Obaid H Hashmi

Frederic Capet

Marc Visseaux
email: marc.visseaux@univ-lille.fr

Yohan Champouret
email: yohan.champouret@univ-lille.fr

Homoleptic and Heteroleptic Substituted Amidomethylpyridine Iron Complexes: Synthesis, Structure and Polymerization of rac-Lactide

The synthesis of iron complexes bearing amidomethylpyridine ligands, one homoleptic [2- [(2,6- Py and R = Me, 3 Py ), are described together with their application as catalysts for the Ring-Opening Polymerization (ROP) of rac-lactide (rac-LA). The determination of the crystal structure of the three iron complexes is reported as well as their behavior in solution by means of 1 H NMR studies.

All of these complexes have shown to be active for the polymerization of rac-LA at 50 °C in toluene, producing, among others, high molecular weight polylactides (PLA) according to SEC analysis, along with a portion of low molecular weight cyclic polymer (cPLA) on the basis of MALDI spectra.

Introduction

The versatility of substituted amidomethylpyridine ligands has attracted significant attention in recent years, given the ease of modification of the steric and electronic properties of the ligand skeleton and its ability to coordinate with diverse transition and rare-earth metals. The well-defined complexes that result from such association are readily available and suitable for a large number of investigations from fundamental coordination chemistry 1 and homogeneous catalysis 2 to materials science. 3 In addition, a variety of substituted amino-(protonated ligand of the form HL) and amido-methylpyridine (anionic ligand of the form L -) metal complexes have been successfully employed in a wide range of polymerization reactions such as i) Ni, Pd complexes for ethylene (co)polymerization, 4 ii) rare-earth and group 4 towards olefins and conjugated dienes, 5 iii) Zn and Cu for lactide Ring-Opening Polymerization (ROP) 6 and iv) the polymerization of methyl methacrylate with complexes containing Pd, Co, Cu, Zn and Cd metals. 7 Within these families of complexes, the coordination chemistry of iron salts containing this type of ligand framework remains underdeveloped, despite the fact that this metal is abundant, inexpensive, has negligible environmental impact and is present in biological systems. 8 As such, iron dichloride complexes bearing protonated substituted aminomethylpyridine ligands were prepared by the group of Wang and the group of Zubris for their application in the polymerization of conjugated dienes 9 and atom transfer polymerization of styrene 10 , respectively, while Westerhausen and coll. disclosed the reactivity of substituted amidomethylpyridine ligands with (thf)Fe[N(SiMe3)2]2Cl and FeCl2 precursors. 1a,11 On the other hand, Bontemps, Pichon and coll. described the synthesis of two homoleptic amidomethylpyridine iron complexes along with a thorough investigation of their magnetic properties. 12 Thus, we noted that no examples of this type of easily tunable iron complexes have been described in the literature for their applications in the polymerization of cyclic esters.

Indeed, environmental problems arising from the exploitation of petroleum resources and the accruement of plastic waste have led the research communities to develop sustainable and eco-compatible alternatives for polymer applications. 13 In this context, synthetic aliphatic polyesters have generated considerable interest as biodegradable, biocompatible and eco-friendly substitutes of oil-based materials. 14 Within this family, polyesters are among the most widely used biodegradable materials since they can be applied in a broad range of applications, from food packaging to medicine. 15 For example, polylactide (PLA) is produced from lactide (LA), a cyclic dimer of lactic acid present in the form of three different stereoisomers (L-LA, D-LA and meso-LA) and is obtained from abundant renewable supplies. 16 It is now generally assumed that the most effective method for obtaining high molar masses polyesters in a controlled fashion is the Ring-Opening Polymerization (ROP) of their related cyclic ester monomers, which proceeds mainly through a coordination-insertion mechanism process when well-defined metal-based or organic catalysts bearing an active ligand/initiator are used. [START_REF] Byers | Catalytic Systems for the Production of Poly(Lactic Acid). In Synthesis, Structure and Properties of Poly(lactic acid)[END_REF] While a plethora of organocatalytic systems 18 and metal-based complexes 19 have been effectively applied to the ROP of cyclic esters, industrial-scale production of PLA relies predominantly on tin octanoate as a catalyst, despite its established toxicity. 20 On the other hand, less attention has been given to the development of iron-based catalysts for the ROP of cyclic esters in the early stages; 21 former studies were based on FeX2 or FeX3 salts (X = Cl, Br, ClO4) 22 and derivatives, 23 but displaying a low level of control and activity compared to tin octanoate. More recently, the design of well-defined complexes bearing suitable auxiliary ligands 24 has led to improved activity and selectivity, however, these latter do not reach the constraints imposed for industrial applications in terms of activity and ease of implementation, although a recent example described by the group of Herres-Pawlis, based on an iron complex supported by a guanidine ligand, is currently showing great promise. 25 Here we report the preparation and characterization of one homoleptic iron amidomethylpyridine complex and two heteroleptic iron bis(trimethylsilyl) amidomethylpyridine complexes, including a thorough 1 H NMR study of the interaction in solution of pyridine and rac-LA with one of the heteroleptic complexes and their evaluation as catalysts for the polymerization of rac-LA.

Results and Discussion

Ligand synthesis and characterization

The preparation of the two racemic aminomethylpyridine pro-ligands (L 1 H -L 2 H), each comprising a 2,6diisopropylphenyl substituent on the N-aryl group, was carried out (Scheme 1) by adapting procedures that have been previously described in the literature for related compounds. The known L 1 H and new L 2 H pro-ligands were synthesized in two steps starting from acid-catalyzed condensation reaction of 2-acetylpyridine or 2-acetyl-6-methylpyridine, respectively, with 2,6-diisopropylaniline in absolute ethanol under reflux conditions to obtain the imino-26 and 6-imino-methylpyridine precursors. 27 The iminomethylpyridine compounds were then subjected to reduction of the imine group with an excess of NaBH4 in a CH2Cl2/MeOH mixture at room temperature to give the L 1 H and L 2 H pro-ligands in better overall yield than previously described for the former (75-80%). The 1 H NMR spectrum of the known aminomethylpyridine L 1 H was similar to that found in the literature (ESI, Figs. S1-S2). 4b,5c The new compound L 2 H was characterized by 1 H, 13 C NMR spectroscopies and high resolution mass spectrometry (HRMS), which was consistent with the formation of 2,6-diisopropyl-N-(1-(6methylpyridin-2-yl)ethyl)aniline pro-ligand (ESI, Fig. S3-S7). In particular, the 1 H NMR spectrum features two doublets corresponding to a pair of diastereotopic isopropyl methyls at  = 1.26 and 1.13 ppm, each of them integrating for 6H, along with a septet at  = 3.55 ppm corresponding to the -CH( i Pr)2 protons, this being similar to that found for L 1 H.

Complexes synthesis and characterization

Synthesis and 1 H NMR studies. Both homoleptic and heteroleptic amidomethylpyridine iron complexes were targeted by reacting Fe[N(SiMe3)2]2 with the appropriate amount of pro-ligand. The dropwise addition of a colorless THF solution containing 2 eq. of L 1 H pro-ligand to a green THF solution of 1 eq. of Fe[N(SiMe3)2]2 immediately produced a deep red mixture at room temperature (Scheme 2, up). The solution was concentrated under vacuum and the crude product was subjected to recrystallization in pentane at -20 °C to produce, after few days, dark pinkpurple crystals (yield = 50%).

Scheme 2. Synthesis of homo-and heteroleptic amidomethylpyridine iron(II) complexes 1-3 complexes

The obtained crystals were characterized by elemental analysis, which was consistent with the formation of an homoleptic iron complex of formula (L 1 )2Fe (1) and single crystals were subjected to X-ray diffraction studies (vide infra). In addition, the 1 H NMR spectrum of complex 1 was recorded in C6D6 and shows typical paramagneticallyshifted peaks for a high-spin iron d 6 complex, ranging from  = -43 to +134 ppm (Figure 1.a and ESI, Fig. S9). At room temperature, this 1 H NMR spectrum reveals for 1 a C2 symmetry in solution with only eleven broad resonances for the hydrogen atoms of both ligands. We can suggest that the ortho-pyridyl protons (H 6-py ) of the coordinated ligands, which are nearest to the paramagnetic iron center, were found to constitute the broadest singlet downfield at  = 134 ppm. In contrast, the attempted synthesis of the homoleptic iron analogue with L 2 H under the same experimental conditions failed, even after prolonging the reaction time and increasing the temperature up to 70 °C.

The unsuccessful formation of homoleptic (L 2 )2Fe could be most likely due to the steric inhibition imparted by the presence of methyl group on the  position of the pyridine moiety, which prevents the simultaneous coordination of the two deprotonated amidomethylpyridine chelates to the iron center.

On the other hand, reaction of 1 eq. of L 1 H, or L 2 H, ligand in pentane with 1 eq. of Fe[N(SiMe3)2]2 led to the formation of the heteroleptic counterpart (L 1 )Fe[(N(SiMe3)2] (2), or (L 2 )Fe[(N(SiMe3)2] (3) complexes, respectively (Scheme 2, down), along with the presence of a small amount of 1 in the case of the reaction involving L 1 H.

