S Sourisseau
email: samuel.sourisseau@oniris-nantes.fr

E Chantoiseau

C Toublanc

M Havet

Computing radiative heat transfers in greenhouses: a methodology coupling analytical and numerical approaches for view factors assessment

Keywords: Radiative heat transfer, View factor, Energy efficiency, PyVista, PyViewFactor

Soilless cultivation under heated greenhouses is a common practice for Northwestern Europe tomato producers. However, it comes with an important energy consumption that shall be more than ever addressed, especially because it still relies in a large proportion on fossil fuels. Modelling the greenhouse climate and its effects on the crop growth and yield can help to evaluate existing and innovative heating systems in relation with the associated energy consumption. Radiative heat transfers in particular shall be studied because of their magnitude compared with convection in greenhouses. However, computing view factors, which characterize how surfaces "see" each other, is complex for greenhouse where the configuration does not correspond to simplified analytical cases, with obstacles and surfaces arrangement varying with time as the crop grows.

In this work, a methodology is presented to evaluate view factors in a 1000 m² soilless tomato experimental greenhouse fitted with three different heating pipe networks and air ducts below the gutters. The approach uses Free and Open Source Software in conjunction with analytical solutions whenever possible to reduce computation time. The following elements are considered: floor, ducts, gutters, heating pipes, horizontal screens, roof and sidewalls. View factors that depend on the crop growth can thus be formulated to take into account the crop rows height and Leaf Area Index (LAI), from low ones (where radiative energy losses with the roof and the walls might be important to assess for year-round energy consumption studies) to mature ones. The methodology itself can not only be applied to any other greenhouse arrangement, but also more generally to any geometry where radiative transfers occur within a volume that includes obstacles.

INTRODUCTION

Heat can be transferred by conduction, convection or radiation. The first mode occurs in the matter and is driven by the internal temperature gradients: applied to process-based greenhouses climate and energy studies, conductive heat transfers are generally considered only in the soil, using for instance a one-dimension layer-by-layer discretized approach [START_REF] De Zwart | Analyzing energy-saving options in greenhouse cultivation using a simulation model[END_REF][START_REF] Pieters | Condensation and dynamic heat transfer in greenhouses part I: theoretical model[END_REF]. Convection phenomena take place between air (indoor or outdoor one) and any surface: the magnitude depend on their respective temperatures, the geometry and the flow characteristics. Correlations established in laboratories for small freeedge plates are sometimes used, but numerous formulas based on experimental studies in greenhouses are also available in the literature [START_REF] Roy | PA-Precision Agriculture: Convective and Ventilation Transfers in Greenhouses, Part 1: the Greenhouse considered as a Perfectly Stirred Tank[END_REF]. Regarding radiative heat transfers, they result of the fact that a body above the absolute zero temperature naturally emits radiation to its environment, without involving any transfer medium. The radiative transfer rate 𝑄 1→2 (W) between two isothermal, diffuse and homogeneous surfaces 𝑆 1 and 𝑆 2 can be computed in its simplest form:

𝑄 1→2 = 𝜀 1 × 𝜀 2 × 𝐹 12 × A 1 × 𝜎 × (𝑇 1 4 -𝑇 2 4 ) (1)
With 𝜀 1 and 𝜀 2 the emissivity of 𝑆 1 and 𝑆 2 (-), 𝐹 12 the view factor from 𝑆 1 to 𝑆 2 (-), 𝐴 1 the area of 𝑆 1 (m 2 ), 𝜎 the Stefan-Boltzmann constant, 𝑇 1 and 𝑇 2 the surface temperatures of 𝑆 1 and 𝑆 2 (K). The determination of 𝐴 1 is straightforward, while a surface emissivity can either be defined from the literature, equipment datasheet or measurements. On the contrary, 𝐹 12 depends on the surfaces geometries and orientations: it characterized the amount of radiation emitted by 𝑆 1 and directly received by 𝑆 2 . Theoretically, 𝐹 12 is computed by [START_REF] Cengel | Heat transfer A Practical Approach[END_REF]:

𝐹 12 = 1 𝐴 1 ∫ ∫ cos 𝜃 1 × cos 𝜃 2 𝜋 × 𝑟 2 𝑑𝐴 2 𝑑𝐴 1 𝐴 2 𝐴 1 (2)
With 𝑑𝐴 1 and 𝑑𝐴 2 the differential surfaces of 𝐴 1 and 𝐴 2 , 𝑟 the distance between these elements (m), 𝜃 1 (𝜃 2 ) the angle between the normal to 𝑑𝐴 1 (𝑑𝐴 2 ) and the direction 𝑑𝐴 1 𝑑𝐴 2 . 𝐹 21 can be deduced from 𝐹 12 considering the relationship 𝐹 12 × 𝐴 1 = 𝐹 21 × 𝐴 2 with 𝐴 2 the area of 𝑆 2 (𝑚 2 ).

A reasonably precise calculation of radiative heat transfers, including a satisfying determination of the related view factors, is required when the radiative part is non-negligible compared to the convective one, that is generally the case in greenhouses. As far as heated ones are concerned, experimental studies focussed on hot water pipes confirm that attention has to be paid for the assessment of radiative transfers in models. For instance, according to [START_REF] Teitel | Effects of Nonsteady Hot-water Greenhouse Heating on Heat Transfer and Microclimate[END_REF] the radiation heat transfer represents between 41% to 52% of the total heat input depending on the pipes heating phase, providing calculation assumptions for the radiative part.

In the literature dedicated to protected cultivation, alternatives to Equation ( 2) are used to define the view factors. For infinitely large greenhouses, all the elements are modelled as horizontal homogeneous layers which are characterized by an equivalent Far Infra-Red (FIR) transmittance 𝜏 𝐹𝐼𝑅 (-). The latter is either defined by material properties (roof), material properties and position (horizontal screens), a FIR extinction coefficient (canopy) or derived from other view factors [START_REF] De Zwart | Analyzing energy-saving options in greenhouse cultivation using a simulation model[END_REF]. [START_REF] Kempkes | Effect of Heating System Position on Vertical Distribution of Crop Temperature and Transpiration in Greenhouse Tomatoes[END_REF] used a more complex approach by sub-dividing the crop layer into up to 9 sub-layers and by computing the view factor of a pipe to a crop layer using geometric considerations (analytical formulas). Modelling a crop arrangement in rows, [START_REF] Teitel | Radiative Heat Transfer from Heating Tubes in a Greenhouse[END_REF] also used geometry to compute the view factors from a pipe to the other surfaces (canopy of the closest row and the opposite one, floor and roof between two rows), depending on the pipe location. Other authors applied the 1D crossed strings method, for instance to solar greenhouses [START_REF] Liu | A fast and practical one-dimensional transient model for greenhouse temperature and humidity[END_REF][START_REF] Zhang | High resolution 3D simulation of light climate and thermal performance of a solar greenhouse model under tomato canopy structure[END_REF]. However, the infinitely large case approach is not suitable for small greenhouses, and boundaries effects prohibit the use of 1D simplifications. Besides, elements inside the greenhouse (such as the growing gutters) may act as obstacles for the radiation between other surfaces. In such a situation, a proper computation of the view factors must be conducted.

Several numerical methods exist to compute view factors between two surfaces [START_REF] Cohen | Radiosity and realistic image synthesis[END_REF]. In particular, [START_REF] Mazumder | General procedure for calculation of diffuse view factors between arbitrary planar polygons[END_REF] published a general calculation procedure based on the contour method and surfaces vector representation, that can be applied to any planar polygonal surfaces. The contour integral approach consists in applying the Stokes theorem to Equation (2) and thus replacing the two surface-integrals by two contour-integrals. Recently, [START_REF] Bogdan | Calcul des facteurs de forme entre polygones -Application à la thermique urbaine et aux études de confort[END_REF] made use of this methodology and have implemented the free and open-source (FOSS) Python module called PyViewFactor, taking benefit of the PyVista framework [START_REF] Sullivan | PyVista: 3D plotting and mesh analysis through a streamlined interface for the Visualization Toolkit (VTK)[END_REF]. The latter is a FOSS 3-D data visualization and post-treatment tool: it exposes all the necessary features to create, handle and display geometries as well as it facilitates mesh-based calculations. While Bodgan et al. (2022) used PyViewFactor for thermal applications in buildings and urban environment, their implementation is generic and can be applied to the protected cultivation context. In addition to the view factor calculation between any planar polygonal surface elements (cells), the module also includes functions that can be used to check if the cells are oriented in such a way that they can actually see each other, as well as if the visibility is impaired by any obstacle using raytracing.

The present paper describes a procedure mixing analytical as well as numerical approaches using PyVista and PyViewFactor to estimate all the view factors needed to conduct year-round simulation of greenhouse energy and climate models. This methodology has been applied to an experimental heated Venlo glasshouse compartment where a soilless tomato crop was cultivated during a whole production season.

