Encarnación Wei Li

Olaf J Borkiewicz

Henri Groult

Damien Dambournet
email: damien.dambournet@sorbonne-universite.fr

Encarnación Raymundo-Piñero

Synthesis of oxy-hydroxyfluorinated anatase nanoparticles grown on carbon nanotubes

Keywords: sol-gel, fluorolysis, composite materials, carbon nanotubes, inorganic nanocrystals

  L'archive ouverte pluridisciplinaire

Introduction

Sol-gel chemistry relies on the condensation of molecular precursors formed through the reactivity of metal-alkoxide bonds. [START_REF] Livage | Sol-gel chemistry of transition metal oxides[END_REF] In 2003, Kemnitz and colleagues were the first to investigate the reactivity of of metal-alkoxide bonds toward anhydrous HF proposing to name the following reaction M-OR + HF  M-F + R-OH, fluorolysis by analogy with hydrolysis. [START_REF] Kemnitz | Amorphous Metal Fluorides with Extraordinary High Surface Areas[END_REF][START_REF] Rüdiger | The fluorolytic sol-gel route to metal fluorides-a versatile process opening a variety of application fields[END_REF] The reaction system employing titanium alkoxide with aqueous HF treated under solvothermal conditions was initially used to promote the growth of (001)-faceted nanoparticles of anatase TiO 2 . [START_REF] Yang | Anatase TiO2 single crystals with a large percentage of reactive facets[END_REF] In 2015, using a low temperature, we showed that fluoride and hydroxide were partially substituting oxide inducing a charge-compensating titanium vacancy () such as Ti 1-x-y  x+y O 2-4(x+y) F 4x (OH) 4y . [START_REF] Li | High Substitution Rate in TiO2 Anatase Nanoparticles with Cationic Vacancies for Fast Lithium Storage[END_REF] Subsequently, we investigated its formation mechanism revealing a solid-state transformation of a highly defective anatase phase having a hydroxyfluoride composition which further evolves through oxolation reaction into an oxy-hydroxyfluoride phase. [START_REF] Li | Atomic Insights into Nanoparticle Formation of Hydroxyfluorinated Anatase Featuring Titanium Vacancies[END_REF] It further shows that titanium alkoxide precursors can react with HF via a fluorolysis process.

The combination of inorganic compounds with electronic conductors such as carbon nanotubes (CNTs) have been widely employed to prepare composite materials with unique properties in numerous fields such as photocatalysis [START_REF] Dang | Virus-templated self-assembled single-walled carbon nanotubes for highly efficient electron collection in photovoltaic devices[END_REF][START_REF] Liu | Carbon Nanotubes Supported Mesoporous Mesocrystals of Anatase TiO2[END_REF] and energy storage [START_REF] Kavan | Multi-walled carbon nanotubes functionalized by carboxylic groups: Activation of TiO2 (anatase) and phosphate olivines (LiMnPO4; LiFePO4) for electrochemical Li-storage[END_REF]. Numerous methods have been used to prepare CNTs-based nanocomposites. [START_REF] Peng | Carbon nanotube-nanocrystal heterostructures[END_REF] Sol-gel process is effective to grow inorganic nanocrystal onto CNTs. [START_REF] Liu | Carbon Nanotubes Supported Mesoporous Mesocrystals of Anatase TiO2[END_REF] The presence of HF, however, might alter the reactivity and crystal growth of the nanoparticles onto the CNTS. In this work, we investigated the feasibility of growing anatase onto CNTs in a fluorinating medium which could open new opportunities in the fields of energy storage and conversion. [START_REF] Koketsu | Reversible magnesium and aluminium ions insertion in cation-deficient anatase TiO 2[END_REF][START_REF] Scanlon | Band alignment of rutile and anatase TiO 2[END_REF]

Results and discussion

For the synthesis, CNTs and titanium isopropoxide were first mixed together in a solution of isopropanol (see experimental section). Thereafter a solution of aqueous HF diluted in isopropanol was added. The resulting mixed solution was heated treated under solvothermal conditions at 90 °C for 12h.

Influence of the nature of CNTs

Two types of CNTs were used for the synthesis of composite materials that are either pure (denoted raw CNTs thereafter) or functionalized (denoted o-CNTs thereafter) CNTs. The later was obtained by treating raw CNTs with nitric acid to oxidize the surface providing additional active sites and functional groups for favor grafting of the inorganic component.

Influence of the content of o-CNTs

The effect of o-CNTs content on the homogeneity of the composite was studied by using two different content of o-CNTs that are 1 and 10 wt.% (Figure 2). The decrease of the o-CNTs content to 1 wt.% led to full coverage of o-CNTs by nanoparticles. On the other hand, heavy aggregation of nanoparticles was observed which suggest that 1 wt.% o-CNTs does not provide sufficient sites for anchoring Ti precursor, so that a large excess of precursors condensed away from o-CNTs. Although TEM is only a qualitative technique, we observed that increasing the o-CNTs content to 10 wt.% effectively decreases the level of aggregations. Nevertheless, uncovered o-CNTs were observed suggesting that the optimized o-CNTs content lies within 5 to 10 wt.%.

Influence of CNTs on the reactivity of Ti precursor

X-ray diffraction analysis was performed on the composite prepared using different content of o-CNTs (Figure 3a). All the XRD patterns were indexed with a tetragonal cell (space group I41/amd) characteristic of the anatase type structure. The features of the Bragg peaks in terms of position, intensity remain unchanged whatever the content of o-CNTs, indicating that the presence of o-CNTs did not affect the crystallization of anatase network. The absence of the main (002) line of o-CNTs is due to the weak scattering of carbon.

To further confirm the stabilization of oxy-hydroxyfluorinated anatase, we used the pair distribution function (PDF). [START_REF] Billinge | Beyond crystallography: the study of disorder, nanocrystallinity and crystallographically challenged materials with pair distribution functions[END_REF] The PDF of the composite containing 10 wt.% of o-CNTs was successfully fitted using the anatase type structural model (Figure 3b). The refined lattice parameters are a = 3.782(1) Å, c = 9.477(6) Å, which is consistent with previous study. [START_REF] Li | High Substitution Rate in TiO2 Anatase Nanoparticles with Cationic Vacancies for Fast Lithium Storage[END_REF] The scattering PDF domain was 5 nm in agreement with TEM observations. Moreover, the refined titanium occupancy was 0.68(3) which is similar to that obtained for the CNTs free compound. [START_REF] Ma | Lithium Intercalation in Anatase Titanium Vacancies and the Role of Local Anionic Environment[END_REF] The chemical composition assessed by PDF analysis and solid-state 19 F NMR, obtained without carbon additives was indeed determined to be Ti 0.69  0.31 O 0.76 F 0.48 (OH) 0.76 , consistent with the vacancy concentration determined here. This further confirms that the presence of CNTs does not affect the formation of fluorinated anatase but in turns provides a support to grow on it. In order to determine whether CNTs can be fluorinated with aqueous HF, a mixture containing o-CNTs, HF and isopropanol was treated under the same condition as the composite. Under this condition, the O content assessed by X-ray photoelectron spectroscopy (XPS) remains similar before and after the treatment. The absence of reactivity of the o-CNTs vs. fluorination was also confirmed by the absence of F1s peak (Figure 4). To check whether titanium alkoxide can alter the reactivity of the o-CNTs vs. fluorine, we further compared the XPS spectrum of a composite containing 10 wt.% of o-CNTs and that of the o-CNTs treated with HF (Figure 4). The presence of o-CNTs did not influence the binding energies of the Ti 2p and F1s peaks. [START_REF] Ma | Lithium Intercalation in Anatase Titanium Vacancies and the Role of Local Anionic Environment[END_REF] Moreover, the C1s region (Figure 4, inset) was further analyzed to check the formation of any C-F bonds. Table 1 gathers the characteristics of the C 1s peaks. The C1s signals are largely dominated by the sp 2 -hybridized carbon. [START_REF] Lee | A high resolution XPS study of sidewall functionalized MWCNTs by fluorination[END_REF] In the composite, we observed a C 1s peak located at 285.84 eV that we assigned to organic residues coming from the sol-gel precursors. The absence of significant peaks at higher binding energies confirms the absence of C-F bonds in the composite. [START_REF] Tressaud | On the nature of C F bonds in various fluorinated carbon materials: XPS and TEM investigations[END_REF][START_REF] Herraiz | Large-scale synthesis of fluorinated graphene by rapid thermal exfoliation of highly fluorinated graphite[END_REF] 1. Assignments and binding energy of C 1s peaks of the composite and o-CNT treated with HF.

Conclusions

In summary, we synthesized composite materials with CNTs and anatase nanoparticles using a sol-gel process performed in the presence of aqueous HF. Our results showed that CNTs are inactive toward fluorination. On the other hand, titanium isopropoxide reacts via fluoro/hydrolysis reactions to form oxy-hydroxyfluorinated anatase. Of importance is the surface state of the CNTs, with oxidized surface providing sites for nucleation-growth of anatase nanoparticles. As a result, a composite material featuring CNTs decorated by nanocrystals was obtained. This study proves that other fluorinated inorganic compounds could be integrated on carbon substrates to tune material's properties with respect to the targeted applications.

Experimental

Materials synthesis:

Multiwalled carbon nanotubes (CNT) were obtained from Arkema (CCVD, Groupement de Recherches de Lacq, Arkema) with a carbon purity of more than 90wt%. These CNTs are composed of 5 to 15 walls with an outer diameter from few nm to 20 nm and a length ranging from 0.1 to 10 µm.