Meanwhile, the use of polar solvents such as THF or Et2O instead of pentane resulted in an increase in the amount of 1 compared to 2. Several attempts to crystallize the resulting complexes in different solvents and at different temperatures unfortunately failed. 1 H NMR spectra of the crude complexes 2 and 3 were nevertheless recorded at room temperature in C6D6 and showed somewhat analogous paramagnetically-shifted resonances between  = -122 to +128 ppm and  = -107 to +137 ppm, respectively (Figure 1.b, 1.c and ESI, Fig. S10 and S12), with no presence of the resonance at  = 61.1 ppm of the (SiMe3)2 group of the starting Fe[N(SiMe3)2]2 material. The 1 H NMR spectrum of complex 2 displays ten resonances with one broad intense signal at  = 34.6 ppm that can be assigned to the proton resonances of the two SiMe3 substituents. As expected, the addition of 1 eq. of L 1 H to a (ESI, Fig. S14). In contrast, the 1 H NMR spectrum of complex 3 reveals twelve resonances with two broad intense signals, one at  = 40.2 ppm and another one overlapping with the residual benzene signal at  = 8.1 ppm. This observation suggests that a restricted rotation of the two SiMe3 substituents occurs in solution, likely due to the steric hindrance imposed by the presence of the methyl group on the  position of the pyridine skeleton. One broad signal is observed for each complex in the  = 25-40 ppm range that can be ascribed, relative to the integration of all peaks, to the protons of the bis(trimethylsilyl)amide group. Interestingly, the splitted resonance peaks of N(SiMe3)2 in complex 3 were found as a very large single peak in complex 3 Py . This signal was also found to be much sharper with higher intensity in complex 2 Py when compared to 3 Py , emphasizing the influence in solution of the substituent on the  position of the pyridine ring of the amidomethylpyridine ligand. The resonance of the ortho and meta protons of the coordinated pyridine in complex 2 Py could not be assigned since these signals overlap

with the resonance of the N(SiMe3)2 group, whereas the signal at  = 13.90 ppm was attributed to the para protons.

In contrast, the assignment of the pyridine protons in complex 3 Py could be established by comparison of the 1 H NMR spectrum of 3 and the relative intensity of each peak upon addition of an increased quantity of pyridine to a solution of complex 3 Py (vide infra). The larger resonance at  = 37.81 ppm was ascribed to the ortho protons of pyridine as they are located closer to the paramagnetic metal center, while the sharper resonance of the meta and para protons were found at  = 13.66 and 9.68 ppm, respectively.

As the pyridine proton resonances could be properly observed with complex 3 Py in C6D6, we decided to investigate the influence of the concentration on the proton resonances of the coordinated ligand and pyridine by recording the 1 H NMR spectrum of the complex at room temperature at different concentrations, from 7.6 x 10 -3 to 3.5 x 10 -2 mol/L. The 1 H NMR spectra revealed that most of the resonances shifted marginally. In contrast, the signal attributed to the N(SiMe3)2 at  = 21.9 ppm broadened significantly upon decreasing the concentration of the complex while the resonances of the pyridine protons shifted upfield, with the ortho protons being the most impacted likely due to their proximity to the paramagnetic center. For instance, the resonance of the ortho-proton was seen at  = ca. 51.4 ppm at a concentration of 3.5 x 10 -2 mol/L vs.  = ca. 37.7 ppm at 3 x 10 -2 mol/L, with this resonance overlapping with the N(SiMe3)2 signal at lower concentration ([C] ≤ 3 x 10 -2 mol/L) (ESI, Fig. S15 andS16).

From these observations, the influence of the concentration of pyridine on the spectrum of complex The fact that only one set of resonances for each pyridine proton was observed upon incremental addition of pyridine revealed a rapid ligand exchange equilibrium on the NMR timescale (ESI, Fig. S19), as previously observed in the case of the interaction of pyridine with the bis(acetylacetonato)iron(II) complex. 28 These results will be used for comparison with the interaction in solution of rac-LA with complex 3 and 3 Py (vide infra).

complexes 2 Py and 3 Py were further analyzed by X-ray diffraction studies to determine their structure in the solid state. The molecular structure of complexes 1, 2 Py and 3 Py are depicted in Figure 4, 5 and 6, respectively, and selected bond lengths and angles are given in Table 1. The solid-state structural determination of complex 1 reveals a four-coordinated geometry at the metal center with the Fe center surrounded by two monoanionic amidomethylpyridine units, the pyridine and amido nitrogen atoms of the two ligands L 1 acting as chelates. Interestingly, the complexation leads to stereospecific coordination of the same pair of enantiomeric ligands to the iron center, with both ligands being crystallographically equivalent since the iron center lies along a two-fold screw axis, as observed in the 1 H NMR spectrum of complex 1 in C6D6 (vide supra). Depending on the 4 value defined by Houser, 29 the geometry can be best described as a distorted see-saw The solid-state structural determination of complexes 2 Py and 3 Py revealed that these two neutral complexes have a closely related structure in the solid state, with the molecular structure revealing in each case a four-coordinated geometry at the metal. The Fe center is bound to a monoanionic amidomethylpyridine unit in a bidentate fashion.

The coordination sphere is then completed by one nitrogen atom from the pyridine and one monoanionic nitrogen atom from the silylamide fragment. Due to the presence of four distinct substituents on the metal, the iron turns into a stereogenic center. Furthermore, considering the additional chiral carbon (C13, C13A) on the amidomethylpyridine ligand, we therefore expected the formation of four diastereoisomers. In fact, the complexation in the presence of pyridine leads to a stereo-differentiation in the solid state, 32 producing two pairs of enantiomers

Ring-Opening Polymerization of Lactide

The ROP of rac-lactide (rac-LA) was carried out using the initiators 2 Py and 3 Py at 50 °C in toluene, the results of which are displayed in Table 2. In addition, complexes 2 and 3 were prepared in situ, due to their high solubility which did not allow us to isolate them from by-products, and were, despite the presence of HN(SiMe3)2, also screened for the polymerization of rac-LA in order to investigate the influence of pyridine on the polymerization process.

Preliminary results showed that all complexes were active for the polymerization of rac-LA for a ratio of [rac-LA]/[Fe] = 100 or 500; complexes 2 Py and 3 Py were more efficient than their pyridine-free counterparts 2 and 3, respectively (Table 2, entry 1 vs 5 and entry 2 vs 6, respectively). For the same [monomer]/[Fe] ratio and reaction time, complex 2 Py was able to convert more monomer relative to complex 3 Py ; a similar observation could be made for their pyridine-free counterparts 2 vs 3. These results could imply that the presence of a methyl group on the  position of the pyridine ring of the amidomethylpyridine ligand might hinder the access of the monomer to the iron center, thus slowing down the overall polymerization rate. Size-Exclusion Chromatography (SEC) equipped with a triple detection was performed to determine their absolute molar masses (Mn) and dispersity (Ð) (ESI, Figs. S32).

The SEC analyses were carried out in chloroform, as these polylactide (PLA) samples were, surprisingly, poorly soluble in THF. In all cases, the Mn of the resulting PLA were found to be much higher than the expected Mn values, with somewhat similar molar masses between the runs performed with rac-LA/cat = 100/1 vs 500/1, but still showing a relatively narrow dispersity (Mn = 100 -166 x 10 3 g/mol and Ð = 1.4 -1.8), which suggest a faster propagation rate compared to the initiation step. The 1 H NMR spectra of all isolated PLA indicated the absence of the resonance belonging to the expected initiating N(SiMe3)2 group (ESI, Figs. S21-S31), likely due to the high Mn of the resulting PLA samples that prevented the identification of the chain ends. We observed a decrease in the solubility of the PLA samples in solution as a function of time by means of the evolution of the resonance at  ≈ 5.2 ppm in the 1 H NMR spectrum. As an example (ESI, Fig. S33), the 1 H NMR spectrum of the PLA sample prepared in Table 2, Entry 3, revealed that the resolution decreases significantly after remaining for 3 h in a CDCl3 solution. We could notice that this process was irreversible even after heating or applying ultrasonic waves on the NMR tube for several minutes, suggesting that such behavior may be related to a kind of auto-aggregation of the polymer in solution;

however, we were unable to find an explanation in the literature. Nevertheless, the tacticity of PLA could still be determined by performing homonuclear decoupled 1 H NMR of the methine region on a freshly CDCl3 solution of Table 2. Polymerization of rac-lactide at 50 °C for 2 hours using initiators 2 Py , 3 Py , 2 and 3

Entry [a] Cat.

[rac-LA]/[Fe] Conversion (%) [b] Mn,theo (g/mol) [c] Mn (g/mol) [d] Ð [d] 1 PLA sample. Indeed, we could expect a stereoselective polymerization process of rac-LA since the molecular structures of complexes 2 Py and 3 Py revealed that the orientation of the coordinated pyridine on the metal center relied on the absolute configuration of the amidomethylpyridine ligand. As a representative example, the homonuclear decoupled 1 H NMR of the PLA obtained in Table 2, Entry 3 showed no control over tacticity (ESI, Fig. S34). Similar results were found for all the other catalyst systems tested, whether in the absence or presence of pyridine ligand. In all cases, a controlled experiment using L-LA instead of rac-LA showed no epimerization, supporting that the process is indeed non-stereoselective (ESI, Fig. S35). In parallel, we attempted to generate the alkoxide complex derivative by reacting complex 3 Py with benzyl alcohol, because the silylamide group is less nucleophilic than the alkoxide and generally leads to a slow initiation step of the polymerization. 35 Following the reaction by NMR spectroscopy, we observed a drastic change in the 1 H NMR spectrum upon addition of 1 eq. of NMR spectroscopy studies (Table 3). The conversion was determined by 1 H NMR and the obtained polymers were also characterized by 1 H and 13 C NMR spectroscopy studies (ESI, Figs S38-43).