MATERIALS AND METHODS

Study case greenhouse

The case study is a 1037 m 2 (24 x 43.2 m) compartment of a Venlo type glasshouse located close to Nantes (France) at the Centre Technique Interprofessionnel des Fruits et Légumes (CTIFL), made of 6 spans (22° roof slope, 4 m width, 7.78 m high and 6.97 m high below the gutters). The tomato crop was laid-out in 15 rows cultivated on a rock wool medium on growing gutters, with an apex at 3.9 m from the ground for a mature crop (corresponding to a crop height from the growing gutters to the apex 𝐻 𝑐𝑎𝑛 of 2.8 m). Two alleyways are located at the gables (3 m large at the North, 1.2 m large at the South). Above the canopy, at the gutters height, two controlled horizontal screens (thermal and shading) were installed. The greenhouse was heated by three hot water pipe networks:  14 loops of "51" rails on the floor between rows (51 mm diameter)  15 loops of Forcas pipes in the canopy (one per row, 35 mm diameter)  15 loops of PE pipes (one pipe on each side of the growing gutters, 25 mm diameter) Four 0.8 m diameter flexible perforated air-mixing ducts were installed below the growing gutters. Figure 1 illustrates how the greenhouse has been modelled.

While the greenhouse cover and equipment can boil down to simple geometries (planes, parallelepipeds and cylinders), the canopy is more complex, especially in its lower area. Although a more realistic (but significantly complex and time-consuming) geometry could be defined, in the present work opaque parallelepipedic shapes have been considered above the Forcas pipes (number 3 on Figure 1). To account for plant growth with the time, both the height and the width of the parallelepipeds were modulated according to regular plant length measurements over the whole production season. Besides, the presence of the stems, fruits and leaves at the Forcas pipes level is taken into account afterwards: the view factors obtained numerically were post-processed considering that 50% of the view factor from the Forcas pipes to any element below them (floor, growing gutters, ….) is systematically intercepted by the lower part of the canopy. Thus, for instance, the actual view factor between the Forcas tubes and the floor 𝐹 𝐹𝑜𝑟𝑐𝑎𝑠→𝐹𝑙𝑟,𝑎𝑐𝑡 is deduced from the value obtained with the numerical approach 𝐹 𝐹𝑜𝑟𝑐𝑎𝑠→𝐹𝑙𝑟,𝑛𝑢𝑚 as 𝐹 𝐹𝑜𝑟𝑐𝑎𝑠→𝐹𝑙𝑟,𝑎𝑐𝑡 = 0.5 × 𝐹 𝐹𝑜𝑟𝑐𝑎𝑠→𝐹𝑙𝑟,𝑛𝑢𝑚 and the remaining part is added to 𝐹 𝐹𝑜𝑟𝑐𝑎𝑠→𝐶𝑎𝑛,𝑛𝑢𝑚 .

Numerical approach

1. General procedure.

The developed script workflow is shown in Figure 2 and its implementation is available in [START_REF] Sourisseau | Implementation of view factors computation using a discretized approach based on PyVista and PyViewFactor, applied to a greenhouse study case with numerous obstacles[END_REF]:

Figure 2. Script workflow Although it is possible to import meshes from other software with PyVista, this has not been considered for the present work. On the contrary, the geometries and the meshes are built from the code using PyVista methods, based on parameter values defined at the script global level (STEP 1): greenhouse dimensions, elements sizes and position in the greenhouse, mesh type (homogeneous, heterogeneous, rectangular or triangular) and step precision, etc.

In STEP 2, the elements geometries are built. Each element has two meshes, which are exactly superimposed on each other. The first, simple, mesh is used for obstacle detection: the less cells it has, the less obstruction checks are required during the calculation process. The second one is denser and is used for the discretization of the source (𝑆 1 in Equation ( 1)) and the receptor (𝑆 2 in Equation ( 1)) elements from/to which a view factor is computed. Both meshes as well as all the cells normal vectors are displayed to the user at STEP 3, so that they can be carefully verified element per element.

In STEP 4, the user is asked for the selection of view factors to compute: calculations which apply to the same geometries can consequently be started in batch.

The algorithm applied in STEP 5 can be summarized as follow: 1) Distribution of the discretized cell-by-cell view factor calculations, based on the finer mesh defined on STEP 2, on all the available CPU. Parallel computing is indeed possible considering the Equation (3):

𝐹 12 = ∑ 𝐹 1𝑖 𝑁 2 𝑖=1 = ∑ ∑ 𝐴 𝑗 × 𝐹 𝑗𝑖 𝑁 1 𝑗=1 𝐴 1 𝑁 2 𝑖=1 (3)
With 𝑁 1 and 𝑁 2 the number of cells of the element 1 and 2 (respectively), 𝐹 1,𝑖 the view factor from the element 1 to the 𝑖 𝑡ℎ cell of element 2, 𝐴 𝑗 the area of the 𝑗 𝑡ℎ cell of element 1 and 𝐹 𝑗𝑖 the view factor from the 𝑗 𝑡ℎ cell of element 1 to the 𝑖 𝑡ℎ cell of element 2. 2) For each cell of element 2, the elementary view factors with all the cells of the element 1 were then summed. However, an elementary view factor 𝐹 𝑗𝑖 is actually computed only if the following conditions, checked with PyViewFactor, are valid: a. The 𝑖 𝑡ℎ and 𝑗 𝑡ℎ cells are oriented in such a way that they can see each other. b. Among the possible obstacle elements cells (simpler mesh scanning), none is an obstacle to a ray drawn between the 𝑖 𝑡ℎ and 𝑗 𝑡ℎ cells. Once all the cells have been treated, results are displayed and saved (STEP 6).

Combined use of numerical and analytical approaches.

Although the numerical approach appears simple since it "only" requires to create geometries, some drawbacks exist: defining the suitable mesh properties is not a straightforward task similarly to CFD studies, and the necessary computing time may be important. Several ways can be used to mitigate the latter issue:  The analytical formula [START_REF] Howell | Radiative transfer configuration factor catalog: A listing of relations for common geometries[END_REF]) can be kept wherever suitable for simple view factors cases with reasonable assumptions. For instance, the view factor between the screens system and the roof is obvious (two rectangular parallel plates, taking into account the cosinus of the roof slope for the roof area).

 The finer meshes cell numbers can be reduced using 1D discretization (instead of 2D) and may be made heterogeneous.  Instead of using the whole greenhouse geometry, representative ones using fewer cells can be created for the computation of specific view factors.  Using Equation (3), some surfaces might be split into two zones: an area where an analytical approach can be applied (simple configurations without obstacles), and the remaining one. It can be illustrated considering the case of the view factor between the floor 𝐹𝑙𝑟 and the screens system 𝑆𝑐𝑟𝑆𝑦𝑠. Instead of computing it at once using the global geometry (which would require a 2D discretization dense mesh for both elements), the floor was divided into three surfaces: the cultivated one 𝐴 𝑐𝑢𝑙 , and the two alleyways close to the gables at the North and South sides 𝐴 𝑎𝑙𝑙,𝑁𝑜 and 𝐴 𝑎𝑙𝑙,𝑆𝑜 . The resulting view factor is formulated:

𝐹 𝐹𝑙𝑟→𝑆𝑐𝑟𝑆𝑦𝑠 = 𝐴 𝑐𝑢𝑙𝑡 × 𝐹
While the view factors implying the alleyways were computed using analytical formulas with reasonable assumptions, 𝐹 𝐹𝑙𝑟,𝑐𝑢𝑙→𝑆𝑐𝑟𝑆𝑦𝑠 was numerically assessed considering a 1D discretization (across the greenhouse width) for the floor and the screens.

Time varying geometries.

Within the context of year-round greenhouse climate and energy studies, geometries may be time-dependant. For the present case, according to the applied cultivation practices, the canopy rows-based view factors have been calculated for the following estimated mean row dimensions [height x width], in m: [0 x 0], [1 x 0.6], [1.6 x 0.83], [2.2 x 1.06], [3 x 1.36] and [4 x 1.36]. Then, regression expressions linking (measured) 𝐻 𝑐𝑎𝑛 to the corresponding numerically assessed view factors 𝐹 𝑖𝑗,𝑛𝑢𝑚 have be deduced (the last two sets of dimensions, while being over the maximum possible height (2.8 m), are used to derive suitable curves). Besides, some of the view factors that partially result from analytical formula also depend on the LAI, computed by the embedded crop yield model [START_REF] Vanthoor | A methodology for model-based greenhouse design: Part 2, description and validation of a tomato yield model[END_REF].

RESULTS

Typical results with the representative geometry

The Figure 3 illustrates how the computation of a view factor is rendered after it ended, using the case between the floor cultivated area and the screens systems as an example. A representative geometry is used and since the alleyways are treated separately, a 1D discretization is sufficient.

Figure 3. Displayed result for 𝐹 𝐹𝑙𝑟,𝑐𝑢𝑙𝑡→𝑆𝑐𝑟𝑆𝑦𝑠 , using a parametrized representative geometry.

