Xiang Hiba Dandachi

Farah Hong

Houssein Ibrahim

Anaïs Nasrallah

Nada Zulauf

Mohamed Jaber

Emmanuelle Mellah

Schulz

Xiang Hong

Farah Ibrahim

Houssein Nasrallah

Anaïs Zulauf

Nada Jaber

Mohamed Mellah

Emmanuelle Schulz
email: emmanuelle.schulz@u-psud.fr

Chiral salen complexes in polymeric main-chains for heterogeneous asymmetric catalysis -a brief account

Keywords: Asymmetric catalysis, salen complexes, polymers, macrocycles, catalyst recycling

come

INTRODUCTION

Performing organometallic asymmetric catalysis is one of the most efficient routes to produce valuable scalemic compounds in an atom economic manner, as demonstrated by its tremendous development, not only in the academic but also in the industrial sphere, since its discovery there are about now fifty years. [START_REF] Nozaki | Asymmetric induction in carbenoid reaction by means of a dissymmetric copper chelate[END_REF] Chiral ligands of salen types played a major role in this domain since, associated to numerous different metals, they were able to catalyze a large variety of asymmetric reactions, allowing formation of C-C, C-heteroatom or heteroatom-heteroatom bonds. [2] In order to go further in respecting the concepts of green chemistry, immobilization of these efficient enantiopure salen complexes was also the subject of in-depth studies aimed both at an easy recovery of the still precious catalysts and at the same time elimination of potentially toxic metal traces in the targeted products. [3] Furthermore, and in a number of reactions, the excellent performances obtained by salen complexes were due to bimetallic cooperativity of two metallic centers, activating respectively the electrophile and the nucleophile. [START_REF] Nielsen | Mechanistic investigation leads to a synthetic improvement in the hydrolytic kinetic resolution of terminal epoxides[END_REF] In such a context, many efforts have been turned towards heterogeneization methods for which salen complexes were placed in close proximity to favor an ideal location of the active sites for efficient dual activation. Modifications of the ligand structure were thus performed at some distance from the metallic coordinating sites, leading to monomers suitable for polymerization within the main chain. Some examples have been described affording linear coordination polymers or, more recently, metal-organic frameworks including enantiopure salen complexes for asymmetric tandem catalysis. [5] Moreover, and due to the inherent structure of the salen ligand, the most developed procedure for polymerization involves polycondensation reactions, via multiple imine bonds synthesis, between enantiopure diamine and suitably modified disalicylaldehyde derivatives. This field was particularly explored by the group of Kureshy et al. [6] who prepared linear salen containing polymers but also macrocycles, following the seminal work of Jacobsen's group, [7] and studied diverse catalytic reactions for the preparation of chiral drugs. We reported another type of polymerization leading to linear structures that involved either electrochemical or chemical oxidation of suitable salen complexes modified by thiophene groups, as chiral heterogeneous catalysts efficient for diverse reactions. [8] We propose in this account to present our best results concerning the synthesis of enantiopure salen complexes in polymeric main-chains and their activity in various asymmetric catalytic reactions, with a specific emphasis on their reusability and their cooperativity.

Salen-containing polymers by oxidative polymerization

Preparation of heterogenized chiral organometallic catalysts by electropolymerization has been rarely described; some metal containing chiral organic conducting polymers can be cited as electrocatalysts for asymmetric hydrogenation, epoxidation reactions [START_REF] Moutet | Heterogeneous and homogeneous asymmetric electrocatalytic hydrogenation with rhodium(III) complexes containing chiral polypyridyl ligands[END_REF] or even formation of C-C bonds by cyclopropanation of olefins. [START_REF] Ferrand | Asymmetric heterogeneous carbene transfer catalyzed by optically active ruthenium spirobifluorenylporphyrin polymers[END_REF] Towards our goal, we prepared new salen derivatives bearing electropolymerizable functionalities on the fifth position of the phenol rings, leading thus to catalysts with the metallic center directly incorporated into the conjugated polymer backbone. We chose introduction of thiophene moieties as a readily oxidizable monomer [START_REF] Shioya | A reversible resistivity-based nitric oxide sensor[END_REF] and the synthesis occurred straightforwardly via a palladium-catalyzed Suzuki coupling followed by condensation with a chiral diamine affording the desired ligands in good yield (see Scheme 1). [8a,12] Scheme 1. Synthesis of new thiophene-salen complexes and their electropolymerization. Metallic insertion could be performed from various precursors including chromium, nickel, copper, aluminum and cobalt salts, leading to isolation of corresponding complexes in high yields and purity. [8] Their oxidative electropolymerization was successfully performed in an undivided electrochemical cell at a platinum anode and a constant current with nBu 4 NBF 4 or nBu 4 NPF 6 as electrolyte. This resulted in each case in the formation of polymers exhibiting high electrochemical stability as quasi-reversible redox processes when submitted to voltammetric repeated scans, whatever the nature of the metal or the structure of the ligand. Electropolymerization on a platinum grid allowed the recovery of the polymeric species as powders from the electrode surface, which proved insoluble in common organic solvents. They were thus washed with various solvents and dried under vacuum before their use in asymmetric catalysis.

Chromium-containing thiophene-salen polymers have been specifically investigated because of the great versatility of these complexes as demonstrated by the group of Jacobsen. We thus initially investigated the asymmetric hetero Diels-Alder (HDA) [START_REF] Schaus | Asymmetric hetero-Diels-Alder reactions catalyzed by chiral (salen)chromium(III) complexes[END_REF] reaction, leading interestingly to the formation of both a C-C and a C-O bond. We explored the reaction between various aldehydes and 1methoxy-3-[(trimethylsilyl)oxy]-1,3-butadiene (Danishefsky's diene) implying first the thiophenesalen monomers under homogeneous conditions to optimize their efficiency. These catalysts promoted the reaction with high efficiency and enantioselectivity, albeit with lower ee values compared to the best ones previously reported in the literature. [START_REF] Zulauf | Chromium-thiophene-salen-based polymers for heterogeneous asymmetric hetero-Diels-Alder reactions[END_REF] The thiophene withdrawing substituent in the fifth position of the salen replacing the donating tert-butyl group is probably slightly detrimental to the enantiofacial discrimination. Reactions were then successfully conducted under heterogeneous conditions with the polymers, and for instance successive HDA reactions were performed by using cyclohexanecarbaldehyde, affording the targeted product in high yield and enantioselectivity, showing no loss of efficiency for up to 15 cycles (see Scheme 2). At the end of a first transformation, the product solution was indeed easily separated from the catalyst powder by simple filtration; this latter was washed several times with the reaction solvent, dried under vacuum and finally recycled by adding new substrates and solvent to the same reaction vessel. Scheme 2. Heterogeneous HDA reaction between cyclohexanecarbaldehyde and Danishefsky's diene promoted by a chromium-containing thiophene-salen polymer.

Furthermore, the same catalyst was engaged in an original multi-substrate procedure, hardly described in the literature, and used 20 times by adding a structurally different aldehyde in each recycling. Indeed at each catalysts reuse, different targeted pyranones were obtained, in their pure form, without being contaminated by the presence of products traces synthesized during the previous catalytic cycle, with activities and enantioselectivities remaining very stable. [START_REF] Zulauf | Chromium-thiophene-salen-based polymers for heterogeneous asymmetric hetero-Diels-Alder reactions[END_REF] This study was continued to examine the extent of the chromium-containing thiophene-salen polymers application, and we were delighted to show their efficiency to promote the Henry reaction, even if the number of effective recycling remained lower than in the previous example. [START_REF] Zulauf | New chiral thiophene-salen chromium complexes for the asymmetric Henry reaction[END_REF] The reaction occurred for instance between various alkyl or aryl aldehydes and nitromethane in the presence of substoichiometric amount of Hünig's base, and it produced the desired nitroaldols with enantiomeric excesses reaching up to 77% (see Scheme 3). Due to the great versatility of salen-chromium complexes, we were able to study other reactions with this heterogeneous catalytic system. Indeed, we also tested the reaction of epoxides opening by trimethylsilyl azides [START_REF] Martínez | Highly enantioselective ring opening of epoxides catalyzed by (salen)Cr(III) complexes[END_REF] and the addition reaction of dimethylzinc on aldehydes. [START_REF] Cozzi | Highly enantioselective addition of Me 2 Zn to aldehydes catalyzed by ClCr(salen)[END_REF] For both transformations, however, appearance of the expected products occurred but the results obtained in terms of both activity and selectivity values were greatly damaged compared to the ones obtained in the homogeneous cases. The mechanistic pathway, now wellestablished to account for the ring-opening of epoxides, may explain these results; as already mentioned, indeed, a bimolecular mechanism is evidenced involving two catalytic species in a cooperative way to activate both the epoxide ring and the nucleophile. [START_REF] Nielsen | Mechanistic investigation leads to a synthetic improvement in the hydrolytic kinetic resolution of terminal epoxides[END_REF][START_REF] Hansen | On the mechanism of asymmetric nucleophilic ring-opening of epoxides catalyzed by (salen)CrIII complexes[END_REF] The arrangement of the catalytic sites in the electrogenerated (poly)salen-thiophene chromium complex, in a pearl necklace manner showing restricted flexibility, is not favorable for this specified optimized position of the catalytic sites.

Nevertheless, and since the chromium-containing thiophene-salen polymers were able to promote diverse reactions as insoluble catalysts, we reported the first example of a recycling procedure for which the same catalyst batch was involved to promote different asymmetric catalytic reactions in an original successive manner. This multi-reaction procedure was tested by sequentially catalyzing the addition of Me 2 Zn, the HDA reaction, the Henry reaction, and finally the epoxides opening. This series was repeated to deliver exactly the same values than those obtained in the first initial runs, thus demonstrating the high stability and versatility of the new polymeric chromium catalyst (see Scheme 3). Scheme 3. Multi-reactions procedure promoted by a chromium-containing thiophene-salen polymer.

