Rinaldo Poli

François Stoffelbach

Sébastien Maria

José Mata

Dr Sebastien Maria

An Experimental and Computational Study on the Effect of Al(OiPr)3 on Atom Transfer Radical Polymerization and on the Catalyst -Dormant Chain Halogen Exchange

Keywords: Lewis acid catalysis, halide exchange, halogen atom transfer, density functional theory

Compound Al(OiPr)3 is shown to catalyze the halogen exchange process leading from CpMoCl2(iPrN=CH-CH=NiPr) and CH3CH(X)COOEt (X = Br, I) to the mixed-halide complexes CpMoClX(iPrN=CH-CH=NiPr). On the other hand, no significant acceleration is observed on the related exchange between MoX3(PMe3)3 (X = Cl, I) and PhCH(Br)CH3, by analogy to a previous report dealing with the Ru II complex RuCl2(PPh3)3 (T. Ando, M.

Introduction

Atom Transfer Radical Polymerisation (ATRP) was introduced in 1995 [1,2] and has since become the most actively investigated controlled radical polymerization process. With this technique, the bimolecular terminations are reduced in importance relative to the chain propagation by implicating the reactive radicals in a rapid and thermodynamically favourable atom transfer equilibrium by the action of a halogenated spin trap M-X, producing halogenterminated dormant chains R-X (<Scheme 1).

<Scheme 1>

Species M is a transition metal complex that is capable of increasing its formal oxidation state and coordination number by one unit. Since, according to <Scheme 1, it is reversibly regenerated after each sequence of activation, monomer insertion(s), and deactivation processes, its effective role is that of catalyzing the polymer chain growth out of the dormant species. Thus, ATRP depends critically on transition metal catalysis and a number of studies have been devoted to understanding its mechanistic details. [3,4] Of particular interest for the present study, the addition of certain Lewis acids, most notably Al(OiPr)3, in conjunction with a number of catalysts has been shown to results in faster polymerisation. This effect seems rather general, since it was shown for many different complexes based on a variety of different transition metals from Group 6 to Group 11, including CpMoCl2(iPrN=CH-CH=NiPr), [5] MoX3(PMe3)3, [6] ReIO2(PPh3)2, [7] CpFeX(CO)2 (X = Br, I), [8] RuCl2(PPh3)3, [9] NiBr2(PPh3)2, [10] Ni(PPh3)4, [11] and CuBr/bipy. [12] The detailed mechanism of action of this additive, however, is not completely understood. A detailed study on the effect of Al(OiPr)3 for the RuCl2(PPh3)3-catalyzed polymerization of methyl methacrylate demonstrated that the compound does not significantly interact with the monomer, with the initiator/dormant chain, with the catalyst, and with the growing free radical. [9] The only noticeable effect, from a cyclic voltammetric study of the catalyst, seemed to be the stabilization of the oxidized metal complex. In the absence of the Al additive, the Ru II complex shows an irreversible oxidation wave "in most cases", whereas in its presence it becomes reproducibly reversible. It is puzzling, however, that "sometimes a reduction wave appeared", even in the absence of the Al additive. A way to rationalize this irreproducibility is to envisage an irreversible interaction between the oxidation product M + and adventitious Lewis basic impurities, whereas this process would not occur in the presence of the Lewis acidic Al scavenger. We have recently observed similar phenomena for another system. [6] At any rate, the nature of the radical trap in the ATRP process (i.e. MX in <Scheme 1) is different from the one-electron oxidation product observed in the cyclic voltammogram (M + ). Thus, the cyclic voltammetric results for complex RuCl2(PPh3)3 are not directly relevant for the rationalization of the accelerating effect by the Al(OiPr)3 additive.

The apparent radical polymerization rate constant takes the form of Equation 1, which contains the concentrations of the various species participating in the atom transfer equilibrium of <Scheme 1, the propagation rate constant kp, and the atom transfer equilibrium constant KAT. There are only two ways in which compound Al(OiPr)3 (or any other additive, for that matter) can affect this apparent rate constant: (i) by accelerating the propagation rate (increase of kp); (ii) by shifting the atom transfer equilibrium. A previous study has shown that conventional free radical polymerization of MMA is not affected by Al(OiPr)3. [9] As far as shifting the atom transfer equilibrium, since the concentrations of M and RX cannot possibly be increased by the addition of the Al compound, the only remaining possibility is a decrease of the MX concentration, as may be accomplished by involving this species in another equilibrium.

kapp = kpKAT[M][RX]/[MX]

(1)

Equation 1

We have therefore based our working hypothesis on the existence of a selective interaction between Al(OiPr)3 and the product of the atom transfer process, the spin trap MX.

Of relevance to this phenomenon, parallel work has shown that Al(OiPr)3 has no effect on the halogen exchange rate between the RuCl2(PPh3)3 catalyst and a bromide initiator, (CH3)2C(COOMe)CH2C(Br)(COOMe)CH3. [13] This is a relevant observation because the simplest possible halogen exchange mechanism involves the ATRP intermediates, see <Scheme 2. Thus, the fact that Al(OiPr)3 accelerates ATRP but not halogen exchange may in principle appear puzzling. Sawamoto et al. have made the proposition that the atom transfer process leads to an inner sphere radical species, [Ru III -X,R • ], which is capable of undergoing fast halogen exchange but leads to slow monomer insertion, whereas the presence of the aluminium compound would favour the formation of an outer sphere radical species, [Ru III -X/Al + R • ], leading to faster insertion, see and Scheme 3. The nature of the Ru III -X/Al adduct, however, was not further speculated. [9,13] <Scheme 2 and Scheme 3>

Recent work in our laboratory on compound CpMoCl2(iPrN=CH-CH=NiPr) has shown that compound Al(OiPr)3 not only accelerates the ATRP of styrene and acrylates, but also catalyzes halogen exchange between the catalyst molecule and the halide initiator ethyl 2iodopropionate (IEA). This result cannot be easily rationalized on the basis of and Scheme 3. It has been reported in a preliminary communication [14] and tentatively rationalized on the basis of an independent and non-radical pathway for halogen exchange, which can be operative for the 16-electron RuCl2(PPh3)3 system but cannot take place for the 17-electron half-sandwich Mo complex. This alternative halogen exchange mechanism was proposed to consist of an internal nucleophilic substitution (SNi) at carbon, activated by coordination of the initiator molecule to the metal center. For the 17-electron system, coordination of the initiator to the transition metal center is made unfavourable by the unavailability of a metalbased, low energy empty orbital to host the two electrons coming from a halide lone pair.

Thus, the only possibility remaining for the halide exchange pathway is the higher-energy, Al(OiPr)3-catalyzed pathway of <Scheme 2.

The purpose of the present contribution is to: (i) extend the observations of the Al(OiPr)3 catalytic effect (or lack thereof) on halogen exchange to other ATRP catalysts used in our laboratory; (ii) to provide computational support for the initiator coordination to the ATRP catalyst and for the alternative halogen exchange mechanism; (iii) to analyze how compound Al(OiPr)3 may be capable to stabilize the spin trap, thereby shifting the atom transfer equilibrium and accelerating ATRP.

Experimental Section

General. All operations were carried out under an atmosphere of pre-purified argon.

Solvents were dried by conventional methods, deoxygenated and distilled under dinitrogen.

NMR spectra were recorded in CDCl3 on a Bruker 300 Avance instrument. The spectra were calibrated with the residual solvent peaks and all resonances are reported with positive shifts downfield from SiMe4. EPR measurements were carried out at the X-band microwave frequency on a Bruker ESP300 spectrometer. The spectrometer frequency was calibrated with diphenylpicrylhydrazyl (DPPH, g = 2.0037). Compounds CpMoCl2(iPr2dad), [15] MoCl3(PMe3)3 [16] and MoI3(PMe3)3 [17] were prepared following published procedures.

Compounds CH3CH(Br)COOEt (BrEA) and CH3CH(Br)Ph (BEB) were purchased from Aldrich and used as received. Ethyl 2-Iodopropionate (IEA) was obtained according to a previously described synthetic procedure. [18] Reaction between CpMoCl2(iPr2dad) and CH3CH DFT calculations. All DFT calculations were carried out with the Gaussian03 program suite [START_REF] Frisch | Gaussian 03, Revision B.04[END_REF] using the B3LYP functional. [START_REF] Becke | [END_REF] The 6-31G* basis set was used for all C, H, O, N, Cl and Br atoms. The Mo atom was described by the LANL2DZ basis set, to which an f polarization function ( = 0.8) was added. All molecules were optimized with the spin unrestricted formalism. The mean value of the S 2 operator was always very close to the theoretical value of 0.75 for spin doublets, 2 for spin triplets, and 3.75 for spin quartets for all optimized geometries corresponding to energy minima. A major spin contamination was only found for the TS relating the spin doublet MoCl3(PMe3)3(ClMe) to the sum of spin triplet MoCl4(PMe3)3 and spin doublet CH3 (<S 2 > = 1.48), a consequence of the near degeneracy of the parallel (S = 3/2) and antiparallel (S = 1/2) arrangements. The S 2 mean value rapidly tends to 0.75 as the C-Cl distance decreases to that of the MoCl3(PMe3)3(ClMe) minimum.

All geometry optimizations were performed without symmetry constraints (C1 symmetry).

The nature of the resulting stationary points as energy minima was verified by a frequency analysis in each case. All energies were corrected for zero point vibrational energy and for thermal energy to obtain the reaction enthalpies at 298 K.

Results

(a) Interaction between Al(OiPr)3 and various compounds

The CpMoX2(iPr2dad) (X = Cl, I) complexes were shown to be capable of controlling the ATRP of styrene and acrylates. [5,14] Complexes MoX3(PMe3)3 (X = Cl, Br, I) are also capable of controlling the ATRP of styrene. [6,21] For both families of ATRP catalysts, the addition of compound Al(OiPr)3 to the catalytic system was found to speed up the polymerization process by a factor between 3 and 13, depending on the nature of the halides. [5,6] In order to probe the possible effect of the Al additive, the interactions of this compound with catalyst, monomer, and initiator were carried out.

The 17-electron complexes CpMoX2(iPr2dad) (X = Cl, I) are characterized by sharp EPR signals in isotropic solutions at room temperature. [14,15] An EPR study of a toluene solution of the dichloride complex, in the presence of an equivalent amount of the Al(OiPr)3 compound, at 80°C (conditions used for the ATRP of MA) reveals no change of shape nor intensity over 1 h. The 15-electron MoX3(PMe3)3 complexes do not show any EPR activity in isotropic solutions at room temperature. However, in spite of their paramagnetism, they display highly shifted and broad but readily visible resonances in the 1 H NMR spectrum. [22] The addition of one equivalent of the Al(OiPr)3 compound to a MoCl3(PMe3)3 solution (C6D6) at 65°C for 8 h does not yield any visible change of the solution NMR spectroscopic properties.

