Grigorii S Astakhov

Alexey N Bilyachenko

Alexander A Korlyukov

Mikhail M Levitsky

Lidia S Shul'pina

Xavier Bantreil

Frédéric Lamaty

Anna V Vologzhanina

Elena V Shubina

Pavel V Dorovatovskii

Fredeŕic Lamaty

Elena S Shubina

Dmytro S Nesterov

Armando J L Pombeiro

Georgiy B Shul'pin

High-Cluster (Cu 9 ) Cage Silsesquioxanes: Synthesis, Structure, and Catalytic Activity

come

■ INTRODUCTION

Hybrid inorganic-organic molecular materials are the focus of both academia and industry chemists because of their unusual reactivity, structure formation principles, and numerous potential applications. 1 Among the best models for such hybrid materials are derivatives of siloxanes and molecular combinations of inorganic (Si-O-Si) skeletons and organic (R-Si) surroundings. 2 An exclusive feature of monosubstituted siloxanes (silsesquioxanes of the general formula [RSiO 1.5 ] n ) is an ability to form different polyhedral clusters. 3 These well-defined polyhedral structures are the objects of utmost interest as precursors of materials in high demand. 4 In turn, it should be mentioned that silsesquioxanes are suitable matrixes for the production of cagelike metal-containing derivatives. 5 Important data concerning the reactivity and structure formation were reported for the cases when additional ligands (e.g., 1,2,3,4,5-pentamethylcyclopentadiene, 6 silylamido, 7 and triethylamine 8 ) were used along with the silsesquioxane one in the assembly of polyhedral metallacomplexes. In all of these works, the "ligands-synergy" principle was reported when two types of ligands (silsesquioxane and organic ones) coordinated metal ions of the cagelike product.

■ RESULTS AND DISCUSSION

Here we discuss a different scenario of additional ligand involvement in the cage metallasilsesquioxane synthesis. First, the self-assembly reaction of the in situ formed sodium phenylsiloxanolate [(PhSiO 1.5 )(NaO 0.5 )] n and copper(II) chloride in the presence of 1,10-phenanthroline (ethanol media) gave unprecedented cagelike Cu 9 ,Na 6 -silsesquioxane 1 (Scheme 1 and Figure 1, top).

It is worth mentioning that the phenanthroline ligand is not participating in the composition of the final product; nevertheless, its influence on the reaction course could not be denied: early reported "phenanthroline-less" synthesis of Cu,Na-phenylsilsesquioxane in ethanol media gave a Cu 2 Na 2 complex [START_REF] Bilyachenko | New Binuclear Cage-like Copper(II) Silsesquioxane ("Cooling Tower"); Its High Catalytic Activity in Oxidation of Benzene and Alcohols[END_REF] of principally different molecular topology. Thus, it may be concluded that phenanthroline acts as a "silent witness" ligand, and this effect was found to be reproducible (additional runs of this reaction gave an analogue of complex 1, with small differences from 1; see complex 1a; Figures S1 andS2).

In the second stage of work, a similar synthetic procedure but in the presence of another type of N-based ligand, neocuproine, was studied. To our surprise, this reaction provoked the formation of a principally different type of molecular architecture, compound 2 (Scheme 1 and Figure 1, bottom). the best of our knowledge, compound 2 represented a novel principle of metal-containing cage compound design in conditions of the presence of O and N ligands in the reaction mixture. The only example of similar reaction in the presence of P-ligand was presented by some of us in recent work. 5g The product 2 composition could be described as a consequence of unusual Cu ion redistribution between two types of ligands. Namely, neocuproine ligands form two cationic fragments Cu + (neocuproine) 2 , while Cu 9 Na 4 -silsesquioxane plays the role of a negatively (2-) charged component of compound 2. Such a difference of behavior between phenanthroline and neocuproine could be explained by the better σ donation of the latter ligand, as demonstrated in the formation of copper ethylene complexes. [START_REF] Munakata | Studies of copper(I) olefin complexes. Formation constants of copper olefin complexes with 2,2′-bipyridine, 1,10-phenanthroline, and their derivatives[END_REF] The compositions of the silsesquioxane fragments of complexes 1 and 2 are quite different from each other. Namely, compound 1 comprises two acyclic Ph 10 Si 10 O 21 ligands, unprecedented for cage metallasilsesquioxanes (Figure 2, top). In turn, the cage component of complex 2 includes two pairs of ligands: cyclic (PhSiO 1.5 ) 6 and acyclic (PhSiO 1.5 ) 4 (Figure 2, bottom). While (PhSiO 1.5 ) 6 ligands are widespread among metallasilsesquioxane structures, 5a,b we could cite the only work-described (PhSiO 1.5 ) 4 ligands in the composition of cage Ti-silsesquioxanes. [START_REF] Hirotsu | Synthesis and Structures of the First Titanium(IV) Complexes with Cyclic Tetrasiloxide Ligands: Incomplete and Complete Cage Titanosiloxanes[END_REF] This publication, unlike compound 2, reported cyclic (PhSiO 1.5 ) 4 rings, sandwiched between Tibased planes. Detailed topological analysis of the structures of compounds 1 and 2 is presented in Figure S4. Finally, we could mention Mn-based phenylsilsesquioxane [START_REF] Bilyachenko | Cage-like manganesephenylsiloxane with an unusual structure[END_REF] possessing the same [( PhSiO 1.5 ) 6 and(PhSiO 1.5 ) 4 as compound 2. It is worth noting that the resulting structure of the compound from ref 12 is quite unlike complex 2.

The structure and composition of Si-O-Si skeletons in both products 1 and 2 are worth mentioning as new evidence of the role of additional factors during Si-O-Si fragment assembly. It is known that the basic reaction [PhSi(OR′) 3 with NaOH] leads to the cyclic trimer or tetramer product ([PhSi(O)ONa)] n , n = 3 or 4; Figure S3). 5a,13 In turn, numerous further syntheses, using these trimers or tetramers for interaction with transition-metal salts, showed the structural rearrangement of these cycles into ligands of other forms. 5a,14 Keeping in mind that the procedures presented here that allowed the isolation of compounds 1and 2 are even more complicated because of the involvement of additional N ligands, it is not surprising that the silsesquioxane ligand topology in 1 and 2 is far from the classical one.

Regarding the properties of complexes 1 and 2, we could state that both compounds are air-stable, nonhygroscopic, and well soluble in polar organic solvents (e.g., N,N-dimethylformide, diemthyl sulfoxide, or benzonitrile) or in their mixtures with aromatic solvents (benzene and toluene).

