Dr Roxanne Bercy

Jean-Noël Rebilly
email: jean-noel.rebilly@universite-paris-saclay.fr

Christian Herrero

Dr Hélène Maisonneuve

Pr Régis Guillot

Frédéric Banse
email: frederic.banse@universite-paris-saclay.fr

Dr J.-N Rebilly

Beneficial Effect of Acetic Acid on the Formation of Fe III (OOH) Species and on the Catalytic Activity of Bioinspired Nonheme Fe II Complexes

Keywords: nonheme, iron, hydroperoxo, O-O activation, mechanism

Three new amine/pyridine Fe II complexes bearing pentadentate ligand with one, two or three electron enriched 4methoxy-3,5-dimethylpyridine were used as catalysts for the oxidation of small organic molecules by hydrogen peroxide. The distribution of products formed suggests that these ligands are not enough electron donating to promote the O-O heterolytic cleavage of the oxidant in order to generate selective Fe V (O) species. Using acetic acid in the reaction mixtures results in a significant increase of the efficiency of these catalytic systems. Our investigations show that the use of AcOH leads to the protonation/dissociation of a pyridyl moiety and the formation of (N4)Fe II (OAc)(OH) species. These complexes readily react with excess hydrogen peroxide to yield (N4)Fe III (OAc)(OOH) intermediates. These latter intermediates are proposed to evolve into (N4)Fe IV (OAc)(O), which are more efficient than the usual (N4)Fe IV (O) and (N5)Fe IV (O).

Introduction

The development of efficient, selective and sustainable methods for the oxidative transformation of hydrocarbons is required to replace the energy-demanding or harmful technologies used on a large scale by the chemical industry. [START_REF] Teles | Oxidation in Applied Homogeneous Catalysis with Organometallic Compounds: A Comprehensive Handbook in Four Volumes[END_REF] Inspired by nonheme iron enzymes capable to activate O 2 to carry out hydrocarbon oxidations, [START_REF] Kal | [END_REF]3] numerous studies on nonheme Fe II complexes catalyzing the oxygenation of alkanes and alkenes by H 2 O 2 have been reported. [4] In these systems, the steady-state generation of Fe III (OOH) intermediates has been observed [5][6][7] and the homo-or heterolytic nature of their O-O bond cleavage was found to essentially depend on the ligand framework. With tetradentate N 4 ligands leaving two cis labile sites on the metal center, the (N 4 )Fe III (OOH) species cleave heterolytically to yield selective Fe V (O)(OH) oxidizing species. [8] On an other hand, with pentadentate N 5 ligands, the cleavage in (N 5 )Fe III (OOH) species is homolytic which generates pairs of Fe IV (O) / hydroxyl radicals oxidants, [9] [10] frequently resulting in unselective oxidations due to these latter species. Nonetheless, the interplay between Fe IV (O) and HO • is beneficial in aromatic hydroxylation. [11][12][13] Learning from heme enzymes in which the heterolytic cleavage is promoted by the so-called push-pull effect, [14] we were wondering if pentadentate nonheme (N 5 )Fe III (OOH) complexes could switch from a homolytic to a heterolytic cleavage with electron rich ligands (push effect) and in the presence of acetic acid to protonate the distal oxygen atom (pull effect).

We report herein that the electron-donating character of these new ligands is not strong enough by itself to promote the O-O heterolytic cleavage of the resulting Fe III (OOH) species. However, in the presence of acetic acid the catalytic efficiency of these complexes towards the oxidation of alkanes and alkenes by H 2 O 2 is enhanced. In contrast to the expected acid-triggered heterolytic O-O cleavage reported in the literature, [15] we show that the most obvious effect of acetic acid is to enhance the reactivity of the (N 5 )Fe II complexes with hydrogen peroxide which leads to the formation of (N 4 )Fe III (OOH) intermediates. As shown in this paper, the use of acetic acid is beneficial since it helps in forming an intermediate which is known to be precursor to a high-valent Fe(O) species following O-O scission.

Results and Discussion

Ligands and complexes.

The ligands used in this study are derived from the well-known pentadentate ligand L 5 2 by substitution of one, two or three pyridines by electron enriched 4-methoxy-3,5-dimethylpyridine moieties in 1, 2 and 3, respectively (Scheme 1 and Supporting Information). Complexation of these ligands with Fe II (OTf) 2 , as described in the Supporting Information, yielded the complexes [(1)Fe II (OTf)](OTf), [(2)Fe II (OTf)](OTf) and [(3)Fe II (OTf)](OTf). These complexes display a pseudo-octahedral geometry with the ligand bound in a pentadentate fashion and a triflato anion to complete the coordination sphere (Figure 1). For all complexes, the Fe-N distances are ca. 2.2 Å, which is characteristic of high spin (S=2) S1 and Figures S16-S18). The cyclic voltammograms (CVs) of these complexes dissolved in acetonitrile display reversible Fe III /Fe II waves with potential values depending on the number of electron-rich pyridine moieties: the richer the ligand, the smaller the potential value. The E 1/2 values are observed at 0.76, 0.81 and 0.93 V vs SCE with the ligands 3, 2 and 1, respectively (Figure S19 and Table S2). By comparison, the Fe III /Fe II wave of [(L 5 2 )Fe II (MeCN)] 2+ is observed at 0.95 V vs SCE, strongly suggesting that the triflato ligand in S3). Such values are characteristic of low spin (S=0) Fe II complexes, whereas much weaker extinction coefficients values would be observed for high spin Fe II complexes, [5] i.e. with triflato as exogenous ligand.

Catalytic oxidations.

The catalytic activity of [(1)Fe II (OTf)](OTf), [(2)Fe II (OTf)](OTf) and [(3)Fe II (OTf)](OTf) towards aliphatic, aromatic and olefin oxidation was assessed using H 2 O 2 as oxidant and cyclohexane, anisole and cyclooctene as probe substrates (Table 1). All of these catalytic systems display a significant activity toward the tested substrates. With anisole (entries 1, 3, 5), orthomethoxyphenol is observed as the major product with satisfactory yields for all the complexes. Aromatic hydroxylation initiated by non heme Fe III (OOH) intermediates which cleave into a [Fe IV (O) + HO • ] oxidant couple has been shown to essentially occur on the ortho-position. [5,9,[11][12][13][16][17][18] Oxidation of cyclooctene favours epoxidation over dihydroxylation, as generally observed with related catalytic systems. [5] With cyclohexane, a mixture of alcohol and ketone products are observed which is another diagnosis for the transient formation of Fe III (OOH) intermediates which evolve via homolytic cleavage into Fe IV (O) and HO • which will produce long-lived alkyl radicals upon H-abstraction to the substrate. [19] While the results observed with [(1)Fe II (OTf)](OTf) and [(2)Fe II (OTf)](OTf) are very similar, [(3)Fe II (OTf)](OTf) appears significantly more efficient towards epoxidation and alkane hydroxylation. Very efficient O-atom insertion reactions into alkenes or alkanes are observed when Fe V (O) oxidants are generated, i.e. following heterolytic cleavage of the Fe III (OOH) species. [START_REF] Company | Selective Alkane Oxidation[END_REF][START_REF] Serrano-Plana | Inorganic Reaction Mechanisms, Inorganic Reaction Mechanisms[END_REF] However, the yields and distribution of products are not different enough to conclude that a switch in reactivity, from a homolytic O-O cleavage with ligands 1 and 2 to a heterolytic one with 3, is happening. Thus, even with the most electron rich ligand 3 the push electronic effect does not seem to be sufficient to promote a heterolytic O-O cleavage and the generation of a Fe V (O) species. Therefore, we planned to use AcOH to assist the weak push effect with a potential pull effect in order to boost the catalytic activity.The same catalytic assays were performed in the presence of 200 eq. AcOH (entries 2, 4, 6). For anisole oxidation, the results are virtually identical to those in the absence of AcOH. In contrast, for cyclooctene and cyclohexane oxidation, clear enhancement in the formation of the epoxide and the alcohol are observed when AcOH is present. For cyclooctene, the epoxide is the very major product for all catalysts, with [(3)Fe II (OTf)](OTf) yielding 80% conversion. For cyclohexane, the presence of AcOH enhances the total product yield up to 60-70% for [(1)Fe II (OTf)](OTf) and [(2)Fe II (OTf)](OTf) and up to 90% for [(3)Fe II (OTf)](OTf). For all catalysts, cyclohexanol is the major product and the alcohol / ketone (A/K) ratios exceeds 1, whereas it is below 1 in absence of AcOH. A/K values larger than 1 point towards the involvement of a metal-oxo oxidant. Even if this value remains fairly close to 1, it seems evident that the presence of AcOH in the reaction mixture gives rise to a switch in reactivity. Complementary spectroscopic experiments were performed to get mechanistic insights into the mechanism of H 2 O 2 activation induced by the presence of AcOH.