1 H NMR studies have also been carried out on solutions containing compound Al(OiPr)3 and styrene (in C6D6 at 65°C for 8 h, [S]/[Al] = 1/1) as well as on solutions containing compound Al(OiPr)3 and each of the initiators IEA and BEB (in C6D6 at 65°C for

8 h, [initiator]/[Al] = 1/1
). In all these cases, once again, no evidence was obtained for the existence of any interaction engaging the Al(OiPr)3 compound. These results parallel those of analogous studies conducted on the individual components of the RuCl2(PPh3)3-catalyzed MMA polymerization. [9] (b) Halide exchange for the 17-electron CpMoX2(iPr2dad) system

During the ATRP catalyzed by the dichloride complex and using the iodine-containing IEA initiator, the possibility exists for halogen exchange between the Mo catalyst and the initiator molecule. This exchange process was investigated qualitatively by EPR spectroscopy as shown in <Figure 1. Under conditions identical to those of the ATRP of MA, except that no monomer was present, the CpMoCl2(iPr2dad) displays the EPR spectrum shown in <Figure 1(a). No change of this spectrum (including its intensity) is evident after heating for 1 h at 80°C, indicating the absence of halogen exchange with IEA under these conditions. When the same experiment was carried out in the presence of 1 equiv of Al(OiPr)3, on the other hand, a new resonance was evident at a greater g value relative to that of the initial catalyst, see <Figure 1(b). This resonance belongs to the product of halogen exchange, CpMoICl(iPr2dad), as verified by a comparison study with the products of the interaction between CpMoCl2(iPr2dad) and NaI [g = 2.010 for CpMoICl(iPr2dad); 2.042 for CpMoI2(iPr2dad)]. [14] <Figure 1>

An identical experiment was carried out by using ethyl 2-bromopropionate (BrEA), with identical qualitative results, though the exchange process was faster. The results are shown in <Figure 2. Unlike the experiment carried out with IEA, a slow exchange is observed even in the absence of Al additivecf. the spectrum in <Figure 2(a) with that shown in <Figure 1(a).

However, the process is once again greatly accelerated by Al(OiPr)3. The new peak observed at g = 1.991 in <Figure 2(b) is attributed to the mixed halide complex CpMoClBr(iPr2dad), and the broader feature observed at higher g values is certainly due to the double exchange product, CpMoBr2(iPr2dad), showing that a greater extent of exchange has occurred relative to the IEA experiment, cf. with <Figure 1(b). Although neither the mixed-halide nor the dibromide products were isolated, the positive shift of the g parameter upon replacement of Cl with the heavier halides in the order Cl < Br < I parallels exactly the shift reported for the related family of CpMoX2(PMe3)2 complexes (g = 1.982 for X = Cl, 2.006 for X = Br and 2.046 for X = I), [14] whereas the g value for the mixed CpMoICl(PMe3)2 complex is 2.015. [23] <Figure 2>

The results of these experiments demonstrate that compound Al(OiPr)3 catalyzes the halogen exchange between CpMoCl2(iPr2dad) and CH3CHXCOOEt (X = Br, I). As analyzed in the introduction, the most logical mechanistic hypothesis is that an atom transfer occurs, followed by recombination between the free radical and a different halide ligand from the high oxidation state metal complex. Since Al(OiPr)3 does not seem to show any interaction with the catalyst, the monomer, and the initiator, it seems likely that the catalytic action is exerted via a selective thermodynamic stabilization of the spin trap. The way in which this interaction may occur will be examined later in section (e).

(c) Halide exchange for the 15-electron MoX3(PMe3)3 system.

Since the styrene polymerizations catalyzed by the MoX3(PMe3)3 complexes were carried out with the (1-bromoethyl)benzene (BEB) initiator, [6] halide exchange tests on this system were carried out using the same initiator. The exchange process was monitored using both the trichloride and the triiodide complex, in the absence and in the presence of 1 equiv of Al(OiPr)3 per equiv of Mo.

<Figure 3>

The results of these experiments are shown in <Figure 4. Since, for technical reasons (solubility and rate), the reactions could not be carried out under pseudo-first order conditions, and since the second order conditions lead to complications due to the possibility of multiple exchange, no precise kinetic information could be extracted from these monitoring and the results must necessarily be considered only of qualitative value. However, it is rather clear from <Figure 4 that the presence of Al(OiPr)3, contrary to the 17-electron systems analyzed in the previous section, have at best only a very minor effect on the halide exchange rate. For the MoCl3(PMe3)3/BEB system, the exchange process turns out in fact as slightly slower in the presence of the Al additive than in its absence. For the MoI3(PMe3)3/BEB system, on the other hand, the exchange is marginally faster for the experiment carried out in the presence of Al(OiPr)3. The observed variations are probably caused by concentration errors, due to the need to measure rather small quantities of the reagents in each experiment.

<Figure 4>

We can therefore conclude that the halogen exchange process between the 15-electron MoX3(PMe3)3 complexes and BEB is not catalyzed by Al(OiPr)3 and that this process is probably occurring by the same mechanism as for RuCl2(PPh3)3, which is different than the mechanism operating for the 17-electron CpMoX2(iPr2dad) system. The electronic configuration of the 15-electron MoX3(PMe3)3 complexes is different from that of the 16electron RuCl2(PPh3)3 complex, as shown in <Figure 5. The Ru II complex, having a diamagnetic ground state, possesses a metal-based, low-energy empty orbital (which is only tied up in a weak Mo-Cl  interaction), to be used to host the two electrons donated by the RX ligand, see <Figure 5(a). The Mo III complex, on the other hand, is characterized by a spin quartet ground state (pseudo-t2g 3 for the approximately octahedral geometry) and no empty metal orbital is available. However, such necessary orbital may be generated by a process of electron pairing, to yield a spin doublet configuration, see <Figure 5(b).

<Figure 5> (d) Calculations of R-X coordination, atom transfer and halogen exchange with model compounds.

On the basis of the results presented above and our previously formulated hypothesis, [14] the initiator molecule should be able to coordinate to the 15-electron MoX3(PMe3)3 and 16-electron RuCl2(PPh3)3 complexes, but not to the 17-electron CpMoCl2(iPr2dad) complex.

This proposition was investigated with the help of DFT calculations on model compounds.

For the 15-electron species, we used PH3 in place of PMe3 and we limited our study to the trichloride complex. For the 16-electron species, we used again the model PH3 ligand.

Finally, in order to model the 17-electron system, we elected to use again a phosphinesubstituted complex, CpMoCl2(PH3)2. This choice was suggested by the fact that the isoelectronic complex CpMoCl2(PMe3)2 was also shown to catalyze the polymerization of styrene by ATRP, [24] and by the fact that the product of atom transfer, the 18-electron organometallic Mo IV species CpMoCl3(PMe3)2 is also a well characterized compound, [25] whereas analogues containing dad-type ligands have not been reported so far. All optimized compounds were obtained from starting geometries adapted from crystallographically characterized analogues: RuCl2(PPh3)3, [26] MoCl3(PMe2Ph)3, [27] MoCl4(PMe2Ph)3, [28] CpMoCl2(PMe3)2, [29] and CpMoCl3(PMe2Ph)2. [30] No structural data seem to be available for Ru III complexes having the stoichiometry RuX3(PR3)3 with X = any halogen. However, crystallographically characterized RuCl3L3 complexes exist, for instance with L = MeCN [31] or L3 = terpy. [32] The optimized geometrical parameters compared quite well with the experimental data in all cases. Since this work focuses on the conceptual aspect of the Al(OiPr)3 effect in ATRP, we shall not dwell on the detailed comparison of experimental and calculated structures (all optimized geometries are available as supporting information). Our analysis will be limited to the energetic results, whereas the optimized geometric parameters will be commented only when relevant to the mechanistic discussion.

Starting with the Ru II species, we have carried out the coordination study of two models of initiator molecules: compound CH3CHClCOOCH3 (as a model of the real BrEA initiator used in the Ru-catalyzed polymerization) and the simpler model CH3Cl. The addition is found as slightly exoergic in both cases, see <Figure 6. The 2.0 kcal mol -1 difference in favour of CH3Cl is certainly associated to the positive inductive effect of the CH3 group, relative to the CH(CH3)(COOCH3) group, on the C-Cl bond, raising the energy of the Cl lone pair involved in the coordinative bond. Adding the zero point energy and the kT correction to the data, the computed enthalpy changes for the coordination reactions at 298 K become -4.6 and -6.5 kcal mol -1 , respectively. Thus, both reactions are predicted to be slightly exothermic by the calculations. An alternative geometry can be envisaged for the adduct, namely containing the halide initiator ligand in an equatorial position (trans to a Cl ligand). This could be conceived as deriving from the addition to a vacant site in an isomeric molecule where a Cl ligand, rather than a PH3 ligand, occupies the apical position. For both initiator molecules, such isomer is computed as slightly less stable relative to the isomer shown in <Figure 6, the additions to this site being exoergic by only -1.6 and -3.5 kcal mol -1 , respectively.

<Figure 6>

Because of computational expense, representative points at fixed C-Cl distances along the atom transfer coordinate were calculated only for the CH3Cl initiator, as shown in <Figure 6. The transition state along the curve was located at ca. 33.6 kcal mol -1 from the sum of the free RuCl2(PH3)3 and CH3Cl molecules (41.4 kcal mol -1 from the adduct), while the sum of the two reaction products are located at 31.8 kcal mol -1 from the separated reagents (39.6 kcal mol -1 from the adduct). The C-Cl distance at the transition state is 2.80 Å, whereas the Ru-Cl distance is 2.45 Å, closer to the value of the same distance in the final Ru III product (2.39 Å) than to that in the Ru II CH3Cl adduct (2.68 Å). For comparison, the other two Ru-Cl distances (trans to each other) are 2.41 Å in the the final product and 2.46 Å in the Ru II precursor. This transition state is located only 1.8 kcal mol -1 higher in energy than the sum of the free radicals RuCl3(PH3)3 and CH3, thus the reverse atom transfer process (radical deactivation) is predicted to be very fast. This is a necessary condition for good control in ATRP.