It should be noted that the evaluation of the polymerization kinetics of these systems was not performed and the reactions were conducted for the same time of 1.5 h for the complexes in the presence of pyridine and 4 h for their free counterpart. Nevertheless, we could observe that the solution of the ROP of rac-LA with the complexes bearing the amidomethylpyridine ligand became highly viscous after 30 min at 50 °C in toluene. In contrast, the polymerizations with the Fe[N(SiMe3)2]2 precursor required a longer time to achieve the same viscosity, qualitatively Table 3. Polymerization of rac-LA using initiators 2 Py , 3 Py , 2, 3 and Fe[N(SiMe3)2]2 in the absence and presence of 1 eq. of pyridine

Entry [a] Cat. Time (h) Conversion (%) [b] Mn,theo (g/mol) [c] Mn (g/mol) [d] Ð [d] 1 underlining the higher activity of the prepared complexes compared to the Fe[N(SiMe3)2]2 precursor in the presence of pyridine. A similar observation could be made for their free-pyridine counterpart, in addition to a complete conversion of monomer for complexes 2 and 3 while it only reached 90% for the iron precursor after 4 hours of reaction at 50 °C in toluene as shown in Table 3. As observed previously, the experimental Mn were found in all cases to be much higher than the theoretical values with low dispersity (ESI, Figs S44.a-e), again emphasizing a faster rate of propagation compared to the initiation.

Due to the technique used, MALDI-TOF analysis of the isolated polymers prepared from entries 1-6

showed only peaks for low molar masses up to 4 000 g/mol, which precludes the possibility of generalizing the interpretation of these data. Nevertheless, the MALDI-TOF analysis of the isolated polymers prepared from complexes 2 Py and 3 Py showed quasi similar spectra that exhibits multiple sets of peaks with positive charge (ESI, To investigate if the anionic amido arm of the ligand could be involved in the polymerization initiation step, a control experiment was performed using homoleptic complex 1 with 100 eq. of rac-LA in toluene. It was found that the catalyst was poorly active, converting only 10 % of monomer in 2 hours at 50 °C. Interestingly, the MALDI-TOF spectrum of the obtained polymer showed a set of signals attributable to cyclic and linear PLA (ESI, Figs S51), almost similar to those found for complexes 2 Py and 3 Py and thus suggesting that the amidomethylpyridine ligand can also be involved in the polymerization initiation step. Nevertheless, all 1 H and 13 C NMR spectra of these polymer samples showed the absence of resonances characteristic of the polymer chain end bearing a N(SiMe3)2 or amidomethylpyridine group (ESI, Figs. S38-S43).

1 H NMR studies: For a better understanding of the interaction of the monomer with the iron complex, we monitored the 1 H NMR scale reaction of complex 3 Py in the presence of rac-LA (ESI, Fig. S52-S53). The addition of 2 eq. of rac-LA to a solution of complex 3 Py in C6D6 ([C] = 3.5 x 10 -2 mol/L) did not affect significantly the proton resonances of the coordinated amidomethylpyridine L 2 ligand and pyridine, nor the resonance of the N(SiMe3)2 group. The single broad resonance of the proton of rac-LA in the presence of complex 3 Py was found at  = 3.77 ppm, which is a value slightly shifted downfield when compared to free rac-LA in C6D6 ( = 3.66 ppm, COC(H)(CH3)O). This observation suggests a rapid exchange equilibrium between the coordinated rac-LA on the metal center and the free rac-LA. Since the proton resonances of complex 3 Py were barely affected by the presence of 2 eq. of rac-LA and that the proton resonance of rac-LA with 3 Py is very close to the resonance of free rac-LA, we can suggest that rac-LA is weakly coordinated to the iron center when compared to pyridine. Although, we could observe a small resonance at  = 5.13 ppm that belongs to the protons of the polymer chain -[COC(H)(CH3)O]xproduced in the initial growth steps. Monitoring the 1 H NMR spectrum over time, we did not observe any marginal change in the proton resonances other than the increase in intensity of the polymer proton resonances at  = 5.13 ppm alongside the decrease in intensity of the monomer resonance at  = 3.77 ppm.

To confirm that rac-LA is weakly coordinated to the metal center, we studied the 1 H NMR spectra of complex 3 Py upon addition of an incremental amount of [rac-LA]/[Fe] to qualitatively assess the interaction of the monomer with the iron complex, as previously done with pyridine. The 1 H NMR spectrum of complex 3 Py in C6D6

([C] = 3.2 x 10 -2 mol/L) was first recorded, followed by the successive addition of 1 to 30 eq./[Fe] of rac-LA (ESI, Figs S54-S55). It is worth mentioning that at a ratio of [rac-LA]/[Fe] > 15, a fraction of monomer was not dissolved in the C6D6 solution due to the poor solubility of rac-LA in benzene, however, the increase in the amount of monomer relative to the 3 Py complex in solution could be observed in the 1 H NMR spectra. Overall, the resonances of the ortho-, meta-and para-protons of pyridine were negligibly impacted by the increased quantity of rac-LA, with signals at  = 45 -41 ppm (overlapping with the resonance of one proton of the coordinated L 2 ligand),  = 14.7 -15.4 ppm and  = 10.8 -10.6 ppm, respectively. The proton resonance of rac-LA was slightly shifted downfield from  = 3.76 to  = 3.91 ppm upon increasing the quantity of monomer. Furthermore, we carried out the same set of experiments in the presence of complex 3. Firstly, the in situ preparation of complex 3 was performed by mixing an equal amount of Fe[N(SiMe3)2]2 and L 2 H in C6D6 overnight and the 1 H NMR spectrum was recorded followed by the successive addition of 1 to 30 eq./[Fe] of rac-LA (ESI, Figs S56-S57). Again, by increasing the quantity of monomer the 1 H NMR spectrum of complex 3 was barely affected and the single proton resonance of rac-LA was found to be slightly shifted downfield from  3.80 to = 3.91 ppm. These results confirm that rac-LA is weakly coordinated to the iron center and predominantly free in solution on the NMR timescale, in contrast to what was found for pyridine, where the latter remains strongly coordinated to the metal center throughout the experiment. Furthermore, this series of experiments is consistent with complex 3 being less active than 3 Py since no PLA formation was observed when recording the 1 H NMR spectrum of the mixture 3/rac-LA, in contrast to the 1 H NMR spectrum of the mixture 3 Py /rac-LA described previously.

Conclusion

A homoleptic iron complex bearing two amidomethylpyridine ligands (complex 1) as well as two heteroleptic iron complexes supported by a bis(trimethylsilyl)amide group, a pyridine and an amidomethylpyridine ligand (complexes 2 Py and 3 Py ) were successfully prepared and fully characterized by 1 H NMR, X-ray crystallography as well as elemental analyses. The solid-state structure of the three complexes revealed a tetracoordinated environment around the iron center, which displayed a distorted geometry between trigonal pyramidal and see-saw. Additionally, the molecular structure of both heteroleptic complexes indicated that the orientation of the coordinated pyridine on the metal center was dependent on the absolute configuration of the amidomethylpyridine ligand, leading to a stereo-differentiation in the solid state. Evaluation of the coordination of pyridine and rac-LA in solution by 1 H NMR studies with complex 3 Py demonstrated a rapid exchange equilibrium on the NMR timescale, with the pyridine ligand being strongly coordinated to the metal center compared to the rac-LA monomer. All three complexes as well as the pyridine-free counterparts of the two heteroleptic complexes (complexes 2 and 3) and the iron bis(trimethylsilyl)amide were found to be active for the polymerization of rac-LA at 50 °C, leading to the formation of PLA with cyclic structures identified for low Mn. The presence of pyridine within the heteroleptic complexes appeared to increase the rate of polymerization while giving rise to the formation of a broad mixture of polymer structures including low Mn of cyclic PLA. At this stage, it is difficult to propose a mechanism because the polymerization of lactide with these systems could be based on a coordination-insertion or activated monomer process, which could also involve one of the two ligands (e.g. Li, 37 Na 38 or Zn 36 systems). Also, we cannot exclude that the mechanism could occur by zwitterionic ROP (e.g. N-heterocycle carbene, 39 Ce(III)-NHC complexes, 40 ) or a ring-expansion polymerization process (e.g. Ln, 41 Bi, 42 Sn 43 complexes).

Experimental Section

Material and technique: All manipulations were performed using Schlenk techniques and/or a dry/inert glovebox (O2 < 1 ppm, H2O < 1 ppm, Jacomex, Dagneux, France) under an argon atmosphere. Toluene, pentane, diethyl ether, THF were dried by passing through an alumina column from Mbraun SPS (MBraun France, Mérignac, Paul-Sabatier, Toulouse, France. Determination of the number-average molar masses (Mn) and dispersity (Ð = Mw/Mn) of the polylactide samples was achieved by size exclusion chromatography (SEC) in CHCl3 at 35 °C with a rate of 1mL/min. The system was equipped with a triple detection system comprising a differential refractometer detector (Optilab Rex Wyatt), a multi-angle light scattering detector (TREOS Wyatt, laser = 658 nm) and an UV detector (Shimadzu SPD-M20A, = 254 nm). The SEC system was fitted with a pre-column and two Agilent PL gel 5 m Mixed C columns. The differential refractive index increment (dn/dc) value was determined by using a differential refractometer from PSS equipped with a laser fixed at 620 nm and two cells at 35 °C, a reference cell containing the solvent alone and a measuring cell containing the polymer solution. The principle of the measurement is to define the difference in refractive index between the polymer solution and the solvent for different concentrations. The value of dn/dc was measured on one sample, its value was found to be equal to 0.0204 mL/g. This value was used for the mass calculations of all samples provided. Samples were prepared by dissolving the product in CHCl3 and were then filtered with 0.45 m filters before the injection of 50 L of the solution. Matrix

Assisted Laser Desorption and Ionization Time-Of-Flight (MALDI-TOF): MALDI-TOF analyses were performed by Dr. François Stoffelbach, Sorbonne Université, Paris, France. Mass spectra were recorded using a Bruker autoflex III smartbeam mass spectrometer, equipped with the laser that produces pulses at 337 nm using dithranol as a matrix and NaI as cationizing agent. Spectra were recorded in linear mode at an accelerating potential of 20 kV.