We previously reported the activity of cagelike metallasilsesquioxanes (CLMSs) as precatalysts in the homogeneous catalysis of highly relevant organic reactions: amidation and hydrocarbons functionalization. [START_REF] Bilyachenko | Cage-like Copper(II) Silsesquioxanes: Transmetalation Reactions and Structural, Quantum Chemical, and Catalytic Studies[END_REF] Importantly, the use of CLMSs allowed minimization of the catalyst loading (down to 100 ppm of Cu) with retention of the catalytic activity. A quick optimization of the reaction conditions, for the formation of amides directly from alcohol, 16 revealed that compound 1 was as efficient as the previously reported CLMSs because only 100 ppm of Cu was required to catalyze the reaction. When benzyl alcohol was reacted with different amines, protected as their ammonium salt, in the presence of calcium carbonate, tert-butyl hydroperoxide (TBHP) as the terminal oxidant, and 1 as the catalyst, the corresponding amides could be isolated in good yield (Scheme 2). To ensure that compound 1, and not traces of metals, was indeed responsible for the catalytic activity, high-purity calcium carbonate (>99.999% purity) was used. Because low catalyst loading was used, the turnover number and frequency (TON and TOF) were calculated. [START_REF]G. B. Metal-catalyzed hydrocarbon oxygenations in solutions: the dramatic role of additives: a review[END_REF] The reaction was found to be efficient with primary and secondary amines, with TON and TOF values reaching 7700 and 325 h -1 , respectively. These values are slightly lower than those obtained with a Cu 4 -CLMS featuring a globulelike structure (TON and TOF of 9400 and 392 h -1 , respectively). [START_REF] Bilyachenko | Cage-like Copper(II) Silsesquioxanes: Transmetalation Reactions and Structural, Quantum Chemical, and Catalytic Studies[END_REF] However, compound 1 was undoubtedly more efficient than the CuO that was used in our seminal publication on this reaction, with TON and TOF values of 44 and 11 h -1 , respectively. 16e Impressive results were obtained in the catalyzed alkane oxidations with peroxides (H 2 O 2 and m-chloroperoxybenzoic acid). Oxidation of cyclohexane (CyH) with H 2 O 2 , in the presence of 1 as a catalyst and HNO 3 as a promoter either in air (see Figures S10-S15) or under an atmosphere of 18 O 2 (see Figures 3 and4 and S16-S24), afforded cyclohexyl hydroperoxide as the main reaction product, which is selectively reduced with PPh 3 (refs 18) to produce cyclohexanol. The highest incorporation of 18 O into cyclohexyl hydroperoxide was observed at the beginning of the reaction (Figure S16). The chromatogram recorded before the addition of PPh 3 reveals a strong peak of cyclohexyl hydroperoxide, disappearing upon treatment. The pattern of byproducts (Figure 3) is typical for reaction systems oxidizing with hydroxyl radicals, showing isomers of cyclohexanediols and hydroxycyclohexanones. Cyclohexanediols are mainly single 18 O-labeled. Hydroxycyclohexanones are probably only labeled We clearly realize that synthesized complexes are only precatalysts in hydrocarbon oxidations. Under the action of HNO 3 acid, the complex is modified to open fragments, with reaction centers available for approaching both H 2 O 2 and the substrate. As in the case of oxidations catalyzed by polyoxometallates, in our oxidation, a few species with active reaction centers work. Figures S10-S14 demonstrate peculiarities of CyH oxidation with H 2 O 2 in acetonitrile. The maximum yield of the products was 21%. The products of methylcyclohexane oxidation by the same system are shown in Figure S15. Oxidations of n-heptane and n-octane allowed us to measure regioselectivity parameters: 1:5.4:5.4:4.9 and 1:5.4:5.3:4.8, respectively. The reaction of cis-1,2-dimethylcyclohexane with H 2 O 2 catalyzed by compound 1 gave a trans/ cis ratio of 1.0 for tert-alcohols. The interaction of tert-butyl hydroperoxide (1.5 M) with 1-phenylethanol or cyclohexanol (0.5 M) in the absence of HNO 3 afforded after 10 or 5 h at 60 °C cyclohexanone or acetophenone in 63% or 98% yield, respectively.

The system 1/HNO 3 /H 2 O 2 shows low stereoselectivity with 14% retention of the stereoconfiguration (RC), corresponding to a trans/cis ratio of 1.34 of tertiary alcohols. Such a value of the RC index indicates the involvement of free alkyl radical intermediates. The normalized bond selectivity for cis-1,2dimethylcyclohexane is 3°:2°:1°= 90:21:1. Although the ratio 3°:2°= 4.3:1 is expected for the hydroxyl radical range, [START_REF] Nesterov | Homoand heterometallic polynuclear transition metal catalysts for alkane C-H bonds oxidative functionalization: Recent advances[END_REF] the amounts of primary oxygenates are abnormally low. This may be explained by a high reactivity of closely located tertiary H atoms of cis-1,2-dimethylcyclohexane.

Oxidation of cis-1,2-dimethylcyclohexane with m-CPBA, catalyzed by 1, in the presence of a HNO 3 promoter and under an atmosphere of 18 O 2 , afforded much higher stereoselectivity with an RC index of up to 51% (trans/cis ratio of up to 3.1 of tertiary alcohols). The incorporation of 18 O into the tertiary alcohols is also higher than those in the case of H 2 O 2 , reaching 70% after 3 h (Figure 4, bottom). Remarkably, while cis-alcohol accumulated ca. 2 times more 18 O than transalcohol (in percent), the absolute amounts of both 18 O-labeled alcohols are nearly equal (Figure 4, top). The initial reaction rates of 18 O-labeled tertiary alcohols (1.4 × 10 -7 and 1.0 × 10 -7 M s -1 for trans-and cis-alcohols, respectively) are ca. 2 times lower than the overall initial rates (5.0 × 10 -7 and 1.7 × 10 -7 M s -1 for trans-and cis-alcohols, respectively).

Recently, we found that a hexacopper complex with neocuproine (an analogue of complex 2) is a highly efficient homogeneous catalyst in the oxidation of alkanes and alcohols with peroxides. [START_REF] Bilyachenko | Mild and Regioselective Hydroxylation of Methyl Group in Neocuproine: Approach to an N,O-Ligated Cu6 Cage Phenylsilsesquioxane[END_REF] All of the peculiarities of alkane oxidation indicate that the oxidation with H 2 O 2 occurs with the participation of free hydroxyl radicals. The hexacopper complex with neocuproine also efficiently catalyzes the oxidation of alcohols to ketones with TBHP in acetonitrile. The oxidation of alcohols apparently involves the generation of hydroxyl radicals.

■ CONCLUSIONS

In summary, two unique cage Cu,Na-silsesquioxanes were synthesized through self-assembly reactions of sodium siloxanolate species with CuCl 2 . The specific role of N-based ligands (phenanthroline and neocuproine) in this reaction was demonstrated. Namely, the presence of phenanthroline provokes the formation of compound 1, containing no phenanthroline moieties in the composition (phenanthroline as a "silent witness" 21 ligand). In turn, neocuproine-assisted synthesis led to the formation of complex 2 with an unusual redistribution of Cu centers between O (silsesquioxane) and N (neocuproine) ligands. These observations point to the huge influence of various types of ligands, depending on their binding affinity, on the mechanism of cage metallacomplex formation. It is worth noting that small differences in the Nligand structures (phenanthroline vs neocuproine) cause the formation of dramatically different cage products. This indicates that the formation of cage metallasilsesquioxane is complicated, especially in the presence of additional ligands, which could provoke "ligand-to-metal" competitions and metal redistribution. In turn, the isolation of compounds 1 and 2 was performed for the conditions of different solvent media. No doubt, the difference in the solvates' surroundings may also be crucial during the assembly/crystallization of the cage product. In summary, we could state that at least three parameters could influence/control the metallasilsesquioxane formation: (i) ratio between "key elements" in reaction media, namely, Si:-Cu:Na:N; (ii) nature of the ligand; (iii) ability of solvent molecules to play a role of solvating ligands. All of these factors could be of high influence for the process under discussion, and in our future reports, we will try to shed some light on this point. Compound 1 was found to be efficient in the alkane oxidation and oxidative amidation of benzylic alcohols, with a catalyst loading down to 100 ppm of Cu and TON and TOF values of up to 7800 and 325 h -1 . Compound 1 exhibited high activity in the hydroperoxidation of alkanes with H 2 O 2 .