Two halogen exchange processes have been investigated computationally. Once again, in order to limit the computational time factor, the transition states were calculated only for the CH3 containing initiator molecules. The results for both processes are illustrated in <Figure 7. The first process is a degenerate exchange where the axially coordinated CH3Cl ligand in the 18-electron adduct transfers its methyl group to one of the two equatorial Cl ligands, leading to the isomeric RuCl2(PH3)(eq-CH3Cl) adduct. The transition state for this process is located at 40.2 kcal mol -1 relative to the CH3Cl adduct. This is marginally higher in energy relative to the atom transfer products (39.6 kcal mol -1 ), but slightly lower than the transition state leading to them (41.4 kcal mol -1 ). The geometry of this TS, shown in <Figure 7, is significantly different than that of the TS leading to the atom transfer product, which is shown in <Figure 6. Notably, the methyl C atom is exactly at the same distance from the two Cl atoms (2.63 Å), it retains a more pyramidalized geometry (sum the H-C-H angles = 352.1°;

cf. with 359.5° in the other TS), and the Ru coordination geometry is more distorted in order to bring both Cl atoms involved in the exchange closer to the methyl group (Cl-Ru-Cl angle = 82.9°, cf. with 92.1° in the atom transfer TS). The Cl exchange TS structure was verified to contain only one imaginary frequency (380i cm -1 ), which corresponds to the movement of the CH3 group from the axial to the equatorial Cl atom.

<Figure 7>

The second computed exchange process consisted in a closer model of the experimentally observed Cl/Br exchange. Starting from RuCl2(PH3)3, a molecule of CH3Br was added to the vacant apical position to obtain RuCl2(PH3)3(ax-CH3Br), a process which results exoergic by 13.9 kcal mol -1 (exothermic by 12.6 kcal mol -1 at 298 K). Thus, CH3Br is a stronger ligand than CH3Cl for the Ru II center. From this adduct, transfer of a Me group to one of the two equivalent Cl atoms leads to RuClBr(PH3)3(eq-CH3Cl), with the stereochemistry illustrated in <Figure 7, a process which is endoergic by 3.8 kcal mol -1 . This process occurs via a TS located 38.7 kcal mol -1 higher in energy, whose geometric characteristic are quite analogous to those of the TS for the degenerate Cl exchange. The single imaginary frequency has in this case the value of 366.9i cm -1 , while the Cl-Ru-Br angle is 89.4°. The wider value of this angle compared to the TS for the degenerate Cl exchange is related to the larger radius of the Br ligand (Br-C = 2.79 Å; Cl-C = 2.68 Å). This diminished distortion is probably the main reason for the slightly lower activation barrier in this case. We have not carried out, in this case, calculations at frozen Br-C distances along the atom transfer coordinate. However, the final point of this curve (sum of the free RuCl2Br(PH3)3 and CH3 radicals) is up in energy by 38.8 kcal mol -1 relative to the RuCl2(PH3)3(ax-CH3Br) adduct.

Like for the case of the Cl atom transfer from CH3Cl, a slight overbarrier will probably be present along the Br atom transfer pathway. Therefore, the halogen exchange TS is located lower in energy than the atom transfer TS. The methyl group in transit has developed a carbocationic character at the transition state level according to the calculations. An NBO [33] localized charge density analysis shows that the charge of the CH3 group increases from 0.135 in RuCl2(PH3)3(ax-CH3Cl) to 0.304 in the TS, and then decreases back to 0.152 in RuCl2(PH3)3(eq-CH3Cl). For the Cl/Br exchange pathway, the values are 0.076 for RuCl2(PH3)3(eq-CH3Br), 0.280 for the TS, and 0.140 for RuClBr(PH3)3(eq-CH3Br). On the other hand, a spin unrestricted calculation on the transition state geometries yield zero spin densities for all atoms, indicating that neither the Ru complex nor the migrating CH3 group develop any radical character. In conclusion, this computational study validates the hypothesis that coordination of the halide initiator molecule to the Ru II center in RuCl2(PPh3)3

activates the halogen-carrying C atom towards an internal nucleophilic substitution, SNi.

Moving on to the next system, the interaction between the 17-electron model system CpMoCl2(PH3)2 and compound CH3Cl, which was chosen again as a model of an ATRP initiator, did not lead to any stable local minimum for a 19-electron adduct. This is certainly not the result of a steric problem, because compounds with an analogous coordination sphere are known for Mo IV , which has a smaller ionic radius than Mo relative to the starting materials. Therefore, the overbarrier is worth 4.7 kcal mol -1 in this case. This is significantly higher than for the Ru case examined above, meaning that the CpMoCl3(PH3)2 system should lead to a slower deactivation rate than the RuCl3(PH3)3 systems. However, these values are not highly significant, since the substituents in the real systems may have a rather large effect on these barriers.

<Figure 8>

The last system to be examined, the 15-electron MoCl3(PH3)3, gives the results shown in <Figure 9. The coordination of either CH3Cl or CH3CHClCOOCH3 affords a 17-electron species with a spin doublet ground state. Contrary to the 16-electron RuCl2(PH3)3 system, the coordination reaction is endoergic. This is because of the energetically costly electron pairing process that must accompany the ligand addition. Taking this into account, the bond formation is energetically more favourable by 3.2 kcal mol -1 for CH3Cl than for CH3CHClCOOCH3, similar to the difference found for the coordination to RuCl2(PH3)2. In spite of the endoergic character of this coordination process, a stable local minimum is found, contrary to the 17-electron CpMoCl2(PH3)2 system. The geometry of this ligand adduct is related to that of the product of atom transfer, the 16-electron MoCl4

<Figure 9>

Like for the previously investigated systems, the atom transfer process was investigated only for the addition of CH3Cl. Unlike the 16-electron RuCl2(PH3)3, we were unable to locate the TS of the SNi process. All attempts led again to the atom transfer TS shown in <Figure 9.

(e) Calculations of Al(OCH3)3 interaction with model spin traps.

The isopropoxide groups of Al(OiPr)3 were modelled by methoxide. The Al(OiPr)3 compound is known to exist in solution as a mixture of trinuclear and tetranuclear species. [34] However, species with lower nuclearity (dimers and even monomers) are known for analogous molecules with bulkier substituents. In order to learn more about the relative energetics of the various nuclearities, geometry optimizations were carried out on [Al(OMe)3]n with n = 1, 2, 3 and 4. The guess geometries were taken from the X-ray structures of compounds Al[O(C6H2Bu t 2-2,6-Me-4)]3, [35] [Al(OBu t )2(-OBu t )]2, [36] [Al(OCy)2-(-OCy)2]2Al(OCy), [37] and [Al(OiPr)2(-OiPr)2]3Al, [34] The energetic results are shown in <Figure 10. In agreement with the known behavior, the tetranuclear and trinuclear forms were found as the most stable ones, stabilized by 42.

<Figure 10>

Since we are interested in evaluating the energetic effect of Al(OiPr)3 on the atom transfer equilibrium, we have calculated models of the interaction between Al(OiPr)3 and the species carrying the halogen atom involved in the transfer process. As shown in <Figure 10, the interaction of the model Al compound with the model initiator affords an adduct, (MeO)3Al-ClMe, whose energy is lower relative to the sum of the separated species by 7.6

kcal mol -1 . The weakness of this interaction is signalled by the rather long Al-Cl distance -0.392 in the Ru III complex, -0.450 vs. -0.277 in the Mo IV complex, for eq and ax positions, respectively). These charges are greater than in the CH3Cl molecule, as may be predicted from the greater electropositive character of the metal centers. Upon coordination to the Al center, there is again a decrease of natural charge at the Cl atom (from -0.420 to -0.344 for the Ru III complex; from -0.450 to -0.419 for the Mo IV complex). The energetically stronger Lewis acid-base interaction is also consistent with the shorter Al-Cl distance relative to the CH3Cl-Al(OMe)3 adduct [2.337 Å in the Ru III complex, 2.335 Å in the Mo IV complex] and with the greater degree of pyramidalization at the Al center [sum of the three O-Al-O angles = 345.46° in the Ru III complex, 344.29° in the Mo IV complex].

Local minima where the Al centers establishes bonds contemporarily with two Cl ligands could not be located. Any attempt to optimize such type of structure collapsed to one of the isomeric adducts with the Al(OMe)3 molecule bonded to a single Cl ligand. Thus, although the Al center is capable to increase its coordination number up to 6 with bridging OMe ligands, it does not seems favourable to go beyond 4-coordination with more than one bridging Cl ligand. In summary, the calculations support the idea that the atom transfer equilibrium shifts in favor of the higher oxidation state complex plus free radical in the presence of the Lewis acid. This phenomenon is sufficient to explain the catalytic effect of aluminum triisopropoxide on ATRP and on the halogen exchange process for the CpMo III system, which follows the same atom transfer mechanism.

The above picture should be considered only at the qualitative level. There are several phenomena that affect the quantitative value of the atom transfer energetics. Firstly, the aluminum compound is not present in solution in the monomeric form. Even at the high temperatures used for the ATRP process, the entropic factor is not sufficiently high to generate a significant amount of monomeric species at equilibrium (application of ZPVE, kT, and entropy corrections from the gas-phase calculation, the G for the tetramer to monomer conversion affords 31.7 kcal mol -1 at 298 K and 29.1 kcal mol -1 at 398 K). Although the real isopropoxide system certainly has lower values because of the steric effect, the catalytic action is not likely to be exerted by the mononuclear species. However, the Al center is capable of expanding its coordination up to 6 as show by the structures of the trimer and tetramer. Therefore, the lateral, 4-coordinated Al centers in the trimer and tetramer can exert their Lewis acidic action for the energetic stabilization of the atom transfer products, though with a reduced energetic effect, along the same line shown above for the monomer. A DFT calculation for the interaction products of trimer and tetramer was not attempted given its complexity. A second complicating phenomenon is that the Al centers can also interact with the Cl atoms of the transition metal catalyst, the Ru II and Mo III complex, whose natural charge is relatively high (-0.535 and -0.448, respectively). This interaction would play against the catalytic effect. However, analogous additional interactions may also occur with a second Cl

Discussion

The results presented above, which are subject to several approximations and assumptions, point nevertheless to a simple rationalization of the accelerating effect exerted by Al(OiPr)3 on ATRP. The origin of this effect is a shift of the atom transfer equilibrium (<Scheme 1) caused by a stronger interaction between the Lewis acidic Al centers (probably the 4-coordinated centers in the trimer and tetramer) with the halogen atoms in the spin trap, relative to the organic initiator. This is in turn related to the greater polarity of the metal-halogen bond in the spin trap than the carbon-halogen bond in the initiator molecule. Because of the relative weakness of these interactions, the Lewis acid-base adducts is likely to be present in solution in equilibrium with the separated species. This equilibrium is likely to be very rapid. Furthermore, because of the nature of these interactions, which involve primarily Al and Cl atoms, the spectroscopic properties probably do not change dramatically on going from the separate species to the adduct. This explains the difficulty in determining the presence of these interactions experimentally by 1 H NMR or by examining properties that are directly associated to the metal center (e.g. cyclic voltammetry, EPR for the paramagnetic complexes, …).

