Lee Robison

Xinyi Gong

Austin M Evans

Florencia A Son

Xingjie Wang

Louis R Redfern

Megan C Wasson

†∥ Zoha H Syed

Karam B Idrees Zhijie Chen

Karam B Idrees

Timur Islamoglu

Massimiliano Delferro

William R Dichtel

François-Xavier Coudert

Nathan C Gianneschi

Omar K Farha

Transient Catenation in a Zirconium-Based Metal-Organic Framework and Its Effect on Mechanical Stability and Sorption Properties

Keywords: metal-organic frameworks, catenation, transmission electron microscopy, mechanical properties

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Metal-organic frameworks (MOFs) are hybrid materials obtained by the self-assembly of inorganic nodes and organic linkers into periodic multidimensional structures with high surface areas and porosities. [START_REF] Zhou | Introduction to Metal-Organic Frameworks[END_REF][START_REF] Perry Iv | Design and synthesis of metal-organic frameworks using metal-organic polyhedra as supermolecular building blocks[END_REF] Among the thousands of MOFs synthesized to-date, zirconium clusterbased MOFs are particularly robust due to the strength of the Zr(IV)-carboxylate bond. [START_REF] Howarth | Chemical, Thermal and Mechanical Stabilities of Metal-Organic Frameworks[END_REF][START_REF] Yuan | Stable Metal-Organic Frameworks with Group 4 Metals: Current Status and Trends[END_REF][START_REF] Bon | Tailoring of network dimensionality and porosity adjustment in Zr-and Hf-based MOFs[END_REF] As such, Zr-based MOFs have been explored for applications that may require demanding conditions such as catalysis, [START_REF] Jiang | Topology-Based Functionalization of Robust Chiral Zr-Based Metal-Organic Frameworks for Catalytic Enantioselective Hydrogenation[END_REF][START_REF] Wasson | Metal-Organic Frameworks: A Tunable Platform to Access Single-Site Heterogeneous Catalysts[END_REF] water sorption, [START_REF] Wang | A Robust Large-Pore Zirconium Carboxylate Metal-Organic Framework for Energy-Efficient Water-Sorption-Driven Refrigeration[END_REF][START_REF] Chen | Reticular Access to Highly Porous acs-MOFs with Rigid Trigonal Prismatic Linkers for Water Sorption[END_REF][START_REF] Xu | Metal-Organic Frameworks for Water Harvesting from Air, Anywhere, Anytime[END_REF] and gas separations. [START_REF] Chen | Reticular chemistry in the rational synthesis of functional zirconium cluster-based MOFs[END_REF][START_REF] Lin | Exploration of porous frameworks for gas separation and purification[END_REF] Interpenetration is defined by the presence of two or more mechanically interlocked periodic networks where, although no chemical bonds exist between the frameworks, disentanglement can only be achieved by breaking chemical bonds (Fig. 1). [START_REF] Batten | Interpenetrating Nets: Ordered, Periodic Entanglement[END_REF] Interpenetration typically enhances the stability of a supramolecular framework by filling void space, which increases the density and the abundance of repulsive forces that prevent framework collapse. [START_REF] Chen | Interwoven Metal-Organic Framework on a Periodic Minimal Surface with Extra-Large Pores[END_REF] These attributes of interlocked networks increase the mechanical strength of the material, [START_REF] Ferguson | Controlled Partial Interpenetration in Metal-Organic Frameworks[END_REF] although they decrease the surface area and porosity of the structures as compared to their noninterpenetrated counterparts.

Nevertheless, many interpenetrated MOFs exhibit excellent gas separation and selectivity characteristics due to their tunable pore sizes. [START_REF] Fernandez | Switching Kr/Xe Selectivity with Temperature in a Metal-Organic Framework[END_REF][START_REF] Liang | A Tailor-Made Interpenetrated MOF with Exceptional Carbon-Capture Performance from Flue Gas[END_REF][START_REF] Jiang | Interpenetration Control in Metal-Organic Frameworks for Functional Applications[END_REF] In addition to the interpenetrated Zr-based MOF investigated in this study, other interpenetrated Zr-based MOFs have been reported. [START_REF] Zhang | A Flexible Interpenetrated Zirconium-Based Metal-Organic Framework with High Affinity toward Ammonia[END_REF][START_REF] Xu | Isolated π-Interaction Sites in Mesoporous MOF Backbone for Repetitive and Reversible Dynamics in Water[END_REF][START_REF] Duan | Interpenetrated Metal-Organic Frameworks with ftw Topology and Versatile Functions[END_REF][START_REF] Kong | Constructing new metal-organic frameworks with complicated ligands from "One-Pot" in situ reactions[END_REF] Among these, the UiO-66 type interpenetrated MOFs have been well known. [START_REF] Chen | Highly efficient visible-light-driven CO2 reduction to formate by a new anthracene-based zirconium MOF via dual catalytic routes[END_REF][START_REF] Lippke | Expanding the Group of Porous Interpenetrated Zr-Organic Frameworks (PIZOFs) with Linkers of Different Lengths[END_REF][START_REF] Schaate | Porous Interpenetrated Zirconium-Organic Frameworks (PIZOFs): A Chemically Versatile Family of Metal-Organic Frameworks[END_REF][START_REF] Cavka | A New Zirconium Inorganic Building Brick Forming Metal Organic Frameworks with Exceptional Stability[END_REF][START_REF] Marshall | Stereoselective Halogenation of Integral Unsaturated C-C Bonds in Chemically and Mechanically Robust Zr and Hf MOFs[END_REF][START_REF] Doan | Tailoring the Optical Absorption of Water-Stable ZrIV-and HfIV-Based Metal-Organic Framework Photocatalysts[END_REF][START_REF] Zhang | Two interpenetrated metal-organic frameworks: CH4 and CO2 adsorption and in-situ XRD studies[END_REF] Other systems which exhibit interpenetration were found during isoreticular expansion of the linkers and have either ditopic [START_REF] Marshall | Functional Versatility of a Series of Zr Metal-Organic Frameworks Probed by Solid-State Photoluminescence Spectroscopy[END_REF][START_REF] Wang | Adsorption of Fluorocarbons and Chlorocarbons by Highly Porous and Robust Fluorinated Zirconium Metal-Organic Frameworks[END_REF][START_REF] Feng | Lattice Expansion and Contraction in Metal-Organic Frameworks by Sequential Linker Reinstallation[END_REF][START_REF] Xie | Metal-Organic Frameworks for the Capture of Trace Aromatic Volatile Organic Compounds[END_REF][START_REF] Gao | Synthesis, structure and characterization of two solvatochromic metal-organic frameworks for chemical-sensing applications[END_REF] or tetratopic linkers, [START_REF] Yu | Blue-Light-Excitable, Quantum Yield Enhanced, Yellow-Emitting, Zirconium-Based Metal-Organic Framework Phosphors Formed by Immobilizing Organic Chromophores[END_REF][START_REF] Kong | Two interpenetrated metal-organic frameworks with a slim ethynyl-based ligand: designed for selective gas adsorption and structural tuning[END_REF][START_REF] Xu | Highly Stable Mesoporous Zirconium Porphyrinic Frameworks with Distinct Flexibility[END_REF][START_REF] Chen | All Roads Lead to Rome: Tuning the Luminescence of a Breathing Catenated Zr-MOF by Programmable Multiplexing Pathways[END_REF] which makes the relatively short tritopic linker used in this study a unique case. Herein, we explore the transient catenation processes between two different interpenetrations of a 3,8-connected Zr-based MOF, known as NU-1200 and STA-26 in its non-interpenetrated and interpenetrated forms, respectively.

Contemporary framework interpenetration studies rely heavily on bulk characterization techniques. [START_REF] Jiang | Interpenetration Control in Metal-Organic Frameworks for Functional Applications[END_REF] However, investigations into catenation processes via direct imaging at the singleparticle limit have not yet been performed. While single-crystal X-ray diffraction (SCXRD) can be used to study single particles, the indirectly obtained structure is extracted from the average positions of atoms in a crystal. This can become challenging to interpret when positional or substitutional disorder is present. Furthermore, single-crystal X-ray diffraction evaluates individual particles, which may not be representative of the bulk. In this work, we examine the zirconium cluster-based non-interpenetrated NU-1200 MOF and the analogous doubly interpenetrated STA-26 MOF at the single particle level using high-resolution transmission electron microscopy (TEM) coupled with an automated post-processing script that analyzes interpenetration across many images. These single-particle studies are complemented by ensemble characterization techniques such as powder X-ray diffraction (PXRD) and adsorption isotherms with several probe molecules, including a structural mimic for a chemical warfare agent. These investigations reveal that catenation occurs in a near stepwise process within individual particles, which leads to mixtures of pure phases of the interpenetrated and non-interpenetrated structures rather than partially catenated particles. This observation led us to study the thermodynamics and mechanical properties of both pure phases via density functional theory (DFT), in situ synchrotron X-ray diffraction, and nanoindentation experiments. Collectively, these studies enable the reliable characterization of two different interpenetrated and non-interpenetrated Zr-based MOFs and reveal their promise for applications where demanding mechanical stresses are encountered, including the storage of toxic chemical warfare agents. [START_REF] Decoste | Metal-Organic Frameworks for Air Purification of Toxic Chemicals[END_REF][START_REF] Islamoglu | Metal-Organic Frameworks against Toxic Chemicals[END_REF]

RESULTS and DISCUSSION

Herein, we characterize the properties of two distinct zirconium cluster-based, 3,8-connected MOFs (Fig. 2A). Each MOF features 4,4′,4′′-(2,4,6-trimethylbenzene-1,3,5-triyl)tribenzoic acid (TMTB) linkers and Zr 6 (μ 3 -OH) 4 (μ 3 -O) 4 (OH) 4 (OH 2 ) 4 Zr 6 -oxo cluster metal nodes. The tritopic TMTB linkers and 8-connected Zr 6 -oxo clusters assemble to form the non-interpenetrated NU-1200, which possesses 14Å diameter sodalite cages and mesoporous 1D channels that are 20 Å in width. The NU-1200 MOF features the topology (cubic clusters and triangular ligands) and crystallizes in the Pm m space group. [START_REF] Liu | Adding to the Arsenal of Zirconium-Based Metal-Organic Frameworks: the Topology as a Platform for Solvent-Assisted Metal Incorporation[END_REF][START_REF] Wang | Highly Stable Zr(IV)-Based Metal-Organic Frameworks for the Detection and Removal of Antibiotics and Organic Explosives in Water[END_REF] Additional characterization data is available in SI (Fig. 3 S1-S3).

microporous rather than mesoporous as a result of the second lattice being displaced by [ ] from the original non-interpenetrated NU-1200 framework. This displacement means that the vertex of one sublattice is in the exact center of the sodalite cages of the other, while the diameter of the octahedral cages remains the same (14Å). The STA-26 MOF exhibits Im m symmetry. 3

However, we observed that the interpenetration can be initiated post-synthetically, and the degree of interpenetration between these two networks could be controlled by regulating the reaction time.

