M Santoro

Lorenzo Piola

Karl Mc Cabe

Olivier Lhost

Aurélien Katty Den Dauw

Alexandre Vantomme

Laurent Welle

Jean-François Maron

Evgueni Carpentier

Kirillov

Orlando Santoro

Katty Den Dauw

Aurélien Vantomme

Alexandre Welle

Laurent Maron
email: laurent.maron@irsamc.ups-tlse.fr

Jean-François Carpentier
email: jean-francois.carpentier@univ-rennes1.fr

Evgueni Kirillov
email: evgueni.kirillov@univ-rennes1.fr

François Carpentier

Long-Chain Branched Polyethylene via Coordinative Tandem Insertion and Chain-Transfer Polymerization Using rac-EBTHIZrCl2/MAO/Al-alkenyl Combinations: An Experimental and Theoretical Study

HAL is

INTRODUCTION

Behavior of polymers during processing is known to be strongly dependent on their properties such as molecular weight (M w ), polydispersity (M w /M n ) and, more generally, their molecular architecture (linear, branched or cross-linked). The development of single-site catalyst technology has allowed the production of polyethylene (PE) with high molecular weight and narrow polydispersity. 1 Although these characteristics confer excellent mechanical properties, such polymers suffer from poor processability due to their low elasticity and melt viscosity. In the 1990s, it has been shown that these properties can be improved by introducing long-chain branching (LCB) to linear PEs. 2,3,4 Generally, LCB-polymers can be produced by post-reactor technologies such as high-energy electron beam irradiation, 5,6 peroxide curing [START_REF] Parent | Structure-rheology relationships of long-chain branched polypropylene: Comparative analysis of acrylic and allylic coagent chemistry[END_REF][START_REF] Wang | Chemical modification of polypropylene with peroxide/pentaerythritol triacrylate by reactive extrusion[END_REF][START_REF] Tian | The preparation and rheology characterization of long chain branching polypropylene[END_REF][START_REF] Stanic | Influence of Different Types of Peroxides on the Long-Chain Branching of PP via Reactive Extrusion[END_REF] and grafting. [START_REF] Lu | Synthesis of long chain branched polypropylene with relatively well-defined molecular structure[END_REF] Yet, because of the radical mechanisms involved in these treatments, ample control over LCB formation is rather impossible; hence, polymers with complex structures are obtained in those cases. In-reactor technologies such as copolymerization of ethylene with either in situ-formed or previously isolated macromonomers, [START_REF] Chum | Materials properties derived from INSITE metallocene catalysts[END_REF][START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] or with a non-conjugated α,ω-diene 14,[START_REF] Pietikäinem | Copolymerization of ethylene and nonconjugated dienes with Cp 2 ZrCl 2 /MAO catalyst system[END_REF][START_REF] Pietikäinem | Copolymerization of ethylene and non-conjugated dienes with Cp 2 ZrCl 2 /MAO catalyst system: effect of polymerization temperature on the copolymer structure[END_REF][START_REF] Naga | Copolymerization of ethylene and 1,7-octadiene, 1,9-decadiene with zirconocene catalysts[END_REF][START_REF] Williamson | G Alternating copolymers of ethylene and diolefins containing pendent functional groups[END_REF][START_REF] Das | Controlled Synthesis, Characterization, and Flow Properties of Ethylene-Diene Copolymers[END_REF][START_REF] Liu | A Comprehensive Review on Controlled Synthesis of Long-Chain Branched Polyolefins: Part 3, Characterization of Long-Chain Branched Polymers[END_REF][START_REF] Wang | Copolymerization of propylene with Si-containing α,ω-diolefins: how steric hindrance of diolefins affects long chain branch formation[END_REF][START_REF] Uzawa | Direct Synthesis of High-Melt-Strength Polypropylene Using the Fourth Generation Heterogeneous Ziegler-Natta Catalyst and Commercial Production Process[END_REF][START_REF] Liu | Assessing 1,9-Decadiene/Propylene Copolymerization with Ziegler-Natta Catalysts to Long-Chain-Branched Polypropylene[END_REF] have also been developed. Nevertheless, only a limited number of catalyst systems have been shown to efficiently generate and incorporate macromonomers, while the second approach is limited by the rather scarce availability of α,ω-dienes and by the uneven distribution of LCB in the final polymers.

The detection of LCB (quantification, nature) is a difficult task, [START_REF] Shroff | Assessment of NMR and Rheology for the Characterization of LCB in Essentially Linear Polyethylenes[END_REF] for which three main different techniques have been employed: (1) solution [START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] C NMR spectroscopy, (2) size exclusion chromatography coupled with multi-angle laser light scattering (SEC-MALLS), and

(3) melt rheological measurements. NMR spectroscopy suffers from the limited sensitivity (largely beyond that where rheological effects can be observed) and the hard/impossible discrimination between short-and long-chain branches, while costly apparatus are required for the second technique. [START_REF] Stadler | Influence of type and content of various comonomers on long-chain branching of ethene/α-olefin copolymers[END_REF] On the other hand, melt rheological measurements proved to be very sensitive, enabling to evidence the presence of quite low contents of LCB. [START_REF] Stadler | Detecting very low levels of long-chain branching in metallocene-catalyzed polyethylenes[END_REF][START_REF] Takeh | Analytical rheology of metallocene-catalyzed polyethylenes[END_REF][START_REF] Stadler | Correlations between the Characteristic Rheological Quantities and Molecular Structure of Long-Chain Branched Metallocene Catalyzed Polyethylenes[END_REF] Recently, researchers at ExxonMobil have claimed in a patent the use of the hydroalumination product from diisobutylaluminum hydride (DIBAL-H) and 1,7-octadiene (referred to as "Al-octenyl" (Al-1)) as a promoter in the zirconocene-catalyzed polymerization of ethylene. [START_REF] Holtcamp | Aluminum alkyls with pendant olefins for polyolefin reactions[END_REF] Formation of LCB-PE was surmised, although no direct correlation with the Al promoter was made. In such a scenario, it can be assumed that LCB may originate from the (1,2-) insertion of a pending oct-7-enyl moiety into the ZrC bond of a growing polymer chain, followed by AlZr transmetallation, eventually generating a growing side-chain (Scheme 1). On the contrary, protonolysis of the intermediate species (during quenching/workup) would lead to the formation of n-hexyl short-chain branches (SCB).

Although industrially-promising, this approach suffers from drawbacks to decipher the nature of possible LCB and its mechanistic origin. In fact, the co-reactant consists of a mixture of mono-, di-and trisubstituted Al-oct-7-enyl species rather than of only one discrete species and, due to the similar nature of the alkenyl chains, discrimination between LCB and SCB in the final polymer cannot be performed by means of [START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] C NMR spectroscopy. In a very recent study, various bis(alkenyl)-Zn reagents were used as chain-transfer agents (CTA) in {pyridylamido}-hafnium-catalyzed ethylene/-olefin copolymerization, affording the corresponding LCB products. [START_REF] Lee | Synthesis of Long-Chain Branched Polyolefins by Coordinative Chain Transfer Polymerization[END_REF] Yet, the methodology remains widely unexplored since, to our knowledge, excepting the above example, no reports are available in the open literature concerning neither Al-based reagents nor other Malkenyl compounds.

Herein, we report on the synthesis of LCB-PEs mediated by iBuAl(oct-7-en-1-yl) 2 (Al-1) as co-reactant and their characterization by [START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] C NMR spectroscopy and melt rheology. We highlight the efficiency of Al-alkenyl reactants as LCB promoters, which operate via a tandem mechanism involving both coordination/insertion and transmetallation reactions, as corroborated by extensive DFT computations.

RESULTS AND DISCUSSION

Synthesis of Al-octenyl reagents. In order to prepare a mixed, well-defined isobutyl/oct-7-en-1-yl aluminum derivative, the hydroalumination reaction between DIBAL-H and a large excess of 1,7-octadiene was studied under the reported conditions (Scheme 2). [START_REF] Holtcamp | Aluminum alkyls with pendant olefins for polyolefin reactions[END_REF] The average composition of the reaction mixture was assessed by 1 H NMR spectroscopy, through integration of the signals corresponding to the Al-CH 2 (-octenyl) and Al-CH 2 (-i Bu) groups ( H 0.45 and 0.25 ppm, respectively; Figure S1). A series of signals corresponding to side-products was also detected. The principal side-product, identified as 7methylenepentadeca-1,14-diene, can result from the 1,2-insertion of an octenyl moiety into AlC bond followed by β-H elimination. (entry 4). The low M w value of the polymer is likely due to the capability of both H 2 and Al-1 to act as chain-transfer reagents, while the formation of crosslinks was probably suppressed due to the higher temperature, compared to entries 2 and 3 (60 and 40 ºC, respectively).

ZrCl 2 In order to study the effect of Al-1 on the formation of LCB, the soluble samples (PE1 and PE4) were analyzed by [START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] C NMR spectroscopy. A signal compatible with the CH-group at the branching point of the LCB ( CH 38.22 ppm) [START_REF] Hsieh | Ethylene-l-Butene Copolymers. 1. Comonomer Sequence Distribution[END_REF] was observed in the spectrum of the sample synthesized with MAO only (Figure S3). Based on the integration of this signal, the LCB density was estimated to be 0.3/10,000C. However, some resonances related to the sample oxidation (that allegedly occurred under the analytical conditions, i.e. 135 °C) were also detected. The presence of these resonances may affect the accuracy of LCB quantification, hence the value found must be considered as indicative. The 13 C NMR spectroscopic analysis of the PE prepared in the presence of Al-1 proved purposeless; indeed, the technique is unable to distinguish the tertiary carbon atoms of side-chains longer than 6 carbons (Scheme 1; Figures S4 andS5). [START_REF] Stadler | Detecting very low levels of long-chain branching in metallocene-catalyzed polyethylenes[END_REF][START_REF] Hsieh | Ethylene-l-Butene Copolymers. 1. Comonomer Sequence Distribution[END_REF] Hence, the unambiguous discrimination between LCB and n-hexyl-branches, possibly derived from the insertion of Al-1 followed by hydrolysis, could not be performed. Moreover, [START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] C NMR spectroscopy is poorly suited to low LCB content (vide infra). [START_REF] Stadler | Detecting very low levels of long-chain branching in metallocene-catalyzed polyethylenes[END_REF] In order to overcome these limitations, both samples were further examined by melt rheology. For this purpose, the trend of the modulus of the complex viscosity (| * |) as a function of the angular frequency () was first investigated (Figure 2a). In the whole range of frequencies considered (0.1250 rad.s 1 ), the viscosity of the PE synthesized with MAO only (PE1, entry 1) was higher than that of the sample prepared in the presence of Al-1 (PE4, entry 4). The van Gurp-Palmen (vGP) plot was next considered (Figure 2b). In this representation, curves of purely linear polymers tend to a phase angle () of 90 ° at low |G*|, while lower angles are observed upon increasing the LCB density and/or broadening of the molecular weight distribution (MDW). 32,33 The curves for both PE1 and PE4 were found to be far from the linear case, which is consistent with the above [START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] C NMR spectroscopy results and a significant presence of LCB. However, the polymer prepared with Al-1 showed lower phase angle than that of the PE synthesized with MAO only. Given that the polydispersity values of these two polymers were found to be rather similar (M w /M n = 3.3-3.6), the drop of  could be ascribed only to an increase of LCB content. The presence of LCB also in the PE produced in the absence of the Al-alkenyl promoter is not surprising, since the pre-catalyst herein employed is known to promote the formation of branches to a certain extent via macromonomer insertion. 34 For comparison purposes, a benchmark polymerization was carried out with only TIBAL as scavenger (PE5). Ethylene polymerizations were performed by increasing progressively the amount of Al-1 (PE6-PE11) and also by using mixtures of TIBAL/Al-1 (PE12-PE14). All polymerizations showed the same productivity, regardless of the amount of Al-1 used. In addition, thermal properties of the PEs were found to be similar. No direct correlation was found between the amount of AVTA introduced and the molecular weight (and MWD) of the resulting polymer, thus arguing against chain-transfer processes operating efficiently. On the other hand, remarkably, the rheological fingerprints of the samples were found to be highly affected by the use of Al-1. In fact, all samples prepared with the co-reactant exhibited higher