The appealing feature of this interpretation is that it is generally applicable to all transition metal catalysts, since M-X bonds in the spin trap will always be more polarized than the corresponding C-X bonds in the initiator. On the basis of this interpretation, it can be easily predicted that the accelerating effect of Al(OiPr)3 should be greater when the catalyst has more polarized M-X bonds. For the same halogen X, this in general occurs for higher oxidation state complexes. Indeed, the calculated strength of the Al(OMe)3 interaction is greater with CpMoCl3(PH3)2, a Mo IV complex, than with RuCl3(PH3)3, a Ru III complex.

The selective catalytic action exerted by Al(OiPr)3 on the halide exchange reaction between the ATRP catalyst and the initiator, which is function of the metal electronic configuration, see <Scheme 4, has been interpreted by the presence of two alternative halide exchange pathways, one involving atom transfer, the second involving coordination of the initiator to an available coordination site of the transition metal complex, followed by an internal nucleophilic substitution (SNi), Scheme 5. The absence of a catalytic effect by Al(OiPr)3 on the SNi mechanism is not suprising. The Lewis acidic Al(OiPr)3 can in principle interact only with nucleophilic centers, such as the negatively polarized halogen ligands. The SNi pathway involves transfer of electron density from the initiator molecule to the metal center, whose sterically crowded coordination sphere does not facilitate an electronically stabilizing interaction with the Lewis acid. The halide lone pairs need to be available to accomplish the SNi process, thus their involvement in a Lewis acid-base interaction with Al(OiPr)3 would, if anything, exert a negative effect on the halide exchange process. Finally, the sterically exposed migrating alkyl group has carbocationic character and is not expected to interact with the Lewis acid.

<Scheme 4 and Scheme 5>

According to our interpretation, when the SNi pathway occurs at lower energy than the ATRP pathway, the halide exchange process is faster than the atom transfer process and is not catalyzed by Al(OiPr)3. Conversely, when the SNi pathway occurs at higher energy than the ATRP pathway, both the halide exchange and the atom transfer process follow the same mechanism and are both accelerated by Al(OiPr)3. The proposed SNi mechanism is expected to take place at two conditions: (i) the initiator molecule must be capable to enter the coordination sphere of the transition metal complex; (ii) the SNi pathway must have a lower energy transition state than the atom transfer pathway. The first condition cannot be satisfied by all transition metal complexes: those having a 17-electron configuration are unlikely to coordinate weakly coordinating ligands such as organic halides and lead to 19-electron adducts, as we have verified here by DFT calculations for the model CpMoCl2(PH3)2 compound. The same situation is expected to occur for 18-electron complexes, which generally do not yield stable 20-electron adducts. However, 18-electron complexes cannot function as ATRP catalysts either, because they cannot accept the transfer of a halogen atom (a 1-electron ligand), unless another ligand dissociates first. The second condition, also, is not necessarily valid in general. It is expected to depend on the relative proximity of the two halogen atoms that are involved in the exchange (i.e. on the coordination stereochemistry) and on the nucleophilicity of the M-X bond that attacks the C atom (i.e. on the M-X bond polarity). It may be safely assumed that, if the halide exchange process is not catalyzed by Lewis acids, as is the case here for the MoX3(PMe3)3 species, then this condition is satisfied.

This scheme appears more satisfying than the outer sphere-inner sphere mechanism proposed by Sawamoto et al. ( and Scheme 3). According to the latter, it would be necessary to assume that the "inner-sphere" {M-Y,R • } species has a sufficiently long lifetime to accomplish the uncatalyzed halide exchange, before being transformed to the "outer sphere" counterpart under the action of Al(OiPr)3, only for the 16-electron MoX3Y(PMe3)3 and the 17-electron RuCl2Y(PPh3)3, but not for the 18-electron CpMoCl2Y(iPr2dad). A notable difference between those species is that the 18-electron CpMo IV complex is diamagnetic, whereas the others have radical character (S = ½ for the 17-electron Ru III complex and S = 1 for the 16-electron Mo IV tetrahalide complex). Thus, one would need to assume that the organic radical escapes the transition metal rapidly when this has no spin density and slowly when it has one or more unpaired electrons. In addition, one would have to assume that halogen exchange, like monomer insertion, is faster for the "outer sphere" complex, but only when using the 17-electron ATRP catalyst. Although we have no concrete experimental data to exclude this scenario, we cannot find any simple argument to support it. The energetic comparison of the computed transition states for atom transfer and for SNi halide exchange for the RuCl2(PH3)3 model system is persuading us that the uncatalyzed halide exchange occurs via the SNi pathway.

It is evident that, according to the SNi mechanism, the halide exchange should be accompanied by 100% retention of the absolute configuration, whereas exchange via a radical species should lead to partial loss of the enantiomeric excess, depending on the radical lifetime in the "inner sphere complex". Unfortunately, we could not test the stereochemistry of this halide exchange using optically active molecules such as CH3CH(X)Ph or CH3CH(X)COOEt because of the need to carry out these processes under conditions (i.e.

65°C) in which racemization would easily occur, at least for the more fragile bromide systems, according to the literature. [38,39] It is interesting to note that, in a previous investigation on a Cu I system, Matyjaszewski et al. found that the rates of halogen exchange, activation and racemization were exactly the same. [40] This observation is perfectly in line with our proposed model, since the electronically saturated Cu I complex would not allow easy access to the initiator molecule (a weak 2-electron neutral ligand) to open access to the SNi pathway. Thus, halogen exchange can only occur via the atom transfer pathway (<Scheme 2).

Conclusions

Our calculations on model compounds suggest that the accelerating effect exerted by the Al(OiPr)3 additive on ATRP is related to the establishment of a stronger Lewis acid-base interaction with the halogen atom after this is transferred from the organic initiator to the more electropositive transition metal catalyst. As a result, the atom transfer equilibrium is slightly shifted toward the formation of the active radical and the spin trap. This phenomenon is also responsible for the acceleration of the halide exchange process between the initiator and the catalyst, when the latter is a 17-electron half-sandwich Mo III compound.

This acceleration phenomenon has been experimentally verified for the halide exchange between CpMoCl2(iPrN=CH-CH=NiPr) and the CH3CH(X)COOEt (X = Br, I) initiators. On the other hand, the absence of an acceleration by Al(OiPr)3, which was reported previously [13] for the halide exchange between the 16-electron RuCl2(PPh3)3 and (CH3)2C(COOMe)CH2C(Br)(COOMe)CH3, is also found for the halide exchange between the 15-electron MoX3(PMe3)3 (X = Cl, I) complexes and PhCH(Br)CH3. A rationalization of this dichotomy has been presented on the basis of a non-radical, competitive halide exchange mechanism, which is favored over the atom transfer mechanism depending on the electronic structure of the transition metal complex. The new mechanism consists of an internal nucleophilic substitution (SNi), which requires activation of the initiator C-X bond by coordination to the metal center. The feasibility of this coordination process for the RuCl2(PPh3)3 and MoCl3(PMe3)3 complexes (as well as its unfeasibility for the 17-electron half-sandwich Mo III complex) was verified by DFT calculations on model systems.

Furthermore, the reaction profile of the SNi process has been identified, including the geometry and energy of the transition state. This mechanistic dichotomy for the halide exchange process is at variance with the previous proposition [13] of an inner-sphere/outersphere equilibrium, and appears to be broadly applicable to the interpretation and prediction of the Al(OiPr)3 effect (as well as to the effect of related Lewis acidic additives) on atom transfer and halide exchange reactions involving transition metal halides and organic halides.

Captions for Figures

  The 1 H NMR resonances of the Mo complexes are not sufficiently dependent on the nature of the halides and the possibility of multiple exchange to afford all possible MoXnY3-n(PMe3)3 species complicates further the NMR monitoring of the transition metal complex resonances. The exchange reactions could be conveniently monitored, in this case, by following the methyne and methyl proton resonances of the initiator by 1 H NMR spectrometry, since these resonances are dependent on the halide nature (for instance, the methyne resonance is at  = 4.85 for X = Cl, 4.75 for X = Br and 5.02 for X = I), see an example in <Figure 3.

  III . Any attempt to place the CH3Cl ligand either in the axial position or in an equatorial position of a putative CpMoCl2(PH3)(CH3Cl) molecule, using a starting geometry adapted from CpMoCl3(PH3)2, inevitably led to the expulsion of the CH3Cl molecule far away from the coordination sphere, ultimately yielding energy and geometries corresponding to the sum of free CpMoCl2(PH3) + CH3Cl.The energetic pathway corresponding to the Cl atom transfer process is shown in <Figure 8. In this case, stretching of the CH3-Cl bond must start when the initiator is still quite far away from the metal center. The curve therefore acquires a rather repulsive character while the Mo-Cl distance is still rather long. The best approach of the initiator is found from the same side as the Cp ring. An abrupt change in orientation occurs when the CH3-Cl bond is stretched beyond 2.35 Å, corresponding to the incipient formation of the new Mo-Cl bond. The highest energy along the atom transfer pathway (44.5 kcal mol -1 ) corresponds to a C-Cl distance of 2.4 Å, whereas the final products are at 39.8 kcal mol-1

  (PH3)3 complex, namely a capped octahedron where one Cl atom acts as the capping ligand of a triangular (PH3)3 face. The initiator molecule occupies one of the positions of the opposite, uncapped triangular face together with the other two Cl atoms. The Mo-Cl distances to the coordinated initiator molecule (3.02 Å for CH3CHClCOOCH3) are significantly longer than the other Mo-Cl distances (2.40-2.56 Å), but still compatible with a genuine coordinative interaction.

  3 and 38.6 kcal mol -1 (per Al atom) relative to the monomer. The greatest energetic gain is experienced on going from monomer to dimer, where the coordination number around the Al center goes from 3 to 4. This is because the electronic unsaturation of the Al center in the monomer (relieved only by the O lone pairs via a -donation mechanism) is eliminated by the formation of a fourth Al-O  bond per each Al ion in the dimer. On going from dimer to trimer, one Al ion out of three increases further its coordination number from 4 to 5, whereas the other two Al ions maintain the same coordination environment. On going from dimer to tetramer, one Al ion out of four increases its coordination number from 4 to 6, whereas the other three Al atoms remain 4coordinated. It is to be kept in mind that, for steric reasons, the energetic gains associated to the oligomerization of Al(OiPr)3 are probably smaller relative to those calculated for Al(OMe)3.

( 2 .