Characterization of reaction intermediates.

Addition of 100 eq. H 2 O 2 to a 1 mM solution of complex in MeOH at 20°C was followed by stopped flow UV-visible spectroscopy (Figure 2). For [(1)Fe II (OTf)](OTf) and [(2)Fe II (OTf)](OTf) an absorption band at ca. 530 nm progressively develops, characteristic of the hydroperoxo-to-Fe III charge transfer in (N 5 )Fe III (OOH) intermediates. Such species obtained with parent complexes display extinction coefficient for this band of ca. 1000 M -1 cm -1 . [5,6,16] Thus, the conversion into a (N 5 )Fe III (OOH) intermediate can be estimated to be between 50 and 70% for [(1)Fe II (OTf)](OTf) and [(2)Fe II (OTf)](OTf), respectively. Surprisingly, no such spectroscopic evolution is observed when [(3)Fe II (OTf)](OTf) is reacted with H 2 O 2 under the same conditions (Figure 2C) indicating that the expected (N 5 )Fe III (OOH) intermediate does not accumulate with the more electrondonating ligand 3. Aliquots were taken from these reaction mixtures at the apex of the 530-nm absorption before its decay (or when the development

C

of the absorption at ca. 500 nm ceased for [(3)Fe II (OTf)](OTf)) and the samples were studied by EPR (Figure 3). For the three complexes, EPR resonances are observed around g = 2, characteristic of the presence of low spin Fe III species and only residual signals at g = 4.3 due to ubiquitous high spin Fe III (Figures 3 andS21).

Table 1. Oxidation of substrates by hydrogen peroxide catalyzed by the three complexes at room temperature in the absence or presence of 200 eq. AcOH. [a] Yields are given as % based on starting H2O2.

Anisole [b] Cyclooctene Cyclohexane S21), [9,11,22][23] or a mixture of (N 5 )Fe III (OOH) and (N 4 )Fe III (OOH) species. [24,25] The more rhombic set of values are assigned to (N 5 )Fe III (OMe/OH) species, which are the precursors of the Fe III (OOH) intermediates. [START_REF] Martinho | [END_REF] In contrast, for [(3)Fe II (OTf)](OTf), the major species is a (N 5 )Fe III (OMe/OH) with g = 2.34, 2.15, 1.93 and only weak resonances are observed at g = 2.22 and 1.97 indicating that (N 5 )Fe III (OOH) is hardly accumulated in this case, in agreement with the UV-visible experiments.

entry o-MeO-PhOH Cyclooctene-oxide cyclooctane-1,2-diol cyclohexanol cyclohexanone -ol / -one [c]
Reaction of the three complexes with H 2 O 2 in acetonitrile only led to a slight increase in the absorption at ca. 500 nm as seen by visible absorption spectroscopy (Figure 4) indicating that the (N 5 )Fe III (OOH) species do not accumulate to a large extent in these conditions, as generally observed with parent complexes. [26][5,6] This conclusion is confirmed by EPR spectroscopy which shows that, for all three complexes, formation of (N 5 )Fe III (OH) species is prevalent over the formation of the (N 5 )Fe III (OOH) species (Figures 5 andS22). Interestingly, when the reactions were performed in the presence of 200 eq AcOH, development of an absorption band around 510-515 nm was observed for the 3 complexes suggesting the accumulation of Fe III (OOH) intermediates (Figure 4).

F

with AcOH EPR analysis of these samples confirmed this former assumption with the detection of intense resonances at g ca. 2.22, 2.16, 1.97/1.96 for the three complexes which can be ascribed to Fe III (OOH) species (Figure 5). However these EPR signatures are significantly different to those observed in MeOH in the absence of AcOH (Figure 3). In particular, they do not display two resolved g max values as described above. This observation strongly suggests that distinct Fe III (OOH) species are generated depending on the absence or presence of AcOH. The ca. 500 cm - 1 shift of the absorption maxima (Figures 2 and4) is also consistent with the generation of different Fe III (OOH) intermediates. Interestingly, the EPR spectra observed for all three complexes in the presence of AcOH is almost identical (2.20, 2.16, 1.96) to the one reported for a (N 4 )Fe III (OOH) intermediate supported by the tetradentate ligand N,N'-dimethyl-N,N'-bis(2-(3,5-dimethyl-4-methoxy-pyridylmethyl)ethane-1,2-diamine, i.e. the electron-rich version of the bpmen ligand with two methyl and one methoxy groups on the pyridines. [27] Note that, numerous (N 4 )Fe III (OOH) intermediates display similar EPR g values. [15,24,28] Thus, it can be proposed that (N 4 )Fe III (OOH) species are formed when the (1)Fe II , (2)Fe II or (3)Fe II precursors are reacted with hydrogen peroxide in the presence of AcOH. This proposition implies the dissociation of a pyridyl arm.

To test this hypothesis, we compared the electronic absorption spectra of the three Fe II complexes in the absence and presence of AcOH. In all cases, a similar evolution is observed upon addition of the acid, with a persistence of the d  → *(pyridines) charge transfer at nearly the same position but with a 25-35% decrease of its extinction coefficient indicating a structural change has occurred (Figure S20 and Table S3). The decrease in intensity suggests the conversion of a small amount of the initial low spin Fe II complex into a high spin one, which is characterized by a lower extinction coefficient (1000-2000 M -1 .cm -1 ). [5,16,29,30] Protonation of a pyridyl arm would result in its dissociation and the resulting vacant site could be occupied either by the nascent acetate ion or an acetonitrile solvent molecule. However, in this latter case, the Fe II complex would remain low spin and the intensity would not be significantly affected. In contrast, (N 5 )Fe II complexes with an exogenous anionic ligand to complete the coordination sphere are generally HS (S = 2), [5,29,31] in agreement with the X-ray structure of these complexes (Figure 1). Thus, it is plausible that the use of AcOH results in the protonation/dissociation of a pyridyl arm and subsequent coordination of acetate on the vacant position to form a (N 4 )Fe II (OAc) complex, which would likely be HS. However, the formation of this new species is not quantitative as indicated by the moderate decrease of the UV-visible absorption, which can be ascribed to the very weak acidity of AcOH (pK a = 23.5 in acetonitrile). Nonetheless, the formation of the (N 4 )Fe II (OAc) complex promotes the reaction with hydrogen peroxide and the formation of the (N 4 )Fe III (OOH) intermediate. Que and coworkers recently reported on the beneficial effect of AcOH in the formation of (BnTPEN)Fe III (OOH) (BNTPEN is the pentadentate ligand Nbenzyl-N,N',N'-tris(2-pyridylmethyl)-1,2-diamino-ethane)). [32] They proposed that AcOH protonates the low spin (N 5 )Fe III (OH) formed upon oxidation of the (N 5 )Fe II precursor by H 2 O 2 to convert it in a putative high spin (N 5 )Fe III (OH 2 ) more reactive towards hydrogen peroxide, and goes on to form (BnTPEN)Fe III (OOH). In our case, we observe the formation of a (N 4 )Fe III (OOH), and thus our mechanistic proposal is different.

