The first example of a centro-symmetrical bis(imido)-bridged dinuclear cobalt(III) complex: synthesis via oxidative dehydrogenation and phenoxazinone synthase activity Anangamohan Panja* a,b,c and Philippe Guionneau a,

Introduction

Imines are useful intermediates in the synthesis of biologically active nitrogen-containing organic compounds and pharmaceutical substances; thus the transition metal-promoted oxidation of amines has attracted extensive attention. [START_REF] Murahashi | Transition Metals for Organic Synthesis[END_REF][START_REF] Gómez | [END_REF] Substantial progress has been made in recent years in the development of oxidation reactions for the synthesis of imines from secondary amines; notable examples include stoichiometric oxidation with N-tert-butylphenylsulfinimidoyl chloride 3 or 2-iodoxybenzoic acid (IBX), 4 metal-catalyzed oxidation with H 2 O 2 , t BuOOH, PhIO, or K 2 S 2 O 8 , 5 iridium-catalyzed dehydrogenation, 6 along with osmium, ruthenium, cobalt, copper, and iron catalyzed aerobic oxidation. [7][8][9][10][11] However, comparatively little attention has been paid to the oxidation of primary amines, probably because the corresponding imines usually constitute intermediate products that are rapidly dehydrogenated to nitriles due to the reactivity of the second α-amino hydrogen. 12 Although the oxidative dehydrogenation of amines by the assistance of coordinated metal centres involving a variety of mono-, bi-and multi-dentate ligands has been described, the reports on crystal structures of the imine coordinated metal complexes are scarce. 10,11,13 Therefore, it remains a challenge to structurally characterize the imine-coordinated metal complexes that are expected to disclose some relevant information for the further development of metal-assisted dehydrogenation of coordinated amines.

The design of dinuclear metal complexes incorporating suitable bridging ligands has attracted considerable research interest in recent years, 14,15 primarily due to their relevance for biological processes, molecular electronics and for theoretical studies of electron transfer processes. [16][17][18] As far as biochemistry is concerned, the enzyme phenoxazinone synthase, 19,20 a multicopper oxidase is found naturally in the bacterium Streptomyces antibioticus and catalyzes the oxidative coupling of two molecules of a substituted 2-aminophenol to the phenoxazinone chromophore in the final step for the biosynthesis of positive) spectra were recorded on a MICROMASS Q-TOF mass spectrometer.

Synthesis of [Co 2 (amp) 2 (µ-imp) 2 Cl 2 ]Cl 2 •2H 2 O (1)

A mixture of CoCl 2 •6H 2 O (238 mg, 1.0 mmol) and 2-aminomethylpyridine (216 mg, 2.0 mmol) was mixed together in 40 mL of methanol. The resulting red solution was treated with a mixture of H 2 O 2 and NaOH (4.0 mmol) in 1 mL of water with stirring for 10 min at room temperature, during which time the mixture turned brownish red. The brown mass of complex 1 was isolated by evaporation of solvent under reduced pressure. Recrystallization from methanol afforded crops of brownish red blocks of the complex and were isolated via suction filtration, washed with diethyl ether, and air dried. Yield: 151 mg (42%

X-ray crystallography

A suitable single crystal of complex 1 was coated with N-paratone oil, picked up with a nylon fibre and mounted in the cold nitrogen stream operating at 250 K of a Nonius Kappa CCD diffractometer equipped with a Mo-target rotating anode X-ray source and a graphite monochromator (Mo-Kα, λ = 0.71073 Å). Cell refinement, indexing, and scaling of the data set were performed using the programs DENZO-SMN and SCALEPACK. [START_REF] Otwinowski | Methods in Enzymol[END_REF] The structures were solved by direct methods and subsequent Fourier analyses, and refined by the full-matrix least-squares technique based on F 2 with the SHELXL97 program using all the reflections. [START_REF] Sheldrick | SHELXL97, Program for Crystal Structure Refinement[END_REF] All the non-hydrogen atoms were refined using anisotropic thermal parameters. Hydrogen atoms belonging to the complex were placed at calculated positions and refined as riding atoms with isotropic displacement parameters. All the calculations were performed using the WinGX System, Ver. 1.70.01. [START_REF] Farrugia | [END_REF] Crystal data and details of data collection and refinement for complex 1 are summarized in Table 1. CCDC 883590 contains the supplementary crystallographic data for complex 1.