 2 572 Å) and by the small pyramidal distortion of the Al(OMe)3 unit (sum of the three O-Al-O angles = 355.7°). The Cl natural charge decreases only slightly from -0.083 in free CH3Cl to -0.037 in the adduct with Al(OMe)3, reflecting the weak charge transfer upon coordination to the Al center. The interaction of Al(OMe)3 with the models of the spin traps RuCl3(PH3)3 and CpMoCl3(PH3)2 affords adducts that are stabilized by 17.3 and 20.0 kcal mol -1 , respectively. The stronger interaction is experienced with the equatorial Cl ligands for both complexes, as shown in <Figure 10, the interaction at the axial positions being exoergic only by 9.0 and 7.8 kcal mol -1 , respectively. This choice reflects the different polarization of the different M-Cl bonds, as indicated by the natural charges in the free Ru III and Mo IV complexes: (-0.420 vs.

  ligand in the atom transfer product. At constant number of Al centers, therefore, the effect favours the atom transfer products. Thirdly, the use of a simplified model for the DFT calculations may affect the value of the interaction energy. As a notable example, the close inspection of the optimized geometries of the Ru III and Mo IV adducts with Al(OMe)3 reveals the presence of one close contact between a H atoms of a PH3 ligands and an O atom of a methoxy group[2.143 Å in the Ru III complex; 2.033 Å in the Mo IV complex]. However, the arrangement of the Al(OMe)3 moiety (M-Al-O angles, M-Al-O-C dihedral angles) and PH3 ligands (M-P-H angles, M-P-H-O dihedral angles) do not give any clear evidence of distortions that may be associated to the formation of a H-bond. Therefore, these interactions probably do not contribute significantly to the stabilization energy of the Al(OMe)3 adducts, which seems to be mostly related to the strengths of the new Al-Cl bonds. The steric effect associated to the ligand simplification is probably generating a greater energetic change on going from the computed model compounds to the real systems.

Figure 1 .

 1 Figure 1. EPR study of complex CpMoCl2(iPr2dad) (1.4 . 10 -2 M) in toluene. (a) after 1 h at 80° with 1 equiv of IEA (spectrum of the dichloride complex); (b) same conditions as in (a), except for the presence of 1 equiv of Al(OiPr)3.

Figure 2 .

 2 Figure 2. EPR study of complex CpMoCl2(iPr2dad) (1.6 . 10 -2 M) in toluene. (a) after 50 min at 80° with 1 equiv of BrEA; (b) same conditions as in (a), except for the presence of 1 equiv of Al(OiPr)3.

Figure 3 .

 3 Figure 3. Examples of 1 H NMR spectra obtained during the halide exchange between MoCl3(PMe3)3 (4 . 10 -2 M) and BEB (1 equiv) at 65°C in C6D6. (a) after 20 min; (b) after 450 min.

Figure 4 .

 4 Figure 4. Results of the halogen exchange monitoring in the absence (solid line, squares) or presence (broken line, diamonds) of Al(OiPr)3 at 65°C in C6D6. (a) MoCl3(PMe3)3 (4 . 10 -2 M) and BEB (1 equiv). (b) MoI3(PMe3)3 (1.4 . 10 -2 M) and BEB (1 equiv).

Figure 5 .

 5 Figure 5. Schematic representation of the electronic structure evolution during the process of RX coordination to RuCl2(PPh3)3 (a) and MoX3(PMe3)3 (b).

Figure 6 .

 6 Figure 6. DFT energetic results for the coordination of CH3CHClCOOCH3 and CH3Cl to RuCl2(PH3)3, and for the Cl atom transfer from CH3Cl.

Figure 7 .

 7 Figure 7. DFT energetic results for the halogen exchange process between RuCl2(PH3)3 and CH3Cl (degenerate exchange, left) or CH3Br (right).

Figure 8 .

 8 Figure 8. DFT energetic results for the Cl atom transfer from CH3Cl to CpMoCl2(PH3)2.

Figure 9 .

 9 Figure 9. DFT energetic results for the Cl atom transfer from CH3Cl to MoCl3(PH3)3.

Figure 10 .

 10 Figure 10. DFT energetic results for the interaction between [Al(OCH3)3]x and various compounds.

  Fig. 1

Fig. 2

 2 Fig.2

  Fig. 3

Fig. 5

 5 Fig. 5

Fig. 6

 6 Fig. 6

Fig. 7

 7 Fig. 7

Fig. 8

 8 Fig. 8

Fig. 9

 9 Fig. 9

Fig. 10

 10 Fig. 10

Reaction between MoCl3(PMe3)3 and CH3CH(Br)Ph. (a) With Al(OiPr)3. MoCl3

	(b) Without Al(OiPr)3. MoCl3(PMe3)3 (12.7 mg, 2.95 . 10 -2 mmol) and BEB (4 L, 5.42 mg,
	2.93 . 10 -2 mmol) were dissolved in 0.6 mL of C6D6. The resulting solution was transferred
	(I)COOEt. A solution containing into an NMR tube and introduced into the NMR probe maintained at 65°C. The progress of
	CpMoCl2(iPr2dad) (8.1 mg, 2.1 . 10 -2 mmol) and CH3CH(I)COOEt (3.0 L, 5.0 mg, 2.2 . 10 -2 the reaction was followed by monitoring the iPr methyne resonance (PhCHXCH3:  4.85 and
	mmol) in 1.5 mL of toluene was split into two equal portions. To one of them was added 4.75 for X = Br and Cl, respectively).
	Al(OiPr)3 (2.4 mg, 1.2 . 10 -2 mmol). The two solutions were transferred into 4 mm Pyrex tubes,
	which were flame sealed and then placed together in a heated oil bath (80°C). The progress
	of the reaction was monitored by EPR spectroscopy (see Results section).
	progress of the reaction was
	monitored by EPR spectroscopy (see Results section).
	(PMe3)3 (10.7 mg, 2.49 . 10 -2 mmol), BEB (3.4 L, 4.61 mg,
	2.49 . 10 -2 mmol), and Al(OiPr)3 (5.0 mg, 2.45 . 10 -2 mmol) were dissolved in 0.6 mL of C6D6.
	The resulting solution was transferred into an NMR tube and introduced into the NMR probe
	maintained at 65°C. The progress of the reaction was followed by monitoring the decrease of
	the iPr methyne resonance for BEB (PhCHBrCH3,  4.85) against the increase of the methyl
	resonance for the chloride analogue (PhCHClCH3  1.58). This was necessary because the
	methyne resonance for PhCHClCH3 overlapped with one of the iPr methyne resonances of
	Al(OiPr)3.

Reaction between CpMoCl2(iPr2dad) and CH3CH(Br)COOEt. Following a procedure analogous to that described above, a solution containing CpMoCl2(iPr2dad) (90 mg, 0.24 mmol) and CH3CH(Br)COOEt (31.2 L, 43.4 mg, 0.24 mmol) in 15 mL of toluene was split into two equal portions. To one of them was added Al(OiPr)3 (24.7 mg, 0.12 mmol).

Portions of the two solutions were transferred into 4 mm Pyrex tubes, which were flame sealed and then placed together in a heated oil bath (80°C). The

Reaction between MoI3(PMe3)3 and CH3CH(Br)Ph. (a) With Al(OiPr)3. MoI3

	(PMe3)3 (6.1 mg, 8.6 . 10 -3 mmol), BEB (1.2 L, 1.63 mg, 8.8 . 10 -3
	mmol), and Al(OiPr)3 (2.0 mg, 9.8 . 10 -3 mmol) were dissolved in 0.6 mL of C6D6. The
	resulting solution was transferred into an NMR tube and introduced into the NMR probe
	maintained at 65°C. The progress of the reaction was followed by monitoring the iPr
	methyne resonance (PhCHXCH3:  4.85 and 5.02 for X = Br and I, respectively).
	(b) Without Al(OiPr)3. MoI3(PMe3)3 (11.2 mg, 1.59 . 10 -2 mmol) and BEB (2.2 L, 2.98 mg,
	1.61 . 10 -2 mmol) were dissolved in 0.6 mL of C6D6. The resulting solution was transferred
	into an NMR tube and introduced into the NMR probe maintained at 65°C. The progress of
	the reaction was monitored by 1 H NMR spectroscopy as described in part (a) above.

Acknowledgements. We are grateful to the CNRS for financial help. FS and SM are grateful to CNRS and to the Conseil Régional de Bourgogne for BDI fellowships, and JM thanks the Spanish Ministerio de Educacion y Ciencia for a post-doctoral fellowship.

Text for the Table Of Contents

The co-catalytic effect of Al(OiPr)3 on ATRP originates from a stronger Lewis acid-base interaction with the halogen atom in the oxidized metal complex, relative to the initiator molecule. The process of halogen exchange between one specific ATRP catalyst and the initiator molecule is also shown to be accelerated by Al(OiPr)3. The absence of this effect for other ATRP catalyst systems is rationalized on the basis of an alternative and more facile halogen exchange mechanism based on internal nucleophilic substitution (SNi).

Graphic for the Table Of Contents


Claire Livet
email: claire.livet@irisa.fr

Théo Rouvier
email: theo.rouvier@ensam.eu

Georges Dumont
email: georges.dumont@irisa.fr

Charles Pontonnier
email: charles.pontonnier@irisa.fr

An automatic and simplified approach to muscle path modelling

The current paper aims at proposing an automatic method to design and adjust simplified muscle paths of a musculoskeletal model. These muscle paths are composed of straight-lines described by a limited set of fixed active via points and an optimization routine is developed to place these via points on the model in order to fit moment arms and musculotendon lengths input data. The method has been applied to a forearm musculoskeletal model extracted from the literature, using theoretical input data as an example. Results showed that for 75% of the muscle set, the relative root mean square error between literature theoretical data and the results from optimized muscle path was under 29.23% for moment arms and of 1.09% for musculotendon lengths. These results confirm the ability of the method to automatically generate computationally efficient muscle paths for musculoskeletal simulations. Using only via points lowers computational expense compared to paths exhibiting wrapping objects. A proper balance between computational time and anatomical realism should be found to help those models being interpreted by practitioners.