As synthesis of stable cobalt-containing polymers by electropolymerization was also successful, these new catalysts were likewise engaged in the well-known powerful hydrolytic kinetic resolution (HKR) of epoxides. [START_REF] Tokunaga | Asymmetric catalysis with water: efficient kinetic resolution of terminal epoxides by means of catalytic hydrolysis[END_REF] This reaction has been largely explored, also with active species allowing for easy catalysts recycling. Nevertheless and most frequently, reengaged catalyst batches had to undergo a reactivation step to keep their efficiency; the bimetallic activation indeed required triggering the epoxide by a Lewis acidic cobalt site (for instance a Co III -OAc), whereas a Co III -OH species, generated consequently to the nucleophilicity of the acetate counteranion, activates the water. By catalyst reuse, more and more essential Co III -OAc sites are transformed into Co III -OH ones, leading to deactivation with thus a mandatory catalyst treatment with acetic acid for an efficient further recycling. [START_REF] Nielsen | Mechanistic investigation leads to a synthetic improvement in the hydrolytic kinetic resolution of terminal epoxides[END_REF][START_REF] Jain | Importance of counterion reactivity on the deactivation of Co-salen catalysts in the hydrolytic kinetic resolution of epichlorohydrin[END_REF] The electropolymerization method solved this problem; the formation of the polymer was indeed performed from the direct oxidation of a Co II -thiophene-salen complex in a CH 2 Cl 2 /AcOH/nBu 4 NBF 4 electrolytic mixture. Under these conditions, the cobalt center was oxidized along with an increase in the peak currents in both the anodic and cathodic waves, consistent with an electroactive polymer growth at the electrode surface (see Scheme 4). [START_REF] Hong | Electrogenerated polymers as efficient and robust heterogeneous catalysts for the hydrolytic kinetic resolution of terminal epoxides[END_REF] EDS and elemental analysis of the obtained polymer were performed indicating representative signals for fluorine in the sample, proving that BF 4 -was introduced, to some extent, together with AcO -as axial ligands to the Co III center. As a consequence, the obtained polymer was less dependent on reactivation because of the presence of non-hydrolyzable Co III -BF 4 units. The benchmark reaction implied the dynamic HKR of epibromohydrin; [START_REF] Furrow | Practical access to highly enantioenriched C-3 building blocks via hydrolytic kinetic resolution[END_REF] as it readily underwent rapid racemization under mild conditions, it revealed to be an ideal substrate for studying the HKR reaction under heterogeneous conditions because a complete conversion could be achieved and the ee of the resulting diol did not vary during the resolution process. The electropolymerized catalyst afforded the expected diol (up to > 99 % yield and 87 % ee) and was easily recovered by filtration. It was effectively reused without the need for a reactivation step and was highly stable, even after eleven consecutive catalytic runs (See Scheme 4). In this case also, a multisubstrate recovery procedure was realized with various terminal epoxides, thereby demonstrating its usefulness for the transformation of diverse substrates.

Scheme 4.

Electrochemical polymerization of a cobalt-containing thiophene-salen polymer and use in the dynamic HKR of epibromohydrin.

Suitable chemical oxidative polymerization could also be performed for easily reaching larger quantities of these heterogeneous thiophene-salen polymers. For instance, FeCl 3 or NOBF 4 delivered insoluble polymeric species in nearly quantitative yield, from Co II -containing monomers, under an air atmosphere, or directly from Co III -precursors. [START_REF] Hong | Chemical-promoted oxidative polymerization of modified-cobalt salen complexes, efficient catalysts for the dynamic HKR[END_REF] Accordingly, the polymers proved efficient catalysts for the dynamic HKR of epibromohydrin, leading to the expected scalemic diol with high yield and enantioselectivity. The polymerized catalyst could be recovered and reused leading to stable values of ee but large variations in the conversion went along with the recycling, as a proof for its chemical alteration during the HKR. Thus a polymer containing both nucleophilic and non-nucleophilic counter anions was prepared by chemical co-polymerization in the presence of FeCl 3 , of a thiophene salen Co III -OAc complex and an analogous Co III -BF 4 complex (see Scheme 5). The resulting polymer was also recovered as an insoluble powder after methanol washings in 90% yield. Elemental analyses were performed for the determination of the Co, B and F contents, and the percentage of Co III -BF 4 in the copolymer was determined to be 49 %. Delightfully, excellent results were obtained in the benchmark reaction and the stability of this insoluble catalyst was demonstrated in a flow procedure. Immobilization occurred in the presence of octadecyl carbon chain-bonded silica in a reversed phase mode to facilitate release of the diol resulting from the HKR, thus preventing fast reactor saturation. The solid support-catalyst mixture was confined in a short HPLC-type pre-column that was used as a flow reactor. The reaction was realized under diluted conditions in THF to adopt the flow conditions. Under these conditions, the repetitive preparation of the scalemic diol with stable high enantiomeric excesses could be performed, with no more noticeable yield fluctuation. In this case, since the polymer catalyst possesses both nucleophilic and non-nucleophilic active sites, a deactivation is presumably prevented to allow its repeated efficient use. Chemical oxidative polymerization has been thus demonstrated as a powerful and practical procedure for the preparation of chiral insoluble cobalt salen complexes in high yield.

Scheme 5. Chemical oxidative procedure for the preparation of a cobalt-containing thiophenesalen polymer engaged in a flow procedure.

Salen-containing polymers by polycondensation

already mentioned, insertion of salen complexes directly in the polymer main-chains targeted not only the preparation of catalysts with easily controlled solubility but also the close location of the active sites. This strategy on the other hand was to make it possible to obtain catalytic solids with high density of active sites, avoiding the use of large masses, usually necessary for catalysts supported on organic or inorganic solids. Following our results obtained by oxidative polymerization leading to linear polymers, we envisaged synthesizing oligomeric salen complexes through multiple imine bonds formation between structurally diverse salicylaldehyde derivatives and chiral diamines. Towards this aim, and for matching at best the polymers structures that were prepared through oxidation, a thiophenedisalicylaldehyde was prepared in two steps (see Scheme 6). Subsequent condensation with an enantiopure diamine yielded a polymeric compound which was demonstrated to possess a cyclic structure with two to five repetitive units. [START_REF] Zulauf | New chiral calixsalen chromium complexes: recyclable asymmetric catalysts[END_REF] This polymer was analyzed by [START_REF] Nozaki | Asymmetric induction in carbenoid reaction by means of a dissymmetric copper chelate[END_REF] H NMR but also MALDI-TOF spectroscopy, proving undoubtedly the presence of different macrocycles with the loss of a water molecule. The mixture reacted with chromium salts to deliver calixsalen catalysts that were tested for their ability to promote asymmetric catalysis. Scheme 6. Synthesis of thiophene-salen chromium calix-catalysts.

With this insoluble powder, the nucleophilic ring opening of epoxides with trimethylsilylazide [START_REF] Martínez | Highly enantioselective ring opening of epoxides catalyzed by (salen)Cr(III) complexes[END_REF] was studied under heterogeneous conditions; this reaction was chosen as benchmark test for studying the influence of the position of the catalytic sites to each other, as a critical feature for an efficient cooperative bimetallic pathway. In such a context, the thiophene-salen chromium complex was engaged in the transformation as homogeneous catalyst, but also the polymer generated by electrochemical oxidation and the calixsalen derivative as a mixture, for comparison (see Scheme 7). Scheme 7. Homogeneous and heterogeneous epoxide ring opening by thiophene-salen chromium complexes.

For instance, ring-opening of 3,6-dioxa-bicyclo[3.1.0]hexane occurred in homogeneous conditions in the presence of the thiophene-salen chromium monomer, with a satisfying activity and a high enantioselectivity value that nevertheless does not reach the values described by Jacobsen, as already noticed. By testing the electrogenerated catalytic species, a dramatic drop in enantioselectivity to 24 % occurred, probably due to the linear structure of the polymer. This value pleasingly reached again 62 % thanks to the use of the macrocyclic catalyst. We assume that this improvement is due to the ability of the cyclic calixsalen structure to dispose the position of the catalytic sites in a favorable manner. As an insoluble species in the solvent of the reaction, the catalyst was easily separated from the products by simple filtration and reused in 5 consecutive runs. Such a behavior was confirmed by evaluating the three catalysts in other transformations (i.e. the hetero Diels-Alder reaction, the Henry reaction and the addition of dimethylzinc to aldehydes), leading to the same conclusion for those reactions involving a bimolecular activation mechanism.

The scope of application of these new calixsalen complexes was easily demonstrated since the introduction of the metal, leading to catalytically active sites, occurred at the last synthesis step. Thus, the chromium-complexes but also the corresponding Co II -species were shown to promote the Henry reaction. [START_REF] Ibrahim | Recoverable salen-based macrocyclic chiral complexes; catalysts for enantioselective Henry reactions[END_REF] These insoluble catalysts were recovered and reused in this transformation; they nevertheless did not lead to constant results in terms both of yield and selectivity, demonstrating a poor stability under the basic conditions necessary to the reaction. We therefore considered testing the ability of the corresponding reduced salen derivatives to act likewise as potential ligands to promote this reaction, with the aim of preparing more stable complexes possessing increased nitrogen basicity and framework flexibility (see Scheme 8). [26] Scheme 8. Copper-calix-tetrahydrosalen complex and use as heterogeneous catalyst in Henry reaction.