T m b [°C] T cryst b [°C] M w c [×10 3 ] M w /M n c 1 PE5 1,000 (350)
|*| at low frequencies and stronger shear thinning as compared with those PE5 (Figure 3a,b). Moreover, the magnitude of such variations was consistent with the amount of Al-1 employed. Since, in this case, M w effect can be excluded, the different rheological behaviors can be confidently attributed entirely to LCB formation. This assumption was confirmed by the vGP plots (Figure 3c,d). It has to be noted that all polymers have a decreasing delta towards the points with the lowest frequency. We assume to be in proximity of a minimum which cannot be observed due to the limited range of frequencies investigated. The presence of a minimum in vGP plots has been already reported in the literature, for example by Stadler et al. [START_REF] Stadler | Correlations between the Characteristic Rheological Quantities and Molecular Structure of Long-Chain Branched Metallocene Catalyzed Polyethylenes[END_REF]35,36,37 At the same time, no meaningful information could be obtained upon trying to correlate the double logarithmic derivative of the viscosity function (dlog| * |/dlog ) with the phase angle  (Figure S6). 38 The effect of H 2 on the process and polymer properties was also investigated. For this purpose, a second series of polymerization tests was carried out in the presence of a fixed amount of Al-1 and variable amounts of H 2 (Table 3). In order to achieve higher activity, 1-Page 12 of 30 hexene was added as co-monomer (cf. entries 3 and 4). 39 Under these reaction conditions, a drop of productivity and a decrease of the polymer molecular weight were observed, as the amount of H 2 in feed was increased; in fact, if too much H 2 is added, it rather behaves as a chain-terminating agent. On the other hand, the T m and polydispersity values were not significantly altered (except for the run conducted in the absence of the co-monomer), spanning in the ranges of 120122 °C and of 2.83.3, respectively. Excessively increasing the amount of H 2 had also a detrimental effect on LCB formation, as highlighted by their corresponding vGP plots (Figure 4). 40 In that case, the phase angle  increased with the amount of H 2 introduced, suggesting a drop of LCB density. 38,41 This could be attributed to the occurrence of side processes (chain-transfer, termination) competing at higher H 2 concentrations with monomer insertion and transmetallation reactions, preventing both the growth of the main PE chain and LCB formation. The polymers isolated in entries 3 and 4 were further analyzed by [START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] DFT studies on the LCB formation. Usually, the influence of an organoaluminium reagent in the polymerization process is not taken into account by a standard computational model. This is due to the assumption that such a co-catalytic species simply delivers the alkyl group to initiate the polymerization. In this DFT study, we aimed at rationalizing the mechanism of formation for long-chain branches during polymerization of ethylene with rac-{EBTHI}ZrCl 2 carried out in the presence of iBuAl(oct-7-en-1-yl) 2 (Al-1) (Scheme 3). Scheme 3. An outline of the stepwise protocol produced to gain further insights into possible mechanisms of branching in polyethylene. Three main investigations (highlighted steps in green) were conducted, in respective order, concerning: 1) the initial coordination of the co-catalyst to the catalyst.

T m b [°C] M n c [×10 -3 ] M w c [×10 -3 ] M w /M n c 1 PE15 TIBAL [
2) The resulting effects on ethylene polymerization.

3) The control the co-catalyst may introduce on the mechanism of insertion.

The first three ethylene insertions were computed with the cationic form of the active catalyst [rac-{EBTHI}ZrMe] + (Figure S8). The reaction was found to be favored both kinetically and thermodynamically (highest barrier of 6.8 kcal•mol 1 for the first insertion and exothermic by ~20 kcal•mol 1 each step). This is in line with the experimental ability of the EBTHI-Zr catalyst to efficiently polymerize ethylene. However, the overall situation is different when Al-1 is introduced in the polymerization due to the presence of, not only an alkyl but also, alkenyl chains on the co-catalyst. It was then suggested that Al-1 could coordinate to the [rac-{EBTHI}ZrMe] + catalyst generating an Al/Zr heterobimetallic species.

Depending on the nature of the 'bridge' formed between the Zr and Al atoms, upon the binding of Al-1 to [rac-{EBTHI}ZrMe] + , three different isomers can be distinguished (Scheme 4). A comparison of the various coordination affinities between the three isomers and the parent [rac-{EBTHI}ZrMe] + system showed isomer A to be the most stabilized, with the bimetallic complex gaining 4.8 kcal•mol 1 . Although the stabilizing effect is rather small, the influence of the co-catalyst coordination on the ethylene polymerization needed to be investigated. It was found that two pathways for insertion of an ethylene monomer exist: after dissociation (Scheme 5) or whilst Al-1 remained coordinated (Figure S9). As a result, the insertion of ethylene via a systematic dissociation of the co-catalyst is kinetically preferable (Figures S10 andS11). Interestingly, regardless of the pathway ethylene is inserted, the re-coordination of the co-catalyst is thermodynamically most favorable, affording the same [rac-{EBTHI}Zr(n-propyl)-Al(oct-7-en-1-yl) 2 iBu] + bimetallic species. 2.

Scheme 6. The dissociation of Al-1 respective of: (1) Alalkyl (1) or (2) Al(oct-7-en-1-yl) bond cleavage. Note that the latter mechanism involves the transmetallation of the alkyl and alkenyl chains between Al and Zr centers.

Cleavage of the Al(oct-7-en-1-yl) bond (pathway 2) owes to an alkenyl chain-shuttling from the co-catalytic species to the catalyst system forming the [rac-{EBTHI}Zr(oct-7-en-1-yl)] +

species. An investigation was conducted into the required energy input needed to cleave either the Almethyl or Alalkenyl bonds. It was concluded that the Alalkenyl bond was slightly stronger, with a dissociation energy of 6 kcal•mol 1 relative to 4.8 kcal•mol 1 for the Almethyl bond. A difference of 1.2 kcal•mol 1 gives little concrete evidence of one bond cleavage being definitively favorable from the another but may offer insight into a competitive environment that allows the co-catalytic species to dissociate. An Alalkenyl group transfer to form [rac-{EBTHI}Zr(oct-7-en-1-yl)] + species brings about a Zrpolymeryl moiety with a terminal vinyl group after one step (considering the formation of the bimetallic species as a reference).

Considering the length of the alkenyl chain, an intramolecular cyclisation was considered and investigated. It was found that subsequent insertions of ethylene into the [rac-{EBTHI}Zr(oct-7-en-1-yl)] + chain are plausible (Scheme 7) and, although a cyclisation is Page 19 of 30 feasible, it is kinetically less favorable than the adduct-less insertions of ethylene (with a corresponding transition state barrier difference of 7.2 kcal•mol 1 ; Figure S12). This gives sufficient evidence to suggest that an Alalkenyl chain transfer provides a significantly more facile one-step precursor to longer chain formations. In comparison to a cyclisation reaction, the ethylene insertions are far more likely and the adduct-less nature of this owes to a faster generation of a linear Zr-polymeryl species.

Zr Al 7.9

Scheme 7. The profile for an insertion of ethylene monomer to [rac-{EBTHI}Zr(oct-7-en-1-yl)] + formed via the cleavage of the Alalkenyl bond and subsequent oct-7-en-1-yl group transfer. Note, the adduct-less interaction of the monomer suggests a more rapid insertion of ethylene.

The cleavage of the Alalkyl bond results in a competition between the insertion of ethylene monomer and dissociation of the Al-1 moiety. Therefore, the subsequent insertions will occur in a three-step process: 1) dissociation of co-catalyst 2) insertion of ethylene 3) recoordination of Al-1 to form the more stable bimetallic species. Thus far in this study, we have highlighted that the dissociation of Al-1 is preferable prior to the insertion of ethylene.

Hence, the next comparison to be made was the potential effects that subsequent insertions induce and examine whether the length of the alkyl/polymeryl chain R on the [rac-{EBTHI}ZrR] + system affects either the incorporation of Al-1, the insertion of ethylene, or both. The computed coordination energies for the growing polymer chain generated via first, second and third ethylene insertions are summarized in Table 4. The presence of Al-1 provides a stabilizing interaction with [rac-{EBTHI}ZrR] + throughout each insertion step. Generally, the extent to which the co-catalyst stabilizes the bimetallic species seems to decrease as the ZrR chain length increases. In contrast, ethylene coordination provides little, if any, stabilizing effects and thus a competition between the alkenyl moiety of the co-catalyst and monomer becomes more prevalent as the polymer chain grows.

Since Al-1 possesses two alkenyl chains with a terminal vinyl group that may compete with the ethylene coordination, once the co-catalyst has dissociated, the coordination and subsequent insertions of the vinyl group from oct-7-en-1-yl was investigated (Scheme 8). The adducts produced by either monomer or co-catalyst coordination differ greatly, whereby the initial interaction of ethylene is destabilizing (+4.0 kcal•mol 1 ) and the vinyl group coordination could arguably be in an equilibrium with the bimetallic resting state.