■ ASSOCIATED CONTENT * S Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.8b01496.

Details of catalytic experiments (PDF)

Accession Codes

CCDC 1834447-1834449 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge via www.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.

■ AUTHOR INFORMATION

Corresponding Authors *E-mail: bilyachenko@ineos.ac.ru. *E-mail: frederic.lamaty@umontpellier.fr. *E-mail: shulpin@chph.ras.ru.

Figure 1 .

 1 Scheme 1. Reaction Scheme of the Synthesis of 1 and 2

Figure 2 .

 2 Figure 2. Silsesquioxane ligand structures in 1 (top) and 2 (bottom).

Scheme 2 .a

 2 Scheme 2. Catalytic Activity of Compound 1 in Oxidative Amidation a

Figure 3 .

 3 Figure 3. Fragments of the chromatograms showing the main reaction products and byproducts (inset) in the course of CyH oxidation with H 2 O 2 , in the presence of HNO 3 , catalyzed by 1 (for conditions, see Figure S16), before (top, green) and after (bottom, red) the addition of solid PPh 3 to the reaction sample. Percentages indicate the incorporations of 18 O into the respective oxygenates measured after the addition of PPh 3 . The incorporation of 18 O into the main products is shown in Figure S16, right.

Figure 4 .

 4 Figure 4. Top: Accumulations of all and 18 O-labeled trans and cis tertiary alcohols in the course of oxidation of cis-1,2-dimethylcyclohexane (0.14 M) with m-CPBA (0.18 M), in the presence of HNO 3 (0.05 M), catalyzed by 1 (5 × 10 -4 M), in CH 3 CN at 50 °C; measured after the addition of solid PPh 3 to the samples. Bottom: Dependence of the 18 O percentage in the tertiary alcohols.

DOI: 10.1021/acs.inorgchem.8b01496 Inorg. Chem. 2018, 57, 11524-11529

ORCID

Alexey N. Bilyachenko: 0000-0003-3136-3675 Xavier Bantreil: 0000-0002-2676-6851 Anna V. Vologzhanina: 0000-0002-6228-303X Elena S. Shubina: 0000-0001-8057-3703 Pavel V. Dorovatovskii: 0000-0002-2978-3614 Georgiy B. Shul'pin: 0000-0002-7053-4812

■ ACKNOWLEDGMENTS

The publication was prepared with the support of RUDN University Program 5-100 and funded by RFBR according to Research Projects 16-03-00254, 16-53-150008, 16-29-05180, and 17-03-00993, French Embassy in Moscow, University of Montpellier, CNRS (Programme de Recherche Conjoint). Synchrotron single-crystal diffraction measurements were performed at the unique scientific facility Kurchatov Synchrotron Radiation Source supported by the Ministry of Education and Science of the Russian Federation (Project code RFMEFI61917X0007). This work was also supported by the State within the frameworks of the state task for ICP RAS 0082-2014-0007. The number of state registration is AAAAA-A18-118012390374-3. Support from the Fundacaõ para a Ciencia e a Tecnologia, Portugal (Projects UID/QUI/00100/ 2013 and PTDC/QEQ-QIN/3967/2014 and Fellowship SFRH/BPD/99533/2014 (to D.S.N.)], is also acknowledged.

Author Contributions

The manuscript was written through contributions of all authors.

Notes

The authors declare no competing financial interest.


Yanfeng Yang

Gabriela Vincze

Cyrille Baudouin

Hocine Chalal

Tudor Balan
email: tudor.balan@ensam.eu.

Strain-path dependent hardening models with rigorously identical predictions under monotonic loading

Accurate sheet metal simulation often requires advanced strain-path dependent material models, in order to predict the material response under complex loading conditions, including monotonic, reverse and orthogonal paths. More and more flexible models imply higher and higher costs in terms of parameter identification, computer implementation and simulation time, and robust comparison is often compromised by the inconsistent predictions of advanced models under monotonic loading. In this paper, a simple and general approach is proposed for the alteration of advanced hardening models in order to make them rigorously identical to each other under monotonic loading. This objective was reached without any drawback other than the addition of the corresponding equations. On the contrary, the flexibility and accuracy of the selected models was improved, and the parameter identification procedure became simpler, more accurate and more robust. Three material models of increasing complexity were selected to demonstrate the interest of this approach with respect to a complete set of characterisation experiments for a DP600 sheet steel.

Introduction

The accuracy and reliability of sheet metal forming numerical simulation depends heavily on the finite element (FE) code ability to suitably describe the material behavior [START_REF] Hu | A new modified algorithm for elasticplastic FEM: Quasi-chord modulus method[END_REF], [START_REF] Flores | Model identification and FE simulations: effect of different yield loci and hardening laws in sheet forming[END_REF]. The success of the numerical results relies on the quality of the adopted constitutive model [START_REF] Yoshida | A model of large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation[END_REF]- [START_REF] Clausmeyer | Modeling and finite element simulation of loading-pathdependent hardening in sheet metals during forming[END_REF], and on the identification of the material parameters [START_REF] Haddadi | Towards an accurate description of the anisotropic behaviour of sheet metals under large plastic deformations: modelling, numerical analysis and identification[END_REF].

During sheet metal forming processes, the material is generally subjected to complex loading conditions, including monotonic, reverse and orthogonal strain paths. The work hardening behaviour of the material becomes more complicated when the deformation path involves several steps [START_REF] Bambach | Characteristics of testing conditions for constitutive models in metal plasticity[END_REF]. In sheet metal forming simulation, the choice of hardening model has a strong influence for final prediction accuracy. The complexity of the material models generally means that their experimental identification requires a greater number of material characterization tests [START_REF] Bouvier | Characterization of the strain-induced plastic anisotropy of rolled sheets by using sequences of simple shear and uniaxial tensile tests[END_REF]. Regarding the mechanical characterization of sheet metal following complex deformation paths, the tests with strainpath change have a great importance because they highlight the kinematic hardening, linked to the Bauschinger effect and to the phenomena of stagnation of work hardening [START_REF] Wang | On the modeling of hardening in metals during nonproportional loading[END_REF]- [START_REF] Sun | Proportional and non-proportional hardening behavior of dual-phase steels[END_REF]. Taking these phenomena into account in the model leads to a better prediction of certain defects observed in the forming of the sheet metal [START_REF] Haddag | Investigation of advanced strain-path dependent material models for sheet metal forming simulations[END_REF]- [START_REF] Qin | Modeling strainpath changes in aluminum and steel[END_REF].