Mechanistic investigation on the effect of AcOH.

Using the three Fe II complexes, we monitored the formation of the Fe III (OOH) species by stopped-flow UV-visible spectrophotometry.

In each case, 100 eq. H 2 O 2 vs Fe and different amounts of AcOH (from 2 to 200 eq) were used. Even as low as 2 eq. AcOH vs Fe are enough to promote the formation of the (N 4 )Fe III (OOH) species (Figure S23). Interestingly, the time traces for the formation of the (N 4 )Fe III (OOH) intermediates show that the rate depends on the concentration in AcOH for low [AcOH] but appears independent for high concentrations. As revealed by the initial slopes of the time traces for [(1)Fe II (OTf)](OTf) and [(2)Fe II (OTf)](OTf), an obvious increase of the formation rate is observed with increasing amounts of AcOH for 2, 5, 10, 15 and 20 eq. AcOH while the rate is unchanged for larger amounts of AcOH (Figures 6A,6C). For [(3)Fe II (OTf)](OTf), a similar evolution is observed but the differences between the time traces are less clear (Figure 6E). These observations suggest that the formation of the (N 4 )Fe III (OOH) species follows saturation kinetics depending on [AcOH]. Plot of the pseudo-first order rate constants as a function of [AcOH] confirm this assumption (Figures 6B,6D and 6F; Figures S24). Note that for [(3)Fe II (OTf)](OTf), the slight decrease in k observed when AcOH exceeds 20 eq. may reveal some inhibition of the formation of the (N 4 )Fe III (OOH) species by AcOH. [33] However, the difference in the k values (for 10 eq. ≤ AcOH ≤ 200 eq.) being close to the error bars, this potential inhibition effect was not considered in the following. In contrast, for the degradation of the Fe III (OOH) species, no dependence on [AcOH] is observed when comparing the initial slopes (Figure S25). The same behaviour was observed by Que et al. which, in their case, was interpreted by the inability of AcOH to interact with the Fe III (OOH) species. [32] Interestingly, the decay of the Fe III (OOH) intermediates seems to be sensitive to the electron-donating character of the ligand : the more donating the ligand, the faster the degradation.

Discussion.

The beneficial effect of AcOH on the catalytic efficiency of [(1)Fe II (OTf)](OTf), [(2)Fe II (OTf)](OTf) and [(3)Fe II (OTf)](OTf) has been observed above and the distribution in products suggests a switch in reactivity from, essentially, a radical-based reactivity in the absence of AcOH, to the involvement of metal-based oxidant in its presence. Addition of AcOH promotes the reaction of the iron complexes with hydrogen peroxide, rendering its accumulation possible in MeCN. Our investigations show that the acid binds the Fe II center by protonating/substituting a pyridine moiety. However, it does not clearly influence the depletion of the (N 4 )Fe III (OOH) intermediate. This latter point contrasts with the carboxylic acidassisted mechanism reported in the literature where it was observed that the decay of Fe III (OOH) intermediates supported by genuine tetradentate (N 4 ) ligands such as TPA or BPMEN is promoted by the addition of acetic acid. [34] In this mechanism, binding of acetic acid to the metal center assists in the heterolytic O-O cleavage of the Fe III (OOH) by protonation of the distal Oatom, thereby facilitating the release of water and the generation of a formal Fe V (O)(OAc) species (Scheme 2, dashed frame). [35][36][37] Thus, the non-dependence of the (N 4 )Fe III (OOH) on AcOH we observe (i) suggests that the protonation of the distal O-atom of the hydroperoxo ligand is not favoured, and (ii) points towards a specific role played by the pending pyridine moiety in our systems. Based on our observations, a possible mechanism is shown on Scheme 2, plain frame. Coordination of AcOH is accompanied by protonation/dissociation of a pyridine group to form a (N 4 )Fe II (OAc) complex. From there, generation of the Fe III (OOH) intermediate proceeds via a two-step process. [10,16] Oxidation by, formally, 0. [38] which cannot play the same role. In this mechanism, the initial equilibrium (coordination of AcOH) is fast compared to the formation of the Fe III (OOH) species, which accounts for the saturation behaviour observed. The kinetic model for this mechanism is given in the Supporting Information (Scheme S1). Interestingly, it provides association constants (1/K D ) which depend on the electronic properties of the ligand. The binding of AcOH is indeed improved when the number of 4methoxy-3,5-dimethylpyridines increases (1 < 2 < 3), thus when the basicity of the ligand increases. This observation is consistent with the concerted proton transfer/pyridine dissociation upon AcOH coordination (Scheme S1, step (i)). Additionally, the rate constants for the formation of the Fe III (OOH) species decrease when the number of 4-methoxy-3,5-dimethylpyridines increases (k 2 obs decrease in the order 1 > 2 > 3). When going from 1 to 3, the probability to decoordinate a more basic 4-methoxy-3,5dimethylpyridine is more and more important. Thus, the conjugated dangling pyridinium is less and less acid and less and less prone to transfer its proton to the neighbouring hydroxo for releasing water and forming the Fe III Adapted from Olivo et al. [39] Considering the decay of the (N 4 )Fe III (OOH) species, while it does not obviously depend on AcOH, it appears to be accelerated when the electron-donating properties of the ligand increase (compare the initial slopes for the decay for the 3 complexes in Figure S25). The formation of the expected high-valent iron-oxo species may thus be favoured with ligand 3, which is in line with the catalysis results given in Table 1. While formation of an Fe V (O) oxidizing species could initially be expected, the distribution in products is not acute enough to support it. [39] Finally, AcOH helps in making a better use of the hydrogen peroxide oxidant by the Fe II precursors. The (N 4 )Fe II [40] No attempt has been made at this time to trap this species.

Conclusion

Three novel nonheme Fe II complexes with electron-rich pentadentate ligands were used as catalysts for the oxidation of small molecules by hydrogen peroxide. The distribution of products formed suggests that these ligands are not enough electron donating to promote the O-O heterolytic cleavage of the oxidant in order to generate selective Fe V (O) species. Using acetic acid results in a significant increase of the efficiency of these catalytic systems. This is due to the binding of AcOH to the metal center which leads to the protonation/deccordination of a pyrydine moiety. The resulting (N 4 )Fe II (OAc) species are more reactive towards the oxidant to yield (N 4 )Fe III (OAc)(OOH) intermediates. In contrast to the well-known carboxylic acidassisted mechanism, there is no proton available to assist in the heterolytic O-O cleavage since it is located on the pending pyridinium, which is a much weaker acid than coordinated carboxylic acid. Thus, we propose that the (N 4 )Fe III (OAc)(OOH) intermediates formed under these conditions evolve into (N 4 )Fe IV (OAc)(O), which appear more efficient than the usual (N 4 )Fe IV (O) and (N 5 )Fe IV (O).

Experimental Section

Experimental procedures and additional characterization data are provided in the Supporting Information.

Scheme 1 .

 1 Scheme 1. Structures of the ligands mentioned in this article.

  centers (Supporting Information, Table

[( 1 )

 1 Fe II (OTf)](OTf), [(2)Fe II (OTf)](OTf) and [(3)Fe II (OTf)](OTf) is, as expected, displaced by MeCN upon dissolution. If triflato remained bound, the charge of the complexes would remain +1 and the potential values would be much more affected.

Figure 1 .