To obtain oxidized CNTs (o-CNTs), 2 g of CNTs were stirred in 60 mL solution of HNO 3 10 mol L -1 at 80°C during 4 hours. Subsequently, o-CNTs was extensively washed with water and dried overnight at 120°C. The synthesis of CNT-anatase composite was performed using a modified protocol found in reference [START_REF] Li | High Substitution Rate in TiO2 Anatase Nanoparticles with Cationic Vacancies for Fast Lithium Storage[END_REF]. A defined amount of CNTs was firstly suspended in 14.8 mL isopropanol inside a Teflon-lined cup. The resulting mixture was sonicated for 1 h to favor the CNTs dispersion. Thereafter 4 mL of titanium isopropoxide was added and the mixture was stirred vigorously for 30 min. A solution of 10 mL isopropanol containing 1.2 mL aqueous HF (40 wt.%) was subsequently poured into the system under stirring. The mixture was sealed in a stainless steel autoclave and activated solvothermally at 90 °C for 12 h. After cooling down to room temperature, the precipitate was separated from the solution using centrifugation (4400 rpm) and washed three times with ethanol. The solid was finally dried at 80 °C for 2 h.

Material characterization:

Powder X-ray diffraction data were collected using a Rikaku diffractometer, equipped with Cu Kα radiation, in a Bragg-Brentano geometry. The Transmission Electron Microscopy analysis was performed using a JEOL 2010 UHR microscope operating at 200 kV equipped with a TCD camera.

Total scattering data were collected at the 11-ID-B beamline at the Advanced Photon Source at Argonne National Laboratory, using high energy X-rays (λ = 0.2128 Å) with high values of momentum transfer Qmax = 22 Å -1 . [START_REF] Chupas | Applications of an amorphous silicon-based area detector for high-resolution, high-sensitivity and fast time-resolved pair distribution function measurements[END_REF][START_REF] Chupas | Rapid-acquisition pair distribution function (RA-PDF) analysis[END_REF] One-dimensional diffraction data were obtained by integrating the raw 2D total scattering data in Fit2D. [START_REF] Hammersley | Two-dimensional detector software: From real detector to idealised image or two-theta scan[END_REF] PDFs, G(r), were extracted from the background and Compton scattering corrected data following Fourier transform within PDFgetX2. [START_REF] Qiu | PDFgetX2: a GUI-driven program to obtain the pair distribution function from X-ray powder diffraction data[END_REF] The PDFs were subsequently modeled using PDFgui. [START_REF] Farrow | PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals[END_REF] X-ray photoelectron spectroscopy (XPS) was performed using a VG ESCALAB 250 spectrometer with an Al Kα monochromatic source (15 kV, 15 mA) and a multidetection analyzer, under 10 -8 Pa residual pressure.

Figure 1

 1 presents representative TEM images of the composites containing 5 wt.% raw CNTs or o-CNTs. It shows that raw CNTs are not effectively covered by nanoparticles that formed separated aggregates (Figure1a). In contrast, surface functionalization allows nanoparticles to grow on the o-CNTs which are almost fully covered (Figure1b). High magnification images (Figure1c,d) better evidences the effective grafting of nanocrystals on the o-CNTs surfaces. The nanoparticles present lattice fringes of 0.350 nm, characteristic to the spacing of (101) planes in anatase. The particle sizes were ~8 nm, consistent with the sample prepared without CNTs. The homogeneity of o-CNTs-based nanocomposites was also confirmed by the recovery of more homogeneous precipitate.

Figure 1 :

 1 Figure 1: Representative transmission electron microscopy images of composite materials obtained using raw CNTs (a,c) and o-CNTs (b,d). White circles represent single crystals.

Figure 2 :

 2 Figure 2: Representative transmission electron microscopy images of composites prepared using 1 wt.% (a) and 10 wt.% of o-CNTs.

Figure 3 :

 3 Figure 3: (a) XRD patterns of pristine Ti 0.69  0.31 O 0.76 F 0.48 (OH) 0.76 , and composites prepared with different content of o-CNTs. The Bragg peaks of the composites are indexed with anatase type structure. (b) Fit of the PDF data of the composite prepared with 10 wt.% o-CNTs.

Figure 4 :

 4 Figure 4: XPS spectra of composite and HF treated o-CNT. Inset: reconstructed C 1s spectra.

Acknowledgements

We thank S. Casale for HRTEM analyses. The work done at the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.


Pierre Thuéry

Youssef Atoini

Jack Harrowfield

Chiral Discrete and Polymeric Uranyl Ion Complexes with (1R,3S )-(+)-Camphorate Ligands: Counterion-Dependent Formation of a Hexanuclear Cage

INTRODUCTION

The generation of polymeric, oligomeric or oligonuclear cage rather than simple chelate structures by complexation of metal ions with multidentate ligands is to some extent dependent upon the nature of the particular metal ion and associated anions but, more importantly, is open to control through an appropriate choice of the ligand. [START_REF] Tranchemontagne | Reticular Chemistry of Metal-Organic Polyhedra[END_REF][START_REF] Young | Structural Design Principles for Self-Assembled Coordination Polygons and[END_REF] Dicarboxylates, as a major subset of polycarboxylates, provide an example of a class of ligands which has been widely applied in the synthesis of both coordination polymers and oligonuclear cages, [START_REF] Harrowfield | Solid State Luminescence and π-Stacking in Crystalline Uranyl Dipicolinates[END_REF][START_REF] Wang | Extended Structures and Physicochemical Properties of Uranyl-Organic Compounds[END_REF][START_REF] Li | Synthesis, Structures and Luminescent Properties of New Pb(II)/M(I) (M = K, Rb and Cs) Frameworks Based on Dicarboxylic Acids: a Novel Icosahedral Pb6-M6 SBU[END_REF][START_REF] Andrews | Uranyl Bearing Hybrid Materials: Synthesis, Speciation, and Solid-State Structures[END_REF][START_REF] Ahmad | Discrete Metal-Carboxylate Self-Assembled Cages: Design, Synthesis and Applications[END_REF][START_REF] Loiseau | The Crystal Chemistry of Uranium Carboxylates[END_REF][START_REF] Su | MOFs of Uranium and the Actinides[END_REF][START_REF] Thuéry | Structural Variations in the Uranyl/4,4ʹ-Biphenyldicarboxylate System. Rare Examples of 2D → 3D Polycatenated Uranyl-Organic Networks[END_REF][START_REF] Thuéry | Uranyl Ion Complexes with Long-Chain Aliphatic α,ω-Dicarboxylates and 3d-Block Metal Counterions[END_REF][START_REF] Jayasinghe | Synthesis and Characterization of Heterometallic Uranyl Pyridinedicarboxylate Compounds[END_REF][START_REF] Thuéry | Recent Advances in Structural Studies of Heterometallic Uranyl-Containing Coordination Polymers and Polynuclear Closed Species[END_REF] the latter species being of particular interest because of the possibility that they may define cavities utilisable as selective reaction vessels. [START_REF] Wang | Extended Structures and Physicochemical Properties of Uranyl-Organic Compounds[END_REF][START_REF] Ahmad | Discrete Metal-Carboxylate Self-Assembled Cages: Design, Synthesis and Applications[END_REF] Two carboxylate substituents may be found posed on scaffolds of an enormous variety ranging from flexible polymethylene chains compatible with many different separations between the carboxylate groups to highly rigid cyclic structures imposing a relatively precise limit of separation. Even with the most rigid of scaffolds, however, some flexibility remains in that there is facile rotation about the C-CO2 -bond which can allow some variation in the O⋅⋅⋅O separations between the two carboxylate centres.

The versatility of the carboxylate unit as a donor species, where one or both of the oxygen atoms may be bound in both direct or bridging modes, is another important factor to be taken into consideration. Thus, on a rigid scaffold such as 1,3-disubstituted adamantane, for example, the complexation of uranyl ion [START_REF] Rusanova | A New Adamantanecarboxylate Coordination Polymer: Poly[[(µ3-Adamantane-1,3-dicarboxylato)aquadioxidouranium(VI)] monohydate][END_REF][START_REF] Thuéry | Uranyl and Uranyl-3d Block Cation Complexes with 1,3-Adamantanedicarboxylate: Crystal Structures, Luminescence and Magnetic Properties[END_REF] (our particular interest) does not involve the formation of 8-membered chelate rings involving one oxygen of each carboxylate but does involve bridging through formation of 4-membered chelate rings (κ 2 O,Oʹ mode) from one or both carboxylate units as well as through binding of another uranyl ion to such a chelate and through separate interactions of each oxygen of a given carboxylate with two different uranyl ions (µ2-κ 1 O: κ 1 Oʹ). All these are associated with the formation of polymeric arrays of various dimensionality depending on other constituents of the lattice. On a less rigid scaffold with adjacent carboxylate substituents as is found with cis-1,2-cyclohexanedicarboxylate, formation of 7-membered chelate rings does occur in some instances despite the apparently divergent axial-equatorial geometry of the carboxylate units, although such rings are only part of more extended bridged arrays. [START_REF] Thuéry | Crown Ethers and Their Alkali Metal Ion Complexes as Assembler Groups in Uranyl-Organic Coordination Polymers with cis-1,3-, cis-1,2-, and trans-1,2-Cyclohexanedicarboxylates[END_REF][START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF] It is thus a little surprising that on the chiral scaffold of (1R,3S)-(+)-camphoric acid (H2cam), [START_REF] Gu | Chiral Chemistry of Metal-Camphorate Frameworks[END_REF] which is certainly sufficiently flexible to allow the carboxylate substituents to adopt the diaxial conformation bringing them into close proximity, as seen in the structure of the anhydride, [START_REF] Kumaradhas | The Low-Temperature Study of D-and DL-Camphoric Anhydride[END_REF] there is seemingly no tendency to form 8-membered chelate rings in its metal complexes. More significantly, while in its general coordination chemistry with separate metal cations camphorate is characterized by bridging roles where each carboxylate interacts with different metal ion units or a metal ion and a proton, as shown by a survey of the crystal structures reported in the Cambridge Structural Database (CSD, Version 5.39), [START_REF] Groom | The Cambridge Structural Database[END_REF] and in particular by some uranyl ion complexes, [START_REF] Thuéry | Solvothermal Synthesis and Crystal Structure of Uranyl Complexes with 1,1-Cyclobutanedicarboxylic and (1R,3S)-(+)-Camphoric Acids -Novel Chiral Uranyl-Organic Frameworks[END_REF] introduction of a second cation, metallic or nonmetallic (including the proton), along with uranyl ion has been found to be an effective method for formation of a pseudo-cubic octanuclear cage species. [START_REF] Thuéry | A Nanosized Uranyl Camphorate Cage and its Use as a Building Unit in a Metal-Organic Framework[END_REF][START_REF] Thuéry | Chiral One-to Three-Dimensional Uranyl-Organic Assemblies from (1R,3S)-(+)-Camphoric Acid[END_REF] To understand better the origin of this effect, we have therefore extended our studies of the influence of large cations on the structure of heterocationic uranyl complexes [START_REF] Thuéry | Crown Ethers and Their Alkali Metal Ion Complexes as Assembler Groups in Uranyl-Organic Coordination Polymers with cis-1,3-, cis-1,2-, and trans-1,2-Cyclohexanedicarboxylates[END_REF][START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Uranyl-Organic Coordination Polymers with trans-1,2-, trans-1,4-, and cis-1,4-Cyclohexanedicarboxylates: Effects of Bulky PPh4 + and PPh3Me + Counterions[END_REF][START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF] to the camphorate system.