Kinetics of the oxidation of 2-aminophenol

The phenoxazinone synthase activity of the dinuclear cobalt(III) complex was studied by the reaction with 2-aminophenol at 25 °C in methanol using UV-vis spectroscopy. The kinetics of the aerobic oxidation of 2-aminophenol catalyzed by complex 1 was measured by monitoring the change in the absorbance as a function of time at 433 nm (ε = 24 × 10 3 M -1 cm -1 ), which is characteristic of 2-aminophenoxazin-3-one in methanol. In a first test series, varying amounts of the catalyst, (5-100) × 10 -5 M, were added to a fixed concentration (1 × 10 -2 M) of substrate in methanol (2.0 mL) saturated with dioxygen. In a second test series, a 5 × 10 -5 M solution of the complex was mixed with varying concentrations of substrate (0.5-20 mM) in dioxygen saturated methanol (2.0 mL). The initial rate method was applied to determine the rate of a reaction, and the average initial rate over three independent measurements was Scheme 1 Schematic representation of the synthesis of the dinuclear complex. actinomycin D. 21 This is used clinically for the treatment of Wilms' tumor, gestational choriocarcinoma, and other tumors. 22,23 The synthesis and investigation of functional model complexes for metalloenzymes that mimic the oxidase (oxygenase) activity is therefore of great importance in the development of new and efficient catalysts for oxidation reactions. Only a few functional models based on cobalt have been reported which show phenoxazinone synthase activities. 24 We have recently reported the study of the catechol oxidase/ dioxygenase activity of mononuclear Cu(II) and iron(III) complexes and also stoichiometric oxidation of 2-aminophenol by a mononuclear cobalt(III) complex. 25,26 The catalytic inefficiency of the reported cobalt(III) complex is mainly due to failure of regeneration of the catalyst during the course of the reaction. 26 In order to explore better functional models for the phenoxazinone synthase, herein, we report the synthesis, physicochemical and spectral properties of the first structurally characterized bis(imido)-bridged dinuclear cobalt(III) complex, [Co 2 (amp) 2 (µ-imp) 2 Cl 2 ]Cl 2 •2H 2 O(1) [amp = 2-aminomethylpyridine; imp = 2-iminomethylpyridine anion], derived from the oxidative dehydrogenation of half of the coordinated amine ligands under mild conditions (Scheme 1). The synthesized dinuclear complex shows phenoxazinone synthase-like activity, and the detailed kinetic investigations concerning this activity are described.

Experimental section

Materials and physical measurements

Chemicals such as 2-aminomethylpyridine, 2-aminophenol and cobalt(II) chloride hexahydrate were purchased from Aldrich and Alfa Aesar. All other chemicals and solvents were of reagent or analytical grade and were used without further purification.

Elemental analyses for C, H, and N were performed in a Thermo Fischer Flash EA1112 analyzer. The infrared spectra of the sample in KBr disks were recorded in the 400-4000 cm -1 range on a PerkinElmer Infrared spectrometer. Electrochemical data were collected on an EG&G Princeton Applied Research potentiostat model 263A with a Pt working electrode, glassy-carbon electrode, and Ag/AgCl reference electrode. UV-vis spectra of the complex were measured using a PerkinElmer Lamda-35 spectrophotometer with a 1 cm pathlength quartz-cell. Electrospray ionization mass (ESI-MS such as hydrogen peroxide, is required in some systems to start the reaction and different imines were produced as a result of the oxidative dehydrogenation reactions. Based on the above observations, we envisioned that alkaline hydrogen peroxide may produce a peroxo-adduct of the cobalt(III) complex coordinated to 2-aminomethylpyridine at room temperature, which may rapidly generate radical species that may be responsible for the oxidative dehydrogenation of coordinated amine. Accordingly, the present bis(imido)-bridged dinuclear cobalt(III) complex was synthesized by the reaction of cobalt(II) chloride with 2-aminomethylpyridine in a 1 : 2 molar ratio in methanol, followed by treatment with alkaline H 2 O 2 in aqueous medium at room temperature (Scheme 1). Brownish red crystals of complex 1 were isolated by the slow evaporation of methanol solution in moderate yield. The use of a strong alkaline base helps to deprotonate the resulting imine and to produce the bis(imido)-bridged dinuclear complex. The complex was characterized by elemental analysis, and various spectroscopic and physicochemical methods. The infrared spectral data of the complex are reported in the Experimental section with tentative assignments. The IR spectrum of the complex shows a broad band observed at 3448 cm -1 assigned to solvent water molecules that illustrates the involvement of strong hydrogen bonding by water molecules in the crystal packing. The most significant feature of the IR spectra is the presence of a band at 1643 cm -1 , attributed to ν CvN for imine functionalities, which is in good agreement with the X-ray crystal structure of the complex (vide infra).

Description of the crystal structure of complex 1

The molecular structure of the dinuclear cobalt(III) complex was determined by the single crystal X-ray diffraction technique. Crystallographic data and the details of data collection and refinement for the complex are assembled in Table 1. The important bond lengths and angles are summarized in Table 2. The compound crystallizes in the triclinic P1 ˉspace group, where the asymmetric unit corresponds to half of the molecule. The perspective view of the complex cation of 1 along with the selected atom numbering scheme is depicted in Fig. 1. The complex cation is formed by a bis(imido)bridged Co III dimer, and in the molecular structure two metal

Symmetry code: # = 1 -x, -y, 1 -z.
determined by linear regression from the slope of absorbance versus time plot.