In order to improve the consistency of hardening model comparison, the rigorous separation of the isotropic and kinematic components of the hardening model was attempted in the literature [START_REF] Carbonniere | Comparison of the work hardening of metallic sheets in bendingunbending and simple shear[END_REF], [START_REF] Chun | Modeling the Bauschinger effect for sheet metals, part I: theory[END_REF]. The advantage of this modelling approach is that the model predictions under monotonic loading are independent of the kinematic hardening model and parameters. However, this approach could only be applied to the Armstrong-Frederick kinematic hardening model.

The objective of this work was to propose an alternative approach for the self-compensation of non-isotropic hardening terms in elasto-plastic constitutive models, in order to obtain identical predictions under monotonic loading and thus allow for consistent model comparison. The paper is structured as follows: The selected material, material characterization tests and corresponding results are shown in Section 2. Section 3 presents three hardening models of increasing complexity selected for comparison. Selfcompensated versions of these models are developed in Section 4, and their parameters are identified. Finally, the conclusion of this work is summarized in Section 5.

Material characterization

A 1.2 mm thick DP600 AHSS sheet steel was used in this work. The aim of the selected characterization tests is to reveal the hardening behaviour of the materials under large monotonic deformations and during strain-path change. In particular, a reverse loading sequence was provided since this sequence frequently occurs in sheet metal forming processes. Rheological tests carried out consist of: • UT -uniaxial tension in rolling direction (RD), transverse direction (TD) and diagonal direction (DD);

• SS -simple shear in RD; RS -reversal shear in RD;

• OR -uniaxial tension followed by simple shear in RD. Fig. 1 summarizes the main material characterization results for DP600. The tensile tests under different directions clearly illustrate the very weak plastic anisotropy of DP600, confirmed by the anisotropy coefficients 0.85, 0.80 and 1.0. A very good experimental repeatability is also observed. The simple shear test revealed the material behavior at large deformation, as shear strains up to 0.7 could be reached. Reverse shear tests performed with shear pre-strain levels between 0.1 and 0.3 exhibit an important Bauschinger effect, fast transient behaviour with hardening stagnation and further resumption, under cumulated shear strain up to 1. However, in uniaxial tension followed by simple shear test (OR), the cross-hardening effect was not significant. Therefore, the modeling of the orthogonal effect was not considered important in the current investigation.

Constitutive framework and models

The frame-invariance of the material model is insured by formulating the constitutive model in a convenient rotating frame, such that simple material time derivatives can be used. With this approach, classically adopted in most finite-element implementations of plasticity models, the material model is form-identical to a small-strain formulation, without loss of accuracy. During the plastic deformation process, the elasticity domain evolves in the stress space in a complex way: it can change its size, translate, rotate and even distort. Here, the shape of the yield surface is supposed fixed, while its size and center location may evolve according to an arbitrary hardening model. Given the isotropy of the tested material, and the paper's focus on hardening modeling, the von Mises yield surface is adopted throughout the paper.

The total strain rate tensor can be decomposed in elastic strain rate and plastic strain rate . The hypo-elastic law expressed as

: : 1
is considered, where is the Cauchy stress rate and 2 ⨂ is the fourth-order tensor in isotropic linear elasticity. and can be calculated with using Young's modulus and Poisson's ratio . The components of second order unit tensor are the Kronecker deltas ( !" # !" ). While is the fourth-order symmetric deviatoric unit tensor and its components are $%!" 1 2 ⁄ # $! # %" # $" # %! 1 3 ⁄ # $% # !" . The plastic strain rate tensor ( is given by the associated flow rule

( ) • +, - + ) • ., 2
where . is the flow direction defined by the normality rule, , -0 is the equivalent stress using von Mises function and denotes the deviatoric part of the Cauchy stress tensor. The second order tensor 0 designates the backstress and 1 describes the size of the yield surface; their evolution is formally modeled by generic equations

1 2 3 • ) , 3 0 4 0 • ) , 4
where 2 3 and 4 0 represent isotropic hardening and kinematic hardening. The plastic multiplier ) is calculated with equation

) .: : .: : . .: 4 0 2 3 , 5

deriving from the consistency condition. This generic modeling framework can be particularized to various yield functions, and various hardening models. According to the characterization tests of DP600, the hardening behavior of this material exhibits several specificities after reverse loading: early reyielding (Bauschinger effect), fast hardening followed by stagnation, and further hardening resumption at larger reverse strains. Consequently, this material was selected to investigate the ability of different types of hardening models to predict these features. In contrast, DP600 exhibits almost no anisotropy, and almost no crosshardening effect on the stress-strain curves; thus these features of plastic behavior will not be investigated. More specifically, three representative hardening models were confronted, belonging to three categories widely encountered in the literature, with increasing levels of complexity: isotropic hardening, combined isotropickinematic hardening and a microstructure-based advanced hardening model.

Swift-Voce isotropic hardening

Generally, the size of the yield surface 1 is decomposed into an initial yield stress 1 and an isotropic hardening variable 7, 1 1 7. 6

The isotropic hardening models are the oldest, and the most commonly used in industry still nowadays. This type of model can only describe accurately monotonic loading, for example uniaxial tension (UT) or simple shear (SS). Thus the backstress is set to zero and the so-called Swift-Voce isotropic hardening model is adopted, which is a classical combination of a power law and a saturating law:

7 : 2 ; < • ) = ; • 7 >? 7 : • ) , 7 7 2 ; A • ) B • C D ⁄ E D C D ⁄ • ) , 8
where = ; and 7 >? are the parameters of Voce isotropic hardening described by variable 7 : , and , E , B are the parameters of Swift isotropic hardening 7 . The resulting isotropic hardening is given by 1 7 7 : 7 , with 1 E D ; 9 2 3 2 ; < 2 ; A . 10

The Swift-Voce combined law was used as it is repeatedly identified in the literature as one of the most adequate isotropic hardening law to describe large strain hardening behavior of sheet metals, in particular steels [START_REF] Sun | Proportional and non-proportional hardening behavior of dual-phase steels[END_REF], [START_REF] Venet | Direct usage of the wire drawing process for large strain parameter identification[END_REF].

Chaboche combined hardening

The so-called combined hardening models describe isotropic and kinematic hardening with two internal state variables 7 and 0. The isotropic hardening 7 uses the same equations ( 7)- [START_REF] Bambach | Characteristics of testing conditions for constitutive models in metal plasticity[END_REF] as in the previous model. Two backstress tensors are used to describe kinematic hardening with Armstrong-Frederick type equations,

0 N 4 0N • ) = OC • P >?C • Q 0 N • ) , 11 0 R 4 0R • ) = OS • P >?S • Q 0 R • ) , 12
where = OC , P >?C , = OS , P >?S are material parameters. Q 0 , -⁄ designates the offset deviatoric stress direction. The rate of total kinematic hardening 0 is calculated with

0 0 N 0 R , 13
4 0 4 0N 4 0R , 14
with initial values usually equal to zero (annealed material). The use of two backstress tensors is not compulsory, but is was shown to improve the predictions with respect to the classical Frederick-Armstrong model with only one term; this was also the case in this study. More than two backstress tensors are rarely proved necessary.