Samples were prepared by dissolving the polymer in CHCl3 at a concentration of 2-5 mg/mL. A 10 L aliquot of this solution was mixed with 20 L of matrix solution and 10 L of NaI solution (both at 20 mg/mL in CHCl3). Standards (polyethylene glycol of known structure) were used to calibrate the mass scale. Single-crystal X-rays measurements were performed at 100 K. The data were collected using an Apex II DUO CCD 4K Bruker diffractometer (λ = 0.71073 Å). The structures were solved using SHELXT 45 and refined by least-squares procedures on F2 using SHELXL, 46 the refinement data can be found in ESI, Part 12. CCDC 2126558 (for complex 1), 2126559 (for complex 2 Py ) and 2126560 (for complex 3 Py ) include the additional crystallographic data for this article. This information can be acquired free of charge from The Cambridge Crystallographic Data Centre.

Crystal and Refinement data for complexes 1, 2 methylpyridin-2-yl)ethan-1-imine (588 mg, 2 mmol) was dissolved in 8 mL of CH2Cl2 in a round bottom flask equipped with a condenser followed by the addition of a large excess of NaBH4 (7.57 g, n = 0.2 mol, 100 eq.) to afford a white yellowish slurry. MeOH (40 mL) was added by portion to this mixture (caution exothermic reaction) and the solution was stirred at room temperature for 24 hours. The evolution of the reaction was monitored by TLC (eluent: petroleum ether/ethylacetate (EtOAc) = 20/1) and, if necessary, new addition of NaBH4 and MeOH was carried out until the starting material disappeared. The solution was quenched by careful addition of deionized water after which the organic phase was extracted with CH2Cl2 and dried over Na2SO4. After filtration, the solvent was evaporated and the resulting oil was further purified by column chromatography (SiO2; petroleum ether/EtOAc = 30/1) to afford a colorless solid after evaporation of the eluent. Yield = 80 %, m = 474 mg. 1 H NMR (300 MHz, C6D6, 25 °C) δ (ppm): 7.13 -7.03 (m, 3H, H Ar ), 6.88 (dd, 3 J = 7.7 Hz, 3 J = 7.7 Hz, 1H, H 4-py ), 6.53 (d, 3 J = 7.7 Hz, 1H, H 3- py or H 5-py ), 6.53 (d, 3 J = 7.6 Hz, 1H, H 3-py or H 5-py ), 4.56 (br s, 1H, HN), 4.29 (q, 3 J = 6.7 Hz, 1H, NCH(CH3)), 3.23 (m = 400 mg, 1 previously in 5 mL of pentane, at room temperature. The solution turned red-yellow to dark-red after stirring for 1 h 30. The solution was filtered through celite and reduced under vacuum to afford a red solid (complex 2). 1 H NMR (300 MHz, C6D6, 25 °C, δ in ppm): 128.3, 118.2, 69.9, 52.5, 37.6, 34.65 (broad, SiMe3), -27.3, -31.3, -35.0, -122.0.

The crude complex 2 (considering quantitative yield, 1.06 mmol) was dissolved in 6 mL of pentane and filtered to a solution of pentane (2 mL) containing 4 eq. of pyridine (V = 340 L). After 30 min, red crystals appeared, which were extracted from the solution and dried under vacuum. The red crystals were consistent with the formation of for complex 2. Under argon, the pro-ligand L 2 H (m = 334 mg, n = 1.13 mmol) was placed in 5 mL of pentane followed by the addition of a solution of Fe[N(SiMe3)2]2 (m = 425 mg, 1 eq.), previously dissolved in 5 mL of pentane, at room temperature. The solution turned red-yellow to dark-red after stirring for 1 h 30. The solution was filtered through celite and reduced under vacuum to afford a red solid (complex 3). 1 H NMR (300 MHz, C6D6, 25 °C, δ in ppm): 137.4, 110.7, 70.1, 54.6, 40.2 (broad, SiMe3), 33.7, 24.2, 8.12 (broad, SiMe3), -20.4, -33.9, -35.4, -107.3.

The crude complex 3 (considering quantitative yield, 1.13 mmol) was dissolved in 6 mL of pentane and filtered to a solution of pentane (2 mL) containing an excess of pyridine (V = 340 L). After 30 min, red crystals appeared, which were extracted from the solution and dried under vacuum. The red crystals were consistent with the formation of Typical polymerization of rac-LA: A glass reactor was charged with the appropriate iron catalyst solution (250 µL, 2.5 µmol, 1 eq., 0.01M in toluene) via syringe and toluene (1 mL) was added subsequently to obtain a red solution. The solution was stirred for 1 min inside the glove box before adding purified rac-LA (1.25 mmol, 500 eq.,

[C] = 1 mol/L). The reactor was brought outside the glove box and the mixture was stirred at 50 °C for appropriate time. The reaction was quenched with acidifed toluene and the mixture was diluted with chloroform to homogenize the medium before withdrawing an aliquot. The aliquot was dried to evaporate the solvent and the conversion was determined by 1 H NMR spectroscopy in CDCl3. The polymer solution in chloroform was added dropwise to a large excess of ethanol (150 mL) that induced the precipitation of PLA. The polymer was dried overnight under vacuum before being subjected to further characterization.

Scheme 1 .

 1 Scheme 1. Synthesis of aminomethylpyridine pro-ligands L 1 H and L 2 H

Figure 1 .

 1 Figure 1. 1 H NMR spectra in C6D6 of complexes a) 1, b) 2 (# represents the N(SiMe3)2 group,* represents the residual peaks of the homoleptic complex 1) and c) 3

Figure 2 .

 2 Figure 2. 1 H NMR spectra in C6D6 of complexes a) 2 Py and b) 3 Py ([C] = 3 x 10 -2 mol/L) -ortho-(o), meta-(m) and para-(p) protons of the pyridine, # represents the Ni(SiMe3)2 group

3

 3 Py was evaluated by means of 1 H NMR techniques by adding different amounts of pyridine from 0 to 30 eq. to a solution of the complex in C6D6 ([3 Py ] = 3 x 10 -2 mol/L). Increasing the pyridine concentration had a drastic change on the overall spectrum, with the resonances of the ligand protons being strongly affected (ESI, Figs. S17-S18) as well as the signal of the pyridine protons. For example, the most downfield resonance at  = 124.1 ppm of the amidomethylpyridine ligand in complex 3 Py was continuously shifted upfield upon increasing the pyridine concentration, reaching a value of  = 82.7 ppm with a [pyridine]/[Fe] ratio of 30. On the other hand, the most upfield resonance at  = -87.8 ppm was gradually shifted downfield upon increasing the concentration of pyridine, which attained a resonance at  = -26.2 ppm for [pyridine]/[Fe] = 30.

Figure 3 .

 3 Figure 3. Stacked 1 H NMR spectra (two regions) in C6D6 of complexes 3 Py [C] = 3 x 10 -2 in the presence of various amounts of pyridine with a) Py/Fe = 1, b) Py/Fe = 2, c) Py/Fe = 4, d) Py/Fe = 8, e) Py/Fe = 15, f Py/Fe = 30. The assignment of the resonances of the ortho-(o), meta-(m) and para-(p) protons of the pyridine Overall, this study confirmed the proposed assignments of the ortho-, meta-and para-proton resonances (vide supra), as the signals were found to be shifted upfield and became much sharper with higher intensity upon increasing the pyridine concentration. As expected, the resonances of the ortho-protons were the most impacted due to their closer proximity to the iron center. Nevertheless, even at a ratio of [pyridine]/[Fe] = 30, the resonance of the pyridine protons did not match the typical value of the free ligand in a diamagnetic field ( 1 H NMR of pyridine recorded at room temperature in C6D6, ortho-H  = 8.53 ppm, meta-H  = 6.98 ppm and para-H  = 7.00 ppm).

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Molecular structures of complex 1, with ellipsoid drawn at 50% probability level (hydrogen atoms have been omitted for clarity)

(

  4 = 0.67). The Fe-N bond distances[Fe-Namido = 1.9323(9) and Fe-Npyridine = 2.1205(9) Å] and the N1-Fe1-N2 bite angle [81.61(4) °] of the N^N chelate were found in the similar range as observed in related homoleptic iron amidomethylpyridine complexes reported in the literature (Fe-Namido ≈ 1.92 Å, Fe-Npyridine ≈ 2.12 Å and Namido^Npyridine ≈ 79 -82 °).11a,12,30 The lengths of the C6-C5 [1.5131(15) Å] and C6-N2 [1.4542(13) Å] bonds fall within the typical intervals of C(sp 3 )-C(sp 3 ) and C(sp 3 )-N bond distances, respectively.31 Finally, the N-aryl group deviates from orthogonality with respect to the pyridine plane [tors. C6-N2-C8-C13 = 77.1(1)°], while the amidomethylpyridine chelating unit is out of planarity [tors. N2-C6-C5-N1 = -16.5(1)°].

RC13,

  RFe and SC13,SFe in the unit cell of the molecular structure of the two complexes 2 Py and 3 Py . Because of the steric repulsion induced by the bis(trimethylsilyl) amide and the isopropyl groups of the N-aryl substituent, the methyl C14(A) points in the same direction as the pyridine with respect to the amidomethylpyridine chelate N1-C13(A)-C15-N2. The geometry at the iron center displays an intermediate geometry that can be best described as distorted from either trigonal pyramidal or see-saw,  value ranging from 0.69-0.73. The amidomethylpyridine chelating framework is almost planar [N1-C13-C15-N2 ranging from 2.4(2)-12.1(3)°] while the N-aryl group slightly deviates from orthogonality to this plane [C13-N1-C1-C6 ranging from 103.6(2)-105.0(2)°]. The Fe1-N1 distance of the Namido group [1.959(1)-1.961(1) Å] is shorter than the Fe1-N2 distance of the pyridine fragment of the binucleating ligand [2.144(1)-2.155(1) Å], the latter being quasi-identical to the Fe1-N3 distance of the pyridine ligand [2.155(1)-2.159(1) Å]. In addition, the Fe1-N4 bond length [1.979(1)-1.988(1) Å] is longer than the terminal Fe-N distances in [Fe(N(SiMe3)2)2]2 [ca 1.925(3) Å] 33 and lies in the upper range of other four-coordinated bis(trimethylsilyl)amide iron complexes reported in the literature (1.95-1.99 Å). 34 The lengths of the C13-C14 [1.530(2)-1.520(3) Å] and C13-N1 [1.451(2) -1.461(4) Å] bonds fall within the typical intervals of C(sp 3 )-C(sp 3 ) and C(sp 3 )-N bond distances, respectively.