Nevertheless, the vinyl coordination is much more stabilizing, and therefore favored, in comparison to the ethylene monomer. This concept is also in agreement with the weak coordination/dissociation proposal of the Al-1 co-catalyst. Kinetically, comparing the transition state barriers in Scheme 9, one observes that the vinyl 1,2-coordination/insertion pathway is in direct competition with the insertion of ethylene. The vinyl group 1,2-insertions generally become more viable as the chain length increases, referring to the competitive kinetics of the first, second and now third insertions of ethylene (Figures S12 andS13).

Although the stabilities of the products differ, it is an interesting propensity that the vinyl coordination to the catalyst produces a more stable adduct, relative to the ethylene coordination/insertions. Therefore, our model suggests a growing competition that kinetically favors the formation of a branched product over a linear polymer.

CONCLUDING REMARKS

The Al-alkenyl compound Al-1 has been synthesized and employed as a LCB-promoter in the Though rheological studies were employed as the main analytical technique in this study, they were not aimed at quantifying the amount of LCBs. Given the fact that, to reach such a quantitative analysis, a wider range of frequencies should be collected, only a trend for a very restricted range of samples was examined with a wish to understand the role of Al-1 co-catalyst as an efficient promoter of LCBs.

Because the rac-{EBTHI}ZrCl 2 /MAO system has the intrinsic capability to generate LCB (minimal but noticeable amounts), forthcoming investigations shall be directed toward the evaluation of Al-1, as a co-catalyst and promoter of LCBs, in the presence of a catalyst system that, by itself, does not auto-generate LCB. Hence, experimental and computational studies on the analogous {n-BuCp} 2 ZrCl 2 /metal-alkenyl systems are currently under way in our research groups to provide a complete rationale of the different reactivity of such Malkenyl co-reagents; the results of these investigations will be published in due course.

ASSOCIATED CONTENT

Supporting information is available free of charge on the ACS Publications website at DOI: synthetic procedures, NMR and rheological analyses of polymers.

AKNOWLEDGMENTS

This work was gratefully supported by Total SA and Total Research and Technology Feluy (postdoctoral fellowships to OS, LP and KMG). The dlog| * |/dlog  vs.  plots for several samples, namely PE8, PE10, PE11, PE14, showed a certain deviation from linear relation (Figure S6). This may stem from some machine errors at low torque, although the raw data was manually checked and no anomalies were detected. Another explanation is that the true Newtonian viscosity value, which we could get from the approach proposed, is apparently underestimated; see:

REFERENCES AND NOTES

Scheme 1 .

 1 Scheme 1. Possible generation pathway and[START_REF] Weng | Synthesis of long-chain branched propylene polymers via macromonomer incorporation[END_REF] C NMR spectroscopic fingerprint of short-(SCB) and long-chain branching (LCB) obtained in the presence of Alalkenyl co-reactant (Pol= polymeryl chain).[START_REF] Hsieh | Ethylene-l-Butene Copolymers. 1. Comonomer Sequence Distribution[END_REF]

Figure 1 .

 1 Figure 1. Structure of rac-{EBTHI}ZrCl 2 (Zr-1).

Figure 2 .

 2 Figure 2. Rheological properties of PE samples obtained with Zr-1/MAO/Al-1 systems: (a) complex viscosity (|*|) vs angular frequency ( plot; (b) van Gurp-Palmen plot. Shear strain (γ) 10%, angular frequencies (ω) from 0.1 to 250 rad . s 1 , T = 190°C.

Figure 3 .

 3 Figure 3. Rheological properties of the PEs obtained with supp-Zr-1/TIBAL/Al-1 systems: (a, b) complex viscosity (|*|) vs angular frequency ( plot, (c, d) van Gurp-Palmen plot. Shear strain (γ) 10%, angular frequencies (ω) from 0.1 to 250 rad . s 1 , T = 190 °C.

  25 g; T pol = 85 °C; P(ethylene) = 25 bar (6 wt%); 1-hexene 2.44 wt%; time = 60 min; supp-Zr-1 = 5.0 mg ([Zr] 0 = 2.0 μM). b Determined by DSC from the second run. c Determined by SEC. d No 1-hexene added.

Figure 4 .

 4 Figure 4. Van Gurp-Palmen plot of the PEs obtained with supp-rac-{EBTHI}ZrCl 2 /TIBAL/Al-1 system with different amounts of H 2 . Shear strain (γ) 10%, angular frequencies (ω) from 0.1 to 250 rad . s 1 , T = 190°C.

Scheme 4 .

 4 Scheme 4. Isomers of Al/Zr heterobimetallic intermediates obtained upon coordination of Al-1 to [rac-{EBTHI}ZrMe] + . Each isomer can be considered a rotamer with respect to the positioning of the alkyl and alkenyl chains.

Page 16 of 30 IsomerScheme 5 .

 305 Scheme 5. The energy profile for the coordination of Al-1 to [rac-{EBTHI}ZrMe] + followed by the insertion of ethylene via a dissociative pathway. The re-coordination of the co-catalyst species is shown, forming the thermodynamically stable [rac-{EBTHI}Zr(nPr)] + .

Figure 4 .

 4 Figure 4. Transition state geometries for the first ethylene insertion: (top) the dissociation of Al-1 prior to ethylene insertion, (bottom) the non-dissociative transition state.

Scheme 8 .Scheme 9 .

 89 Scheme 8. The two-step mechanism for the re-coordination and insertion of a vinyl group from the Al-1 cocatalyst.

  polymerization of ethylene catalyzed by homogeneous rac-{EBTHI}ZrCl 2 /MAO or heterogeneous MAO on silica-supported-rac-{EBTHI}ZrCl 2 /TIBAL catalytic systems. The use of this Al-based co-reagent induces a marked change in the rheological behavior of the PEs, unambiguously consistent with LCB generation. While the nature of LCB could not be unambiguously corroborated by 13 C NMR spectroscopy, it probably constitutes a regular Cbranched LCB microstructure, and most likely arises from efficient tandem insertion and transmetallation processes between Al-and Zr-species. The intermediacy of Al/Zr heterobimetallic species, resulting from reversible coordination of Al-1 to [rac-{EBTHI}ZrR] + , was established and for the first time scrutinized in details by DFT computations.

{EBTHI}ZrCl 2 /MAO/Al-alkenyl systems. For

			Procedures
	+ DIBAL-H	A and B	iBu n Al	3	m	+	6	4
	excess			(n + m = 3)
				Al-1 (m/n = 2)
	Procedure A : (1) 110 °C, 1 h; (2) 70 °C, 16 h
	B : 85 °C, 16 h
	Scheme 2. Synthesis of Al-1.
	The amount of side-product was determined from the ratio of the signal area at  H 4.82
	ppm (terminal =CH 2 protons) to the total area of the CH 2 Al signals. Complete removal by a
	high-vacuum distillation of the side-product failed. Therefore, the original reaction conditions
	were optimized to minimize the formation of the undesired compound. Thus, by performing
	the reaction under slightly modified conditions (Procedure B: 85 °C, reaction time 16 h), the
	targeted product Al-1 exhibiting the octenyl/ i Bu ratio (m/n) close to 2 was obtained in a
	selective and reproducible manner (ESI, Figure S2).
	Polymerization	of	ethylene	catalyzed	by		homogeneous	rac-
				this investigation, rac-{EBTHI}ZrCl 2 (Zr-1,
	Figure 1) was selected as pre-catalyst. The polymerization of ethylene in the presence of Zr-1
	/MAO/Al-1 system was studied (Table 1). Compared to the benchmark reaction conducted
	without Al-promoter (entry 1), significantly lower productivities were achieved for the
	reactions performed at 40 °C (entries 2 and 3). The polymers isolated in these runs (PE2 and
	PE3) exhibited poor solubility in 1,3,5-trichlorobenzene at 135 °C and lower T m and T cryst
	values (126128 °C and 114 °C, respectively). The insolubility of these materials was
	attributed to the alleged formation of crosslinks promoted by Al-1. A soluble sample (PE4)
	characterized by SEC was obtained by carrying out the reaction at 60 °C in the presence of H 2

Table 1 :

 1 Polymerization of ethylene in the presence of rac-{EBTHI}ZrCl 2 /MAO/Al-1. a

	Entry	Sample ID	Al-1 [Al/Zr]	T polym [°C]	Productivity [kg Pol /kg Zr ]	T m c [°C]	T cryst c [°C]	M w d [10 3 ]	M w /M n d
	1 e	PE1	-	60	8,170	135.8	117.8	141.3	3.3
	2	PE2	500	40	3,330	126.8	114.6	Insoluble
	3 f	PE3	500	40	5,170	128.5	114.5	Insoluble
	4 f	PE4	500	60	7,670	127.7	115.5	94.0	3.6
	a Polymerization conditions: 300 mL-high pressure glass reactor; solvent: toluene, 150 mL; P(ethylene) = 4 bar;
	time = 15 min; [Zr] 0 = 10 μM, [Al MAO ]/[Zr] 0 = 4,500. c Determined by DSC from second run. d Determined by
	SEC. e [Al MAO ]/[Zr] 0 = 5,000. f H 2 = 800 ppm.

Table 2

 2

). Page 9 of 30

Table 2 .

 2 Polymerization of ethylene in the presence of supp-Zr-1/TIBAL/Al-1. a

	Entry	Sample ID	TIBAL [ppm] (Al/Zr)	Al-1 [ppm] (Al/Zr)	Productivity [kg Pol /kg cat ]

Table 3 .

 3 Effect of H 2 concentration on the polymerization of ethylene in the presence of supp-

	Zr-1/Al-1. a
	Entry	Sample ID	Co-reactant [ppm]	H 2 [ppm]	Productivity [kg Pol /kg cat ]

Table 4 .

 4 A comparison between the respective coordination/re-coordination energies of the co-catalyst and monomer species.

  Elastic substantially linear olefin polymers. U.S. Pat. 5,272,236, 1993 (to Dow Chemical Co.). Canich, J. A. M.; Dias, A. J.; Bamberger, R. L.; Licciardi, G. G.; Henrichs, P. M. Long chain branched polymers and a process to make long chain branched polymers. WO Pat. 94/07930, 1994 (to ExxonMobil Chemical co.). Kim, Y. S.; Chung, C. I.; Lai, S. Y.; Hyum, K. S. Melt rheological and thermodynamic properties of polyethylene homopolymers and poly(ethylene/α-olefin) copolymers with respect to molecular composition and structure. J. Appl. Polym. Sci. 1996, 59, 125137. Trinkle, S.; Friedrich, C. Van Gurp-Palmen-plot: a way to characterize polydispersity of linear polymers, Rheol. Acta 2001, 40, 322-328. 33 Trinkle, S.; Walter, P.; Friedrich, C. Van Gurp-Palmen Plot II -classification of long chain branched polymers by their topology Rheol. Acta 2002, 41, 103-113. 34 Malmberg, A.; Kokko, E.; Lehmus, P.; Löfgren, B.; Seppälä, J.V. Long-Chain Branched Polyethene Polymerized by Metallocene Catalysts Et[Ind] 2 ZrCl 2 /MAO and Et[IndH 4 ] 2 ZrCl 2 /MAO, Macromolecules 1998, 31, 8448-8454. Correlation between the shape of viscosity functions and the molecular structure of long-chain branched polyethylenes, Macromol. Mater. Eng. 2009,

	294, 25-34.