Microstructure-based hardening

On the basis of the microstructural evolution after twostage non-proportional loading at finite strains, a microstructural model was developed by Teodosiu and Hu [START_REF] Teodosiu | Evolution of the intragranular microstructure at moderate and large strains: modelling and computational significance[END_REF]. This advanced hardening model is able to reproduce not only the Bauschinger effect but also other transient hardening phenomena during non-monotonic loading process. It is based on physical considerations, mainly the description of the evolution of the so-called planar persistent dislocation structures (PPDS) and their contribution to the hardening of the material. Proper description of PPDS evolution mechanisms reproduces the elementary transient phenomena observed on stress-strain evolution curves. This model involves four internal state variables: 7, 0, T, U. The variable 7 is a scalar, while 0 and U are second-order tensors and T is fourth order tensor.

The yield surface size is given as function of 7 and T by

1 1 7 V|T|, 15
where 7 describes the contribution of the randomly distributed dislocations to the isotropic hardening. The term V|T| represents the effect of PPDS on isotropic hardening, where T describes the directional strength of the PPDS and V is a material parameter.

The kinematic hardening evolution law, described by the back-stress variable, is given by Eq. [START_REF] Carbonniere | Comparison of the work hardening of metallic sheets in bendingunbending and simple shear[END_REF]. Nevertheless, the saturation value P >? is no longer a material parameter in Teodosiu-Hu's model, but a function of the internal state variable T . This dependency of P >? on the T variable is assumed of the form

P >? P 1 V |T|X 1 Y Z S , 16
where P is the initial value of P >? and is a material parameter in Eq. ( 16). The ratio Y Z [ \ |T| ⁄ is a measure of the change in the orientation of the current strain rate tensor with respect to the PPDS. This parameter is therefore considered to be an indicator of strain-path change. It evolves between 0 (orthogonal loading) and 1 (monotonic or reverse loading).

Experimental observations indicate that the PPDS associated with the current direction of the strain rate evolve quite differently from the rest of the PPDS during strain-path change. The variable T is therefore decomposed into two parts: [ \ (scalar) representing the strength associated with the currently active slip systems, and T ] (fourth-order tensor), associated with the latent part of the PPDS. The decomposition of T takes the following form:

[ \ ^: T: ^, T ] T [ \ ^⨂^, 17
where ^ _ | _ | ⁄ represents the plastic strain rate direction. The evolution laws of [ \ and T ] are given by

[ \ 2 Z\ • ) = Z\ `a [ >? [ \ ℎ[ \ c • ) , 18
where [ >? and = Z\ govern the saturation value and evolution rate of [ \ , and

[ d 4 T] • ) = Zd e |T ] | [ >? f D g T ] • ), 19
where = Zd and B d characterize the saturation rate of T ] . The functions a and ℎ in Eq. ( 18) have been introduced in order to further describe transient hardening after a change in strain-path. Their assumed mathematical forms are In Eq. ( 20), B ( is a material parameter and U is the internal state variable describing the polarity of the PPDS. Its evolution law is given by

a ⎩ ⎨ ⎧ 1 = k = Z\ = k , if U: ^≥ 0 1 U: ^ D m1 = k = Z\ = k • [ \ [ >? n , otherwise 20
U = k ^ U • ) , 22
where = k characterizes the polarization rate of the PPDS in Eq. ( 22). From Eqs. ( 17)-( 19), one can obtain the time derivative of the norm of the T tensor as [START_REF] Haddag | Investigation of advanced strain-path dependent material models for sheet metal forming simulations[END_REF]

|T| 2 |T| • ) 1 |T| o2 Z\ [ \ = Zd e |T ] | [ >? f D g |T ] | S p • ) , 23
where |T| q|T ] | S [ \ S . Thus, the scalar function 2 3 in Eq. ( 10) is deduced for this model as

2 3 2 ; V2 |T| . 24
Remarkably, this advanced model reduces to the generic form assumed in Eqs (1)-( 5) [START_REF] Haddag | Investigation of advanced strain-path dependent material models for sheet metal forming simulations[END_REF], [START_REF] Balan | On the numerical implementation of elasto-plastic constitutive equations for metal forming[END_REF], which simplifies its numerical implementation. Nevertheless, it involves a number of additional state variables, including second and fourth order tensors, interconnected through several rate and algebraic equations. It is used here as a representative candidate for advanced models that can reproduce complex transient behavior after strain path change (for example, hardening stagnation followed by hardening recovery at larger strains).

Parameter identification of hardening models

The material parameter identification is seen as an inverse problem. The gap between the experimental stress-strain curves and the simulated ones is quantified via an objective function V rs% , which is minimized with respect to the material model's parameters [START_REF] Chaparro | Material parameters identification: Gradient-based, genetic and hybrid optimization algorithms[END_REF], [START_REF] Andrade-Campos | Novel criteria for determination of material model parameters[END_REF]. We are looking for min (V rs% ), with

V rs% t 1 u $ t v , $,% w( , $,% $x , $,% w( y S v ℎ $,% w( ℎ $,% $x ℎ $,% w( y . z { %|C } $|C 25
In order to improve the accuracy of the identification, a combined cost function is used [START_REF] Haddadi | Towards an accurate description of the anisotropic behaviour of sheet metals under large plastic deformations: modelling, numerical analysis and identification[END_REF], which considers the work hardening rate ℎ along with the stress values , . ~ is the number of tests, u $ is the number of measurement points of the • ?€ test. The upper indices 'sim' and 'exp' designate calculated values and experimental values, respectively.

A Fortran program was developed to simulate the rheological tests (UT, SS, RS, OR) with the various models, independently of any commercial software. The optimization is performed with SciLab software, which provides an interactive call to the Fortran user program which allows data to be exchanged between SciLab and the Fortran program.

Fig. 2 shows simulation results of rheological tests (UT, SS and RS) by using the identified hardening parameters. It is clear that the selected isotropic hardening model can well predict monotonic loading test (UT, SS) up to large strains. However, this model failed to describe reverse shear because isotropic hardening model cannot capture the Bauschinger effect. The combined hardening is able to improve the predictions in reverse shear due to kinematic hardening. However, the accuracy of the combined hardening model is relatively smaller in UT and SS, compared with isotropic hardening. This inconsistency is a very serious drawback for material model comparison and selection. Finally, the microstructure-based hardening captured not only the Bauschinger effect in reversal tests, but also hardening stagnation and resumption at large deformation. Nevertheless, compared with isotropic hardening, this model also exhibits accuracy loss in monotonic loading (UT, SS). Obviously, this loss can be reduced by modifying the relative weight of monotonic tests in the cost function, yet it is not possible to rigorously eliminate it without seriously compromising the benefits observed under strain-path change. In the next section, a rigorous solution is proposed to eradicate this inconsistency.