  2.5 µmol of complex; [rac-LA] = 1.0 M in toluene, T = 50 °C, Time = 2 h; b determined by 1 H NMR spectroscopy (ESI Figs. S20-S30); c Mn,theo = 144.13 x rac-LA/[Fe] x conversion; d determined by Size-Exclusion Chromatography (SEC) analysis in chloroform using a triple detection system.

  BnOH. The addition of BnOH induced a partial disappearance of the paramagnetic resonances of the protons assigned to the -N(SiMe3)2 group as well as the coordinated ligand, leading to the emergence of proton resonances of HN(SiMe3)2 and L 2 H in the diamagnetic region (ESI, Fig.S36-S37). These results demonstrated that the protonation of the amidomethylpyridine ligand was competitive with the substitution reaction of the N(SiMe3)2 group by benzyloxide OBn. Increasing the concentration of BnOH from 1 eq. to 4 eq./[Fe] resulted in a clear increase of the proton resonances of HN(SiMe3)2 relative to L 2 H that slowly rises along with complete loss of the proton resonances of the paramagnetic complex, similar results were found with isopropanol.Determination of polymer chain-end: Subsequently, we performed a series of polymerization reactions with a ratio of [rac-LA]/[Fe] = 20 in order to carefully analyze the polymer chain-end by MALDI-TOF spectrometry and

Figs

  Figs. S45-S46). A series of major peaks at 72n + 23 [i.e. (C3H4O2)n + Na + ], together with a series of smaller peaks at 72n + 1 [i.e. (C3H4O2)n + H + ], can be identified as cyclic polymers with even-and odd-number lactyl units indicative of the occurrence of transesterification reaction. Moreover, a series of peaks at 72n'+ 18 + 23 [i.e. H-(C3H4O2)n'-OH + Na + ] can also be attributed to linear polylactide with hydroxyl and carboxylic acid end groups that could arise from the of the cyclic PLA polymer. Although, we could not exclude that these peaks can also be ascribedto [(SiMe3)2N-[COC(H)(Me)O](n-2)-H + H + ] (160.38 + 72(n-2) + 1 + 1),as the molar mass of -N(SiMe3)2 is close to the molar mass of a lactide unit terminated by an hydroxyl group -[C(H)(Me)-C(O)]2-OH (M = 161.13 g/mol). 36 A series of peaks at m/z = 56 + 72m and a peak at m/z = 1360 could not be assigned in the analysis window enlargement. In contrast, in the absence of pyridine, complexes 2 and 3 unambiguously led to the formation of cyclic polymers, with the MALDI-TOF spectra exclusively showing a minor series of peaks at 72n + 23 [i.e. (C3H4O2)n + Na + ] along with a major population at 72n + 1 [i.e. (C3H4O2)n + H + ] (ESI, Figs. S47-S48). An experiment using Fe[N(SiMe3)2]2 as initiator for the polymerization of rac-LA (20 eq./[Fe]) was performed and the MALDI-TOF spectrum of the obtained polymer revealed only one set of peaks at 72n + 1 [i.e. (C3H4O2)n + H + ] (ESI, Fig. S49) attributed to cyclic polymers like for complexes 2 and 3. Addition of 1 eq. of pyridine to a solution of Fe[N(SiMe3)2] in the presence of 20 eq. of rac-LA resulted to the predominant formation of cyclic polymers with peaks at 72n + 23 [i.e. (C3H4O2)n + Na + ]} along with an unidentified minor series of peaks at m/z = 56 + 72m (ESI, Figs S50), similar to that found for complex 3 Py .

France) and stored

  over 4 Å molecular sieves in a glove box before use. The reagents 2-acetylpyridine, 6-methyl-2-acetylpyridine, 2,6-diisopropylaniline, sodium borohydride, pyridine, lithium bis(trimethylsilyl)amide as well as iron dichloride were purchased from Sigma-Aldrich or Fisher Scientic S.A.S. and used as received. L-LA and rac-LA were acquired from Sigma-Aldrich, recrystallized twice in toluene before being sublimed and stored in the glove box before use. Benzene-d6 was dried over sodium, distilled and stored over 4 Å molecular sieves in a glove box.Chloroform-d was used as received. The compounds N-(2,6-diisopropylphenyl)-1-(pyridin-2-yl)ethan-1-imine, Erreur ! Signet non défini. 2,6-diisopropyl-N-(1-(pyridin-2-yl)ethyl)aniline (L 1 H) Erreur ! Signet non défini.b,Erreur ! Signet non défini.c and N-(2,6-diisopropylphenyl)-1-(6-methylpyridin-2-yl)ethan-1-imine27 were prepared following modified procedures described in the literature (see ESI, Part 2 to 4). The synthesis of iron bis(trimethylsilyl)amide was carried out by using a procedure described in the literature (see ESI, Part 6).44 Characterization: Nuclear Magnetic Resonances (NMR): 1 H-and 13 C NMR spectra were recorded on an Avance 300 instrument (Bruker France, Palaiseau France) at 298 K. All 1 H chemical shifts (reported in [ppm]) were calibrated by using residual signals of the deuterated solvents. Elemental analyses were performed by Céline Delabre on a Micro Cube apparatus Elementar Vario (Elementar France SARL, Lyon, France) at UCCS, University Lille Nord de France. In the case of polymers analyses, the conversion as well as the composition of the polymers and the microstructural contents were calculated by mean of 1 H-and 13 C-NMR spectroscopy analysis. Gel Permeation Chromatography-Size Exclusion Chromatography (GPC-SEC): The GPC-SEC was performed by Pascale Saint-Aguet and Dr. Régis Laurent at the Technoplym platform, Institut de Chimie de Toulouse, Université

(

  sept, 3 J = 6.8 Hz, 2H, CH(CH3)2), 2.41 (s, 3H, CH3 py ), 1.58(d, 3 J = 6.7 Hz, 3H, NCH(CH3)), 1.26(d, 3 JHH = 6.8 Hz, 6H, CH(CH3 A CH3 B )), 1.13(d, 3 JHH = 6.8 Hz, 6H, CH(CH3 A CH3 B )).13 C NMR (75 MHz, C6D6, 25 °C) δ (ppm):162.8, 157.9, 142.4, 142.1, 136.0 (C 4-py ), 123.5 (C Ar ), 123.3 (C Ar ), 121.0 (C 5-py or C 3-py ), 118.5 (C 5-py or C 3-py ), 61.0 (NCH(CH3), 27.7 (CH(CH3)2), 24.0 (CH(CH3)2), 24.0 (CH3 py ), 22.1(NCH(CH3)). HRMS ESI(+): m/z (%) calcd. 297.2331 [C20H29N2] + ; found 297.2484 [C20H29N2] + .Preparation of complex 1 {(L 1 )2Fe}:The pro-ligand L 1 H (150 mg, 0.53 mmol) was placed in 2 mL of THF in a Schlenk flask and a solution of Fe[N(SiMe3)2]2 (m = 100 mg, n = 0.27 mmol in 3 mL of THF) was added dropwise (3 drops/second). The color of the solution immediately turned from yellow to wine. The mixture was left stirring for 4 hours and the solvent was removed under vacuum. The residue was dissolved in a minimum of pentane and filtered over celite. The solution was placed at -40°C to afford red/wine crystals after 2 days. Yield = 50 %, m = 168 mg. The red/wine crystals were consistent with the formation of complex 1 -(L 1 )2Fe by elemental analysis.Anal. Calcd. for C38H50FeN4: C, 73.77; H, 8.15; N, 9.06; found: C, 73.23; H, 8.54; N, 9.00. 1 H NMR (300 MHz, C6D6, 25 °C, δ in ppm): 134.6 (Δν1/2 = 1415 Hz, 1H), 87.9 (Δν1/2 = 516 Hz, 1H), 47.1, 46.5 and 43.3 (overlapping peaks, 5H), 37.4 (Δν1/2 = 91 Hz, 1H), 30.7 (Δν1/2 = 96 Hz, 1H), 16.6 (Δν1/2 = 175 Hz, 3H), -22.6 (Δν1/2 = 394 Hz, 6H), -37.3 (Δν1/2 = 706 Hz, 3H), -43.8 (Δν1/2 = 120 Hz, 1H). Total of 22 protons for one ligand, 3 protons resonances are missing. Under argon, the pro-ligand L 1 H (m = 300 mg, n = 1.06 mmol) was placed in 5 mL of pentane followed by the addition of a solution of Fe[N(SiMe3)2]2

complex 2 Py

 2 {L 1 Fe[N(SiMe3)2](Py)} by elemental analysis. Yield = 47%, m = 290 mg. Anal. Calcd. for C30H48FeN4Si2: C, 62.48; H, 8.39; N, 9.71; found: C, 62.03; H, 8.49; N, 9.50. 1 H NMR (300 MHz, C6D6, 25 °C, δ in ppm): 102.5 (Δν1/2 = 870 Hz, 1H), 75.9 (Δν1/2 = 1540 Hz, 2H), 52.5 ( Δν1/2 = 386 Hz, 1H), 35.2 (Δν1/2 = 140 Hz, 1H), 34.2 (Δν1/2 = 106 Hz, 1H), 26.2 (Δν1/2 = 670 Hz, 18H), 22.7 (Δν1/2 = 323 Hz, 3H), 14.0 (Δν1/2 = 153 Hz, 3H), 6.2 (Δν1/2 = 120 Hz, 2H), -23.3 (Δν1/2 = 360 Hz, 3H), -37.3 (Δν1/2 = 82 Hz, 1H), -65.6 (Δν1/2 = 2218 Hz, 2H). Total of 38 protons, Similar procedure to that described