Mesh step and computing time

For a given view factor, the required computation time highly depends on the mesh characteristics as well as on the quantity of cells of the possible obstacles. Considering the same example as Figure 3 (𝐹 𝐹𝑙𝑟,𝑐𝑢𝑙𝑡→𝑆𝑐𝑟𝑆𝑦𝑠 ), Figure 4 shows the effect of the discretization step across the greenhouse width for both the floor and the screens in the result accuracy. Using the value obtained with the smallest step as a reference (20 mm), the absolute difference (-) globally decreases from 0.2 m to 90 mm, but does not vary significantly from 90 mm to 20 mm. Applied to greenhouses, where the emissivity of the surfaces, precise equipment location (such as for the flexible PE heating pipes along the growing gutters) and canopy geometry (to name but a few) are subject to uncertainty, it does not really make sense to claim an error lower than 0.01 (-) for 𝐹 𝐹𝑙𝑟,𝑐𝑢𝑙𝑡→𝑆𝑐𝑟𝑆𝑦𝑠 . However, since the scattering appears to be unpredictable for steps larger than 0.1 m, this value is a good compromise, in the present case, between the accuracy and calculation time.

Computations based on the same geometry but without obstacle between the floor and the screens showed that the points scattering on Figure 4 is due to the obstacle detection, not to the source and receptor surfaces discretization. Indeed, the no-obstacle absolute error is lower than 5.5.10 -7 (-) for all the step cases (200  20 mm) compared with the analytical solution of two limited rectangular parallel plates.

The indicated computation times are representative of what can be expected from the implementation in [START_REF] Sourisseau | Implementation of view factors computation using a discretized approach based on PyVista and PyViewFactor, applied to a greenhouse study case with numerous obstacles[END_REF] with a standard laptop (for the present work, a 4cores i7-1165G7 CPU running Python 3.8.10, PyVista 0.41.1, PyViewFactor 0.0.16 and associated dependencies versions). However, possible implementation improvements have been identified but not yet investigated for the present study.

Application to a greenhouse climate and energy model

The methodology, mixing numerical approach and analytical one, has been applied to assess the view factors required to compute the thermal radiation heat transfers in a climate and energy model of the greenhouse compartment. As an example, Figure 5 (a) shows the cumulated view factors implying the screens system from the beginning of the production season (estimated canopy height 𝐻 𝑐𝑎𝑛 of 0.3 m and LAI of 0.5 m 2 .m -2 ) to the mature crop stage (2.8 m height and a LAI of 4.85 m 2 .m -2 , simulated by the model). Obviously, it is expected that the view factors sum equals 1: however, the use of an analytical approach for some of the view factors as well as the scaling of the numerical results from the representative geometry to the global greenhouse one necessarily induced deviations. In the present case, the cumulated error 𝛿 remains between [+0.07:-0.02] (-) while 11 view factors are involved, and 𝛿 is mainly limited to the first weeks of the production season (the crop reached 1.3 m high 5 weeks after the planting). Starting the 11 th week, the plant reached its mature high (2.8 m) for which the cumulated error appears negligible. Once all the necessary view factors are computed (should they be constants such as for 𝐹 𝐹𝑙𝑜𝑜𝑟→51 𝑟𝑎𝑖𝑙𝑠 , or functions of the crop height and LAI), the last step of the methodology consists in the normalization. For a given element, 𝛿 can be proportionally split between the view factors that have the highest expected inaccuracy due to the hypotheses that have been considered: the other view factors should not be modified. Applied to a source element 𝑋, a normalized view factor from 𝑋 to the element 𝑌 is computed:

𝐹 𝑋→𝑌,𝑛𝑜𝑟𝑚 = 𝐹 𝑋→𝑌 + 𝛿 × 𝐹 𝑋→𝑌 ∑ 𝐹 𝑋→𝑘 (5)
With 𝑘 being an item in the list of all the elements on which the normalization is done (in the present case the floor, the growing gutters, the canopy and the gables (Figure 5 (b)). In dynamic models computing some of the variables on which view factors formulas depend (LAI, crop height), the normalization shall necessarily be embedded in the model itself.

CONCLUSION

Whereas radiative heat transfers in greenhouses have to be taken into account in dynamic climate and energy models, the view factors assessment may be complex due to the presence of obstacles and because of the dependency on changing crop properties during the production season. A methodology mixing analytical and numerical approaches has been presented and illustrated using a 1000 m 2 heated compartment study case. While a full numerical approach would be the most accurate providing suitable geometries and meshes are used, the numerous view factors to compute (at different crop heights) may lead to an important calculation time on standard computer at least with proposed implementation. However, it has been shown that applying analytical formulas from the literature in conjunction with the discretization and ray-tracing based approach can provide satisfactory view factor expressions, which may in turn be used in dynamic models. Future work would consist in implementing improvements, since moving towards more numerical resolution should alleviate the need for a normalization, The methodology will also be used to evaluate prospective technologies based on a system approach for a new fossil energy-free greenhouse concept.

Figure 1 .

 1 Figure 1. Study case greenhouse compartment. The global geometry (A) and its detailed view (B) correspond to 2.2 m high x 1.06 m width crop rows. C) is a picture taken while the crop was at a mature stage (2.8 m high x 1.36 m width rows).

Figure 4 .

 4 Figure 4. Absolute differences between 𝐹 𝐹𝑙𝑟,𝑐𝑢𝑙𝑡→𝑆𝑐𝑟𝑆𝑦𝑠 computed with a discretization step across the greenhouse width of 20 mm for the floor and the screens, for a range of step values between 0.2 m to 20 mm.

Figure 5 .

 5 Figure 5. Cumulated view factors between the screens system and all the other elements in the greenhouse, before (a) and after (b) normalization

  𝐹𝑙𝑟,𝑐𝑢𝑙𝑡→𝑆𝑐𝑟𝑆𝑦𝑠 + 𝐴 𝐴𝑙𝑙,𝑁𝑜 × 𝐹 𝐴𝑙𝑙,𝑁𝑜→𝑆𝑐𝑟𝑆𝑦𝑠 + 𝐴 𝐴𝑙𝑙,𝑆𝑜 × 𝐹 𝐴𝑙𝑙,𝑆𝑜→𝑆𝑐𝑟𝑆𝑦𝑠 𝐴 𝑐𝑢𝑙𝑡 + 𝐴 𝐴𝑙𝑙,𝑁𝑜 + 𝐴 𝐴𝑙𝑙,𝑆𝑜

ACKNOWLEDGMENTS

This study is part of SERRES+ project, funded by the regional councils of Brittany and Pays-de-Loire and supported by the Vegepolys Valley precompetitive cell. The authors would like to thank M. Bogdan for having provided PyViewFactor and for the interesting discussions and advices about view factors and obstacle detection.


Bouchaib Mouhsine

Abdallah Karim

Clement Dumont

Isabelle Suisse

Mathieu Sauthier
email: mathieu.sauthier@univ-lille.fr.

Clément Dumont

Efficient and Clean Nickel Catalyzed α-Allylation Reaction of Nitriles

Keywords: Allylation, Nickel, Green chemistry, Atom economy allylation, nickel, nitrile, allylic alcohol

come

Introduction

Allylic alkylation reactions remain one of the most powerful tools for the formation of homo (C-C ) and hetero (C-O, C-N, C-S) bonds. [1] The introduction of the allylic moiety is particularly attractive as it allows subsequent chemical transformations based on the C=C bond reactivity. [2] Commonly used allylation reagents bear good leaving groups like acetate, carbonate or halide. Such reagents are however limited to the low atom economy related with their use that often results in the production of large amounts of side products such as salts. In comparison to other reagents, allylic alcohols could be advantageously used. At a first stand, it is noteworthy that most of the commonly used allylation reagents are synthesized from allylic alcohols. The direct use of allylic alcohols in allylation reactions is more straightforward and consistent with a step economy. It is also noteworthy that their use generates only water as byproduct. In such case, this is related to the fact that the hydroxyl group acts as a base toward the nucleophile. On the point of view the accessibility, allyl alcohol is industrially produced from propene oxydation [3] and potentially accessible form vegetal feedstock like glycerol. [4] The main limitation with the use of allylic alcohols is related to the low reactivity of this reagent as the hydroxyl group is recognised as a weak leaving group. To activate allylic alcohols, catalytic amounts of transition metals are generally needed. [START_REF] Tamaru | For reviews of palladium-catalyzed allylation of nucleophiles by directly using allylic alcohols, see a) Y[END_REF] We reported very simple protocols of Nicatalyzed allylic alkylation with allylic alcohol for the transformation of nucleophiles such as amides, [6] ketoesters and diketones, [7] malonates, [8] aldehydes [9] and ketones. [10] Beyond the use of more precious metals such as palladium, [11] iridium, [12] ruthenium, [13] platinum [14] and rhodium, [15] the use of nickel has opened up interesting prospects. This metal is abundant and inexpensive and it is noteworthy that nickel based catalysts prevent the addition of bases or other additives. [16] The highly efficient -allylation of carbonyl compounds obtained with nickel-based catalysts [7][8][9][10] opens the question of the reactivity of nitriles (Scheme 1). Nitriles are industrially used as precursors to many other functional groups such as carboxylic acids, aldehydes, ketones, and amines. [START_REF] Larock | Comprehensive Organic Transformations[END_REF] They are also available in many natural products and medicinal compounds. [18] In order to perform the alpha-allylation of this family of compounds, palladium-based catalysts have been commonly used. As examples, the direct coupling of allylic alcohols with weakly acidic pronucleophiles has been efficiently performed using catalytic Pd(PPh 3 ) 4 via activation of C-OH bonds with CO 2 . [19] Allylic alcohols were also used as allylic reagents during the palladium catalyzed deacylative allylation of ketone enolates, αarylcyanoacetones and α -arylcyanoacetic esters. [20] Other metals were likewise used as rhodium in the enantioselective metal-catalyzed allylic alkylation of 2 nitriles with allyl benzoate [21] or iridium during allylation of cyanoacetates with allyl carbonate. [22] Herein, we wish to report the first nickel catalyzed allylation of nitriles with allyl alcohol performed under neutral conditions (Scheme 1).