Index terms-Moment arm, Musculotendon length, Musculoskeletal model, Forearm

Nomenclature n q ∈ R number of degrees of freedom actuated by a muscle A i ⊂ R range of motion of the ith degree of freedom qi ∈ A i ith degree of freedom actuated by a muscle q ∈ ∏ n q i=1 A i vector of degrees of freedom actuated by a muscle n s ∈ R number of segments on the muscle path. n s may be greater than n q n m ∈ R number of joint with single degree of freedom p( j) ∈ N predecessor segment of joint j s( j) ∈ N successor segment of joint j C k ∈ R 3 cylinder representing the shape of the segment k X k j ∈ C k via point vector position belonging to segment k acting on joint j, with k ∈ p( j), s( j). X∈ ∏ n s k=1 C k vector of all via points L input ( q) ∈ R input musculotendon path length L( q) ∈ R model musculotendon path length R qi input ( q) ∈ R input moment arm associated to qi R qi ( q, X) ∈ R model moment arm associated to qi

O input ∈ R 3 input origin point O ∈ R 3 model origin point I input ∈ R 3 input insertion point I ∈ R 3 model insertion point λ ∈ R homothety factor θ ∈ R n m angles of rotation 1 Introduction
Muscle moment arms and musculotendon lengths play an important role in the muscle forces sharing problem in musculoskeletal simulation. Indeed, the potential contribution of a muscle to a joint motion depends on its geometrical implementation -its geometrical path, with regard to the joint. This path will directly define the capacity of the muscle to actuate a joint via the definition of its moment arm and its inner length [START_REF] Erdemir | Model-based estimation of muscle forces exerted during movements[END_REF]. It is therefore important to properly implement this path to provide a realistic muscle actuation to a musculoskeletal model.

Imaging techniques [START_REF] Herrmann | Moment arm and force-generating capacity of the extensor carpi ulnaris after transfer to the extensor carpi radialis brevis[END_REF][START_REF] Garner | Musculoskeletal model of the upper limb based on the visible human male dataset[END_REF][START_REF] Garland | Wrist tendon moment arms: Quantification by imaging and experimental techniques[END_REF] and anatomical landmarks [START_REF] Murray | Variation of muscle moment arms with elbow and forearm position[END_REF][START_REF] Gonzalez | How muscle architecture and moment arms affect wrist flexion-extension moments[END_REF][START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] can be used to estimate and model muscle paths. Such data can be very helpful to model objects to lead the muscle path. Via points (VP) can be used in straight-lines or centroidlines methods [START_REF] Jensen | An investigation of muscle lines of action about the hip: A centroid line approach vs the straight line approach[END_REF]. Obstacles, like spheres, cylinders, ellipsoids (generally denominated wrapping objects) [START_REF] Garner | Musculoskeletal model of the upper limb based on the visible human male dataset[END_REF][START_REF] Murray | Variation of muscle moment arms with elbow and forearm position[END_REF][START_REF] Gonzalez | How muscle architecture and moment arms affect wrist flexion-extension moments[END_REF], can simulate wrapping around bone prominences for obstacleset methods. These via points and wrapping objects affect moment arms and musculotendon lengths as well. Wrapping objects parameters are sometimes adjusted manually [START_REF] Gonzalez | How muscle architecture and moment arms affect wrist flexion-extension moments[END_REF][START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF][START_REF] Song | Model-based sensorimotor integration for multi-joint control: Development of a virtual arm model[END_REF], however this work is long and tedious [START_REF] Gatti | Optimization of muscle wrapping objects using simulated annealing[END_REF]. The creation of these models requires technical [START_REF] Garner | The obstacle-9 set method for representing muscle paths in musculoskeletal models[END_REF][START_REF] Favre | Automated muscle wrapping using finite element contact detection[END_REF] and experimental expertise in data fitting and muscle modeling, while increasing the cost of computation during simulations [START_REF] Favre | Automated muscle wrapping using finite element contact detection[END_REF] since wrapping trajectories must be computed at each time step.

Several methods using experimental moment arms data to automatically define muscle path parameters have been proposed [START_REF] Gatti | Optimization of muscle wrapping objects using simulated annealing[END_REF][START_REF] Favre | Automated muscle wrapping using finite element contact detection[END_REF][START_REF] Agarwal | A novel framework for virtual prototyping of rehabilitation exoskeletons[END_REF]. The paths resulting from these methods save fitting time but can contain many wrapping objects. Moreover, the methods developed by [START_REF] Gatti | Optimization of muscle wrapping objects using simulated annealing[END_REF][START_REF] Agarwal | A novel framework for virtual prototyping of rehabilitation exoskeletons[END_REF] require to define a priori the obstacles to be placed for each muscle. Lastly, the geometric parameters of multiple wrapping objects prevent simple scaling from a generic model to a subject-specific model [START_REF] Favre | Automated muscle wrapping using finite element contact detection[END_REF] since scaling rules (homothetic [START_REF] Puchaud | Accuracy and kinematics consistency of marker-based scaling approaches on a lower limb model: a comparative study with imagery data[END_REF], radial [START_REF] Lund | Scaling of musculoskeletal models from static and dynamic trials[END_REF]) may not be relevant for such objects.

These studies show that there is a lack of methods able to: a) automatize muscle path modelling from experimental data; b) reduce the computation time for muscle lengths and moment arms during moment arm computation for muscle forces estimation in musculoskeletal simulation; c) facilitate the geometrical scaling of the musculoskeletal model.

The current paper aims at proposing an automatic method to model and adjust simplified muscle paths of a musculoskeletal model. The muscle paths are composed of a limited set of fixed active via points and optimization routines are developed to fit moment arms and musculotendon lengths input data. The method can be used to describe a generic model as well as for a subject-specific model. As an example, the method has been applied to a forearm musculoskeletal model issued from the literature [START_REF] Pennestrì | Virtual musculo-skeletal model for the biomechanical analysis of the upper limb[END_REF], using theoretical input data [START_REF] Rankin | Musculotendon lengths and moment arms for a three-dimensional upper-extremity model[END_REF].

The aim of the following method is to automatically set via points to obtain moment arms and musculotendon lengths values as close as possible to moment arms and musculotendon lengths inputs. The needed muscle data inputs are: moment arms data, musculotendon lengths data, muscle origin and insertion points. Input data can be issued from experiments or from literature models. Fitting procedures based on an optimization scheme are performed: a first one minimizing the difference between the moment arms issued from the input data and those computed by the model, a second one between lengths issued from the input data and those computed by the model, with spatial positions of via points as decision variables. The first procedure also takes into account spatial constraints, so that the final muscle is contained within a consistent body volume, which represents human flesh volume where muscles lie. Since muscles actuate joints, the method relies on an osteoarticular model to compute moment arms and musculotendon lengths. The same osteoarticular model and body volumes can be used for all muscles.

The outline of the paper is as follow: firstly, we describe the muscle path model and the method we use to tune it from generic input data. An overview of this method is illustrated in Figure 1, describing inputs and outputs as well as successive optimization stages. The application case on an upper limb model for a middle size subject is then developed and results are presented and discussed. The computational and anatomical realism of the results are discussed with regard to the aforementioned objectives.

Materials and Methods

Muscle path model definition Moment arm definition

Among all the possible definitions of the muscle moment arm described in [START_REF] Sherman | What is a moment arm? Calculating muscle effectiveness in biomechanical models using generalized coordinates[END_REF], the definition given in [START_REF] An | Determination of muscle orientations and moment arms[END_REF] is chosen, where R qi is the muscle moment arm, L( q) the musculotendon path length and qi the functional degrees of freedom:

R qi ( q) = ∂L( q) ∂ qi (1)
Indeed, they are the coefficients representing the contribution of muscle efforts in the equations of dynamics. This definition is relevant to properly actuate functional degrees of freedom of the human body.

Muscle types

In this method, muscles are divided in two categories because the problem to solve in moment arm optimization depends on these categories. Muscles categories are:

• Type 1: the muscle actuates at least a joint with only one degree of freedom (DOF) • Type 2: the muscle actuates only joints with more than one DOF

Geometrical path

Every muscle path is composed of straight lines systematically built using only muscle origin, muscle insertion, and fixed active via points. Muscles with multiple heads are artificially divided into independent muscles.

We propose to use only two via points for each joint, i.e. one via point per segment articulated by the joint. Examples of muscles paths illustrate this in Figure 2, where the number of via points (black circles) changes with the number of segments. Using two via points per joint enables us to compensate possible misplacement of origin and insertion points.

For a muscle actuating a single joint, one via point on each side of the joint guarantees the independence of moment arms from origin and insertion points. Indeed, the distance between pairs of via points and origin or insertion points remains constant with the variation of joint parameters, so origin and insertion points have no impact on moment arm values.

Adding a third via point has no impact on the moment arm on the joint, because two of them would belong to the same segment. Their distance would remain constant while the joint angles are varying, so the moment arm does not change.

Using only one via point enables the distance between the via point and the origin or insertion points to vary with the joint parameters. Therefore, the position of origin or insertion points would have an impact on moment arm, which is against the hypothesis of the independence of moment arm from origin and insertion points.

For a muscle actuating two serial joints, its path is also composed of two via points per joint. With this rule, two via points belong to the same segment between the two joints. The first pair of via points impacts only on the first joint moment arm, independently of the second pair of via points, and inversely.

Spatial boundaries

Cylinders define consistent body volumes by delimiting spatial via point positions. The muscle path is entirely contained between origin and insertion points, with no possible u-turn, as it is physiologically observed. Cylinders are delimited in length by origin and insertion points. The origin frame of each segment, taken from the osteoarticular model, delimits the length too. Points delimiting cylinders length are called reference points. The cylinders radii represent the maximum distance from the bone to the skin.

Two examples of cylinders are shown in Figures 3 and4. Figure 3 illustrates a case of a muscle actuating three consecutive segments. For instance, humerus, radius and hand segments can be represented by that serial chain. Figure 4 represents a case when a muscle actuates two non-consecutive segments who share the same predecessor, like the ulna and radius share the humerus as predecessor.

Fitting moment arms 2.2.1 Type 1 muscles

A first approach to automatically place via points is to directly fit the model moment arm to an input moment arm. However, as it can be seen in Figure 5, there are an infinite number of via point positions giving the same moment arm for a type 1 muscle.

In Figure 5, two different positions of via points are considered. Via points A1 and B1 belong to the segment 1, while via points A2 and B2 belong to the segment 2. The B triangle is a rotation of the A triangle by the angle θ. When the segment 2 rotates around the joint, A2 and B2 will move of the same angle, and the two triangles will still be the same triangle rotated of angle θ. We conclude that there is an infinity of via points positions which lead to the same moment arm, and those positions can all be deduced from one given position and an angle θ.

Therefore, for a type 1 muscle, a set of via point positions should be obtained by choosing the shortest musculotendon length L( q, X) under body volume constraints and moment arm fitting. In fact, musculotendon length is the shortest path from origin to insertion [START_REF] Sherman | What is a moment arm? Calculating muscle effectiveness in biomechanical models using generalized coordinates[END_REF] while circumventing obstacles.

For each type 1 muscle, the equation to solve is:

min X L( q, X) subject to        ∀ j ∈ [[1, n s ]], ∀k ∈ {p( j), s( j)} : X k j ∈ C k X ∈ arg min n q ∑ i=1 R qi input ( q) -R qi ( q, X) 2 R qi input ( q) 2 (2)
(2) is an optimization to find the shortest musculotendon path that leads to the closest moment arm to the input data, under a spatial physiological constraint.

