Φ (0) > 0. If f in is compactly supported, (2.4) and (2.5) hold and u in < Φ 0 < ρ then every outflow solution of (1.1)-(1.3) is local in time.

Proof. Thanks to the hypotheses we can find a concave, strictly increasing profile Φ c such that Φ(x) ≥ Φ c (x) for every x ∈ [0, ∞) and Φ c (0) = Φ 0 . Then, notice that

u ≥ ∞ 0 a(x)(Φ c (x) -u(t))f (t, x) dx .
(2.6)

Since Φ c is concave we have

Φ c (x) ≥ Φ(0) + Φ c (z) -Φ(0) z x, x ∈ [0, z) (2.7)
-recall that Φ(0) = Φ inf = Φ 0 . Now we shall choose z taking into account that we can control the spatial support of the solution. Namely, if supp f in ⊂ [0, f in ) then thanks to (2.4) there exists C > 0 such that supp f (t) ⊂ [0, f in + tC] for every t ≥ 0 in the life span of the solution. That enables us to use (2.7) with z(t) = f in + tC.

It follows that u ≥ (Φ 0 -u(t))a z(t) 0 f (t, x) dx + a Φ c (z(t)) -Φ 0 z(t) (ρ -u(t)) .
(2.8)

Note that both terms at the right hand side are positive. We drop the first one; to handle the second we argue by contradiction and assume that u(t) < Φ 0 for every time. In that way we get to

u(t) ≥ u in + (ρ -Φ 0 )a t 0 Φ c (z(τ )) -Φ 0 z(τ ) dτ .
The former integral is strictly positive, and so is ρ -Φ 0 by assumption. Since

|Φ c (z(t)) -Φ 0 | ≥ |Φ c (z(0)) -Φ 0 | the time integral diverges logarithmically.
Then u(t) reaches Φ 0 in finite time, say u(t * ) = Φ 0 . Moreover, from (2.8) we can also deduce that lim t t * u (t) > 0.

Prolongability and global solutions

Our next result gives a sufficient condition to merge inflow solutions with nucleation kinetics and outflow solutions. Proposition 3.1. Given u in = Φ 0 , assume that u (0) < 0 (resp. u (0) > 0). Then there is some δ > 0 such that (1.1)-(1.3) has an outflow solution (resp. (1.1)-(1.4) has an inflow solution with nucleation kinetics) for t ∈ (0, δ), agreeing continuously with the initial data and such that u(t) < Φ 0 (resp. u(t) > Φ 0 ) for t ∈ (0, δ).

Proof. Let us deal first with the case of negative starting derivative. We consider the auxiliary problem given by (1.3) coupled with ∂f ∂t

+ ∂ ∂x [((Φ 0 -|u(t) -Φ 0 |)a -b)f ] = 0, (3.1)
in [0, 1) × R + , say. Note that for this problem the characteristics at x = 0 are never pointing inwards, so there is no need for a boundary condition. Having Lipschitz rates grants uniqueness of the characteristic curves; moreover, it enables us to apply a suitable modification of Proposition 2.4 for the above flow in the specific case u in = Φ 0 . This provides us with some δ > 0 -which can be estimated from f, a and b-such that: (i) there is a pair u ∈ C 1 (0, δ) and f ∈ C([0, T ), w -L 1 ((0, ∞), (1 + x) dx)) which matches continuously the initial data, (ii) the pair solves (3.1) coupled with (1.2) and (1.3), (iii) the given solution satisfies u(t) < Φ 0 on (0, δ). Then, we notice that on (0, δ) problems The proof for the second claim is nearly the same as for the first one. This time we consider the following auxiliary problem:

∂f ∂t + ∂ ∂x [((Φ 0 + |u(t) -Φ 0 |)a(x) -b(x))f (t, x)] = 0, (3.2)
for which characteristics at zero are never pointing outwards. This is complemented with (1.2) and the following boundary condition replacing (1.4):

((Φ 0 + |u(t) -Φ 0 |)a(0) -b(0))f (t, 0) = n(u(t)) .
The remaining steps can be performed as before.

There are some particular cases in which we can show a priori that t * = +∞ in Proposition 2.4 based on the information we have about the temporal evolution of the monomer concentration. The existing mathematical results for Ostwald ripening dynamics [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF][START_REF] Laurencot | Weak solutions to the Lifshitz-Slyozov-Wagner Equation[END_REF][START_REF] Niethammer | On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening[END_REF][START_REF] Niethammer | Well-posedness for measure transport in a family of nonlocal domain coarsening models[END_REF] are based on a set of assumptions on the rates that guarantee that the transport field will point outwards for every positive time. Here we provide another result in the same line but with a different way of formulating the outflow assumptions. Proposition 3.2. Assume that Φ 0 = Φ sup . If u in < Φ 0 then the outflow solution of (1.1)-(1.3) satisfies t * = ∞ and thus it is globally defined.