We have recently shown that the structure-directing effects exerted by PPh4 + and PPh3Me + cations promoted the formation of a discrete uranyl octanuclear cage with cis-1,2cyclohexanedicarboxylate, [START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF] and of a discrete tetranuclear metallatricycle with 1,3adamantanediacetate, [START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF] so that their impact in the case of a ligand already known for its propensity to give cage complexes, such as (1R,3S)-(+)-camphorate, was worth investigating.

Cages and metallacycles are still relatively rare in uranyl chemistry, apart from the family of uranyl peroxides, [START_REF] Hickam | Oxo Clusters of 5f Elements[END_REF] and, in addition to those already mentioned, only about a dozen are known, [START_REF] Thuéry | A Metal-Organic Molecular Box Obtained from Self-Assembling around Uranyl Ions[END_REF][START_REF] Thuéry | Self-Assembly of an Octa-Uranate Cage Complex with a Rigid bis-Catechol Ligand[END_REF][30][31][32][33][34][35][36][37][38][START_REF] Lee | Uranyl Dication Mediated Photoswitching of a Calix[4]pyrrole-Based Metal Coordination Cage[END_REF] most of them containing carboxylate ligands. Reported herein are five uranyl ion complexes synthesized from (1R,3S)-(+)-camphoric acid, three of which include bulky 3d-block metal-containing countercations, and the other two PPh4 + and PPh3Me + cations. These complexes, which have been characterized by their crystal structure and, for some of them, their emission spectrum in the solid state, crystallize as one-or two-dimensional (1D or 2D) coordination polymers, or as discrete species, one of them a hexanuclear cage. A potentially important property of all these complexes is that due to their incorporation of enantiomerically pure (1R,3S)-(+)-camphorate ligands, their crystals belong to chiral lattices. [START_REF] Gu | Chiral Chemistry of Metal-Camphorate Frameworks[END_REF]

EXPERIMENTAL SECTION

Syntheses. Caution! Uranium is a radioactive and chemically toxic element, and uranium-containing samples must be handled with suitable care and protection. UO2(NO3)2•6H2O (depleted uranium, R. P. Normapur, 99%) was purchased from Prolabo, and (1R,3S)-(+)-camphoric acid was from Aldrich. R,S-Me6cyclam (meso isomer of 7(R),14(S)-5,5,7,12,12,14-hexamethyl-1,4,8,11-tetraazacyclotetradecane) was prepared as described in the literature. [START_REF] Tait | Inorganic Syntheses[END_REF] N(R,S,R,S)-[M(R,S-Me6cyclam)(NO3)2] (M = Cu, Ni) were synthesized as previously described. [START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] Resolution of [Co(en)3] 3+ (en = ethylenediamine) by R,R-tartrate has been known for long, [START_REF] Werner | Zur Kenntnis des Asymmetrischen Kobaltatoms[END_REF][START_REF] Broomhead | Resolution of the Tris(Ethylenediamine)Cobalt(III) Ion. Inorganic Syntheses[END_REF] and Λ-[Co(en)3](H2R,R-tart)Cl⋅2H2O, where H4R,Rtart is R,R-tartaric acid, was obtained as described. Elemental analyses were performed by MEDAC Ltd. at Chobham, UK. For all syntheses of uranyl ion complexes, the mixtures in demineralized water/organic solvent were placed in 10 mL tightly closed glass vessels and heated at 140 °C under autogenous pressure.

[Cu(R,S-Me6cyclam)][UO2(Hcam)2(HCOO)2] (1). (1R,3S

)-(+)-Camphoric acid (20 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and N(R,S,R,S)-[Cu(R,S-Me6cyclam)(NO3)2] (24 mg, 0.05 mmol) were dissolved in water (0.7 mL) and DMF (0.2 mL).

Purple crystals of complex

[Ni(R,S-Me6cyclam)][UO2(cam)(HCOO)2] (2). (1R,3S

)-(+)-Camphoric acid (20 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and N(R,S,R,S)-[Ni(R,S-Me6cyclam)(NO3)2] (23 mg, 0.05 mmol) were dissolved in water (0.5 mL) and DMF (0.2 mL). Orange crystals of complex 2 were obtained in low yield within two weeks. (3). (1R,3S)-(+)-Camphoric acid (20 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and Λ-[Co(en)3](H2R,R-tart)Cl⋅2H2O (23 mg, 0.05 mmol) were dissolved in water (0.6 mL) and acetonitrile (0.2 mL). Orange crystals of complex 3 were obtained in low yield within three days.

[Co(en)3][(UO2)4(cam)(R,R-tart)2(OH)]⋅3H2O
[PPh4][UO2(cam)(NO3)] ( 4). (1R,3S)-(+)-Camphoric acid (20 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh4Br (42 mg, 0.10 mmol) were dissolved in water (0.5 mL) and acetonitrile (0.2 mL). Yellow crystals of complex 4 were obtained within four days (50 mg, 82% yield based on U). Anal. Calcd for C34H34NO9PU: C, 46.96; H, 3.94; N, 1.61.

Found: C, 46.30; H, 3.76; N, 1.64%.

[PPh3Me]3[NH4]3[(UO2)6(cam)9] (5). (1R,3S)-(+)-Camphoric acid (20 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh3MeBr (36 mg, 0.10 mmol) were dissolved in water (0.9 mL), acetonitrile (0.2 mL) and DMF (0.2 mL). Yellow crystals of complex 5 were obtained within three weeks (22 mg, 46% yield based on the acid). Anal. Calcd for C147H192N3O48P3U6: C, 41.16; H, 4.51; N, 0.98. Found: C, 40.66; H, 4.63; N, 1.80%. The discrepancy between calculated and found values may be due to the presence of lattice solvent molecules (water, acetonitrile or DMF) which could not be resolved in the crystal structure (see below).

Crystallography. The data were collected at 100(2) K on a Nonius Kappa-CCD area detector diffractometer [START_REF] Hooft | [END_REF] using graphite-monochromated Mo Kα radiation (λ = 0.71073 Å). The crystals were introduced into glass capillaries with a protective coating of Paratone-N oil (Hampton Research). The unit cell parameters were determined from ten frames, then refined on all data. The data (combinations of ϕand ω-scans with a minimum redundancy of 4 for 90% of the reflections) were processed with HKL2000. [START_REF] Otwinowski | Processing of X-Ray Diffraction Data Collected in Oscillation Mode[END_REF] Absorption effects were corrected empirically with the program SCALEPACK. [START_REF] Otwinowski | Processing of X-Ray Diffraction Data Collected in Oscillation Mode[END_REF] The structures were solved by intrinsic phasing with SHELXT, [START_REF] Sheldrick | SHELXT -Integrated Space-Group and Crystal-Structure Determination[END_REF] expanded by subsequent difference Fourier synthesis and refined by fullmatrix least-squares on F 2 with SHELXL-2014. [START_REF] Sheldrick | Crystal Structure Refinement with SHELXL[END_REF] All non-hydrogen atoms were refined with anisotropic displacement parameters. The hydrogen atoms bound to oxygen and nitrogen atoms were retrieved from difference Fourier maps when possible (see details below), and the carbonbound hydrogen atoms were introduced at calculated positions. All hydrogen atoms were treated as riding atoms with an isotropic displacement parameter equal to 1.2 times that of the parent atom (1.5 for CH3, with optimized geometry). Large voids in the lattices of compounds 3 and 5 indicate the presence of unresolved solvent molecules, whose contribution to the structure factors was taken into account with PLATON/SQUEEZE. [START_REF] Spek | SQUEEZE: a Tool for the Calculation of the Disordered Solvent Contribution to the Calculated Structure Factors[END_REF] Crystal data and structure refinement parameters are given in Table 1. The molecular plots were drawn with ORTEP-3, [START_REF] Farrugia | WinGX and ORTEP for Windows: an Update[END_REF] and the polyhedral representations with VESTA. [START_REF] Momma | VESTA: a Three-Dimensional Visualization System for Electronic and Structural Analysis[END_REF] The topological analyses were conducted with TOPOS. 51 Special details, when present, are as follows.