1 Resconi, L.; Cavallo, L.; Fait, A.; Piemontesi, F. Selectivity in propene polymerization with metallocene catalysts. Chem. Rev. 2000, 100, 12531346. 2 Lai, S.-Y.; Wilson, J. R.; Knight, G, W.; Stevens, J. C.; Chum, P.-W. 3 Brant, P.; 4 35 Stadler, F. J.; Munstedt, H. 36 Stadler, F. J.; Chen, S.; Chen, S. On "modulus shift" and thermorheological complexity in polyolefins, Rheol. Acta 2015, 54, 695-704. 37 Yan, Z-C.; Stadler, F. J. Classification of thermorheolegical complexity for linear and branched polyolefins, Rheol. Acta 2018, 57, 377-388. 38

  Stadler, F. J.; Munstedt, H. Numerical description of shear viscosity functions of longchain branched metallocene-catalyzed polyethylenes, J. Non-Newtonian Fluid Mech.

	2008, 151, 129135.

39 Yang, Q.; Jensen, M. D.; McDaniel, M. P. Alternative View of Long Chain Branch Formation by Metallocene Catalysts, Macromolecules 2010, 43, 88368852.

It can be noted that the quality of the rheological data reported in Fig. 3 and4 is remarkably different. The analyses have been performed in two different laboratories, albeit with the same type of Rheometer and under the same operative conditions. This may be the origin of the difference observed.

41

The dlog| * |/dlog  vs.  plots for samples PE15PE22 appeared to be generally linear (Figure S7).

Page 30 of 30


Christopher Merlhe
email: christopher.merlhe@irisa.fr

Cedric Gueguen
email: cedric.gueguen@irisa.fr

Dynamic Cell-Less Radio Access Network Meta-Scheduler for High System Capacity Increase

Keywords: Wireless Network, Interference management, Cell-less, C-RAN, Opportunistic Scheduling

Radio resources are limited and subject to interferences, multi-path fading, shadowing and path-loss which highly impact the transmission efficiency. Optimizing the usage of these resources is the main issue in wireless networks. The resource management can be performed at different levels: in intracellular context, it is carried out by scheduling algorithms while in multi-cellular context, it is done by interference management strategies. This paper aims to merge these two steps. The proposed solution is compatible with the most acknowledged schedulers in order to optimize spectrum usage in intra-cell domain while efficiently decreasing the interferences in multicell domain. An innovative strategy is to consider the network as an "hyper-cell" rather than a sum of cells quite independent. This strategy often called "Cell-less" is especially effective in a Cloud Radio Access Network (C-RAN) architecture because it allows to centralize the decision making. This is more suitable for interference management as it provides a better flexibility to the system by retrieving and managing data from several cells that allows an overall performance increase. The proposed solution called Dynamic Cell-less Radio Access Network Meta-Scheduler (DC-RAN-MS) dynamically handles for each cell the management of radio resources depending on the interferences potentially experienced by users. Performance evaluation shows that the DC-RAN-MS offers an increased system capacity by optimizing the usage of bandwidth while reducing the magnitude of interferences received.

I. INTRODUCTION

Guaranteeing a good Quality of Service (QoS) and Quality of Experience (QoE) to the users is challenging in wireless networks. The continuously growing needs in term of throughput, system capacity and delay requirements require to optimize transmission efficiency particularly in ultra dense networks. In this way, scheduling algorithms are a crucial issue to achieve this goal.

Traditional solutions like Round Robin (RR) [START_REF] Minelli | Scheduling Impact on the Performance of Relay-Enhanced LTE-A Networks[END_REF][START_REF] Kuurne | Weighted Round Robin scheduling strategies in (E)GPRS radio interface[END_REF] and Random Access (RA) schedulers were inherited from the wired context. These solutions do not take into account the physical layer information. Thus, they neglect the channel state attenuations due to the path loss, the shadowing and the multipath fading. This last induces high variations of throughput over the time and on each sub-carrier. Consequently, RR and RA solutions provide a poor throughput and system capacity, making them unsuitable for the wireless network specificities [START_REF] Minelli | Scheduling Impact on the Performance of Relay-Enhanced LTE-A Networks[END_REF].

Consequently, many works have been done to design opportunistic scheduling algorithms (i.e. using the physical layer information). For instance, Maximum Signal to Noise Ratio (MaxSNR) takes benefits from the multi-user and frequency diversities by allocating the Resource Units (RUs) to mobiles with the best Signal-to-Noise Ratio (SNR) [START_REF] Wong | Multiuser OFDM with adaptive subcarrier, bit, and power allocation[END_REF][START_REF] Bechir Dadi | Scheduling performance's study for LTE downlink system[END_REF]. Its good throughput, high system capacity and its ease of implementation make it one of the most acknowledged scheduler.

However, its unfair behavior regarding the distance of mobiles from the access point (due to the path loss) leads to develop different resource allocation algorithms like the Proportional Fair (PF) and PF-based schedulers [START_REF] Minelli | Scheduling Impact on the Performance of Relay-Enhanced LTE-A Networks[END_REF][START_REF] Viswanath | Opportunistic beamforming using dumb antennas[END_REF][START_REF] Hamouda | A Cross-Layer Downlink Scheduling Scheme for Balancing QoS in[END_REF][START_REF] Ge | Cdf-based scheduling algorithm for proportional throughput fairness[END_REF]. These scheduling algorithms perform a fairer resource allocation which increases the global QoS of the system.

Nevertheless, these solutions have the similarity of being built for an intra-cellular context. It means that they does not manage efficiently Inter-Cell Interference (ICI). In this way, interference management solutions are another level of the performance control. Depending on the interference management algorithm chosen, the global QoS of a system can highly vary. Classical approaches like Reuse 1 and Sliced Bandwidth (for instance the Reuse 3) [START_REF] Elayoubi | Performance evaluation of frequency planning schemes in OFDMA-based networks[END_REF], make cells to squarely ignore interference management or to limit the usage of their bandwidth. Usually, with these classical solutions, cells are managed independently of each other. The Reuse 1 solution uses the completeness of the bandwidth without avoiding ICI. This scheme uses all the available resources but the interference impacts on the radio transmission quality is high, especially for mobiles in cell edges. Thus, the most distant mobiles from their access point will be highly penalized considering their poor SNR (due to the path loss) and the magnitude of interferences received (regarding their proximity to the neighboring cells).

The classical ICI avoidance solution is the Sliced Bandwidth. This scheme splits its frequency band according to the number of neighboring cells. In this way, there is no interference and the mobiles in cell edges are protected. However, there is a high unused bandwidth leading to a poor system capacity. In [START_REF] Qian | Adaptive Soft Frequency Reuse Scheme for Wireless Cellular Networks[END_REF][START_REF] Kumar | Impact of Sub-Band Correlation on SFR and Comparison of FFR and SFR[END_REF], the Fractional Frequency Reuse (FFR) and Soft Frequency Reuse (SFR) solutions are presented. These classical approaches perform a Sliced Bandwidth strategy in cell edges while in inner cell, a Reuse 1 strategy is used. In addition, SFR solution provides different power levels between the inner cell and edges, leading to a better usage of the bandwidth. Despite that the performances are increased compared to Reuse 1 and Sliced Bandwidth solutions, SFR and FFR schemes can not adapt their behavior to the mobile distribution as frequencies allocated to each cell are fixed beforehand.

All these classical solutions have a static behavior and perform their ICI only according to the mobile locations. Although the decision attributing resource blocks is taken in multi-cell domain, the allocation is performed in intracell domain which prevents the processing of special cases. For instance, they do not take into account the Signal-to-Interference-plus-Noise Ratio (SINR) as well as the information that a mobile could be not interfered at a given time. This leads to inaccurate allocations and a poor flexibility regarding the channel state and the mobile distribution in the cell. Thus, designing an interference management approach in a C-RAN architecture seems to be more effective as it allows to centralize the decision making and to retrieve data from several access points.

Traditionally in C-RAN architecture [START_REF] Checko | Cloud Radio Access Network architecture. Towards 5G mobile networks[END_REF][START_REF] Park | Robust and Efficient Distributed Compression for Cloud Radio Access Networks[END_REF], base stations are split into three parts : the Remote Radio Head (RRH), the Baseband Unit (BBU) and the optical link with highperformance, low delay and high bandwidth front-haul which connects the RRH to the cloud pool where BBUs are gathered. In C-RAN, most of the computational tasks can be considered as moved to BBU pool such as the channel state information estimation, the centralized signal processing management of the RAN etc [START_REF] Wang | Joint Energy Minimization and Resource Allocation in C-RAN with Mobile Cloud[END_REF]. On the other hand, the RRHs include radio antennas with their associated amplifier and are dispatched among several remote sites [START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for Cloud Radio Access Networks[END_REF]. Thus, It is possible to distinguish different C-RAN oriented approaches.

The first is the one-to-one logical mapping (Fig. 1(a)) where one BBU is assigned to one RRH. Although BBUs are gathered in the same BBU pool which can reduce the maintenance costs, this mapping only allows to manage one RRH at the same time. This makes ICI management solutions to have a static behavior. Consequently, it is not suitable to react to the channel state variations and to the user mobility.

Another approach is the one-to-many mapping (Fig. 1(b)), where one BBU is assigned to several RRHs. It allows to open new prospects. For instance, an innovative approach is to consider the network as "Cell-less" [START_REF] Li | Edge Cloud and Underlay Networks: Empowering 5G Cell-Less Wireless Architecture[END_REF][START_REF] Han | 5G Converged Cell-Less Communications in Smart Cities[END_REF]. The network is seen as an "hyper-cell" rather than a sum of independent cells. The cells become a set of access points and the decision making for this cluster is fully centralized. In this way, the "Cell-less" approach with the one-to-many mapping enables to manage several cells with only one entity. Thus, ICI can be fully controlled in a multi-cellular oriented approach which allows to share frames between RRHs providing a better usage of the bandwidth while efficiently reducing the magnitude of interferences received.