 1 Figure 1. ORTEP diagrams of the molecular cations [(1)Fe II (OTf)] + (left), [(2)Fe II (OTf)] + (middle) and [(3)Fe II (OTf)] + (right). Hydrogen atoms have been omitted for clarity.

Figure 2 .

 2 Figure 2. UV-visible spectra of 1 mM solutions in MeOH at RT of [(1)Fe II (OTf)](OTf) (A), [(2)Fe II (OTf)](OTf) (B), and [(3)Fe II (OTf)](OTf) (C), in the presence of 100 eq. H2O2.

Figure 4 .

 4 Figure 4. UV-visible spectra of 1 mM solutions in MeCN at RT of [(1)Fe II (OTf)](OTf) (A), [(2)Fe II (OTf)](OTf) (B), and [(3)Fe II (OTf)](OTf) (C), in the presence of 100 eq. H2O2; and [(1)Fe II (OTf)](OTf) (D), [(2)Fe II (OTf)](OTf) (E), and [(3)Fe II (OTf)](OTf) (F), in the presence of 100 H2O2 and 200 AcOH.

Figure 5 .

 5 Figure 5. X-band EPR spectra at 90 K of solutions in MeCN of [(1)Fe II (OTf)](OTf) (A), [(2)Fe II (OTf)](OTf) (B), and [(3)Fe II (OTf)](OTf) (C), in the presence of 100 eq. H2O2; and [(1)Fe II (OTf)](OTf) (D), [(2)Fe II (OTf)](OTf) (E), and [(3)Fe II (OTf)](OTf) (F), in the presence of 100 H2O2 and 200 AcOH.

Figure 6 .

 6 Figure 6. Time traces for the formation of the Fe III (OOH) species generated by reaction of [(1)Fe II (OTf)](OTf) (A), [(2)Fe II (OTf)](OTf) (C), and [(3)Fe II (OTf)](OTf) (E), with100 eq. H2O2 and various amounts of AcOH, in MeCN at RT. Plots of the rate constants for the formation of the Fe III (OOH) intermediates vs AcOH for [(1)Fe II (OTf)](OTf) (AB, [(2)Fe II (OTf)](OTf) (D), and [(3)Fe II (OTf)](OTf) (F). Fiting to a saturation kinetic behaviour is represented by the red lines.

«

  carboxylic acid-assisted mechanism » O-O activated by the strong coordinated AcOH no assistance in O-O cleavage by the weak acid

  Fe II (OTf)](OTf) are very similar. For [(1)Fe II (OTf)](OTf), two sets of resonances are observed at g = 2.32, 2.12, 1.93 and g = 2.21/2.18, 2.15, 1.97, whereas for [(2)Fe II (OTf)](OTf), they are observed at g = 2.34, 2.12, 1.93 and g = 2.22/2.19, 2.15, 1.97 are observed. The sets of the more intense, less rhombic, resonances are characteristic of low spin Fe III (OOH), with two different g max values revealing either the presence of two rotation isomers of (N 5 )Fe III (OOH) species, as frequently observed for such species (Figure

	1	[(1)Fe II (OTf)](OTf)	43.1	37.1	3.7	4.9	26.0	0.19
	2	+ 200 AcOH	42.5	64.4	3.5	37.7	20.0	1.89
	3	[(2)Fe II (OTf)](OTf)	41.5	37.0	3.5	3.7	23.2	0.16
	4	+ 200 AcOH	42.6	49.9	3.4	39.3	32.5	1.21
	5	[(3)Fe II (OTf)](OTf)	35.1	64.0	4.2	18.2	24.8	0.73
	6	+ 200 AcOH	37.8	80.5	4.2	47.4	41.1	1.15
	The	spectra	observed	for	[(1)Fe II (OTf)](OTf)	and
	[(2)

[a] Fe / H2O2 / anisole : 1 / 20 / 3000; Fe / H2O2 / cyclooctene: 1 / 20 / 800; Fe / H2O2 / cyclohexane : 1 / 20 / 800; all assays were performed under aerobic conditions.

[b] The other possible products of anisole oxidation, i.e. Phenol, p-MeO-phenol, m-MeO-phenol, were not detected. [c] -ol / -one = cyclohexanol / cyclohexanone. Figure 3. X-band EPR spectra at 90 K of solutions in MeOH of [(1)Fe II (OTf)](OTf) (blue), [(2)Fe II (OTf)](OTf) (red), and [(3)Fe II (OTf)](OTf) (green), in the presence of 100 eq. H2O2.

  5 eq. H 2 O 2 , yields an (N 4 )Fe III (OAc)(OH) with an internal H-bond between the protonated pyridine and an exogenous ligand. Reaction with an extra H 2 O 2 molecule to form the (N 4 )Fe III (OOH) intermediate is then promoted due to the presence of water as a good leaving group, which is easily released under these conditions. To account for the AcOHindependant decay of the (N 4 )Fe III (OOH) intermediate, we propose that the proximal proton of the incoming H 2 O 2 is Hbonded between the remote pending pyridine group and the acetato ligand, more nucleophilic than the distal O-atom of the hydroperoxo. Thus, this proton is not available to protonate the distal O-atom and assist in the O-O cleavage. By comparison with the carboxylic acid-assisted mechanism where the protonation of the distal O-atom is ensured by acetic acid whose acidity is significantly enhanced upon coordination, the neighbouring pyridinium in our complexes is a weak acid (pK a ≥ ca.13.3 in MeCN)

  (OAc) complex formed reacts more easily with H 2 O 2 to generate a (N 4 )Fe III (OAc)(OOH) intermediate. Our observations indicate that the decay of this latter species yields a more reactive oxidizing species, which we tentatively formulate as (N 4 )Fe IV (OAc)(O), as reported by McKenzie et al. for a Fe III (OOH) supported by a tetrazacarboxylato ligand.

Acknowledgements

This work was financially supported by the ANR (LABEX CHARMMMAT, ANR-11-LABX-0039; and ANR-22-CE07-0037-01).


Juan F Torres

Collin H Oi

Ian Moseley

Nabila El-Sakkout

Brian J Knight

Jason Shearer
email: jshearer@trinity.edu

Ricardo García-Serres
email: ricardo.garcia@cea.fr

Joseph M Zadrozny
email: joe.zadrozny@colostate.edu

Leslie J Murray
email: murray@chem.ufl.edu

Dinitrogen Coordination to a High Spin Diiron(I/II) Species

Keywords: Electronic structure, Mixed-valent compounds, Multiiron complexes, Nitrogen fixation Converting

Dinitrogen coordination to iron centers underpins industrial and biological fixation in the Haber-Bosch process and by the FeM cofactors in the nitrogenase enzymes. The latter employ local high spin metal centers; however, iron-dinitrogen coordination chemistry remains dominated by low valent states, contrasting the enzyme systems. Here, we report a high-spin mixed-valent cis-(μ-1,2dinitrogen)diiron(I/II) complex [(FeBr) 2 (μ-N 2 )L bis ] -(2) where [L bis ] -is a bis(β-diketiminate) cyclophane. Field-applied Mössbauer spectra, dc and ac magnetic susceptibility measurements, and computational methods support a delocalized S= 7 / 2 Fe 2 N 2 unit with D=-5.23 cm -1 and consequent slow magnetic relaxation.