Results and discussion

Synthesis and general characterizations

The reaction of 2-aminomethylpyridine (amp) with a cobalt(II) salt generally yields bis(amp) or tris(amp) chelated cobalt(III) complexes by the oxidation of molecular dioxygen. 30 It is well established that peroxo-bridged dinuclear cobalt(III) complexes having various nitrogen donor ligands are usually formed under aerobic conditions which thereafter produces imine functionalities of the coordinated amines in basic media by the degradation of the peroxo-intermediate. 31,32 In most cases, an elevated temperature was employed in order to facilitate such oxidative dehydrogenation of amines. Based on this view, we intended to produce an imine functionality by the reaction of 2-aminomethylpyridine with cobalt(II) chloride in alkaline media. The spectrum of the resulting blue solution is consistent with the cobalt(II) species 33 (Fig. S1 †) and on standing for at least half an hour no significant spectral modification was noticed. Thus, under the present reaction conditions either the reactive cobalt-dioxygen species was not formed or it was too weak to assist the oxidative dehydrogenation of the coordinated primary amine. However, peroxo-adducts of cobalt(III) may be produced by the direct treatment of hydrogen peroxide with the cobalt(II) complex in the presence of a base at low temperature. 34 It is also reported that an additional oxidant, octahedral geometry in which the coordinated nitrogen atoms of primary amine were not oxidatively dehydrogenated. 30 In the present complex, the main feature of the structure is the modification of the coordinated ligand: the C1-N4 distance [1.263(5) Å] indicates the presence of an imine moiety, which is consistent with the spectroscopic studies (cf. below). The structure of complex 1 can only be compared with an imidomalonato-bridged binuclear Co III complex which was synthesized by the reaction of CoCl 2 (tren)Cl•H 2 O [tren = tris-(2-aminoethyl)amine] with the α-aminomalonate anion in the presence of triethylamine at elevated temperature. In the crystal structure, an imido-bridge is stabilized by the simultaneous coordination of malonato groups to the metal centres. 35 But in the present structure, the complex is simply stabilized by the two imido-bridges, and this remarkable stability arises mainly from the strong Co-N imino bonding, indicative of the significant cooperation of two imido-bridges in the molecule. To the best of our knowledge, complex 1 is the first example of a doubly imido-bridged cobalt(III) complex.

Dehydrogenation of coordinated 2-aminomethylpyridine

Dehydrogenation of the coordinated amine ligand was further established by UV-vis and IR spectroscopic methods. For this purpose, shortly after mixing cobalt(II) chloride and 2-aminomethylpyridine, the UV-vis spectrum of a methanol solution was recorded. It displays an absorption maximum at 252 nm (ε = 5.6 × 10 4 ) and a shoulder at 368 nm. Upon addition of an aqueous solution of alkaline hydrogen peroxide, a vigorous reaction had been observed and ultimately resulted in a dark brown solution. The UV-vis spectroscopy revealed two new absorption bands at 344 nm (ε = 1.2 × 10 4 ) and 541 nm (ε = 2.4 × 10 3 ) with isosbestic points observed at 282, 381 and 497 nm, clearly supporting the formation of a new product (Fig. 2). The band at 541 nm can be assigned to the deprotonated imine to metal charge transfer; the facile charge transfer is consistent with the crystal structure of the dinuclear complex where imine nitrogen atoms are firmly bound to the metal centres.

In order to get further insight into the solution species, the mass spectroscopy (ESI, positive mode) of the resulting solution was carried out. The base peak at m/z = 307.0298 observed in the mass spectra can be assigned to the monocationic halfmolecule of complex 1, [Co(amp)(imp)Cl] + , confirmed by the isotopic distribution patterns (Fig. S3 †). In addition, a weak peak observed at m/z = 613.1391 corresponds to the monocation of dimer, [1-H] + , consistent with the isotopic distribution patterns. The mass spectroscopic results unambiguously suggest that the major species produced by the oxidative Symmetry code: # = 1x, -y, 1z.

centres are related to each other through an inversion centre which bisects Co1 and Co1a and also N4 and N4a. A C 2 rotation about this point of origin transposes the coordinates of half of the atoms in the molecule to their 'a' equivalents.

The geometry of each of the Co III ions can be best described as a distorted octahedral structure; the CoN 5 Cl coordination environment comprises two imido-bridged nitrogen atoms, one amine nitrogen atom, two pyridyl nitrogen atoms, and one chloride ion. Importantly, the imino ligand lies on the same plane along with the Co III centre as expected from their electronic structure, and the maximum deviation from the mean plane defined by N4-C1-C2-C3-C4-C5-N3 observed for N3 is 0.012 Å. The C-N imino bond distance of 1.263(5) Å is significantly shorter than the C-N amine bond distance of 1.478(5) Å as expected. 35 The Co-N amine , average Co-N imino , mean Co-N pyridine , and Co-Cl bond lengths are 1.947(3), 1.895(3), 1.965(3) and 2.264(1), respectively, which are in good agreement with those found in the related cobalt(III) complexes. 30,36 The Co-N imino bond length is shorter than the Co-N amine bond length, indicating that the deprotonated bridging imino-N coordinates more strongly to the metal centre than the primary amine group. The average bite angle of the bi-dentate ligands, 82.54( 14)°, deviates significantly from 90°, which is the main reason for the structural distortion of complex 1 from the ideal geometry. The intramolecular separation between the two imido-bridged Co III centres is 2.925(1) Å, which is comparable to the metal-metal separations found in many multinuclear metalloenzymes showing oxidase activity. 37 The molecular packing of the complex is stabilized by several π⋯π stacking interactions between the aromatic rings (Fig. S2 †), and the moderately strong hydrogen-bonding interactions involving primary amines, solvent water molecules and Cl -ions (Table 3). Co III ions in mononuclear complexes with 2-aminomethylpyridine have been reported to be six-coordinate with peroxide in basic medium at low temperature. 34 UV-vis spectral data suggest that molecular dioxygen is unable to start oxidative dehydrogenation of the coordinated primary amine, and an external activator, such as hydrogen peroxide, is required for this chemical transformation (Fig. S1 †). As with all the reported cobalt complexes which show oxidative dehydrogenation of amines, 31,32,[38][39][40][41] in the present study, the peroxoadduct rapidly decomposes and subsequently produces bis-(imido)-bridged dinuclear cobalt(III) complex 1.