R N R N + OH H 2 O neutral conditions [Ni]
Scheme 1. The nickel-catalyzed -allylation of nitriles with allyl alcohol.

Results and discussion

The optimization of the reaction parameters was carried out with phenylacetonitrile 1a and allyl alcohol 2a (Table 1). The reaction led to the mono 3a and the bisallylated 4a compounds (Scheme 2 gave similar results with a mixture of mono (36 %) and bis-allylated (35 %) products. It is noteworthy that 3a was partially hydrolyzed in the corresponding amide (11 %) that has been isolated and fully characterized by 1 H and 13 C NMR whereas no product of hydrolysis of 4a was never observed (Entry 3). Significantly better yields were obtained with the ferrocenyl ligand L4 (dppf) that led to 25 % of 3a and 60% yield of 4a (Entry 4). In order to improve the conversion of the nitrile, the reaction temperature was increased to 100°C and 120°C (Entries 5 and 6). However, a higher temperature did not lead to better results but favored the hydrolysis of 3a. Moreover the increase of the temperature up to 120°C resulted to a lower catalytic efficiency probably due to a catalyst decomposition at that temperature (Entry 6). As observed in previous studies that involved ketones and aldehydes as nucleophile,14, 15 the use of aprotic solvents such as toluene (Entry 7) or solvent-free conditions (Entry 8) led to reduced yields along with a larger part of hydrolysis. A remarkable yield of 80 % in 4a was expectedly observed with sodium hydroxide as additive (Entry 9). The addition of a base probably favorably increased the concentration of the deprotonated nucleophile. On the other hand, it is thus not surprising that poor results were obtained in the presence of a strong acid such as PTSA (Entry 10). Finally, we varied the amount of the allyl alcohol 2a. Reducing the number of allyl alcohol equivalents to 1 did not promote the monosubstitution in respect to the disubstitution but only contributed to reduce the overall conversion of 1a (Entry 11). By increasing the number of equivalents up to 3, a comparable conversion of 1a was observed (90 %) with yields in 3a and 4a of 20 % and 70 % respectively (Entry 12). Nevertheless, the yield in 4a remained limited to 70 % with 2 equivalents of allylic alcohol (entry 14). In order to achieve a full conversion of the starting material and a very high selectivity in 4a, a combination of higher catalyst loading up to 1.5 mol % and 3 equivalents of allyl alcohol proved to be necessary (Entry 15). [c] L4 80 MeOH 1 36 40 14 [c] L4 80 MeOH 2 30 70 15 [c] L4 Isolated yield of the major product (%) [b] GC Yield (%) 57(3k) [a] Reaction conditions: 1a-k (1.8 mmol), 2a (5.4 mmol), Ni(cod) 2 (0.027 mmol), dppf (0.054 mmol), MeOH (0.5 mL), 17 h, T = 80°C in a sealed Schlenk tube. [b] Isolated yield after silica gel chromatography.

+ OH MeOH, 8O°C, 17h C N R R 1 CN R 1 1a 2a R = H 3a-k R = H 4a-k 1a
Using the optimized conditions, we then examined the possibility to perform this reaction with alphamonosubstituted acetonitriles bearing aromatic rings (Table 2). After isolation of the product, phenylacetonitrile 1a led to 87 % of 3a yield (Entry 1, Table 2). The presence of a methoxy group on the ortho position oriented the synthesis toward the sole formation of the monoallylated derivative 3b with a yield of 51 % (Entry 2). Good yields into the bisallylated product were observed with 3,4dimethoxyphenylacetonitrile 1c and 3,4,5trimethoxyphenylacetonitrile 1d, 70 and 62 % respectively (Entries 3 and 4). These results showed that the presence of methoxy groups on the meta and para positions of the aromatic ring did not bring significant steric hindrance. 2-cyanophenylacetonitrile 1e and 2fluorophenyl acetonitrile 1f were efficiently diallylated and the compounds 3e and 3f were thus isolated with 70 % and 60 % yields respectively (Entries 5-6).

Table 3: Reaction scope of the nickel catalyzed direct allylation of the , disubstituted acetonitriles with allyl alcohol. [a]

[a] Reaction conditions: 5a-f (1.8 mmol), 2a (5.4 mmol), Ni(cod) 2 (0.027mmol), dppf (0.054mmol), MeOH (0.5 mL), 17 h, T = 80°C in a sealed Schlenk tube. [b] Isolated yield after silica gel chromatography.

Very similar results were obtained with the fluorinated phenylacetonitrile 1g (Entry 7). The 3methylphenylacetonitrile 1h and 4-biphenyl acetonitrile 1i were converted in derivatives 3h and 3i with yields of 60 % and 65 % respectively (Entries 8-9). A very good yield was observed with naphthylacetonitrile 1j of 89 % (Entry 10) while 7-methoxy-2- The higher steric hindrance due to the presence of the methoxy group likely impeded further substitution on the carbon atom.

+ Ni(cod) 2 -1.5 mol% dppf -3 mol% MeOH, 80°C, 17 h 2a N R 1 N R 1 R 2 5a-f OH 6a-f
Using the same conditions, the scope of the nitriles was extended to -disubstituted acetonitriles. In that case, only monoallylation could be performed. It is noteworthy that in all cases reported in Table 3, complete conversions of the starting materials were observed. High isolated yields were obtained with diphenylacetonitrile 5a and methyl-2naphthylacetonitrile 5b of 94 % and 85 % respectively (Entries 1 and 2). A slight decrease of yield with substrates bearing an electrodonor group was observed after isolation: -methylphenylacetonitrile 5c, ethylphenyl acetonitrile 5d and 1,2,3,4tetrahydronaphthalene-1-carbonitrile 5e were converted with yields of 73 %, 65 % and 61 % respectively (Entries 3, 4 and 5). The 4-benzoyl-αmethylbenzeneacetonitrile 5f was isolated with an excellent yield of 91 % (entry 6). 62 [a] Reaction conditions: 5a (1.8 mmol), 7a-d (5.4 mmol), Ni(cod) 2 (0.027mmol), dppf (0.054mmol), MeOH (0.5 mL), 17 h, T = 80°C in a sealed Schlenk tube. [b] Isolated yield after silica gel chromatography. [c] 2 mol% of Ni(dppf) 2 .

We further investigated different allylic alcohols with diphenylacetonitrile 5a as substrate (Table 4). High yields were obtained with the allylic linear alcohols such as crotyl alcohol 7a and octa-2,7-dien-1-ol 7b, 85 % and 80 % respectively (Entries 1 and 2). A very similar result was obtained with 2-methyl-2-propen-1-ol 7c as a branched alcohol (Entry 3). Cinnamyl alcohol (entry 4) could also be converted in the allylated derivative 8d with 62% isolated yield. The linear allylic structure was selectively obtained and the branched isomer was not observed. However, the reaction with a more substituted alcohol such as 3-methyl-2buten-1-ol 7d did not give the allylated nitrile. The fact that this alcohol is more substituted disfavors the oxidative addition to a low valent Ni(0) intermediate.

As we previously reported for the allylation of ketones, 15 diallylether 9a could also been involved in that reaction. As example with 2,2-diphenylacetonitrile 5a, a good isolated yield of 80 % was then obtained by using 1 equivalent of diallylether 9a (Scheme 3). The diallylated nitriles can be further used in other transformations that involve either the allylic or the nitrile moieties. The reduction of the nitrile in the corresponding primary amine was accomplished with lithium aluminum hydride in ether leading to the diallylamine 11a. The ruthenium catalyzed ring closing metathesis was also performed in high yield in refluxing toluene to form the cyclopentene ring along with the release of ethylene (Scheme 4). In conclusion, we have developed an efficient and very clean nickel-catalyzed process for the direct allylic alkylation of -substituted nitriles with allyl alcohol. The reaction was carried out under neutral conditions giving only water as side product. A large variety of allylic nitriles were synthesized and different allylic alcohols were also investigated. Similar results were observed with diallylether in place of allyl alcohol. The thus obtained allylated derivatives can be further chemically modified through nitrile reduction or olefin metathesis reactions to generate a primary amine or a cyclopentene ring respectively. We believed that this methodology is relevant to straightforwardly produce new allyl compounds and derivatives.

Experimental Section

-Allylation of phenylacetonitrile 1a with allyl alcohol 2a: In a Schlenk tube were placed first the catalytic precursor Ni(cod) 2 (1.5 mol %), then the ligand (3 mol %), the nitrile (1.8 mmol, 1 equiv.). Under nitrogen, freshly distilled and degassed allyl alcohol (3 eq.) or diallyl ether (1 eq.) were then added. Degassed MeOH (0.5 mL) was added under nitrogen and the reaction mixture was stirred at 80°C for 18 h. The reaction mixture was then concentrated under reduced pressure and the product was purified by silica gel column chromatography with petroleum ether/ethyl acetate (90/10) as eluent.

  only the monoallylated product 3k with an isolated yield of 57 % (Entry 11).