The tetrahydro-calixsalen derivative was synthesized from the corresponding calixsalen by reducing the imine functionalities in the presence of the borane-tetrahydrofuran complex. Cleavage of the B-N bond in the presence of palladium on charcoal in methanol was performed and subsequent copper addition delivered a new cyclic catalyst. The Henry reaction was conducted in the absence of a base, since the ligand played the role, as expected, in this case. Excellent results were obtained terms of both activity and enantioselectivity, the reduced form of derivative being much more efficient than its precursor. The classical recovery procedure (with a prior pentane addition, in order to ensure their complete precipitation) followed by filtration made it possible to reuse the catalyst numerous times (see Scheme 8) leading to the target product with a steady high yield and selectivity. The absence of catalyst leaching was verified by a control experiment, implying addition of new substrates to the supernatant solution recovered from the catalyst by filtration, with no further conversion. The usefulness of this new copper complex was also confirmed in an original multisubstrate procedure in which it was engaged after a simple filtration, in a new reaction with a different substrate at each reuse. As all catalysts were prepared from a mixture of macrocyclic species which contains a different number of monomer units, we then evaluated the influence of the cavity size on the catalytic results. More detailed analyses such as SEC and DOSY experiments were performed on calixsalen mixtures issued from numerous disalicylaldehyde derivatives spaced by different aryl groups and various steric bulk. Precise quantitative analyses allowed assignment of each macrocycle according to their size, a mixture mainly composed from dimers to tetramers. [START_REF] Ibrahim | Synthesis and characterization of original calix-salen type ligands[END_REF] Furthermore, the reaction conditions towards their synthesis were optimized and different dilutions favored either the isolation of the pure dimeric species (diluted media) or of the tetrameric one (concentrated media). Therefore, we investigated and compared the efficiency of pure trimer or tetramer of calixsalen Co III complexes to promote the dynamic HKR of epibromohydrin. This benchmark reaction is particularly sensible towards cooperative bimetallic interactions and thus suitable to evaluate the effect of the cyclic structure size of the macrocyclic catalysts . [START_REF] Dandachi | Chiral calix-salen cobalt complexes, catalysts for the enantioselective dynamic hydrolytic kinetic resolution of epibromohydrin[END_REF] Disalicylaldehyde with a benzene ring as spacer was used for the synthesis of the corresponding macrocycles with the preparation of either the dimer or the tetramer as major product according to the reaction concentration. Each calixsalen derivatives was isolated in its pure form, after chromatographic separation on silicagel. Corresponding Co III complexes were prepared, and the mixture, the pure trimer and the pure tetramer led all to the epoxide hydrolysis with activities and selectivities far better than those previously reported for the corresponding oligomeric linear analogues. The isolated Co III -tetramer delivered the best values, together with the higher initial rate, but not far enough from those obtained by the macrocycles mixture to justify the supplementary purification step towards its isolation (see Scheme 9). Scheme 9. Dynamic HKR of epibromohydrin with Co III -calixsalen oligomers.

Since the obtained catalysts were easily recovered from the reaction mixture by simple filtration, they were reused in another run to produce the targeted diol with steady enantioselectivity but decreasing yield. As already reported for cobalt acetate complexes delivering gradually nonnucleophilic Co III -OH species, a reactivation step with acetic acid was needed to recover a high activity. Another strategy was then developed to avoid this reactivation step; we indeed reported, under homogeneous conditions, that a heterobimetalic dual-catalyst system proved very efficient to catalyze this transformation. Thus, a combination of cobalt-salen and manganese-salen complexes, generated from ligands with the same configuration and introduced in a equimolar quantity, hydrolyzed not only epibromohydrin but also other terminal epoxides and the more challenging cyclohexene oxide in a more efficient way than cobalt complexes used alone. [START_REF] Hong | Heterobimetallic dual-catalyst systems for the hydrolytic kinetic resolution of terminal epoxides[END_REF] Kinetic studies showed that only one salen Co III complex was involved in the rate-determining step of the HKR in this case, which supported a heterobimetallic highly enantioselective pathway based on the crucial existence of in situ generated Co III -OH species, as previously postulated in the literature. The role of additional Mn-complexes was ascribed to epoxide activation as a beneficial effect, inhibiting the alternative less enantioselective monometallic reaction pathway.

In this context, we also prepared manganese-containing calixsalen complexes; whereas they showed no activity in the hydrolysis of meso epoxides, the use of a 1/1 mixture of Mn-and Comacrocycles delivered the targeted diol with a higher enantioselectivity than that obtained in the presence of the unique cobalt complex (see Scheme 10). [START_REF] Dandachi | Mixing and matching chiral cobalt-and manganese-based calix-salen catalysts for the asymmetric hydrolytic ring opening of epoxides[END_REF] Scheme 10. Hydrolysis of cyclohexene oxide by Co-and Mn-calixsalen complexes.

Indeed cooperativity between the Co-and Mn-containing catalysts was obvious, at least when the introduced species possessed the same configuration at the stereogenic centers. As can be seen in Scheme 10 for instance, opening of cyclopentenoxide occurred with both a large activity and selectivity enhancement with the dual activation, as a match combination. If the Mn salen used arose from the ligand enantiomer with the opposite configuration however, poorer results were obtained as a proof of the involvement of the manganese complex in the selectivity-determining transition state. Moreover, the mixture of catalysts could be recovered by simple filtration and used in new runs with fresh substrates with improved results, compared to the use of cobalt complexes only. Several recycling have been made in the case of the dynamic HKR of epibromohydrin, with no need for reactivation step. We assume this proves a heterobimetallic activation with the Mn-complex substituting the through time disappearing cobalt-acetate complex, for activating the epoxide.

CONCLUSION

Immobilization of structurally slightly modified salen derivatives through polymerization led thus to linear or cyclic insoluble oligomers, with the active catalytic sites as part of the main chain. Heterogeneous catalysis was therefore performed with these new catalysts and excellent results in terms of activity and enantioselectivity were obtained in different transformations, especially those involving C-C or C-O bonds formation, with Cr-or Co-complexes. Robustness of the catalysts was evaluated in multi-substrates or even multi-reactions procedures, hardly described in the literature. The catalysts maintained their efficiency at each recycling and targeted products were recovered free from any traces of the compound prepared in the preceding cycle. Specifically, synthesis of cyclic oligomers led to highly efficient catalysts for those reactions needing a bimetallic activation, with the active sites placed in an optimal spatial shape, necessary for obtaining high enantioselectivities. We managed also the synthesis of new catalysts dedicated to avoid reactivation steps needed, for example, for epoxides ring opening reactions; this implied heterogeneization of two salen catalysts bearing different active sites. We are currently pursuing our efforts towards this direction, with the preparation of insoluble heteropolymetallic salen catalysts obtained by polymerization methodologies, for their use in asymmetric multicatalytic tandem reactions.

Acknowledgements

The authors thank MENSR, Université Paris Sud, CNRS and Charm3At LabEx (ANR-11-LABEX-0039) for financial support. Our research has been also funded with the support of the National Council for Scientific Research in Lebanon CNRS-L and Lebanese University.


M.-A A Pinault-Thaury
email: marie-amandine.pinault-thaury@uvsq.fr

F Jomard

Christine Mer-Calfati

Nicolas Tranchant

Michal Pomorski

Philippe Bergonzo

Jean-Charles Arnault

J.-C Arnault

Front and back side SIMS analysis of borondoped delta-layer in diamond

Keywords: Both sides SIMS analysis, backside, depth profiling, ion mixing, diamond, boron-doped delta-layer

come

Introduction

Due to its unusual physical and chemical properties, diamond is a promising material for many applications related to color centers with nitrogen-vacancy centers [1][2][3][4] in the field of photonics and spintronics, as well as to high breakdown voltage / high temperature electronic devices [5,6] in the development of next generation high power devices based on boron doping. During the last ten years, an intensive work was performed on its physical properties and its synthetic growth by chemical vapour deposition (CVD). The improvement in diamond epitaxy allows growing ultra-thin boron doped structures and achieving diamond MESFET fabrication [7,8]. Such structures consist in a nanometric layer of highly boron doped diamond (P+ layer with [B] > 5x10 20 at/cm 3 ) sandwiched between two low boron doped diamond layers (P-layers with [B] ~ 1x10 16 at/cm 3 ). The challenge to achieve such multilayer structure is related to the growth of the P+ layer as thin as possible (around 2 nm) and to controlled sharpness of both P+/P-interfaces. According to simulations, this may allow carriers delocalization within P-layers which should lead to higher mobility [9]. Nevertheless, electrical characterizations of such boron-doped delta-layers in diamond structures didn't show mobility enhancement by confinement [START_REF] Chicot | Electronic and physico-chemical properties of nanometric boron delta-doped diamond structures[END_REF]. Such P-/ P+ / P-multilayers were initially obtained by insertion of a boron rod in the CVD plasma ball [START_REF] Kohn | Concepts for diamond electronics[END_REF][START_REF] El-Hajj | Characteristics of boron δ-doped diamond for electronic applications[END_REF]. However, the reproducibility of this method was perfectible. More recently, this multilayer structure was achieved thanks to a precise control of the diamond growth rate at high boron doping level and to the in situ etching process with oxygen [3].

Another method, used in this study, is based on an injection system developed to shorten the residence time of active species during CVD growth [START_REF] Volpe | Ultrasharp boron interfaces for delta doped diamond structures[END_REF]. As a result, several groups are currently able to grow excellent and reproducible thin heavily boron doped diamond layers surrounded by P-layers [START_REF] Chicot | Electronic and physico-chemical properties of nanometric boron delta-doped diamond structures[END_REF][START_REF] Mer-Calfati | Sharp interfaces for diamond delta-doping and SIMS profile modelling[END_REF][START_REF] Vikharev | Novel microwave plasma-assisted CVD reactor for diamond delta doping[END_REF].