Self-compensated hardening models

When only isotropic hardening is used to simulate a monotonic loading mode, say uniaxial tension, then variable 1 describes the evolution of the tensile stress component , • , 1 ≡ , • for isotropic hardening, 26

which is experimentally measured and directly used for the hardening model parameter identification. However, when kinematic hardening is added, this equality does not hold anymore: one obtains , • 1 P * , where P * is a scalar quantity corresponding to the tensile component of 0 under monotonic tensile loading [START_REF] Yang | Prediction of the yield surface evolution and some apparent non-normality effects after abrupt strain-path change using classical plasticity[END_REF]. This implies that any change to the model will impact the overall predictions under monotonic loading, even if the isotropic hardening part of the model is unchanged. An alternative to circumvent this drawback consists in explicitly modelling , • and further calculating the size of the yield surface as This requires the explicit calculation of term P * , which selfcompensates the contribution of kinematic hardening under monotonic loading. Several authors have derived and used its explicit expression for the particular case of Armstrong-Frederick kinematic hardening:

P * P >? • 1 "… † = O • E̅ ( , 28
where E̅ ( is the equivalent plastic strain. However, the need for an algebraic expression of the selfcompensating term prevented the extension of this approach to more complex models, so this approach was only applied to the Armostrong-Frederick model in the literature. In this paper, the self-compensating terms are determined using rate equations, which provides a much more versatile approach.

Self-compensated combined hardening

In the proposed approach, the isotropic part of the hardening model describes the apparent flow stress under monotonic loading , • by the Swift-Voce rate equation,

, • 2 ˆ‰ • ) , 2 ˆ‰ 2 ; < 2 ; A .
29

The size Y of the yield surface is determined by Eq. ( 27), where P * is also described by rate equations, corresponding to the tensile component of 0 under monotonic tensile loading: The rate form of Eq. ( 27) writes:

P C 2
2 3 2 ˆ‰ 2 O * . 34
In this alternative method, the monotonic flow curve is described independently of the kinematic hardening model and its parameters, which makes it more robust and especially convenient for model comparison. Eqns. ( 30)-( 33) are directly inspired from the Eqns. ( 11)-( 14) of the kinematic hardening model. Although it gives the same results as the closed-form approach (28), this approach based on rate equations is much more suitable for generalization to more complex hardening models.

Self-compensated microstructure-based hardening model

A self-compensated version of the Teodosiu-Hu model is introduced here for the first time. Again, , • is governed by Eq. ( 29) and the size Y of the yield surface is determined by

‹ * 2 k * • ) , 2 k * = k 1 ‹ * . 41
The rate form of Eq. ( 35) can be rewritten as

2 3 2 ˆ‰ 2 * , 2 * 2 O * V2 Z\ * . 42
Of course, the core of the Teodosiu-Hu model -Eqns. ( 15)-( 22) -remains unchanged.

Compared with the classical version of the microstructure-based hardening model, the selfcompensated version's predictions under monotonic loading are rigorously identical to the selected isotropic hardening model, without being affected by any further change in the parameters of kinematic hardening and cross hardening of this model. In addition to this significant improvement in consistency, it also brings some advantages in terms of parameter identification. The isotropic part of the hardening model can be identified by using only the available monotonic tests, and then its parameters can be fixed once and for all, since the monotonic simulations will be strictly identical. Therefore, any isotropic model can be adopted and identified independently. Then, in the second step, only the rest of the parameters need to be determined with respect only to the remaining experiments, without any loss in accuracy.

Parameter identification of self-compensated hardening models

The two self-compensated hardening models discussed above were implemented in the developed Fortran program and their parameters were identified with the help of the identification method presented in Section 3.4, and the twostep procedure discussed above. The identified parameters of all models, classical and self-compensated, are summarized in Table 1. As expected, the first six parameters of the selfcompensated models are identical to those of the isotropic Swift-Vice (SV) model. Fig. 3 shows the rheological test predictions by using the two self-compensated hardening models. Comparison to the isotropic hardening predictions in Fig. 2(a) shows that both self-compensated hardening models predict identical responses in monotonic loading. This improvement came with no degradation of the reverse loading predictions, which on the contrary could improve due to the rigorous separation of the parameter identification sequences. In particular, the Teodosiu-Hu model almost perfectly reproduces the entire set of experimental curves. Even the self-compensated Chaboche model provides excellent predictions of reverse shear for moderate pre-strains values -up to 0.1. Furthermore, any isotropic hardening can also be adopted in place of the Swift-Voce one, without impact on the nonmonotonic predictions: the two parameter identification steps are truly independent of each other.

Conclusions

In this paper, a novel approach was applied to make hardening models of various types identical under monotonic loading conditions. The proposed approach is more general than former closed-form attempts, and could be extended to an advanced microstructure-based hardening model. The resulting models have several advantages: • Their predictions are rigorously identical to the (arbitrarily) selected isotropic hardening model; • Their parameter identification is completely split in two independent sequences, one for the isotropic hardening part and one for the remaining parameters, without interference;

• The isotropic hardening part of each model can be freely selected and further modified, independently of the rest of the model; • The self-compensated version of any model can be implemented in FE codes without additional programing: only the isotropic part of the hardening model is concerned, which can be simply implemented in tabular form. Thus, any available model can be "compensated" non-intrusively. Three categories of classical and self-compensated models were selected, from simple isotropic hardening to advanced microstructure-based hardening. Their parameters were identified with respect to an extended set of monotonic and sequential characterization tests up to cumulated strains of 1. The superior robustness of the self-compensated models was clearly demonstrated in terms of simultaneously improved consistency, accuracy and robustness. In their selfcompensated versions, the three models exhibit a clear progression in accuracy from isotropic to combined and finally to "advanced" hardening models, with no inconsistency. This approach provides a convenient ground for the subsequent material model comparison and selection with respect to specific applications. Future work will explore the relative accuracy of the proposed models and parameter sets in springback predictions for sheet metal forming applications.

Fig. 1 .

 1 Fig. 1. Material characterization results of DP600: (a) uniaxial tension (UT) in RD (black curves), DD (green curves) and TD (orange curves); (b) simple shear (SS) in RD; (c) reverse shear (RS) in RD; (d) uniaxial tension followed by simple shear (OR) in RD.

Fig. 2 .

 2 Fig. 2. Rheological predictions (solid line) of classical type hardening models: (a) Isotropic; (b) Combined; (c) Microstructural, and their comparisons of experimental results (symbols).

Fig. 3 .

 3 Fig. 3. Rheological predictions (solid line) of selfcompensated type hardening models: (a) Combined; (b) Microstructural; symbols designate experimental results.

Table 1 .

 1 Identified parameters for the classical and selfcompensated hardening models for DP600.

			Classical		Self-compensated
		S-V	Chaboche	Teodosiu	Chaboche	Teodosiu
	44.8 7 >? (MPa) 136.9 = ; (MPa) 664.1	62 84 449	30.8 93 -		44.8 136.9 664.1
	B E = OC P >?C (MPa) = OS P >?S (MPa) = O P (MPa) [ >? (MPa) = Z\ = Zd = k B d B ( V	0.32 0.004 --------------	0.5 0.001 157.5 184 28.7 79 ----------	------65.1 168.2 405.3 2.66 1.2 0.58 0 609.4 0.04 0.8	90.2 198.4 5.9 124.7 ----------	0.32 0.004	----65.1 168.2 565.6 2.85 1.2 0.67 0 890 0.79 0.8

Acknowledgements

The first author is grateful to the China Scholarship Council for providing him a PhD grant during the preparation of this work. GV acknowledges support from COMPETE 2020, in its FEDER/FNR component and the Portuguese Foundation of Science and Technology (FCT), in its State Budget component (OE) through projects POCI-01-0145-FEDER-032466. The authors thank Dr Leopold Wagner from voestalpine Stahl GmbH, Austria for fruitful discussions and for providing the DP600 material.