Scheme 3 .

 3 Scheme 3. The nickel-catalyzed -allylation of ,diphenylacetonitrile with diallyether.

  Scheme 4: Transformations of diallylnitrile 4a.

  catalysts allow the alpha-allylation of nitriles under neutral conditions (no added base or coreagent)

Table 1 . Optimization of the nickel catalyzed allylation of phenylacetonitrile with allyl alcohol. [a]

 1

	PPh 2 PPh 2	Fe

Table 4 : Scope of the Ni-catalyzed allylation of diphenyl acetonitrile 5a with various allylic alcohols.

 4

Acknowledgements

We acknowledge the Ministère de l'Enseignement supérieur, de la Recherche et de l'Innovation, the Institut Universitaire de France and the CNRS for their financial support. We also would like to thank Céline Delabre for her technical support (GC analysis of the crudes).


Beate Hoffmann

Emilie Aubry

Anne Marmagne

Sylvie Dinant

Fabien Chardon

Rozenn Le Hir

Linking sugar transport and nitrogen remobilization and use efficiency in Arabidopsis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction Methodology

Beate Hoffmann 1, , Emilie Aubry 1 , Anne Marmange 1 , Sylvie Dinant 1 , Fabien Chardon 1 and Rozenn Le Hir 1

SWEET transporters are sugar facilitators

Reverse genetic approach with double and quadruple sweet mutant lines

Multiscale approach from plant to cell level

Main results

The SWEET proteins were discovered as sugar facilitators able to transport glucose, fructose and/or sucrose between cells (Chen et al.

1

  Université Paris-Saclay, INRAE, AgroParisTech, Institute Jean-Pierre Bourgin (IJPB), 78000 Versailles, France Sugars availability in the cell is controlled by transporters activity at the plasma membrane and the tonoplast SWEET11/SWEET12 and SWEET16 transport hexoses and sucrose at the plasma membrane or at the tonoplast and SWEET17 is a tonoplastic fructose-specific transporter (Chen et al., 2010; Klemens et al., 2013; Chardon et al., 2013; Le Hir et al., 2015) cytosol vacuole cytosol vacuole

  , 2010). They are involved in plant development and plant response to abiotic and biotic stresses (for review see Xue et al., 2022). (Ji et al., 2022) SWEET genes modulate plant biomass, yield and NUE Nitrogen remobilization efficiency is partly dependent from SWEET genes intermediates (e.g., 2oxoglutarate) are required for ammonium assimilation -Nitrogen is needed for Rubisco synthesis -Amino acid metabolism is essential to generate C skeletons, energy and reductants But what about the connexion between sugar transport and N allocation and vice versa? Carbon skeletons (Hoffmann et al., 2022)Mutation in SWEET genes affects vascular bundle developmentConclusionsIntracellular sugar exchanges mediated by SWEET16 and SWEET17 specifically impact the vascular system development but not the N distribution swt11swt12 is impacted in vascular development and displays an improved NUE and nitrogen remobilization to the seeds.Xylem defects in the stem constitute a bottleneck for the amino acids transfer to the seeds (increased amino acids content in stem)Finally, less C reached the seeds most probably because C is not loaded or remobilized properly in the phloem sap from the stem and accumulates as soluble sugars and starch.


Piotr Paluch

Ēriks Kupče

Julien Trébosc

Olivier Lafon

Jean-Paul Amoureux
email: jean-paul.amoureux@univ-lille.fr

Hadamard acquisition of 13 C-13 C 2D correlation NMR spectra

Keywords: solid-state NMR, selective excitation, Hadamard encoding, 2D spectra, 13 C-13 C HOMCOR

We show that a multi-selective excitation with Hadamard encoding is a powerful tool for 2D acquisition of 13 C-13 C homo-nuclear correlations. This method is not designed to improve the sensitivity, but rather to reduce the experiment time, provided there is sufficient sensitivity. Therefore, it allows fast acquisition of such 2D spectra in labeled molecules. The technique has been demonstrated using a U-13 C-15 N histidine hydrochloride monohydrate sample allowing each point of the build-up curves of the 13 C-13 C cross-peaks to be recorded within 4 min 35 s, which is very difficult with conventional methods. Using the U-13 C-15 N f-MLF sample we have demonstrated that the method can be applied to molecules with fourteen 13 C resonances with a minimum frequency separation of 240 Hz.

I.

Introduction

Most modern NMR (Nuclear Magnetic Resonance) multi-dimensional experiments are based on the recording of an ensemble of free induction decay (FID) signals that are excited by a sequence of hard non-selective radio-frequency (rf) pulses. The frequency spectra are then obtained by applying a Fourier Transform (FT) along all dimensions. However, this type of signal acquisition requires accumulating numerous FIDs, especially in the case of well crystallized solid compounds producing narrow resonances that are spread over large frequency ranges. For example, this is the case when one of the indirect dimensions encodes the 13 C isotope. Indeed, on one hand the step in this indirect dimension, ∆t1, which is usually constant, must be small enough to observe the large frequency range, ∆0,max (∆t1 < 1/∆0,max), and on the other hand the maximum evolution time along the indirect dimension must be long enough to achieve a sufficient resolution to distinguish between close resonances separated by ∆0,min (t1,max  1/∆0,min). Globally, this type of acquisition requires numerous steps in the indirect dimensions, and hence very long accumulation times. In the case of big molecules, such as proteins, the main way to decrease the acquisition time is to use non-linear sampling of the indirect dimensions, which allows gaining approximately a factor of two in time for each indirect dimension. [START_REF] Sibisi | Maximum entropy signal processing in practical NMR spectroscopy[END_REF][START_REF] Kazimierczuk | Non-uniform sampling: post-Fourier era of NMR data collection and processing: Non-uniform sampling[END_REF][START_REF] Barna | Conventional and exponential sampling for 2D NMR experiments with application to a 2D NMR spectrum of a protein[END_REF] However, for small molecules with a limited number of resonances (i.e. sparse spectra), another type of multi-dimensional acquisition can be used. In this case, the broadband excitation of all species in the indirect dimensions is replaced with multiple-selective excitations combined with some sort of frequency encoding, e.g. Hadamard or phase encoding [START_REF] Kupče | Fast multi-dimensional Hadamard spectroscopy[END_REF][START_REF] Kupče | Multisite correlation spectroscopy with soft pulses. A new phase-encoding scheme[END_REF]. Obviously, this approach requires prior knowledge of the resonance frequencies along the indirect dimensions After this selective frequency excitation and acquisition of the individual spectra, the full multi-dimensional spectrum may then be re-constructed by using this ensemble of spectra and the prior knowledge of the resonance frequencies and line-shapes along the indirect dimensions. [START_REF] Kupče | Hadamard NMR spectroscopy[END_REF]. Up to now this selective methodology has been mainly applied to 2D experiments in liquids. [START_REF] Kupče | Fast multi-dimensional Hadamard spectroscopy[END_REF][START_REF] Kupče | Multisite correlation spectroscopy with soft pulses. A new phase-encoding scheme[END_REF][START_REF] Kupče | Hadamard NMR spectroscopy[END_REF][START_REF] Kupče | Two-dimensional Hadamard spectroscopy[END_REF]. It has been demonstrated that in the case of sensitive experiments performed on small molecules presenting a limited number of resonances, m, this selective excitation technique allows recording fast 2D spectra because of the much smaller number of 1D FIDs to be acquired: m instead of 2t1,max/∆t1 = 2∆0,max/∆0,min with conventional 2D methods. The factor of 2 comes from the sign-discrimination required with the FT methods. This problem does not occur with selective excitation experiments because of the prior knowledge of the resonance frequencies. An additional gain in S/N ratio of up to m is obtained by replacing the m sequential individual single-selective pulses (SSP) with n multi-selective pulses (MSP, n ≥ m). Most of the time, the two types of selective pulse sequences (SSP and MSP) are related through an encoding Hadamard square matrix of size n (called Hn hereafter), consisting of 0 and 1 (or 1) coefficients. The FIDs corresponding to all SSP can afterwards be retrieved from those recorded with MSP with the same Hn matrix. The optimum size n of the Hn matrix is the multiple of 4 which is just larger than the number m of resonances. It must be noted that with respect to the conventional broad-band excitation, the gain in S/N per unit of time afforded by the use of SSP is limited to 2, due to sign-discrimination. Therefore, the gain in time provided by the use of much less t1 steps is mainly useful in the case of sensitive experiments. In this 'sampling limited' case, it allows obtaining new information with an acceptable S/N that is hardly accessible with the conventional methods due to too long acquisition times, e.g. those related to build-up curves. In the case of 'sensitivity limited' experiments, numerous accumulations are always required, independently of the number of t1 steps used, and this type of information is hardly accessible anyway. The 2D acquisition with Hadamard encoding has been previously used to record at moderate magic-angle spinning (MAS) speeds (R = 15-19 kHz) either single-quantum to singlequantum (SQ-SQ) DARR 13 C HOMCOR (Homo-nuclear Correlation) or 1 H 13 C CPMAS HETCOR (Hetero-nuclear Correlation) spectra of U-13 C amino-acids. [START_REF] Ashida | Hadamard NMR spectroscopy in solids[END_REF] It has also been used more recently to record 13 C- 15 N HETCOR spectra of U-13 C-15 N f-MLF at R = 13 kHz [START_REF] Kupče | Recording 13C-15N HMQC 2D sparse spectra in solids in 30 s[END_REF] or to measure 13 C-13 C scalar couplings. [START_REF] Kakita | Solid-state Hadamard NMR spectroscopy: Simultaneous measurements of multiple selective homonuclear scalar couplings[END_REF] In this communication, we show how this Hadamard encoded acquisition can also be used to record 13 C-13 C 2D HOMCOR spectra at the much higher spinning speed of R = 62.5 kHz.

