Carole Lecoutre-Chabot

Samuel Marre

Yves Garrabos

Daniel Beysens

Inseob Hahn

C Lecoutre
email: carole.lecoutre@icmcb.cnrs.fr

Near-critical density filling of the SF6 fluid cell for the ALI-R-DECLIC experiment in weightlessness

Keywords: slightly off-critical sulfur-hexafluoride, liquid-gas density diameter, liquid-gas coexisting densities

Introduction

Thermodynamic and transport properties show singularities asymptotically close to the critical points of many different systems. The current theoretical paradigm on critical phenomena using renormalization group (RG) approach [START_REF] Wilson | The renormalization group: Critical phenomena and the Kondo problem[END_REF] has ordered these systems in well-defined universality classes [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF] and has characterized the asymptotic singularities in terms of power laws of only two relevant scaling fields [START_REF] Fisher | Correlation Functions and the Critical Region of Simple Fluids[END_REF] in order to be conform to the scaling hypothesis. Simple fluids are then assumed similar [START_REF] Garrabos | Crossover equation of state models applied to the critical behavior of xenon[END_REF] to the O(1) symmetric (Φ 2 ) 2 field theory and the N=1vector model of three-dimensional (3D) Ising-like systems ( [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF], [START_REF] Barmatz | Critical phenomena in microgravity: Past, present, and future[END_REF]). Their study in weightlessness condition is well-recommended to test the two-scalefactor universality approaching their critical point. However, for the case of the gas-liquid critical point of simple fluids, some additional difficulties can occur as the order parameter -the fluctuating local densityshows a noticeable asymmetry, as for instance the well-known rectilinear diameter form of the liquid-gas coexisting density curve first evidenced by Cailletet and Mathias [START_REF] Cailletet | Recherches sur les densités des gaz liquéfiés et de leurs vapeurs saturées[END_REF]. This linear asymmetry was largely confirmed in the subsequent literature (see for instance Ref. [START_REF] Singh | Rectilinear diameters and extended corresponding states theory[END_REF]). Such asymmetrical effects cannot be accounted for from the symmetrical uniaxial 3D Ising model and its induced standard fluid-like version, i.e., the symmetrical lattice-gas model.

An alternative theoretical way to introduce the fluid asymmetry nature in the scaling approach consists in extending the number of the physical fields contributing explicitly to the relevant scaling fields, the so-called complete scaling phenomenological hypothesis ( [START_REF] Fisher | The Yang-Yang anomaly in fluid criticality: experiment and scaling theory[END_REF]- [START_REF] Wang | Nature of vapor-liquid asymmetry in fluid criticality[END_REF]). For example, in a recent work [START_REF] Cerdeirina | Soluble model fluids with complete scaling and Yang-Yang features[END_REF], Yang-Yang and singular diameter critical anomalies arise in exactly soluble compressible cell gas models where complete scaling includes pressure mixing. The predictions of complete scaling have been tested against experiments and simulations at finite distance from the critical point, increasing the complexity in the fundamental quest of a true asymptotic fluid behavior. The latter remains a conundrum to the scientists who have for objective to check it by performing an experiment closer and closer to the critical point with the required precision. De facto, the asymmetrical contributions, the analytical backgrounds, and the classical-to-critical crossover behavior due to the mean-field-like critical point, further hindered the test of the asymptotic Ising-like fluid behavior. Such difficulties are intrinsically ineludible, even along the true critical paths where the crossover contribution due to one additional non-relevant field [START_REF] Wegner | Corrections to scaling laws[END_REF] can be accounted for correctly in the field theory framework ([15]- [START_REF] Garrabos | Master crossover functions for one-component fluids[END_REF]).

Moreover, the experiments are never exactly on these critical paths, adding paradoxically a new opportunity to investigate the theoretical expectations related to the non-symmetrical behaviors. Indeed, even though the temperature can be made very close to 𝑇𝑇 𝑐𝑐 , the mean density of the fluid cell is never at its exact critical density value [START_REF] Lecoutre | Weightless experiments to probe universality of fluid critical behavior[END_REF]. The error-bar related to this latter critical parameter was never contributing to the discussion of the Earth's based results in terms of true experimental distance to the critical point. Nevertheless, from the above experimental facts and the theoretical expectations, it appears that the related non-symmetrical effects can be unambiguously viewed in a slightly off-critical (liquid-like) cell. Indeed, in such a closed liquid-like cell, it is expected that the meniscus position crosses the median volumetric plane at a single finite temperature distance below the coexistence temperature. From the symmetrical lattice-gas model, we recall that the meniscus of any liquid-like cell is expected to be visible always above this median volumetric plane in the two-phase temperature range.

Therefore, an academic interest to use gravity field acceleration to horizontally stabilize the position of the liquid gas meniscus inside eight different cell positions is precisely investigated during the pre-flight determination of the off-critical mean density of a fluid cell before its use under weightlessness environment. More specifically, we would like to check if SF6 remains well similar -or not -to the 1974 standard SF6 fluid ([18]- [START_REF] Ley-Koo | Revised and extended scaling for coexisting densities of SF6[END_REF]) which support fluid asymmetry resulting from complete scaling hypothesis ([10]- [START_REF] Wang | Nature of vapor-liquid asymmetry in fluid criticality[END_REF]). Our experimental challenge is then to detect the previously observed significant hook (of 0.5% amplitude) in the rectilinear density diameter when the relative uncertainty in the filling density value is controlled with 0.1% precision along a non-critical path which exceeds from ∼0.2% the exact critical isochore. This experimental challenge is illustrated in Fig. 1. The high optical and thermal performances of the ALI-R insert used in the DECLIC facility Engineering Model allow the observation of the meniscus position behavior precisely due to the density diameter behavior, as the temperature of highly symmetrical test cells is changed. Each of these test cells consists in a quasi-perfect disk-shaped cylindrical fluid volume observed in light transmission, surrounded by two opposite, small, and similar dead volumes. The latter volumes define the single remaining transverse (non-cylindrical) axis of the fluid volume due to the cell in-line filling setup. Here, the selected test cell ALIR5 [START_REF]ALIR5 test cell was selected among a series of 10 identical ALIR n cell (with n=1 to 10). The series have provided statistical evaluation of the fluid volume and fluid mass uncertainties (0.05% and 0.1%, respectively)[END_REF] was filled at a liquid-like mean density 〈𝛿𝛿𝜌𝜌 �〉 = (𝜌𝜌 𝜌𝜌 𝑐𝑐 ⁄ ) -1 very close to the critical density ρc of SF6. The relative off-critical density 〈𝛿𝛿𝜌𝜌 �〉 = +0.20 -0.04 +0.04 % of ALIR5 was measured with great accuracy from our Earth's-based filling and checking processes (see below § 6 and Ref. [START_REF] Morteau | Proceedings of 2nd European Symposium Fluids in Space[END_REF]). The fluid under study is SF6 of electronic quality, corresponding to 99.995% purity (from Alpha Gaz -Air Liquide). The meniscus behavior could be analyzed in eight cell configurations. Such analyses provide an accurate experimental evaluation of the relative effects of (i) the complete cell design, (ii) the cell displacement in front of the CCD camera, (iii) the meniscus optical observations through gravitational stratification and liquid wettability, (iv) the cell filling mean density, and finally, (v) the coexisting density diameter behavior. Only the evaluations of (iv) and (v) are treated hereafter.

Experimental set-up and methods

Highly-symmetrical cell design

The essential characteristic of the ALIR5 cell (see Fig. 2(a)) is its highly symmetrical design with respect to any median plane of the observed cylindrical fluid volume. The main part of the fluid sample consists in a fluid layer of thickness 𝑒𝑒 𝑓𝑓 = (2.510 ± 0.002) mm and diameter 𝑑𝑑 𝑓𝑓 = 2𝑅𝑅 = (10.606 ± 0.005) mm. This fluid layer is confined between two flat, parallel, and transparent sapphire windows of thickness 𝑒𝑒 𝑤𝑤 = 8.995 mm and external diameter 𝑑𝑑 𝑤𝑤 = 12 mm. An engraved circle of 10 mm diameter, 30 μm thickness, is deposited on each sapphire external surface. Such a pancake cell design [START_REF] Zappoli | Heat Transfers and Related Effects in Supercritical Fluids[END_REF] The experiment is then performed in four cell directions 𝜃𝜃 = {-23.2°; 0°; +22.9°; +90°} regarding the above single fill-line positions with respect to two reverse orientations of the earth's gravity vector. That permits to analyze potential systematic errors associated with the cell dead volume. Throughout the paper, each cell configuration is labeled ⟦𝑖𝑖, 𝑋𝑋⟧ where the digit i represents two reverse gravity orientations (g ↓ for i=1 and g ↑ for i=2) and the letter X describes four directions of the fill-line axis of the fluid cell (X=H for 𝜃𝜃 = 0 °, X=V for 𝜃𝜃 = +90°, X=T for 𝜃𝜃 = 22.9°, and X=Z for 𝜃𝜃 = -23.2°). The corresponding cross-sectional shape of the cell is schematically pictured in Fig. 2(b), illustrating the relative positions of the meniscus and the dead fluid volumes with respect to the earth's gravity vector.

Phase transition temperature.