Proof. This is clearly true in the case Φ 0 = Φ sup > ρ, since mass conservation implies that lim t t * u(t) = Φ 0 will never happen. Therefore, we focus on the case Φ 0 = Φ sup ≤ ρ. Let (f, u) be the solution pair provided by Proposition 2.4. For any t < t * we have that

d dt (u(t) -Φ sup ) = u (t) ≤ (Φ sup -u(t)) ∞ 0 a(x)f (t, x) dx.
This implies that

u(t) -Φ 0 ≤ u in -Φ 0 exp - t 0 ∞ 0 a(x)f (τ, x) dxdτ ≤ u in -Φ 0 exp -At ρ + ∞ 0 f in (x) dx < 0.
Thus, if we assume t * < ∞ we arrive to the fact that lim t t * u(t) < Φ 0 , which is a contradiction. Therefore we must have t * = ∞.

The proof of the former result is very similar to that of the dual statement for inflow solutions with nucleation kinetics given in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF], Theorem 1.9; we recall it here for the reader's convenience:

Proposition 3.3. Assume that Φ 0 = Φ inf . If u in > Φ 0 then the inflow solution of (1.1)-(1.
3) with nucleation kinetics (1.4) satisfies t * = ∞ and thus it is globally defined.

3.1. Hints on finite-time crossings. The purpose of this paragraph is to analyze some situations for which local-in time solutions can be extended to global matching solutions. Intuitively speaking, a sufficient condition for prolongability would be as follows: u(t) should reach Φ 0 with non-vanishing derivative. Let us show that this is enough to extend those local solutions constructed in Proposition 2.6, which exhibit a change from outflow to inflow behavior. This was announced in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF] and the proof follows similar ideas to those used to prove Theorem 1.9 in that reference. Proposition 3.4. Under the same assumptions of Proposition 2.6, there is a matching solution in the sense of Definition 2.3 which is globally defined and such that S is a singleton and lim inf t→∞ u(t) ≥ Φ 0 .

Proof. Proposition 2.4 ensures that there is an outflow solution in some nonempty time interval [0, t * ) in the sense of Definition 2.2, with initial pair (u in , f in ). Actually, t * < ∞. In fact, Proposition 2.6 shows that lim t (t * ) -u(t) = Φ 0 , lim t (t * ) -u (t) < 0.

Next we use Proposition 3.1 with the initial pair (u((t * ) -), f ((t * ) -). This produces an inflow solution with nucleation kinetics in some nonempty interval [t * , T ). Here T can be made arbitrary large after Proposition 3.3. Therefore, the concatenation of the outflow solution in [0, t * ) and the inflow solution with nucleation kinetics in [t * , ∞) produces a globally defined matching solution.

Remark 3.5. The matching solution constructed above is actually unique. This is due to the fact that matching solutions as introduced in Definition 2.3 are actually such that u(t) is a C 1 function of time, check Proposition 2.4. However, this does not suffice to disregard nonuniqueness phenomena in a general scenario, where we have to consider the possibility that u (t) might vanish as we approach t * .

For completeness we state a dual result, which covers the case of some solutions whose behavior changes from inflow to outflow. This can be proved using the same techniques as the previous proposition and extends the scope of the result already given in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF].

Corollary 3.6. Assume that lim sup x→∞ Φ(x) < Φ 0 = Φ sup < ∞ and that Φ sup is only attained at x = 0. Assume that (2.4) is fulfilled. If f in is compactly supported and u in > Φ 0 then there is a matching solution of (1.1)-(1.4) which is globally defined and such that S is a singleton and lim sup t→∞ u(t) ≤ Φ 0 .

  (1.1)-(1.2)-(1.3) and (3.1)-(1.2)-(1.3) are equivalent for this solution, thus proving our first claim.

Acknowledgments

J. C. is partially supported by the spanish MINECO-Feder (grant RTI2018-098850-B-I00) and Junta de Andalucía (grants PY18-RT-2422 and A-FQM-311-UGR18). E. H. and R. Y. have been supported by ECOS-Sud project n. C20E03 (France-Chile) and acknowledge financial support from the Inria Associated team ANACONDA.