In [START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for Cloud Radio Access Networks[END_REF], the Hybrid Static (HS) strategy is presented. It is based on the analyze that giving resources to mobiles from different cells, at the same time and on the same frequency is better than protected them when the magnitude of interferences received is not significant. Thus, the HS splits its bandwidth into two slices. In the first slice, a Reuse 1 strategy is used and it is allocated to mobiles in inner cell. On the other hand, the second slice is allocated to mobiles in cell edges where a Sliced Bandwidth strategy is performed. In this slice, the HS compares the channel state of mobiles chosen by the intracell scheduler and allocates the resource to the one with the best SNR. Thus, this slice is fully shared between the RRHs which allows to optimize the resource allocation regardless of the number of mobiles in the edges of cells considered. Consequently, the HS solution makes a better usage of the spectrum than SFR/FFR approaches. However, with a static boundary between the two slices, the HS solution relies on an almost evenly distribution of mobiles between the inner and the edges of the cell. This makes this solution to have a static behavior which can hardly adapt to the context. Moreover, depending on the mobile distribution, one slice reaches the congestion before the other. This paper aims to make a better usage of the one-to-many mapping by proposing the Dynamic Cell-less Radio Access Network Meta-Scheduler (DC-RAN-MS). The DC-RAN-MS relies on the following analysis:

• ICI management performed in intra-cell domain often induces a waste of bandwidth due to frequencies allocated beforehand. For instance, classical solution does not take into account that a mobile could not be interfered at a given time. • Static solutions can not react to mobile distribution and to the channel state. In this way, they neglect that mobiles in cell edges could have an enough significant SINR to perform a Reuse 1 strategy (and conversely).

Consequently, the ICI management should coordinate with the schedulers in order to provide a more accurate resource allocation. Thus, the DC-RAN-MS aims to merge the intra-cell scheduling and the ICI management by controlling a cluster of RRHs. The DC-RAN-MS acts as a meta-scheduler by allowing or preventing schedulers to allocate resources according to: the channel state (i.e. the SINR/SNR), the magnitude of interferences experienced and the number of mobiles available to transmit in each RRH. Unlike the previously described solutions, the usage of a Reuse 1 or Sliced Bandwidth strategy does not rely on the position of mobiles in cells but on interferences themselves. The entire bandwidth is shared between the cluster of RRHs considered and resources unallocated by a RRH can be used by another. This provides a more accurate and dynamic ICI management which reduces the bandwidth waste and the magnitude of interferences experienced.

This paper is organized as follow: section II describes the proposed solution and performance evaluation is explained in section III. This paper ends with the conclusion in section IV.

II. DYNAMIC CELL-LESS RADIO ACCESS NETWORK META-SCHEDULER (DC-RAN-MS)

A. Cell-Less model

In this work, the global available bandwidth is assumed divided in sub-frequency bands called sub-carriers. The radio resources are distributed in the time domain in frames where each frame is divided in Time Slot (TS) of constant duration. A Resource Unit is defined as a pair (sub-carrier, time slot). The channel gain between the RRH i and the user k on the sub-carrier n is given by :

G i k,n = h × 10 Xσ 10 × ( 1 d k,i ) α ( 1
)
where h represents the Rayleigh multi-path fading, which is modeled by and exponential distribution, X is a standard Gaussian random variable, σ is the standard derivation of shadowing in dB, d k,i is the distance between the mobile k and the RRH i and α is the path loss exponent. We denote C, the set of L neighboring RRHs of the RRH i considered mapped to the BBU of the DC-RAN. Thus, the Signal-to-Interference-plus-Noise Ratio of user k on sub-carrier n associated to RRH i is given by :

γ i k,n = P i n G i k,n B sub N 0 + L (j=1,j∈C) P j n G j k,n (2)
where P i n and P j n are respectively the transmitted power on sub-carrier n of RRH i and the transmitted power on subcarrier n of the interfering RRH j that belongs to the set C. G j k,n is the channel gain between the mobile k and RRH j, the parameter N 0 is the thermal noise power density, and B sub is the sub-carrier spacing [START_REF] Ezzaouia | Autonomous and Dynamic Inter-Cell Interference Coordination Techniques for Future Wireless Networks[END_REF].

To compute the spectral efficiency η k,n of mobile k on subcarrier n associated to RRH i, the Shannon's formula is used such as:

η i k,n = log 2 (1 + γ i k,n Γ ) (3)
with Γ, a SNR correction factor that takes into account the difference between the information-theoretic performances and the practical implementation of the MCS [START_REF] Seo | A proportional-fair power allocation scheme for fair and efficient multiuser OFDM systems[END_REF] defined as follows:

Γ = - ln(5.E) 1.5 (4)
where E is a BER Target.

B. The proposed solution

The DC-RAN-MS relies on a one-to-many mapping within the scope of a "Cell-less" approach. In this way, it manages several RRHs at the same time and aims to coordinate with the intra-cell schedulers. The proposed solution is compatible with the most acknowledged schedulers while emphasizing their specific features (throughput and system capacity for the MaxSNR, system capacity and fairness for the PF etc.). The mobile m is defined as the mobile chosen by the scheduler on the sub-carrier n in the RRH i. For instance, with a MaxSNR allocation, m is computed such as:

m = argmax k (η i k,n ) (5)
The DC-RAN-MS avoids ICI when their magnitude is high by restricting the usage of a sub-carrier n for a given TS. Otherwise, the proposed solution uses the entirety of the spectrum. This decision is taken according to the channel state of the mobile m in each RRH. The η i m,n interf is defined as the η i m,n while the mobile m of the RRH i is interfered on the sub-carrier n. In this way, the DC-RAN-MS performs its resource allocation following this inequality:

L (i=1,i∈C) η i m,n interf ≥ max η i m,n , ∀i ∈ C (6)
Thanks to this inequality, the DC-RAN-MS can deduce the number of mobiles available to transmit (i.e. if they are interfered). Depending on their channel states, the proposed solution performs different frequency reuses. When there is no interference or their magnitudes are too high the DC-RAN-MS uses a Sliced Bandwidth strategy. Otherwise, if the interferences are not significant, it performs a Reuse 1 scheme. Consequently, the DC-RAN-MS does not rely on a criteria in correlation with interferences (i.e the distance like state of the art solutions) but directly on the interferences themselves.

C. Operation of the DC-RAN-MS

The figure 2 illustrates some solutions previously described. For each solution, we consider two frames belonging respectively to the RRH i and the RRH j. Within the RRHs, the scheduling algorithm performed is a MaxSNR. The RUs are allocated TS per TS and denoted by (sub-carrier, Time Slot). A mobile is embodied by a specified color. The rate of color filling illustrates the spectral efficiency variations regarding the interferences received. To provide a realistic scenario, the mobiles have different needs in term of application requirements which leads to different RU allocations between the mobiles. The green and purple mobiles are considered in cell edges while the others are in the inner. In addition, the red mobile has the most packets to transmit.

The Reuse 1 strategy (Fig. 2(a)) uses the entirety of the bandwidth without attempting to avoid ICI. Consequently, the green and purple mobiles are highly interfered and their spectral efficiency is significantly downgraded. This makes them to need much more RUs than usual to end their transmission. As the resource allocation is performed TS per TS in this example, the red mobile takes advantage that others mobiles have ended their transmission to transmit on the sub-carriers they previously used (RUs : [START_REF] Wong | Multiuser OFDM with adaptive subcarrier, bit, and power allocation[END_REF][START_REF] Park | Robust and Efficient Distributed Compression for Cloud Radio Access Networks[END_REF], [START_REF] Viswanath | Opportunistic beamforming using dumb antennas[END_REF][START_REF] Park | Robust and Efficient Distributed Compression for Cloud Radio Access Networks[END_REF], [START_REF] Wong | Multiuser OFDM with adaptive subcarrier, bit, and power allocation[END_REF][START_REF] Wang | Joint Energy Minimization and Resource Allocation in C-RAN with Mobile Cloud[END_REF], [START_REF] Viswanath | Opportunistic beamforming using dumb antennas[END_REF][START_REF] Wang | Joint Energy Minimization and Resource Allocation in C-RAN with Mobile Cloud[END_REF], [START_REF] Kuurne | Weighted Round Robin scheduling strategies in (E)GPRS radio interface[END_REF][START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for Cloud Radio Access Networks[END_REF] and ends on the RU [START_REF] Wong | Multiuser OFDM with adaptive subcarrier, bit, and power allocation[END_REF][START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for Cloud Radio Access Networks[END_REF]). However, considering that the red mobile is likely to do not have a good SNR on these sub-carriers and interferes mobiles of the RRH j, it is not always efficient that the red uses these RUs. Depending on the channel state of this mobile and the number of RUs available, it could be more profitable for the system that the red waits for transmitting on sub-carriers 1 and 4. Notice that users from the RRH j are not interfered on the RUs (4,14), [START_REF] Viswanath | Opportunistic beamforming using dumb antennas[END_REF][START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for Cloud Radio Access Networks[END_REF] and [START_REF] Hamouda | A Cross-Layer Downlink Scheduling Scheme for Balancing QoS in[END_REF][START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for Cloud Radio Access Networks[END_REF] as well as on the TS 15 since users from the RRH i have ended their transmission.

The classical ICI avoidance Sliced Bandwidth strategy (Fig. 2(b)), splits its bandwidth into two parts. The Sliced Bandwidth strategy protects the green mobile which is highly interfered. However, due to the MaxSNR allocation, the purple mobile does not have any resource as the channel state of brown and orange mobiles is better. In addition, the half of the bandwidth is unused on each RRH which leads to a poor system capacity.

The HS solution (Fig. 2(c)) divides the bandwidth into two slices. The left slice is allocated to mobiles in inner cell where a Reuse 1 strategy is performed. In the right slice, a Sliced Bandwidth scheme is used to protect cell edge mobiles. On this side, the HS solution compares the η i green,n with the η j purple,n and allocates the resource on the sub-carrier n to the mobile with the best value. Although this solution attempts to provide a better fairness among mobiles by protecting the most affected from interferences, its static behavior is not well suitable for wireless networks specificities to be efficient. In this example, the Reuse 1 slice is overloaded and the RUs unused in the Sliced bandwidth part can not be allocated to the Reuse 1 one. Indeed, this solution relies on an ideal distribution between mobiles in inner cell and cell edge ones in order to avoid a load asymmetry between the two slices. In addition, even with a moving boundary, as the radio conditions highly vary, it is not enough flexible to react to the channel state of mobiles and to their mobility.