Earth. However, scission of dinitrogen is a kinetically-limited reaction. [1] The Haber-Bosh process (HB) for industrial production of NH 3 employs the iron-based Mittasch catalyst and high temperatures and pressures to achieve reductive cleavage of N 2 . [2] Contrastingly, nitrogenase enzymes in biological systems effect N 2 reduction to NH 3 under ambient conditions utilizing Fe 7 M (M=Mo, V or Fe) cofactors with local high-spin Fe centers. [3] Whereas iron reactive sites in the Mittasch catalyst are predominantly in reduced states, [2] the nitrogenase cofactors are proposed to employ a cluster with minimal, if any, low valent iron character for N 2 conversion to NH 3 . [4][5][6][7] However, N 2 coordination to high-spin Fe centers remains dominated by low valent states. [1] Motivated by these observations, we explored reduction of a diiron(II) complex supported by a dinucleating cyclophane in the presence of dinitrogen, resulting in the most oxidized dinitrogen-bridged iron complex with local high spin centers to date. This complex also displays the cis-μ-1,2 coordination mode, which remains rare in diiron chemistry with only two other examples reported (Figure 1). Notably, prior examples were obtained for highly-reduced compounds or using strong-field ligands contrasting the ligand field imposed here. [8,9] Previously, trinuclear 3d metal complexes of a tris(β-diketimine) cyclophane (H 3 L Et/Me ) were shown competent for N 2 activation. [10][11][12] Analogous to the synthesis of H 3 L Et/Me , Schiff base condensation of 1,3-bis(aminomethyl)-2,4,6-triethylbenzene with the monoketal of acetylacetone affords H 2 L bis (Scheme S2), which can be deprotonated and metalated with FeBr 2 to give Fe 2 Br 2 L bis (1) (92% yield). The solid-state structure of 1 is analogous to reported (di-μ-halido)diiron(II) cores and the Fe 2 (μ 3 -Br)(μ-Br) fragment of Fe 3 Br 3 L Et/Me (Figure S17). [13] Complex 1 is distinct from previous reports, however, in the dihedral angle of 150.6(1)° between the β-diketiminate (or BDI) planes, which is more acute than analogous aggregates of mononuclear metal complexes.

Reaction of Fe 2 Br 2 L bis with 1 equiv. of [K(18-crown-6)(THF) 2 ](C 10 H 8 ) in THF at -35 °C under Ar affords a green solution that decomposes over several hours at low temperature. Attempts to crystallize this species were unsuccessful, yielding red crystals of the formally diiron(II) complex [(FeBr) 2 (μ-Br)L bis ] -(Br 3 ) (Figure S18). In situ reaction of the reduced green species with dinitrogen rapidly afford the dark red compound [K(18-crown-6)(THF) 2 ] [(FeBr) 2 (μ-N 2 )L bis ] (2-crown) in excellent yield (> 90% by 1 H NMR), albeit only 19% was isolated as crystalline solid. The N-N stretching mode at 1932 cm -1 shifts to 1870 cm -1 when 15 N 2 is used, consistent with the predicted value (1866 cm -1 ) and confirming N 2 incorporation (Figure S11). Comparable results were obtained when complex 1 was reduced at -35 °C under a N 2 atmosphere (Scheme 1). Compound 2-crown decomposes above 40 °C and under reduced pressure, presumably from N 2 dissociation based on loss of the assigned N-N stretching mode in IR spectra.

Despite numerous attempts, we were only able to determine a connectivity structure of 2-crown by single-crystal X-ray diffraction (SCXRD) (Figure S19). Employing [K(crypt-222)][C 10 H 8 ] as the reductant, however, yields 2-crypt of which single crystals of sufficient quality for a structure solution by SCXRD could be grown. The [(FeBr) 2 (μ-N 2 )L bis ] -complex ion is isostructural in 2-crown and 2-crypt with no apparent effect of the non-interacting countercation, based on the connectivity structure solutions (Figure S21), IR spectra (Figures S10 andS12), paramagnetic 1 H NMR spectra (Figures S4 andS6), and calculated solution magnetic moments (μ eff =7.64 μ B and 7.77 μ B for 2-crown and 2crypt, respectively). Each Fe center in 2-crypt resides in a pseudo-tetrahedral coordination environment with τ 4 values of 0.94 and 0.93 for Fe1 and Fe2, respectively (viz. τ 4 =1 for tetrahedral). [14] The dinitrogen ligand in 2-crypt adopts a cis-μ-1,2 mode with Fe-N-N angles of 167.0(2) and 164.8(2) (Figure 2). The diiron core undergoes substantial structural rearrangement from 1 to 2 (viz. Δd Fe-Fe > 1 Å, Figure S28). Notably, 2-crypt has a longer Fe-Fe distance (4.7562(4) Å) than one predicted for the observed Fe-N-N bond angle based on structures reported (Figure S22).

Few mixed-valent dinitrogen-bridged iron complexes have been characterized, [15][16][17][18] with 2 being the most oxidized to date. The zero-applied field 80-K Mössbauer spectrum of 2-crown displays a single quadrupole doublet, implying valence delocalization across both iron centers (Figure 3a). The isomer shift for the metal centers in 2-crown (δ = 0.81 mm/s) is comparable to those observed for tetrahedral high-spin Fe(II) BDI complexes, with only a slight decrease with respect to 1 (δ 1 = 0.86 mm/s and δ 1 = 0.90 mm/s, Figure S25). Variable temperature Mössbauer spectra (Figure S26) reveal slow relaxation behavior with intermediate relaxation behavior from 8-20 K and features from the fast and slow relaxation limits persist throughout the swept temperature range (2.35 K to 30 K). Applied-field spectra below 8 K or above 20 K were simulated as composites of fast-and slow-relaxing fractions calculated with the same S = 7 / 2 spin Hamiltonian (Figure 3b). Simulations of data recorded between 5.8 and 35 K provide estimates for the zero-field splitting parameters of D = -5 cm -1 (±1 cm -1 ) and 0.2 < E/D < 1 / 3 (Figure S27). These values are in good agreement with those determined from magnetic susceptibility measurements (vide infra and Table S2).

Variable-temperature direct-current (dc) magnetic susceptibility (χ M T) measurements corroborate the ground-state spin observed for 2-crown in the Mössbauer data (Figure 3c). At room temperature, the value of χ M T is 8.12 cm 3 K/mol, which remains relatively constant with decreasing temperature to 50 K. The χ M T value at room temperature compares well with the effective moment measured by Evans method and that expected for an S = 7 / 2 system (7.85 cm 3 •K/mol, g = 2.00) rather than weakly coupled S = 2 Fe(II) and S = 3 / 2 Fe(I) ions (~4.9 cm 3 •K/mol with g = 2.00). [19] Below 50 K, χ M T values decrease rapidly to 5.83 cm 3 K/mol at 5 K due to zero-field splitting. Modeling this temperature dependence of χ M T yields best fit values consistent with a rhombic S = 7 / 2 system of g x = 2.05, g y = 2.10, g z = 1.97, and zero-field splitting terms of D = -5.23 cm -1 and |E| = 1.53 cm -1 (see SI for additional fitting details). The magnitude of D is in the expected range for N 2 bridged dinuclear weak-field ligated Fe complexes. [17] Alternating current susceptibility measurements of 2-crown at zero-applied field reveal a frequency-dependent out-of-phase ac susceptibility (χ M ") signal near 11 Hz below 6.0 K (Figure 3c inset,S24). Applying a static magnetic field shifts the peaks to lower frequency, evidencing a slowing of the magnetic relaxation process and possible disruption of a contribution from a tunneling relaxation mechanism. [20] Relaxation times (τ) determined with a Debye model support Arrhenius-like behavior with an effective activation energy for relaxation (U eff ) of 16.3 cm -1 and an attempt time of 1.9 × 10 -6 s (Figure S23). [21] Although the limited temperature range investigated here prevents conclusive assignment of the relaxation processes present, these observations show slow magnetization dynamics arising from a negative D parameter, consistent with the χ M T and Mössbauer data. [20] Turning to DFT calculations, the geometry optimized structure of [(FeBr) 2 (μ-N 2 )L bis ] - using BP86/def2-tzvp with Grimme's D3 dispersion correction and S = 7 / 2 ground state agrees with the crystallographic data for 2-crypt and 2-crown, including the cis-μ-1,2 coordination mode of the dinitrogen ligand (Table S4). Vibrational analysis (B2PLYP/ def2-tzvp) yields an N-N stretching frequency of 1944.7 cm -1 , which agrees with the experimental values of 1932 cm -1 . Similarly, calculated Mössbauer parameters of δ calc =0.81 mm/s and ΔE Q,calc =2.76 mm/s are in excellent agreement with the experimental values (viz. δ exp =0.81 mm/s and ΔE Q,exp =2.14 mm/s). The calculated zero-field splitting parameters reproduce the negative value of D and E/D ~0.3; however, the magnitudes of the zerofield splitting parameters were approximately twice that derived from the Mössbauer and magnetic susceptibility data. From the B2PLYP/def2-tzvp/CP(PPP) calculations, the seven singly occupied orbitals are predominantly Fe 3d in character with Fe 3d compositions ranging from 78-98% (Figure S29). The extent of activation of the N 2 ligand is readily reflected in the presence of only one frontier molecular orbital of appreciable Fe d π to N 2 π*, which is predominantly N 2 π* in character.