The laser light transmission measurements of the EM-DECLIC facility [24] and the wide field-of-view observation of the fluid sample are combined to observe the phase separation process during the cell cooling down. Each temperature quench step crossing the transition temperature (noted 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ) is -1mK. The exact value of 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is not essential for the following discussion. The temperature results for each experimental configuration are then reported from reference to the lowest temperature (noted 𝑇𝑇 1𝜑𝜑 ) of the monophasic range. The resulting true SF6 coexistence temperature 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is such as 0 < 𝑇𝑇 1𝜑𝜑 -𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 < 1 mK, noting in addition the high reproducibility (2mK range) of 𝑇𝑇 1𝜑𝜑 (here from 318.721 K to 318.723 K) for the eight experimental runs. Moreover, the 𝑇𝑇 𝑐𝑐 -𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ≃ 0.4 µK shift [START_REF]For ALIR5, 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 𝑐𝑐 = 0.20% and[END_REF] due to the off-density criticality of the test-cell is neglected, i.e., 𝑇𝑇 𝑐𝑐 ~𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 . Finally, wide field-of-view imaging of each meniscus position data is recorded when thermal equilibration is achieved at each temperature difference (𝑇𝑇 1𝜑𝜑 -𝑇𝑇)~(𝑇𝑇 𝑐𝑐 -𝑇𝑇). Then 𝑇𝑇 𝑐𝑐 -𝑇𝑇 follows a logarithmic-like scale to cover the experimental temperature range 0 < 𝑇𝑇 𝑐𝑐 -𝑇𝑇 ≤ 15000 m°C (here with 𝑇𝑇 1𝜑𝜑 >𝑇𝑇 𝑐𝑐 = 45573 ± 1 m°C).

Cell imaging and Image processing

Cell imaging of the meniscus position.

The liquid-gas meniscus is observed from optical transmission imaging through the cell, using LED illumination and cell view observation with a CCD camera (1024×1024 pixels). The pixel size corresponds to 12 μm in the wide field-of-view imaging, well controlled from the two engraved circles on the external surface of each sapphire window. Additional small field-of-view (microscopy) imaging with 1.0 μm -pixel resolution of a typical object area 1×1 mm 2 in the central part of the fluid sample are also performed but not reported here. The cell images are also made by tuning the focal plane between the two window internal surfaces to control small optical perturbative effects related to any nonlinear light bending situations, e.g., due to a nonparallelism between tilted windows, wetting layerlensing effects, compressibility effects, or displacement of the optical axis of imaging lenses versus the exact center axis of the cylindrical fluid cell volume.

Image processing of the cell position.

Before the determination of the meniscus position, the image processing needs the exact pixel coordinates of the viewed fluid cell volume to be determined inside the images recorded for each ⟦𝑖𝑖, 𝑋𝑋⟧ configuration. The picture given in Fig. 3 for the ⟦1, 𝑉𝑉⟧ configuration at 𝑇𝑇 = 45473 m°C is chosen to briefly summarize the method that uses the line profile analysis provided by the NI Vision Assistant 2012 software. Each pixel point is characterized by its x (horizontal)-y (vertical) raw coordinates where the axis origin takes place on the top-left corner of the picture. Therefore, the line profiles provide the x-y coordinates of the selected borderline points between the fluid and body cell (see A, B, C, T, & R points in Fig. 3). The resulting position of the cell borderline (here a quasi-circle of ∼10.380-10.464 mm diameter, i.e., ∼865-872 pixels) can be controlled by the comparison with the position of the two engraved circles (10±0.01 mm or 833.3/833.4 pixels of diameter) on the external surface of the input and output windows. As an essential result, the (horizontal and vertical) pixel coordinates of the apparent center point O are the intrinsic characteristics parameters of the fluid volume position whatever each cell picture. The resulting estimation of the maximum error on the absolute position of any characteristic point of each profile line of each picture is ± 0.5 pixels. The last step optimizes the matching of the selected characteristic points for two reversed similar configuration (⟦1, 𝑉𝑉⟧ and ⟦2, 𝑉𝑉⟧ for the chosen case). Indeed, the changes of the facility positions under the Earth's gravitational acceleration field induce small mechanical relative displacements due to the intrinsic clearance between the different (optical and mechanical) components. The present concern involves the cell (housed in the insert) in front of the video camera (located in the optical box of the facility). Therefore, this cell image matching step leads to the determination of the (horizontal and vertical) pixel shifts (∼2 to 6 pixels, typically) between two reversed images of the viewed fluid cell volume.

Image processing of the meniscus position.

For each ⟦𝑖𝑖, 𝑋𝑋⟧ case, the line profile analyses are then applied to the horizontal (or vertical) lines that are closer to the related O point (see for example the line DE in Fig. 3). The details of these analyses are not reported here, and only are illustrated in Fig. 4 the main results of these line profile analyses (for ⟦1, 𝑉𝑉⟧ and ⟦2, 𝑉𝑉⟧ configurations). The line profiles along DE give access to the position and shape of the meniscus at each temperature. Taking then reference from the x-y position of a characteristic point of the viewed cell volume (such as the point B in the selected case of Fig. 3), the bare pixel distance of the meniscus position can be estimated. The temperature dependences of these bare distances are reported in Fig. 4 which illustrates (i) the well-defined crossing of the meniscus at finite temperature distance from the transition temperature, and (ii) the well-defined position of the volumetric median plane of the fluid cell.

Additional important results are also obtained, as the amplitude and shape of the capillary rising effect, the symmetrical matching of the meniscus position by fine tuning (±0.1 pixel, typically) of the apparent median plane for the two (slightly shifted) apparent cells, and the resulting noticeable symmetrical behavior of the capillary rising on the complete temperature range. Finally, the essential feature of the image analyses reflects the combination of the highly symmetrical cell design, the small off-density criticality of the cell filling and the wide field-of-view cell imaging. Such a combination leads to estimate the absolute position of the volumetric median plane of the cell and the meniscus position with the ±0.5 pixel (i.e., ±6 μm) resolution. Such a resolution is obtained whatever the ⟦𝑖𝑖, 𝑋𝑋⟧ configurations, thanks to the similarity of the meniscus behavior and temperature crossing for two reverse positions under the gravity field acceleration. One noticeable remark concerns the (gas or liquid) filling of one half part of the dead volume (i.e., 7.0 mm 3 ). Such a non-viewed fluid volume is thus similar to a viewed fluid median layer of 〈𝛿𝛿ℎ〉 𝑓𝑓𝑓𝑓 = (1 2 ⁄ )𝑉𝑉 𝑓𝑓𝑓𝑓 𝐴𝐴 𝑓𝑓𝑓𝑓 ≃ 263 ⁄ µm thickness (i.e., ≃ 21.91 pixels). The possible non-symmetrical effects related to the phase behavior in each windowless fluid volume can then easily be detected from the related viewed change of the meniscus position (see § 4), while its minimum ±0.5 pixel variation only corresponds to ±0.0675% of the total fluid volume, conform to the density precision requirements.

Results

Figure 4 shows that the pixel coordinate of the symmetrized meniscus positions, i.e., one half part noted ℎ 𝑖𝑖,𝑋𝑋 of the differences between the related bare pixel distances, can be estimated from reference to the volumetric median plane of the cell in the selected configurations. The temperature behaviors of ℎ 𝑖𝑖,𝑋𝑋 are reported in Fig. 5 for the eight ⟦𝑖𝑖, 𝑋𝑋⟧ configurations. Except for both ⟦𝑖𝑖, 𝑇𝑇⟧ and ⟦𝑖𝑖, 𝐻𝐻⟧ cases (see below), the temperature crossing of the volumetric median plane of the cell occurs in the range 44673 ≤ 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (m°C) ≤ 44973 (i.e., 𝑇𝑇 = 𝑇𝑇 𝑐𝑐 -{600; 900} m°C (accounting for ±0.5 pixel ~�𝛿𝛿ℎ 𝑖𝑖,𝑋𝑋 � ≤ ±6 µm uncertainty). The meniscus behaviour for ⟦𝑖𝑖, 𝑇𝑇⟧ cases is clearly affected by a significant non-symmetrical effect of wetting liquid phase inside the dead cell volume. Indeed that is the only configuration where the expected gas-like dead volume appears on a position which can easily be connected to the liquid side of the cell by the capillary effects. Accounting for the above remark about 〈𝛿𝛿ℎ〉 𝑓𝑓𝑓𝑓 , it seems that 1/3 of this gas-like dead volume can be filled by liquid around 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 . Obviously, this over liquid trapping decreases with temperature since the meniscus position is more and more lowered below the corresponding fill-in channel.

Such plausible non-symmetrical liquid wetting effects can also occur for ⟦𝑖𝑖, 𝐻𝐻⟧ cases, particularly observed at low temperatures ( 𝑇𝑇 ≃ 43573 m°C) where capillary condensation can exist in the small fill-in channel. Conversely only a very small part (2/10) of the dead volumes seems to be responsible of the ℎ 𝑖𝑖,𝐻𝐻 differences compared to the ⟦𝑖𝑖, 𝑉𝑉⟧ or the ⟦𝑖𝑖, 𝑍𝑍⟧ configuration cases.

Modeling

The following modeling starts from the initial result given in Ref. [START_REF] Morteau | Proceedings of 2nd European Symposium Fluids in Space[END_REF] for an ideal constant cylindrical volume of the fluid sample with radius 𝑅𝑅 filled at a small liquid-like off-critical density 〈𝛿𝛿𝜌𝜌 �〉 > 0 . The horizontal position ℎ ≪ 𝑅𝑅 of the liquid-gas meniscus from reference to the horizontal cell median plane is written as follows

ℎ 𝑅𝑅 = 𝜋𝜋 4 〈𝛿𝛿𝜌𝜌 �〉-∆𝜌𝜌 � 𝑑𝑑 (∆𝜌𝜌 �) 𝐿𝐿𝐿𝐿 (1)
where

∆𝜌𝜌 � 𝑑𝑑 = 𝜌𝜌 𝐿𝐿 +𝜌𝜌 𝐿𝐿 2𝜌𝜌 𝑐𝑐 -1 (2)
(∆𝜌𝜌 �) 𝐿𝐿𝐿𝐿 = 𝜌𝜌 𝐿𝐿 -𝜌𝜌 𝐿𝐿 2𝜌𝜌 𝑐𝑐 (3)
𝜌𝜌 𝐿𝐿 and 𝜌𝜌 𝐿𝐿 are the coexisting liquid and vapor densities at temperature 𝑇𝑇 < 𝑇𝑇 𝑐𝑐 , respectively. In this ideal cylindrical cell, the fluid compressibility and capillary effects are neglected, while only simple geometrical considerations are used to define the liquid-vapor distribution which results from the fluid mass conservation at any 𝑇𝑇.