Complex 3. Restraints on bond lengths, angles and displacement parameters were applied for the atoms of the badly resolved camphorate molecule. The hydrogen atom bound to O25 was found on a difference Fourier map, but not those of the lattice water molecules, one of which was given an occupancy factor of 0.5 in order to retain an acceptable displacement parameter.

Complex 5. The crystals, and consequently the diffraction data were of very poor quality and the refinement required the use of extensive restraints on bond lengths, angles and displacement parameters. The hydrogen atoms of the NH4 + cations, which are disordered and were given occupancy parameters of 1/3 for charge balance and to retain acceptable displacement parameters, were neither found, nor introduced. Luminescence Measurements. Emission spectra were recorded on solid samples using a Horiba-Jobin-Yvon IBH FL-322 Fluorolog 3 spectrometer equipped with a 450 W xenon arc lamp, double-grating excitation and emission monochromator (2.1 nm/mm of dispersion; 1200 grooves/mm) and a TBX-04 single photon-counting detector. The powdered compounds were put into a quartz tube and pressed to the wall of the tube, and the measurements were performed using the right-angle mode. An excitation wavelength of 420 nm, a commonly used point although only part of a broad manifold, was used in all cases and the emission was monitored between 450 and 650 nm. The quantum yield measurements were performed by using a Hamamatsu Quantaurus C11347 absolute photoluminescence quantum yield spectrometer and exciting the samples between 300 and 400 nm.

RESULTS AND DISCUSSION

Synthesis. Crystals of complexes 1-5 were grown under solvo-hydrothermal conditions, at a temperature of 140 °C, with the organic cosolvent being N,Ndimethylformamide (1 and 2), acetonitrile (3 and 4), or a mixture thereof (5). The crystals were deposited directly from the pressurized and heated reaction mixtures and not as a result of subsequent cooling. The uranium/dicarboxylate ligand ratio was 7:10 in all cases, so as to favour the formation of an anionic species, but the expected ratio of 2:3 is retained in complex 5 only, while a 1:2 ratio is found in 1, 1:1 in 2 and 4, and 4:1 in 3. Coligands (formate, tartrate, nitrate) are present in all the last cases, while 5 is homoleptic. The composition of the crystalline materials isolated in the present work provides further illustration of the vagaries of solvothermal synthesis. Thus, in four of the five complexes the camphorate dianion cam 2-is present but in complex 1 the Hcam -monoanion is found, giving rise to a different stoichiometry to that of complex 2, despite the identical isomeric form and very closely similar size of the cations [Cu(R,S-Me6cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as well as the use of identical reaction conditions. The formate ligands present in both complexes arise from hydrolysis of DMF under the reaction conditions, [START_REF] Thuéry | Uranyl-Organic Coordination Polymers with trans-1,2-, trans-1,4-, and cis-1,4-Cyclohexanedicarboxylates: Effects of Bulky PPh4 + and PPh3Me + Counterions[END_REF][START_REF] Thuéry | Ag I and Pb II as Additional Assembling Cations in Uranyl Coordination Polymers and Frameworks[END_REF] a process that also gives rise to dimethylammonium cations that, in many other related systems but not here, are found as components of the crystalline products. Despite the use of DMF as a cosolvent in the synthesis of complex 5, neither formate nor dimethylammonium is present but ammonium ion, presumably derived from the acetonitrile cosolvent, [START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF]38 is. It is not, however present in either complex 3 or complex 4, both of which were deposited from aqueous acetonitrile solvent. It is also notable that although nitrate was present in all the reaction mixtures, it appears in complex 4 only.

Complex 3 is the only one to contain an hydroxo ligand, possibly in reflection of the weak basicity induced by the presence of tartrate anion. Thus, while our efforts to associate different large cations with anionic uranyl camphorate species were successful in every case, it cannot be said that the nature of the different cations must have been the sole influence upon the composition of the isolated crystals. The syntheses therefore demand structural studies to define in detail the interactions that determine the form of the crystal lattices. coordination polymer parallel to the c axis (Figure 2). The nickel(II) centre is bound to the four nitrogen atoms [Ni-N 1.938(6)-1.953(7) Å], but it is not involved in axial coordination interactions and the macrocycle complex unit functions essentially as an hydrogen bond donor.

Crystal Structures

Here, there is a parallel with the behaviour of the cations in the complex [Ni(R,S- [START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF] The ability to prepare a reaction mixture from enantiomerically pure components, however, was expected to limit the number of possible product forms and the crystals obtained, albeit in low yield, appeared to be uniform. The asymmetric unit contains four independent uranium atoms, each of them being chelated by one carboxylate and one hydroxylic group from one R,R-tart 4-ligand (Figure 3), a coordination mode previously found in [UO2(H2tart)(H2O)],

Me6cyclam)][Ni(R,S-Me6cyclam)(H2O)2][(UO2)2(t-1,2-chdc)2(O)]2 (t-1,2-chdc = trans-1,2- cyclohexanedicarboxylate),
the only other uranyl tartrate complex reported, [START_REF] Thuéry | Reaction of Uranyl Nitrate with Carboxylic Diacids under Hydrothermal Conditions. Crystal Structure of Complexes with L(+)-Tartaric and Oxalic Acids[END_REF] and also in complexes with citrate, malate and citramalate ligands. [START_REF] Thuéry | Uranyl Ion Complexes with Chiral Malic and Citramalic, and Prochiral Citric and Tricarballylic Acids: Influence of Coligands and Additional Metal Cations[END_REF] The coordination sphere is completed differently for each uranium atom.

U1 connectivity results in the formation of a tetra-nodal 2D assembly parallel to (010), which has the point (Schläfli) symbol {4.6 2 }2{4 2 .6 2 .8 2 }{4 2 .6 3 .8}2{4 2 .6}2 and is represented in Figure 3d. 2.36(2)-2.51(3) Å], but here the bridging nature of all ligands and their curved geometry result in the formation of a hexanuclear cage with threefold rotation symmetry. This cage differs from the octanuclear, pseudo-cubic species which is frequently found in uranyl camphorate complexes (Figure 6), [START_REF] Thuéry | A Nanosized Uranyl Camphorate Cage and its Use as a Building Unit in a Metal-Organic Framework[END_REF][START_REF] Thuéry | Chiral One-to Three-Dimensional Uranyl-Organic Assemblies from (1R,3S)-(+)-Camphoric Acid[END_REF] and it is in fact the first hexanuclear uranyl carboxylate cage, the previous examples being either dinuclear (helicate), 36 tetranuclear, [START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF][37][38][START_REF] Lee | Uranyl Dication Mediated Photoswitching of a Calix[4]pyrrole-Based Metal Coordination Cage[END_REF] octanuclear, [START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | A Nanosized Uranyl Camphorate Cage and its Use as a Building Unit in a Metal-Organic Framework[END_REF][START_REF] Thuéry | Chiral One-to Three-Dimensional Uranyl-Organic Assemblies from (1R,3S)-(+)-Camphoric Acid[END_REF][START_REF] Thuéry | A Metal-Organic Molecular Box Obtained from Self-Assembling around Uranyl Ions[END_REF]32,34,35 or even of higher nuclearity. 32,35 The [(UO2)6(cam)9] 6-moiety in 5 has C3 point symmetry, but the uranium atoms alone define a trigonal prismatic motif of D3h symmetry, which results in two sets of U⋅⋅⋅U distances, the shorter ones being within the trigonal bases [8.9093(18) . All these differences in dihedral angles are in keeping with the smaller size and higher curvature of the cage in 5. The three well-ordered PPh3Me + counterions, all with threefold rotation symmetry, are in close association with the cage unit, two of them outside and with their methyl group pointing toward the prism basis centre, and one inside the cavity and with its phenyl rings directed towards the spaces between camphorate ligands (Figure 7).

When viewed down the c axis (Figure 5d), the packing displays two kinds of channels: those corresponding to the cage cavities and containing the PPh3Me + counterions, and those between cages and containing the ammonium cations generated in situ, which are highly disordered around threefold rotation axes (see Experimental Section), as well as probably disordered solvent molecules (the KPI with disordered ammonium cations excluded amounts to 0.58). The cages are arranged in sheets parallel to (0 0 1), which are separated from one another by layers of unincluded PPh3Me + counterions. Complex 5 provides another example where the presence of both ammonium and phosphonium cations in the lattice is associated with the crystallization of a closed oligomeric uranyl-containing polyanion, [START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF] and the remarkable difference from complex 4 is probably due to the influence of this combination. In other cases where PPh4 + or PPh3Me + is the sole additional cation associated with an anionic uranyl species, relatively minor structural differences have been found. [START_REF] Thuéry | Uranyl-Organic Coordination Polymers with trans-1,2-, trans-1,4-, and cis-1,4-Cyclohexanedicarboxylates: Effects of Bulky PPh4 + and PPh3Me + Counterions[END_REF] In contrast to the octanuclear cage in [START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF] where ammonium ion is included, it is the much larger PPh3Me + ion that is included in the hexanuclear cage here. The exact role of ammonium ions in the lattice is unclear due to their disorder and lack of hydrogen atom locations but all lie at points compatible with NH⋅⋅⋅O interactions involving carboxylate oxygen atoms [N⋅⋅⋅O 2.81(4)-2.97( 6) Å] and thus appear to have a role in the assembly. The form of the cages is compatible with the threefold symmetry of the included PPh3Me + cation in that the C3 axes are collinear, and the Hirshfeld surface for the cation indicates a single CH⋅⋅⋅O interaction beyond dispersion of each phenyl ring with a uranyl oxo group directed toward the interior of the cage [C⋅⋅⋅O 3.57(2) Å, H⋅⋅⋅O 2.62 Å, C-H⋅⋅⋅O 174°]. The chirality of the cage has an influence in that the included cation has a chiral conformation, being that of a right-handed helix. This influence extends to the other two inequivalent PPh3Me + cations which are poised above the trigonal faces of the cage in that they adopt chiral conformations but these are opposite, so that the cations containing P1 (included) and P2 are of the same chirality and enantiomeric to that containing P3. Because of this, the sheets of unincluded cations that lie between those formed by the cages are racemic, since they include both P2-and P3-containing counterions. These unincluded cations appear to be involved in triple CH⋅⋅⋅O interactions with carboxylate oxygen atoms bound to uranyl [C⋅⋅⋅O 3.42(2) and 3.47(3) Å, H⋅⋅⋅O 2.55 and 2.62 Å, C-H⋅⋅⋅O 152 and 149°], but there is no evidence of "embrace" interactions 60 beyond dispersion within their sheets.