Electrochemical studies

The function of a redox active metalloenzyme is to catalyze a redox process; therefore, a study of the electrochemical properties of their models is of profound importance especially under identical experimental conditions as the kinetic studies. The electrochemical data of the complex have been recorded in methanol containing 0.1 M tetrabutylammonium perchlorate as a supporting electrolyte at a glassy-carbon working electrode and using a Ag/AgCl reference electrode. The cyclic voltammogram of complex 1 shows an electrochemically irreversible peak at -0.70 V both under aerobic and inert conditions which is assigned to the reduction of the cobalt(III) centres to cobalt(II) (Fig. 3). The irreversible nature of the electrochemical process may be due to the instability of Co II species on the electrode surface presumably due to the loss of a labile chloride ion from the coordination sphere. Moreover, the Co II species is formed at a quite negative potential which indicates that complex 1 may not be a suitable catalyst for oxidation reactions although the regeneration of catalyst is quite favourable under aerobic conditions. An irreversible oxidation response at 0.54 V attributed to the oxidation of 2-aminophenol under identical conditions supports this fact (Fig. S5 †). But the electrochemical scenario of the reactants may be Fig. 2 UV-vis spectra of cobalt(II) chloride with 2-aminomethylpyridine in a 1 : 2 molar ratio in methanol (dashed line) and upon addition of alkaline hydrogen peroxide (solid line). dehydrogenation of coordinated 2-aminomethylpyridine in solution is 1.

Infrared spectroscopy is also a good tool for monitoring the conversion of dehydrogenation of an amine. The IR spectrum of a mixture of cobalt(II) chloride and 2-aminomethylpyridine in a 1 : 2 molar ratio in methanol shows a band at 1613 cm -1 that might be due to the CvN stretching of pyridyl moieties [Fig. S4 †]. Upon addition of the alkaline hydrogen peroxide the peak was shifted to a higher frequency with a band maximum at 1640 cm -1 , which clearly indicates the conversion of a primary amine to an imine via an oxidative dehydrogenation process. The ease of oxidative dehydrogenation of the coordinated 2-aminomethylpyridine is probably due to the conjugated structure of the resulting imine, which is consistent with the crystal structure of complex 1. The spectroscopic results, together with the crystal structure of the complex, clearly demonstrate the oxidative dehydrogenation of the coordinated primary amine to imine functionalitythe metal ion associated with hydrogen peroxide in basic media plays a significant role in this oxidative dehydrogenation process. The presence of an alkali metal cation is strong enough to deprotonate the imine group of coordinated ligands, resulting in the formation of the imido-bridged dinuclear cobalt(III) complex.

Most of the previous reports on metal ion promoted dehydrogenation of the coordinated amines by cobalt complexes describe that the key intermediate for the oxidation of amines is the peroxo-adduct of the cobalt(III) complex. 31,32,[38][39][40][41] In the presence of a base, the cobalt(II) species activates molecular dioxygen for the catalytic dehydrogenation of the coordinated amine groups. The elevated temperature was used in most cases to facilitate the dehydrogenation process. However, a similar intermediate may be generated by the treatment of a coordinatively unsaturated cobalt complex with hydrogen chromophore formed over time. The product was isolated after 24 h of reaction, and analyzed by GC, which showed the formation of 2-aminophenoxazine-3-one as the sole product with 100% conversion.

The reaction kinetics of the oxidation of OAPH was determined by the method of initial rates by monitoring the growth of the 433 nm band of the product 2-aminophenoxazine-3-one. The concentration of the substrate OAPH was always kept at least 10 times higher than that of the complex to maintain the pseudo-first-order condition. All the kinetic experiments were carried out at a constant temperature of 25 °C, monitored with a thermostat. In order to explore the dependence of the rate on the substrate concentration, a fixed concentration of complex 1 (5 × 10 -5 M) was treated with varying concentrations of the substrate from 5 × 10 -4 M to 1 × 10 -2 M, and the formation of 2-aminophenoxazine-3-one was monitored over time at 433 nm. Initial rates were determined by linear regression from the slope of the absorbance versus time plot for each concentration of the substrate and each experiment was repeated three times with a variation of ±4%. The average rate of the reactions shows that the rate is first order at the region of low concentrations of the substrate OAPH and zero order at its higher concentrations (Fig. 5). This observation indicates that the oxidation of OAPH proceeds through the formation of a relatively stable intermediate, a complex-substrate adduct, presumably formed by the coordination of substrate to catalyst by the replacement of a chloride ion, followed by the irreversible substrate oxidation in the rate-determining step of the catalytic cycle. This type of saturation rate dependency on the concentration of the substrate may be explained by considering the Michaelis-Menten model, originally developed for the enzyme kinetics. Analysis of the experimental data yielded an dramatically changed if a catalyst-substrate adduct is formed during the course of a redox transformation, and in such a situation facile oxidation of the coordinated 2-aminophenol by aCo III centre through an inner-sphere electron transfer pathway would be possible. In fact, the rate saturation kinetic profile (cf. below) strongly suggests the formation of a complex-substrate adduct prior to the electron transfer process.