C Monteiro

C Maechling

C Hervé Du Penhoat

Monitoring the non-specific interactions of catechin through diffusion measurements based on pulsed-field gradients

The self-association of aqueous catechin as a function of concentration was monitored through variationsin 1H chemical shifts, proton T1 and T2 data and translational diffusion coefficients obtained with the pulsed-field gradient spin-echo method. The latter approach is very efficient and it is not restricted to aromatic compounds. Equilibrium constants were estimated for various models of self-association and the apparent enthalpy of dissociation was measured with isothermal titration calorimetry. Comparison of the latter parameter with thermodynamic data reported for various types of non-specific interactions suggests that such phenomena could be studied using this approach.

INTRODUCTION

Non-specific interactions play a major role in the aggregation phenomena that govern the physical state of molecular mixtures. This is particularly true of plant extracts where control of colloid formation and/or precipitation is a major industrial concern. Althoughmuch effort has been devoted to devising NMRexperiments for studying specific interactions such as ligand binding, non-specific interactions have received less attention 1,2. The aggregation phenomena of surfactants 3,4, nucleotides 5,6, caffeine 7,8, polyphenol s8-13 and aromatic drug molécules [START_REF] Davies | [END_REF][15][16] and the non-specific stacking interactions of caffeine with methyl gallate 8, polyphenols with peptides 10,13,17 and aroma substances12 are notable exceptions. The majority of these NMR studies of self-or hetero-association involve aromatic compounds and have relied on chemical shift titration versus concentration to quantify the equilibrium constants 7,12,14-16. However, in several cases no significant chemical shift variations were observed in spite of the detection of line broadening and/or intermolecular NOEs that indicated hetero-association [START_REF] Hatano | Plant Polyphenols 2: Chemistry, Biology, Pharmacology[END_REF]16,17 .

For many types of molecules, where non-specific interactions are not expected to produce significant chemical shift changes, monitoring variations in molecular motion (either rotational18 or translational diffusion3-6)wouldseemtobeanattractivealternative.Measurementofself-diffusioncoefficientsisrapid19,20 andpredictionofthevariationsintranslationalmotiononthebasisofhydrodynamictheory21isstraightforward as long as themolecules arewell represented by the Stokes-Einstein-Debye model ('stick regime') 22. In this work,theself-associationofcatechinwasstudiedtoexplorethefeasibilityofsuchanapproach. Catechin(2,3-trans)andepicatechin(2,3-cis)(Fig. 1)arethebasicmonomericunitsofcondensedtanninswhich are ubiquitous secondary metabolites of higher plants. Tannins are thought to participate in plant defense by rendering the tissues of leaves, bark and fruit unpalatable for herbivores and omnivores. These polyphenols display a strong propensity to form complexes with other biomolecules such as proteins, polysaccharides and alkaloids. This phenomenon is responsible for the astringency (precipitation of salivary proteins) and poor nutritionalquality(irreversiblecomplexationofgutenzymesanddietaryproteins,inhibitionofironabsorption) of polyphenol-rich foods 10.The dimerization of catechin in silico has afforded both parallel and anti-parallel orientationsofthetargetmolécules23corroboratingthenonspecificnatureofcatechinself-association.

Figure 1. Structure of catechin with atom numbering

RESULTS AND DISCUSSION

Chemical shift titrations of catechin as a function of concentration have been investigated previously11,12 and selfassociationconstants(4.3<Ka<7.4)wereestablishedfromthelatterdatafortheisodesmicmodel.12This modelofsoluteself-associationisbasedontheassumptionthatthesolute molecules associate to form stacks (dimers, trimers, etc.) with an equilibrium constant, Ka, that is the same for each step. This leads to simple expressions for the fractional populations of the various species as a function of the monomer concentration [C1] and Ka:

The limiting value of the Taylor series for the sum of the fractional populations yields the following expression for the total concentration [C0] as long as the product Ka x [C1] is <1:

A single chemical shift is observed for each spin, indicating that the various species are undergoing fast exchange. As the distances between vertically stacked molecules are in the 3-4 Å range and the ring current magnetic field giving the main contribution to nuclear shielding decreases rapidly with the distance (∼1/r3), it is assumed that the upfield chemical shifts on complexation are due only to the influence of the neighboring molecule (within the limits of this model of molecular association all such contributions are assumed to be equal). The chemical shift of the Cn species and the chemical shift dependence on concentration for the isodesmic model are given by Eqs ( 5) and ( 6), respectively 14 :

A very similar expression is obtained for the dependence of δobs on concentration in the case of dimerization 14 :

If it is assumed that each successive stack of the catechin monomer affords the same decrease in Dobs t , ΔDt, the dependence of Dobs t on total concentration will be analogous to that of the chemical shifts for both the isodesmic (Eq 6) and dimerization (Eq 7) models. The average value of Ka (3.3 M _1 using all of the data for H-3, H-4˛ and H-4β) is slightly lower than the average value (5.6 M _1 ) reported by Dufour and Bayonove 12 for catechin dissolved in D2O-H2O-ethanol-d6 (8 : 1 : 1) but the spread in Ka parameters (±50%) for the various spins is very similar. Comparison of the chemical shifts of the acetic acid methyl moiety with respect to those of the HOD resonance for the 10 samples listed in Table 1 yields an average value of 2.084 ± 0.008 ppm in spite of a digital resolution of ±0.001 ppm. Unfortunately, in the present case, it is not feasible to acquire data for a much larger range of concentrations owing to the poor signalto-noise ratio at lower concentrations and precipitation at higher concentrations. Hence it is difficult to improve the convergence of data from chemical shift titrations and the adjustable parameter Δδmax cannot be used to validate the simulations as it is not easily calculated. Owing to deuterium exchange with the solvent, the intensity of the catechin H-6 and H-8 singlets evolved with time. Therefore, the attenuation of the H-2' signal was used to establish the D obs t values, but its intensitycouldnotbeevaluatedwithprecisionowingtothenearbyH-5'andH-6'multiplets(thehigh-fieldH-3, H-4˛andH-4ˇsignalswereevenlesssuitableowingtophasedistortioninducedbyJ-modulationduringtheecho sequence), with the result that standard deviations for D obs t were almost an order of magnitude larger (±0.4 x 10 _6 cm 2 s _1 )thanthosereferredtoabove.AKavalueof∼8M_1wasobtainedbyleastsquaresminimizationforboth isodesmicanddimerizationmodelsofself-association.