The electronic structure of 2 differs from the three-spin model and the ferromagnetically coupled metal centers model commonly used to explain the magnetic properties of M-N 2 -M fragments. [22] The absence of significant unpaired-spin contribution on the N 2 ligand suggests that the observed delocalization is likely mediated by a double-exchange mechanism analogous to that reported for mixed-valent divanadium complexes. [23] Electron delocalization across a neutral bridging dinitrogen was previously proposed to explain the magnetic properties of a covalent μ-1,2-N 2 dicobalt(I) BDI complex. [24] Complex 2 demonstrate that cis-dinitrogen bridged iron complexes can display valence delocalization (Figure S28), affording Mössbauer parameters nearly indistinguishable from formal iron(II) centers, and suggest that such reduced state can be present in the enzymes cofactors.

In conclusion, we report the synthesis and characterization of the first S= 7 / 2 mixed-valent cis-(μ-1,2-dinitrogen)diiron(I/II) complex. The observed N 2 -coordination mode remains rare for μ-N 2 -dirion complexes due to the expected decrease in π-backbonding vs. a linear mode. Here, [L bis ] 2-limits the accessible dihedral angles between the two BDI planes resulting in the preferred bent mode. This compound demonstrates that polynuclear Fe species are competent to bind N 2 with electron loadings, formal oxidation states, local spin states, and coordination modes feasible for the nitrogenase enzyme system. b) Mössbauer spectra of 2-crown recorded at 2.35 K (top spectrum) or 5.8 K in variable magnetic fields applied parallel to the direction of the gamma-rays. The simulation (solid lines) was performed assuming only one iron site in the slow relaxation limit, with admixture of 18% of the absorbers in the fast relaxation limit. Fitting parameters are given in table S3. c) Variable temperature, dc susceptibility data for 2-crown taken under a 1000 Oe applied field. The black line is the best fit obtained, full fitting details can be found in the SI. (inset) Ac susceptibility data measured at 1.8 K under zero applied field (red) and a 1000 Oe applied field (blue). Reaction scheme for synthesis of complex 2-crown. Yield evaluated by 1 H NMR in parenthesis.

Reported cis-(μ-1,2-dinitrogen)diiron complexes

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2. Crystal structure of the anion [(FeBr) 2 (μ-N 2 )L bis ] -from complex 2-crypt with the thermal ellipsoids at the 65% probability level (left). All H atoms are omitted for clarity. C, N, Br, and Fe atoms depicted as gray, blue, pink, and orange ellipsoids, respectively. Core of [(FeBr) 2 (μ-N 2 )L bis ] -anion from 2-crypt (right). Selected bond lengths (Å): Fe1-Fe2 4.7562(4), N5-N6 1.164(6). Selected angles (°): Fe1-N6-N5 164.8(2), Fe2-N5-N6 167.0(2).

Figure 3 .

 3 Figure 3. a) Zero field Mössbauer spectrum of 2-crown recorded at 80 K. The solid line represents a simulation with δ = 0.81 mm s -1 , ΔE Q = 2.14 mm s -1 , and Γ = 0.44/0.48 mm s -1 .b) Mössbauer spectra of 2-crown recorded at 2.35 K (top spectrum) or 5.8 K in variable magnetic fields applied parallel to the direction of the gamma-rays. The simulation (solid lines) was performed assuming only one iron site in the slow relaxation limit, with admixture of 18% of the absorbers in the fast relaxation limit. Fitting parameters are given in tableS3. c) Variable temperature, dc susceptibility data for 2-crown taken under a 1000 Oe applied field. The black line is the best fit obtained, full fitting details can be found in the SI. (inset) Ac susceptibility data measured at 1.8 K under zero applied field (red) and a 1000 Oe applied field (blue).

  Scheme 1.

Angew Chem Int Ed Engl. Author manuscript; available in PMC 2023May 23.

Acknowledgements

Authors thank Dr. S. N. MacMillan and A. L. Rheingold for the XRD measurements and the National Institutes of Health (R01-GM123241: J.F.T., C.H.O, B.J.K, L.J.M.), Labex ARCANE and CBH-EUR-GS (ANR-17-EURE-0003: N.E.S., R.G.S.), National Science Foundation (CHE-1854854: J.S.), and the National Science Foundation Graduate Research Fellowship Program (006784-00002: I.M., J.M.Z.).

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.


Magali Batteau

Florian Delrue

René De Vaumas

Karine Faure

Développement d'une méthode LC x SFC -MS/MS hors ligne pour l'analyse de phytostérols issus des microalgues

Industrie des nutraceutiques

• Recherche de nouvelles sources de lipides bioactifs

Culture des microalgues



  même rétention LC  même fragmentation  même chaine latérale  ≠ positions de la double liaison (noyau) 1 seul isomère répond à ces critères Développement d'une méthode LC x SFC -MS/MS hors ligne pour l'analyse de phytostérols issus des microalgues Magali Batteau 1 , Florian Delrue 2 , René de Vaumas 3 , Karine Faure 1 1 Institut des Sciences Analytiques, CNRS UMR 5280 5, rue de la Doua, 69100 Villeurbanne 2 MicroAlgae Processes Platform-CEA, CEA Tech Région Sud, 13108 Saint-Paul-lez-Durance 3 Extrasynthèse, Impasse Jacquard, 69730 Genay Développer une méthode selon les critères suivants : • Séparation de molécules neutres • Séparation de composés analogues / isomères de position (grand degré d'orthogonalité et sélectivité suffisante) • Mise en oeuvre de la détection MS/MS LC x SFC -MS/MS hors ligne Objectif Contexte Les phytostérols sont des composés analogues à haute valeur ajoutée qui se différencient par le nombre et la position d'insaturations dans le cycle ou sur la chaine latérale. Cette grande similitude structurelle avec un certain nombre d'isomères de position rend la séparation et l'identification relativement difficile.