The SIMS analysis provides the depth distribution of the boron concentration. SIMS measurements are usually performed from the front side of the sample: the sputtering goes from the sample surface to the substrate. When the layer is very thin, typically lower than 10 nm, a broadening effect of SIMS profiles cannot be neglected due to the interactions between incident ions and the studied material. Indeed, ion mixing and roughness effects led to artefacts in SIMS profiles such as asymmetry and broadening [START_REF] Hayashi | SIMS study of depth profiles of delta-doped boron/silicon alternating layers by low-energy ion beams[END_REF][START_REF] Takano | Evaluation of SIMS depth resolution using delta-doped multilayers and mixingroughness-information depth model[END_REF]. This is inherent to the SIMS technique.

In this work, we compare the SIMS profiles of the boron concentration from the front and back sides of a boron-doped delta-layer in diamond sample using SIMS parameters commonly used for diamond. We focus on artefacts occurring during SIMS analysis: ion mixing and roughness effects. The present study allows to highlight the limitations of SIMS measurements to characterize nanometric doped layers. Some previous studies performed on boron-doped silicon samples show the effectiveness of back side SIMS analysis [START_REF] Bilger | Copper-indium-gallium-diselenide/molybdenum layers analyzed by corrected SIMS depth profiles[END_REF][START_REF] Laugiera | Backside and frontside depth profiling of B delta doping, at low energy, using new and previous magnetic SIMS instruments[END_REF][START_REF] Hongo | Accurate depth profiling for ultra-shallow implants using backside-SIMS[END_REF]. It appears that combination of front and back side SIMS depth profiling can help to exclude ion beam induced atomic mixing. Such analysis provides sharp dopant distribution profiles suitable for analysing ultra-shallow layers. Thereby front and back sides SIMS analysis should allow the identification of the broadening effects and a more accurate description of the dopant distribution profile.

Experimental procedure

Boron-doped delta-layer in diamond sample

The diamond sample was realized in five steps (see figure 1).

Firstly (figure 1a), a 3x3x0.5 mm 3 (100) type Ib HPHT diamond substrate is implanted with oxygen ions (3 MeV with 10 17 ions/cm² dose [START_REF] Tran Thi | Ultra-smooth single crystal diamond surfaces resulting from implantation and lift-off processes[END_REF][START_REF] Mer-Calfati | High surface smoothening of diamond HPHT (100) substrates[END_REF]). This implantation creates a graphitized zone localized at a depth of 2 µm from the substrate surface as confirmed by 2D confocal Raman spectroscopy (not shown here). After conventional chemical cleaning, the substrate is introduced in a metallic CVD reactor.

The second step (figure 1b) consists in growing the diamond boron-doped delta-layer structure by microwave plasma CVD. The gas mixture is composed of methane (CH4), hydrogen (H2), trimethylboron (TMB). In the case of the P-layer, oxygen is added to the gas phase to decrease boron content below 5x10 16 at/cm 3 [START_REF] Volpe | Boron incorporation issues in diamond when TMB is used as precursor: Toward extreme doping levels[END_REF]. For the P+ layer, a new injection system has been developed in the CVD reactor to incorporate the gas mixture in the vicinity of the diamond substrate. This innovative configuration permits to obtain high boron content up to 3x10 20 at/cm 3 and very sharp interfaces between the P+ and P-layers [START_REF] Volpe | Ultrasharp boron interfaces for delta doped diamond structures[END_REF]. The growth temperature, the pressure and the microwave power were fixed at 850°C, 120 mbar and 600W, respectively. The substrate temperature is measured by a pyrometer (disappearance of filament). The growth parameters and the sample characteristics are summarized in Table 1.

After the second step, SIMS measurements were performed on the front side of the sample, just after the growth of the delta-layer structure.

For handling purpose, the sample is thickened (see figure 1c). As third step, a 30 µm undoped diamond layer is grown on the top of the delta-layer structure. The fourth step consists in the lift-off of the substrate (figure 1d). It is achieved using bipolar (called Marchywka effect [START_REF] Marchywka | United State Patent "Electrochemical process and product therefrom[END_REF]) electrochemical etching method in the implanted region on the edges of the diamond substrate. The sample is placed in ultra-pure water between two platinum rod-electrodes. The etching process is realized by applying a high voltage (700V -1000V). 42h of continuous process is needed to realize this lift-off. The last step is the etching back of the 2 µm residual diamond substrate (see figure 1e) with an Ar/O2 plasma in a Physical Vapour Deposition (PVD) system with magnetron sputtering. The magnetic field enhancing sputtering yield was blocked with thick mu-metal layer and the diamond sample was placed at the sputtering target place. The following parameters were set during the process: gas flows of 32 sccm for both Ar and O2, pressure of 11.2 mbar, RF power of 200 W and DC bias of 1200 V. For these parameters, the diamond etching rate was about 1 µm/h. A first etching was stopped at a few hundred of nanometers into the residual substrate before the P-/P+/P-structure. A second etching allows stopping in the 1 rst first P-layer (partially etched layer), before a few hundred of nanometers of the P+/P-diamond layers. After each etching steps, SIMS measurements were carried out from the back side of the sample.

The previous steps increase the difficulty in handling together with a sample thinning. The sample size goes from a square of 3x3x0.3 mm 3 to an isosceles triangle with ~2 mm in side and ~40 µm thick. The surface morphology of both front and back sides was investigated by Atomic Force Microscopy (AFM). A Molecular Imaging PicoLE AFM was used in contact mode. The roughness average value (RMS) of the diamond surface was systematically estimated from topographic AFM images. The RMS of the front side (1.5 nm) and of the back side (2.0 nm) measured on 5x5 m 2 surfaces are comparable (see figure 2). As boron-doped delta-layer in diamond sample (P-/P+/P-structure) presents a similar roughness on both sides, the roughness effect taking place during the SIMS sputtering might be identical for all the following SIMS results.

SIMS parameters

Secondary Ion Mass Spectrometry (SIMS) has been performed in order to measure the depth distribution of the boron concentration [B]. Specific care was taken to reach a vacuum limit of ~10 -8 mbar in the analysis chamber. We use a CAMECA IMS4f equipment. The depth of the resulting 150x150 µm² SIMS crater was measured using a Dektak8 step-meter. The analysed zone is restricted to a diameter of 33 µm to limit the crater edge effects. The secondary ions of masses 12 (carbon) and 11 (boron) were detected. The analysis have been done at a low mass resolution, M/M = 300, to have a maximum sensitivity.

As the SIMS technique is mostly used in diamond to detect impurities like hydrogen or boron, measurements are performed by using parameters allowing high sensitivities. Namely the Cs + /M-configuration is classically employed (positive primary ions with a Cs + source and detection of negative secondary ions of mass M). The energy of the Cs + primary beam is set to 10 keV. Secondary ions are detected in the negative mode (the sample is biased to -4500 V) leading to an interaction energy of the primary ions of 14.5 keV and an incidence angle of 27° with respect to the normal of the sample. Note that SIMS analysis of diamond is usually performed from the front side of the sample. Figure 3 shows an example of the boron distribution in the P-/P+/P-diamond structure extracted from our SIMS analysis from the front side. We have plotted the boron depth profile of the P+ delta-layer (≤ 7 nm) sandwiched between two low boron doped diamond layers (P-).

We observe that the sensitivity reaches four decades on the intensity. The up slope, Λup, characterises the interface between the second P-layer and the P+ layer while the down slope, Λdown, characterises the interface between the P+ layer and the first P-layer (front of boron incorporation during growth). The front and back side analyses allow estimating the stop and the onset of boron incorporation in diamond during growth of the delta-layer structure, respectively. In the following, we will study these two slopes.

The interaction mechanisms are governed by the kinetic energy, the ion nature and the incidence angle of the primary beam. As diamond analysis requires high sensitivity for impurity detection, our SIMS analysis conditions optimize the sensitivity at the expense of the depth resolution. In this study, the current of the Cs + ion beam is set at 40 nA allowing a sputtering rate of 0.2 nm/s.

Results and discussion

Figure 3 shows the boron depth profiles of our boron-doped delta-layer in diamond for the front side analysis (classical way of measurement). We observe that the delta-layer peak position is located at 335 nm in depth. The delta-layer presents an up slope, Λup, of 2.5 nm per decade (nm/dec) and a down slope, Λdown, of 7.3 nm/dec. Those values are significantly different: Λup is ~3 times lower than Λdown. For this analysis, Λup characterises the interface between the second P-layer and the delta-layer. This value is in agreement with our recent report [START_REF] Mer-Calfati | Sharp interfaces for diamond delta-doping and SIMS profile modelling[END_REF]. Such a low value suggests that the stop of the boron doping during growth is particularly efficient.

Figure 4 shows the boron depth profiles of the boron-doped diamond delta-layer for the two intermediate back side analyses: after the first etching with a few hundred nanometers of residual diamond substrate and after the second etching with partially etched first P-layer (figure 1 e). The delta-layer peak positions of the two back side analyses are 574 nm and 238 nm in depth for the first and the second etchings respectively. The delta-layer presents an up slope, Λup, of 4.5 and 5.8 nm/dec and a down slope, Λdown, of 7.6 and 8.1 nm/dec for the first and second etchings respectively. We observe that Λup values are close to 5 nm/dec. It represents the interface quality between the first P-layer and the delta-layer. This Λup denotes a good control of the front of boron incorporation during growth. The sharpness of the boron delta-layer interfaces confirms the good quality of the boron-doped delta-layer in diamond.