We initiated the interpenetration of NU-1200 to STA-26 by exposing 20 mg of thermally activated NU-1200 to a solution of DMF:HCOOH that is 2.5:1 by volume at 120 °C for 40, 110 or 180 min, and referred as NU-1200-x or STA-26-x where x indicates the time that MOF particles spend in DMF:HCOOH solution. We found that this transient catenation process was complete after 110 min. We monitored this transition using PXRD (Cu Kα1 radiation, λ = 1.54056 Å) by tracking the disappearance of the peak at 2θ = 3.1°, which corresponds to the NU-1200 (100) Bragg feature, and the growth of the 4.42° feature, which corresponds to the (110) reflection of STA-26 (Fig. 2B). We posit a mechanism for this process, but it is preliminary as we do not have comprehensive experimental data to explain the process. [START_REF]hard acid Zr 4+ ions and the hard base carboxylate oxygens of the tritopic TMTB linker by protonation of the linker followed by a reformation of the framework when the MOF solution was placed within the oven at 120[END_REF] Wright, Prasad, and co-workers reported that the same TMTB linker and Zr 6 -oxo cluster metal node produces STA-26, a doubly interpenetrated analogue of NU-1200. [START_REF] Bumstead | Modulator-Controlled Synthesis of Microporous STA-26, an Interpenetrated 8,3-Connected Zirconium MOF with the the-i Topology, and its Reversible Lattice Shift[END_REF] The authors synthetically targeted the structure by changing the identity and concentration of the modulating species present during the synthesis. The STA-26 MOF possesses the same the topology but is

TEM Imaging and Automated Interpenetration Mapping

Despite the numerous investigations into framework catenation processes, [START_REF] Jiang | Interpenetration Control in Metal-Organic Frameworks for Functional Applications[END_REF] there still exists a limited understanding of how this process occurs within single particles. Recently, high-resolution transmission electron microscopy (HR-TEM) hardware and imaging techniques have been developed to study beam-sensitive materials, such as MOFs. [START_REF] Gong | Insights into the Structure and Dynamics of Metal-Organic Frameworks via Transmission Electron Microscopy[END_REF] Here, we combine those advances with an automated post-processing Fourier transform mapping technique to explore the transition of NU-1200 to STA-26 at the single-particle limit ( NU-1200 and STA-26, respectively (Fig. S5). Due to this difference in symmetry, and accompanying difference in electron density contrast, these two lattices can be resolved by evaluating the relative intensity ratios of the FFT features at 0.36 nm -1 and 0.49 nm -1 that correspond to real spacings of 2.8 nm and 2.0 nm, respectively. Using this understanding, we developed a post-processing script, which automatically rasters a small region-of-interest across an image, extracts the Fourier transform of that sub-image, and then assigns the dominant interpenetration within that region based on the relative intensity ratios of the FFT features mentioned above (Fig. S6). This method allows us to spatially resolve the interpenetration of entire crystallites across a series of images. When this method was applied to pure crystal phases, we found that the script could reliably disambiguate the two phases (Fig. S7-S8).

We then applied the same technique to microtome-cut intermediate timepoint samples, NU-1200 40 min and STA-26 110 min, which we obtained along the course of the interpenetration transition.

This approach allows us to statistically investigate the process of transient catenation. In particular, we investigated whether interpenetration occurs gradually across all crystals within a sample or whether two predominantly pure phases are present at all times, which would be challenging to resolve using bulk techniques such as powder X-ray diffraction or nitrogen adsorption at 77 K. From our post-imaging analysis, we predominantly observed that pure phases were present at all times, which suggests that once an interpenetration transition is initialized it occurs rapidly and completely. Across several images of NU-1200 40 min (Fig. S9-S10), we found that nearly all particles were obtained as pure non-interpenetrated NU-1200 (Fig. S13). However, a select number of particles were obtained as the pure STA-26 phase. In contrast, we found the images of the STA-26 110 min sample (Fig. S11-S12) dominated by the interpenetrated STA-26 crystallites (Fig. S14). In rare instances, we observed minor, residual NU-1200 non-interpenetrated domains at the fringes of these crystallites (Fig. S15-S18), which may account for the minor X-ray diffraction features observed at prolonged reaction times. The non-interpenetrated lattice being confined at the edge of these intermixed particles indicates that edges are the final location to interpenetrate. Taken together, these results suggest that catenation occurs rapidly from NU-1200 to STA-26 within single particles. This contrasts with the possibility that the catenation process occurs gradually across all crystallites, which would lead to the prevalent observation of intermixed phases within single particles. This finding has implications for the physical properties of samples undergoing catenation, which in this case are likely to behave similarly to physical mixtures of pure phases. More investigation is needed to resolve the thermodynamic and kinetic underpinnings of this transition.

Physical Mixtures of Multiple Phases

From our findings that intermediate samples were predominantly single-phase particles, we decided to compare our ensemble measurements performed on intermediate samples with those of physical mixtures of pure NU-1200 and pure STA-26 phases. We observed the same decrease in nitrogen sorption capacity, reduction in the differential pore volume and pore size, and change in PXRD pattern for the physical mixtures (Fig. S19) as MOF samples measured during the interpenetration process (Fig. 2).

Our ensemble and direct imaging findings lead us to conclude that the interpenetration of NU-1200 occurs very quickly, with some percentage of crystals in intermediate samples being doubly interpenetrated STA-26 or non-interpenetrated NU-1200. This contrasts with our initial hypothesis that we synthesized MOFs exhibiting partial interpenetration. Combined with our TEM mapping results, we conclude that the decrease in adsorbed nitrogen and concurrent decrease in pore volume dominantly arises from different ratios of the two phases. The total uptake of N 2 from the nitrogen sorption isotherms of the physical mixtures (Fig. S19) display a linear relationship (R 2 = 0.96) from non-interpenetrated to doubly interpenetrated (Fig. S20). Additionally, we observed a linear relationship (R 2 = 0.96) between the total pore volume (cc g -1 ) plotted and the ratio of pure crystals that are mixed (Fig. S21). These observations are largely consistent with our observation of a stepwise transition from the NU-1200 to the STA-26 phase, rather than gradual framework interpenetration within single particles.

Reinforced Mechanical Strength of Interpenetrated Lattices

Previous work in the MOF field has established interesting pressure-induced behavior in framework materials, including discovery of new phases, polymorphism, negative linear compressibility and single crystal to single crystal phase transitions, among others. [START_REF] Moggach | The Effect of Pressure on ZIF-8: Increasing Pore Size with Pressure and the Formation of a High-Pressure Phase at 1.47 GPa[END_REF][START_REF] Widmer | Rich Polymorphism of a Metal-Organic Framework in Pressure-Temperature Space[END_REF][START_REF] Gagnon | MOFs Under Pressure: The Reversible Compression of a Single Crystal[END_REF][START_REF] Li | Negative Linear Compressibility of a Metal-Organic Framework[END_REF][START_REF] Tan | Chemical structure, network topology, and porosity effects on the mechanical properties of Zeolitic Imidazolate Frameworks[END_REF] While it is generally understood that physical properties of MOFs are affected by interpenetration, very few studies have explored the differences between mechanical strength of differentially interpenetrated chemically identical networks. [START_REF] Ferguson | Controlled Partial Interpenetration in Metal-Organic Frameworks[END_REF] To our knowledge this is the first study to combine DFT computations and experimental work to determine the effect of interpenetration on the hydrostatic, uniaxial, and shear stress on a MOF structure. Studying the bulk mechanical properties of differently interpenetrated structures [START_REF] Redfern | Mechanical Properties of Metal-Organic Frameworks[END_REF] is crucial for the use of MOFs in commercial applications which require that powdered MOF samples be processed into shaped constructs such as pellets, extrudates, or as composite materials. [START_REF] Peterson | Effects of Pelletization Pressure on the Physical and Chemical Properties of the Metal-Organic Frameworks Cu 3 (BTC) 2 and UiO-66[END_REF][START_REF] Purewal | Increased Volumetric Hydrogen Uptake of MOF-5 by Powder Densification[END_REF][START_REF] Kitao | Hybridization of MOFs and polymers[END_REF] We investigated NU-1200 and STA-26 using density functional theory (DFT) to obtain values for the bulk modulus (K), the Young's modulus (E), and the shear modulus (G) (Fig. 4). [START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF] The bulk modulus (K) of a material is a measure of the elastic resistance to hydrostatic compression and related to the ratio of volumetric stress over the volumetric strain (K = -V dP/dV) in an isothermal process. The Young's modulus (E) is a measure of a material's ability to deform under uniaxial constraint (tension or compression). The Young's modulus is equivalent to the tensile stress over the tensile strain (E = σ/ε). The shear modulus (G = Fl/AΔx) is the measure of deformation of one surface of a material while an opposite face of the material experiences an opposing force. The shear modulus is the ratio of shear stress to shear strain. Our DFT results show that interpenetration nearly doubles the value of the bulk modulus and increases the Young's and shear moduli by 60%. These calculations reveal that the interpenetrated STA-26 framework is stiffer and more mechanically robust than the NU-1200 non-interpenetrated structure.