  ± 0.19 min (modulation) • 2 tr ± 4 s (3) • Exact mass ± 10ppm • Fragmentation MS/MS Analyse ciblée des standards par LC x SFC -MS/MS stigmasterol (5) & spinasterol (6)  même rétention LC  même fragmentation  Même chaine latérale  ≠ positions de la double liaison (noyau) fucosterol (2) & stigmasterol (5)  même rétention SFC  ≠ fragmentations  ≠ chaine latérale  même position de la double liaison (noyau) -MS hors ligne -somme des EIC Luna Omega C18 150 mm x 4 6 mm, 3 µm, A: ACN B: IPA, 20°C, gradient 10-12,8% B en 5,53 min, 1,8 mL/min, V inj 60 µL, collecte toutes les 0,19 min Torus 1-AA, 100 mm x 3 mm, 1,7 µm, 40°C, BPR 140 bar, gradient 5-12% [MeOH] 5,44min, 1,2 mL/min V inj 15 µL, 46 fractions dans heptane APCI + , 50-1000 m/z, 1 spectre/sec, tension capillaire 1.0 kV, courant Corona 4 µA, fragm. 100 V, collision 20eV, make-up MeOH LC x SFC-MS hors ligne -BPC Luna Omega C18 150 mm x 4 6 mm, 3 µm, A: ACN B: IPA, 20°C, gradient 10-12,8% B en 5,53 min, 1,8 mL/min, V inj 60 µL, collecte toutes les 0,19 min Torus 1-AA, 100 mm x 3 mm, 1,7 µm, 40°C, BPR 140 bar, gradient 5-12% [MeOH] 5,44min, 1,2 mL/min V inj 15 µL, 46 fractions dans heptane APCI + , 50-1000 m/z, 1 spectre/sec, tension capillaire 1.0 kV, courant Corona 4 µA, fragm. 100 V, collision 20eV, make-up MeOH Analyse non ciblée Chlorophycea Scenedesmus par LC x SFC -MS/MS

molécules cibles, 7 isomères de ces molécules et de 4 autres stérols

		Phytostérols
		• Régulation du taux de
	• Accès à des		cholestérol
	phytostérols peu	• Fabrication de
	présents dans les		nombreuses hormones
	plantes terrestres		(stéroïdes, vitamine D)
		Echantillon Fraction 1 D	RPLC SFC => Stratégie	UV UV	Collecte de fractions MS	Conclusion Détection de 7 Préparation d'échantillon Séchage Saponification Résidu


Nathan Szwarcberg

Tom Colinot

Christophe Vergez

Michaël Jousserand

Amplitude-dependent modal coefficients accounting for localized nonlinear losses in a time-domain integration of woodwind model

 were carried out. Simulations show that the model reproduces these experimental results qualitatively and quantitatively.

Introduction

Sound synthesis by modal decomposition is based on an input impedance measurement, which captures the passive acoustical response of a real instrument. This method thus has the advantage of capturing the acoustical subtleties that differentiate two clarinets by means of a single measurement. Other methods, such as delay lines [START_REF] Guillemain | Real-time synthesis of clarinet-like instruments using digital impedance models[END_REF], waveguides [START_REF] Paul | An acoustic analysis of single-reed woodwind instruments with an emphasis on design and performance issues and digital waveguide modeling techniques[END_REF][START_REF] Mignot | Réalisation en guides d'ondes numériques stables d'un modèle acoustique réaliste pour la simulation en temps-réel d'instruments à vent[END_REF] or spatial discretization [START_REF] Bilbao | Numerical sound synthesis: finite difference schemes and simulation in musical acoustics[END_REF][START_REF] Ernoult | Full waveform inversion for bore reconstruction of woodwind-like instruments[END_REF] would require more effort on the geometrical description of the resonator to render such acoustical subtleties.

Furthermore, sound synthesis by modal decomposition requires little RAM compared to such other methods, which can be beneficial for embedded devices. Indeed, in modal decomposition synthesis, the temporal integration scheme requires to keep only a few previous iterations (precisely two for [START_REF] Guillemain | Real-time synthesis of clarinet-like instruments using digital impedance models[END_REF]) to compute a new one. For the delay line synthesis, it is necessary to keep in memory all the temporal iterations during a round trip, i.e. during 2LF s /c 0 iterations for an academic cylindrical resonator without lateral holes. Although this is not a problem for most embedded processors, fine modeling of viscothermal losses in the waveguide formalism also requires additional computations involving fractional derivatives [START_REF] Bilbao | Finite difference time domain simulation for the brass instrument bore[END_REF]. Finally, the ODE formalism is particularly well suited to the bifurcation analysis, which is not the case of the PDE [START_REF] Bilbao | Finite difference time domain simulation for the brass instrument bore[END_REF] or of the DDE formalisms [START_REF] Guillemain | Digital synthesis models of clarinet-like instruments including nonlinear losses in the resonator[END_REF].

The input impedance is a measure of the linear frequency response of the resonator, for a low-amplitude excitation. However, in the case of woodwind instrument playing, the measured acoustic pressure and velocity in the resonator can be very high. For instance, [START_REF] Bergeot | Response of an artificially blown clarinet to different blowing pressure profiles[END_REF] measure a pressure in the mouthpiece of 4 kPa, i.e. 163 dB for a monochromatic wave. According to [START_REF] Chaigne | Acoustics of musical instruments[END_REF]Chap. 8.4.5], from an acoustic speed of roughly 1 m/s, jet separation phenomena appear. At the level of a geometrical discontinuity, such as the open end of a pipe [START_REF] Disselhorst | Flow in the exit of open pipes during acoustic resonance[END_REF] or a lateral orifice [START_REF] Ingård | Acoustic circulation effects and the nonlinear impedance of orifices[END_REF], vortices are observed. A part of the kinetic energy of the jet is absorbed by these vortices and is dissipated as heat by friction. These losses are accounted for through a nonlinear relationship derived from Bernoulli's law, as demonstrated by the implementation in waveguide modeling or delay lines of [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF], [START_REF] Guillemain | Digital synthesis models of clarinet-like instruments including nonlinear losses in the resonator[END_REF], [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF] and [START_REF] Taillard | Theoretical and experimental study of the role of the reed in clarinet playing[END_REF].

Following [START_REF] Ingård | Acoustic circulation effects and the nonlinear impedance of orifices[END_REF], the works from Atig [2, 3] and Dalmont [START_REF] Dalmont | Experimental Determination of the Equivalent Circuit of an Open Side Hole: Linear and Non Linear Behaviour[END_REF][START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF] model localized nonlinear losses through a resistive impedance Z t . This impedance is in series with the radiation impedance Z R , and depends on the amplitude of the acoustic velocity v 0 at the location of the discontinuity:

Z t = 4c d 3π |v 0 | c 0 Z c , (1)
where Z c = ρ 0 c 0 /S is the characteristic impedance of the medium for plane waves, and c d ∈ [0; 3] is a parameter depending on the geometry of the termination. The larger the radius of curvature at the output, the smaller this coefficient is. The coefficient c d is difficult to predict theoretically, and easier to determine experimentally [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF]. As shown by [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF], nonlinear losses have a significant influence on the playing range of a clarinet, hence models of sound production in woodwinds should include such effects. Models of woodwind instruments taking into account localized nonlinear losses at the end of the resonator have never been developped in the framework of modal synthesis. This is the main contribution of this work.

First, a method proposed by [START_REF] Diab | Nonlinear broadband time-domain admittance boundary condition for duct acoustics. application to perforated plate liners[END_REF] to account for nonlinear losses by modal decomposition is presented. It is then applied to a cylindrical tube with nonlinear losses located at the open end. From this resonator, a selfoscillating clarinet-like system is defined and simulated by time integration. These simulations are finally compared to the experimental results published by [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF] and [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF].

2 Method of modal decomposition accounting for localized nonlinear losses

The definition of a "nonlinear impedance" (i.e. adjusting the impedance so that the link between acoustic flow and pressure stays valid in nonlinear conditions) opens the way to consider localized nonlinear losses by modal decomposition. In [START_REF] Diab | Nonlinear broadband time-domain admittance boundary condition for duct acoustics. application to perforated plate liners[END_REF], the evolution of the surface impedance of perforated plates is computed by temporal simulation for a broadband excitation, with respect to the RMS amplitude of the acoustic velocity at the level of the hole, noted v RM S . From the decomposition of the impedance in the linear domain into a sum of N modes, two methods are studied for increasing values of v RM S : the interpolation of the impedance, on one hand, and the regression of the poles and complex residues (s n , C n ) on the other hand. This second method will be considered hereafter.