II. 13 C-13 C HOMCOR sequences

We have chosen implementing the Hadamard encoding/decoding with multi-selective pulses (MSP) and three different 13 C HOMCOR sequences with SQ-SQ observation, usable at ultrafast MAS: RFDR,[11-13] CP 3 ,[14-19] and DQF-BR2 2 1 (double-quantum-filtered BR2 2 1 ). [START_REF] Hu | Double-quantum NMR spectroscopy of 31P species submitted to very large CSAs[END_REF][START_REF] Edén | Improved double-quantum NMR correlation spectroscopy of dipolarcoupled quadrupolar spins[END_REF][START_REF] Schnell | Dipolar recoupling in fast-MAS solid-state NMR spectroscopy[END_REF] The sequences are shown in Fig. 1 and for each of them the part corresponding to the 1 H channel is only shown once at the top as it is the same for the conventional and the Hadamard versions. It must be noted that the CP 3 sequence (Figs. 1e,f) can only be used with protonated samples, whereas the two other schemes can always be used, with or without the initial CPMAS signal enhancement and the 1 H decoupling. In theory, for each 13 C resonance one may optimize the pulse selectivity according to its closest 13 C resonance. However, this would lead to different selective pulse lengths, and hence to different signal attenuations during the selective pulses for the various resonances. Moreover, even in this setup, the total duration of the encoding waveform would still be defined by the longest encoding element. Therefore, we see such an approach as hardly practical and we have used a common selective pulse length, sp, corresponding to the most selective component. For each single-selective pulse (SSP), we have used a Gaussian lineshape, truncated at 10 % to limit the losses occurring during these long excitations. The multiselective pulses (MSP) have been generated by the Bruker WaveMaker software, which also allows calculating the multi-excitation profiles of the rf-pulse-shapes.

II.1. RFDR

In Fig. 1a-d, we show the RFDR (Radio-Frequency-Driven Recoupling) [START_REF] Shen | Broadband finite-pulse radiofrequency-driven recoupling (fp-RFDR) with (XY8)41 super-cycling for homo-nuclear correlations in very high magnetic fields at fast and ultra-fast MAS frequencies[END_REF][START_REF] Zhang | Phase cycling schemes for finite-pulse-RFDR MAS solid state NMR experiments[END_REF][START_REF] Paluch | Analysis of local molecular motions of aromatic sidechains in proteins by 2D and 3D fast MAS NMR spectroscopy and quantum mechanical calculations[END_REF] pulse sequences used in this article; either with conventional acquisition (Fig. 1a) or with Hadamard encoding (Fig. 1b-d). In Fig. 1a, after the 1 H 13 C CPMAS initial transfer, the 13 C magnetization is transverse and can evolve during t1. Then, this magnetization must be stored along z because RFDR is a zero-quantum recoupling. This scheme consists of rotorsynchronized -pulses, and at ultra-fast MAS the (XY8)4 1 super-cycling has been shown to be optimized for 13 C-13 C interactions. [START_REF] Shen | Broadband finite-pulse radiofrequency-driven recoupling (fp-RFDR) with (XY8)41 super-cycling for homo-nuclear correlations in very high magnetic fields at fast and ultra-fast MAS frequencies[END_REF] There are several ways of introducing a selective excitation with RFDR, which implies replacing the t1 delay and the following /2 storing hard-pulse (HP) on the 13 C channel. The simplest one employs a /2-MSP (Fig. 1b), and the second involves a /2-HP positioned either before (Fig. 1c) or after (Fig. 1d) the -MSP. In Fig. 2 we show the six rows of histidine hydrochloride monohydrate, recorded with the three versions of 2D H8-RFDR sequences. One observes that the two most efficient methods use a -MSP for either inversion or refocusing purposes (Fig. 1c andd), respectively. Sequences using a /2-MSP are less efficient than those using a -MSP because their selective pulse length is twice longer for the same selectivity, here sp = 3.6 and 1.8 ms respectively for 500 Hz bandwidth, which leads to larger losses. By comparing in detail the spectra shown in Fig. 2, one notices that the selective inversion is on average slightly more efficient than the refocusing. Therefore, in the following we have chosen to use only selective inversion -MSP, as shown in Fig. 1c,f,h.

II.2. CP 3

This HOMCOR sequence uses three CPMAS 1 H- 13 C transfers (Fig. 1e), and hence is called CP 3 or CHHC when describing the transfer of magnetization. [START_REF] Mulder | Spin diffusion with 13 C selection and detection for the characterization of morphology in labeled polymer blends with MAS NMR[END_REF][START_REF] Lange | Structural Constraints from Proton-Mediated Rare-Spin Correlation Spectroscopy in Rotating Solids[END_REF][START_REF] De Boer | 2D 13C-13C MAS NMR correlation spectroscopy with mixing by true 1H spin diffusion reveals long-range intermolecular distance restraints in ultra-high magnetic field[END_REF][START_REF] Aluas | CHHC and 1H-1H magnetization exchange: Analysis by experimental solid-state NMR and 11spin density-matrix simulations[END_REF][START_REF] Kobayashi | Measuring long-range 13 C-13 C correlations on a surface under natural abundance using Dynamic Nuclear Polarization-Enhanced Solid-State Nuclear Magnetic Resonance[END_REF][START_REF] Wilhelm | 2D CP/MAS 13C isotropic chemical shift correlation established by 1H spin diffusion[END_REF] After the initial CP1 transfer, the total 13 C magnetization evolves in the xy plane during t1. A second CP2 step transfers the t1 modulated magnetization back to the protons. The 1 H magnetization is then stored along the B0 magnetic field by a /2-pulse. The distribution of 1 H longitudinal magnetization is then allowed to equilibrate during a spin diffusion mixing period, τmix. With another /2-pulse, the 1 H polarization is restored in the xy plane and a final CP3 step is applied for high-resolution 13 C detection. It is important to note that the 1 H transverse magnetization that is not transferred through the 13 C channel during t1 must be perfectly purged before the CP2 step. Therefore, just after CP1 the residual transverse spin-lock 1 H magnetization is sent back to the z-axis with the 2 nd 1 H /2-HP. However, due to rf-inhomogeneity this pulse is not perfect, and the transverse 1 H magnetization remaining after the 2 nd /2-HP must be cancelled by cycling the phase of the initial 1 H /2-HP, relative to the phase of the 1 H spin-lock pulse of CP2. So, generally the efficient polarization transfers between 1 H nuclei can be combined with the superior spectral resolution of 13 C nuclei in a 2D HOMCOR correlation spectroscopy experiment. The conventional CP 3 sequence is shown in Fig. 1e, and the selective version in Fig. 1f.

II.3. DQF-𝐁𝐑𝟐 𝟐 𝟏

The main disadvantage of the two previous SQ-SQ experiments is the fact that the signal contributions on the diagonal have no physical meaning, because they are the sum of the autocorrelation signals and of those arising from spins which do not take part in transfer processes. The second contributions are usually much larger than the first ones, especially for short mixing times. Practically, this means that spatial proximities are impossible to analyze between carbons that are of the same species but in different molecules and difficult with species with close resonance frequencies. These limitations can be overcome by using DQ-SQ experiments, in which the DQ-correlated states are detected along the indirect dimension (e.g. Fig. 1g). [START_REF] Hu | Double-quantum NMR spectroscopy of 31P species submitted to very large CSAs[END_REF][START_REF] Edén | Improved double-quantum NMR correlation spectroscopy of dipolarcoupled quadrupolar spins[END_REF][START_REF] Schnell | Dipolar recoupling in fast-MAS solid-state NMR spectroscopy[END_REF] Unfortunately, it is difficult to perform a selective excitation in such DQ-SQ experiments. Indeed, each excitation should be double-selective to only excite one pair of spins, not a single spin as with previous SQ-SQ experiments. Moreover, the number of such double-selective excitations should be (n-1)/2 higher than with SQ-SQ experiments: n(n-1)/2 instead of n. Therefore, we have chosen using a compromise between pure SQ-SQ and DQ-SQ experiments, and used a SQ-SQ one with DQ-filtering. The filter ensures that all signals observed in the 2D spectrum arise from spins that have passed through a DQ coherence in the course of the recoupling procedure. In this way, signal contributions from spins which have not at all been involved in transfer processes are removed from the diagonal of the spectrum. However, the auto-correlation signal located on the diagonal is superimposed with signal contributions from other correlations. We have used the BR2 2 1 recoupling (Fig. 1h), but it must be mentioned that other such sequences have been proposed, such as the Sp ones (p = 2-4). [START_REF] Teymoori | Low-power broadband homonuclear dipolar recoupling without decoupling: Double-quantum 13C NMR correlations at very fast magic-angle spinning[END_REF][START_REF] Teymoori | Low-power broadband homonuclear dipolar recoupling in MAS NMR by two-fold symmetry pulse schemes for magnetization transfers and double-quantum excitation[END_REF] It should be noted that the BR2 2 1 recoupling scheme can also be applied to halfinteger spin quadrupolar nuclei. [START_REF] Wang | Double-quantum homonuclear NMR correlation spectroscopy of quadrupolar nuclei subjected to magic-angle spinning and high magnetic field[END_REF][START_REF] Yu | Insights into the dealumination of zeolite HY revealed by sensitivity-enhanced 27Al DQ-MAS NMR spectroscopy at high field[END_REF] Fig. 2. Rows from the 2D RFDR spectra of U-13 C labeled histidine hydrochloride monohydrate recorded with the 3 different MSP used with H8-RFDR and a selective excitation bandwidth of BWFWHM = 500 Hz, which corresponds to a pulse length of sp = 1.8, 1.8 and 3.6 ms for: 180° inversion (Fig. 1c), 180° refocusing (Fig. 1d) and 90° excitation (Fig. 1b), respectively. B0 = 18.8 T, R = 62.5 kHz, mix = 8 ms, NS = 64, RD = 2 s, and Texp = 18 min 25 s for each 2D.