agrees with other large-pore interpenetrated frameworks [START_REF] Ferguson | Controlled Partial Interpenetration in Metal-Organic Frameworks[END_REF] and reveals that the non-interpenetrated phase is metastable compared to its denser interpenetrated analogue (see supporting information for additional computational details). This finding is in line with other additional classes of porous materials (mesoporous silicas and siliceous zeolites) [START_REF] Le | Energetics of formation of alkali and ammonium cobalt and zinc phosphate frameworks[END_REF][START_REF] Trofymluk | Energetics of Mesoporous Silica: Investigation into Pore Size and Symmetry[END_REF] along with other MOF frameworks. [START_REF] Hughes | Thermochemistry of Zeolitic Imidazolate Frameworks of Varying Porosity[END_REF][START_REF] Bennett | Reversible pressure-induced amorphization of a zeolitic imidazolate framework (ZIF-4)[END_REF] Table 1. Experimental Properties of non-interpenetrated NU-1200 and interpenetrated STA-26. We determined the bulk modulus (K) for the NU-1200 and STA-26 MOFs using in situ synchrotron PXRD using a diamond anvil cell (DAC) pressure apparatus at the 17-BM beamline (λ = 0.45418 Å) at the Advanced Photon Source at Argonne National Laboratory. The PXRD peaks of the two MOFs shift to higher angles of diffraction upon the application of modest pressures, which we applied up to 0.55 GPa (S28-S29) and indicate compression along all crystallographic axes of the MOF sample. By first extracting the unit cell volume from the location of our diffraction features and then using a second-order Birch Murnaghan equation of state, we determine the bulk modulus of each MOF (Fig. 5A). The plots of the unit cell volumes vs. pressure reveal the interpenetrated STA-26 MOF has a higher bulk modulus (K = 21.1 ± 0.5 GPa) than the non-interpenetrated NU-1200 (K = 5.7 ± 0.3 GPa) (Fig. S27). The experimental data for the interpenetrated STA-26 MOF aligns well with the second-order model, even though the computationally derived value for the bulk modulus of the interpenetrated MOF is 6.0 GPa lower than the experimental value. The difference between the values for the bulk modulus of NU-1200 is only 2.4 GPa. However, the experimental data exhibits deviations from the best fit using a second order Birch Murnaghan equation of state (Fig. S27). This discontinuity at low pressures may indicate mechanically induced phase transitions. [START_REF] Widmer | Pressure Promoted Low-Temperature Melting of Metal-Organic Frameworks[END_REF][START_REF] Chapman | Pressure-Induced Amorphization and Porosity Modification in a Metal-Organic Framework[END_REF] Indeed, the computationally derived elastic tensors support that phase transitions may occur at low pressures in NU-1200. We found that the tetragonal shear modulus, C 11 -C 12 , value of NU-1200 is the lowest eigenvalue of all calculated elastic tensors for both structures, which suggests that this system is the least robust to elastic mechanical deformation and is therefore prone to phase transitions. In particular, C 11 -C 12 is 0.6 GPa for the non-interpenetrated NU-1200 phase, meaning the system will likely undergo a phase transition upon the application of modest amounts of pressure (Fig. S25). Due to the large coordination number of NU-1200, it remains mechanical stable at ambient pressure, while other highly porous MOFs with low shear moduli have been shown to be unstable under guest removal. [START_REF] Yan | Non-Interpenetrated Metal-Organic Frameworks Based on Copper(II) Paddlewheel and Oligoparaxylene-Isophthalate Linkers: Synthesis, Structure, and Gas Adsorption[END_REF] Therefore, the tetragonal shear is the softest mode of deformation. each indentation and obtained the Young's modulus and hardness as a function of indentation depth using the method proposed by Oliver and Pharr. [START_REF] Oliver | Measurement of Hardness and Elastic Modulus by Instrumented Indentation: Advances in Understanding and Refinements to Methodology[END_REF] By averaging measurement over five indentations, we assign the Young's modulus of NU-1200 as 2.9 GPa with a hardness of 100 MPa and the Young's modulus of STA-26 as 4.6 GPa with a hardness of 300 MPa. These values agree well with those determined using computational methods, which also reveal that the noninterpenetrated NU-1200 is softer than the STA-26 under uniaxial compression. Collectively, our experimental and computational findings reveal that STA-26 is considerably more structurally robust than NU-1200. These findings demonstrate that interpenetrated MOFs are likely more stable to all forms of mechanical stress, including those listed here (hydrostatic, uniaxial, and shear) than their non-interpenetrated, chemically identical analogues. This indicates that if a sample includes a mixture of interpenetrated and non-interpenetrated crystals, processing conditions are limited by the less stable MOF. Moreover, this observation suggests that if you have mixed phases, they are more likely to deform under mechanical stress.

MOF

Complementary Gas Sorption of Physically Mixed MOF Systems

One potential application for MOFs is the capture and detoxification of chemical warfare agents such as a potent blistering agent, mustard gas. [START_REF] Bobbitt | Metal-organic Frameworks for The Removal of Toxic Industrial Chemicals and Chemical Warfare Agents[END_REF][START_REF] Son | Uncovering the Role of Metal-Organic Framework Topology on the Capture and Reactivity of Chemical Warfare Agents[END_REF] Since the interpenetrated STA-26 and noninterpenetrated NU-1200 MOFs exhibit different pore structures and N 2 uptake capacities, we hypothesized that they would likely exhibit different adsorption characteristics for n-hexane, which we used as a structural mimic for mustard gas due to similarity in size and hydrophobicity of these two molecules. [START_REF] Chen | Understanding the Loading Dependence of Adsorbate Diffusivities in Hierarchical Metal-Organic Frameworks[END_REF] The uptake trends we report can only be directly applied to n-hexane; however, we can use this hydrocarbon as a model to begin to understand more complex compounds, such as mustard gas. We collected n-hexane adsorption isotherms in both pure phase MOFs and variable mixtures of the two pure phases. We observed a much greater uptake of nhexane at lower partial pressure in the interpenetrated microporous STA-26 MOF than in the mesoporous non-interpenetrated NU-1200 MOF. The 100% STA-26 sample reaches saturation at 0.01 P/P 0 while the 100% NU-1200 sample reaches saturation at 0.05 P/P 0 (Fig. 6). This data suggests that the interpenetrated MOF will exhibit better performance for low concentration capture of mustard gas but the non-interpenetrated NU-1200 will have an overall higher capacity for toxic gas capture. To support this observation, we plotted the total uptake of hexanes (cc g -1 )

plotted against each sample (Fig. S23) and found a linear relationship (R 2 = 0.98) between the two.

This indicates that benefits may exist by combining different interpenetrations of MOF crystallites within a single capture device.

CONCLUSIONS

In summary, we have investigated the interpenetration of the zirconium cluster-based mesoporous NU-1200 MOF to the chemically identical microporous STA-26 MOF at the bulk and singleparticle limits. Using bulk methods, one may propose that we have obtained partially interpenetrated crystallites. However, we find that our X-ray diffraction, gas adsorption, and transmission electron microscopy measurements better describe our system as statistical mixtures of crystallites with integral values of interpenetration, rather than fractionally occupied phases.

This suggests that interpenetration, once initialized, occurs rapidly. Experimental and computational evaluation of the mechanical properties for each framework revealed that the interpenetrated phase is more mechanically robust and thermodynamically stable than its noninterpenetrated counterpart. Finally, we find that these two phases exhibit radically different uptake behavior for n-hexane. Isotherms of mixed-phase systems show intermediate uptake behavior, which suggests that an opportunity exists to systematically tune adsorption characteristics by mixtures of variably interpenetrated crystallites, which we have shown can be obtained by de novo synthetic methods. Future studies should aim to explore mechanistic processes and physical characteristics related to interpenetrated MOFs more broadly, which we suspect will be an important area of study for the commercial deployment of these materials.

ASSOCIATED CONTENT

Supporting Information

Physical methods and instrumentation, syntheses of TMTB linker and Zr-MOFs, PXRD data, NMR data, SEM and TEM images, mapping script, computational methods, and mechanical properties measurements.

Figure 1 .

 1 Figure 1. Schematic of non-interpenetrated frameworks and the doubly interpenetrated analogue.

  Nitrogen isotherms obtained along the course of this transition demonstrated a decrease in gravimetric adsorption capacity consistent with the interpenetrated framework decreasing the total void space of the MOF. Similarly, the extracted pore sizes calculated from the nonlocal density functional theory (NLDFT) model for Pillared Clay reveal that during this transition the mesopore of NU-1200 (20 Å) disappears, and after 110 min of soaking in formic acid/DMF solution, only the STA-26 micropore (10 Å) is observed. To determine if the phase transition to the denser STA-26 interpenetrated phase could be reversed to the mesoporous NU-1200, which could imply entropic control between the two, we applied the original synthesis conditions to activated STA-26. However, we found that we could not reverse the interpenetration trend we observe (FigS4).

Figure 2 .

 2 Figure 2. (A) NU-1200 and its interpenetrated analogue STA-26. (B) PXRD patterns of noninterpenetrated NU-1200 transiently transforming to interpenetrated STA-26 over variable reaction times. (C) Nitrogen sorption isotherms and (D) NLDFT-calculated pore size distributions of NU-1200 crystals transforming to STA-26 crystals over variable reaction times.

Fig 3 ).

 3 We first imaged samples of the noninterpenetrated NU-1200 and interpenetrated STA-26. Fast Fourier transforms (FFTs) of MOF particles on the [100] zone axis showed the expected lattice symmetries of Pm m and Im m for 3 3

Figure 3 .

 3 Figure 3. High-resolution transmission electron micrographs of pure-phase (A-B) NU-1200 and (D-E) STA-26. Fourier transforms of (C) NU-1200 and (F) STA-26. (G) Transmission electron micrograph obtained after 40 min of reaction revealing a predominant non-interpenetrated structure. Lattice-resolution image of the blue boxed region in G, showing a NU-1200 structure. (H) Transmission electron micrograph obtained after 110 min of reaction revealing a predominant interpenetrated structure. Lattice-resolution image of the red boxed region in G, showing an STA-26 structure. Grey regions indicate void space, lacey carbon substrate, or damaged crystallites.

Figure 4 .

 4 Figure 4. Illustrations of elastic properties studied in this manuscript: (A) bulk modulus (K): measure of elastic resistance to hydrostatic compression (B) Young's modulus (E): measure of resistance in length during uniaxial tension or compression (C) shear modulus (G): measure of the resistance when subjected to opposing shear forces. Computational mechanical properties of noninterpenetrated NU-1200 and interpenetrated STA-26.

Figure 5 .

 5 Figure 5. (A) Relative lattice compression of interpenetrated STA-26 obtained using in-situ synchrotron PXRD to determine the bulk modulus. Line represents the second-order Birch-Murnaghan equation-of-state fit to the data. (B) Example of Force vs Displacement curves of interpenetrated STA-26 sample obtained using single crystal nanoindentation to determine the Young's modulus.

Figure 6 .

 6 Figure 6. (A) n-hexane adsorption isotherms for physical mixture of pure-phase samples.(B) Blow up of 0-0.1 P/P 0 region for n-hexane adsorption isotherms for physical mixture of pure-phase samples.

ACKNOWLEDGEMENTS O.K.F. gratefully acknowledges the financial support from the Air Force Research Laboratory (FA8650-15-2-5518) for the MOF materials synthesis. O.K.F. gratefully acknowledges support from the Defense Threat Reduction Agency (HDTRA1-19-1-0007) for the physical stability sorption measurements. O.K.F. and N.C.G. gratefully acknowledge support from National Science Foundation's MRSEC program (grant number NSF DMR-1720139). N.C.G acknowledges the Army Research Office for support of the development of TEM for MOFs (W911NF-15-1-0189). M.D. acknowledges the financial support from the Inorganometallic Catalyst Design Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under award no. DE-SC0012702. F.-X.C. is supported by the Agence Nationale de la Recherche (project MATAREB, ANR-18-CE29-0009-01), and access to HPC platforms was provided by a GENCI grant (A0090807069). X.G. gratefully acknowledges the support of the Northwestern University Ryan Fellowship granted by the International Institute of Nanotechnology and The Graduate School at Northwestern University. M.C.W. and Z.H.S are supported by the NSF Graduate Research Fellowship under grant DGE-1842165. A.M.E. is supported by the NSF Graduate Research Fellowship under grant DGE-1324585. F.A.S. is supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program. The authors thank Dr. Yuyang Wu and Dr. Yongbo Zhang for assistance in acquiring and interpreting the solid state NMR data reported herein. Use was made of the IMSERC X-ray facility at Northwestern University, which has received support from the Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-1542205), the State of Illinois, and the International Institute for Nanotechnology (IIN). Use was made of NMR-Hg400-solids at Northwestern University, which has received support from National Science Foundation (CHE-9871268) and International Institute for Nanotechnology (IIN).This work made use of the SPID facility of Northwestern University's NUANCE Center, which has received support from the Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-1542205); the MRSEC program (NSF DMR-1720139) at the Materials Research Center; the International Institute for Nanotechnology (IIN); the Keck Foundation; and the State of Illinois, though the IIN. This work made use of the EPIC facility of Northwestern University's NUANCE Center, which has received support from the Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-1542205), the MRSEC program (NSF DMR-1720139) at the Materials Research Center, the International for Nanotechnology (IIN), the Keck Foundation, and the State of Illinois through the IIN. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract no. DE-AC02-06CH11357.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: nathan.gianneschi@northwestern.edu and o-farha@northwestern.edu

Author Contributions

# L.R. and X.G. contributed equally. The manuscript was written though contributions of all authors. All authors have given approval to the final version of the manuscript.