For the present ALIR5 case, the additional Γ-like symmetrical windowless fluid volume is accounted for rewriting the total volume 𝑉𝑉 𝑓𝑓 as 𝑉𝑉 𝑓𝑓 = 𝜋𝜋𝑅𝑅 2 𝑒𝑒 𝑓𝑓 (1 + 𝑥𝑥) , with 𝑥𝑥 = 𝑉𝑉 𝑓𝑓𝑓𝑓 𝑉𝑉 𝑓𝑓𝑓𝑓 ⁄ ≃ 0.063 . As a direct consequence, the fluid (gas or liquid) filling of the half part of the dead volume is measured by the ratio 〈𝛿𝛿ℎ〉 𝑓𝑓𝑓𝑓 𝑅𝑅 ⁄ = ±𝜋𝜋𝑥𝑥 4 ⁄ ≃ ±0.0496 . 〈𝛿𝛿ℎ〉 𝑓𝑓𝑓𝑓 is the above viewed change of the meniscus position around the fluid median plane (i.e., ≃±21.91 pixels or ≃±263 μm).

The thermal effects are accounted for by exchanging the 〈𝛿𝛿𝜌𝜌 �〉 term of Eq. ( 1) by

〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 = 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 𝑐𝑐 (1 + 3𝛼𝛼 𝑇𝑇 𝑇𝑇 𝑐𝑐 ∆𝜏𝜏 * ) + 3𝛼𝛼 𝑇𝑇 𝑇𝑇 𝑐𝑐 ∆𝜏𝜏 * (4)
The above temperature dependence of 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 is obtained from a linear change of the cell mean density written as 𝜌𝜌 𝑇𝑇 = 〈𝜌𝜌〉 𝑇𝑇 𝑐𝑐 (1 + 3𝛼𝛼 𝑇𝑇 𝑇𝑇 𝑐𝑐 ∆𝜏𝜏 * ) . 𝛼𝛼 𝑇𝑇 = 1.8 × 10 -6 K -1 is the thermal dilatation coefficient of the CuCo2Be alloy. In the temperature range 𝑇𝑇 𝑐𝑐 -𝑇𝑇 ≤ 2 K, the cell thermal dilatation effect is lower than 5.5%. These effects reach 29% at 𝑇𝑇 𝑐𝑐 -𝑇𝑇 ≃ 10 K. Such effects need to be accounted for in the cell filling process at the laboratory temperature (they are ≃60 % in the temperature range at 𝑇𝑇 𝑙𝑙𝑙𝑙𝑓𝑓 ≃ 20 -25°C.

The liquid wettability effects are estimated as the form of an equivalent height down 〈𝛿𝛿ℎ〉 𝑐𝑐𝑙𝑙 of the meniscus position

〈𝛿𝛿ℎ〉 𝑐𝑐𝑙𝑙 ∝ 𝑙𝑙 𝑐𝑐𝑙𝑙 2 �1 - 𝜋𝜋 4 � 2𝑅𝑅+𝑐𝑐 𝑓𝑓 𝑅𝑅𝑐𝑐 𝑓𝑓 (5)
〈𝛿𝛿ℎ〉 𝑐𝑐𝑙𝑙 corresponds to the thickness of a horizontal liquid planar layer having similar volume to the total wetted liquid volume on sapphire and cell body. To derive Eq. ( 5), it is assumed an ellipsoidal shape for the meniscus capillary rising such that the product of characteristic size parameters are proportional to the squared capillary length 𝑙𝑙 𝑐𝑐𝑙𝑙 2 (so-called the capillary constant 𝑎𝑎 2 ) i.e., 𝑙𝑙 𝑐𝑐𝑙𝑙 2 = 𝑙𝑙 0 2 |∆𝜏𝜏 * | 2ν-𝛽𝛽 , with 2ν -𝛽𝛽 = 0.935 and asymptotic amplitude 𝑙𝑙 0 2 ≃ 3.84 mm 2 [START_REF] Garrabos | Master singular behavior for the Sugden factor of one-component fluids near their gas-liquid critical point[END_REF]. The 𝑙𝑙 𝑐𝑐𝑙𝑙 2 behavior compares well with the effective singular behavior 𝑎𝑎 2 = (3.94 𝑚𝑚𝑚𝑚 2 )|∆𝜏𝜏 * | 0.944 [27] [START_REF] Moldover | Capillary rise, wetting layers, and critical phenomena in confined geometry[END_REF]. The validity of 𝑙𝑙 𝑐𝑐𝑙𝑙 2 ∝ 𝑎𝑎 2 is controlled from the apparent size of the meniscus thickness due to the meniscus capillary rising. Finally only the proportional amplitude (of value 1.44, see § 6) of Eq. ( 5) remains as the adjustable parameter at large temperature distances from 𝑇𝑇 𝑐𝑐 . However, in the temperature range 𝑇𝑇 𝑐𝑐 -𝑇𝑇 ≤ 3 K, it is noticeable that 〈𝛿𝛿ℎ〉 𝑐𝑐𝑙𝑙 remains lower than one half pixel (<6 µm), i.e., 〈𝛿𝛿ℎ〉 𝑐𝑐𝑙𝑙 𝑅𝑅 < 1.2 × 10 -3 ⁄ . In such a case, it is also important to note the large value of the ratio 〈𝛿𝛿ℎ〉 𝑓𝑓𝑓𝑓 〈𝛿𝛿ℎ〉 𝑐𝑐𝑙𝑙 ⁄ ≃ 50. The final functional form of ℎ then writes as follows (1 + 𝑥𝑥) -〈𝛿𝛿ℎ〉 𝑐𝑐𝑐𝑐 𝑅𝑅 [START_REF] Cailletet | Recherches sur les densités des gaz liquéfiés et de leurs vapeurs saturées[END_REF] where only the fluid compressibility effects still remain neglected. These latter effects can be observed from the grid deformation and the related local turbidity on the both sides of the vapor-liquid meniscus and are only noticeable in the temperature range 𝑇𝑇 𝑐𝑐 -𝑇𝑇 ≤ 5 m°C.

Discussion

When the capillary rise effects are negligible (i.e. in the temperature range 𝑇𝑇 𝑐𝑐 -𝑇𝑇 ≤ 3 K), Eq. [START_REF] Cailletet | Recherches sur les densités des gaz liquéfiés et de leurs vapeurs saturées[END_REF] shows that the meniscus behavior in a cell with finite positive value of 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 crosses the cell median plane at a single temperature 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 where ℎ = 0 , i.e., 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = ∆𝜌𝜌 � 𝑑𝑑 . A first approach considers the linear functional form of 𝜌𝜌 � 𝑑𝑑 such as

𝜌𝜌 � 𝑑𝑑 = 1 + 𝑎𝑎 𝑑𝑑 |∆𝜏𝜏 * | (7)
where the value 𝑎𝑎 𝑑𝑑 = 0.84 ± 0.015 of the slope of rectilinear diameter results from the coexisting density data on the complete two-phase range. The above central estimation of 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 44823 m°C then leads to 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ≅ 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 𝑐𝑐 = 0.20% . Earth's-based visualization of the meniscus behavior in the ALIR5 cell is well an academic benchmark experiment where the resolution in the image processing at the pixel level is of prime interest for accurate determination of the mean filling density of the fluid cell. This experiment authorizes a preliminary checking (without accounting for the compressibility effects) of the validity of the expected singular top shape of the coexisting density curve and its related singular diameter presumably satisfying the different theoretical functional forms issued from the literature. The singular top-shape of the coexistence curve (∆𝜌𝜌 �) 𝐿𝐿𝐿𝐿 (|∆𝜏𝜏 * |) for |∆𝜏𝜏 * | ≤ 10 -2 can be predicted without adjustable parameter [START_REF] Garrabos | Crossover equation of state models applied to the critical behavior of xenon[END_REF] from the theoretical master crossover functions estimated from the massive renormalization scheme. Nevertheless, any other effective power laws to describe (∆𝜌𝜌 �) 𝐿𝐿𝐿𝐿 (|∆𝜏𝜏 * |) of SF6 (such as for instance (∆𝜌𝜌 �) 𝐿𝐿𝐿𝐿 = 1.7147|∆𝜏𝜏 * | 0.3271 + 0.8203|∆𝜏𝜏 * | 0.8215 -1.4396|∆𝜏𝜏 * | 1.2989 from Ref. [START_REF] Ley-Koo | Revised and extended scaling for coexisting densities of SF6[END_REF]) do not modify the following analysis, especially considering the temperature range 0.03 ≤ 𝑇𝑇 𝑐𝑐 -𝑇𝑇 < 3 K where �𝛿𝛿ℎ 𝑖𝑖,𝑋𝑋 � ≤ 100 µm (i.e., �𝛿𝛿ℎ 𝑖𝑖,𝑋𝑋 � ≤ 8 pixels.