In order to dynamically react to the context, the DC-RAN-MS performs an ICI management that does not rely on the locations of mobiles. The figure 2(d) illustrates the resource management of the proposed solution. For instance, in accordance with the inequality ( 6), the result of the (η i green,2 interf , η j orange,2 interf ) is not enough significant to perform a double allocation. In this way, as the green mobile has a better channel state than the orange, it starts to transmit on the RU (2,1) and ends on the RU (2,7). On the other hand, the purple mobile also highly interfered, can transmit on the same TS and subcarrier than the blue as (η i blue,3 interf , η j purple,3 interf ) is greater than max η i blue,3 , η j purple,3 . This allocation is not possible with state of the art solutions as they does not take into account the channel state but only the locations of mobiles. In addition, the red mobile takes advantage that blue and green mobiles have ended their transmission to use the following RUs: (2,8) to (2,13) and [START_REF] Viswanath | Opportunistic beamforming using dumb antennas[END_REF][START_REF] Park | Robust and Efficient Distributed Compression for Cloud Radio Access Networks[END_REF]. As the sum of η i red,n interf and η j orange,n interf on sub-carriers 2, 5 and 6 is greater than their respective η m,n , red and orange mobiles are allowed to transmit at the same time. However, red mobile can not transmit on the same TS than the purple (i.e. RU [START_REF] Wong | Multiuser OFDM with adaptive subcarrier, bit, and power allocation[END_REF][START_REF] Wang | Joint Energy Minimization and Resource Allocation in C-RAN with Mobile Cloud[END_REF]) due to the high interferences experienced by this last and to the poor channel state.

To conclude, The DC-RAN-MS performs a wise resource allocation by providing a more accurate interference management according to: the possibility that a mobile could not be interfered, the magnitude of interferences experienced and their η m,n , η m,n interf . This leads to an optimized decision making: either allocating the resources to all mobiles when the interferences are not significant or only allowing to one mobile to transmit when the channel state is poor. Thereby, the DC-RAN-MS optimizes the usage of the bandwidth, reduces the interferences experienced and protects the most affected users. This leads to increase the system capacity and the QoS.

III. PERFORMANCES EVALUATION

In this section, the DC-RAN-MS is compared to the classical C-RAN strategy, the HS solution, as well as to the classical ICI management strategies: the Reuse 1 and Sliced Bandwidth schemes.

In the simulations, a small network of 2 adjacent cells is considered. The area 1 is a zone where the mobiles are likely to be highly interfered as they are closer to the neighboring cell than mobiles in area 2 (where the interferences are less significant). The figure 3 illustrates an example of the position of these areas. The scheduling algorithms of each cell are the same which allows to only study the influence of the interference management solutions. The scheduler chosen is the MaxSNR for its ability to increase the system capacity and as it is one of the most acknowledged scheduler. In addition, the mobiles are at the same distance from their respective RRH to neglect the unfair behavior of this scheduler. The traffic generated by sources is considered realistic and variable which produces high volume of data with important sporadic and tight delay requirements [START_REF] Tanwir | A survey of VBR video traffic models[END_REF][START_REF] Tamimi | Modeling and prediction of high defninition video traffic: A real-world case study[END_REF][START_REF] Heyman | The GBAR source model for VBR videoconferences[END_REF]. This significantly complicates the task of schedulers. Simulations parameters are described in the table I.

The performances evaluation is composed of 2 scenarios. For each scenario, only one parameter varies. The first examines when users are equally distributes between area 1 and 2. Thus, the traffic load increases until the congestion with this particular distribution. On the other hand, the second scenario analyzes the performances of the different strategies when the repartition of mobile between the area 1 and 2 varies.

A. Scenario 1 : Influence of the traffic load increase

In this scenario, considering the two cells, the mobiles are equally distributed between the area 1 and 2 such as: 25% are located in area 2 of the RRH i, 50% in the area 1 and 25% in area 2 of the RRH j. In this way, mobiles are added two per two in each cell (one per area). 1) Spectral efficiency: Figure 4(a) shows the spectral efficiency obtained with each solution for different traffic load in the system. The spectral efficiency presented takes into account the two cells considered. It means that it is the average of bit per RU consumed. A RU is considered as consumed either when it is used by the both RRHs or when an ICI management prevents a RRH from allocating the RU in order to protect a mobile in the other RRH. Computing the spectral efficiency only on RUs allocated is misleading because it does not take into account RUs unused to avoid interferences. For instance, given a RRH i and a RRH j which have only one mobile, respectively the mobile m i and the mobile m j . In this example it is considered that they both get the same η m,n and η m,n interf equal to 14 and 6.5. With a Reuse 1 solution their classical spectral efficiency (bits/RU) is equal to 6.5 while it is equal to 14 with a Sliced Bandwidth scheme as only the mobile m i is allowed to transmit (according to the frequency reuse). However, this last value does not take into account the RU of the RHH j unused. To provide a more accurate indicator of the bandwidth waste, the average of bits per RU consumed seems to be more appropriate to compute the spectral efficiency. Thus, with the previous example, the spectral efficiency (bits/RU consumed) of m i and m j is equal to 6.5 with a Reuse 1 solution while for m i it is equal to 7 with a Sliced Bandwidth strategy (14 ÷ 2, where 2 is the number of RUs consumed).

The figure 4(b) shows the percentage of non-interfered allocation for different traffic load. An interference appears when a mobile of the RRH i and a mobile of the RRH j are chosen to transmit on the same frequency and time. In underloaded context, non-interfered allocations occur more frequently compared to a loaded system. Thus, this figure shows that the mobiles become more and more interfered as the traffic load increases. Indeed, it is rare that only one mobile is chosen to transmit on a given RU among the RRHs since their bandwidth become more and more filled. Notice that when the percentage of non-interfered allocation reaches zero, it means that there is no more RU available without being interfered.

The figure 4(c) shows the type of bandwidth usage regarding the total number of available RUs for different traffic load. It illustrates the usage ratio of the different ICI management (either Reuse 1 or Sliced Bandwidth) performed by the HS and DC-RAN-MS solutions.

On the figure 4(a), the solutions adopt at low traffic load, a typical behavior of a MaxSNR allocation which benefits from the multi-user and frequency diversities. In this way, their spectral efficiency increases with the traffic load. However, depending on the ICI management, interference effects on this metric highly vary.

The Sliced Bandwidth avoids interferences and keeps the opportunistic MaxSNR behavior. Nevertheless, as the part of unused bandwidth is high, it provides a poor spectral efficiency. Notice that with a classical spectral efficiency computation (Bits/RU used) the result is significantly different.

From 12 users to 16, the Reuse 1 solution experiences a high degradation of its spectral efficiency due to the decrease of the non-interfered allocation number (Fig. 4(b)). At 16 users, the number of possible interferences has reached its peak: all the RUs are interfered and the system is overloaded. Since there are two phenomenons that face each other (the MaxSNR opportunistic behavior and the magnitude of interferences), from 16 users the MaxSNR allocation can counterbalance interference effects as it favors the less interfered users. This leads to increase the spectral efficiency of the Reuse 1 solution.

On the other hand, HS and DC-RAN-MS solutions are less affected by interferences as they provide a better ICI management.

The HS solution puts the cell edges mobiles in its right part (Fig. 2(c)) where a Sliced Bandwidth is performed. Consequently, only users in inner cells are affected by interferences. As they are less interfered, the magnitude of ICI experienced is not significant making the spectral efficiency less affected by interferences (Fig. 4(a)).

The DC-RAN-MS dynamically performs its ICI management thanks to appropriate allocations according to the channel state (Inequality ( 6)). In addition, unlike the HS solution, the DC-RAN-MS does not restrict the usage of the ICI management schemes (Reuse 1 or Sliced Bandwidth) to 50% of its bandwidth (Fig. 4(c)). This provides a better usage of the bandwidth which leads to a high spectral efficiency (Fig 4(a)).

2) HS and DC-RAN-MS behaviors and system capacities:

The system capacity provided by a solution does not only rely on its spectral efficiency. For instance, the HS solution has a better spectral efficiency at a traffic load of 18 mobiles (Fig 4(a)). As only mobiles in inner cell are affected by interferences, the percentage of mono allocation reaches 0% at a traffic load of 20 mobiles (Fig. 4(b)). However, this is not relevant from the system capacity provided by the HS solution. The figure 4(c) shows that the interference avoidance (i.e. Sliced Bandwidth) part of the HS solution is overloaded since a while (it reaches 50% of the available bandwidth at a traffic load of 16 mobiles). Thus, as the boundary of the HS solution is static, it can not take advantage of the unused RUs in the Reuse 1 slice to allocate them to the Sliced Bandwidth part. This leads to an asymmetry load between mobiles in inner and edges of the cell.

Unlike the HS solution, the DC-RAN-MS can adapt its behavior to the context. When the system is underloaded, the DC-RAN-MS performs more Sliced Bandwidth allocation than Reuse 1 ones as only one mobile is allowed to transmit with this kind of frequency reuse (Fig. 4(c)). As the DC-RAN-MS does not rely on a boundary, it can allocate the RUs unused by the Reuse 1 strategy to perform its Sliced Bandwidth allocations. This provides a better system capacity than the HS solution for mobiles in inner cell and edges without an asymmetry.

3) Mean packet delay: A crucial indicator of the QoS experienced by users is the latency. Figure 4(d) represents the mean packet delay in the system in milliseconds.

The Sliced Bandwidth strategy provides a poor system capacity as a half of the bandwidth is unused. Thus, even if the mobiles are not interfered, they will quickly experience an increase of their mean packet delay due to the poor available RUs number as the traffic load rises.

The Reuse 1 solution uses the total available bandwidth. However, as there is no interference management, the most interfered mobiles are highly penalized which leads to provide a poor global QoS.

The HS strategy attempts to protect the most interfered mobiles by dividing the bandwidth. Nevertheless, as only one mobile is allowed to transmit on the Sliced Bandwidth part, this side is quickly overloaded (Fig. 4(c)). As the boundary is static, the Reuse 1 part can not be used to provide a better system capacity and the RUs unused stays wasted until this part is overloaded too. Consequently it can better handle the QoS of mobiles than the two previously described solutions, but overall, the system collapses at the same traffic load than others solutions (14 mobiles).

The DC-RAN-MS optimizes the spectrum usage (Fig. 4(a)) and allows to reduces the magnitude of interferences. The proposed solution performs a Sliced Bandwidth strategy only when its necessary (Fig. 4(c)). This reduces the waste of bandwidth and provides a more accurate and fairer resource allocation. Consequently a better QoS is experienced and the system can handle the traffic load longer (16 users) than others solutions (Fig. 4(d)).

B. Scenario 2 : influence of users distribution

This scenario analyzes the behavior of the solutions when users distribution varies between the areas 1 and 2 for a traffic load of 14 mobiles. First, mobiles are located in the area 1. Then they move one by one to the area 2.