Sophie Hameury

Dr Laure Kayser

Dr Roberto Pattacini

Dr Patrick Rosa
email: patrick.rosa@icmcb.cnrs.fr

Anne-Laure Barra

Pierre Braunstein
email: braunstein@unistra.fr

Dr A.-L Barra

Synthesis, Structures and Single Molecule Magnet Behaviour of High Nuclearity Ni(II) Dicubane-type Complexes with Pyridyl-Alcohol Ligands

The condensation reaction between two chemically different tetranuclear cubane-like clusters [Ni(3-Cl)(Cl)(HL)]4 (1) (HL = 2-methyl-1-(pyridin-2-yl)propan-2-ol) and [Ni(3-OH)(Cl)(HL)]4 (2) is shown to provide a new access to higher nuclearity complexes, as shown with the synthesis of the octanuclear complex [Ni8(3-Cl)3(3-OH)5(-Cl)2Cl6(HL)7] (3). In one instance, we also isolated the heptanuclear complex [Ni7(3-Cl)2(3-OH)6Cl4(H2O)2(HL)6]Cl2 (4). Magnetic properties measured by SQUID susceptometry evidenced S = 8 and S = 5 ground states for complexes 3 and 4, respectively. In particular, ac susceptometry evidenced slow relaxation of the magnetization for complex 3, and suggests a similar behaviour at sub-kelvin temperatures for complex 4.

Introduction

The coordination chemistry of ligands possessing significantly different donor groups, also called hybrid ligands, is attracting increasing interest in diverse areas of chemistry. Their successful applications in homogeneous catalysis results from the possibility of a fine tuning of the catalytic properties through chemical modifications of the donor functions. Furthermore, their different affinities for the metal centre can lead to enhanced reactivity and hemilabile behaviour. [1] Such ligands may also provide access to metal complexes with unusual structures, possibly endowed with interesting catalytic and/or magnetic properties. This has been observed, in particular, with pyridinealcohol or pyridine-alcoholate ligands, where N,O chelation and O-bridge formation can result in polynuclear complexes with unpredictable structures. The rich chemistry of 2-pyridone and of deprotonated pyridine-2-ylmethanolate (LMe -), derived from pyridine-2-ylmethanol (HLMe Scheme 1) is well documented. [2] Scheme 1.

Motivated by the versatility of this type of ligands, we previously focused our interest on the synthesis of Ni(II) complexes for applications in the catalytic oligomerization of ethylene. [3] Further work showed that deprotonation of the alcohol function resulted in unexpected mixed nickel/sodium polynuclear complexes. Thus, coordination clusters such as [Ni7(LMe)12]Cl2, [Na3Ni4(LMe)9(3-OH)]Cl and [NaNi6(LMe)12]Cl (Figure 1) were isolated and fully characterized when NaH was used as a base. Their formation depends on the reagents stoichiometry. [4] Figure 1. View of the structure of the cation in [NaNi6(LMe)12]Cl. The structural diversity observed with pyridine-alcoholate ligands encouraged us to explore further the coordination chemistry of pyridine alcohol ligands and we isolated, in the absence of NaH, tetranuclear, cubane-like transition metal dihalido complexes [Ni(3-Cl)Cl(HL)]4 (1). [5] Another cubane-type complex, [Ni(3-OH)Cl(HL)]4 (2), was prepared by reaction of 1 with an excess of highly dispersed NaOH, prepared in situ by hydrolysis of NaH, and the hydroxide anion selectively substituted the chloride corners without deprotonating the 2-methyl-1-(pyridin-2yl)propan-2-olligand (HL) (Scheme 2). The magnetic properties of these cubanes were investigated and unveiled a Single-Molecule Magnet (SMM) behaviour. [5] Scheme 2. For the known cubane-type complexes 1 and 2, one N,OH chelating ligand is explicitly depicted but only the chelate cycle is displayed for the remaining three.

[5]

The chemistry of complex 1 proved to be even richer, since in the course of these studies, we managed to isolate the new, higher nuclearity clusters [Ni8(3-Cl)3(3-OH)5(-Cl)2Cl6(HL)7] ( 3) and [Ni7(3-Cl)2(3-OH)6Cl4(H2O)2(HL)6]Cl2 (4). We studied the formation of the octanuclear complex 3 and were able to rationalize it. We also describe here the magnetic behaviour of both complexes.

Results and Discussion

Syntheses and Structures.

The ligand 2-methyl-1-(pyridin-2-yl)propan-2-ol (HL) was synthesized according to published procedures, by condensation of lithiated 2-picoline with acetone. [6] In the course of numerous attempts aimed at rationalizing the synthesis of [Ni(3-OH)Cl(HL)]4 (2) by varying the NaOH/HL ligand stoichiometry, we isolated the octanuclear cluster [Ni8(3-Cl)3(3-OH)5(-Cl)2Cl6(HL)7] (3) in low yields. The structure of 3 was determined by single crystal X-ray diffraction (Figure 2, Table 1). Each Ni(II) centre is hexacoordinated and shows a slightly distorted octahedral coordination geometry. Five of the metal 3), Ni1-N1 2.097(4), Ni1-N2 2.098(4), Ni1-Cl1 2.474(1), Ni1-Cl2 2.405(1), Ni2-O8 2.034(3), Ni2-O9 2.017(3), Ni2-Cl2 2.443(1), Ni2-Cl1 2.423(1), Ni2-Cl3 2.469(1), Ni2-Cl4 2.463(1), Ni3-O7 2.057(4), Ni3-O9 1.994(3), Ni3-N7 2.081(4), Ni3-Cl4 2.586(1), Ni3-Cl5 2.420(1), Ni3-Cl6 2.431(1), Ni4-O6 2.066(4), Ni4-O8 2.007(3), Ni4-N6 2.080(4), Ni4-Cl4 2.587(1), Ni4-Cl6 2.506(1), Ni4-Cl7 2.370(1), Ni5-O8 2.041(3), Ni5-O9 2.063(3), Ni5-O10 2.034(3), Ni5-O11 2.045(3), Ni5-O12 2.061(3), Ni5-Cl6 2.478(1), Ni6-O3 2.102(4), Ni6-O10 2.030(3), Ni6-O12 2.067(3), Ni6-N3 2.089(4), Ni6-Cl8 2.407(1), Ni6-Cl10 2.463(1), Ni7-O4 2.115(4), Ni7-O10 2.056(3), Ni7-O11 2.054(3), Ni7-N4 2.068(4), Ni7-Cl10 2.500(1), Ni7-Cl11 2.405(1), Ni8-O5 2.086(4), Ni8-O11 2.034(3), Ni8-O12 2.039(3), Ni8-N5 2.081(4), Ni8-Cl9 2.394(1), Ni8-Cl10 2.511(1). For selected bond angles, see centres (Ni3, Ni4, Ni6, Ni7 and Ni8) are chelated by HL through the O and N donors, while one is doubly chelated (Ni1). The remaining two nickel centres are not coordinated by HL (Ni2 and Ni5). A heptanuclear dicubane core can be recognized within the structure. The two cubane moieties share a common vertex represented by a Ni cation (Ni5). Within this subunit, the metal centres are capped by chloride and hydroxide anions, in an asymmetric fashion. One of the cubanes (containing Ni6-Ni8, Figure 3) is associated with one Cl -and three OH -triply bridging anionic ligands, while the second (containing Ni2-Ni4, Figure 3) contains two chlorides and two hydroxide ligands. Six terminal halides are coordinated to six of the seven nickel centres. Two bridging chlorides connect a non N,O chelated metal centre (Ni2) of the Ni7 dicubane with a mononuclear fragment which contains the only doubly chelated Ni(II) cation (Ni1). Multiple intramolecular hydrogen bonds are present between the alcohol functions and neighbouring terminal chlorides. The nearest nickel centre to Ni1 in another molecule is Ni7, at a distance of 8.998(1) Å. This distance is too long to influence significantly the magnetic properties of 3 (see below). To the best of our knowledge, only two examples of dicubane Ni(II) structures sharing a common vertex have been reported to date. [7] We first hypothesized that 3 could be an intermediate in the formation of 2 from the reaction of 1 with NaOH. Indeed 3 displays an OH/Cl ratio (5/11) lower than that observed in 2. Equations ( 1) and ( 2) could account for this reaction sequence. However, the reaction of 1 with a stoichiometric amount or an excess of NaOH pellets (eq 1) did not lead to 2 and only 3 was observed when the latter reaction was monitored by IR spectroscopy. This suggested that when 2 is formed, it reacts rapidly with unreacted 1 to give 3 (Scheme 3). The reaction 3/4 1 + 5/4 2 → 3 would thus be faster than the reaction 1 + 4 NaOH → 2, preventing detection of the latter cluster in the course of the reaction since it is consumed almost as soon as formed. However, cluster 2 could be obtained in ca. 30% yield when highly dispersed NaOH, prepared in situ by hydrolysis of NaH, was reacted with 1. [5] We previously found that 3 could be obtained when NaOH pellets were used in excess (4.2 equiv. based on Ni), owing to the lower reactivity of NaOH in this form and consequently slower Cl/OH rate of substitution (see above). This favours the reaction 1 + 2 → 3 vs. the reaction 1 + 4 NaOH → 2. We have now found that the direct reaction of 1 with 2 gave 3 in better yield (Scheme 3). Whereas the condensation between two identical nickel(II) tetranuclear cubanes has been previously observed, [7b] the reaction between 1 and 2 represents, to the best of our knowledge, the first example of condensation between two chemically different cubanes. Such a synthetic approach should have a high potential in view of its versatility.

In the course of reactions leading to the formation of complex 3, we isolated once a new complex [Ni7(3-Cl)2(3-OH)6Cl4(H2O)2(HL)6]Cl2 (4). Its structure was determined by single crystal X-ray diffraction (Figures 4 and5 and Table 1). 3), Ni2-O1 2.030(4), Ni2-O5 2.071(4), Ni2-N2 2.073( 5), Ni2-O2 2.075(4), Ni2-Cl3 2.430(2), Ni2-Cl1 2.499(2), Ni3-O3 1.996(4), Ni3-O6 2.078(4), Ni3-O2 2.089(4), Ni3-N3 2.094(4), Ni3-O7 2.106(4), Ni3-Cl1 2.457(2). For selected bond angles, see Table S1 in ESI.

In the centrosymmetric heptanuclear structure of 4, each Ni(II) metal centre is hexacoordinated and in a slightly distorted octahedral coordination environment.