The second approach thus introduces the singular functional forms of ∆𝜌𝜌 � 𝑑𝑑 as follows

∆𝜌𝜌 � 𝑑𝑑 = 𝐴𝐴 𝛽𝛽 |∆𝜏𝜏 * | 2𝛽𝛽 +𝐴𝐴 𝛼𝛼 |∆𝜏𝜏 * | 1-𝛼𝛼 +𝐴𝐴 1 ∆𝜏𝜏 * +𝐴𝐴 ∆ |∆𝜏𝜏 * | 𝑥𝑥 ∆ 1+𝑙𝑙 ∆ |∆𝜏𝜏 * | ∆ (8)
Equation ( 8) results from the various complete field mixing (CFM) models predicting the singular asymmetry with adjustable amplitudes. The amplitude sets obtained from the Weiner's data fitting [START_REF] Kim | Singular coexistence-curve diameters: Experiments and simulations[END_REF] given in Table 1 with 𝛼𝛼 = 0.109 , 𝛽𝛽 = 0.326 , ∆= 0.52, and 𝑥𝑥 ∆ = 1 -𝛼𝛼 + ∆. Any adjustment of (at least three) free amplitudes appears Weiner-like compatible whatever the used additive forms of the power laws and exponents involved in Eq. ( 8). The corresponding estimations of (ℎ 𝑅𝑅 ⁄ ) as a function of 𝑇𝑇 𝑐𝑐 -𝑇𝑇 are illustrated in Fig. [START_REF] Cailletet | Recherches sur les densités des gaz liquéfiés et de leurs vapeurs saturées[END_REF] where the value 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 𝑐𝑐 = 0.20% is fixed. Only the experimental results for the ⟦𝑖𝑖, 𝑉𝑉⟧ (full blue circles) and ⟦𝑖𝑖, 𝑍𝑍⟧ (full green triangles) configurations are used. For the rectilinear diameter case, the dotted and full blue curves are for the use of Eqs. ( 6)-( 7), without or with capillary correction term, where are introduced the corresponding predictions of Ref. [START_REF] Garrabos | Crossover equation of state models applied to the critical behavior of xenon[END_REF] (see above). For the critical hook case, the brown, green, and pink full curves are for the use of Eqs. ( 6) and ( 8) with parameters of columns 2, 3, and 5 (without visible difference between the green curve (column 3 case) and green circles (column 4 case). In addition, published 𝜌𝜌 𝐿𝐿 𝜌𝜌 𝑐𝑐 ⁄ and 𝜌𝜌 𝐿𝐿 𝜌𝜌 𝑐𝑐 ⁄ Weiner's data (Table X of [START_REF] Weiner | Breakdown of the law of rectilinear diameter[END_REF]), earlier supporting the diameter deviation on SF6, can directly be used to estimate the various terms of Eq. ( 6). In Fig. 6, the orange full diamonds are for the corresponding meniscus positions at the Weiner's experimental 𝑇𝑇 𝑐𝑐 -𝑇𝑇 values. Clearly, only the (ℎ 𝑅𝑅 ⁄ ) calculations for the linear density diameter case are in good agreement with the experimental data, especially in the two temperature decades 25 ≤ 𝑇𝑇 𝑐𝑐 -𝑇𝑇 < 2500 mK of prime interest regarding the neglected effects. In addition to an intrinsic questioning of the Weiner's measurements of the SF6 diameter deviation [START_REF] Moldover | Capillary rise, wetting layers, and critical phenomena in confined geometry[END_REF], the noticeable inconsistency observed in Fig. 6 can be attributed to a non-realistic estimation of the uncertainty on (𝜌𝜌 𝐿𝐿 𝜌𝜌 𝑐𝑐 ⁄ ) + (𝜌𝜌 𝐿𝐿 𝜌𝜌 𝑐𝑐 ⁄ ) Weiner values (at least onedecade larger than the maximum amplitude (0.5%) of the hook-like deviation), especially close to critical temperature (see Fig. 1). More generally, the systematic large data dispersion on the complete temperature range of the Weiner's data mainly seems due to the Weiner's values of the critical parameters 𝜌𝜌 𝑐𝑐 (density), 𝜀𝜀 𝑐𝑐 (dielectric constant), and then 𝐶𝐶𝐶𝐶 𝑐𝑐 = (1 𝜌𝜌 𝑐𝑐 ⁄ ) (𝜀𝜀 𝑐𝑐 + 1) (𝜀𝜀 𝑐𝑐 + 2) ⁄ (Clausius-Mossotti constant), which are significantly different ( -1.5% , -10.9% , and -3.3% , respectively) from the literature ones [START_REF]Weiner's values are 𝜌𝜌 𝑐𝑐 = (0.731 ± 0.001) g.cm -3 , 𝜀𝜀 𝑐𝑐 = 0.262 ± 0.010[END_REF].

Conclusions

The (ℎ 𝑅𝑅 ⁄ ) modeling from Eqs. ( 6) and ( 7) is comparable (in amplitude and uncertainty) with the Earth's-based measurements. Along the off-critical thermodynamic path of 〈𝛿𝛿𝜌𝜌 �〉 = +0.20 -0.04 +0.04 % , the careful imaging analysis of the SF6 two-phase domain appears well understood without the supplementary addition of any singular hook-shaped deviation in the rectilinear density diameter. The main part of the uncertainty in the rectilinear density diameter remains due to the actual level of precision (0.21%) for the SF6 critical density value. In such an uncertainty range, the cell thermal dilatation, the fluid compressibility, the fluid coexisting densities and the liquid wettability effects can be well-controlled from the highly symmetrical ALIR5 cell design. The slope of the SF6 linear density diameter seems the only remaining adjustable parameter, leading to questionable applicability of the complete field mixing to the simple fluid case. Future modeling approaches [START_REF] Garrabos | Liquid-vapor rectilinear diameter revisited[END_REF] will be focused on the estimation of the fluid compressibility effects using an upgraded version of the universal parametric equation of state. Moreover ongoing experimental works will be performed to account for the eventual contribution of the SF6 purity on the critical parameters before to close in a sure manner the debating situation about the density diameter behavior close to its critical point.

Fig. 1 .

 1 Fig. 1. Experimental critical hook of the linear density diameter of SF6 reported in [10] and related temperature 𝑇𝑇 𝑐𝑐 -𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 where the meniscus position must be collapsed on the volumetric median plane of a cylindrical cell filled at the liquid-like off-critical density of 0.20%. Green lines: Expected from the singular hook of the density diameter. Red lines: expected from the rectilinear density diameter

  leads to the ALIR5 viewed cylindrical fluid volume of 𝑉𝑉 𝑓𝑓𝑓𝑓 = 𝜋𝜋𝑅𝑅 2 𝑒𝑒 𝑓𝑓 = (221.7 -0.70 +0.20 ) mm 3 and the ALIR5 cross-sectional area of 𝐴𝐴 𝑓𝑓𝑓𝑓 = 2𝑅𝑅𝑒𝑒 𝑓𝑓 = (26.621 ± 0.020) mm 2 for any viewed median volumetric plane, except around the direction of the fill-line setup, as detailed below. The cell body is made of a machined CuCo2Be parallelepipedic block of external dimensions 𝐿𝐿(= 25) × 𝑙𝑙(= 27) × ℎ(= 24) mm 3 . This body contains two similar fill-line dead volumes -each one in the Γ-like form of two perpendicularly crossed cylindrical holes -located in the median plane of the fluid thickness. These two Γ-like volumes are symmetrical from the central optical axis used as the rotation axis of the cell. The resulting windowless fluid volume is (1 2 ⁄ )𝑉𝑉 𝑓𝑓𝑓𝑓 = (7.0 ± 0.2) mm 3 on each side of the observed cylindrical fluid volume. Therefore, 𝑉𝑉 𝑓𝑓 = 𝑉𝑉 𝑓𝑓𝑓𝑓 + 𝑉𝑉 𝑓𝑓𝑓𝑓 = (235.70 -1.0 +0.5 ) mm 3 is the total fluid sample volume. All the above dimensional values are from mechanical measurements performed at 20°C. The common axis of the two small opposite cylindrical holes opened in the main cylindrical fluid volume defines the single particular direction of the common median plane. In this latter plane occurs the maximum fluid area ( 𝐴𝐴 𝑓𝑓𝑓𝑓 + 𝐴𝐴 𝑓𝑓𝑓𝑓 = 𝐴𝐴 𝑓𝑓𝑓𝑓 + (17.8 ± 1.0) mm 2 ) crossing the complete fluid volume. The horizontal position of this median plane is chosen as the zero angle ( 𝜃𝜃 = 0 °) of the cell rotation (or 𝜃𝜃 = 180 °, equivalently, for the opposite configuration of the cell versus the direction of the gravity vector). From reference to this cell direction, the maximum tilted angle 𝜃𝜃 𝑚𝑚 that overlaps the Γ-like configuration of the dead volume is 𝜃𝜃 𝑚𝑚 ≳ 28°. The ±𝜃𝜃 𝑚𝑚 -directions are not equivalent versus the liquid (or gas) gravity positionning inside each dead volume (see below § 6).

Fig. 2 .

 2 Fig. 2. (a) Picture of the ALIR5 cell. (b) Schematic cross section of the ALIR5 cell where are illustrated the four relative directions of the meniscus 𝜃𝜃 = {-23.2°; 0°; +22.9°; +90°} and the direction 𝜃𝜃 𝑚𝑚 ≳ 28° that overlaps the Γ-like forms of the two symmetrical dead volumes associated to the fill-line direction. Red area: fluid volume; blue area: diffusion windows, filling screws and stoppers; green area: cell body; external circle : dimensional scale .

Fig. 3 .

 3 Fig. 3. Video picture of the ALIR5 cell for the ⟦1, 𝑉𝑉⟧ configuration at temperature 𝑇𝑇 = 45473 m°C. Selected borderline points A, B, C, T, & R between the fluid and cell body are used in the image processing to define the fluid sample cell position (especially the apparent center point O) inside the picture. Vertical line DE is used to analyse the meniscus position and shape as functions of temperature. The next step compares similar line profiles obtained at different temperatures to probe the absence of thermal effect at the pixel level during the temperature timeline on each facility configuration.The last step optimizes the matching of the selected characteristic points for two reversed similar configuration (⟦1, 𝑉𝑉⟧ and ⟦2, 𝑉𝑉⟧ for the chosen case). Indeed, the changes of the facility positions under the Earth's gravitational acceleration field induce small mechanical relative displacements due to the intrinsic clearance between the different (optical and mechanical) components. The present concern involves the cell (housed in the insert) in front of the video camera (located in the optical box of the facility). Therefore, this cell image matching step leads to the determination of the (horizontal and vertical) pixel shifts (∼2 to 6 pixels, typically) between two reversed images of the viewed fluid cell volume.