1) Spectral efficiency and system capacity: The Sliced Bandwidth strategy has a constant spectral efficiency (Fig. 5(a)) as this solution manages the interferences by dividing its bandwidth. In this way, its spectral efficiency depends only on the traffic load and not on users distribution (i.e magnitude of interferences experienced)

The Reuse 1 solution provides a poor spectral efficiency (Fig. 5(a)) when mobiles are highly interfered (i.e located in area 1). Thus, mobiles need much more RUs than usual to end their transmission which leads to consume all the RUs available and an overloaded system as the ratio of non-interfered allocation reaches 0% (Fig. 5(b)). Like a virtuous circle, the rise of the spectral efficiency increases the percentage of non-interfered allocation. As the number of RUs interfered decreases, it provides a better spectral efficiency and so on.

The HS and DC-RAN-MS schemes provides a better ICI management leading to a better spectral efficiency. However, as the HS solution is static, there is an asymmetry between the Sliced Bandwidth part and the Reuse 1 one (Fig. 5(c)). Until 42% of mobiles in the area 2, the Sliced Bandwidth part is overloaded and the spectral efficiency increases thanks to the opportunistic MaxSNR behavior. Then, the number of users interfered rises which decreases the percentage of non interfered allocation. At 71% of mobiles in area 2, the Reuse 1 slice is overloaded (Fig. 5(c)) which means that the influence of interferences has reached its peak. Thus, the MaxSNR prioritizes less interfered mobiles and counterbalances the ICI effects. In this way, the spectral efficiency increases from 71% until 100% of mobiles in area 2. Notice that the percentage of non-interfered allocations reaches 0% only when mobiles are all located in area 2 as before, RUs are still allocated in the Sliced Bandwidth part.

Unlike the HS solution, the DC-RAN-MS has been designed to dynamically react to the context. The DC-RAN-MS does not take into account the location of mobiles but only their channel state. This allows to be even more accurate than a solution with a dynamic boundary. When the mobiles are mostly in the area 1 (i.e highly interfered) it uses mainly the Sliced Bandwidth strategy (Fig. 4(c)). Nevertheless, it can performs a Reuse 1 strategy provided the radio conditions of mobiles are enough significant. The same analyze can be apply when mobiles are mostly located in area 2 (i.e. less interfered) where the percentage of Sliced bandwidth strategy used is still important. Consequently, the DC-RAN-MS provide a better spectral efficiency than other solutions (Fig. 5(a)) and a better system capacity.

2) Mean packet delay:

The Sliced bandwidth provides a constant delay (Fig. 5(d)) as its spectral efficiency is the same whatever the distribution of mobiles.

The mean packet delay provided by the Reuse 1 solution highly depends on the mobile locations (i.e. magnitude of interference received). In this way, the mobiles have a better QoS when they are located in the area 2 than in the area 1.

The HS solution is designed with a static boundary. This leads to a poor QoS experienced by mobiles except when they have an ideal distribution (between 42% and 61%).

The DC-RAN-MS dynamically adapt its behavior to the channel state of mobiles regardless of their location. Thanks to a better spectral efficiency and system capacity provided by the inequality [START_REF] Hamouda | A Cross-Layer Downlink Scheduling Scheme for Balancing QoS in[END_REF], the proposed solution outperforms state of the art schemes on this metric.

IV. CONCLUSION

The continuously growing needs in term of throughput, system capacity and delay requirements lead to search new manner to optimize transmission efficiency particularly in ultra dense networks. In this way, the contribution of this paper named DC-RAN-MS, aims to merge the ICI management and the scheduling processes to optimize the resource allocation. Thanks to a "Cell-less" approach based on the C-RAN architecture, the proposed solution allocates resources for a cluster of RRHs. The proposed solution does not rely on a criteria in correlation with ICI (like the distance) but directly on the interferences themselves. According to the channel state of mobiles, the proposed solution performs dynamically its ICI management either by: allowing the usage of the entire bandwidth or by preventing schedulers from allocating resources when the magnitude of interferences experienced are too high. This leads to optimize the usage of the spectrum which decreases the global ratio of unused bandwidth while efficiently reducing the magnitude of interferences. This results in higher spectral efficiency, higher system capacity and a QoS increased.

Fig. 1 .

 1 Fig. 1. Example of C-RAN approaches

  (a) Reuse 1 strategy. (b) Sliced Bandwidth strategy. (c) Hybrid Static strategy. (d) DC-RAN-MS strategy.

Fig. 2 .

 2 Fig. 2. Interference management solutions with a MaxSNR allocation.

Fig. 3 .

 3 Fig. 3. Conceptual representation of the simulation context.

  DC-RAN-MS and HS behavior regarding the traffic load.

Fig. 4 .

 4 Fig. 4. Scenario 1.

  DC-RAN-MS and HS behavior regarding the mobile distribution.

Fig. 5 .

 5 Fig. 5. Scenario 2


Maryna Kudinova

Géraud Bouwmans

Rémi Habert

Stéphane Plus

Karen Baudelle

Remy Bernard

Bertrand Chazallon

Andy Cassez

Hicham El Hamzaoui

Olivier Vanvincq

Rémy Bernard

Johann Troles

Laurent Bigot
email: laurent.bigot@univ-lille.fr

Hundreds of meter-long low-loss silicon-core optical fiber

Keywords: Hybrid optical fibers, Semiconductor optical fibers, Silicon-core optical fibers

INTRODUCTION

Silica-based optical fibers are characterized by ease of use, exceptional guiding properties -especially a very low attenuation -and well-mastered manufacturing processes that led to their widespread use in Telecom, sensors and light sources application domains. These qualities, together with the wide panel of optical components available today to interface them with light sources, detectors and special optically active/passive systems, explain why when people are considering the implementation of new functions in fiber optics components, they try to make them as much compatible as possible with the existing silica fibers. This is typically the philosophy of hybrid fibers that are constituted by a silica basis combined with a more exotic material such as a soft glass, a polymer, a metal or a semiconductor. [START_REF] Schmidt | Hybrid optical fibres -an innovative platform for in-fibre photonic devices[END_REF] Silica/silicon optical fibers are part of this special branch of the optical fiber family and have been the subject of numerous works since 2006. [START_REF] Sazio | Microstructured optical fibers as high-pressure microfluidic reactors[END_REF][START_REF] Peacock | Silicon optical fibres -past, present, and future[END_REF] These works are motivated by the attractive optical and electrical properties of silicon, very different from the silica's ones, that allow to target optical functions such as light sources in the Mid-Infrared (Mid-IR) or in-fiber optoelectronic integration. The very different properties of silica and silicon are also observed in the thermo-structural domain, which is illustrated, for example, by the melting point of silicon which is well below that of silica. Such a difference makes the combination of the two materials quite complicated in the optical fiber geometry. However, Ballato and co-workers demonstrated in the past that a quite simple approach such as the Molten Core Method (MCM) permits, quite contra-intuitively, to manufacture such exotic fibers using conventional fiber drawing equipments. [START_REF] Ballato | Silicon optical fiber[END_REF] These works and those based on fibers developed by High Pressure Chemical Vapor Deposition (HPCVD) method have demonstrated the potentialities of silicon-core fibers and also raised some limitations for those applications, such as the difficulty to reach a small core size together with the obtention of reasonable background losses. [START_REF] Healy | A review of materials engineering in silicon-based optical fibres[END_REF] Hence, besides impressive results already published, there is still a need today for a better understanding of the link between optical properties and manufacturing technique. It is the aim of this work to present some of the results recently obtained by our group in this context.

ALTERNATIVE APPROACHES FOR THE REALIZATION OF SILICON-CORE FIBERS

As has been previously recorded, one challenge when trying to synthesize a silica/silicon hybrid fiber lies in the very different thermal behaviors of the two materials, which can be at the origin of cracks, inhomogeneous distribution of the two materials, bubbles and uncontrolled structural properties along the fiber length. When using solid materials (silicon rod and silica tubes) as starting materials, these unwanted phenomena are supposed to be facilitated by the air gap existing between the silicon and silica materials. We hence developed a first original approach so as to try to limit the presence of such a gap.

Assembly with in a Sol-Gel -made porous silica material

The basic idea of the first technique consists in obtaining an all-solid silicon/silica preform without heating the materials at a temperature higher than the one of the silicon melting point (1414 • C). So as to reach this goal, the proposed approach consists in the sintering of a porous silica material onto a silicon rod. The starting silica material is a porous rod synthesized by the Sol-Gel polymeric route, following the general protocol described in ref. [START_REF] Hamzaoui | From porous silica xerogels to bulk optical glasses: The control of densification[END_REF] This porous rod is drilled in order to obtain a non-through hole in which a commercially-available silicon rod can be inserted. In the present example, the hole is about 3 mm in diameter whereas the silicon rod is 2 mm in diameter. Once the core material is inserted, the whole assembly is processed between 900 and 1350 • C under controlled atmosphere so as to remove hydroxyl groups from the porous silica network and consolidate it into a transparent glass. During the last part of this process, the silica network shrinkages onto the silicon rod making, at the end, an all-solid silicon core preform available for drawing. Such a drawing can then be done on a conventional drawing tower at a temperature close to 2000 • C. The resulting fiber obtained in the example proposed here is about 1 mm in diameter and the core diameter is about 150 µm. A Scanning Electron Image (SEM) of such a fiber is illustrated on Fig. 1. One can note the good quality of the silicon/silica interface, which indicates that the silica cladding has been efficiently sintered and shrinked on the silicon rod. The structural nature of the core and cladding materials has been studied by Raman spectroscopy in a confocal configuration using a 514.5 nm laser line. Examples of Raman spectra recorded at different positions of the fiber facet are reported on Fig. 2. The crystalline nature of the core is deduced from the well-structured Raman spectrum recorded in the core region, particularly evidenced by the peak centered around 520 cm -1 whose Full Width at Half the Maximum (FWHM) suggests a good degree of crystallization (see Fig. 2a). At the opposite, the cladding presents the well-known signature of amorphous silica (see Fig. 2b). Further studies indicate the polycrystalline nature of the core material. The large core dimension obtained here is however hardly compatible with any application in optics making a second drawing mandatory, which should be facilitated by the good quality of the obtained structure. This work is still under development today but represents a really novel route when compared to the techniques already reported in the literature. The second manufacturing technique developed in this context is based on an approach derived from the MCM and based on the knowledges and skills accumulated during the last twenty years in the context of Photonic Crystal Fibers (PCFs) manufacturing. This technique will hereafter be referred as the rod-in-stack method because it consists in inserting a silicon core rod in the center of a stack made of silica capillaries, this stack then being itself sleeved into a tube. In the case of this work, the starting silicon material will be the same as in section 2.1. As is commonly done for silica PCFs, the stack is first drawn into a cane and then turned into a fiber after a second high temperature drawing. The point here is that it seems that this approach, in the context of a silicon/silica assembly, tends to limit the occurring of cracks during drawing. This is attributed to the air/silica cladding structure that may facilitate the absorption of the deformations that appear in the structure due to the different thermal coefficients of the two materials. One remarkable consequence of the good stability of the structure is the possibility to perform continuous drawing without interruption due to fiber breaks. It has then been possible to manufacture hundreds of meters of several fiber spans with different micrometer-sized core dimensions, the smallest one being about 800 nm in diameter. A view of some fibers is presented on Fig. 3. In the case of those fibers, the collapsing of the air holes of the cladding during fiber drawing permitted to obtain an all-solid structure. Such fibers, polymer coated and with good geometrical homogeneity, can be managed similarly to conventional optical fibers for optical characterizations. As an example, a cut-back measurement performed on a 1.46 m-long starting piece of 3.4 µm core fiber (then reduced to 36 cm) is presented on Fig. 4. This measurement has been conducted using a conventional supercontinuum source injected into a high numerical aperture fiber butt-coupled to the fiber under test, the output light being collected by a multimode fiber connected to an Optical Spectrum Analyzer (OSA). One can note that the background losses are below 0.25 dB/cm over a wide spectral range in the Near Infrared (NIR) region, i.e. similar to the best values reported for Silicon-On Insulator (SOI) waveguides. Additional measurement performed with a different set-up indicates that the mean attenuation is below 2 dB/cm in Mid-IR region and that the fiber transmits light up to at least 4 µm wavelength. This measurement has been possible thanks to the use of a Mid-IR supercontinuum based on a fluoride optical fiber from "Le Verre Fluoré" and compatible monochromator and detector. These remarkable optical properties are attributed to the good structural quality of the core material and of the core/cladding interface, as is illustrated by Fig. 5.