Phenoxazinone synthase activity and kinetic studies

The catalytic activity of complex 1 for the oxidation of 2-aminophenol (OAPH) by molecular O 2 was evaluated spectrophotometrically in dioxygen-saturated methanol at 25 °C. The reactivity studies were performed in methanol because of good solubility of the complex, substrate and their product. In order to avoid the autoxidation of the substrate by air the catalytic activity was checked in the absence of an added base. Before going into the detailed kinetic study, it is necessary to check the ability of the complex to oxidize 2-aminophenol. For this purpose, a 5 × 10 -5 M solution of complex 1 was treated with 100 equivalents of OAPH under aerobic conditions. The course of the reaction was followed by UV-vis spectroscopy by monitoring the increase in the absorbance at 433 nm, corresponding to the formation of the phenoxazinone chromophore. The time dependent spectral change for the period of 2 h after the addition of OAPH to the methanol solution of the complex is depicted in Fig. 4. The spectral scan reveals the increase of peak intensity at 433 nm characteristic of the phenoxazinone Fig. 4 Time resolved UV-vis spectral changes for the oxidation of 2-aminophenol (5 × 10 -3 mol dm -3 ) catalyzed by the complex (5 × 10 -5 mol dm -3 ) for up to 2 h of reaction in dioxygen-saturated methanol at 25 °C. equilibrium constant (K M ) value of 1. 57 × 10 -3 and a V max value of 1.91 × 10 -7 M -1 s -1 ; these are also verified by the Lineweaver-Burk plot. The k cat value is obtained by dividing the V max by the concentration of the complex used, and is found to be 3.82 × 10 -3 s -1 . Although the turnover rate of 13.75 h -1 is moderate enough, the reactivity cannot be compared to the reported functional models of phenoxazinone synthase as, in most of the cases, turnover numbers are not evaluated. 42,43 Moreover, for a particular substrate concentration, varying the complex concentration, a linear relationship for the initial rates was obtained, which shows a first-order dependence on the complex concentration (Fig. S6 †). Based on the above results, a possible mechanism for OAPH oxidation catalyzed by complex 1 is proposed (Scheme 2). At the first step OAP forms an adduct with the dinuclear cobalt(III) complex, which yields an OAP radical in the rate determining step. The OAP radical may generate o-benzoquinone monoamine (BQMI) in many ways including the disproportionation of the OAP radical itself. Ultimately, 2-aminophenoxazine-3-one is produced via several oxidative dehydrogenation processes involving OAPH, O 2 , and BQMI as shown in Scheme 2.

Conclusions

A dinuclear cobalt(III) complex was synthesized by the oxidative dehydrogenation of coordinated 2-aminomethylpyridine in the presence of alkaline hydrogen peroxide. The X-ray crystal structure reveals an unprecedented bis(imido)-bridged dinuclear cobalt(III) complex in which both the octahedrally coordinated metal centres are related to each other through an inversion centre. The spectroscopic results suggest that the alkaline hydrogen peroxide plays a significant role in the oxidative dehydrogenation of the amine under mild conditions. The complex was found to be an excellent functional model of phenoxazinone synthase, catalyzing the oxidative coupling of 2-aminophenol to the corresponding phenoxazinone chromophore in a dioxygen saturated methanol medium via a catalyst-substrate intermediate. An apically coordinated labile chloride ion apparently helps in the formation of a catalystsubstrate intermediate, and thereby increases the catalytic efficiency of the complex. The metal-metal separation of 2.925 Å in the molecule is comparable to the separations found in many multinuclear metalloenzymes, and thus the present report would encourage chemists to synthesise imidobridged di-or multi-nuclear complexes for the development of functional models of various metalloenzymes.

Fig. 1

 1 Fig. 1 ORTEP diagram of complex cation 1 showing the atom numbering scheme. Ellipsoids are drawn at the 50% probability level. H atoms are omitted for clarity. Symmetry code: a = 1x, -y, 1z.

Fig. 3

 3 Fig.2UV-vis spectra of cobalt(II) chloride with 2-aminomethylpyridine in a 1 : 2 molar ratio in methanol (dashed line) and upon addition of alkaline hydrogen peroxide (solid line).Fig.3Cyclic voltammogram of the complex in methanol containing TBAP as the supporting electrolyte at a scan rate of 100 mV s -1 .

Fig. 5

 5 Fig.5Plot of the initial rates at the OAPH concentrations for the oxidation reaction catalyzed by the complex at a fixed concentration in dioxygen-saturated methanol at 25 °C. The inset shows the reciprocal Lineweaver-Burk plot.

  ). Anal Calcd for C 24 H 30 Cl 4 Co 2 N 8 O 2 : C, 39.91%; H, 4.19%; N, 15.51%. Found: C, 40.08%; H, 4.22%; N, 15.36%. IR (KBr, cm -1 ): 3448 (br; ν O-H ); 3204 (w; ν N-H ); 3112-2972 (w; ν C-H ); 1643 (s; ν CvN ).