Fig. 4 .

 4 Fig. 4. Bare pixel distances of the meniscus as functions of the temperature for the ⟦1, 𝑉𝑉⟧ (open red circles) and ⟦2, 𝑉𝑉⟧ (open blue circles) configurations. Related bare pixel distance of the volumetric median plane of the cell. From reference to the y pixel coordinate of the B point in Fig. 3.

Fig. 5 .

 5 Fig. 5. Temperature dependence of the symmetrical pixel shift of the meniscus position from reference to the corresponding volumetric median plane, for the eight ⟦𝑖𝑖, 𝑋𝑋⟧ configurations. 1pixel = 12µm.

Fig. 6 .

 6 Fig. 6. Comparison between experimental and modelling results of (ℎ 𝑅𝑅 ⁄ ) as functions of 𝑇𝑇 𝑐𝑐 -𝑇𝑇 , fixing 〈𝛿𝛿𝜌𝜌 �〉 𝑇𝑇 𝑐𝑐 = 0.20%.

Table 1 .

 1

		SF6 parameters for Eq. (8)
		[7]	[7]	[7]	[9]
	Aβ	0	1.124	1.0864	0.46028392
	Aα	6.365	-9.042 -7.990	-0.6778981
	A1	-10.13	11.37	9.770	0.13516245
	A∆	8.080	-3.354 0	0
	a∆	0	0	3.318	0

Acknowledgements

We thank all the CNES, CNES-CADMOS, NASA, and associated industrial teams involved the DECLIC facility project. CL, SM, YG, DB, are grateful to CNES for the financial support. They are also grateful to Philippe Bioulez and Hervé Burger for their operational support at CADMOS. The research of I.H. was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.


Elisa Louyriac

Eva Laur

Alexandre Welle

Aurélien Vantomme

Olivier Miserque

Jean-Michel Brusson

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

Experimental and Computational Investigations on Highly Syndioselective Styrene-Ethylene Copolymerization Catalyzed by Allyl ansa-Lanthanidocenes

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Syndiotactic polystyrene (sPS) is an attractive engineering plastic potentially usable for many industrial applications due to its fast crystallization rate, low permeability to gases, low dielectric constant and good chemical and temperature resistance. [START_REF] Ishihara | Stereospecific Polymerization of Styrene Giving the Syndiotactic Polymer[END_REF][START_REF] Malanga | Syndiotactic Polystyrene Materials[END_REF][START_REF] Schellenberg | Syndiotactic Polystyrene: Process and Applications[END_REF] However, its high melting point (270 °C) and its brittleness are the two main drawbacks limiting its processability. To tackle this issue, several strategies have been envisaged: blending or postmodification of sPS, polymerization of functionalized styrene derivatives, or copolymerization of styrene with other monomers. [START_REF] Zinck | Functionalization of Syndiotactic Polystyrene[END_REF][START_REF] Jaymand | Recent Progress in the Chemical Modification of Syndiotactic Polystyrene[END_REF] The latter approach was found effective and versatile to fine-tune the properties of sPS, [START_REF] Laur | Engineering of Syndiotactic and Isotactic Polystyrene-Based Copolymers via Stereoselective Catalytic Polymerization[END_REF] more particularly via syndioselective copolymerization of styrene with ethylene. [START_REF] Rodrigues | Groups 3 and 4 Single-Site Catalysts for Styrene-Ethylene and Styrene-α-olefin Copolymerization[END_REF] The copolymerization of those two monomers is quite challenging due to their strikingly different reactivity. As a result, most of the group 4 catalysts active for sPS production only provided "ethylene-styrene interpolymers" (ESI), featuring no stereoregularity and amounts of incorporated styrene below 50mol%. Those issues were overcome by the development of group 3 catalysts, independently disclosed by our group [START_REF] Rodrigues | Allyl ansa-lanthanidocenes: Single-Component, Single-Site Ccatalysts for Controlled Syndiospecific Styrene and Styrene-Ethylene (Co)Polymerization[END_REF] and by Hou and co-workers. [START_REF] Luo | Scandium Half-Metallocene-Catalyzed Syndiospecific Styrene Polymerization and Styrene-Ethylene Copolymerization: Unprecedented Incorporation of Syndiotactic Styrene-Styrene Sequences in Styrene-Ethylene Copolymers[END_REF] Yet, the number of effective catalytic systems for sPSE synthesis remains quite limited to date. [START_REF] Li | Aluminum Effects in the Syndiospecific A c c e p t e d m a n u s c r i p t 21[END_REF] Very recently, we reported on the synthesis and catalytic investigations of a new series of neutral ansa-lanthanidocene catalysts for the production of sPS; 11 a thorough DFT study of these systems highlighted the different factors governing the formation of sPS. 11 In this new contribution, we describe the syndioselective copolymerization of styrene with ethylene using this latter series of complexes and demonstrate that some of them feature improved catalytic performances as compared to the current state-of-the-art (Scheme 1). For the first time, the parameters that control syndioselective styrene-ethylene copolymerization were investigated also by DFT computations. These calculations contributed to a better understanding of the

Results and Discussion

Styrene-Ethylene Copolymerizations Catalyzed by Allyl Ansa-Lanthanidocenes.

Styrene/ethylene copolymerizations catalyzed by complexes 1-Nd-K-allyl, 2-Nd7-Nd, 2-Sc, 2-La, 2-Sm and 2-Pr were first screened under similar conditions (Table 1, entries 111). As already described for styrene homopolymerization, 11 the reactions were best conducted using a few equiv of (nBu)2Mg as scavenger, to prevent catalyst decomposition by trace impurities, especially at low catalyst loading and high temperature (vide infra). This dialkylmagnesium appeared to be a poor chain transfer agent under those conditions and did not affect the reaction mechanism nor the properties of the produced sPSE copolymers. showed lower productivities and afforded sPSE copolymers with a higher ethylene content (thus affecting the calculation of syndioselectivity which appeared, at first sight, lower due to more abundant St-E enchainments) (entries 2 and 67). The latter observation suggests that introduction of substituents bulkier than tBu, namely cumyl or Ph2MeC-, at the 2,7-positions of the fluorenyl ligand favors insertion of a small monomer ethylene rather than styrene.

The nature of the metal center played also a key role. Complex 2-Sc was nearly inactive whereas 2-Pr and 2-La afforded sPSE copolymer with productivities of ca. 300 kg•mol(Ln) 1 •h 1 . Under those non-discriminating conditions, full styrene conversion was reached when using complex 2-Sm, as observed with its neodymium analogue 2-Nd. Substantial improvement of the productivity values was obtained under more forcing and demanding copolymerization conditions (entries 1217). Increasing both the temperature of polymerization up to 140 °C and the monomer-to-catalyst ratio up to 40 000 allowed to reach productivities above 1,000 kg(sPSE)•mol(Ln) of the copolymer significantly dropped ([r] 5 = 3235%). Such a marked discrepancy between the stereoselectivity of 2-Nd and 2-Sm was not observed for the copolymerizations performed at 60 °C (compare entries 2, 10 and 11 with entries 1217). Overall, these results are in line with those already described for syndioselective styrene homopolymerization, 11 and highlight the remarkable stability of 2-Ln catalytic systems under such drastic conditions (thanks to (nBu)2Mg as scavenger). The productivities of these systems are comparable with those of the most active cationic scandium-based systems reported for syndiospecific styrene/ethylene copolymerization. [START_REF] Rodrigues | Allyl ansa-lanthanidocenes: Single-Component, Single-Site Ccatalysts for Controlled Syndiospecific Styrene and Styrene-Ethylene (Co)Polymerization[END_REF][START_REF] Li | Aluminum Effects in the Syndiospecific A c c e p t e d m a n u s c r i p t 21[END_REF] The most productive and syndioselective catalyst, 2-Nd, was tested on a 10-fold larger production scale (i.e., on a half-kg styrene) in bulk conditions at 100 °C in a closed reactor;

five different experiments with variable amounts of ethylene (vide infra) were conducted and returned improved productivities in the range 2 7305 430 kg(sPSE)•mol(Nd) 1 •h 1 (entries 1822). Under these bulk conditions, molecular weights of the resulting copolymers were somewhat higher than those obtained at a lower (bench) scale in 50:50 v/v mixtures of styrene/hydrocarbon solvent (Mn = 43 00062 000 g•mol 1 vs. Mn = 33 000 g•mol 1 , respectively) and the polydispersities were also narrower (ÐM = 1.42.5). These data highlight the significant impact of the process conditions on both the catalytic system productivities and characteristics of the polymers.

A c c e p t e d m a n u s c r i p t The initial styrene-to-ethylene ratio was also varied by changing the amount of ethylene introduced at the beginning of the polymerization (entries 18-22; see Experimental part). The ethylene content in the copolymer can be hence easily tuned, allowing the Those 13 C{ 1 H} NMR spectra were recorded using an inverse-gated-decoupling sequence in order to accurately determine the amount of ethylene incorporated. As only isolated units of ethylene were detected, the amount of ethylene incorporated was determined integrating the signal of the ipso carbon (polystyrene sequences, δ 145.8 ppm) and the signals at 3738 ppm corresponding to the secondary carbons Sαγ.