Spin

The D obs t coefficient of 7.9 x10 _6 cm 2 s _1 for the most dilute solution is much higher than the value of 3.6 x10 _6 cm 2 s _1 predicted on the basis of hydrodynamic theory21 (molecular models of catechin could be assimilated to a cylinder 14 Å in length with a radius of 3.8 Å), indicating that a microviscosity correction factor, f, of 0.45 would be necessary. The simulation protocol afforded ΔDt values of -6.7 x10_6 and -4.4 x10 _6 cm 2 s _1 for the isodesmic and dimerization models of self-association, respectively. As the catechinmonomerdoes not obey the Stokes-Einstein-Debye model, it is not possible to predict this adjustable parameter accurately. However, upper and lower limits for the microviscosity factor are known (0.45 < f < 1.0) and the Dt value for the dimer (1.2 x10_6 cm2 s_1) that can be extrapolated from the ΔDt parameter of the isodesmicmodel is clearly too low (with f = 1.0 this would correspond to a sphere of radius 16.6 Å , where Dt = kT/6_fr0), whereas that of the dimerization model (3.5 x10 _6 cm 2 s _1 ) is reasonable.

A plot of the apparent enthalpy of dilution as a function of catechin concentration is shown in Fig. 3.

Dissociation is endothermic and can be detected even at the lowest concentration (open squares). The apparent molar enthalpy of dilution (6.5 kJ mol_1) includes both dilution and dissociation contributions and has not been treated quantitatively in this study.26 Plots of theoretical monomer and dimer fractional populations (%) for both the isodesmic and dimerization models as a function of total catechin concentration are displayed in Fig. 4. The Ka value estimated from the translational diffusion profile (8 M_1) was used to establish the fractional [C1] and [C2] populations. For both models of self-association, the predicted dimer population of the 2.3 mM sample is compatible with the reduced (3/410% of that of a 23 mM solution) apparent enthalpy of dilution that is observed experimentally. It appears that dimerization is followed by precipitation without significant accumulation of aggregates higher than dimers. In conclusion, it has been demonstrated that selfassociation of catechin can be monitored with translational diffusion measurements based on the pulsed-field gradient spin-echo method. The thermodynamic data for various non-specific interactions involving sugars (carbohydrate-surfactant and carbohydrate-carbohydrate interactions) has been reviewed recently 27 and comparison with results from the present work suggests that these systems should be amenable to study using translational diffusion measurements. In favorable cases (isolated NMR signals), the association constants should be more accurately defined than those obtained from chemical shift titrations and this approach is not limited to aromatic molecules. An added advantage of the Dt approach, in the case of moderately sized molecules (for example, tetrasaccharides [START_REF] Monteiro | [END_REF], is that Dt can be related to the dimensions of the molecular assembly. As pointed out be a referee, other data treatment such as biexponential fitting 20 associated with the BPPLED pulse sequence28 (here the composite pulses provide selfcompensation of induced eddy currents) might be useful for extracting the diffusion coefficients of the monomers and the dimers.

EXPERIMENTAL Sample preparation

Various amounts of catechin (Sigma) were added to 1.75 mM solutions of acetic acid (SDS, purex 99.8%) in D2O to afford solutions (0.75 ml) containing the total concentrations, [C0] indicated in Table 1. The NMR tubes were sealed under argon after vacuum removal of dissolved oxygen.

Isothermal titration calorimetry

Calorimetric data were obtained using a VP-ITCmicrocalorimeter (MicroCal, USA). Experiments were performed at 298 K. Solutions were degassed under vacuum just prior to the experiment. Aliquots of 15 µl of catechin solution were automatically injected from a microsyringe at 5 min intervals into the sample cell (1.4192 ml) containing acetic acid solution.Efficientmixing of the solutionwas achievedby a paddle on the end of the syringe rotated at constant speed (310 rpm). The power versus time plot shows a peak as heat is absorbed upon equilibrium displacement by dilution, and then returns to baseline as thermal equilibrium is reached. The area under the peaks gives the amount of heat absorbed. The data were analysed using Microcal Origin software 29,30.

NMR

1H NMR pulsed-gradient spin-echo experiments were conducted on a Bruker Avance 400 spectrometer at 298 K. The stimulated spin-echo sequence31 was used to measure the translational self-diffusion coefficients as described previously. [START_REF] Monteiro | [END_REF] The gradient duration (δ) was varied from1 to 20 ms while keeping its strength fixed at 9.15 G cm _1 . The intergradient delay (Δ) and gradient recovery delay were 5 ms and 50 µs, respectively. The translational self-diffusion coefficients were obtained by fitting the intensities of the H-2' signal in spectra acquired with various lengths of the gradient pulses (8-15 data points) to the Stejskal-Tanner equation with inhouse software : All experiments were run at least four times. Spin-lattice relaxation times were measured with the inversion-recovery sequence. The recycle time was greater than 6xT1 and data were collected for 10-20 τ values which varied from 5 ms to 2xT1. Spin-spin relaxation times were obtained with the Carr-Purcell-Meiboom-Gill sequence and data were collected for roughly 20 spectrawith various numbers of echos (total echo duration, 2nτ, varied from 2 ms to 1 s). The integrals of the peaks were fitted to a three-(T1) or two-(T2) parameter exponential function using spectrometer system software and inhouse software, respectively, and all relaxation experiments were also run at least twice.

Figure 2 . 2 .

 22 Figure 2. Plots of T1 (♦), T2 () and Dt for various total concentrations of catechin measured at 298 K and 400 MHz. Experimental error bars are indicated.

Figure 3 .

 3 Figure 3. Plots of the apparent enthalpy change (J mol _1 ) on diluting aqueous solutions of catechin [2.3 mM () and 23.3 mM ()] versus total catechin concentration [C0] in the cell.

Figure 4 .

 4 Figure 4. Plots of the relative percentage of monomer ([C1]) and dimer ([C2]) concentrations as a function of the total concentration ([C0]) of catechin for both dimerization (♦, x) and isodesmic (, ) models characterized by an equilibrium constant, Ka, of 8.

Table 1 .

 1

	b	2.3	4.6	8.0	11.5	17.2	23.0	27.6	34.4	41.3	45.9
	H-2	-c	-c	-c	-c	-c	4.77	4.77	4.73	4.73	4.74
	H-3	4.28	4.28	4.28	4.27	4.27	4.23	4.23	4.23	4.23	4.20
	H-4a	2.90	2.90	2.90	2.90	2.90	2.89	2.88	2.89	2.88	2.87
	H-4b	2.58	2.58	2.58	2.58	2.58	2.55	2.55	2.54	2.54	2.52
	H-20	6.89	6.89	6.89	6.89	6.89	6.88	6.89	6.88	6.88	6.86

1H NMR chemical shifts in (υobs, ppm) for various concentrations of catechina Catechin concentration (mM) a Catechin (Sigma) was added to 1.75 mM solutions of acetic acid (SDS, purex 99.8%) in D2O to afford solutions (0.75 ml) containing the above total concentrations, [C0]. Both the HOD and acetic acid methyl signals were evaluated as internal references for the chemical shifts (δHOD = 4.800 ppm was the internal standard for the data given above). b See Fig.

1

for the numbering. c Overlapping with the HOD signal.

Table 2 .

 2 Self-association constants of catechin, Ka, for isodesmic (I) and dimerization (D) models

1H NMR chemical shifts of catechin as a function of [C0] are given in Table1. The H-3, H-4α and H-4β signals display the most significant variations in δobs with increasing concentration. Δδmax and Ka values were calculated by leastsquares minimization of the experimental data (δobs _ δC1 ) with respect to the theoretical values in Eqs (6) and (7) using the simplex method 24 and the modeled data are given in Table2.