We observe that the delta peak position of the front side analysis is between the values of the two front side analysis (238 nm < 335 nm< 574 nm). Ideally, the precise way to compare the delta layer from the front and back side analysis would be to have the position of the delta layer at the same depth from the sample surface in each case. However, diamond etching is difficult to control at tens nm level. As a result, the position of the delta-layer is different from an etched sample to the other. Nevertheless, the boron depth profiles after the first and the second etchings present delta-layer peak positions (back side SIMS analysis) that frame the delta-layer peak position of the front side SIMS analysis. Thus it is possible to directly compare the profiles of front and back side SIMS analysis.

In figure 5, we superimpose the profile of the front side analysis to each profile of the back side analysis. The position of the delta-layer is shifted by -239 nm and +97 nm for the profile analysis of the back side after the first and the second etchings respectively. We observe that the down slopes, Λdown, superimpose perfectly. On the contrary, the up slopes, Λup, differ significantly. The lowest Λup value is reached with the front side SIMS analysis (2.5 nm/dec).

The slopes of the delta-layer peak plotted in figures 3 and 4 are reported in Table 2. We remind here that for the front side analysis, the Λup and Λdown slopes represent the interfaces between the delta-layer and the second P-layer and between the delta-layer and the first P-layer, respectively. Conversely, for the back side analysis, the Λup and Λdown slopes represent the interfaces between the delta-layer and the first P-layer and between the delta-layer and the second P-layer, respectively.

The Λdown values are all in the range of 7.5 nm/dec. According to AFM measurements (Figure 2), the roughness effect occurring during the sputtering might be identical whatever the analysed side. The similar values of Λdown are explained by the fact that primary ions interact with sample atoms. A collision cascade develops in the material during the sputtering process.

In our case, this effect occurs in about 10 nm in depth according to mixing-roughnessinformation depth (MRI) model [START_REF] Mer-Calfati | Sharp interfaces for diamond delta-doping and SIMS profile modelling[END_REF][START_REF] Hofmann | he MRI-Model in Sputter Depth Profiling: Capabilities, Limitations and Recent Progress[END_REF]. This leads to the implantation of boron atoms from the delta-layer into the neighbouring sputtered P-layer. This phenomenon is the so called "ion mixing".

The interface between the second P-layer and the P+ layer of the delta-layer structure is characterized by (i) the up slope of the front side SIMS analysis and (ii) the down slope of the back side the SIMS analysis. The values are 2.5 nm/dec for the front side analysis and systematically in the range of 7.5 nm/dec for the back side analysis. Such increase between the front and back side values clearly shows that the last value is due to an instrumental limit.

Such limit is inherent to the SIMS technique. It restricts the correct description of the front of boron incorporation in the direction P-/P+ (first P-layer/delta-layer structure) for classical SIMS analysis performed from the front side of the sample. This effect was supported by our previous comparison between old and new generation of CAMECA IMS equipments (namely IMS4f and IMS7f respectively) on the SIMS characterization of the same P-/ P+ / Pmultilayer. Indeed, the use of lighter ions with lower kinetic energy (O2 + ions at 2 keV, unreachable with a CAMECA IMS4f equipment) led to a lower Λdown value due to the reduction of ion mixing (4 nm/dec instead of 7 nm/dec) [START_REF] Mer-Calfati | Sharp interfaces for diamond delta-doping and SIMS profile modelling[END_REF].

Conclusion

In standard SIMS analysis, the measurement is performed from the surface to the substrate (front side). The depth resolution is sensitive to the burial depth of the delta-layer: the deltalayer should be as closer as possible to the sample surface. As a result, the only significant measurement in a delta-layer structure is the up slope giving access to the interface between the delta-layer and the second P-layer. In the present study, the use of a lift-off process allows to perform depth profiles of boron concentration from the substrate to the surface (back side).

We have shown that the front of boron incorporation (interface between the first P-layer and delta layer) is steeper than suggested by classical SIMS analysis performed from the front side of the sample. We observe sharp interfaces of the delta layer on both sides. According to MRI model [START_REF] Hofmann | he MRI-Model in Sputter Depth Profiling: Capabilities, Limitations and Recent Progress[END_REF], we evidence the ion mixing occurring in about a depth of 10 nm in classical diamond SIMS analysis (front side). Moreover, nowadays commercialized SIMS equipment has better specifications (impact energy as low as 500 eV). In the future, the use of lighter incident ions and /or lower kinetic energy should allow a better understanding of the physical phenomena occurring during the SIMS analysis of diamond delta-layer structures. The synthesis of boron-doped delta-layers in diamond is now well controlled. As a result, it is now possible to compare boron-doped delta-layers in diamond to boron-doped delta-layers in silicon (under progress). Such structures could be further investigated for photonics, spintronics or electronics applications but could be also an alternative to silicon structures to allow measurements in conditions that silicon cannot tolerate.

Table captions

Table 1: Growth parameters for P-/P+/P-multilayers. The thickness and the boron content of the layers are measured by front side SIMS analysis.

Table 2: Up and down slopes of the delta structures plotted in figures 3 and 4.

Figure captions

Figure 1: Schematic realization of the P-/ P+ / P-sample (five steps) with (a) the ion implantation of the HPHT (100) diamond substrate followed by (b) the growth of the delta structure for the SIMS analysis on the front side, then (c) the growth of a diamond bulk layer for handling purpose followed by (d) a lift-off and (e) an etching of the diamond substrate allowing the SIMS analysis of the back side. Table 1 Pinault-Thaury et al.

Figure 2 :

 2 Figure 2: AFM images of the sample surfaces (a) after the second step (front side) and (b)after the second etching of the 5 th step (back side) of the P-/P+/P-sample.

Figure 3 :

 3 Figure 3: SIMS profiles of 11 mass (boron) centred around the boron delta-doped diamond layer sandwiched between two low boron doped diamond layers. The profile has been performed from the front side of the sample.

Figure 4 :

 4 Figure 4: Boron SIMS profiles of the boron delta doped diamond sample from the back side analyses performed a) after the 1 rst etching (with a few hundred of residual substrate) and b) after the second etching (with partially etched first P-layer).

Figure 5 :

 5 Figure5: Zoom in the boron SIMS profiles by overlaying the front side analysis to each back side analysis (on the left, after the first etching and on the right, after the second etching). For convenience to the eye, the delta peak positions of the back side profiles have been shifted to coincide to the one of the front side profile.

Figure 1 Pinault-

 1 Figure 1

Figure 2 Pinault-

 2 Figure 2

Figure 3 Pinault-

 3 Figure 3

Figure 4 Pinault-

 4 Figure 4

Figure 5

 5 Figure 5

Acknowledgement

The authors are grateful to José Alvarez for his scientific support on Raman analyses. The sample was provided thanks to the Deltadiam-ANR-08-BLAN-0195 contract.


Nicolas Reiminger
email: nreiminger@air-d.fr

Xavier Jurado

José Vazquez

Cédric Wemmert

Nadège Blond

Jonathan Wertel

Matthieu Dufresne

Methodologies to assess mean annual air pollution concentration combining numerical results and wind roses

Numerical models are valuable tools to assess air pollutant concentrations in cities which can be used to define new strategies to achieve sustainable cities of the future in terms of air quality.

Numerical results are however difficult to be directly compared to air quality standards since they are usually valid only for specific wind speed and direction while some standards are on annual values. The purpose of this paper is to present existing and new methodologies to turn numerical results into mean annual concentrations and discuss their limitations. To this end, methodologies to assess wind speed distribution based on wind rose data are presented first.

Then, methodologies are compared to assess mean annual concentrations based on numerical results and on wind speed distributions. According to the results, a Weibull distribution can be used to accurately assess wind speed distribution in France, but the results can be improved using a sigmoid function presented in this paper. It is also shown that using the wind rose data directly to assess mean annual concentrations can lead to underestimations of annual concentrations. Finally, the limitations of discrete methodologies to assess mean annual concentrations are discussed and a new methodology using continuous functions is described.

Introduction

Over the past decades, outdoor air pollution has become a major issue, especially in highly densified urban areas where pollutant sources are numerous and air pollutant emissions high.

In order to protect people from excessive exposure to air pollution, which can cause several diseases [START_REF] Anderson | Clearing the Air: A Review of the Effects of Particulate Matter Air Pollution on Human Health[END_REF][START_REF] Kim | A review on the human health impact of airborne particulate matter[END_REF], the World Health Organization (WHO) have recommended standard values that must not be exceeded for different pollutants such as nitrogen dioxide (NO2) and particulate matter (EU, 2008;[START_REF] Who | Evolution of WHO air quality guidelines past, present and future[END_REF] to protect population health, and the European Union (EU) decided to respect the same or other standards depending on the air pollutants. Among the different types of values given as standards, studies have shown that annual standards are generally more constraining and harder to reach than the other standards [START_REF] Chaloulakou | Compliance with the annual NO2 air quality standard in Athens. Required NOx levels and expected health implications[END_REF][START_REF] Jenkin | Analysis of sources and partitioning of oxidant in the UK-Part 1: the NOX-dependence of annual mean concentrations of nitrogen dioxide and ozone[END_REF].