Rod-in-Stack technique

First, several Raman spectra have been recorded at different positions in the fiber core using a confocal Raman microscope with a 532 nm excitation laser line. As was the case for the fiber obtained using Sol-Gel sintering, the good crystallinity of the core material is highlighted by the presence of a clear peak centered around 520 cm -1 and only observed in the core region. A deeper investigation of the fiber structure has been performed using High-Resolution High Angle Annular Dark Field (HR-HAADF) imaging in Scanning Transmission Electron (STEM) microscopy of a fiber sample prepared by Focused Ion Beam (FIB) technique. In this case, a net contrast between the core and the cladding is put in evidence when analyzing a fiber facet. At this stage, it seems that the core material is poorly affected by a diffusion of oxygen from the cladding. Moreover, more information on the structure of the fiber materials is illustrated by Fig. 5 (right), which demonstrates the amorphous structure of the cladding whereas close to the core/cladding interface and in the core, monocrystalline (112) structure of silicon is clearly observed.

CONCLUSION

In this paper, we reported on two techniques developed in our group so as to facilitate the obtention of good quality silicon-core optical fibers. The first one is based on the sintering of a Sol-Gel -made porous silica matrix onto a silicon rod. It permits to obtain an all-solid silicon/silica preform at a temperature smaller than the melting point of silicon. It has been shown that such a preform can be drawn into a fiber. The second technique consists in a method derived from the stack-and-draw method used to manufacture PCFs. By using a commercially available silicon rod as the central element of the stack, it has been shown that long lengths of all-solid micrometer-sized core silicon-core fibers can be obtain. These fibers exhibit very low background losses explained by the good quality of the core material and core/cladding interface, making them good support for photonics applications like broadband light sources in the Mid-IR wavelength range which represents the current trend in the field. [START_REF] Ren | Low-loss silicon core fibre platform for mid-infrared nonlinear photonics[END_REF][START_REF] Ghosh | Chalcogenide-glass polarization-maintaining photonic crystal fiber for mid-infrared supercontinuum generation[END_REF]

Figure 1 .

 1 Figure 1. SEM image of a silicon-core fiber obtained by sintering of a Sol-Gel -made silica cladding onto a silicon rod and drawing the resulting preform into a fiber.

7

 7

Figure 2 .

 2 Figure 2. Raman spectra obtained when probing (a) the silicon core or (b) the silica cladding materials of the hybrid-fiber drawn from the all-solid silicon/silica preform.

Figure 3 .

 3 Figure 3. SEM image of two fibers obtained by the rod-in-stack technique.

Figure 4 .

 4 Figure 4. Attenuation curve measured on a 3.4 µm diameter core silicon-core fiber obtained by the rod-in-stack method.

Figure 5 .

 5 Figure 5. Left: Raman spectra recorded when probing the core of the 3.4 µm core fiber. Right: TEM images recorded on a facet of a 3.4 µm core fiber obtained by the rod-in-stack method.

ACKNOWLEDGEMENTS

This work was supported by the French Agence Nationale de la Recherche: SPICY project (ANR-16-CE24-0015), the LABEX CEMPI (ANR-11-LABX-0007) and the Equipex Flux (ANR-11-EQPX-0017); The Ministry of Higher Education and Research, the Hauts de-France Regional Council and the European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER Photonics for Society, P4S). This work was developed at the IRCICA (USR CNRS 3380, www.ircica.univ-lille1.fr).


A French classification of permanent grasslands at national level to evaluate their forage and environmental services

Keywords: permanent grassland, classification, production, environmental value

To demonstrate and quantify the forage and environmental services provided by permanent grasslands, a research program (2008)(2009)(2010)(2011) was performed, including together INRA, Institut of Elevage and 23 extension services. A national network of 190 permanent grasslands was sampled in a database of 1500 grasslands, to represent the diversity of management and agroclimatic conditions, and has been studied for two years. On each plot, a botanical survey was carried out (2009). Grass production, functional composition of the grassland and forage nutritive value for ruminants were measured at four dates during the growing season (2009, 2010). Data of functional and botanical composition were analyzed to define 19 types of grasslands identifiable from a simple decision key (agro-climatic conditions, management). For each type, quantitative references of seasonal evolution of functional composition, biomass production and nutritive value are provided. The contribution of the different types of grasslands to ecosystemic services was evaluated.

Introduction

The contribution of permanent grasslands in the ruminant feeding is considerable and the economic interest of these surfaces in the fodder systems is now recognized. Their environmental interest is also important, e.g. fauna flora biodiversity [START_REF] Farruggia | Quels intérêts de la diversité floristique des prairies permanentes pour les ruminants et les produits animaux?[END_REF] or carbon sequestration [START_REF] Soussana | Temperate grassland and global atmospheric change: a review[END_REF]. However, these surfaces are decreasing in France for 50 years and are mainly replaced by maize for silage or more profitable annual crops. This decrease can be partly explained by a deficit of technical and scientific knowledge about the agronomic, ecological value and the management of the permanent grasslands. To revalue permanent grasslands in the forage systems, a R&D Program (funded by the French Ministry of Agriculture) was set up (2008)(2009)(2010)(2011). The objective of this program was to build a reference tool (a classification) reporting the diversity of agronomic and environmental value and of some ecosystemic services of French permanent grasslands. This tool was intended for education and actors of development with the aim of managing better and valuing these surfaces in the fodder systems.

Material and methods

A set of 190 permanent grasslands was selected from 1500 listed by a survey conducted in 2008 on 78 farms (in which permanent grasslands represented more than 50% of forage area) distributed in the main lowland and mountain grassland areas of France except the Alps and the Mediterranean area. Information concerning management practices and the characteristics of the permanent grasslands of the network (location, type of soil) were listed. Botanical composition (exhaustive inventory of the species and the estimation of their dominance) was determined in spring 2009 on each grassland in a homogeneous plant community (Michaud et al. 2011). The seasonal dynamic of forage production and nutritive value was assessed during 2009 and 2010 on the dominant homogeneous plant community of each grassland. The proportions of grasses, legumes and forbs were estimated visually, according to volume, and on a sub-sample of 40 tillers grasses the proportion of functional types were determined according [START_REF] Cruz | Typologie fonctionnelle ed graminées fourragères perennes : une classification multitraits[END_REF]. On dried samples, the nutritive value of the herbage (organic matter digestibility and crude protein content) was estimated using NIRS [START_REF] Michaud | Links between functional composition, biomass production and forage quality in permanent grasslands over a broad gradient of conditions[END_REF]. Principal component analysis based on six criteria of vegetation (proportion of legumes in the beginning and at the end of spring, proportion of forbs at the end of spring, proportion of function type C in the beginning and at the end of spring, proportion of function type B at the end of spring) contributed to build a classification of permanent grasslands. A first classification based on this criteria of vegetation and criteria of nutritive value allowed to identify 15 types of permanent grasslands related to forage services. At the same time, a second classification based on criteria of the vegetation and criteria relevant for the prediction of the environmental value (floristic composition and dominance of entomophilous species) contributed to identify 12 types of grasslands related to environment services. The crossing of these two classifications ended in the classification in 19 defined types each by a particular combination of six characteristics of the vegetation retained. A key of determination allowing to recognize the types of grasslands was established by leaning mainly on the nature of the environment (height …) and management of practices.

Results and discussion

The 19 types of permanent grasslands are distributed in four groups according to the characteristics of climate and height (Figure 1): mountain permanent grasslands (5 types), plains and hills with semi continental climate permanent grasslands (6 types), plains and hills oceanic permanent grasslands (5types), atlantic coast permanent grasslands (3 types).The proportion of legumes in biomass contributes to discriminate semi continental permanent grasslands. Management practices and particularly fertilization contribute to discriminate all types of permanent grasslands. The 19 types are described in a book [START_REF] Launay | Prairies Permanentes : des références pour valoriser leur diversité[END_REF]. For each grassland type, an index card describes the geographical conditions, type of soil, management practices and the botanical characteristics (presence and dominance species). Proportion of grasses, legumes and forbs and stages of development, dry matter yield and nutritive value are given for the various periods of measures (Figure 2). These values allowed to estimate some ecosystemic services (biomass production, biodiversity, pollination). This tool field. It ambition is to bring references of agronomic values and ecosystemic services provided by various types of permanent grasslands.

Conclusion

This national classification reports the diversity of French permanent grasslands and so the diversity of ecosystemic services provided. This classification of permanent grasslands constitutes a tool used in agronomic and environmental assessments. The methodology could be used in other European countries.

Figure 1

 1 Figure 1 Distribution of the types of permanent grasslands ( Mountain permanent grasslands, Plains and hills with semi continental climate permanent grasslands, Plains and hills oceanic permanent grasslands, Atlantic coast permanent grasslands). A circle represents a partner farm of the project.

Figure 2 .

 2 Figure 2. Example of references brought by the classification: a) biomass production marks for each type of permanent grasslands in t DM/ha/year), b) specific richness in each type of permanent grasslands (in number)

We thank the financial institution of CASDAR program, advisors who worked on this program and farmers, which made available permanent grasslands.