III.

Tests on U- 13 C histidine hydrochloride monohydrate.

The experiments have been recorded at 18.8 T with a spinning speed of R = 62.5 kHz ( = 1.3 mm). Full details of experiments setup have been provided in ESI. The 13 C shifts are referenced with respect to TMS. The main use of Hadamard 13 C- 13 C HOMCOR 2D spectra is to record fast the build-up curves of labeled samples. To test this possibility, we have first recorded a large series of 2D spectra on U- 13 C histidine hydrochloride monohydrate. This compound contains 6 different carbon species, and hence we have used an H8 Hadamard encoding matrix. In the following, the peak intensities are normalized with respect to the highest signal in the series of spectra.

III.1. Optimization of the selective pulse-length

The first parameter to optimize is the length, sp, of the selective pulses. Indeed, this pulse must be long enough to selectively excite only one resonance at a time, but not too long because of the relaxation losses occurring during this pulse. It must be reminded that the excitation bandwidth is inverse proportional to the pulse-length, BWFWHM = A/sp, with A = 0.9 or 1.8 in the case of 10% truncated Gaussian line-shapes for  or /2 pulses, respectively. An easy such optimization of the pulse-length is obtained by (i) fixing the mixing time to a very small value to theoretically only observe SQ-SQ diagonal peaks, and (ii) recording a 1D spectrum with selective excitation of one of the two carbons that are with the minimum frequency separation. In Fig. 3, we show six such 1D H8-RFDR spectra for the C of histidine hydrochloride. Due to the decrease of the pulse-length and the associated losses, the autocorrelation peak amplitude increases with the bandwidth up to BWFWHM = 500 Hz. One or two negative cross-peak are observed with BWFWHM = 750, 1000 or 1500 Hz, respectively, which means that one or two very close resonances are also partially excited. In the following, we have chosen BWFWHM = 500 Hz. It must be noted that when they are excited, the closest resonances always appear with negative sign because of the off-resonance irradiation.

Therefore, in the case of very large losses, which may occur with slow spinning speeds, for which using the shortest as possible selective pulse length is recommended, one way to gain in S/N ratio is to use a BWFWHM value slightly larger that optimum, and to only take into account the cross-peaks with positive amplitudes. For example, at slow spinning speed using BWFWHM = 1500 Hz should allow to decrease the pulse length to p = 600 s.

III.2. Build-up curves

We have then recorded a series of 2D SQ-SQ spectra, as the three ones shown in Fig. S1, for the three different HOMCOR schemes, versus the mixing time up to mix = 13 ms. For such a labeled compound, each Hadamard 2D spectrum was acquired in only 4 min 35 sec. In Fig. 4 we show the build-up curves corresponding to C (Fig. 4a-c) or C2 (Fig. 4d-f) carbon atoms. These build-up curves present different evolutions, which specify their use.

* For RFDR and CP 3 experiments (Figs. 4a,b,d,e), the amplitudes of the diagonal peaks are largely predominant for small mixing times. However, the analysis of these diagonal peaks is useless because a large part of their intensities has no physical meaning and is not related to inter-nuclear distances. Only the cross-peak build-up curves can thus be used for structural analyzes. This limitation impedes the use of SQ-SQ experiments for species with close or equal resonance frequencies. This constraint is emphasized by the use of selective excitations which can hardly select very close resonances.

* For DQF-BR2 2 1 experiments, all build-up curves start from zero, because they all are related to dipolar transfers. However, only those of the cross-peaks are directly related to internuclear distances. Indeed, the build-up curves of diagonal peaks (e.g. C) are related to auto (C-C) and hetero (C-C and C-CO) correlations. Nevertheless, the auto-correlation build-up curves can be retrieved qualitatively by subtracting the hetero-correlation signals to the diagonal one. Moreover, it must be noted that the cross-peaks of nearby resonances, which are close to the diagonal, are easier to detect than with the two other methods because the diagonal peaks are much smaller, especially for small mixing times. However, the limitation related to the selective excitation of species with close resonance frequencies remains. * For RFDR and CP 3 experiments, the contact between the various carbons can be obtained through the 13 C-13 C (RFDR) or 1 H-1 H (CP 3 ) spin-diffusion process. Because the homonuclear dipolar interactions are ca. 16 times smaller with 13 C than with 1 H, the build-up curve variations are faster with CP 3 than with RFDR (compare Fig. 4a andb). For example, in Fig. 4b,e they mainly occur up to mix  500 s for CP 3 , and afterwards only plateaus are observed for the cross-peaks so that a slow decrease is observed for the diagonal peaks. * One observes that the amplitudes of some CP 3 cross-peaks are not zero with mix = 0 (e.g. C-C in Fig. 4b). This signal is related to the 1 H-1 H spin-diffusion occurring during the three CPMAS transfers. * The DQF-BR2 2 1 evolution is faster than that with RFDR.

* As a result, the analysis of the build-up curves of these three 13 C- 13 C sequences can be limited to e.g. mix  3-4 ms.

We have tried to verify the build-up curves observed with Hadamard MSP excitation with those obtained in a conventional way. We have used the DQF-BR2 2 1 recoupling and tried to record the 2D spectra corresponding to the 24 mixing times used in Figs. 4c andf in the limited total time of 24 hrs. However, to respect this time limitation we had to largely limit the indirect evolution time, hence leading to a large broadening of the resonances, which limited the number of cross-peaks that could be resolved. We show in Fig. 4c the results that could be exploited for C. The other mixing times lead to too small and too broad resonances to be used. It was not possible to obtain any build-up curve for C2 due to limited resolution. Actually, a roughly 10 times longer evolution time would have been required to obtain a usable resolution, thus leading to ca. 10 hours for each mixing time (Fig. S2). Nevertheless, these conventional results shown in Fig. 4c demonstrate the correctness of the build-up curves obtained with Hadamard MSP excitation in a much smaller experiment time: 4 min 35 s instead of several hours. As a conclusion, the CP 3 scheme should be used to analyze mostly the proximities of CH, CH2 or CH3 moieties, with the high-resolution of carbons, which is advantageous at moderate MAS speeds. The two other methods, RFDR and DQF-BR2 2 1 , directly analyze the 13 C-13 C proximities. RFDR is approximately twice more efficient than BR2 2 1 , but it does not allow analyzing auto-correlations and distances between nuclei with very close resonance frequencies; especially with selective excitation.

IV. Tests on U-13 C/ 15 N f-MLF

As a second test sample, we have chosen a bigger molecule, the labelled f-MLF tri-peptide, N-formyl-Met-Leu-Phe (U-13 C-15 N f-MLF). The molecule and the 1D CPMAS 13 C spectrum are shown in Fig. 5. The sample contains 19 different 13 C species and their isotropic chemical shifts, measured with an unlabeled sample, are given in Table .S1. [START_REF] Mao | Directly and indirectly detected through-bond heteronuclear correlation solidstate NMR spectroscopy under fast MAS[END_REF][START_REF] Althaus | Indirectly detected heteronuclear correlation solid-state NMR spectroscopy of naturally abundant 15N nuclei[END_REF] It must be noted that in Fig. 5b, most of the 13 C resonances are broadened by the 1 J13C-13C couplings in this quasifully labeled sample, and e.g. L 1 and L  resonances overlap, whereas they are discernible with unlabeled samples. Moreover, the F myl formyl carbon is not observed because it is not labelled in our sample. Finally, aromatic carbons (F  and F  ) are broadened by 180° flips of the phenyl ring and hence they cannot be cleanly separated at room temperature, and furthermore, they overlap with the F  resonance. As a result, only 14 resonances are well resolved in Fig. 5b, and hence we have used an H16 Hadamard encoding matrix for the multiselective excitation.