In the meantime, recent studies have shown that the indoor air quality is strongly correlated with the outdoor one: while for nitrogen dioxide a 5% increase in indoor air pollutant concentrations can be expected for only a 1% increase in outdoor concentrations [START_REF] Shaw | The association between indoor and outdoor NO2 levels: A case study in 50 residences in an urban neighbourhood in New Zealand[END_REF], for particulate matters such as PM2.5 the outdoor concentration can contribute from 27% to 65% of the indoor concentration [START_REF] Bai | Investigation of yearly indoor/outdoor PM2.5 levels in the perspectives of health impacts and air pollution control: Case study in Changchun, in the northeast of China[END_REF]. Being able to assess outdoor pollutant concentrations is therefore a necessity to improve air quality in the outdoor built environment, but also in the indoor one (Ścibor et al., 2019).

Annual concentrations can be assessed using both on-site monitoring and numerical modeling.

On site monitoring requires measurements over long periods to be able to assess mean annual concentrations of pollutants, although a recent study has shown that mean annual concentration of NO2 can be assessed using only one month of data [START_REF] Jurado | Assessment of mean annual NO2 concentration based on a partial dataset[END_REF], which significantly reduces the measurement time required. Monitoring nonetheless has other limitations: it does not allow assessing the future evolution of the built environment or pollutant emissions, thus, DOI : 10.1016/j.scs.2020.102221 3/26 limiting its applicability to achieve the smart sustainable cities of the future as defined by [START_REF] Bibri | Smart sustainable cities of the future: An extensive interdisciplinary literature review[END_REF]. Numerical modelling can overcome these limitations and can help define new strategies to improve air quality in cities combining wind data, various air pollution scenarios and urban morphologies [START_REF] Yang | Air pollution dispersal in high density urban areas: Research on the triadic relation of wind, air pollution, and urban form[END_REF]. Among the several models currently available, Computational Fluid Dynamics (CFD) has shown great potential for modeling pollutant dispersion from traffic-induced emissions by including numerous physical phenomena such as the effects of trees [START_REF] Buccolieri | Review on urban tree modelling in CFD simulations: Aerodynamic, deposition and thermal effects[END_REF][START_REF] Santiago | On the Impact of Trees on Ventilation in a Real Street in Pamplona, Spain[END_REF][START_REF] Vranckx | Impact of trees on pollutant dispersion in street canyons: A numerical study of the annual average effects in Antwerp, Belgium[END_REF] and heat exchanges [START_REF] Qu | Numerical study of the thermal effects of buildings on low-speed airflow taking into account 3D atmospheric radiation in urban canopy[END_REF][START_REF] Toparlar | A review on the CFD analysis of urban microclimate[END_REF][START_REF] Wang | Thermal Effect on Pollutant Dispersion in an Urban Street Canyon[END_REF] on the scale of a neighborhood. However, this type of numerical result cannot be directly compared with the annual standards. Methodologies designed to assess mean annual concentrations based on numerical results, such as described by [START_REF] Solazzo | A novel methodology for interpreting air quality measurements from urban streets using CFD modelling[END_REF], are thus required and further work is required to improve these methodologies and assess their limits.

The aim of this study is to provide tools and methodologies to assess mean annual concentrations based on numerical results and wind rose data to improve air quality in built environment and cities. It is firstly to evaluate whether it is possible to assess continuous wind speed distributions based on wind rose data. To do so, a statistical law called Weibull distribution is compared with a new sigmoid-based function built for the purpose of this study.

Secondly, it is to present and compare a discrete methodology usually used to assess mean annual concentrations based on numerical results with a continuous methodology built for the purpose of this study, and to discuss their respective advantages and limitations. The data used for the wind speed distribution assessments, the area modeled and the CFD model used for illustration purposes are presented in Section 2. Then, the description and the comparison of the different methodologies are presented in Section 3 and, finally, a discussion is provided in Section 4.

Material and methods

Meteorological data

Data location

This work uses wind velocity and wind direction data from four cities in France. These cities were chosen to cover most of France to obtain representative results and include the cities of Strasbourg (Grand-Est region), Nîmes (Occitanie region), Brest (Bretagne region) and Lille (Hauts-de-France region). In particular, the data were obtained from the stations named Strasbourg-Entzheim, Nîmes-Courbessac, Brest-Guipavas and Lille-Lesquin, respectively. The location of these stations and their corresponding regions are presented in Fig. 1.

Data availability and data range

The data used in this work were provided by Météo-France, a public institution and France's official meteorology and climatology service. The data are mainly couples of wind velocity and wind direction over a twenty-year period from 1999 to 2018, except for the Strasbourg-DOI : 10.1016/j.scs.2020.102221

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Entzheim station where it is a ten-year period from 1999 to 2008. The data were obtained via a personal request addressed to Météo-France and were not available on open-access. A summary of the information of the stations is presented in Table 1, with the time ranges of the data and the number of data available (the coordinates are given in the World Geodetic System 1984). 1999 -2008 29,199 25 All the data were monitored from wind sensors placed 10 meters from the ground and the wind frequencies are available for each wind direction with 20° steps for two distinct wind discretizations: a "basic" discretization giving wind frequencies for 4 velocity ranges (from 0 to 1.5 m/s, 1.5 to 3.5 m/s, 3.5 to 8 m/s and more than 8 m/s), illustrated in Fig. 2. (A); and a "detailed" discretization giving wind frequencies by 1 m/s steps except between 0 and 0.5 m/s, illustrated in Fig. 2. (B). The "basic" discretization is a common format mostly found in wind roses (possibly with different velocity ranges) while the "detailed" data are less common and more expensive.

Interpolation functions

A two-parametric continuous probability function, the Weibull distribution, mainly used in the wind power industry, can be used to describe wind speed distribution [START_REF] Kumar | Wind Energy Potential Assessment by Weibull Parameter Estimation Using Multiverse Optimization Method: A Case Study of Tirumala Region in India[END_REF]; [START_REF] Mahmood | Wind characteristic analysis based on Weibull distribution of Al-Salman site[END_REF]. The equation of the corresponding probability density function is given in (1).

𝑓(𝑣) = 𝑘 𝜆 ( 𝑣 𝜆 ) 𝑘-1 𝑒 -(𝑣/𝜆) 𝑘 (1)
where 𝑣 is the wind velocity, 𝑘 is the shape parameter and 𝜆 is the scale parameter of the distribution, with 𝑘 and 𝜆 being positive.

For the purpose of this study, an original 5-parametric continuous function was built to determine the "detailed" wind discretization based on the "basic" 4-velocity-range wind discretization. This function, called Sigmoid function, based on the composition of two sigmoid functions, is given in (2). The two functions will be compared in the results section.

𝑓(𝑣) = 𝛼. (-1 + 1 1 + 𝛽 1 . 𝑒 -𝛾 1 .𝑣 + 1 1 + 𝛽 2 . 𝑒 𝛾 2 .𝑣 ) (2)
where 𝛼, 𝛽 1 , 𝛽 2 , 𝛾 1 and 𝛾 2 are positive parameters.

Numerical model

Simulations were performed using the unsteady and incompressible solver pimpleFoam from OpenFOAM 6.0. A Reynolds-Averaged Navier-Stokes (RANS) methodology was used to solve the Navier-Stokes equations with the RNG k-ε turbulence model, and the transport of particulate matter was performed using a transport equation. This solver was validated previously in [START_REF] Reiminger | CFD evaluation of mean pollutant concentration variations in step-down street canyons[END_REF].

The area chosen to illustrate the methodologies discussed in this paper is located in Schiltigheim, France (48°36'24", 7°44'00"), a few kilometers north of Strasbourg. This area, as well as the only road considered as an emission source in this study (D120, rue de la Paix), are illustrated in Fig. 4. (A). PM10 traffic-related emissions were estimated at 1.39 mg/s using daily annual mean traffic and were applied along the street considering its length in the numerical domain (200 m), its width (9 m) and an emission height of 0.5 m to take into account initial dispersion.

The recommendations given by [START_REF] Franke | Best practice guideline for the CFD simulation of flows in the urban environment[END_REF] were followed. In particular, with 𝐻 being the highest building height (16 m), the distances between the buildings and the lateral boundaries are at least 5𝐻, the distances between the inlet and the buildings as well as for the outlet and the buildings are at least 5𝐻 and the domain height is around 6𝐻. An illustration of the resulting 3D sketch is presented in Fig. 4. (B). A grid sensitivity test was performed and showed that hexahedral meshes of 1 m in the study area and 0.5 m near the building walls are sufficient, leading to a more comparable resolution than other CFD studies [START_REF] Blocken | Computational Fluid Dynamics for urban physics: Importance, scales, possibilities, limitations and ten tips and tricks towards accurate and reliable simulations[END_REF] and leading to a total number of around 800,000 cells. The resulting mesh is illustrated in Fig. 5. ) )

one o interest DOI : 10.1016/j.scs.2020.102221

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No-slip conditions (U = 0 m/s) were applied to the building walls and ground, and symmetry conditions to the lateral and the top boundaries. A freestream condition was applied to the outlet boundary, and neutral velocity, turbulent kinetic energy and turbulent dissipation profiles suggested by [START_REF] Richards | Appropriate boundary conditions for computational wind engineering models revisited[END_REF] were applied to the inlet boundary.

A total of 18 simulations were performed using the same wind velocity (U10 m = 1.5 m/s) but with different wind directions from 0° to 340° using a 20° step. Since the simulations were performed in neutral conditions and without traffic-induced turbulence, the dimensionless concentration 𝐶 * given in ( 3) is a function only of the wind direction [START_REF] Schatzmann | Issues with validation of urban flow and dispersion CFD models[END_REF]. In other words, this means that considering the previous hypothesis, and for a given emission and building configuration (leading to constant 𝐻. 𝐿 𝑞 ⁄ ratio), only one simulation is needed for each wind direction simulated. The pollutant concentrations for a non-simulated wind velocity 𝑢 can therefore be computed using (4).