Table 1

 1 Crystal data and structure refinement of dinuclear complex 1

	Empirical formula	C 24 H 30 Cl 4 Co 2 N 8 O 2
	Formula weight	722.22
	Temperature (K)	250(2)
	Wavelength (Å)	0.71073
	Crystal system Space group	Triclinic P1 ā
	(Å)	8.5370(10)
	b (Å)	8.9420(10)
	c (Å)	10.680(2)
	α (°)	70.914(6)
	β (°)	75.052(6)
	γ (°)	75.559(7)
	Volume (Å 3 )	732.14(18)
	Z D calc (g cm -3 ) Absorption coefficient (mm -1 )	2 1.638 1.536
	F(000)	368
	Crystal size (mm)	0.10 × 0.05 × 0.03
	θ Range for data collection (°)	4.12-26.37
	Limiting indices	-01 ≤ h ≤ 8, -11 ≤ k ≤ 11,
		-13 ≤ l ≤ 13
	Reflections collected/unique	4893
	Independent reflection/R int	2970/0.0493
	Observed reflections	2046
	Completeness to 2θ (%)	99.1
	Data/restraints/parameters	2970/0/181
	Goodness-of-fit on F 2	1.011
	Final R indices [I > 2σ(I)]	R 1 = 0.0478, wR 2 = 0.1046
	R indices (all data) Largest diff. peak/hole (e Å -3 )	R 1 = 0.0832, wR 2 = 0.1187 0.937/-0.436

Table 2

 2 Selected bond lengths (Å) and bond angles (°) for 1

	Co1-N1	1.981(3)	Co1-N4	1.884(3)
	Co1-N2	1.947(3)	Co1-N4#	1.907(3)
	Co1-N3	1.950(3)	Co1-Cl1	2.2645(11)
	Co1-Co1#	2.9255(11)
	N4-Co1-N1	173.84(13)	N2-Co1-N3	91.04(14)
	N4-Co1-N1#	96.72(13)	N4-Co1-N4#	79.00(14)
	N2-Co1-N1	82.10(14)	N4-Co1-Cl1	90.97(10)
	N3-Co1-N1	100.25(14)	N4-Co1-Cl1#	92.37(10)
	N4-Co1-N2	93.40(13)	N2-Co1-Cl1	175.38(10)
	N4-Co1-N2#	89.93(13)	N3-Co1-Cl1	87.93(10)
	N4-Co1-N3	83.98(14)	N1-Co1-Cl1	93.64(10)
	N4-Co1-N3#	162.98(14)	Cl1-Co1-Co1#	92.17(4)

Table 3

 3 Important hydrogen bonds (Å, °) for dinuclear complex 1

	D-H⋯A	D -H	H ⋯A	D ⋯A	< D -H⋯A
	N2-H2B⋯Cl2#	0.91	2.302(2)	3.166(4)	158.3(2)
	O2-H2O⋯Cl2	1.00	2.239(2)	3.234(5)	173.7(3)
	O2-H1O⋯Cl2#	0.94	2.295(2)	3.219(5)	167.4(3)

Acknowledgements

The authors are grateful to the University Grants Commission, India for the financial support of this research (Sanction No. PSW-173/11-12). The instrumental facility from Jadavpur University, India is also gratefully acknowledged.


La loi de Henry et la disparition de la phase gazeuse

Résumé : Pour un système diphasique (liquide-gaz) à deux composants (eauhydrogène) on considère la formulation de la dissolution éventuelle de l'hydrogène dans la phase aqueuse. On montre comment la loi de Henry entre dans un diagramme de phase. Le problème est alors formulé comme un ensemble d'équations aux dérivées partielles non-linéaires avec des contraintes de comp[lémentarité.

Mots-clés : Milieu poreux, écoulement diphasique, dissolution, stockage profond de déchets nucléaires

Introduction

Recently the Couplex-Gas benchmark (http://www.gdrmomas.org/ex qualifications.html) was proposed by Andra and MoMAS in order to improve the simulation of the migration of hydrogen produced by the corrosion of nuclear waste packages in an underground storage. This is a system of two-phase (liquid-gas) flow with two components (hydrogen-water). This is similar to a black-oil model but the benchmark was proposing to use in Henry's law to model the dissolution of hydrogen. This benchmark generated some interest and engineers encountered difficulties in handling the appearance and disappearance of the phases.

Recently two other papers [START_REF] Bourgeat | Two phase partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository[END_REF], [START_REF] Abadpour | Method of negative saturations for multiple compositional flow with oversaturated zones[END_REF] adressed the formulation of this problem.

In this paper, for the sake of the simplicity of exposition, we will assume that only hydrogen is in the gas phase (water does not vaporize). We discuss how to extend Henry's law to the case where the gas phase disappears. This is done by showing how Henry's law fits in a phase diagram. The resulting formulation is a set of partial differential equations with complementarity constraints. A similar formulation is also presented in [START_REF] Bourgeat | Two phase partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository[END_REF] and it also can be viewed as a variation of the formulation of the black oil model presented in [START_REF] Chavent | Mathematical Models and Finite Elements for Reservoir Simulation[END_REF].

Mathematical formulation

We first describe the equations and some simplifying hypotheses. Since we consider a problem where the gas phase can disappear while the liquid phase cannot disappear we will consider a formulation using the saturation and the pressure of the liquid phase as the two main unknowns.