[NH4][PPh4][(UO2)8(c-1,2- chdc)9(H2O)6]•3H2O (c-1,2-chdc = cis-1,2-cyclohexanedicarboxylate),
Luminescence properties. Emission spectra under excitation at 420 nm were recorded for complexes 1, 4 and 5 in the solid state (Figure 8), the other compounds being unavailable in sufficient quantity and purity. Whereas complex 5 is non-luminescent, both 1 and 4 display the usual fine structure associated with the vibronic progression corresponding to the S11 → S00

and S10 → S0ν (ν = 0-4) electronic transitions, [START_REF] Brachmann | Study of Uranyl(VI) Malonate Complexation by Time Resolved Laser-Induced Fluorescence Spectroscopy (TRLFS)[END_REF] although emission in 4 is more intense than in 1, and its spectrum better resolved. Uranyl emission in 1 may be partially quenched due to the presence of copper(II) cations, as frequently observed. [START_REF] Kerr | Postsynthetic Rearrangement/Metalation as a Route to Bimetallic Uranyl Coordination Polymers: Syntheses, Structures, and Luminescence[END_REF] The main maxima in the spectrum of 4 are at 465, 483, 503, 524, 548 and 574 nm, the four well-resolved maxima in the spectrum of 1 being blue-shifted by about 2 nm with respect to their counterparts in 4. These positions match those generally found for hexacoordinated uranyl species. 63 Solid-state photoluminescence quantum yields (PLQYs) of 2 and 23% (with a standard deviation of ca. ±2.5%) have been measured for complexes 1 and 4, respectively. The low value for compound 1, possibly due to partial quenching by copper(II), is comparable to values recently found in a series of uranyl complexes with aliphatic α,ω-dicarboxylates. [START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] Both [Cu(R,S-Me6cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ have absorptions in the visible region [START_REF] Curtis | Some Cyclic Tetra-Amines and their Metal-Ion Complexes. Part I. Two Isomeric Hexamethyltetra-azacyclotetradecanes and their Copper(II) and Nickel(II) Complexes[END_REF] which overlap with that of [UO2(H2O)5] 2+ , [START_REF] Burrows | Applications and Limitations of Uranyl Ion as a Photophysical Probe[END_REF] and which also overlap with the uranyl emission bands, so that the effect of [Cu(R,S-Me6cyclam)] 2+ in complex 1 may be due to both preferential absorption of the incident light as well as energy transfer from any excited uranyl centres to the complex cations (a similar effect can be expected for compound 2, for which no spectrum could be recorded). While this argument is based on solution spectroscopic data, it is known that the solid state spectra of the Ni II and Cu II complexes of unfunctionalised cyclam closely match those in solution. [START_REF] Makowska-Janusik | Mesoporous Silica Functionalized by Cyclam-Metal Groups: Spectroscopic Studies and Numerical Modeling[END_REF] The PLQY for complex 4 is similar to that of 24% we have measured under the same conditions for uranyl nitrate hexahydrate, and is one of the largest measured up to now in the family of uranyl carboxylate complexes, to be compared for example with 13% for

[PPh4]2[(UO2)2(t-1,4-chdc)3]•4H2O (t-1,4-chdcH2 = trans-1,4-cyclohexanedicarboxylic acid), 24
12% for [PPh4][UO2(C6)(NO3)] (H2C6 = adipic acid), [START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] and 9% for [H2NMe2][PPh3Me][(UO2)2(ADA)3]•H2O (H2ADA = 1,3-adamantanediacetic acid). [START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF] However, larger PLQYs are known, such as 42% measured for the previously described Rb2[UO2(dipic)2] (dipic = dipicolinate), 3 49% for [NMe4]2[(UO2)4(C2O4)4(C4)] (H2C4 = succinic acid), [START_REF] Xie | Highly Sensitive Detection of Ionizing Radiations by a Photoluminescent Uranyl Organic Framework[END_REF] and 58% for [UO2(HL)(H2O)] (H3L = benzene-1,3,5-tricarboxylic acid). [START_REF] Wang | Emergence of Uranium as a Distinct Metal Center for Building Intrinsic X-ray Scintillators[END_REF] We have discussed elsewhere [START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] the possibility that energy transfer between close uranyl centres might prolong excited state lifetimes and thus that U⋅⋅⋅U separations within lattices may be important criteria in relation to PLQY values but the shortest U⋅⋅⋅U distance does not seem to be a relevant parameter in the case of complex 4 since, at 8.0460(3) Å (interchain), it is indeed larger than the values measured in the two most strongly emitting complexes cited above (5.511 and 5.232 Å, respectively), but comparable to distances in some much more weakly emitting species. [START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] We have also considered that phosphonium cations may act as quenching centres and thus that U⋅⋅⋅P separations may be significant but the shortest U⋅⋅⋅P distance in 4, 6.750(3) Å, is within the range found in a series of complexes with smaller PLQYs, [START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] which suggests that this is not the dominant factor here. This also appears to be the case for the non-emissive complex 5, where the shortest U⋅⋅⋅P separation [U1⋅⋅⋅P1 6.251(6) Å] is longer than that in emissive relatives. [START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF]

CONCLUSIONS

We have reported here the synthesis and crystal structure of five uranyl camphorate complexes including five different bulky counterions, as well as the luminescence properties of three of them. Three of the counterions are 3d block metal ion complexes, [M(R,S-Me6cyclam)] 2+ (M = Cu, Ni), which have been previoulsy used in association with cis-and trans-1,2cyclohexanedicarboxylate, and aliphatic α,ω-dicarboxylate ligands, [START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] and [Co(en)3] 3+ , for which only one example of use as a counterion for an anionic uranyl pimelate complex is known. [START_REF] Thuéry | Counterion-Induced Variations in the Dimensionality and Topology of Uranyl Pimelate Complexes[END_REF] The camphorate complexes with these counterions are either discrete, 1D or 2D, respectively, the last having R,R-tartrate coligands, and its lattice thus containing three enantiomerically pure chiral species. The fact that a change in interaction energy as small as 10 kJ mol -1 is sufficient to change an equilibrium constant such as a solubility product by a factor of 100 is an obvious explanation of the variability observed in our efforts to systematically explore the effect of heterocations on the nature of crystalline uranyl ion complexes. In just the present examples, the balance between coordinative and hydrogen bonding interactions of [M(R,S-Me6cyclam)] 2+ complexes is clearly rather delicate and while the use of [Co(en)3] 3+

eliminates the competition of coordination, the hydrogen bonding sites on this cation appear to be of more restricted access than those of the macrocycle complexes and to favour in this case interactions with small species such as water molecules rather than direct interaction with carboxylate groups.

The other two cations used are PPh4 + and PPh3Me + , the latter associated with NH4 + counterions formed in situ. While the complex with PPh4 + counterions is a 1D coordination polymer (with nitrate coligands), the complex with PPh3Me + /NH4 + is a discrete, anionic hexanuclear cage, different from the octanuclear uranyl camphorate species previously described, [START_REF] Thuéry | A Nanosized Uranyl Camphorate Cage and its Use as a Building Unit in a Metal-Organic Framework[END_REF][START_REF] Thuéry | Chiral One-to Three-Dimensional Uranyl-Organic Assemblies from (1R,3S)-(+)-Camphoric Acid[END_REF] thus providing an example in which the size of the cage can be modulated through variation of the structure-directing counterions. That the combination of ammonium and phosphonium cations seems to favour the formation of closed uranyl dicarboxylate oligomers, as observed here and in previous work with different carboxylates, [START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF] is encouraging, especially given that in the present case it leads to inclusion of a relatively large cation, PPh3Me + (NH4 + only was found to be included in one of the previously reported cages [START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF] ), but the number of known structures is still too small to be sure of ways to exploit this combination. This complex belongs to the yet somewhat restricted family of uranyl-organic cage compounds, with the added benefit from the use of the (1R,3S)-(+)-camphorate ligand that the cage formed is chiral and enantiomerically pure. It is disappointing, however, to find that this complex is non-emissive, a somewhat surprising observation given that such a degree of quenching has not been found in other uranyl ion complexes involving phosphonium countercations, [START_REF] Thuéry | Uranyl-Organic Coordination Polymers with trans-1,2-, trans-1,4-, and cis-1,4-Cyclohexanedicarboxylates: Effects of Bulky PPh4 + and PPh3Me + Counterions[END_REF][START_REF] Thuéry | Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α[END_REF] although none of these have involved inclusion of the cation in a cage.

ASSOCIATED CONTENT Accession Codes

CCDC 1874636-1874640 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge via www. Five uranyl ion complexes with (1R,3S)-(+)-camphorate ligands incorporating diverse bulky counterions were synthesized. While four of the complexes crystallize as 0D (mononuclear), 1D and 2D species, the complex with PPh3Me + and NH4 + counterions is a homochiral and homoleptic hexanuclear cage with a cavity of sufficient size for inclusion of one phosphonium cation.

Figure 1 .