As shown in Figure 5, one only needs to find one configuration of X that minimizes the distance between the input moment arm and the model moment arm, and then use a rotation of θ around the single DOF joints as a new variable.

Once an initial vector position 0 X is found respecting (2) constraints, the problem to solve can be written as:

min θ L( q, θ) subject to ∀ j ∈ [[1, n s ]], ∀k ∈ {p( j), s( j)} : X k j (θ, 0 X) ∈ C k (3)

Type 2 muscles

For a type 2 muscle, fitting model moment arm and input moment arm can be done directly. With more than one degree of freedom, there is not an infinity of via point positions leading to the same moment arm.

For each type 2 muscle, the equation to solve is:

min X n q ∑ i=1 R qi input ( q) -R qi ( q, X) 2 R qi input ( q) 2 subject to ∀ j ∈ [[1, n s ]], ∀k ∈ {p( j), s( j)} : X k j ∈ C k (4)
(4) is the generalization of (2) constraints for moment arms. The moment arm error is normalized to give the same weight to each moment arm of the path, despite their variation range.

Fitting musculotendon length

Since the moment arm of the new muscle path is the derivative of musculotendon length, a constant still remains between model length and input length. The final step is to decrease this constant. The origin and insertion points are linearly scaled to perform this, since they don't influence the moment arm.

The scaling principle is represented in Figure 6. Via points are represented with black circles while origin and insertion points are represented by full color disks. Input origin and insertion points have a lighter color than the new origin and insertion points. Those new points must belong to the half-line starting from the nearest via point and passing by the input origin and insertion points.

λ L input ( q) -L( q, λ) 2 subject to OX p(1) 1 = λO input X p(1) 1 IX p(n s ) n s = λI input X p(n s ) n s (5)

Performance criteria

For each degree of freedom qi actuated by the considered muscle, root mean square error between the input data moment arm and the model is computed, in absolute (RMSE) and relative (rRMSE) terms. rRMSE is obtained by dividing the RMSE by the RMS value of the input data moment arm. This gives an estimate of the solution quality resulting from the method. The balance between two angles for a type 2 muscle is checked by the rRMSE.

Correlation and rRMSE value for musculotendon lengths comparisons are considered as performance criteria. Correlation shows the global impact of moment arm on musculotendon length.

Use case

To evaluate its performance, the method has been applied to a forearm model. The use case is a comparison between reference polynomial values modeling muscles created with an obstacle-set method and data coming from muscle paths generated with the method described above. Input data for this use case are:

• Osteoarticular model: closed-loop forearm model proposed by [START_REF] Pennestrì | Virtual musculo-skeletal model for the biomechanical analysis of the upper limb[END_REF]; • Body volumes: complete description is given in Table 1; • Origin and insertions points: taken from the musculoskeletal upper limb model proposed by [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF]; • Moment arm input data: [START_REF] Rankin | Musculotendon lengths and moment arms for a three-dimensional upper-extremity model[END_REF] provides polynomial coefficients to model moment arm values from [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] muscles; • Musculotendon length input data: [START_REF] Rankin | Musculotendon lengths and moment arms for a three-dimensional upper-extremity model[END_REF] provides polynomial coefficients to model musculotendon length values from [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] muscles.

The model proposed in [START_REF] Pennestrì | Virtual musculo-skeletal model for the biomechanical analysis of the upper limb[END_REF] was linearly scaled to obtain an osteoarticular model 170cm tall, similarly to the model described in [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF].

The input moment arm and musculotendon length proposed in [START_REF] Rankin | Musculotendon lengths and moment arms for a three-dimensional upper-extremity model[END_REF] interpolate muscle moment arms and musculotendon length described in [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] as polynomial functions of degrees of freedom. These polynomials were used to generate both moment arms and musculotendon lengths input data.

Anatomical segments are described in Table 1. The range of motion for each degree of freedom was taken from [START_REF] Rankin | Musculotendon lengths and moment arms for a three-dimensional upper-extremity model[END_REF]: The axes relative to the aforementioned degrees of freedom follow the ISB recommendations [START_REF] Wu | ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion -Part II: Shoulder, elbow, wrist and hand[END_REF].

Table 1: Description of segments of interest for a man (170cm tall, same as [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF]'s model) with a weight of 70 kg (standard body mass index). Mass values were adapted from [START_REF] Dumas | Adjustments to McConville et al. and Young et al. body segment inertial parameters[END_REF], density values were taken from [START_REF] Yeadon | The simulation of aerial movement-II. A mathematical inertia model of the human body[END_REF] and length values were adapted from [START_REF] Pennestrì | Virtual musculo-skeletal model for the biomechanical analysis of the upper limb[END_REF]. The cylinder radius was deduced from mass, density and length data. The method was applied on muscles listed in Table 2. A comparison of muscle paths from the original model [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] and the proposed model is also given in Table 2, because a goal of the proposed method was to simplify muscle path description. For instance, the type 1 muscle Anconeus was modeled by only one obstacle in the original model while the proposed method used two via points instead. The type 2 muscle Pronator Teres was modeled by one obstacle and three via points in the original model, while the proposed method simplified the path with only two via points. Pronator Teres (PT) 1 3 2

Segment Mass Density

The method has been implemented in CusToM [START_REF] Muller | CusToM: a Matlab toolbox for musculoskeletal simulation[END_REF], an open-source MATLAB ® toolbox.

The moment arm optimization (4) was computed using the Global Search solver, with an interior-point method. When the reference points -origin and insertion points issued from the input data -respected the constraints of lying inside the defined cylinders, they were used as initial guesses. Otherwise, initial points which respect the cylinder constraints were chosen randomly.

The other optimization problems (3) and ( 5) were solved with a sequential quadratic programming algorithm, with a null initial guess. A synthesis of the results is given in Table 3. The median of moment arm rRMSEs was of 13.27%. This result is acceptable in regards to the discrepancy in moment arms values reported in the literature. For Brachioradialis, Extensor Carpi Radialis Longus, Brachialis, Pronator Teres and Triceps muscles, [START_REF] Murray | Scaling of peak moment arms of elbow muscles with upper extremity bone dimensions[END_REF] reports a maximum inter-specimen variability of 18% (achieved for Pronator Teres). For wrist muscles (Flexor Carpi Radialis, Flexor Carpi Ulnaris, Extensor Carpi Radialis Longus, Extensor Carpi Radialis Brevis and Extensor Carpi Ulnaris muscles), [START_REF] Garland | Wrist tendon moment arms: Quantification by imaging and experimental techniques[END_REF] reported maximum inter-specimen variability of 45.5% (achieved for Extensor Carpi Ulnaris). So, the method is relevant for 75% of the muscles (Table 3), regarding the moment arms. Some initial complex muscle models, such as the Flexor Carpi Radialis muscle, can be accurately represented using only two via points per joint. For all muscles, the original number of obstacles had no significant impact on the results.

Correlation between lengths are shown in Figure 7. The minimal correlation was 0.8435, reported for the Extensor Carpi Ulnaris muscle. Figure 7 shows that type 1, Brachioradialis and Pronator Teres muscles were all above 0.999 of correlation. All these muscles have only two via points on their path (Table 2). They were also among those with the lowest musculotendon length rRMSE. It was deduced that for muscles actuating only one joint, a muscle path composed of origin, insertion, and two via points was enough to accurately estimate their moment arms and inner lengths.

Musculotendon lengths RMSEs were below 11.18%. A comparison of musculotendon lengths before and after length optimization is given in Figure 8. Extensor Carpi Radialis Brevis muscle can be considered as an outlier, regarding the fact that all other musculotendon length rRMSEs were below 2.93%. [START_REF] De Monsabert | Importance of Consistent Datasets in Musculoskeletal Modelling: A Study of the Hand and Wrist[END_REF] reported a maximum of 53% error difference between their model and experimental data for Extensor Carpi Radialis Brevis muscle. Compared to [START_REF] De Monsabert | Importance of Consistent Datasets in Musculoskeletal Modelling: A Study of the Hand and Wrist[END_REF], musculotendon lengths resulting from our model were acceptable. The relevance of the length optimization could be questionned for Supinator Brevis, Triceps Lateral, Extensor Carpi Radialis Longus, Flexor Carpi Ulnaris, Flexor Carpi Radialis, Palmaris Longus and Pronator Teres muscles. Be-fore length optimization, the model was already under 10% rRMSE in musculotendon length fitting. Depending on the accuracy requested for a given simulation, this step in the proposed method could be skipped.

As it can be seen in Table 3, a significant correlation (third quartile 0.9998) is obtained for musculotendon length while moment arm rRMSEs for WDV and WFE axis were around 40%. WDV and WFE axis moment arm were ten times shorter than EFE and PS axis moment arm, so their impact was less significant on musculotendon length. This also meant that having a significant musculotendon length correlation did not imply having low moment arm rRMSE.

Muscle path resulting from this use case may be taken with caution: input data is a polynomial model [START_REF] Rankin | Musculotendon lengths and moment arms for a three-dimensional upper-extremity model[END_REF] depicting another model [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF], which was itself manually adjusted to fit experimental data. For instance, for the Extensor Carpi Ulnaris muscle, the polynomial data have a rather high error (86.63% error for WDV moment arm). It is therefore not surprising that the method had difficulty to approximate a model that is poorly representing a muscle path. In addition, a model fitting this polynomial perfectly would be very far from [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] model, because of the initial high error rate. Using experimental data as input data should improve the realism of the generated muscle paths.

The model used in the example was a 170cm tall generic male model. Generally, in a musculoskeletal simulation, it is from this musculoskeletal model that the scaling is performed to fit the specific anatomy of the subject. The interest of using a model with only via points is to go beyond the problem of scaling wrapping objects. Scaling wrapping objects is more complex since it requires to scale an object defined by multiple parameters. There is not any obvious scaling rules for these parameters, since literature often reports user intervention [START_REF] Favre | Automated muscle wrapping using finite element contact detection[END_REF][START_REF] Lund | Scaling of musculoskeletal models from static and dynamic trials[END_REF].

Method limitations

Computation time is gained as soon as there are more than two obstacles in the input data model (Table 2). Muscle moment arms are derived from musculotendon lengths, so they can be computed through a musculotendon length discretization. The critical computation time in moment arm computation for muscle forces estimation is therefore located in the computation time of the musculotendon length at a given angle. However, to compute a muscle path length with at least two wrapping surfaces, it is necessary to go through an optimization routine [START_REF] Garner | The obstacle-9 set method for representing muscle paths in musculoskeletal models[END_REF]. For via points, the musculotendon length is the norm between all the points of the path. Computational cost of via points is far less expensive than computational cost of obstacles [START_REF] Favre | Automated muscle wrapping using finite element contact detection[END_REF]. The method did not provide any computational time gain for the Brachialis, which already had only two via points, as it can be seen in Table 2. For all the other muscles with a single obstacle, the gain in terms of complexity and computation time still has to be quantified to draw a conclusion as to the usefulness of the method for them.