Fluid phases

ℓ and g are the respective indices for the liquid phase and the gas phase. Darcy's law reads

q i = -K(x)k i (s i )(∇p i -ρ i g∇z), i = ℓ, g, (2.1)
where K is the absolute permeability. Assuming that the phases occupy the whole pore space, the phase saturations s i , i = ℓ, g satisfy

0 ≤ s i ≤ 1, s ℓ + s g = 1.
The mobilities are denoted by k i (s i ) = k r i (s i ) µ i with k ri the relative permeability, and µ i the viscosity of the phase i = ℓ, g, assumed to be constant. The mobility k i is an increasing function of s i such that k i (0) = 0, i = ℓ, g. The phase pressures are related through the capillary pressure law

p c (s ℓ ) = p g -p ℓ ≥ 0. (2.2)
assuming that the gas phase is the nonwetting phase. The capillary pressure is a decreasing function of the saturation s ℓ such that p c (1) = 0.

In the following we will choose s ℓ , p ℓ as the main variables since we assume that the liquid phase cannot disappear for the problem under consideration.

Fluid components

We consider two components, water and hydrogen denoted by the indices j = w, h. The mass density of the phase i is

ρ i = ρ i w + ρ i h , i = ℓ, g,
where ρ i w and ρ i h are the mass concentrations of water and of hydrogen in the phase i, i = ℓ, g. The mass fractions are

χ i h = ρ i h ρ i , χ i w = ρ i w ρ i , i = ℓ, g.
Obviously we have χ i w + χ i h = 1, i = ℓ, g. We assume that the liquid phase may contain both components, while the gas phase contains only hydrogen, that is the water does not vaporize. In this situation we have

ρ g w = 0, ρ g = ρ g h , χ g h = ρ g h ρ g = 1, χ g w = 0.
A third main unknown will be χ ℓ h .

Conservation of mass

We introduce the molecular diffusion fluxes j ℓ i , i = w, h for the diffusion of component i in the liquid phase. They must satisfy j ℓ h + j ℓ w = 0 and we have

j ℓ h = -φs ℓ ρ ℓ D ℓ h ∇χ ℓ h . (2.
3)

The coefficients D ℓ h is a molecular diffusion coefficient. Conservation of mass applied to each component, water and hydrogen, gives

φ ∂ ∂t (s ℓ ρ ℓ χ l w ) + div(ρ ℓ χ l w q ℓ + j ℓ w ) = Q w , φ ∂ ∂t (s ℓ ρ ℓ χ ℓ h + s g ρ g ) + div(ρ ℓ χ ℓ h q ℓ + ρ g q g + j ℓ h ) = Q h . (2.4)
We assume also that the gas is slightly compressible, that is ρ g = C g p g with C g the compressibility constant, and that the liquid phase is incompressible, that is ρ ℓ is constant.

3 Phase equilibrium

Phase diagram

A phase diagram is a figure like that shown in Figure 3.1 where a curve f (P, C) = 0 separates two zones, a first zone defined by f (P, C) > 0 where the systeme is monophasic -liquid or gas -and a second zone defined by f (P, C) < 0 where the system is diphasic -liquid + gas. It gives the repartition of two components, like hydrogen and water, into the two phases, liquid and gas. For the example we are concerned with, that is the eventual disappearance of the gas phase, we are interested in the left part of the phase diagram. There we have either s l = 1 and f (P, C) ≥ 0 in the liquid zone, or s l < 1 and f (P, C) ≤ 0 in the diphasic zone. This can be written in terms of the following set of complementarity constraints

(1 -s l )f (P, C) = 0, 1 -s l ≥ 0, f (P, C) ≥ 0.
The variable C is the total concentration of hydrogen, that is in the diphasic zone C = ρ ℓ h + ρ g (since we assumed that only hydrogen can be present in the gas phase), in the liquid zone liquide C = ρ ℓ h and in the gas zone C = ρ g since we assumed that the water cannot vaporize.

The variable P is called the "gas pressure", P = p g in the gas and diphasic zone. To extend this definition to the liquid zone we express p g in terms of the capillary pressure, p g = p ℓ + p c (s ℓ ), and define P = p ℓ + p c (s ℓ ). Thus in the liquid zone we have s ℓ = 1, p c (s ℓ ) = 0 et P = p ℓ . The definitions of the variables P and C are shown in Table 1. liquid liquid + gas gas RR n 6891

s ℓ = 1 0 < s ℓ < 1 s ℓ = 0 P = p ℓ p g = p ℓ + p c (s ℓ ) p g C = ρ ℓ h ρ ℓ h + ρ g ρ g f (P, C) ≥ 0 ≤ 0 ≥ 0

Extension of Henry's law

In the following we focus on the interface between the liquid zone and the diphasic zone. In the presence of the gas phase Henry's law reads

Hp g = ρ ℓ h (3.1)
with H the Henry constant. We can integrate this relation into a phase diagram formulation which includes the situation with no gas phase. This formulation includes a set of complimentarity constraints. Henry's law is defined only in the diphasic zone. In this zone we have HP -C = Hp g -(ρ ℓ h + ρ g ) = -ρ g < 0, and when we move to the curve separating the liquid and diphasic zones we have HP -C = ρ g → 0. Above this curve is the liquid zone HP -C = Hp ℓ -ρ ℓ h > 0. Thus using Henry's law is actually replacing the part of the curve f (P, C) = 0 which separates the liquid and diphasic zones by the straight line HP -C = 0. This is shown in Figure 3.2 where only the left part of the phase diagram is shown with the line separating the liquid zone and the diphasic zone.