𝐶 * = 𝐶. 𝑈. 𝐻. 𝐿 𝑞 (3)
where 𝐶 * is the dimensionless concentration, 𝐶 is the concentration, 𝑈 the wind velocity, 𝐻 the characteristic building height and 𝑞/𝐿 the source strength of emission.

𝐶 𝑢 = 𝑈 𝑟𝑒𝑓 . 𝐶 𝑟𝑒𝑓 𝑢 (4)
where 𝐶 𝑢 is the pollutant concentration for the wind velocity 𝑢 not simulated and 𝐶 𝑟𝑒𝑓 the pollutant concentration for the simulated wind velocity 𝑈 𝑟𝑒𝑓 .

Results

Wind data interpolation

Comparison between the Weibull distribution and the sigmoid function

The best fitting parameters of the two functions were determined for the whole dataset using a non-linear solver and the "basic" 4-velocity-range wind data. The solver was set up to solve equation ( 5) for the four-velocity ranges [0, 1.5[, [1.5, 4.5[, [4.5, 8[ and [8, +∞[ for both the Weibull and the sigmoid functions. This equation reflects that the sum of the frequencies between two wind velocities (i.e. the area under the curve) must be equal to the frequency given in the "basic" 4-velocity-range wind data. Since the sigmoid function has five parameters, a fifth equation to be solved was added only for this function and corresponds to (6). With this equation, it is assumed that the wind frequency tends toward 0% when the wind speed tends toward 0 m/s, as for the Weibull distribution. According to these figures, the two functions generally give the same trends, and both appear to give a good estimation of the "detailed" wind data. However, depending on the case, the Weibull function can provide improvements in comparison to the sigmoid function, as in Fig. 33.1% respectively); a lower first quartile (8.1% and 9.5% resp.); a lower third quartile (13.8% and 14.5% resp.); a lower mean (11.7% and 13.5% resp.); and a lower median (10.6% and 12.4% resp.). The differences are, however, small and may not be significant, especially for the median because the notches slightly overlap. These differences between the Weibull distribution and the sigmoid function are also location dependent, with for example better prediction of the wind distribution in Strasbourg using the sigmoid function and an equivalent prediction in Brest. Finally, it should be noted that both functions can lead to underestimations of the lower wind velocity frequencies, as shown in Fig. 6. (A) and (D).

∫ 𝑓(𝑣

According to the previous results, the Weibull distribution and the sigmoid function can accurately reproduce the "detailed" wind distribution based on a "basic" 4-velocity-range discretization with an average error of around 12% over the four stations considered in France.

They can nonetheless lead to underestimations of the low wind velocity frequencies, for which the highest pollutant concentrations appear.

Optimization of the sigmoid function interpolation for low wind velocities

The parametrization of the sigmoid function, called standard sigmoid function, was modified to improve the estimation of the low wind velocity frequencies in order to avoid underestimating pollutant concentrations.

Based on all the meteorological data considered in this study, it was found that the underestimation of low wind velocity frequencies occurs mostly when the frequency of the first velocity range is lower than the frequency of the second velocity range. In this specific case, the optimized sigmoid function still needs the equation ( 5) for the four-velocity ranges given in the "basic" wind data, but equation ( 6) is replaced by equation ( 7); otherwise, the previous parametrization using equations ( 5) and ( 6) is kept. The improvements with the optimized sigmoid function compared to the standard function were assessed and the results are presented in Fig. 7. (C). For this comparison, only the wind directions where the optimized function was applied are considered and the errors compared to the "detailed" 18-velocity-range data were calculated for the low wind velocity frequencies (between 0 and 3.5 m/s According to the previous results, using the optimized sigmoid function can improve the reproduction of the "detailed" wind distribution based on a "basic" 4-velocity-range compared to the standard sigmoid function, especially for low wind velocities.

Mean annual concentration assessment

Discrete methodology with intermediate velocities

Initially, mean annual concentrations based on the CFD results can be calculated using a discrete methodology. This methodology considers that the mean annual concentration at a given location is composed of several small contributions of different wind velocities and wind directions. The mean concentration over one wind direction can be calculated with equation ( 8)

and the mean annual concentration with equation (9). A similar methodology can be found in [START_REF] Solazzo | A novel methodology for interpreting air quality measurements from urban streets using CFD modelling[END_REF].

where 𝐶 ̅ 𝑑 is the mean concentration over one wind direction, 𝐶 𝑑,𝑟 is the concentration for a given wind direction 𝑑 and a given wind velocity range 𝑟, 𝑓 𝑑,𝑟 is the frequency for a given wind direction and a given wind velocity range, 𝐶 𝑏𝑔 is the background concentration, 𝐶 ̅ is the mean annual concentration and 𝑓 𝑑 the total frequency of a given wind direction.

With this methodology, it is necessary to choose a wind velocity in each velocity range for which the concentration will be calculated based on the CFD result. Initially, it can be seen that using the "basic" 4-velocity-range data leads to an underestimation of the concentrations compared to the case using "detailed" 18-velocity-range data by around 19%. hen calculating the "detailed" wind velocity distribution based on the "basic" data with the sigmoid function, the difference is reduced to 12.9%. Finally, the best results are obtained when using the optimized sigmoid function with an underestimation of 3.4%. According to these results, using the "basic" 4-velocity-range frequencies can give an estimation of the mean annual concentrations but is not sufficient to reach good accuracy compared to the mean annual concentration calculated with the "detailed" wind velocity distribution. However, using the sigmoid function and especially the optimized variant significatively improves the results, DOI : 10.1016/j.scs.2020.102221 17/26 leading to almost the same results as those obtained with the "detailed" wind velocity distribution.

Discrete methodology with representative velocities

The previous methodology used to compute annual concentrations, which was easy to set up, nonetheless has certain weaknesses that mostly concern the choice of the wind velocity for which the concentrations will be calculated, based on the CFD results. Using an intermediate velocity 𝑣 𝑖 corresponding to the average between the minimal and the maximal value of the velocity range can lead to underestimations of the mean annual concentrations. Indeed, in doing so, it is implicitly assumed that the concentration is constant with the wind velocity in a given wind velocity range. However, according to equation ( 4), this assumption is wrong because the concentration evolves hyperbolically with velocity. The representative velocity over one velocity range, considering the hyperbolic evolution of the concentration, is given in (11) as a result of ( 10) and (4).

𝑣 𝑟 = √ 2 1 𝑣 𝑚𝑎𝑥 2 + 1 𝑣 𝑚𝑖𝑛 2 (11)
where 𝑣 𝑚𝑎𝑥 and 𝑣 𝑚𝑖𝑛 are respectively the maximal and the minimal velocities of the velocity range, 𝑣 𝑟 is the representative velocity of the velocity range and 𝑐(𝑣) the equation describing the evolution of the concentration as a function of the wind velocity, i.e. equation (4).

The representative velocities 𝑣 𝑟 were calculated with equation ( 11) and compared to the intermediate velocities 𝑣 𝑖 . It is noteworthy that for a velocity range with a minimal velocity of 0 m/s, it is mathematically not possible to compute the representative velocity due to the domain DOI : 10.1016/j.scs.2020.102221 18/26 definition of the function. A choice is therefore required; for the purpose of this study, the same ratio 𝑣 𝑟 /𝑣 𝑖 as for [0.5, 1.5[ was considered.

According to the results summarized in Table 2. for wind velocities ranging from 0 to 6.5 m/s, the intermediate velocity can be much higher than the representative velocity for low velocities.

For example, for wind velocities ranging from 0.5 to 1.5 m/s, the intermediate velocity of 1 m/s is almost twice as high as the representative velocity of 0.67 m/s. For higher velocity ranges, such as [2.5, 3.5[ or more, the differences can be neglected. This last statement is true for 1 m/s steps between the minimal and the maximal velocities of the velocity range but can become wrong for higher velocity steps.

Table 2. Comparison between the intermediate velocity 𝑣 𝑖 and the representative velocity 𝑣 𝑟 (*: the representative velocity was calculated considering the same ratio 𝑣 𝑟 /𝑣 𝑖 as for [0.5, 1.5[ ).

𝑣 𝑚𝑖𝑛 [m/s] 0 0.5 1.5 2.5 3.5 4.5 5.5

𝑣 𝑚𝑎𝑥 [m/s] 0.5 1.5 2.5 3.5 4.5 5.5 6.5

𝑣 𝑖 [m/s] 0.25 1.00 2.00 3.00 4.00 5.00 6.00

𝑣

Continuous methodology using the sigmoid function

For the last approach, mean annual concentrations based on CFD results can be calculated using a continuous methodology. This methodology is a combination of equation ( 4 where 𝐶 ̅ is the mean annual concentration, 𝑐(𝑣) is the function describing the evolution of the concentration with the wind velocity, 𝑓(𝑣) is the function describing the evolution of the wind velocity frequency with the wind velocity, and 𝐶 𝑏𝑔 is the background concentration.