 1 Figure 1. (a) View of compound 1. Displacement ellipsoids are drawn at the 50% probability level. Carbon-bound hydrogen atoms are omitted and hydrogen bonds are shown as dashed lines. (b) and (c) Two views of the packing with uranium coordination polyhedra colored yellow and copper(II) ions shown as blue spheres, and hydrogen atoms omitted.

Figure 2 .

 2 Figure 2. (a) View of compound 2. Displacement ellipsoids are drawn at the 30% probability level. Carbon-bound hydrogen atoms are omitted and hydrogen bonds are shown as dashed lines. Symmetry codes: i = 3/2 -x, 1 -y, z -1/2; j = 3/2 -x, 1 -y, z + 1/2. (b) and (c) Two views of the packing with uranium coordination polyhedra colored yellow and nickel(II) ions shown as blue spheres, and hydrogen atoms omitted.

Figure 3 .

 3 Figure 3. (a) View of compound 3. Displacement ellipsoids are drawn at the 30% probability level. Carbon-bound hydrogen atoms are omitted. Symmetry codes: i = x, y, z + 1; j = x + 1/2, 1/2 -y, -z; k = x -1/2, 1/2 -y, 1 -z; l = x, y, z -1; m = x + 1/2, 1/2 -y, 1 -z; n = x -1/2, 1/2 -y, -z. (b) View of the 2D network. (c) View of the packing with layers edge-on and cobalt(III) ions shown as orange spheres. (d) Nodal representation of the 2D network (yellow, uranium; dark blue, tartrate ligands; light blue, camphorate ligands).

The slightly undulating layersFigure 4 .

 4 Figure 4. (a) View of compound 4. Displacement ellipsoids are drawn at the 30% probability level and counterions are omitted. Symmetry codes: i = 2 -x, y + 1/2, 1 -z; j = 2 -x, y -1/2, 1 -z. (b) View of the packing with chains viewed side-on. Hydrogen atoms are omitted in both views.

Figure 5 .

 5 Figure 5. (a) View of compound 5. Displacement ellipsoids are drawn at the 10% probability level. Counterions and hydrogen atoms are omitted. Symmetry codes: i = 1 -y, x -y, z; j = y -x + 1, 1 -x, z. (b) View of the hexanuclear cage and the three associated PPh3Me + counterions. (c) The hexanuclear cage and the included PPh3Me + counterion viewed down the threefold rotation axis. (d) View of the packing. Hydrogen atoms and disordered ammonium ions are omitted.

Figure 6 .

 6 Figure 6. Simplified views of the hexanuclear cage in complex 5 (a) and the octanuclear cage in previous uranyl camphorate complexes (b). Yellow, uranium; red, oxygen; blue, camphorate ligand.

  and 9.012(3) Å for U1 and U2, respectively], and the longer ones being along the prism height [9.5041(14) Å]. These distances are comparable to those in the octanuclear camphorate cages [9.2606(8)-9.7845(7) Å], and they are larger than those found in cages based on other polycarboxylates such as Kemp's triacid [5.01-5.63 Å], 34 cis-1,2cyclohexanedicarboxylate [5.37-5.61 Å], 17 or trans-1,2-cyclohexanedicarboxylate [5.82-6.95 Å]. 37,38 The dihedral angles between the two carboxylate groups in each ligand are 104(3), 107(4) and 108(3)°, somewhat smaller than in the octanuclear cages [113.8(13)-119.3(13)°]. The dihedral angles between the equatorial planes of camphorate-bridged uranyl cations are 95.1(4) and 94.2(5)° for the cations in the trigonal faces, and 103.0(5)° for the cations pertaining to different faces; these values are also smaller that those in the larger cage [96.32(14)-117.05(17)°]

Figure 7 .

 7 Figure 7. Spacefill representations of the cage in complex 5, viewed down the trigonal axis (a), sideways (b), and sideways with the included and two adjoining counterions in green (c). Yellow, uranium; red, oxygen; blue, carbon; hydrogen atoms are omitted.

Figure 8 .

 8 Figure 8. Emission spectra of compounds 1, 4 and 5 in the solid state at room temperature, under excitation at a wavelength of 420 nm.

  1 were obtained within two weeks (36 mg, 65% yield based on the acid). Anal. Calcd for C38H68CuN4O14U: C, 41.25; H, 6.19; N, 5.06. Found: C, 41.31; H, 6.05;

	N, 5.59%.

Table 1 . Crystal Data and Structure Refinement Details

 1

		1	2	3	4	5
	chemical formula	C38H68CuN4O14U	C28H52N4NiO10U	C24H49CoN6O28U4	C34H34NO9PU	C147H192N3O48P3U6
	M (g mol -1 )	1106.53	901.47	1880.74	869.62	4290.12
	cryst syst	triclinic	orthorhombic	orthorhombic	monoclinic	hexagonal
	space group	P1	P212121	P21212	P21	P63
	a (Å)	8.2223(4)	9.1167(3)	17.3555(8)	8.0460(3)	15.9876(7)
	b (Å)	11.4446(3)	18.4978(7)	21.4656(13)	16.2439(10)	15.9876(7)
	c (Å) α (deg) β (deg) γ (deg)	13.2749(6) 68.345(3) 86.979(2) 76.794(3)	19.5856(5) 90 90 90	13.8492(6) 90 90 90	12.9430(8) 90 100.564(4) 90	39.925(3) 90 90 120
	V (Å 3 )	1129.62(8)	3302.90(19)	5159.5(5)	1662.96(16)	8837.8(12)
	Z	1	4	4	2	2
	reflns collcd	61854	110279	88767	55097	143094
	indep reflns	8491	6244	9766	6280	9817
	obsd reflns [I > 2σ(I)]	8482	6005	8176	5924	4703
	Rint	0.025	0.022	0.038	0.020	0.101
	params refined	536	407	577	419	611
	R1	0.029	0.026	0.047	0.034	0.063
	wR2	0.074	0.066	0.117	0.088	0.208
	S ∆ρmin (e Å -3 ) ∆ρmax (e Å -3 )	1.055 -0.72 0.57	1.070 -1.26 0.70	1.038 -1.38 1.19	1.030 -1.74 0.76	1.022 -2.11 1.21
	Flack parameter	0.014(7)	0.015(9)	0.053(17)	0.010(11)	0.09(2)

For Table of Contents Use Only Chiral Discrete and Polymeric Uranyl Ion Complexes with (1R,3S)-(+)-Camphorate Ligands: Counterion-Dependent Formation of a Hexanuclear Cage

  ccdc.cam.ac.uk/data_request/cif, or by emailing data_request@ccdc.cam.ac.uk, or by contacting The Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: +44 1223 336033. 30. Thuéry, P.; Villiers, C.; Jaud, J.; Ephritikhine, M.; Masci, B. Uranyl-Based Tetrahydrofurantetracarboxylic Acid. J. Am. Chem. Soc. 2004, 126, 6838-6839. 31. Charushnikova, I. A.; Grigor'ev, M. S.; Krot, N. N. Synthesis and Crystal Structure of New U(VI) and Np(VI) Benzoates, K11(AnO2)23(OOCC6H5)57(H2O)18+x. Radiochemistry 2010, 52, 138-144. 32. Pasquale, S.; Sattin, S.; Escudero-Adán, E. C.; Martínez-Belmonte, M.; de Mendoza, J. Giant Regular Polyhedra from Calixarene Carboxylates and Uranyl. Nature Commun. 2012, 3, 785-791. 33. Unruh, D. K.; Gojdas, K.; Libo, A.; Forbes, T. Z. Development of Metal-Organic Nanotubes Exhibiting Low-Temperature, Reversible Exchange of Confined "Ice Channels". J. Am. Chem. Soc. 2013, 135, 7398-7401. 34. Thuéry, P. A Highly Adjustable Coordination System: Nanotubular and Molecular Cage Species in Uranyl Ion Complexes with Kemp's Triacid. Cryst. Growth Des. 2014, 14, 901-904. 35. Thuéry, P. Increasing Complexity in the Uranyl Ion-Kemp's Triacid System: From One-and Two-Dimensional Polymers to Uranyl-Copper(II) Dodeca-and Hexadecanuclear Species. Cryst. Growth Des. 2014, 14, 2665-2676. 36. Thuéry, P.; Harrowfield, J. A New Form of Triple-Stranded Helicate Found in Uranyl Complexes of Aliphatic α,ω-Dicarboxylates. Inorg. Chem. 2015, 54, 10539-10541. 37. Thuéry, P.; Harrowfield, J. Coordination Polymers and Cage-Containing Frameworks in Uranyl Ion Complexes with rac-and (1R,2R)-trans-1,2-Cyclohexanedicarboxylates: Consequences of Chirality. Inorg. Chem. 2017, 56, 1455-1469. 38. Thuéry, P.; Harrowfield, J. Tetrahedral and Cuboidal Clusters in Complexes of Uranyl and

	Metallamacrocycles:	Tri-	and	Tetranuclear	Complexes	with	(2R,3R,4S,5S)-
	Alkali	or	Alkaline-Earth	Metal	Ions	with	rac-	and	(1R,2R)-trans-1,2-

Cyclohexanedicarboxylate. Cryst. Growth Des. 2017, 17, 2881-2892. Pierre Thuéry, Youssef Atoini and Jack Harrowfield

AUTHOR INFORMATION

Corresponding Authors

*E-mail: pierre.thuery@cea.fr. (P.T.) *E-mail: harrowfield@unistra.fr. (J.H.) ORCID Pierre Thuéry: 0000-0003-1683-570X Youssef Atoini: 0000-0003-4851-3713 Jack Harrowfield: 0000-0003-4005-740X

Notes

The authors declare no competing financial interest.