Indeed, either the gas phase exists, 1 -s ℓ > 0, Henry's law applies and H(p ℓ + p c (s ℓ )) -ρ ℓ h = 0, or the gas phase does not exist, s ℓ = 1, p c (s ℓ ) = 0 and Hp ℓ -ρ ℓ h > 0 which says that for a given pressure p ℓ the concentration is too small for the hydrogen component to be partly gaseous, or conversely for a given concentration ρ ℓ h the pressure p ℓ is too large for the hydrogen component to be partly gaseous. These cases can be written as the complementarity constraints Finally we end up with a system of nonlinear partial equations -conservation equations and Darcy laws -with nonlinear complementarity constraints describing the transfer of hydrogen between the two phases, the unknowns being s ℓ , p ℓ , χ ℓ w :

(1 -s ℓ ) H(p ℓ + p c (s ℓ )) -ρ ℓ h = 0, 1 -s ℓ ≥ 0, H(p ℓ + p c (s ℓ )) -ρ ℓ h ≥ 0. ( 3
φ ∂ ∂t s ℓ ρ ℓ (1 -χ ℓ h ) + div ρ ℓ (1 -χ ℓ h )q ℓ -j ℓ h = Q w φ ∂ ∂t s ℓ ρ ℓ χ ℓ h + (1 -s ℓ )C g (p ℓ + p c (s ℓ )) + div ρ ℓ χ ℓ h q ℓ + C g (p ℓ + p c (s ℓ ))q g + j ℓ h = Q h q ℓ = -K(x)k ℓ (s ℓ )(∇p ℓ -ρ ℓ g∇z) q g = -K(x)k g (1 -s ℓ )(∇(p ℓ + p c (s ℓ )) -C g (p ℓ + p c (s ℓ ))g∇z) j ℓ h = -φs ℓ ρ ℓ D ℓ h ∇χ ℓ h (1 -s ℓ ) H(p ℓ + p c (s ℓ )) -ρ ℓ χ ℓ h = 0, 1 -s ℓ ≥ 0, H(p ℓ + p c (s ℓ )) -ρ ℓ χ ℓ h ≥ 0. (4.
1) This formulation is valid whether the gas phase exists or not. A similar system was presented in [START_REF] Bourgeat | Two phase partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository[END_REF]. We note that when the gas phase disappears, i.e. s ℓ = 1, the second equation in (4.1) becomes a standard transport equation of hydrogen as a solute in the liquid phase.

Assume that we discretize problem (4.1) with cell centered finite volumes and we denote by N , the number of degrees of freedom for s ℓ , p ℓ , χ ℓ h which is equal to the number of cells. We introduce x ∈ R 3N , the vector of unknowns for s ℓ , p ℓ , χ ℓ h , H : R 3N → R 2N , the vector of the discretized component conservation equations, F : R 3N → R N , the discrete vector for 1 -s ℓ , G : R 3N → R N , the discrete vector for H(p ℓ + p c (s ℓ )) -ρ ℓ χ ℓ h .

Then the problem can be written in compact form

H(x) = 0, F (x) ⊤ G(x) = 0, F (x) ≥ 0, G(x) ≥ 0.
Building a solver to solve properly such a problem is our current task. This task is undertaken in the field of numerical optimization. For a survey on complementarity problems see [START_REF] Facchinei | Finite Dimensional Variational Inequalities and Complementarity Problems[END_REF].

There are clearly alternative ways to solve this problem without using such a solver. See for recent examples [START_REF] Bourgeat | Two phase partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository[END_REF], [START_REF] Abadpour | Method of negative saturations for multiple compositional flow with oversaturated zones[END_REF]. However nonlinear problems with complementary constraints occur in many circumstances as for example in a black oil model [START_REF] Chavent | Mathematical Models and Finite Elements for Reservoir Simulation[END_REF] or in problems with dissolution-precipitation. In [START_REF] Kraütle | General Multi-Species Reactive Transport Problems in Porous Media: Efficient Numerical Approaches and Existence of Global Solutions[END_REF] this latter problem was solved using a semi-smooth Newton method inspired by [START_REF] Kanzow | Inexact semismooth Newton methods for large-scale complementarity problems[END_REF]. Therefore we believe that it is worthwhile to build a general purpose appropriate solver for nonlinear problems with complementarity constraints, and the problem of a liquid-gas system with dissolution of hydrogen is a very good test problem.

Conclusion

We showed how to extend Henry's law to the case where the gas phase disappears by plugging it into a phase diagram formulation. This results into the formulation of a two-phase (liquid-gas) two-component (water-hydrogen) system with possible dissolution of hydrogen as a system of nonlinear partial differential equations with a set of nonlinear complementarity constraints.

Figure 3 . 1 :

 31 Figure 3.1: A standard phase diagram.

Figure 3 . 2 :INRIA 4

 324 Figure 3.2: Part of the phase diagram separating the liquid and diphasic zones when Henry's law is used.

Table 1 :

 1 Definition of the variables P et C in the phase diagram.

RR n 6891

* This work was partially supported by GdR MoMaS (PACEN/