complex 3 Py {L 2

 2 Fe[N(SiMe3)2](Py)} by elemental analysis. Yield = 54%, m = 220 mg. Anal. Calcd. for C31H50FeN4Si2:C, 63.02; H, 8.53; N, 9.48; found: C, 62.62; H, 8.78; N, 9.09. 1 H NMR (300 MHz, C6D6, 25 °C, δ in ppm): 124.1 (Δν1/2 = 440 Hz, 1H), 94.3 (Δν1/2 = 1040 Hz, 3H), 64.0 ( Δν1/2 = 142 Hz, 1H), 44.6 (Δν1/2 = 220 Hz, 4H), 37.7 (broad, Δν1/2 = 1170 Hz, 1H, should integrate for 2H, o-Py), 32.7 (Δν1/2 = 80 Hz, 1H), 30.5 (Δν1/2 = 66 Hz, 1H), 21.9 (broad, Δν1/2 = 1670 Hz, > 10H, N(SiMe3)2), 13.7 (Δν1/2 = 102 Hz, 2H, m-Py), 9.7 (Δν1/2 = 90 Hz, 1H, p-Py), 7.9 (Δν1/2 = 130 Hz, 1H), 2.7 (Δν1/2 = 100 Hz, 3H), -29.5 (overlapping peaks, 6H), -36.5 (Δν1/2 = 76 Hz, 1H), -87.7 (Δν1/2 = 600 Hz, 3H). Total of 40 protons, 10 protons resonances are missing.

Table 1 .

 1 Selected bond distances (Å) and angles (°) for the amidomethylpyridine iron complexes 1, 2 Py and 3 Py

	Parameters	Complex 1	Parameters	Complex 2 Py	Complex 3 Py
			Distances (Å)
	Fe1-N1	2.1205(9)	Fe1-N1	1.961(1)	1.959(1)
	Fe1-N2	1.9323(9)	Fe1-N2	2.144(1)	2.155(1)
	C6-N2	1.4542(13)	Fe1-N3	2.155(1)	2.159(1)
	C6-C5	1.5131(15)	Fe1-N4	1.988(1)	1.979(1)
			C13(A)-N1	1.451(2)	1.461(4)
			C13(A)-C14(A)	1.530(2)	1.538(4)
			Angles (°)
	N1-Fe1-N2	81.61(4)	N1-Fe1-N2	79.21(5)	80.06(5)
	N1-Fe1-N1 i	108.76(5)	N1-Fe1-N3	118.99(5)	113.95(5)
	N2-Fe1-N2 i	144.59(6)	N1-Fe1-N4	132.69(5)	130.93(5)
	N1-Fe1-N2 i	120.01(4)	N2-Fe1-N3	87.87(5)	89.11(5)
			N2-Fe1-N4	124.45(5)	131.13(5)
			N3-Fe1-N4	103.56(5)	104.51(5)
			Torsions angles (°)
	C7-C6-N1 i -C1 i	1.62(9)	C13(A)-N1-C1-C6	105.0(2)	103.6(2)
	C6-N2-C8-C13	77.1(1)	N2-C15-C13(A)-N1	2.4(2)	-12.1(3)
	N1-C5-C6-N2	-16.5(1)

Acknowledgments: This research work was financially supported by the University of Lille and partially by the CNRS. Dr Eric Buisine, Patrick Daubias and Dr Xavier Trivelli are thanked for their help in the NMR spectroscopy studies. Pascale Saint-Aguet and Dr Régis Laurent are acknowledged for performing the SEC analysis and Céline Delabre for the elemental analysis. Professor Philippe Zinck is thanked for the useful discussions the characterizations of cyclic PLA. Miss Nelly Kervarec from the "Service Général des Plateformes technologiques" of Brest is acknowledge for the preliminary assessment of the samples by MALDI-TOF. Finally, M.V. and Y.C. would like to express their warmest gratitude to Dr François Stoffelbach for performing the MALDI-TOF measurements and the thorough analysis of all spectra.

Keywords: amidomethylpyridine ligand, iron complexes, lactide, NMR, ring-opening polymerization.

Structure and Polymerization of rac-Lactide

Dr. Obaid H. Hashmi, Dr. Frederic Capet, Prof. Marc Visseaux, Dr. Yohan Champouret The synthesis and characterization of iron complexes bearing amidomethylpyridine ligands are described as well as their application as catalysts for the ringopening polymerization (ROP) of raclactide (rac-LA).


A Tinzefte

Y Le Menach

LAMEL, L2EP, USTL F Piriou

Iterative Solvers For Singular Symmetric Linear Systems in Low Frequency Electromagnetics

Keywords: Numerical analysis, Eddy currents, Magnetostatics

In this paper, several methods based on Krylov methods are proposed to solve the singular linear systems from Finite Element Method. Indeed, in the magnetostatic case, for Aformulation the system to solve is singular but it is auto-gauged by Krylov methods. However, due to the computation of residual vector all the methods (CG, MRTR, SQMR,MINRES) do not present the same behaviour. Moreover, these methods are applied to eddy current problem. The numerical behavior are compared and analyzed.

I. INTRODUCTION

For the Finite Element Method (FEM) applied to electromagnetism, we are led to solve linear system of equations, there are different methods for solving efficaciously such systems. Given sufficient memory and reasonable performance, direct solvers would be the method of choice for solving linear systems which have a regular matrix. The Krylov subspace iterative methods such as Conjugate Gradient (CG) [START_REF] Golub | Matrix Computations[END_REF] require less memory than direct solvers. Consequently, they are preferred for very large sparse linear systems that cannot fit into memory using direct solvers. Also, these methods represent a good computation tool for solving singular systems. This is the case in magnetostatic when we use the formulation in term of the vector potential without gauge condition. The gauge condition can be introduced with the help of an edge tree; alternative gauges are possible. With the gauge condition the system is well defined but the convergence of iterative method is very slow, so the gauging is impracticable [START_REF] Ren | Influence of the R.H.S on the Convergence Behaviour of the Curl-Curl Equation[END_REF].

In the case of non-gauged formulation, the properties of iterative solvers can be used to auto-gauged the problem [START_REF] Ren | Influence of the R.H.S on the Convergence Behaviour of the Curl-Curl Equation[END_REF]. So, the system converges and has an infinity of solutions (potential) if the matrix equation is compatible. In the case of vector formulation (curl-curl equation), the compatibility implies that the right hand side (the current density) should be divergence free. We can see, that all solutions in term of potential give the same magnetic field. The same finding is observed for eddy current problem, when we use the potential formulations.

The CG is reliable on positive-definite systems. Although, we notice that, in the case of singular systems, this method can breakdown and his convergence is irregular. In this paper, we propose to use other methods of Krylov subspace type, which have a much more stable behaviour. An example is proposed to illustrate our results. France (e-mail: yvonnick.le-menach@univ-lille1.fr).

II. MODEL

A. magnetostatic Case

The magnetostatic assumptions introduced into Maxwell's equation and with the help of vector potential and the magnetic relationship, we obtain the classical A-formulation.

curl 1 µ curlA = J (1)
With µ the magnetic permeability, J the current density and A the magnetic vector potential. According to the classical works [START_REF] Bossavit | Computational Electromagnetism[END_REF] [14] concerning the discretisation of magnetic quantities, the vector potential A is defined on the edges (circulation) and the current density J on the facets (flux). But to define the discrete form of the constitutive law we must introduce the duality conditions between two series of spaces. In these conditions, if we consider a primal mesh on which the vector potential (noted A e ) then the current density (noted J f ) will be expressed on facets of the dual mesh M. Moreover, using the properties of the incidence matrix we can define the discrete operators of gradient, curl and divergence. These operators are denoted on the primal mesh: G, C and D and respectively G, C and D on the dual mesh. It can be noted that we have the following properties: G T = -D and C = C T . Consequently, the discretized form of A formulation (1) can be written

C T M ν ff CA e = J f (2)
where M ν ff is the material matrix including the permeability and the metric information. Usually, the current density are known on primal mesh then a matrix (noted M ef ) containing only geometric data must be introduced to transfer the current density on dual mesh.

C T M ν ff CA e = M ef J f ( 3
)
The system is singular. Indeed an infinity of solution can verify the equation (1) if M ef J f is in the range of C T M ν ff C (divergence of J must be free [START_REF] Abe | MRTR method: An iterative method based on the three-term recurrence formula of CG-type for non symmetric matrix[END_REF]). In fact, the system is oversized by the number of nodes minus one equation. To ensure the uniqueness of A e there are two possibilities : the first one consists to use a spanning edge tree to eliminate the useless unknowns. The second one consists to impose the new constraint on the A e adding the coulomb gauge DA e = 0. This gauge can be write such that -G T A e = 0. As the size of G T is the product of the node number by edge number, and to ensure the solution one potential must fixed. Consequently, the system becomes well-defined by adding this gauge condition.

In the case of non-gauged formulation, Z. Ren [START_REF] Ren | Influence of the R.H.S on the Convergence Behaviour of the Curl-Curl Equation[END_REF] clarified the coulomb gauge are weakly imposed with the algorithm of conjugate gradient (CG).

B. eddy current problem

Many formulations can be used to compute the eddy current (H-formulation, T-Ω formulation and A-ϕ formulation). In this paper, the potential vector introduced in the magnetostatic case is kept. Then we present only the A-ϕ formulation. In Maxwell-Ampere equation, the current density represents the eddy current. In this conditions, with the help of the electrical scalar potential the equation (3) becomes

curl 1 µ curlA + σ( ∂A ∂t + gradϕ) = 0 (4)
Moreover, the conservation of the eddy current must verified.

divσ( ∂A ∂t + gradϕ) = 0 (5)
The discretisation can be carried out with the same scheme which present in the previous paragraph.