Static Magnetic Properties.

Variable-temperature dc magnetic susceptibility (M) data were measured on microcrystalline powders of complexes 3 and 4 between 2 and 290 K in a 10 kOe magnetic field (Figure 6). Both complexes show Curie-Weiss behaviour, above 20 K for complex 4 (see Figure S1 in ESI), above 100 K for complex 3 (see Figure S2 in ESI). We find resulting Curie constants of 11.38(2) cm 3 Kmol -1 and 8.90(4) cm 3 Kmol -1 , respectively. This is in good agreement with 8 and 7 uncorrelated Ni(II) cations, with average Landé factors g of 2.38 and 2.25, respectively, as expected for d 8 Ni(II) ions with a 3 A2 ground state because of spin-orbit coupling. The Weiss temperatures of +13.1(1) K and +6.8(1) K show the likely occurrence of ferromagnetic interactions. Accordingly, upon cooling, the value of MT continuously increases and reaches a maximum of 31.1 cm 3 Kmol -1 at 3.0 K and 14.9 cm 3 Kmol -1 at 3.5 K, respectively. For complex 3, the former value supports either a S = 7 ground state with a g = 2.11 value (31.2 cm 3 Kmol -1 ), or even a S = 8 ground state with significant anisotropy. For complex 4, the value of 14.9 cm 3 Kmol -1 is compatible with the spin-only value expected for a S = 5 ground state with a g = 2.0 value (15.0 cm 3 Kmol -1 ). For complex 4, MT decreases slightly at lower temperatures to reach 13.4 cm 3 Kmol -1 at 1.8 K, showing that there are either intra-or intermolecular antiferromagnetic interactions and/or single-ion/cluster magnetic anisotropy. For both complexes an unambiguous determination of the ground state S value proved to be quite difficult, as was recently observed in a similar Ni7 cluster, [7b] and contrary to what we observed for complex 2. [5] Starting with the more symmetrical complex 4, the magnetic topology is nevertheless not so trivial, mixing NiO6, NiO4NCl, cis-and trans-NiO3NCl2 chromophores, with magnetic interactions mediated through 3-OH and 3-Cl bridges. For 3-OH bridges, magneto-structural correlations are known with ferromagnetic (antiferromagnetic) interactions for Ni-O-Ni angles close to 90° (above about 100°), [8] which is evidenced by an abundant literature on oxygen-bridged polynuclear nickel complexes. [9] For 3-Cl bridges data are much more scarce and not very conclusive. [8b,10] Indeed, even for simpler Ni2(2-Cl)2 moieties, no clear magneto-structural correlations could be extracted, though magnetic interactions are usually small (<<10 cm -1 ), in agreement with long Ni-Ni distances. [11] The crystal packing of compound 42CH2Cl2 shows that the complexes are isolated, with no significant intermolecular contacts and the shortest distances to neighbouring molecules are above 1 nm. In the light of these considerations, we can expect (see Table 1 and Table S1 in ESI) ferromagnetic interactions between Ni02 and Ni1-Ni3, and, neglecting the chlorine-mediated interactions, antiferromagnetic interactions between external Ni(II) cations. We simplified the spin topology considering two equilateral N1-Ni2-Ni3 triangles equivalent through the inversion centre with isotropic spins, with the following spin Hamiltonian where = 1 to 6 and is are spin operators corresponding to the 6 peripheral Ni(II) ions and the total spin, respectively. Following the Kambe approach, [12] we derived an analytical solution [13] in the form of a polynomial expression of variables (see Figure S3 in ESI). Fitting the MT experimental values showed the J, J' (and g) values to be heavily correlated, with very close ground states with S = 1, favoured by antiferromagnetic interactions between external Ni(II), and S = 7, favoured by ferromagnetic interactions between the apical Ni02 and external Ni(II) cations. We checked the obtained values with softwares that directly diagonalizes the temperature-and field-dependent Hamiltonian. [14] , which showed that there is likely close competition between antagonist interactions in compound 4, this without taking into account anisotropic effects which are usually found in Ni(II) complexes. The best fitting set of values yields ferromagnetic interactions between external Ni(II) cations, and a very weak antiferromagnetic interaction between apical Ni02 and external Ni(II) cations (J = +2.8(1) cm -1 , J' = -0.016(1) cm -1 , and g = 2.233(7), see Figure S3 in ESI), in apparent contradiction with conclusions drawn from magnetostructural correlations on Ni-O-Ni angles. This model supports thus a S = 5 ground state for the complex, in agreement with isothermal magnetization curves (see Figure S4 in ESI), the small antiferromagnetic interactions with the central Ni(II) inducing low-lying excited states of higher multiplicity. The combination of those low-lying excited states with magnetic anisotropy induces a dependence of the maximum value observed for the MT product on the magnetic field applied for the measurement, making the estimation of the ground spin state more difficult. The small magnitude of the ferromagnetic coupling constant obtained between peripheral ions is striking when considering the higher value obtained in complex 2 (J = 11.5 cm -1 ), [5] in which the Ni-Cl-Ni angles are similar (on average 85.5° vs. 83.6° for complex 4), but the Ni-Ni distances are much longer (on average 3.5783(3) Å vs. 3.351(1) Å for complex 4). The intermediate Ni-Ni distance in 4 supports competition between reinforced chlorine-mediated ferromagnetic interactions and weakened hydroxo-mediated antiferromagnetic interactions, resulting in an overall weak ferromagnetic interaction. Simultaneous simulations of susceptibility and isothermal magnetization curves proved to be inconclusive, certainly due to the smallness of magnetic interactions combined with the presence of magnetic anisotropy.

For complex 3, almost all Ni(II) coordination spheres differ: NiO5Cl (Ni5), cis-and trans-NiO3NCl2 (Ni6-Ni8), cis-NiO2Cl3 (Ni2), cis-NiO2Cl4 (Ni3-Ni4) and trans-NiO2N2Cl2 (Ni1), with no less than 7 different magnetic interactions mediated through 3-OH, 3-Cl and 2-Cl bridges, the latter corresponding to the longest Ni-Ni distances (see Table 1). No convenient symmetry allowed us to try for complex 3 with its 3 8 = 6561 microstates the modelling we used for complex 4. Again molecules of 3 are isolated from each other by the organic ligands, and the crystal packing reveals no strong or outstanding intermolecular contacts, the shortest distances are all above 0.9 nm. The S = 7 or S = 8 ground state evidenced by the MT product maximum at low temperatures is surprising given the numerous Ni-O-Ni angles close to or above 100° (see Table S1 in ESI), which would rather be associated with antiferromagnetic interactions. Clearly and similarly to what was evidenced for complex 4, the increase in Ni-Ni distances caused by the chlorine bridges weakens significantly the antiferromagnetic interactions mediated by the hydroxo bridges. The reduced isothermal magnetization curves show a less pronounced nesting behaviour than for compound 4 (see Figure S5 in ESI). The measurement at 1.8 K reaches a value of 18.3 B without saturating, a value that precludes the S = 7 ground state, and rather supports a S = 8 ground state that is reached only at the lowest temperatures. Indeed, simulations with the simple spin Hamiltonian show a definite change in behaviour below 5 K, where magnetizations at 3 and 1.8 K are fairly well reproduced for a S = 8 ground state with g = 2.3 and D = -0.25 cm -1 , while for the data at 5 K, the simulated curve lies clearly above the experimental one. Disagreements may be due to an expected anisotropy in g, and the possible occurrence of a rhombic component of Zero-Field Splitting, which are impossible to unambiguously determine on a measurement performed on powder. Moreover, as seen above, this S = 8 ground state is clearly not well isolated from lower spin excited states. This was confirmed by High-Field EPR measurements which did not show features attributable to such a S = 8 spin system down to 5 K. Usually this technique allows a precise determination of Zero-Field Splitting parameters, [16] as performed previously with a related Ni7 cluster. [7b] The few features seen are rather in agreement with contributions from either isolated ions or the coupling of only a few spins within the molecule in the temperature range of the EPR measurements (5 to 30 K).

Dynamic Magnetic Properties.

Ni(II) shows consistently high magnetic anisotropy, as demonstrated by previous examples of Ni-based Single Molecule Magnets, as was evidenced previously for complex 2, [5] and more specifically a Ni7 cluster that shares some common features with compound 4: [7b] centrosymmetry at the apical Ni, low-lying excited states as demonstrated by High Field EPR spectroscopy. Since centrosymmetry at the apical Ni(II) insures that the single-ion magnetic anisotropy contributions of the external cations are mathematically cancelled, leaving only the apical Ni(II) contribution to the overall cluster magnetic anisotropy, it is not surprising that this Ni7 cluster showed magnetic hysteresis, proving thus Single-Molecule Magnet behaviour. [7b] Consequently, we checked whether some slow relaxation of the magnetization could be detected in the out-ofphase magnetic susceptibility " for both complexes. At 1.5 K, from 0.5 to 70 kHz, the ac magnetic susceptibility of compound 4 shows no significant deviation from zero of its dispersion component " (see Figure S6 in the ESI). The 'T product (see Figure S7 in ESI) is stable at 1.5 K at the 15.0 cm 3 Kmol -1 value observed previously in the dc measurement (see Figure 6), supporting thus the S = 5 ground state, and that the decrease observed previously for the MT product below 3.5 K is due to anisotropy effects. Applying a small dc magnetic field shifts energy levels and by lifting energy degeneracy suppresses quantum tunneling of the magnetization between ±MS states, [15] allowing thus to observe slow relaxation if present. The same ac measurement performed under 500 Oe shows a definite increase of " above 30 kHz below 2 K up to some 10% of " (see Figure S8 in ESI). Though maxima were not observed to evidence whether this behaviour is frequency-dependent or not, this result suggests that some slow relaxation behaviour may be present at lower temperatures. For complex 3, the ac susceptibility measurement (0.5-70 kHz, 1.5-3.9 K, see Figure 7) shows the occurrence of a signal in dispersion below 3.3 K, with frequency-dependent maxima. The 'T product plateau at 32 cm 3 Kmol -1 above 3.3 K (see Figure S9 in ESI) is in good agreement with the dc measurement. The '' maxima values are about 25% of the in-phase ', resulting in flattened Argand plots (see Figure S10 in ESI). Nevertheless, these Argand plots could be satisfactorily fitted below 3 K with an extended Debye model considering a distribution of relaxation times, [17] with , being the width of the temperaturedependent  distribution, going from 0.2 at 3 K to 0.5 at 1.5 K (see equations and results in Figure S10 The Arrhenius plot of ln (T) vs. 1/T shows the expected linear dependence (Figure 8), with 0 = 8.3(27)10 -9 s -1 and  = 9.4( 5) cm -1 . The 0 value is in line with those usually observed for SMMs (10 -10 -10 -8 ), while the energy barrier  is lower than the |D|S 2 value of 16 cm -1 for the supposed ground state, as is usually the case for SMMs. [17b] Moreover, complex 3 as we saw has a ground state that is not clearly isolated from low-lying excited states of different spin multiplicity. Application of a small 500 Oe magnetic field results in a shift to higher temperature of the out-of-phase signals " (see Figures S11 and S12 in ESI). The Arrhenius plot (Figure 8) yields 0 = 3.3(7)  10 -9 s -1 and  = 11.7(3) cm -1 . As expected, the dc magnetic field increases the energy barrier, but also impacts 0 which decreases. The value of  varies similarly to the zero-field situation from 0.1 at 3.5 K to 0.5 at 1.7 K.