Nicolas Pierre Boileau

Converging Lines: Needlework in English Literature and Visual Arts

1

In the context of the #metoo movement, some have expressed a long-running fear that feminism and the cause of women's rights may enhance the opposition between men and women instead of bridging the gap between them, and create division instead of harmony. Martine Monacelli's collection of texts written by men in favour of women's rights could hardly have come at a better time to redefine the contours of the extent of men's political engagement throughout the 20 th century for women's rights. The question of men and women's equality is one that concerns everyone. Published in 2017, this collection is structured around four sections preceded by a preface and an introduction, written by Martine Monacelli, that frame the selected texts and provide information and landmarks that historicise these extracts. Men are restored to their key roles in the debate that, throughout the 19 th century, brought the question of women's rights and equality to the fore in relation to the emergence of other political ideologies, and public debates such as the military, education, civil rights and intimate, sexual practices. Women's rights were then seen as the pre-conditions for social and political improvements and progress of a larger scale.

2

In her preface, Martine Monacelli defines the historiographic positioning of her work by looking at almost everything but the question of the suffrage, which was soon to become the emblem of the feminist movement and that by which it is remembered to have started. Reminding us that work on men's activism in the defence of women's rights has been fairly neglected since the mid-1980s, she points out that recent works by Schwartz and Griffin have underlined the intermingling of feminist questions and 'other political traditions' (xi), in which it becomes clear -together with the title chosen -that Monacelli sees a distinction between the question of women's rights and feminism. This, although unexplored, is partly due to the time frame of Monacelli's work. Similarly, by addressing the question of women, men are presented as addressing conceptions and misconceptions of virility and manhood. Martine Monacelli therefore wants to suggest that male activists were far more varied and influential than is often thought. The book does not seek to upstage women's role but to qualify how males have often been perceived in relation to the public debate about women's emancipation.

3

The introduction is a quick, useful and dense overview of the various political stances and radical factions at play in the long 19 th century, and the place these debates came to grant to women's rights. First of all, women were mainly defined in relation to the religious imprint of gender roles and norms of the times. The Evangelicals, by defining women in relation to morality, produced new forms of women's involvement in the public sphere, especially by having them educated to ensure their moral rectitude (6): women's mission became the instrument of women's emancipation. Martine Monacelli goes through the emerging Socialist movement, Chartism and Unitarian arguments that similarly played a role in renegotiating the role of women, notably in the functions women were meant, or asked, to play within marriage. Amongst the many anti-secularist groups, and before the emergence of Socialism as a popular movement, the Freethinkers worked towards making beliefs irrelevant and prejudices a source of discontent ( 12). In drawing these different lines, some philosophical background to political activism such as secularism is given: Comte's positivism ( 15) is said to have paradoxically prompted a reflection on the place of women. Readers will find the section on the "Broad Church Movement" (15-23) particularly interesting. It follows in the footsteps of well-known spokesmen such as Arnold, and lesser-known but currently much worked-upon figures such as Josephine Butler 1 , to delineate the trajectory that led to Christian Socialism as a reaction both against secularism and to socialism at the same time, and their work towards a greater education for all. This section of the introduction finishes with a historiographic bibliography of the relation between religion and women's rights.

4

In what can be regarded as a second part, M. Monacelli moves on to exploring the 'Social turn' of the reforms that took place between 1869 and 1886. She investigates here the function of parties, and associations such as the Social Science Association, in order to analyse influences and pressure that fashioned the politics of the second half of the century. The last two sections focus on the control of private sexual practices and the involvement of men in the policing and thinking about such practices, in particular through the analysis of the case of Charles Bradlaugh (36), as well as women's suffrage. Careful to pay tribute to the role played by women in the progress of their rights in spite of the current object of her study, M. Monacelli concludes with these words: 'Belittling women's achievements was never the intention of this book. The history of women's emancipation is becoming more securely grounded in its intellectual context; it is only fair that its male protagonists should have their share of the limelight." (43) 5 After these two preliminary, scholarly chapters, the book is divided into four sections that each contain nine extracts. Each extract is titled after a quote from the author cited and the primary source is introduced by a biographical note that furthers the contextualisation provided in the introduction. Each section is organized chronologically, which enables the readers to follow the evolution and progress of certain ideas through the century. In section 1, 'Comrades in Struggle', texts are extracted from the works of William Thompson (1775-1833), Reginald John Richardson (1803-1861), John Stuart Mill (1806-1873), W. T. Stead (1849-1912), Edward Aveling (1851-1898) and Eleanor Marx Aveling (1855-1898), George Jacob Holyoake (1817-1906), Robert Percival Downes (1842-1924), William Moore Ede (1849-1935) and Walter Lionel George (1882-1906).

6

In Section 2, 'Provisions to be Made for the Education of Women', the excerpts are from The Rev. Sydney Smith (1771-1845), the Rev. F. D. Maurice (1805-1872), Richard Whately Cooke Taylor (1842-1918), Charles Kingsley (1819-1875), Sir Alexander Grant (1826-1884), William Forsyth (1812-1899), The Rev. John Llewelyn Davies (1826-1916), David George Ritchie (1853-1903) and The Rev. Cecil Grant (1870-1946).

7

Section 3 is entitled 'The Vindication of Women's Civil Rights' and comprises texts by William Bridges Adams (1797-1872), William Johnson Fox (1786-1864), Henry Peter Brougham (1778-1868), Arthur Hobhouse (1819-1904), Henry Fawcett (1813-1884), Karl Pearson (1857-1936), Henry Nevinson (1856-1941), Victor Duval (1885-1945) and Joseph Clayton (1868-1943).

8

The last section is about "Towards a New Sexual Culture" and has another nine texts introduced by a biographical and contextual notice: Richard Carlile (1790-1843), William Cobbett (1763-1835), Robert Owen (1771-1858), George Robert Drysdale (1824-1904), Sir John Robert Seeley (1834-1885), Montague Hughes Cookson (1832-1913), Edward Carpenter (1844-1929), The Rev. Frederick Augustus Morland Spencer (1878-1962) and Henry Havelock Ellis (1859-1939). 9 Amongst the themes and elements that M. Monacelli's book enables readers to highlight, the following examples can give readers a glimpse into some of the challenging terms of the debate. Ritchie's Darwinist feminism offers a radical and powerful argument in favour of the evolution of women's rights, along lines that have not often been commented upon (112-115); Carlile's challenging of gender norms (158) and Nevinson's association of class, social etiquette and the necessity for women to be given equal rights (145) will also strike the readers as sophisticated arguments revealing the dedication of these men to finding new ways of thinking about the equality between men and women; lastly, the presentation of Drysdale as a precursor of Freud in pointing out sexual instincts will surely engage readers in new appreciations of the issue of women's sexuality (169)(170).

10 This book is thus a valuable addition to the debate on the origin of what is now regarded as the feminist movement. By choosing to focus on men's voices on women's rights, the book may be criticised for downplaying the voices of those men that expressed their opposition, if not their hostility, to any progress of the women's cause, but Martine Monacelli, like others, knows that this aspect of anti-feminism has been fairly documented, notably by Lucy Delap. Her work, in addition to providing primary sources and well-structured extracts, gives us an insight into the most recent and seminal works published on the question.

NOTES


Farah Cherfaoui

Valentin Emiya

Liva Ralaivola

Sandrine Anthoine

Frank-Wolfe Algorithm for the m-EXACT-SPARSE Problem

In this paper, we study the properties of the Frank-Wolfe algorithm to solve the m-EXACT-SPARSE reconstruction problem. We prove that when the dictionary is quasi-incoherent, at each iteration, the Frank-Wolfe algorithm picks up an atom indexed by the support. We also prove that when the dictionary is quasi-incoherent, there exists an iteration beyond which the algorithm converges exponentially fast.

Introduction

Given a dictionary of a large number of atoms, the sparse signal approximation problem consists of constructing the best linear combination with a small number of atoms to approximate a given signal. Sparse signal reconstruction is a sub-problem of the sparse signal approximation problem. In the latter case, we suppose that the given signal has an exact representation with m or less atoms from this dictionary. We say that the signal is m-sparse. This subset of atoms is indexed by a set called the support. In this paper, we only consider the sparse signal reconstruction problem, which is called the m-EXACT-SPARSE problem.

Several algorithms have been developed to solve or approximate the m-EXACT-SPARSE problem. The Matching Pursuit algorithm (MP) [START_REF] Stéphane | Matching pursuits with time-frequency dictionaries[END_REF] and Orthogonal Matching Pursuit algorithm (OMP) [START_REF] Chandra Pati | Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition[END_REF] are two fundamental greedy algorithms used for solving this problem. Tropp [START_REF] Tropp | Greed is good: Algorithmic results for sparse approximation[END_REF] and Gribonval and Vandergheynst [START_REF] Gribonval | On the exponential convergence of matching pursuits in quasiincoherent dictionaries[END_REF] proved that, if the dictionary is quasi-incoherent, then at each iteration the MP and OMP algorithms pick up an atom indexed by the support. They also proved that these two algorithms converge exponentially fast. In fact, Tropp in [START_REF] Tropp | Greed is good: Algorithmic results for sparse approximation[END_REF] demonstrates that OMP converges after exactly m iterations, where m is the size of the support. We study in this paper the properties of the Frank-Wolfe algorithm [START_REF] Frank | An algorithm for quadratic programming[END_REF] to solve the m-EXACT-SPARSE problem. The Frank-Wolfe algorithm [START_REF] Frank | An algorithm for quadratic programming[END_REF] is an iterative optimization algorithm designed for constrained convex optimization. It has been proven to converge exponentially if the objective function is strongly convex [START_REF] Guélat | Some comments on Wolfe's 'away step[END_REF] and linearly in the other cases [START_REF] Frank | An algorithm for quadratic programming[END_REF]. The atom selection steps in Matching Pursuit and Frank-Wolfe are very similar. This inspired for example Jaggi and al. [START_REF] Locatello | A unified optimization view on generalized matching pursuit and Frank-Wolfe[END_REF] to use the Frank-Wolfe algorithm to prove the convergence of the MP algorithm when no conditions are made on the dictionary.