C T M ν ff CA e + M σ ee ( ∂A e ∂t + Gϕ n ) = 0 G T M σ ee ( ∂A e ∂t + Gϕ n ) = 0 (6)
where M σ ee is the material matrix including the conductivity and the metric information. This system is singular; indeed the number of these unknowns is the sum of the nodes and edges, and the rank of the system matrix is equal to the number of edge. A gauge condition must be applied to obtain the uniqueness of the solution. Although, the auto-gauged of the CG is not proved for A-ϕ formulation.

III. ITERATIVE METHODS

In the previous section, we have to solve a system of linear equations Ax = b,

where A ∈ IR N ×N is symmetric semi-definite, x and b ∈ IR N . Two cases can be distinguished:

1) The case where A is nonsingular and consequently positive definite. 2) The case where A is singular. Many results are known about the first case [START_REF] Golub | Matrix Computations[END_REF], where the CG is a successful iterative method. We are interested by the second case where A is singular, here, the system (7) has a solution if, and only if, b is in the range of A. In that case the solution is not unique. Indeed, Let x ∈ IR N be a solution of [START_REF] Golub | Matrix Computations[END_REF], then x = x + y is a solution for every y in the Kernel space of A.

An important class of iterative methods available for solving the system [START_REF] Golub | Matrix Computations[END_REF] are the so-called Krylov subspace methods. An iterative scheme for solving linear systems ( 7) is called a Krylov subspace method if it produces approximate solutions of the form

x k ∈ x 0 + K k (A, r 0 ), k = 1, 2, . . . ( 8
)
where x 0 is an arbitrary initial guess with the corresponding residual vector r 0 = b -Ax 0 and K k (A, r 0 ) is the kth Krylov subspace defined by

K k (A, r 0 ) = span{r 0 , Ar 0 , . . . , A k-1 r 0 } (9)
There are two main steps in designing a Krylov subspace method. The first step is the construction of suitable vectors v 1 , v 2 , . . . , v k that span K k (A, r 0 ). Then, setting

V k = [v 1 , v 2 , . . . , v k ] (10)
we parameterize the kth iteration (8) as follows :

x k = x 0 + V k z k , where z k ∈ IR k (11)
Therefore, as a second step, it remains to specify the choice of z k in [START_REF] Saad | GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems[END_REF]. Various strategies in the above two steps lead to various Krylov subspace methods. When A is nonsymmetric, there are Krylov subspace type iterative solvers based on biorthogonality, such as the BICG method [START_REF] Fletcher | Conjugate gradient methods for indefinite systems[END_REF] and its modified versions such as the CGS [START_REF] Sonneveld | CGS: a fast Lanczos-type solver for nonsymmetric linea systems[END_REF], Bicg-stab [START_REF] Van Der | Bi-CGSTAB : a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems[END_REF] and QMR [START_REF] Freund | QMR: a quasi-minimal residual method for non-Hermitian linear systems[END_REF] methods. There are also methods based on minimizing the residual r = b -Ax, such as the Generalized Conjugate Residual (GCR) method [START_REF] Eisenstat | Variational iterative method for nonsymmetric systems of linear equation[END_REF] and the Generalized Minimum Residual (GMRES) method [START_REF] Saad | GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems[END_REF]. This paper is devoted to a symmetric systems. We present some Krylov subspace type iterative methods adapted for this case. The CG is the most popular one. It is effective for systems of the form [START_REF] Golub | Matrix Computations[END_REF] where A is symmetric positive definite. The CG algorithm can be derived from the Lanczos process who construct a basis vectors {v j } for K k (A, r 0 ) and the vector z k is chosen such as the residual r k verifies the Galerkin condition

r k ⊥ K k (A, r 0 ) ⇔ V T k r k = 0. (12)
The Lanczos process iteratively computes vectors v k as follows:

v 0 = 0, β 1 v 1 = r 0 where β 1 = r 0 , α k = v T k Av k , β k+1 v k+1 = Av k -α k v k -β k v k-1
with β k+1 ≧ 0 chosen so that v k+1 = 1. The Lanczos process transforms a symmetric matrix A to a symmetric tridiagonal matrix :

T k :=          α 1 β 2 0 • • • 0 β 2 α 2 . . . . . . . . . 0 . . . . . . . . . 0 . . . . . . . . . . . . β k 0 • • • 0 β k α k          .
In exact arithmetic, the process terminates after m ≤ N steps, i.e. β k = 0 for k = 1, . . . , m and β m+1 = 0. So from now on we can assume that β k = 0, k = 1, . . . , m.

After the kth step, we have

AV k = V k T k + β k+1 v k+1 e (k) k T V T k V k = I = [e (k) 1 , . . . , e (k) k ], V T k v k+1 = 0 (13) then V T k AV k = T k and V T k r 0 = β 1 e (k)
1 . So, [START_REF] Saad | GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems[END_REF] becomes r k = r 0 -AV k z k . And with the choice (12), we have

V T k r k = V T k r 0 -V k T AV k z k = 0
and finally we obtain

T k z k = β 1 e (k)
1 .

The CG attempt to solve this system by applying the Cholesky decomposition to T k . If A is positive definite then so is T k , and hence the Cholesky decomposition

T k = L k DL T k exists.
Here D k is diagonal with positive element, and L k is unit lower bidiagonal. If A is an indefinite symmetric matrix, then the Cholesky factorisation of T k can still be tried, often with success, but it does not always exist and can no longer be relied upon numerically, and the CG became instable numerically. Otherwise, CG may breakdown when α k = v T k Av k = 0 while v k = 0, here the algorithm is stopping prematurely. This is also the case when the matrix A is singular. Because of the problems cited above, the CG is impracticable for singular systems. That is why we propose the use of the following methods which turn out more stable numerically.

MINRES (Minimum Residual Method [START_REF] Paige | Solution of Sparse Indefinite Systems of Linear Equations[END_REF]), SQMR (Symmetric quasi-minimal residual method [START_REF] Freund | A New Krylov-Subspace Method For Symmetric Indefinite Linear Systems[END_REF]) and MRTR (Minimized Residual method based on the three-term recurrence formula of CG type [START_REF] Abe | MRTR method: An iterative method based on the three-term recurrence formula of CG-type for non symmetric matrix[END_REF]) are Krylov subspace methods for solving large symmetric systems, these methods are well adapted for the indefinite systems. they are based on the Lanczos process to construct a basis of K k (A, r 0 ). MINRES, SQMR and MRTR replace the Galerkin condition by the residual minimisation :

r k 2 = min x∈x0+K k (A,r0) b -Ax 2
By using ( 11) and ( 13), we can write

r k = b -Ax k = b -Ax 0 -AV k z k = r 0 -V k+1 Tk z k = β 1 v 1 -V k+1 Tk z k = V k+1 (β 1 e k+1 1 -Tk z k ) ⇒ r k 2 = β 1 e k+1 1 -Tk z k 2
Since V k+1 has orthonormal columns. With . 2 the related Euklidean norms and

Tk = T k 0 • • • 0 β k+1
Therefore, MINRES and SQMR characterize the kth approximate solution as

x k = x 0 + V k z k , where z k ∈ IR k is a solution of the least-squares problem r k 2 = β 1 e k+1 1 -Tk z k 2 = min y∈IR k β 1 e k+1 1 -Tk y 2
Next, for solving this problem, we use a QR-decomposition of Tk by means of plane rotations called Givens rotations.

Let us note that the SQMR is an adaptation of the generale nonsymmetric QMR method to exploit the symmetric of the matrix A. In the implementation of the nonsymmetric QMR, z k is chosen as the solution of the least-squares problem

β 1 e k+1 1 -Tk z k 2 = min y∈IR k β 1 e k+1 1 -Tk y 2
even if V k+1 is not orthonormal in the nonsymmetric case. This is called a quasi-minimisation residual property.

In the MRTR, the Lanczos process is implemented differently than the classical one. And the residual minimization is used to determine the parameters of the residual, see [START_REF] Abe | MRTR method: An iterative method based on the three-term recurrence formula of CG-type for non symmetric matrix[END_REF].

When the system is singular, the CG method and methods based on biorthogonality may diverge when T k is singular. An otherwise, for methods based on minimizing the residual, as shown in [START_REF] Smoch | Some results about GMRES in the singular case[END_REF], the coefficients s k and c k of the Givens rotation matrix verify

r k 2 = |s k | r k-1 2 with c k + s k = 1; and s k = 1 is equivalent to det(T k ) = 0.
So, at the kth step, there will be stagnation of the residual norm associated with the solution x k if and only if s k = 1 (T k singular). Therefore, in exact arithmetic, the residual is expected to decrease monotonically without divergence and stagne if T k is singular.

The convergence rate of all these methods is determined by the spectrum of eigenvalues of the matrix A. An acceleration of the convergence rate can often be achieved by replacing the system (7) by the preconditioned system.

M -1 Ax = M -1 b
The matrix M must be chosen nonsingular and in such a way that the system M z = y is much easier to solve than the original system for every vector y on the right hand side of the equation, and also, so that the matrix M -1 A has a more 'favorable' spectrum of eigenvalues than A.

Let us note that both CG and MINRES require a symmetric positive definite preconditioners while with MRTR and SQMR one can use symmetric indefinite preconditioners for symmetric but indefinite systems.

There are several technique to construct a preconditioners for the symmetric system. The incomplete Cholesky decomposition (IC) [START_REF] Golub | Matrix Computations[END_REF] is a good choice if A is symmetric definite positive. But when A is semi-definite or indefinite, IC may not exist. The SSOR is also a good choice and exist if the diagonal of A is nonzero.