Conclusions

We described herein the synthesis of high nuclearity Ni(II) complexes bearing pyridine alcohol ligands. Depending on the physical properties of the NaOH used with the original intention to deprotonate the ligand, different products were obtained. Indeed, as previously described, the reaction of the pseudocubane [Ni(3-Cl)(Cl)(HL)]4 (1) with highly dispersed NaOH led to the formation of [Ni(3-OH)(Cl)(HL)]4 (2), whereas the use of NaOH pellets afforded [Ni8(3-Cl)3(3-OH)5(-Cl)2Cl6(HL)7] (3). These differences were explained by the lower reactivity of the latter (less active surface). In both cases, the ligand remained protonated but the hydroxide ion acted as a nucleophile and substituted chloride ligands. Subtle changes in the properties of the reactants may thus dramatically influence the nature of the products obtained. The octanuclear complex 3 was more conveniently prepared by the condensation of the two chemically different cubanes 1 and 2. This novel synthetic approach to higher nuclearity clusters should have a promising potential. During the course of the syntheses which led to the isolation of 3, a new heptanuclear complex, [Ni7(3-Cl)2(3-OH)6Cl4(H2O)2(HL)6]Cl2 (4), was isolated once. The magnetic properties of 3 and 4 were investigated and highlight the unexpected ferromagnetic coupling induced by the chlorido bridges, which by lengthening the Ni-Ni distances weaken significantly the competing antiferromagnetic interaction mediated by the capping hydroxyl ligands. Clear evidence of Single-Molecule Magnet behaviour could be found for complex 3, with field-dependent slow relaxation of the magnetization as evidenced by ac susceptometry, even though this complex shows evidence of low-lying excited spin states. Such a behaviour can only be postulated for complex 4 at lower temperatures.

Experimental Section

Ligand HL and complexes 1 and 2 were prepared following published procedures. [5,6] All reactions were performed under dry Argon using standard Schlenk techniques. All solvent used were freshly distilled before use. The FTIR spectra were measured between 500 and 4000 cm -1 with a 1 cm -1 resolution on a diamond GladiATR mounted on a Shimadzu IRAffinity1 FTIR spectrometer. Unless otherwise stated, all products were commercially available and used as received. NaOH pellets were dried under vacuum at 100 °C during 8 h before use.

Preparation of [Ni8(3-Cl)3(2-Cl)2(Cl)6(3-OH)5(HL)7] (3)

a) NaOH (ca. 2 mm spheres, 39.7 mg, 0.99 mmol) was added to a yellow suspension of 1 (268 mg, 0.239 mmol) in dichloromethane (15 mL). The reaction mixture was stirred in an ultrasound bath to activate the NaOH pellets. The slurry was stirred for 9 days at room temperature. After 2 days the reaction mixture colour turned from yellow to green. After 9 days, most of the product was soluble in dichloromethane. A pale green precipitate was separated by filtration. The solution was layered with Et2O, giving green crystals of 3. Yield (based on nickel): 0.050 g, 17%.

b) Solid 1 (200 mg, 0.18 mmol) was added to a solution of 2 (250 mg, 0.24 mmol) in CH2Cl2 (50 mL). The reaction mixture was stirred for 6 h. The solution was filtered and the filtrate was layered with Et2O, which afforded green crystals of 3. Yield (based on Ni): 164 mg, 42%. FTIR (, cm -1 , Figure S13 in ESI): 598.9, 735.4, 758.5, 791.8, 838.1, 879.1, 916.7, 1025.7, 1066.6, 1133.2, 1160.7, 1182.9, 1211.3, 1260.0, 1318.8, 1347.3, 1373.3, 1390.7, 1426.4, 1449.0, 1486.2, 1573.4, 1609.6, 1633.2, 2817.5, 2887.5, 2979.1, 3124.7, 3213.0, 3428.5, 3300.7. Elemental analysis calcd (%) for C63H96Cl11N7Ni8O12: C 37. 78, H 4.83, N 4.90; found: C 37.27, H 5.14, N 4.61.

Characterization of [Ni7(3-Cl)2(3-OH)6Cl4(H2O)2(HL)6]Cl2 (4)

Few single crystals and a microcrystalline powder of complex 4 were obtained during crystallization attempts of 3. FTIR (, cm -1 , Figure S14 in ESI): 608.6, 616.3, 633.6, 648.1, 736.8, 760.0, 791.8, 839.1, 879.6, 917.2, 985.7, 1027.1, 1067.6, 1099.5, 1134.2, 1159.3, 1182.4, 1212.3, 1260.5, 1318.4, 1347.3, 1373.4, 1391.7, 1426.4, 1449.6, 1483.3, 1574.0, 1609.7, 1632.8, 2821.0, 2882.7, 2977.3, 3112.3, 3216.4, 3299.4, 3365.9, 3437.3, 3515.4

Figure 2 .

 2 Figure 2. ORTEP of the molecular structure of 3. Thermal ellipsoids are at 40% probability. Hydrogen atoms are omitted for clarity, except those bound to alcohol oxygen atoms. Hydrogen bonds are depicted with red dashed lines. Selected bond distances (Å) : Ni1-O1 2.065(3), Ni1-O2 2.054(3), Ni1-N1 2.097(4), Ni1-N2 2.098(4), Ni1-Cl1 2.474(1), Ni1-Cl2 2.405(1), Ni2-O8 2.034(3), Ni2-O9 2.017(3), Ni2-Cl2 2.443(1), Ni2-Cl1 2.423(1), Ni2-Cl3 2.469(1), Ni2-Cl4 2.463(1), Ni3-O7 2.057(4), Ni3-O9 1.994(3), Ni3-N7 2.081(4), Ni3-Cl4 2.586(1), Ni3-Cl5 2.420(1), Ni3-Cl6 2.431(1), Ni4-O6 2.066(4), Ni4-O8 2.007(3), Ni4-N6 2.080(4), Ni4-Cl4 2.587(1), Ni4-Cl6 2.506(1), Ni4-Cl7 2.370(1), Ni5-O8 2.041(3), Ni5-O9 2.063(3), Ni5-O10 2.034(3), Ni5-O11 2.045(3), Ni5-O12 2.061(3), Ni5-Cl6 2.478(1), Ni6-O3 2.102(4), Ni6-O10 2.030(3), Ni6-O12 2.067(3), Ni6-N3 2.089(4), Ni6-Cl8 2.407(1), Ni6-Cl10 2.463(1), Ni7-O4 2.115(4), Ni7-O10 2.056(3), Ni7-O11 2.054(3), Ni7-N4 2.068(4), Ni7-Cl10 2.500(1), Ni7-Cl11 2.405(1), Ni8-O5 2.086(4), Ni8-O11 2.034(3), Ni8-O12 2.039(3), Ni8-N5 2.081(4), Ni8-Cl9 2.394(1), Ni8-Cl10 2.511(1). For selected bond angles, see TableS1in ESI.

Figure 3 .

 3 Figure 3. Simplified diagram of the solid state structure of 3.

Scheme 3 .Figure 4 .

 34 Scheme 3. Condensation of two chemically-different Ni(II) cubanes leading to the formation of the octanuclear complex 3.

Figure 5 .

 5 Figure 5. Simplified diagram of the solid state structure of 4 in 42CH2Cl2.

Figure 6 .

 6 Figure 6. Plot of the magnetic susceptibility product with temperature vs. T for complex 3 (triangles) and 4 (squares).

  in ESI). The good agreement found with this extended Debye model supports a single relaxation mode, with distributed characteristic times, rather than the superposition of different modes. Compound 3 behaves thus as a Single-Molecule Magnet (SMM), for which the relaxation time should follow the relation: in which D < 0 is the cluster uniaxial zero-field splitting parameter and S the total spin.

Figure 7 .

 7 Figure 7. Temperature and frequency dependence of in-phase (', top) and out-of-phase (", bottom) components of the ac susceptibility of compound 3. Black lines are guides to the eye.

Figure 8 .

 8 Figure 8. Arrhenius plot derived from the Debye fits of ' and " components of the ac susceptibility of compound 3, under zero Oe (triangles) and 500 Oe (circles) magnetic fields, and corresponding linear fits.

  Cl)(HL)]4 (1) (HL = 2-methyl-1-(pyridin-2-yl)propan-2-ol) and [Ni(µ3-OH)(Cl)(HL)]4 (2) offers a new access to higher nuclearity complexes, such as [Ni8(µ3-Cl)3(µ3-OH)5(µ-Cl)2Cl6(HL)7] (3), which displays Single-Molecule Magnet behaviour. Sophie Hameury, Laure Kayser, Roberto Pattacini, Patrick Rosa,* Anne-Laure Barra and Pierre Braunstein*

  Table S1 in ESI.

Entry for the Table of Contents

	.	Elemental	analysis	calcd	(%)	for

Acknowledgements

We thank Dr. Vitor Rosa for experimental contributions and fruitful discussions and Mélanie Boucher for experimental assistance. We thank the CNRS, the Ministère de l'Enseignement Supérieur et de la Recherche (Paris), the Université de Strasbourg and the International Center for Frontier Research in Chemistry, Strasbourg (icFRC, www.icfrc.fr) for financial support. We thank Professors Roberta Sessoli and Andrea Caneschi in LaMM, INSTM/University of Florence, for acces to the Oxford Instruments MAGLAB platform. Keywords: Coordination clusters • Cubanes • Molecular magnetism • Nickel • N,O ligands

.

X-ray data collection and structure refinement

Suitable crystals for the X-ray analysis were obtained as described above. The intensity data were collected at 173(2) K on a Kappa CCD diffractometer [START_REF]Bruker-Nonius Kappa CCD Reference Manual[END_REF] (graphite monochromated Mo-K radiation,  = 0.71073 Å). Crystallographic and experimental details for the structures are summarized in Table 2. The structures were solved by direct methods (SHELXS-97) and refined by full-matrix least-squares procedures (based on F 2 , SHELXL-97) [START_REF] Sheldrick | [END_REF] with anisotropic displacement parameters for all the non-hydrogen atoms. CCDC 997512-997513 contains the supplementary crystallographic data for this paper that can be obtained free of charge from the Cambridge Crystallographic Data Center via www.ccdc.cam.ac.uk/data_request/cif.

SQUID magnetometry and ac susceptometry

Dc magnetic data were recorded using a MPMS-7XL Quantum Design SQUID magnetometer. Magnetic susceptibilities were measured using a RSO probe between 1.8 and 300 K under 1 or 10 kOe applied field, on polycrystalline samples enclosed in heat-sealed 30 m thick polyethylene bags accurately weighed with a Mettler MX5 microbalance.