Prod. b [kg• mol 1 •h 1 ] C2 inc. c [mol%] Tm d [° C] Tc d [° C] Tg d [° C] Hm d [J•g 1 ] Mn×10 3 [g•mol 1 ] e ÐM e [r]
The analysis of the spectra area corresponding to the resonance of the secondary carbon Sα of PS sequences also allowed quantifying the syndiotacticity at the hexad level (Figure 3). The relative intensity of the rrrrr hexad signal was obtained after deconvolution and integration of all the signals in this area. This means that not only the presence of others hexads mrrrr, rmrrr and rrmrr but also the presence of other unassigned sequences (in particular, those that are the consequence of S-E junctions, and presumably as well hexads with meso diads) were considered for the calculation of [r] [START_REF] Jaymand | Recent Progress in the Chemical Modification of Syndiotactic Polystyrene[END_REF] . The values measured in the present case ([r] 5 = 3278%) are similar to those previously reported in the case of sPSE materials obtained with I (Pr > 0.81, [r] 5 > 35%; depending on the ethylene content). [START_REF] Rodrigues | Allyl ansa-lanthanidocenes: Single-Component, Single-Site Ccatalysts for Controlled Syndiospecific Styrene and Styrene-Ethylene (Co)Polymerization[END_REF] A c c e p t e d m a n u s c r i p t down-si and up-re.

DFT investigation of styrene-ethylene copolymerization catalyzed by {Me2C(C5H4)(Flu)}Nd(C3H5) (I).

Complex I, which is highly effective to copolymerize styrene with ethylene while maintaining a high syndiotacticity, [START_REF] Rodrigues | Allyl ansa-lanthanidocenes: Single-Component, Single-Site Ccatalysts for Controlled Syndiospecific Styrene and Styrene-Ethylene (Co)Polymerization[END_REF] was selected as a benchmark for our theoretical study. Subsequently, it will allow us to highlight the influence of catalyst substituents present on the allyl and fluorenyl ligands on styrene-ethylene copolymerization.

i) First styrene vs. ethylene insertion. Energy profiles were computed for the first ethylene (3-E) and the 2,1-down-re (3d-re) styrene 11 insertions (Figure 4). From a kinetic point of view, transition state 3-E is more stable (by 3.3 kcal mol 1 ) than 3d-re, but this is included within the error range of the method. 14,15 The first insertion is likely more thermodynamically controlled and in favor of styrene insertion by 4.6 kcal mol 1 . The energy difference between 3d-re/3-E is mainly due to the steric hindrance around the metal center (Figure S9). Product 4-E obtained following the intrinsic reaction coordinate is extra stabilized by a resulting interaction between the terminal double bond of the allyl ligand and the metal center (Figure S10). The relaxation of the polymer chain leads to an endothermic product 4-Erelaxed (by 0.2 kcal mol 1 ), which is consistent with the fact that formation of an alkyl-from an allyl-complex is thermodynamically unfavorable.

A c c e p t e d m a n u s c r i p t

ii) Second styrene vs. ethylene insertion. As in the first step, 2,1-down-re styrene insertion is thermodynamically favored, second insertions were computed from the product 4d-re. The energy profiles were calculated for the stationary (6-E) and migratory (6-E') ethylene insertions and for 2,1-up-si (6u-si) stationary styrene insertions (Figure 5). Chart 2. Numbering used for carbon atoms in the allyl ligand.

As regards the 2 nd insertion products, the presence of a π-coordination between the phenyl ring of the first styrene inserted and the metal center further stabilizes the migratory insertion ethylene product 7-E' by 5.0 kcal mol 1 than for stationary insertion product 7-E (Figure S12). Hence, at the second insertion stage, the ethylene monomer will be inserted preferentially, according to the "migratory" mechanism.

ii) Third vs. ethylene insertion. Energy profiles were for the (9-E) ethylene insertion and for 2,1-down-re (9d-re) and 2,1-up-si (9u-si) styrene insertions (Figure 6). At the third insertion step, there is a slight kinetic preference for insertion of ethylene (9d-re/9-E = 4.1 kcal mol 1 ) (dark-green), probably related to the decrease of the steric hindrance around the metal center (Figure S13). In all products, the growing chains feature the same orientation, which may explain the same range of their energies (Figure S14).

To obtain further information about the nature of the resulting copolymer, it was crucial to investigate reactional pathways after insertion of two ethylene units and, more generally, after the two units of the same monomer were consecutively inserted.

Ethylene-Ethylene-Ethylene (E-E-E) vs. Ethylene-Ethylene-Styrene (E-E-S).

The energy profiles were computed for the third ethylene (9-E) and 2,1-down-re (9d-re) styrene

A c c e p t e d m a n u s c r i p t insertions, in case where two ethylene monomers were inserted according to the "migratory" insertion mechanism (Figure 7). At this stage, there is no significant kinetic preference between styrene and ethylene insertions (9d-re/9-E = 3.2 kcal mol 1 ). The energy difference between the both insertion products 10-E and 10d-re is 2.5 kcal mol 1 , which is also within the error range of the method. This is reflected in the product structures in which the growing chains are similarly oriented (Figure S16).

7.

Energetic profiles for the third ethylene insertion in {Me2C(C5H4)(Flu)}Nd(C3H5) (I), after two ethylene insertions. The third 2,1-down-re styrene insertion is plotted in blue.

A c c e p t e d m a n u s c r i p t

similarly performed for the third 2,1-down-re (9d-re) styrene and ethylene (9-E) insertions, after insertion of two styrene monomers (Figure 8). The results match those obtained for the above E-E-E vs. E-E-S study: (i) the energy difference 9d-re/9-E (1.3 kcal mol 1 ) is included within the error range of the method, and (ii) the growing chains appear to be similar in the transition state and product structures (Figures S17 andS18). This is confirmed by the lack of thermodynamic preference between the two monomers (10d-re/10-E = 0.5 kcal mol 1 ).

Overall, the above calculations indicate that, once two units of the same monomer have been inserted, there will be no selectivity at the next stage. In other words, I tends to form random styrene-ethylene copolymers.

A c c e p t e d m a n u s c r i p t (I), after two styrene insertions according to the "stationary" mode. The third 2,1-down-re styrene insertion (the most stable found in homopolymerization case) is plotted in blue.

DFT investigation of styrene-ethylene copolymerization catalyzed by {Me2C(C5H4)(Flu)}Nd(1,3-C3H3(SiMe3)2) (1-Nd).

In order to obtain information on the influence of SiMe3 substituents of the allyl ligand on styrene-ethylene copolymerization as well as on the nature of the sPSE copolymer obtained, the same study as that for I was carried out for the putative 1-Nd catalyst. The computational results are similar to those highlighted for the non-substituted catalyst I (all reaction profiles and structures are available in the Supporting Information; Figures S19-S33): (i) at the first step, a 2,1-down-re styrene insertion is preferred, followed by an ethylene insertion, and then, a slight preference for this latter monomer at the third step; (ii) after insertion of two same monomer units, there is no clear kinetic or thermodynamic preference between the two monomers.

Hence, the above calculations indicate that the presence of the bulky substituents in the allyl initiating group does not affect the chemistry and the nature of the obtained copolymer: the 1-Nd catalyst tends also to form random styrene-ethylene copolymers. This is consistent with an initiating group which is progressively rejected at the end of the growing polymer chain. It should be noted, however, that the bulky substituents on the allyl ligand induce an increase in the energy of the first insertion barriers (for example 24.5 vs. 14.5 kcal mol 1 for the first ethylene insertion), as this has already been observed for styrene homopolymerization. 11 This is again due to charge localization on the "wrong" carbon atom of the allyl ligand, that is the one that ensures the interaction with the metal center and therefore provides the nucleophilic assistance, rather than the one that is involved in the CC coupling.

A c c e p t e d m a n u s c r i p t

Furthermore, it is noteworthy that at the first ethylene insertion step in 1-Nd, the alkyl product 4-E-relaxed is thermodynamically favorable (by 10.3 kcal mol 1 ). This is not consistent with the usual trend that the formation of an alkyl from an allyl compound is disfavored thermodynamically. A charge analysis at the NBO level was then carried out in order to obtain some information about the nature of the allyl ligand in Therefore, this is not a standard allyl in 1-Nd but rather a masked alkyl, explaining why formation of 4-E-relaxed is thermodynamically favorable (10.3 kcal mol 1 vs. +0.2 kcal mol 1 in the case of the (C3H5) allyl in I).

DFT investigation of styrene-ethylene copolymerization catalyzed by {Me2C(C5H4)(2,7-tBu2Flu)}Nd(1,3-C3H3(SiMe3)2) (2-Nd).

It has been experimentally found that complex 2-Nd with tBu groups in 2,7-positions of the fluorenyl ligand exhibits a high productivity of up to 5,430 kg(sPSE)•mol(Nd) 1 •h 1 for styrene-ethylene copolymerization.

The microstructure of the sPSE copolymers shares the same features as those observed for copolymers obtained with I. DFT calculations were performed to rationalize this influence of the 2,7-tBu2 groups on the Flu ligand on the reactivity and on the copolymer obtained.

The first and second styrene insertion were computed (see ESI figures S34 and S35) and it was found that, unlike complex I, the migratory styrene insertion is preferred for the 2-Nd catalyst over the stationary insertion found for I. This can be attributed to the presence of bulky substituents on the allyl ligand that leads to a change in the polymerization mechanism in order to minimize steric repulsion. From a kinetic point of view, there is a clear preference for 3-E which is more stable by 6.8 kcal mol -1 than 3d-re. This energy difference is due to a repulsion between the tBu groups and the Ph ring of the incoming styrene which tends to destabilize 3d-re compared to between the tBu and the SiMe3 groups and cannot ensure the nucleophilic assistance. The C 1 (allyl) carries the negative charge in order to induce a reaction with the carbon atom of the ethylene monomer and maintains the interaction with the metal center (nucleophilic assistance). This implies that ethylene moves away from the metal center and, thus has a CC double bond less activated (CC = 1.40 Å vs.1.42 Å in 1-Nd and in I). This charge localization effect allows decreasing the activation barrier compared to the case of complex 1-

Nd.