Taking equation (4) for 𝑐(𝑣) and equation (2) for 𝑓(𝑣) leads to a mathematical problem. Indeed, 𝑐(𝑣) is not defined for 𝑣 = 0 and the limit of 𝑐(𝑣). 𝑓(𝑣) tends toward infinity when 𝑣 tends DOI : 10.1016/j.scs.2020.102221 20/26 toward 0. To avoid this problem, equation ( 13) is suggested instead of equation ( 12). With this equation, it is considered that a minimal velocity (𝑣 𝑚𝑖𝑛 ) exists for which the pollutant concentration will no longer increase when the wind velocity decreases. This hypothesis can be justified by the additional effects, such as traffic-induced turbulence [START_REF] Vachon | Measurements of Traffic-Induced Turbulence within a Street Canyon during the Nantes'99 Experiment[END_REF] and atmospheric stability [START_REF] Qu | Numerical study of the thermal effects of buildings on low-speed airflow taking into account 3D atmospheric radiation in urban canopy[END_REF] where 𝐶 ̅ is the mean annual concentration, 𝐶 𝑚𝑎𝑥 is the maximal concentration accepted for the calculation, 𝑣 𝑚𝑖𝑛 is the velocity under which 𝑐(𝑣) is considered equal to 𝐶 𝑚𝑎𝑥 , 𝑓(𝑣) is equation

(2), 𝑐(𝑣) is equation ( 4) and 𝐶 𝑏𝑔 is the background concentration.

Discussion

This study provides tools to assess wind velocity distributions based on "basic" data and mean annual air pollutant concentrations based on CFD results. Additional work should be done to improve the methodologies and the major issues are discussed hereafter.

The capability of the Weibull and the sigmoid functions to describe wind velocity distribution was assessed based on wind data from four meteorological stations in France. All of these stations were located in peri-urban environments close to large French cities. It is necessary to take into account that the results, and especially the interpolation-related errors, might be different for other types of stations such as urban and rural stations, and for other countries with different wind characteristics. In particular, the optimization suggested for the sigmoid function may not be suitable for different countries or type of station. Further works are therefore required in this direction.

The mean annual atmospheric pollutant concentrations can be calculated using a discrete methodology. However, this methodology has two major problems. The first concerns the choice of wind velocity for which the pollutant concentrations will be calculated: choosing an ranges: the result depends on the velocity step used, especially for the lower wind velocities for which a decrease in the velocity-step leads to higher mean annual concentrations. To avoid these two problems, a continuous methodology has been proposed. This methodology does not have an intrinsic limitation, but dependent on the function describing the evolution of the concentration as a function of wind velocity. If we consider a hyperbolic evolution of the concentration with wind velocity, it is necessary to choose a minimal value of velocity for which it is considered that lower velocities will not increase the concentrations due to compensatory phenomena (traffic-induced turbulence, atmospheric stability, etc.). The value of the minimal velocity is open to discussion and assessing this value is outside the scope of this paper. Further works are required, for example with infield measurement campaigns and comparisons between mean annual concentrations monitored and calculated with the continuous methodology.

Finally, it should be noted that the methodologies to assess mean annual concentrations were addressed using CFD results implying a neutral atmosphere, but can be used for any numerical results as long as a function describing the evolution of the concentration with the wind velocity is available.

Conclusion

The objectives of this study were to provide methodologies; (1) to assess wind velocity distribution based on "basic" data, and (2) to assess mean annual air pollutant concentrations based on numerical results. Three approaches for each objective were described and compared throughout this paper and the main conclusions are as follows:

(1.a) The Weibull distribution and the sigmoid function can both accurately reproduce "detailed" 18-velocity-range wind distribution based on "basic" 4-velocity-range wind data with an average error of 12%. These functions can nonetheless underestimate the frequencies of low velocities.

(1.b) The optimized sigmoid function improves the wind distribution results over the standard sigmoid function, especially for low wind velocities.

(2.a) Using "basic" 4-velocity-range wind data and the discrete methodology can provide an estimation of the mean annual concentrations but is not sufficient to achieve high precision, leading to a difference of around 19 compared to the use of "detailed" 18-velocity-range wind data. Using the sigmoid function instead, based on the "basic" wind data improves the mean annual concentration results with a global error of less than 4%.

(2.b) When using the discrete methodology to assess mean annual concentrations, it is suggested to use a representative velocity of the function describing the evolution of pollutant concentrations with the wind velocities instead of an intermediate velocity.

The intermediate velocity leads to underestimations of mean annual concentrations, especially when using CFD results with a neutral case hypothesis where the concentration evolves hyperbolically with the wind velocity.

(2.c) Mean annual concentrations can be assessed using a continuous methodology that does not have any of the limitations of discrete methodologies. It is, however, limited by the function describing the evolution of the concentrations with the wind velocities, which leads to the need to choose a minimal velocity when using the sigmoid function. Finally, the methodologies presented in this paper can be used for outdoor air quality study purposes, which is a relevant starting point for improving both outdoor and indoor air quality and, therefore, a key-point to achieve smart sustainable cities.

Fig. 1 .

 1 Fig. 1. Location of the different meteorological stations used.

Fig. 3 .

 3 Fig. 3. Wind roses for each location considered.

Fig. 4 .

 4 Fig. 4. Illustration of (A) the area of Strasbourg modeled with the road considered for the traffic-related emissions (white dashed lines), and (B) the corresponding area built in 3D for the numerical simulations with the emission source (red).

  Fig. 5. Illustration of the meshes in the computational domain with the emission source (red), with 0.5 m meshes near the buildings and 1 m in the study area.

Fig. 6 (

 6 Fig.6 (A-D) shows a comparison between the Weibull distribution, the sigmoid function and

  6. (A), or vice versa, the sigmoid function can provide improvements in comparison to the Weibull function, as in Fig. 6. (D).

Fig. 6 .

 6 Fig. 6. (A-D) Weibull distribution and sigmoid function results compared to the detailed meteorological wind frequency data for one wind direction at each station considered and (E) a notched box plot of the mean error over one wind direction with all stations included for both functions.

  where 𝐹𝑉𝑅 [0,𝛼[ is the wind frequency for the first range of velocities given in the 4-velocityrange data and 𝐹𝑉𝑅 [𝛼,𝛽[ is the wind frequency for the second range of velocities (e.g., in this study 𝛼 = 1.5 and 𝛽 = 4.5).The methodology for the optimized sigmoid function is illustrated in Fig.7. (A-B): when the frequency of the first velocity range is higher than the second, as in Fig.7. (A1), the standard parametrization of the sigmoid function can be used because the low wind velocity frequencies are estimated accurately, as in Fig.7. (A2), when the frequency of the first velocity range is lower than the second, as in Fig.7. (B1), the standard parametrization leads to underestimations of low wind velocity frequencies and the optimized parametrization should be used instead, leading to a better estimation of the frequencies, as shown by the blue curve in Fig.7. (B2) compared to the red curve.

Fig. 7 .

 7 Fig. 7. (A-B) Illustration of the optimized sigmoid function methodology and (C) comparison with the standard sigmoid function results.

  Fig. 8. Mean annual concentrations without background concentration based on ( ) the "basic" 4-velocity-range monitoring data, ( ) the "detailed" 18-velocity-range monitoring data, (C) the sigmoid interpolation data and (D) the optimized sigmoid interpolation data.

Fig. 9 .

 9 Fig. 9. shows a comparison of the mean annual concentrations when using the intermediate

  ), describing the evolution of pollutant concentration with wind velocity, and equation (2), describing the evolution of wind velocity frequency with wind velocity. The equation to compute the mean annual concentrations continuously is given in (12).

Fig. 10 .

 10 Fig. 10. shows a comparison between the discrete methodology with the representative

Table 1 .

 1 Summary of the available data.

		Station				Data availability
	Location	Latitude	Longitude	Altitude	Time range	Number of valid cases	Number of missing cases
	Brest -Guipavas	48°27'00"N	4°22'59"O	94 m	2009 -2018	29,171	45
	Lille -Lesquin	50°34'12"N	3°05'51"E	47 m	2009 -2018	29,185	31
	Nîmes -Courbessac	43°51'24"N	4°24'22"E	59 m	2009 -2018	29,214	2
	Strasbourg -Entzheim	48°32'58"N	7°38'25"E	150 m

  𝑣) is the Weibull or the sigmoid function and 𝐹𝑉𝑅 [𝑎;𝑏[ is the wind frequency given in the 4-velocity-range data for wind velocities ranging from 𝑎 included to 𝑏 excluded.

				DOI : 10.1016/j.scs.2020.102221
	where 𝑓(
	𝑏
	). 𝑑𝑣 =	𝐹𝑉𝑅 [𝑎;𝑏[	(5)
	𝑎
	𝑓(0) = 0	(6)
	11/26

  that may participate in pollutant dispersion for low wind velocities or become preponderant. We suggest applying a constant pollutant concentration for wind velocities ranging from 0 to 𝑣 𝑚𝑖𝑛 and suggest using 𝐶 𝑚𝑎𝑥 = 𝑐(𝑣 𝑚𝑖𝑛 ). The choice of 𝑣 𝑚𝑖𝑛 is particularly important when using the optimized sigmoid function.

	𝐶 ̅ = 𝐶 𝑚𝑎𝑥 .	∫ 0 𝑣 𝑚𝑖𝑛 ∫ 𝑓(𝑣). 𝑑𝑣 𝑓(𝑣). 𝑑𝑣 +∞ 0	+	∫ 𝑣 𝑚𝑖𝑛 +∞ 0 ∫ 𝑓(𝑣). 𝑑𝑣 𝑐(𝑣). 𝑓(𝑣). 𝑑𝑣 +∞	+ 𝐶 𝑏𝑔 (13)

  velocity is a simple approach which can lead to considerable underestimations of pollutant concentrations, and it is better to use a representative velocity instead, as suggested in this paper. Using the representative velocity requires, however, making a choice for the first velocity range. The second problem concerns the velocity step used to build the wind velocity

	DOI : 10.1016/j.scs.2020.102221
	22/26

intermediate

Acknowledgments

We would like to thank the ANRT (Association Nationale de la Recherche et de la Technologie)

for their support and Météo-France for allowing us to use their data for this study.