In this paper, we use the MP algorithm to prove that the Frank-Wolfe algorithm can have the same recovery and convergence properties as MP. We prove that when the dictionary is quasi-incoherent, the Frank-Wolfe algorithm picks up only atoms indexed by the support. Also, we prove that when the dictionary is quasi-incoherent, the Frank-Wolfe algorithm converges exponentially from a certain iteration even though the function we consider is not strongly convex.

The problem and the algorithm 2.1 The m-EXACT-SPARSE problem

For any vector x ∈ R n , we denote by x(i) its i th coordinate. The support of x is the set of indices of nonzero coefficients:

support(x) = {i|x(i) = 0}. Fix a dictionary Φ = [ϕ 1 , . . . , ϕ n ] ∈ R d×n of n unit- norm vectors. Assume that y is m-sparse, then the m-EXACT- SPARSE problem is to find: arg min x∈R n 1 2 y -Φx 2 2 s.t. x 0 ≤ m
where the l 0 pseudo-norm . 0 counts the number of nonzero components in its argument. This problem has been proven to be NP-hard [START_REF] Davis | Adaptive greedy approximations[END_REF] and has been tackled essentially with two kind of approaches. The first one is the local approach, using a greedy algorithm like MP or OMP. The second approach is a global one where one relaxes the problem. A most popular choice is the l 1 relaxation:

arg min x∈R n 1 2 y -Φx 2 2 s.t. x 1 ≤ β (1)
where . 1 is the l 1 norm. We present, in the next parts, the Frank-Wolfe algorithm [START_REF] Frank | An algorithm for quadratic programming[END_REF] for the m-EXACT-SPARSE problem, and then the recovery properties and convergence rate of this algorithm.

The Frank-Wolfe algorithm

The Frank-Wolfe algorithm solves the optimization problem

min x∈C f (x) s.t. x ∈ C
where f is a convex and continuously differentiable function and C is a compact and convex set. In the original version of the Frank-Wolfe algorithm, each iterate x k+1 is defined as a convex combination between x k and s k with s k = arg min s∈C s, ∇f (x k ) .

In the case of the relaxation of the m-EXACT-SPARSE problem (Equation (1)), f (x) = 1 2 y -Φx 2 2 and C = {x : x 1 ≤ β} = B 1 (β) is the l 1 ball of radius β. Noting that B 1 (β) = conv{±βe i |i ∈ {1, . . . , n}} and that ∇f (x) = Φ t (Φxy), we obtain that s k can be calculated as in line 4 and 5 of Algorithm 1. Note also that we initialize x 0 by zero (line 1) and that we select the convex combination parameter γ k as in line 6.

In the analysis of Algorithm 1, we use the residual r k = y -Φx k whose norm is also the minimized objective function

f (x k ) = 1 2 r k 2 2
.

Algorithm 1: Frank-Wolfe algorithm Data: signal y, dictionary Φ = [ϕ 1 , . . . , ϕ n ], scalar β.

1 x 0 = 0 2 k = 0 3 while stopping criterion not verified do 4 i k = arg max i∈{1,...,n} | ϕ i , Φx k -y | 5 s k = -sign( ϕ i k , Φx k -y )βe i k 6 γ k = arg min γ∈[0,1] y -Φ(x k + γ(s k -x k )) 2 2 7 x k+1 = x k + γ k (s k -x k ) 8 k = k + 1 9 end

Recovery property and convergence rate

For a dictionary Φ, we denote by µ = max

j =k | ϕ j , ϕ k | the co- herence of Φ and by µ 1 (m) = max |Λ|=m max i / ∈Λ j∈Λ | ϕ i , ϕ j | the
Babel function. These two quantities measure how much the elements of the dictionary look alike. More details can be found in [START_REF] Tropp | Greed is good: Algorithmic results for sparse approximation[END_REF].

In this section we present our major results. Theorem 1 gives the recovery property for the Frank-Wolfe algorithm. We prove that when the dictionary is quasi-incoherent (i.e. m < 1 2 (µ -1 + 1)), the Frank-Wolfe algorithm reconstructs every m-sparse signal. Theorem 2 shows that when the dictionary is quasi-incoherent, the Frank-Wolfe algorithm converges exponentially. We recall that a sequence (a k ) ∞ k=0 converges exponentially if: ∀ k ∈ {1, . . . , +∞}, a k+1 ≤ qa k with 0 < q < 1.

Theorem 1. Let Φ ∈ R d×n be a dictionary, µ its coherence, and y = Φx * a m-sparse signal (i.e. | support(x * )| = m). If m < 1 2 (µ -1 + 1), then at each iteration, Algorithm 1 picks up a correct atom, i.e. ∀ k, i k ∈ support(x * ).

Sketch of proof.

The proof of this theorem is very similar to the proof of Theorem 3.1 in [START_REF] Tropp | Greed is good: Algorithmic results for sparse approximation[END_REF].

Theorem 2. Let Φ ∈ R d×n be a dictionary, µ its coherence, and y = Φx * a m-sparse signal (i.e. | support(x * )| = m). If m < 1 2 (µ -1 + 1) and x * 1 < β, then there exists a K such that for all iteration k ≥ K of Algorithm 1, we have:

r k+1 2 ≤ r k 2 1 - ǫ 2 (1 -µ 1 (m -1)) 4β 2
where ǫ = 1 2 (βx * 1 ). Sketch of proof. The general idea of the proof can be summarized as follows. The first step will be to prove that if the dictionary is quasi-incoherent, then the step γ k chosen in line 6 of Algorithm 1 is in (0, 1). A consequence of this is that:

γ k = arg min γ∈R y -Φ(x k + γ(s k -x k )) 2 2 (2) = r k , Φ(s k -x k ) Φ(s k -x k ) 2 2 (3)
We can then write the expression of r k+1 2 2 :

r k+1 2 2 = y -Φx k+1 2 2 = r k + γ k Φ(s k -x k ) 2 2 ,
which yields using Eq. (3):

r k+1 2 2 = r k 2 2 - r k , Φ(s k -x k ) 2 Φ(s k -x k ) 2 2
.

The second step is to bound r k , Φ(s kx k ) . Using Theorem 1, we can show that the sequence of x kx * is bounded by the sequence f (x k )-f (x * ). Since the sequence f (x k )-f (x * ) converges to zero, then the sequence of x kx * also converges to zero. Therefore, there exists an iteration K such that for all k ≥ K: x k ∈ B 2 (x * , ǫ) where B 2 (x * , ǫ) is l 2 ball centered in x * and of radius ǫ. As a result,

x k -ǫ ∇f (x k ) ∇f (x k ) ∈ B 2 (x * , 2ǫ). Since x * 1 + 2ǫ < β, we have B 2 (x * , 2ǫ) ⊆ B 1 (β).
By definition of s k :

s k , ∇f (x k ) ≤ x k -ǫ ∇f (x k ) ∇f (x k ) , ∇f (x k ) .
Noting that ∇f (x k ) = -Φ t r k , one obtains

r k , Φ(s k -x k ) ≥ ǫ Φ t r k .
By Theorem 1, r k lies in the linear span of atoms indexed by support(x * ). Since we assume that these atoms are linearly independent, we have

Φ t r k ≥ λ Φ support(x * ) min r k 2 ,
where Φ support(x * ) is the matrix whose columns are the atoms indexed by support(x * ) and λ

Φ support(x * ) min its smallest singular. So, By Lemma 2.3 of [8], λ Φ support(x * ) min ≥ (1 -µ 1 (m -1)
) and we obtain:

r k , Φ(s k -x k ) ≥ ǫ(1 -µ 1 (m -1)) r k 2 .
Finally, we show that Φ(s kx k ) 2 ≤ 2β using the fact that Φ((s kx k )) 2 ≤ s kx k 1 since the ϕ i are of unit norm.

Note that Tropp in [START_REF] Tropp | Greed is good: Algorithmic results for sparse approximation[END_REF] has already proved that if the dictionary is incoherent, then µ 1 (m) + µ 1 (m -1) < 1. As a result, 1µ 1 (m -1) is in (0, 1). We also have that ǫ 2 β 2 < 1 because ǫ < β. Finally, since d is greater that 1, we have that ǫ 2 (1-µ1(m-1)) 4β 2 d is in (0, 1). We conclude that Theorem 2 gives the exponential convergence rate of the residual norm. As f (x k ) = 1 2 r k 2 2 , this implies that this theorem also gives the exponential convergence rate of the objective function beyond a certain iteration.

It is possible to guarantee an exponential convergence from the first iteration if β is big enough. Lemma 1 gives a lower bound of β to obtain this result.

Lemma 1 .Φ

 1 Let Φ be a dictionary of coherence µ, y = Φx * a m-sparse signal (i.e. | support(x * )| = m) and ǫ ∈ (0, 1). support(x * ) max λ Φ support(x * ) min then Algorithm 1 converges exponentially from the first iteration. Here, Φ support(x * ) is the matrix whose columns are the atoms indexed by support(x * ).

We proved in Theorem 2 that when the iterates x k enter the ball B 1 (x * , ǫ), the Frank-Wolfe algorithm converges exponentially. The intuition of this lemma is to grow the value of β compared to x * (then ǫ also grows). This implies that the iterates x k enter the ball B 1 (x * , ǫ) earlier and the exponential convergence starts earlier.

This work was supported by the Agence Nationale de la Recherche under grant JCJC MAD (ANR-14-CE27-0002).