IV. APPLICATIONS

As example of magnetostatic problem, we propose to study with A formulation a Three-phase transformer which the mesh is presented on the figure (1.a). A current density is applied in only one winding. To ensure the divergence free a facet tree technique is used [START_REF] Le Menach | Determination and utilization of the source field in 3D magnetostatic problems[END_REF]. The flux density in the iron core is shown on the figure (1.b). In the case of non-gauged formulation, we obtain a singular system which have 49558 unknowns. We solve this system with CG, MINRES, SQMR and MRTR. In figure (2.a), we remark that all these methods converge up to the 5000 iterations where the residual norme is around 10 -10 , and after that, the CG method diverge while the SQMR, MINRES and MRTR stay with there convergence level. In figure 2.b, the SSOR preconditioner is introduced; we can see the convergence acceleration for all methods up to the 200 iterations. In this figure we can see that their behaviour is the same as in the case without preconditioner. We note that in this two examples, the MRTR, MINRES and SQMR methods have the same behaviour, and give the same result x MR . However, the solution x CG associated to the minimal residual of CG, is different than x MR , and x MRx CG is in the Kernel of the system matrix.

V. CONCLUSION In the case of the singular systems which have the right hand side in the range of the matrix; there exist several solutions. The minimization principle of MINRES, SQMR and MRTR guarantees a regular and smoother convergence than CG. This result is still available when we add a preconditioner to accelerate the convergence. These methods become more interesting than the CG method.

  A. Tinzefte, Y. Le Menach and F. Piriou, are with the Laboratory of Electrotechnics and Power Electronics (L2EP), University of Lille, 59655 Villeneuve dAscq cedex,

Fig. 1 .Fig. 2 .

 12 Fig. 1. (a): Mesh of the transformer (b): The flux density in the iron core of the transformer

Fig. 3 .

 3 Fig. 3. (a): Mesh of NDT problem (b):Current Density in the conductor

Fig. 4 .

 4 Fig. 4. (a):Without preconditioner (b): SSOR preconditioner


Q Guegan

J-N Foulc

Electrical characterization of the solid phase (particles) of electrorheological fluids

Electrorheological (ER) fluids are suspensions of fine particles dispersed in a dielectric liquid. To understand and model the ER phenomenon, it is necessary to take into account the dielectric properties (conductivity and permittivity) of the solid and liquid phases. For the experimental investigation, the main difficulty is to measure the conductivity and the permittivity of the micro-sized (or nano-sized) particles. In this paper, we propose a method based on the arrangement of the particles in columns between electrodes under high applied field. The measurement on a cellulose/silicone oil suspension of the resistive and capacitive parts of the dc and ac time-varying currents allows us to obtain an apparent conductivity and permittivity of the cellulose particles. Finally, we discuss on the interest of the proposed characterization method from the specific conduction properties of the ER fluids.

Introduction

Electrorheological (ER) fluids are suspensions of fine particles dispersed in a dielectric liquid [1]. Upon application of an electric field, the solid phase (particles) fibrillates along the field lines between the electrodes. Thus, the rheological properties of the suspension are drastically modified (strongly dependence of the viscosity of the suspension on the applied field strength). All the models proposed to describe the ER effect take into account the dielectric properties of the host liquid and the suspended particles [2,3]. There are several methods to evaluate the electrical parameters of randomly dispersed particles in a liquid medium [4]. Here we propose an alternative approach, better suited to the electrorheology, based on the characterization of the fibrillated suspension under high electric field to obtain the dielectric properties of the particles. The measurement of the electrical current under dc and ac voltages allows us to calculate the resistive and capacitive parts of the fibrillated ER fluid and then to deduce the conductivity and the permittivity of the particles.

Principle

For the calculation of the conductivity and the permittivity of the powder (fine particles), we assume that the ER fluid subjected to a high electric field is composed of a pure liquid phase (silicone oil) and a solid-like phase (thick cellulose column). Figure 1 describes the system. Thus, we estimate the section areas of the cellulose solid part S S and of the liquid part S L as S S = S T Φ and S L = S T (1-Φ) with S T = S S + S L . Φ denotes the volume fraction of the particles and S T the electrode surface. The thickness e is the same for the solid and the liquid parts. Then each of the two parts of the ER fluid can be represented by a resistance and a capacitance in parallel.

Conductivity

When the particles are fibrillated under dc voltage, the equivalent electrical model (Fig. 1) leads to the following expression:

U I e S e S R R R L L S S L S T = ⋅ + ⋅ = + = σ σ 1 1 1 (1)
where σ S , S S , σ L and S L are the conductivity and the section area of the solid and the liquid parts respectively. I is the measured current, U is the applied voltage and R T is the total resistance of the ER fluid. From Eq. 1, we can deduce the conductivity of the particles:

Φ Φ - ⋅ - = ) 1 ( L S E j σ σ (2)
If the conductivity of the liquid and the volume fraction of the solid phase are known, the measurement of the global current density j = I / S T for a given electric field strength E leads us to obtain the conductivity of the cellulose.

Permittivity

Under ac voltage and in capacitive regime (I ∝ f), we can write (Fig. 1):

ω ε ε ⋅ = ⋅ + ⋅ = + = U I e S e S C C C S S S S L S T (3)
where C T is the total capacitance, ε S and ε L are the permittivities of the solid and the liquid parts respectively and ω = 2πf (f is the frequency). Thus, we obtain:

( ) Φ Φ - ⋅ - ⋅ = 1 L S E j ε ω ε (4)

Experimental set-up

The measurement cell is shown in figure 2. The ER fluids consisted of a suspension of cellulose microcrystalline powder (Avicel, by Merck), with a typical size of 30 µm, relative humidity of 5.2% and bulk density of 1.4 g/cm 3 , dispersed in silicone oil (Rhodia H47V20), with σ L ≅ 10 -13 S/m and relative permittivity ε L ≅ 2.7. Several samples of ER fluid with different values of particle concentration (in g/ml) were prepared and characterized (m grams of solid particles dispersed in 20 ml of silicone oil). To apply the dc or ac high voltage in the cell, we used a generator function (Hameg HM8030) and a high voltage amplifier (Trek 5/80). The current was monitored with a home made pico-ampermeter with an analog output. All measurements were performed at room temperature.

Results and discussion

The j-E and j-f characteristics are show in figure 3. We find a j-E 3/2 law (Fig. 3a) as indicated by the conduction model [3]. It may be noted in figure 3b that the capacitive regime occurs for f > 80 Hz. Therefore, to calculate the particle permittivity we used the data obtained at f = 100 Hz. Figures 4a and4b show that the calculated values of the conductivity and the permittivity of the cellulose are weakly dependents on the concentration (except for the high concentration). We also see in figure 5a that the conductivity of the cellulose increases with the electric field (σ S ∝ E 1/2 ). Figure 5b shows that the permittivity of the cellulose decreases when the frequency increases. We expect that for larger frequencies (in the full capacitive regime) the permittivity tends to a constant value. The conductivity and the permittivity of the cellulose powder greatly depend on its water content. In our case we obtain σ S ≈ 10 -9 S/m and ε S rel ≈ 9 which is rather consistent with the expected values [5].

In previous works [3,[START_REF] Boissy C Atten | Conference on Electrorheological Fluids and Magnetorheological Suspensions IOP[END_REF] we have shown that in dc condition, the conduction properties of the solid and liquid components of the ER fluids determine both the current density and the yield stress of the fluids. More precisely, in the case of two particles (conductivity σ S ) in contact, immersed in dielectric liquid (conductivity σ L < σ S ) and subjected to an electric field, the conduction model highlight the existence of an electrical contact zone of radius δ through which the current escapes from the particle and crosses the liquid. The conductance C S of the electrical contact zone is given by C S = 2 σ S δ with δ = f(σ S /σ L ) and the current flowing the particle chain is I = C S U (U is a particle-particle voltage). Therefore the current flowing the ER fluids (which is essentially the current flowing the particle chains because σ S >> σ L ) depends on both the particles and liquid conductivities. Regarding the characterization specificities of the particles used in ER fluids one can deduce the following remarks: i) for very small particle sizes (< µm) the measurement of the conductivity and the permittivity appears to be very difficult in practice (choice and availability of a specific measurement set-up) ii) an alternative method to obtain the dielectric properties of the particles is that we propose in this paper: fibrillation under dc voltage (with a large dc and a weak ac signals) of the particles immersed in dielectric liquid, measure of the dc and ac values of the current and then calculate the apparent conductivity and permittivity of the particles for a given field strength and volume fraction.

Conclusion

In this paper we proposed a method to obtain the dielectric properties of the particles used in ER fluids. We noted that the electrical characterization of a fine isolated particle is difficult but not essential to determine the efficiency of this material. Our method is based to the characterization of a fibrillated particles immersed in insulating liquid to deduce the apparent conductivity and permittivity of the particles. The results obtained on cellulose powder/silicone oil suspensions are promising and many others ER materials in particular the nanosuspensions need to be tested and a specific measurement procedure detailed before to validate the method.

Figure 1 .

 1 Figure 1. Sketch of the modelled arrangement of the ER fluid in two parts under high field and corresponding electrical circuit.

11thFigure 2 .

 2 Figure 2. Schematic of the measurement cell.

Figure 3 .

 3 Figure 3. (a) Current density vs electric field in dc voltage and (b) current density vs frequency in ac voltage (E = 2 kV/mm) for different particle concentrations.

Figure 4 .

 4 Figure 4. (a) Cellulose conductivity vs particle concentration and (b) cellulose relative permittivity vs particle concentration (E = 2 kV/mm), for different frequencies.

Figure 5 .

 5 Figure 5. (a) Cellulose conductivity vs electric field and (b) cellulose relative permittivity vs frequency (E = 2 kV/mm), for different particle concentration