In terms of thermodynamics, the alkyl product 4-E-relaxed is favorable (by 12.5 kcal mol 1 ) which, as pointed out above for 1-Nd, is related to the charge of the carbon atoms in the allyl ligand [C 1 (allyl) (1.03), C 2 (allyl) (0.23), C 3 (allyl) (1.15)] in 2-Nd. In this case, 4-E-relaxed is more stable by 4.2 kcal mol 1 compared to 4d-re-relaxed; therefore, ethylene would be preferentially inserted.

ii) Second styrene vs. ethylene insertion. Second insertions were computed after a 2,1-downre styrene insertion. The corresponding energy profiles were calculated for the stationary (6-E) and migratory (6-E') ethylene insertions and for 2,1-up-re (6u'-re) migratory styrene insertion (Figure 10). The results for the second insertion with 2-Nd are similar to those obtained for the two previous catalysts. Indeed, after a 2,1-down-re styrene insertion, migratory ethylene insertion is kinetically preferred (6u'-re/6-E' = 10.4 and 6-E/6-E' = 9.5 kcal mol -1 ).

These results are quite similar to those obtained for the 1-Nd and I catalysts, suggesting the formation of random copolymers. This conclusion is further strengthened by the results obtained for the third steps (Figures S40 andS41), as no selectivity was found, in line with the formation of random styrene-ethylene copolymers.

In summary, DFT calculations allowed to rationalize the nature of the copolymer obtained as well as the influence of the substituents of the catalyst. The 1,3-trimethylsilyl substituents on the allyl ligand cause (i) a modification of the distribution of the charges on A c c e p t e d m a n u s c r i p t the allylic carbon atoms, which makes the first ethylene insertion product thermodynamically favorable, and (ii) an increase in the insertion barriers, related to the steric hindrance and the charge distribution. On the other hand, bulky 2,7-tert-butyl groups on the fluorenyl ligand tend to promote ethylene insertion for the second insertion. This is also related to a charge localization effect.

Finally, for the three catalytic systems studied, no modification in the nature of the obtained copolymer is observed, that is the formation of random styrene-ethylene copolymers with a high syndiotacticity in the PS sequences.

Conclusions

The performance of a series of allyl ansa-lanthanidocenes of the general formula {R2C(C5H4)(R'R'Flu)}Ln(1,3-C3H3(SiMe3)2)(THF)x was assessed in styrene-ethylene copolymerization. By using forcing copolymerization conditions, that is a low catalyst loading and relatively high temperature, a high productivity of 5,430 kg(sPSE)•mol(Nd) 1 •h 1 was achieved with 2-Nd on a half-kilogram scale, which is comparable with the most active scandium half-sandwich complexes. [START_REF] Luo | Scandium Half-Metallocene-Catalyzed Syndiospecific Styrene Polymerization and Styrene-Ethylene Copolymerization: Unprecedented Incorporation of Syndiotactic Styrene-Styrene Sequences in Styrene-Ethylene Copolymers[END_REF] The sPSE copolymers thus obtained feature a random microstructure with single ethylene units distributed in highly syndiotactic PS sequences. The ethylene content and thus the thermal properties of the materials can be tuned by the initial comonomer feed.

Theoretical DFT studies allowed rationalizing the random nature of the obtained styrene-ethylene copolymers catalyzed by complexes I, 1-Nd and 2-Nd. The calculations showed that: (i) SiMe3 substituents on the allyl ligand have an influence on the nature of the first insertion product and notably on the stability of the ethylenic product, and (ii) those on the fluorenyl ligand either make the catalyst more ethylene reactive at the second insertion Typical procedure for bench-scale styrene-ethylene copolymerization. In a typical experiment (Table 1, entry 1), a 300 mL glass high-pressure reactor (TOP-Industrie) was charged with 50 mL of solvent (cyclohexane or n-dodecane) under argon flash and heated at the appropriate temperature by circulating water or oil in a double mantle. Under an ethylene A c c e p t e d m a n u s c r i p t flow, styrene (50 mL), a solution of (nBu)2Mg (0.5 mL of a 1.0 M solution in heptane) and a solution of pre-catalyst in toluene (ca. 43 mg in 2 mL) were introduced. The gas pressure in the reactor was set at 2 atm and kept constant with a back regulator, and the reaction media was mechanically stirred. At the end of the polymerization, the reaction was cooled, vented, and the copolymer was precipitated in methanol (ca. 500 mL); after filtration, it was washed with methanol and dried under vacuum at 60 °C until constant weight.

Typical procedure for half-kg-scale styrene-ethylene copolymerizations in a closed reactor. In a typical experiment (Table 1, entry 18), a 1 L high-pressure reactor was charged with 500 mL of styrene (degassed under nitrogen, stored in the fridge on 13X molecular sieves and eluted through an alumina column prior to use) under nitrogen flush and heated at the appropriate temperature by circulating oil in a double mantle. An exact amount of ethylene was introduced in one shot in the reactor using an injecting system equipped with a pressure gauge, followed by a solution of (nBu)2Mg (2.5 mL of a 1.0 M solution in heptane)

and the pre-catalyst (ca. 45 mg). The reactor was closed and the reaction mixture was mechanically stirred. At the end of the polymerization, the reaction mixture was cooled, vented, and the copolymer was precipitated in isopropanol (ca. 2 L); after filtration, it was washed with isopropanol. Polymer samples were dried in under vacuum in an oven heated at 200 °C.

Computational Details. The calculations were performed at the DFT level of theory using the hybrid functional B3PW91. 16,17 Neodymium was treated with a large-core 19,20 Toluene was chosen as solvent. The model that was used to take into account solvent effects is the SMD solvation model. The solvation energies are evaluated by a self-consistent reaction field (SCRF) approach based on accurate numerical solutions of the Poisson-Boltzmann equation. 21 All the calculations were carried out with the Gaussian 09 program. 22 Electronic energies and enthalpies were calculated at T = 298 K. Geometry optimizations were computed without any symmetry constraints and analytical frequency calculations was used to assess the nature of the extrema.

The connectivity of the optimized transition states was determined by performing Intrinsic Reaction Coordinates (IRC) calculations. Activation barriers ΔH # are defined depending on the sign of ΔHcoord (see Figure 11). 13 Electronic charges were obtained by using Natural Population Analysis (NPA) analysis. [START_REF] Reed | Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint[END_REF] NBO analysis [START_REF] Reed | Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint[END_REF]

AScheme 1 .

 1 Scheme 1. Allyl {Cp/Flu} ansa-lanthanidocenes used as single-component catalysts for

  11

A

  c c e p t e d m a n u s c r i p t As we have demonstrated in the previous study, 11 substitution on the fluorenyl moiety of the ligand has a strong influence on copolymerization productivities. Complexes 3-Nd and 5-Nd, bearing bulky substituents at the 3,6-positions of the fluorenyl ring, were not or poorly active. 1-Nd-K-allyl and 4-Nd, which bear no substituents on the fluorenyl ring, exhibited moderate productivities and 2-Nd, which holds tert-butyl substituents on remote 2,7positions, proved to be the most active within the Nd series (non-optimized productivity > 400 kg(sPSE)•mol(Nd) 1 •h 1 , entry 2). Compared to 2-Nd, complexes 6-Nd and 7-Nd

A c c e p t e d m a n u s c r i p t

A c c e p t e d m a n u s c r i p t production of a

  range of sPSE materials containing from 1.1 to 10 mol% of ethylene. DSC measurements showed that the melting transition temperature and the glass transition of those materials are closely related to the quantity of ethylene incorporated, decreasing almost linearly with the quantity of ethylene incorporated (Figure1).

Figure 1 .2457-Nd and 2 -

 12 Figure 1. Melting (Tm) and glass (Tg) transition temperatures of sPSE materials prepared in

Figure 2 .

 2 Figure 2. Aliphatic region of the 13 C{ 1 H} NMR spectra (125 MHz, 130 °C, C6H3Cl3/C6D6)

Chart 1 .

 1 Computational studies. In the previous study,11 DFT calculations including solvent model in the styrene homopolymerization catalyzed by {Me2C(C5H4)(Flu)}Nd(C3H5) (I), the putative {Me2C(C5H4)(Flu)}Nd(1,3-C3H3(SiMe3)2) (1-Nd) and the most effective [{Me2C (C5H4)(2,7-tBu2Flu)}Nd(1,3-C3H3(SiMe3)2)] (2-Nd) allowed to identify the factors which influence the styrene insertion according to the 2,1-pathway (which is the most favored mode). By using Castro et al.13 method, styrene and ethylene insertions were computed in order to evaluate the effectiveness of catalysts I, 1-Nd and 2-Nd in styrene-ethylene copolymerization and the topology of the obtained sPSE copolymer. At each step, the preference between ethylene and styrene insertions has been examined. Moreover, two chainend stereocontrol mechanims were also considered computationally. For the sake of clarity, the following definitions are considered: insertions that occur on the same enantiotopic site of coordination are denoted as "stationary" mechanism whereas "migratory" insertions refer to the switch of coordination site at each step (Chart 1). Nomenclature and orientation modes used for styrene insertion with respect to the ancillary ligand. In this representation, only down-re and up-si styrene coordination modes are depicted, corresponding to the enantiomer of the metal catalyst used for "stationary" A c c e p t e d m a n u s c r i p t insertions. The opposite configurations have been employed for "migratory" insertions, viz.

  A c c e p t e d m a n u s c r i p t

Figure 4 .

 4 Figure 4. Energetic profiles for the first ethylene (black) and 2,1-down-re styrene (blue)

Figure 5 .