+ F  + F  , L  , F  , M  , L  , M  + F  , M  , L  + L 1 , L 2 , M  .
We show the 13 It must be noted that the resolution along F1 is then ca. three times larger than that observed with a 1D CPMAS spectrum. This means that the FIDs were largely truncated along t1, and that the optimum acquisition time should have been ca. 800 min with three times more t1 points. With the H16 matrix, the experimental time was of only Texp = 34 min, and the sensitivity was multiplied by 4 as we used NS = 64 (Fig. 6b-d). With the same sensitivity as the conventional experiment, the 2D acquisition time could have been decreased to Texp = 8 min 30 s. In spite of the much shorter acquisition time, the experimental 2D spectra are fundamentally identical. So, the comparison of the two RFDR spectra implies that for the same resolution, the conventional acquisition requires 3*280/8.5  100 times more experimental time than the H16-RFDR acquisition. This large decrease of Texp allows recording the build-up curves, which may be impossible with the conventional method. The H16-CP 3 acquisition used a short mixing time of mix = 100 s to limit the 1 H-1 H spin diffusion process (Figs. 6c andS12c). However, in spite of a 100-time shorter mixing time, the RFDR and CP 3 cross-peak patterns are not very different (compare Figs. 6b andc andS12b andc). It should be also noted that in the CP 3 spectrum, cross-peaks from protonated carbons are much larger than those from quaternary ones. Finally, in Fig. 6d and S12d we show that a similar 2D spectra can be obtained with the H16-DQF-BR2 2 1 recoupling, with same acquisition time. Actually, the H16-CP 3 and H16-DQF-BR2 2

1 spectra (Figs. 6c andd) only display correlations between directly bonded carbons because they use small mix = 100 and 512 s, respectively, contrary to H16-RFDR with large mix = 10.24 ms (Fig. 6b) with all remote connections.

V. Conclusions

In this article we have shown that a multi-selective excitation with Hadamard encoding is a powerful tool for 2D acquisition of 13 C-13 C homo-nuclear correlations. This method is not designed to improve the sensitivity, but rather to reduce the experiment time, provided there is sufficient sensitivity. Therefore, it allows fast acquisition of such 2D spectra in labeled molecules.

Nevertheless, there are several requirements that must be satisfied. First, the 1D 13 C spectra must be recorded in advance in order to obtain the prior knowledge of the resonance frequencies.

Secondly, this method mainly applies to molecules of small or moderate size which produce a limited number of resonances. Indeed, the main limitation of the Hadamard methods in solids resides in the length of the encoding pulses, which results from a compromise between the selectivity and the sensitivity due to relaxation losses. For this reason, these methods should mainly be used with sparse spectra, profiting from the increased separation of the resonances at high magnetic fields. In the case of 13 C- 13 C HOMCOR experiments, we give a simple rule that allows directly setting the optimum length of the selective pulses, versus the minimum separation of the resonances in the indirect dimension. It cannot be applied to proteins, contrary to the covariance method, which is another way to decrease the acquisition time of HOMCOR spectra in liquid [START_REF] Zhang | Spectral Deconvolution of Chemical Mixtures by Covariance NMR[END_REF][START_REF] Zhang | Indirect Covariance NMR Spectroscopy[END_REF][START_REF] Trbovic | Covariance NMR spectroscopy by singular value decomposition[END_REF][START_REF] Brüschweiler | Covariance nuclear magnetic resonance spectroscopy[END_REF][START_REF] Brüschweiler | Theory of covariance nuclear magnetic resonance spectroscopy[END_REF] or solid samples. [START_REF] Hu | Solid-state NMR covariance of homonuclear correlation spectra[END_REF][START_REF] Lafon | Fast and high-resolution stereochemical analysis by non-uniform sampling and covariance processing of anisotropic natural abundance 2D 2H NMR datasets[END_REF][START_REF] Li | Comparison of various sampling schemes and accumulation profiles in covariance spectroscopy with exponentially decaying 2D signals[END_REF][START_REF] Qian | The dependence of signal-to-noise ratio on number of scans in covariance spectroscopy[END_REF] Thirdly, this method better applies to the case of narrow resonances, that are fully resolved. It should not be used with broad resonances resulting from distribution of surroundings or quadrupolar nuclei. Fourth, its advantage is maximum in the case of narrow resonances that are spread over a large frequency range, because this requires numerous t1 steps with the conventional method, whereas the size of the Hadamard matrix is only related to the number of resonances. One main application of this method is that of 13 C- 13 C spectra of well crystallized samples, especially when they are recorded at very high magnetic fields. Fifth, it must be mentioned that Hadamard experiments require very linear rf-amplifiers in the low rf-regime to allow the simultaneous optimization of the selective -pulses for all resonances and all elements of the Hn Hadamard encoding matrix.

The demonstration has been performed on a U-13 C- 15 N histidine hydrochloride monohydrate. sample, and it allowed recording each point of the build-up curves of the 13 C-13 C cross-peaks within 4 min 35 s, which is very difficult with conventional methods. We have also shown on U-13 C-15 N f-MLF sample that the method can be applied to molecules with 14 13 C resonances with a minimum frequency separation of 240 Hz at 18.8T. It must be noted that for some pairs of carbon atoms (e.g. M  & F  or L  & L 1 ) this difference is much smaller, but in this case each of these pairs has been consider as a single species for the selective excitation. However, it has also to be noted that in these cases the conventional acquisition could not either separate the individual species (Fig. 6a). Supplementary Information. Figures S1-S11, Table S1, and pulse programs for the three HOMCOR sequences.

VI.

Fig. 1 .

 1 Fig.1. 13 C-13 C HOMCOR sequences. (a-d) SQ-SQ RFDR with: (a) only HPs (conventional), (b) /2-MSP, (c) inverting and (d) refocusing -MSP. SQ-SQ CP 3 with: (e) only HPs (conventional), (f) inversion -MSP. (g) DQ-SQ conventional BR2 2 1

Fig. 3 .

 3 Fig.3. U-13 C histidine hydrochloride monohydrate. 1D H8-RFDR spectra recorded with mix = 16 s (negligible 13 C-13 C transfer), for C with bandwidth indicated on the spectra. NS = 16, RD = 2s, Texp = 4min 35s for each 2D.

Fig. 4 .

 4 Fig.4. U-13 C histidine hydrochloride monohydrate. Build-up curves of C (a-c) or C2 (d-f) versus mix, for H8-RFDR (a,d), H8-CP 3 (b,e) and H8-DQF-BR2 2 1 (c,f). BWFWHM = 500 Hz, NS = 16, RD = 2s, Texp = 4 min 35s for each 2D. In (c) we present as squares the intensity of C-C and C-CO cross-peaks recorded in 1hr for each point with the conventional experiment.

Fig. 5 .

 5 Fig.5.13 C,15 N-f-MLF: molecular structure, and13 C CP-MAS spectra observed at 18.8 T with R = 62.5 kHz. The labelling and assignment of13 C and15 N resonances have been given in ref[START_REF] Mao | Directly and indirectly detected through-bond heteronuclear correlation solidstate NMR spectroscopy under fast MAS[END_REF][START_REF] Althaus | Indirectly detected heteronuclear correlation solid-state NMR spectroscopy of naturally abundant 15N nuclei[END_REF]. The 14 resolved resonances used with Hadamard were: L', F', M',F  , F  + F  + F  , L  , F  , M  , L  , M  + F  , M  , L  + L 1 , L 2 , M  .

  C-13 C 2D spectra of f-MLF corresponding either to the aliphatic part or the full frequency range in Figs 6 or S12, respectively. The conventional spectra are shown in Fig.6a and S12a, whereas the Hadamard ones are in Fig.6b-d and S12b-d. The long mixing time used with RFDR (Figs.6a,b and S12a,b), mix = 10.24 ms, leads to the appearance of nearly all cross-peaks, as previously shown in Fig.4a,d. The conventional acquisition used 256 t1steps, and the experimental time was Texp = 280 min with NS = 16.

Fig. 6 .

 6 Fig.6. Aliphatic part of 13 C-13 C 2D spectra of f-MLF recorded at 18.8 T with R = 62.5 kHz, RD = 2 s. (a) Conventional-RFDR acquired with 256 t1-steps in indirect dimension, RFDR = 178 kHz, mix = 10.24 ms = 640TR, NS = 16, Texp = 280 min. (b-d) BWFWHM = 150 Hz, sp = 6 ms, NS = 64, Texp = 34 min. (b) H16-RFDR, mix = 10.24 ms; (c) H16-CP 3 , mix = 100 s, CP1 = 1500, CP2 = CP3 = 200 s; (d) H16-DQF-BR2 2 1 , mix = 512 s.

Acknowledgments

PP is grateful for funding from the MOBILITY PLUS program (grant no. 1630/MOB/V/2017/0). Institut Chevreul (FR 2638), Ministère de l'Enseignement Supérieur et de la Recherche, Région Hauts-de-France and the European Union (FEDER/ERDF) are acknowledged for supporting and partially funding this work. Financial and technical supports from the TGIR-RMN-THC FR 3050 CNRS for conducting the research are gratefully acknowledged. Authors also acknowledge contract CEFIPRA nos. 85208-E, PRC CNRS-NSFC, ANR-14-CE07-0009-01 and ANR-17-ERC2-0022 (EOS). OL acknowledges financial support from Institut Universitaire de France (IUF).