Nevertheless, there is a temporal gain in modeling muscle path. The method is systematic and its parameters are automatically chosen: the user only needs to place origin and insertion points and does not have to choose and adjust obstacles for each muscle path.

The optima found by the algorithm are not guaranteed to be global optima of (4). However, with regards to literature, the obtained optima are relevant enough to be used, as it can be seen in next paragraph.

From model to reality

According to the Zajac-Hill model [START_REF] Zajac | Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control[END_REF], muscle force is related to muscle length. The muscle force generation parameters of a musculoskeletal model, such as the forcelength relationship, can be scaled for a subject by measuring torques at a given position [START_REF] Wu | Subject-specific musculoskeletal modeling in the evaluation of shoulder muscle and joint function[END_REF]. Such parameters can be overfitted to compensate possible errors on moment arm and musculotendon length values, affecting the estimation of muscle forces in musculoskeletal simulation. It is therefore best to properly scale these values, prior to any force generation parameters adjustment.

The model resulting from the method was realistic from a kinematic point of view: the contribution of a muscle to a joint can be realistically computed as long as experimental kinematic data are used as input data of the simulation. However, from an anatomical point of view, no muscle is composed of straight lines. This model can be sufficient as long as interactions between muscles or with other soft tissues, like friction, are not needed in a musculoskeletal simulation. The possible bone penetration was not considered and the body volumes were used only to restrict via point positions, meaning that the whole muscle path may not be contained in it.

Conclusion

In order to simplify tedious muscle modeling, this study proposed an automatic method which creates a systematic muscle path with only via points fitting moment arm and musculotendon length input.

The simplification aimed at three majors goals: avoiding scaling issues of wrapping objects; avoiding specific muscle modeling; and avoiding manual muscle path parameterization. Using only via points achieved the first goal, using a systematic parameterization achieved the second goal, and using an optimization method to automatically find the via point positions achieved the third goal.

Different methods of automatic via point placement were developed for type 1 and type 2 muscles. For type 1 muscles, the method relied on a minimization of musculotendon length while respecting spatial boundaries and moment arm input data data. For type 2 muscles, the method relied only on minimizing the distance between moment arm model and moment arm input data, while respecting spatial boundaries. Origin and insertion points were then moved according to an homothetic rule to fit the musculotendon lengths.

This method was applied to a forearm model from [START_REF] Pennestrì | Virtual musculo-skeletal model for the biomechanical analysis of the upper limb[END_REF], to test its relevance. The input musculotendon length and moment arm were polynomial data from [START_REF] Rankin | Musculotendon lengths and moment arms for a three-dimensional upper-extremity model[END_REF], while initial origin and insertion points were taken from [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF].

Moment arm results lied in literature variation range: for 75% of muscles, rRMSE is under 29.23% for moment arm. EFE and PS moment arms were better approximated than WDV and WFE moment arms. Musculotendon length rRMSE were all upon 11.18%. The method is particularly relevant for muscles actuating a single joint.

The main benefit of this method is its usability: each muscle path can be automatically created from its moment arm, musculotendon length, origin and insertion points data. This functionality is available in the CusToM musculoskeletal simulation library [START_REF] Muller | CusToM: a Matlab toolbox for musculoskeletal simulation[END_REF]. A user can simply gather the proper data, indicate the origin and insertion points to the method and let the muscle paths being automatically computed.

Further studies should investigate the pertinence of different scaling rules to apply to the via points generated using this method, as well as evaluate the computation time gained from using simplified muscles paths. The anatomical realism of the generated path remains a limitation of the proposed method.

In a near future, we plan to apply this method with experimental data for the lower limb, as those available in [START_REF] Buford | Muscle balance at the knee-moment arms for the normal knee and the acl-minus knee[END_REF].

Figure 1 :

 1 Figure 1: Overview of the muscle path design method

Figure 2 :

 2 Figure 2: Examples of geometrical muscle paths depending on the number of segments

Figure 3 :

 3 Figure 3: Cylinders for a three segments open-loop chain. Cylinders are delimited in length by reference points in full color. Via points (black circles) must stay in the cylinder of the same color.

Figure 4 :

 4 Figure 4: Cylinders for two segments with the same predecessor segment. The beginning and end faces of the cylinders are the planes perpendicular to the main axis of the segment containing the origin or insertion of the muscle.

Figure 5 :

 5 Figure 5: Example of positions of via points that lead to the same moment arm. The B triangle is a rotation of the A triangle by the angle θ.

Figure 6 :

 6 Figure 6: Adjusting origin and insertion point to fit input length

  min

•

  Elbow flexion extension (EFE): from 0 • to 100 • • Elbow pronation supination (PS): from -45 • (supination) to 45 • (pronation) • Wrist flexion extension (WFE): from -45 • to 45 • • Wrist deviation (WDV): from -10 • (radial abduction) to 25 • (ulnar adduction)

Figure 7 :

 7 Figure 7: Correlation between musculotendon length input and model. Type 1 muscles, respectively type 2 muscles, are indicated with transparent bars, respectively opaque bars.

Figure 8 :

 8 Figure 8: Musculotendon length rRMSE between musculotendon length input and model, before and after length optimization. Type 1 muscles, respectively type 2 muscles, are indicated with transparent bars, respectively opaque bars.

Figure 9 :Figure 10 :

 910 Figure 9: Example of results for Pronator Quadratus muscle. Maximum moment arm error was 22.1µm (0.29% of maximum moment arm) and maximum musculotendon length error was 2.10µm (0.01% of maximum musculotendon length)

Table 2 :

 2 Comparison of needed data to describe muscles path: figures in obstacles column and via points (VP) column correspond to the number of obstacles and the number of via points, respectively, used to describe a muscle path

	Muscles	Model from [7] Model
		Obstacles VP	VP
	Type 1 muscles
	Anconeus (ANC)	1	0	2
	Brachialis (BRA)	0	2	2
	Pronator Quadratus (PQ)	2	0	2
	Supinator Brevis (SUP)	1	1	2
	Triceps Medial (TRImed)	1	3	2
	Triceps Lateral (TRIlat)	1	3	2
	Type 2 muscles
	Brachioradialis (BRD)	1	1	2
	Extensor Carpi	2	2	4
	Radialis Longus (ECRL)
	Extensor Carpi	1	2	4
	Radialis Brevis (ECRB)
	Extensor Carpi	1	4	4
	Ulnaris (ECU)
	Flexor Carpi	2	1	4
	Radialis (FCR)
	Flexor Carpi	2	2	4
	Ulnaris (FCU)
	Palmaris Longus (PL)	2	2	4

3

  Results and Discussion 3.1 Overall results Complete path descriptions for each muscle are reported in supplementary materials. RMSE and rRMSE for moment arms and rRMSE for musculotendon lengths are reported in supplementary materials. Examples of results are shown for moment arm for Pronator Quadratus and Palmaris Longus muscles in Figures 9 and 10. As the Pronator Quadratus muscle is a type 1 muscle, its moment arm and musculotendon length values are represented as functions of only PS angle. The Palmaris Longus muscle is a type 2 muscle. EFE and PS moment arm are independant from WDV and WFE angles, and inversely.

Table 3 :

 3 Synthesized results for all muscles. Minimum, first quantile, median, third quantile and maximum values are given for moment arm rRMSE of every muscle and for each actuated joint angle. Statistical data are also given for musculotendon length correlation and rRMSE.

	Criterion	Minimum First quantile Median Third quantile Maximum
	Moment arm rRMSE (%)
	EFE axis	2.86	6.41	7.62	12.07	34.80
	PS axis	0.13	5.20	8.88	19.22	57.25
	WDV axis	13.49	20.85	32.57	38.36	41.95
	WFE axis	14.43	18.70	23.57	44.67	51.75
	All axis	0.13	6.59	13.27	29.23	57.25
	Musculotendon length
	Correlation	0.8435	0.9928	0.9992	0.9998	1
	rRMSE (%)	0.00	0.20	0.42	1.09	11.18


Océane Savary

Jérôme Mounier

Anne Thierry

Elisabeth Poirier

Julie Jourdren

Marie-Bernadette Maillard

Emmanuel Coton

Monika Coton
email: monika.coton@univ-brest.fr

Understanding Kombucha fermentations: dynamic follow-up of a lab scale fermentation

Keywords:

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

Introduction

Understanding Kombucha fermentations: dynamic follow-up of a lab scale fermentation Biofilm Bacteria Yeasts

day biofilm sample:

Sample preparation for SEM: biofilm samples were treated with a cacodylate & 2.5% glutaraldehyde fixation solution then dehydrated using an ethanol bath before coating with a thin layer of metal.

Complex network between bacteria and yeast:

-

yeast

•

  in clusters bacterial biofilm network surrounds yeasts biofilm densifies over time Biofilm potentially linked to their symbiotic relationship +2.5 log CFU/ml AAB counts +1 log CFU/ml yeast counts Population stable for LAB All 3 microbial groups well implanted in biofilm +2 log CFU/g in AAB counts AAB dominate in biofilms -biofilm formation at a rate of 0.34g/d for 7d then 0.15g/d up to 27d -rapid decrease in pH -rapid decrease in density ethanol production (<1%) -metabolite production Sample preparation: cells fixed with 1:1 PBS 1X:ice-cold ethanol 96% before hybridization with specific yeast and AAB/LAB FISH probes, in green and red, respectively, blue represents biofilm autofluorescence. Microbial species involved in Kombucha fermentations produce a specific metabolome over time • Clear symbiotic relationship between the different microbial groups exists • Acetic acid bacteria actively participate to biofilm formation which surrounds other bacteria yeast over time Volatile metabolites determined by GC-MS (headspace) Organic acids determined by QTOF LC-MS and enzymatic kits

Figure 1 :Figure 4 :Figure 5 :Figure 6 :

 1456 Figure 1: Different factors impacting Kombucha fermentation

Figure 8 :

 8 Figure 8: Principal component analysis on volatile metabolites during the lab scale kombucha fermentation

Figure 3 :

 3 Figure 3: Microbial counts in biofilm samples during fermentation

Figure 2 :

 2 Figure 2: Microbial counts in tea samples during fermentation

Acknowledgements:

We would like to thank Mr. Philippe Elies of the Microscopy platform at the UBO for the microscopy acquisitions.