 5 Figure 5. Energetic profiles for the second ethylene (stationary, black, and migratory, red)

A c c e p t e d m a n u s c r i p t 6 .

 6 Energetic profiles for the third insertions in {Me2C(C5H4)(Flu)}Nd(C3H5) (I), after a 2,1-down-re styrene first insertion and a migratory ethylene second insertion.

Figure 8 .

 8 Figure 8. Energetic profiles for the third ethylene insertion in {Me2C(C5H4)(Flu)}Nd(C3H5)

1 -

 1 Nd. The charges on the carbon atoms in the (1,3-C3H3(SiMe3)2) allyl ligand are [C 1 (allyl) (1.05), C 2 (allyl) (0.23), C 3 (allyl) (1.11)] in 1-Nd, whereas those obtained in the case of the unsubstituted allyl (C3H5) in I are [C 1 (allyl) (0.79), C 2 (allyl) (0.26), C 3 (allyl) (0.81)]. Thus, the sterically hindered allyl leads to a charge relocalization at the C 3 (allyl) carbon atom.

A c c e p t e d m a n u s c r i p tFigure 9 .

 9 Figure 9. Energetic profiles for the first ethylene (black) and 2,1-down-re styrene (blue)

3 -

 3 E. Moreover, the ethylene insertion barrier is intermediate (ΔH # = 20.4 kcal mol 1 ) to those A c c e p t e d m a n u s c r i p t calculated for I and 1-Nd (ΔH # = 14.5 and 24.5 kcal mol 1 , respectively). Indeed, the incorporation of tBu substituents counteracts the effect of the SiMe3 on the allyl ligand, which reduces the activation barrier and makes the catalyst more reactive towards ethylene. This is reflected in the NBO charge analysis. The charges on the carbon atoms in the allyl ligand in 3-E are [C 1 (allyl) (0.98), C 2 (allyl) (0.29), C 3 (allyl) (0.83)] for 2-Nd, [C 1 (allyl) (0.89), C 2 (allyl) (0.23), C 3 (allyl) (0.95)] for 1-Nd and [C 1 (allyl) (0.66), C 2 (allyl) (0.23), C 3 (allyl) (0.61)] for I. In complex 2-Nd, the carbon C 3 (allyl) is repulsed by an interaction

A c c e p t e d m a n u s c r i p tFigure 10 .

 10 Figure 10. Energetic profiles for the second ethylene (stationary, black, and migratory, red)

A c c e p t e d m a n u s c r i p t ( 2 , 2 -Y, 2 -La, 2 -

 2222 7 substitution) or block the reactivity(3,6 substitution). This last point is essential to explain the good productivity of catalyst 2-Nd for styrene-ethylene copolymerization.Experimental SectionGeneral considerations. All experiments were performed under a dry argon atmosphere, using a glovebox or standard Schlenk techniques. Complexes 1-Nd-K-allyl, 27-Nd, 2-Sc, Pr and 2-Sm were synthesized as reported before. 11 Cyclohexane and n-dodecane were distillated from CaH2 and stored over 3 Ǻ MS. Styrene (Fisher Chemical, general purpose grade, stabilized with 1015 ppm of tert-butylcatechol) was eluted through neutral alumina, stirred and heated over CaH2, vacuum-distilled and stored over 3Ǻ MS at 30 °C under argon. The (nBu)2Mg solution (1.0 M in heptane, Sigma-Aldrich) was used as received. Ethylene (Air Liquide, N35) was used without further purification.Instruments and measurements.13 C{ 1 H} NMR and GPC analyses of sPSE samples were performed at the research center of Total Raffinage-Chimie in Feluy (Belgium). 13 C{ 1 H} NMR analyses were run on a Bruker Avance III 500 MHz equipped with a cryoprobe HTDUL in 10 mm tubes (1,2,4-trichlorobenzene/C6D6, 2:0.5 v/v). GPC analyses were performed in 1,2,4-trichlorobenzene at 135 °C using PS standards for calibration. Differential scanning calorimetry (DSC) analyses were performed on a Setaram DSC 131 apparatus, under continuous flow of helium and using aluminum capsules. Crystallization temperatures were measured during the first cooling cycle (10 °C/min), and glass and melting transition temperatures were measured during the second heating cycle (10 °C/min).

Stuttgart-

  Dresden relativistic effective core potential (RECP) where the 4f electrons are included in core. The RECP was used in combination with its adapted basis set augmented by a set of f polarization function (α = 1.000). 18 A 6-31+G(d,p) double -ζ quality basis set was used for carbon and hydrogen atoms. The Si atoms were described with a Stuttgart-Dresden relativistic effective core potential in combination with its optimized basis set with the A c c e p t e d m a n u s c r i p t addition of a d polarization function (α = 0.284).

Figure 11 .

 11 Figure 11. Definition of ΔH # depending on the sign of ΔHcoord. 13

A c c e p t e d m a n u s c r i p t introduced to scavenge

  the reaction medium, uncontrolled, radical (thermally selfinitiated) polymerization can take place (see ref 11;

A c c e p t e d m a n u s c r i p t

  1

  •h 1 .

	2-Nd gave 1 4001 700
	kg(sPSE)•mol(Nd) 1 •h 1 , affording a highly syndiotactic copolymer ([r] 5 = 5461%) with a
	relatively narrow dispersity value (ÐM = 2.4), despite the elevated polymerization temperature

(entries 12 and 13). Similar results were observed using 2-Pr, even though it appeared to be somewhat less active and stereoselective than its Nd analogue. Better productivities in the range 1 8672 265 kg(sPSE)•mol(Sm) 1 •h 1 were observed with 2-Sm but the syndiotacticity

Table 1 .

 1 Styrene-ethylene copolymerizations catalyzed by 1

-Nd-K-allyl, 27-Nd and 2- Sc,La,Sm,Pr a

	Entry Complex	[St]0 [M]	[St]0/ [Ln]	[Mg] /[Ln]	Tpolym (Tmax) [° C]	Time [min]	Ethylene (bar or g)

Table

  Figure 3. Methylene region of the 13 C{ 1 H} NMR spectrum (125 MHz, 130 °C, C6H3Cl3/C6D6) of a sPSE copolymer (98 mol% styrene; Table 1, entry 9).

	S αα		T ββ
	(SSSS) S αα (SSSE)	(SSSS) (SSSE)
		T βδ
	T δδ	(SSSE)
	(SES)
			rrrrr
		rrmrr	rrrrm	rrrmr

1, entry 2), (bottom) 98 mol% styrene (entry 9).

S αγ (SES) S ββ (SES)

A c c e p t e d m a n u s c r i p t

  of the neodymium system was done by applying Clark et al. method.[START_REF] Clark | DFT study of tris(bis(trimethylsilyl)methyl)lanthanum and samarium[END_REF]

Table 3

 3 , entries 12-13). 13 Castro, L.; Kirillov, E.; Miserque, O.; Welle, A.; Haspeslagh, L.; Carpentier, J.-F.; Maron, L. Are solvent and dispersion effects crucial in olefin polymerization DFT calculations? Some insights from propylene coordination and insertion reactions with group 3 and 4 metallocenes. ACS Catal. 2015, 5, 416-425. 14 Schultz, N. E.; Zhao, Y.; Truhlar, D. G. Benchmarking approximate density functional theory for s/d excitation energies in 3d transition, metal cations. J. Comput. Chem. 2008, Ab initio energy-adjusted pseudopotentials for elements of groups 13-17. Mol. Phys. 1993, 80, 1431-1441.

	29, 185-189.

15 Zhao, Y.; Truhlar, D. G. Density functionals with broad applicability in chemistry. Acc. Chem. Res. 2008, 41, 157-167. 16 Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648-5652. 17 Burke, K.; Perdew, J. P.; Wang, Y. In Electronic Density Functional Theory: Recent Progress and New Directions; Dobson, J. F., Vignale, G., Das, M. P., Eds.; Plenum: New York, 1998. 18 Dolg, M.; Stoll, H.; Savin, A.; Preuss, H. Energy-adjusted pseudopotentials for the rare earth elements. Theor. Chem. Acc. Theory Comput. Model. Theor. Chim. Acta 1989, 75, 173-194. 19 Bergner, A.; Dolg, M.; Küchle, W.; Stoll, H.; Preuss, H. pseudo-potential basis sets of the main group elements Al-Bi and f-type polarization functions for Zn, Cd, Hg. Chem. Phys. Lett. 1993, 208, 237-240.

The presence of atactic PS is likely a result of thermally self-initiated polymerization (Mayo's mechanism). Some of us have previously investigated this process in presence (or absence) of dialkylmagnesium reagents (see: Bogaert, S.; Carpentier, J.-F.; Chenal, T.; Mortreux, A.; Ricart, G. Macromol. Chem. Phys., 2000, 201, 1813-1822) under conditions which are comparable to those reported in the current manuscript, in particular with styrene purified the same way (it is noteworthy that thoroughly purified styrene is much less prone to radical polymerization as there is no "residual" initiator) and reactions performed in bulk styrene. We observed that, at 105 °C, only 18-20% of atactic PS formed after 5 h; with 5 mmol of MgR2 for 175 mmol of styrene, the resulting aPS had a high molecular weight (typically Mn = 150 kg.mol 1 , PDI = 2.4). The reactions reported in the current manuscript were conducted at higher temperatures, but over shorter reactions times. We hence did not expect the formation of significant amounts of atactic PS. This is corroborated by GPC analyses which showed monomodal traces with Mn values in the typical range 12-45 kg/mol; no significant presence of high MW PS was observed (see the Supp. Info.). Yet, the possible presence of a few % of aPS in the essentially sPS(E) materials cannot be discarded. Note that during homopolymerization of styrene with the same neodymocene catalysts, it was noted that if MgR2 is not