Magnetizations were measured at 1.8, 3, 5 and 8 K in the 0-70 kOe range. Data were corrected using multipole expansion and a combination of a diamagnetic and a Curie tail for the polyethylene bag (see details in ref [20]). The diamagnetic contributions were approximated with -1170.10 -6 cm 3 mol -1 (3) and -1040.10 -6 cm 3 mol -1 (4), respectively. Temperature-independent paramagnetism, as usually found for octahedral Ni(II) d 8 compounds, [START_REF]Theoretical foundations of molecular magnetism[END_REF] was corrected at +340.10 -6 (3) and +300.10 -6 cm 3 mol -1 (4) per nickel atom. Data for ac susceptibility were recorded on the MPMS-7XL in the lowest frequency range (0.1-1.5 kHz, 1.8-15 K), and for higher frequency on an Oxford Instruments MAGLAB platform equipped with a homemade probe with a SRS830 Lock-in as both waveform source and detection (0.5-90 kHz, 1.5-4 K).


Anouar Rkhami

Tran Anh Quang Pham

Yassine Hadjadj-Aoul

Abdelkader Outtagarts

Gerardo Rubino

On the Use of Graph Neural Networks for Virtual Network Embedding

Keywords: Virtual Network Embedding, Deep Reinforcement Learning, Graph Neural Networks, 5G

Resource allocation of 5G network slices is one of the most important challenges for network operators. It can be formulated using the Virtual Network Embedding (VNE) problem, which was and remains an active field of studies, also known because of its NP-hardness. Owing to its complexity, several heuristics, meta-heuristics and Deep Learning-based solutions have been proposed. However, these solutions are inefficient either due to their slowness or to not taking into account the structure of data which results in an inefficient exploration of the solutions space. To overcome these issues, in this work we unveil the potential of Graph Convolutional Neural (GCN) networks and Deep Reinforcement Learning techniques in solving the VNE problem. The key point of our approach is modeling of the VNE problem as an episodic Markov Decision Process which is solved in a Reinforcement Learning fashion using a GCNbased neural architecture. The simulation results highlight the efficiency of our approach through an increased performance over time, while outperforming state-of-art solutions in terms of the services' acceptance ratio.

I. INTRODUCTION

With the recent emergence of the fifth generation of mobile networks (5G) and the diversification of the services to be supported, Infrastructure Providers (InPs) are compelled to adapt to these new changes, requiring in particular the deployment of constrained services almost instantaneously [START_REF]Network Functions Virtualisation (NFV): Architectural framework[END_REF].

As hardware solutions are too rigid and unsuitable for real-time deployment, InPs are moving towards new scalable software solutions available with the advent of the Network Function Virtualization (NFV) paradigm [START_REF] Yi | A comprehensive survey of Network Function Virtualization[END_REF]. NFV enables the co-existence of several virtual network requests (VNRs) on top of the same substrate network (SN). The problem of embedding a VNR into the SN, which is known as the Virtual Network Embedding (VNE) problem, is considered as the main resource allocation challenge in network virtualization. Indeed, it has been qualified as a very high computational complexity problem [START_REF] Gupta | A Scalable Approach for Service Chain Mapping With Multiple SC Instances in a Wide-Area Network[END_REF].

One of the most important challenges in VNE is the placement of network services efficiently and automatically, which would make it possible to converge to a fully automated network, also referred to as a zero-touch network [START_REF] Koley | The zero touch network[END_REF].

Several solutions have already been proposed in the literature to address these types of problems [START_REF] Afolabi | Network Slicing and Softwarization: A Survey on Principles, Enabling Technologies, and Solutions[END_REF]. Most existing techniques solve them using classical combinatorial optimization methods. These tools, although effective for very small network instances, reach their limits even for fairly small-scale networks. Moreover, it is difficult to apply them in a real context because of the lack of accurate models [START_REF] Xu | Experience-driven Networking: A Deep Reinforcement Learning based Approach[END_REF] for several metrics, such as latency and loss, which actually result from the placement. Other approaches propose heuristics that can be executed in near real time, at the cost of a very partial exploration of the space of solutions [START_REF] Mechtri | A Scalable Algorithm for the Placement of Service Function Chains[END_REF]. Metaheuristics, and in particular evolutionary algorithms, have also been proposed to attack these difficulties [START_REF] Khebbache | A multi-objective nondominated sorting genetic algorithm for VNF chains placement[END_REF]. These techniques, although efficient, can present some convergence issues and therefore be slow. On the other hand, with these techniques you can solve problems without learning how to solve them, so you don't benefit from past experiences. More recently, techniques based on Deep Reinforcement Learning (DRL) have been used [START_REF] Haeri | Virtual network embedding via Monte Carlo tree search[END_REF]. These techniques show some effectiveness. However, in some cases they rely on a manual feature extraction or they use neural networks which are not suitable to graph-structured data. Moreover, they put particular assumptions on the shape of the substrate and virtual networks. Besides, these techniques are very slow, which could be an obstacle to their adoption to control real networks.

In this work, we propose an end-to-end solution based on DRL and Graph Convolutional Neural networks (GCNs) [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] to train an agent to solve the VNE problem.

The contributions of this paper can be summarized as follows:

• formalization of the VNE problem as an MDP,

• features' extraction automation using GCN-based neural architecture, • proposition of a more efficient placement strategy. The rest of this paper is structured as follows. The formulation of VNR embedding problem is presented in Sec. III. The proposed framework is presented in Sec. IV. The comparison of the proposed framework and other approaches is presented in Sec. V. Finally, we discuss the advantages and disadvantages of the proposed framework in Sec. VI.

II. BACKGROUND

Graph Neural Networks (GNNs), are a type of neural networks dedicated to graph structured data. They were in- troduced in [START_REF] Scarselli | The graph neural network model[END_REF] and several variants of it were proposed [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF], [START_REF] Veličković | Graph attention networks[END_REF], [START_REF] You | Graphrnn: Generating realistic graphs with deep auto-regressive models[END_REF]. The most known one is the Graph Convolutional Networks (GCN), which generalizes the convolution operation from euclidean data (images and grid shaped data) to noneuclidean data (graphs). The key objective of GCNs is to learn a function that generates for each node in a graph a vector representation by aggregating its own features and its neighbors features.

Let G = (V, E) be a graph, where V is the set of its nodes and E the set of edges. We assume that each node n ∈ V is characterized by an input feature vector in n . The graph convolution operation takes as input the feature vector in n and outputs a new vector out n according to the following equation:

out n = f m∈N (n) 1 √ d m d n in m W l (1)
where N (n) represents the set composed of node n and its neighbours, d m is the degree of node m plus one, W l is the weight matrix of the l th GCN layer, and f is an activation function.

III. PROBLEM DEFINITION

In this section, we describe the model of the SN, the VNR and their resources. Then, we define formally the VNE problem. The notations used in this section are shown in Table I.

A. Networks and resources Models

The susbtrate network is defined by an undirected graph G s = (N s , L s ), where N s represents the set of substrate nodes and L s the set of substrate links. The number of the substrate nodes and substrate links are |N s | and |L s | respectively. Each substrate node n s has a computing capacity c n s and each substrate link l s has a bandwidth capacity b l s . Similarly, the VNR is represented by a directed graph

G v = (N v , L v ),

B. VNE Problem

The VNE problem can be divided into two stages: the virtual node mapping VNM and the virtual link mapping VLM. The former is defined as a function f VNM : N v → N s that maps a virtual node to a substrate node. It must be injective, that is, two VNFs of the same VNR can not be hosted by the same substrate network. On the other hand, VLM can be modeled as a function f VLM : L v → P s that maps a virtual link to a physical path (set of physical links).

At the node mapping level, a virtual node is successfully deployed if the substrate node that hosts it has sufficient CPU resources, i.e. if

c n v ≤ c fVNM(n v ) , ∀n v ∈ N v . (2)
At the link mapping stage, a VL is successfully deployed if its virtual nodes are deployed and if each physical link belonging to the physical path on which it is mapped has sufficient bandwidth, i.e, if

b l v ≤ min l s ∈f vlm(l v ) b f VLM(l v )) , ∀l v ∈ L v . (3)
In this work, the aim is to maximize the number of accepted VNRs. A VNR is accepted if and only if the placement of its VNFs and VLs satisfies, respectively, the constraints ( 2) and (3).

IV. PROPOSED METHOD

We formulate the VNE problem as a Reinforcement Learning problem, in which an agent finds iteratively a sub-optimal placement of a VNR on top of the SN. In what follows we present an overview of the Markov Decision Process (MDP) formulation of the VNE problem, the neural architecture of the RL agent we used to solve it and its training process.

A. Deep Reinforcement Learning overview

Reinforcement Learning (RL) is the area of machine learning that deals with sequential decision-making tasks. It is formulated as the interaction of an agent with an environment to optimize a given total amount of return. This interaction can be modelled as a Markov Decision Process (MDP) M = (S, A, P, R) with a state space S, an action space A, transition dynamics P and a reward function R. At each time step t, the agent observes a state s t and performs an action a t , then it receives a reward r t as well as the next state of the environment s t+1 . If the agent receives a special state called terminal, the interaction stops. The objective of the agent is to maximize the expected cumulative discounted return:

R t = E[ t γ t r t ], γ ∈ [0, 1]. (4)
The parameter γ is the discount factor, by which we ensure the convergence of the cumulative return in the infinite horizon setups. The behavior of the agent is defined by a policy π which maps a state s to a distribution over the actions A.

Each state s is associated with a value function V π (s) that maps it to a scalar corresponding to the expected reward the agent will receive in this state and acting according to the policy π:

V π (s) = E π [R t |s t = s]. (5)
In traditional RL, both the policy and the value functions are modeled by tables. However, in most real scenarios, the state and action spaces are large, therefore instead of the tabular representation, the policy and the value functions are approximated by Deep Neural Networks (DNNs). DNNs are proposed for two main reasons: to overcome the slowness of the tabular-based solutions and to extract automatically useful features from the state representation. The combination of RL with DNNs is known as Deep Reinforcement Learning (DRL). To learn the policy function, several DRL algorithms were introduced. They can be divided into three main categories: value-based methods, policy-based methods and actor-critic methods. The policy-based solutions parameterize the policy function directly by a neural architecture and try to find the optimal policy using gradient ascent, while the value-based solutions estimate the value of each state using a DNN and then infer the optimal policy using this estimation. Finally, the actor-critic methods are a combination of the above methods: the agent learns both functions to find the optimal policy. One of the most widely used actor-critic algorithms is A2C [START_REF] Mnih | Asynchronous methods for deep reinforcement learning[END_REF].

B. VNE as MDP

In this section, we describe our MDP formulation of the VNE problem and its main components, namely: the state representation, the actions, the rewards and the transition dynamics.

State Representation: The state s is represented by the graphs (G s , G v ). Each node n s ∈ N s has two features: [START_REF]Network Functions Virtualisation (NFV): Architectural framework[END_REF] available cpu c n s , (2) sum of bandwidth defined as the total available bandwidth of links to which belongs n s . On the other hand each VNF n v ∈ N v is characterized by four features: (1) cpu required by the VNF, (2) total bandwidth required by the links to which belongs the VNF, (3) a flag indicating whether this node is the current node to place, (4) a flag indicating whether the VNF was placed. At the initial state s 0 , the current flag of the first VNF is assigned to 1 and the others to -1, and the placement flag is set to -1 to all VNFs. Action Space: At the time step t in the MDP, the agent has to select a substrate node to host the positive-flagged VNF (+1). Note that the VNFs of the same VNR should not be placed at the same substrate node. Hence the action space at time step t is defined as follows:

A t = N s \S t ( 6
)
where S t represents the set of substrate nodes selected before the time step t. Initially, the action space A 0 is equal to N s . State Transition Dynamics: At time step t, the agent observes the state s t and selects an action a t = n s to place the current VNF n v . The MDP then transits to a new state s t+1 in which the node features of the two graphs experience the following updates:

• If the selected substrate node has sufficient CPU resource, the VNF n v will be placed at the substrate node n s . Then, the CPU resource of n s is updated by c n s := c n s -c n v

• After each successful VNF placement, the VLs related to the VNF and having both head and tail VNFs successfully deployed are considered. The shortest path l s ∈ P s between the head VNF and the tail VNF is computed. If the bandwidth required by this VL is less than the one offered by all physical links belonging to the obtained physical path, then the available bandwidth of each physical link in this physical path is updated as follows:

b l s = b l s -b l v , ∀l s ∈ P s . (7)
• If the previous conditions are satisfied, the placement flag of the current node is set to 1, and another non visited VNF is chosen as the current node. The placement process of the VNR will at most ends in |N v | steps. Reward design: The objective is to maximize the number of accepted requests. To guide the agent to this objective we introduce intermediate and final rewards. When a VNF is placed successfully, it receives an intermediate reward of 0. Final rewards quantify the quality of the final decision made by the agent. Thus, the agent obtains a unit of reward when it successfully places a VNR. Otherwise, it gets a reward of -1 if the VNR is not placed successfully, i.e. if there exists a resource violation at some step in the process.

C. The agent architecture

To solve the MDP defined above, we adopted the actorcritic framework [START_REF] Konda | Actor-critic algorithms[END_REF]. The DRL agent has to approximate the policy and the state value functions. To implement the actor-critic framewrok, we can either use two distinct neural network architectures or one neural network architecture with two heads, one representing the policy function and the other one the value function. We adopt the last approach, because it lets the actor and critic share the same low level features and use them in different ways.

The two functions are parameterized by neural networks. Their inputs must be then a real-valued vector. However, as we stated earlier the state is represented by two graphs (SN and VNR), hence we must encode the graph-structured information of the state into a real-valued vector.

To achieve this, we proposed an end-to-end architecture represented in Fig. 1. It takes the state of the environment as an input and encodes it to a real valued vector which is fed to the two heads described above. Finally, the output corresponds to the prediction of the policy and the value functions. The key idea is that learning a good representation of the state will led to a fast and accurate optimal policy prediction.

Y. Bai [START_REF] Bai | Simgnn: A neural network approach to fast graph similarity computation[END_REF] proposed a GCN-based architecture to solve the graph similarity computation task using supervised learning. We adapt their architecture to our problem for the state encoding steps. The encoding of the state into a real valued vector involves three main steps: each node of the two graphs G s and G v , we use GCNs to aggregate neighborhood information of each node (Eq. ( 1)). After multiple layers of GCNs, we obtain two matrices U s ∈ R |N s |×D and U v ∈ R |N v |×D that represent, respectively, the node-level encoding of SN and VNR. Each row u n represents the encoding of node n and D is an hyperparemeter representing the dimension of u n . Note that we used two different GCNs modules to encode SN and VNR.

• Graph level encoding using attention layers. After getting the node level encoding matrices U s and U v , we can directly use them to encode the graph structure. However, this can be computationally expensive. To overcome this issue and make the representation of the graph independent of its size, we can perform a weighted sum of node representations. The choice of nodes that get higher weights depends on the current node we want to place. Thus, we use an attention mechanism to learn these weights instead of considering fixed ones. For each graph, we first compute a graph context ct, then we compute the attention weight for each node in the graph.

For the VNR we define the context ct v as the node representation of the current VNF followed by a nonlinear transformation, ct v = g v (u c ×W c ), where u c ∈ R D is the node representation of the current VNF, W c ∈ R D×D is a learnable weight matrix and g v is an activation function.

Based on ct v we compute one attention weight a n v for each VNF n v as the inner product between the context ct v and its encoding u n v : a n v = u T nv ct v . Finally the VNR encoding h v is given by

h v = n v ∈N v a n v u n v . (8)
Regarding the SN we define its context ct s by a simple average of nodes encoding followed by a non linear transformation ct s = g s ((

N -1 |N s | n=1 u n )W s ),
where W s is a learnable weight matrix and u n is the encoding of the n th susbtrate node. Similarly we define the encoding of the SN as a weighted sum of the substrate nodes encodings. The weight of each node equals the inner product between its encoding and the context,

h s = n s ∈N s a n s u n s . (9)
• State level encoding using neural tensor networks (NTN). The previous stages let the agent learn how to model the two graphs of the state. The last step consists in learning how to model their relation. Following [START_REF] Socher | Reasoning with neural tensor networks for knowledge base completion[END_REF] and based on the vector representation of SN and VNR, we use Neural Tensor Networks (NTN) to achieve this:

g(h s , h v ) = f 3 (h T v W [1:K] 3 h v + V [ hs hv ] + b 3 ), (10)
where W

[1:K] 3
is a weight tensor, [] denotes the concatenation operation, V is a weight vector, b 3 is a bias, and f 3 is an activation function; K is a hyperparameter.

D. Training process

As stated before, we adopt the actor-critic setup. To update the parameters of the architecture we presented above, we used the Advantage Actor Critic (A2C) algorithm.

V. SIMULATION RESULTS

In this section, we present the simulation setup as well as an analysis of the obtained results.

A. Simulation Setup

To evaluate the performance of the proposed approach, we consider the following setup:

• The substrate network is modeled by the network topology of BtEurope1 . It contains 24 nodes and 37 links.

The capacity of the substrate nodes and links is drawn uniformly from the interval [50, 100]. • To generate the virtual requests, we used the Erdős-Rényi model [START_REF] Erdős | On random graphs[END_REF]. In this model the generated graph is defined by the number of nodes n and the probability p of creating an edge between the nodes. With, for instance, p = 2 ln n/n, the generated graph is in general connected (more precisely, the probability that it is the case goes to one as n → ∞). The requested resources (CPU and BW) of VNRs are drawn randomly following a uniform distribution from the interval [START_REF]Network Functions Virtualisation (NFV): Architectural framework[END_REF][START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF]. The system operates in an episodic manner; during each episode, VNRs come to the system sequentially: once a VNR is processed, the next one arrives. There are 3000 episodes and during each one 30 VNRs come to the system. At the end of each episode VNRs leave the system. We execute 5 runs with different seeds. • For the model architecture, it is written in Python with the Pytorch library2 , and the DGL library 3 . The neural architecture is constructed with the following hyperparameters. We set the numbers of GCN layers to 3, and we use tanh as the activation function. For the NTN layer we set K to 8. We use then 2 fully connected layers and finally we have two heads, each one modeled by a fully connected layer that represent, respectively, the policy network and the value network. The learning rate is set to 10 -3 and the discount factor γ is set to 0.99. To train the model, the Adam optimizer was used [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF].

B. Impact of the hidden units

The number of hidden units used in each layer of the agent's neural architecture impacts both the quality of the solution obtained and the computational cost to get it.

In this section we study the effect of the number of hidden units on the performance of the DRL agent in order to find a trade-off between these two metrics. We consider four different agents: agent16, agent32, agent64 and agent128 which have, respectively, 16, 32, 64 and 128 hidden units in all their hidden layers. The results illustrated in Figures 2, 3 and 4 show, respectively, the performance in terms of number of accepted requests, number of VNFs deployed and number of VLs deployed. As we see in these figures, the agents with 64 and 128 hidden units are better than the other ones and their performance is more stable.

Figure 5 shows the required time to finish one episode for each configuration. The more hidden units we have the more time we need to finish the episode. The time required to finish one episode with the agent with 128 hidden units is greater than the one with 64 hidden units, while the performance is quite the same. Consequently we use the agent with 64 hidden units to compare its performance against other solutions.

C. Performance Benchmarking

To benchmark the performance of our DRL-based solution, we compare it against the following approaches:

• Random agent: It preforms randomly to choose the substrate nodes that will host the VNFs. Based on the obtained VNF mapping, the agent uses the shortest path algorithm to find the link mapping of the Vls, • First Fit: It adopts the First-Fit (FF) algorithm to allocate the VNFs and runs the shortest path algorithm to find the physical path mapping for each VL. For each VNF, FF selects the first substrate node with enough resources. Figures 6, 7 and 8 show, respectively, the number of accepted requests as well as the number of deployed VLs and VNFs. With respect to the three metrics, our approach outperforms the random agent and the First Fit procedures. Unlike the latter algorithm, our approach explores efficiently the possible actions until it finds the optimal ones, while First Fit tries to place the VNFs on the first substrate nodes that have enough resources. Therefore it explores just a part of the possible solutions, resulting in a huge consumption of the residual bandwidth of the physical links, and then in an increased probability of rejection of new arriving requests.

VI. CONCLUSIONS

In this paper, we have proposed a methodology to address the VNE issue. The proposed approach is based on the use of a Deep Reinforcement Learning technique. In particular, we have proposed the use of GCNs, which allow us to take into account the structuring of the data of the problem, and showed how to adapt the tool to the considered VNE problem. The paper also exhibits a new strategy of placement, significantly improving the performance of this type of task. The proposed solution has proven to be very effective compared to existing conventional approaches.

  where N v represents the set of VNFs and L v represents the set of virtual links. The numbers of VNFs and VLs are |N v | and |L v | respectively. Each VNF n v has a computing demand denoted as c n v , and each VL l v has a bandwidth demand b l v .

Fig. 1 .

 1 Fig. 1. An overview illustration of the proposed end-to-end agent neural architecture. Given the state of the environment,the agent has to approximate two functions: the policy function π and the value function V .

Fig. 2 .

 2 Fig. 2. Number of accepted requests vs Number of hidden units.

Fig. 3 .

 3 Fig. 3. Number of accepted VNFs vs Number of hidden units.

Fig. 4 .

 4 Fig. 4. Number of accepted VLs vs Number of hidden units.

Fig. 5 .

 5 Fig. 5. Execution time VS Number of hidden units.

Fig. 6 .

 6 Fig. 6. Number of accepted requests.

Fig. 7 .

 7 Fig. 7. Number of deploed VLS.

Fig. 8 .

 8 Fig. 8. Number of deployed VNFs.

TABLE I NOTATIONS

 I

• Node-level encoding using GCNs. This stage transforms each node of the graph into a vector encoding its features and structural properties. Based on the raw features on

http://www.topology-zoo.org/dataset.html

https://pytorch.org/

https://www.dgl.ai/