Nonlinear effects are taken into account by allowing the poles and residues to vary with respect to v RM S . In the Laplace domain, the modal decomposition of the input impedance is written:

Z(s, v RM S ) = Z c N n=1 C n (v RM S ) s -s n (v RM S ) + C * n (v RM S ) s -s * n (v RM S ) , (2)
where s is the Laplace variable and • * denotes the complex conjugation operation. The relationship between poles and residues and v RM S is assumed to be a rational fraction [START_REF] Diab | Nonlinear broadband time-domain admittance boundary condition for duct acoustics. application to perforated plate liners[END_REF]. In the present work, the approximation is lim-ited to a polynomial of degree N p :

C n (v RM S ) = C (0) n + Np k=1 C (k) n (v RM S ) k , (3)
s n (v RM S ) = s (0) n + Np k=1 s (k) n (v RM S ) k , (4)
where the index • (0) refers to the linear part of the modal coefficient. In [START_REF] Diab | Nonlinear broadband time-domain admittance boundary condition for duct acoustics. application to perforated plate liners[END_REF], the regression coefficients of the poles and residues are then adjusted on a nonlinear surface impedance model Z N L (v RM S ), given by [START_REF] Laly | Acoustical modeling of micro-perforated panel at high sound pressure levels using equivalent fluid approach[END_REF]. It is worth noting that in the present paper, the poles and residues are fitted on the dimensioned quantity of the RMS velocity. Therefore, the unit of coefficients

C (k) n and s (k) n are expressed in rad • s -1 • [m • s -1 ] -k .
The last step consists in computing v RM S , given by:

v 2 RM S (t) = 1 t t 0 v 0 2 (τ )dτ, (5)
where v 0 (t) = u 0 /S is the acoustic velocity at the discontinuity, of cross section S. This definition is rather impractical in the framework of a time-domain simulation. In order to obtain a state formalism, it is replaced by:

∂(tv 2 RM S ) ∂t = v 0 (t) 2 , (6)
which is equivalent. By considering the righthand side of Eq. ( 6) as an input for the ODE solver, tv 2 RM S can be computed, hence v RM S .

Application to a cylindrical tube with nonlinear losses at the open end

The impedance regression technique presented by [START_REF] Diab | Nonlinear broadband time-domain admittance boundary condition for duct acoustics. application to perforated plate liners[END_REF] is applied to the case of a closed-open cylinder of radius R and length L, which is a similar case study to [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF]. Viscothermal losses are taken into account in the propagation:

Γ(s) = s c 0 + (1 + j) η √ s R √ 2jπ , (7)
where η = 3 • 10 -5 s 1/2 . The boundary condition at x = L is given by the radiation impedance Z R , which includes in series the nonlinear impedance Z t defined by Eq. ( 1):

Z R = Z (lin) R + Z t ,
where

Z (lin) R = Z c jk∆l + 1 4 (kR) 2 , Z t = Z c v RM S (L, t) c 0 K N L , (8)
and jk = s/c 0 , ∆l ≈ 0.6R [7, Chap. 12.6.1.3], K N L = 4c d /(3π). In the following, v RM S (L, t) will be denoted as v RM S for reading comfort.

The input impedance is defined by:

z in = tanh ΓL + tanh -1 (z R ) , (9)
introducing the dimensionless notation

z • = Z • /Z c . A linear input impedance z (lin) in is also defined, such that z (lin) in = tanh ΓL + tanh -1 jk∆l + 1 4 (kR) 2 , (10)
which is equivalent to z in for v RM S = 0. Figure 1 illustrates the evolution of z in (computed through Eq. ( 9) and Eq. [START_REF] Colinot | Influence of the "ghost reed" simplification on the bifurcation diagram of a saxophone model[END_REF], in which the Laplace variable is substituted by j2πf ) as a function of the RMS velocity at the open end. It can be observed that taking into account nonlinear losses has a consequence on the amplitude, but not on the frequency of the resonance peaks. Figure 1 b shows that this impact is particularly accentuated on the first resonance peak: its amplitude is 43 % lower for v RM S = 24 m/s than in the case without nonlinear losses. The difference is reduced to 30 % for the second peak, and to 16 % for the sixth peak. Near the antiresonances, the negative relative discrepancies are high, since the impedance values are all close to zero. The absolute deviation remains very low. From the lack of frequency variation of the resonance peaks with v RM S , we could anticipate that nonlinear losses at the end of the pipe would rather impact dynamics than intonation.

Figure 1: Input impedance of a cylinder z in (computed through Eq. ( 9) and Eq. ( 10)) of length L = 64 cm and radius R = 8 mm for different values of the acoustic RMS velocity at the open end. The value of c d has been set to 13/9, which corresponds to a radius of curvature of 0.3 mm, according to [START_REF] Atig | Non-linéarité acoustique localisée à l'extrémité ouverte d'un tube. Mesure, modélisation et[END_REF]. From top to bottom, from left to right : modulus of the input impedance; relative gap between nonlinear and linear definition of z in ; detail view on the first peak; detail view on the fourth peak.

Modal decomposition of the nonlinear input impedance

In order to take into account nonlinear losses in temporal simulation, the modal coefficients of the input impedance must be determined.

In the same way that the nonlinear input impedance depends on the acoustic velocity at the open end, the coefficients C n and s n derived from the modal decomposition of z in also depend on v RM S . To formulate the relationship between the modal coefficients and v RM S , a minor adaptation of the method presented in [7, Chap. 5.5.3] is used. The definition of z R used in [START_REF] Chaigne | Acoustics of musical instruments[END_REF] is substituted by the expression given by Eq. ( 8), taking into account nonlinear losses at the end of the tube.

Expression of the poles s n

Eq. ( 9) can also be written as:

z in = sinh [Γ(s)L + h(s, v RM S )] cosh [Γ(s)L + h(s, v RM S )] , where h(s, v RM S ) = tanh -1 (z R ). (11)
The poles s n (v RM S ) are the solutions of cosh [ΓL + h] = 0, i.e.:

Γ(s n )L+h(s n , v RM S )-j (2n -1)π 2 = 0. ( 12
)
The solutions of Eq. ( 12 )/ℑ(s n )| is decreasing as n increases. This is signaled by the amplitude of peaks of higher index appearing less affected by nonlinear losses, as shown in Figure 1 b.

Expression of the residues C n

It remains to calculate residue C n (v RM S ). In the vicinity of a pole s n , the denominator D of z in can be written through a first-order series expansion of cosh:

D = (s -s n )D ′ (s n , v RM S ) where D ′ = (Γ ′ L + h ′ ) sinh (ΓL + h) , (13)
introducing notation

• ′ = ∂ • /∂s.
According to [7, Chap. 5.5.3], after application of the residues theorem, the modal decomposition of the input impedance for the n-th peak can be written as

Z in,n = P n (s) U(s) = Z c C n s -s n
, where

C n (v RM S ) = 1 Γ ′ (s n )L + h ′ (s n , v RM S ) . ( 14
)
The evolution of coefficients C n with respect to v RM S is plotted on Figure 2b. Real and imaginary parts of C n slightly decrease when v RM S increases. Following [START_REF] Diab | Nonlinear broadband time-domain admittance boundary condition for duct acoustics. application to perforated plate liners[END_REF], it remains to fit the modal coefficients with respect to v RM S , as in the example of Eq. (3).

Polynomial fitting of the modal coefficients

Figure 3a shows the mean relative fitting error for polynomials of different degrees, for the same data as in Figure 2. For linear regression, the mean relative error stays around 10 -7 for each mode n, both for C n and s n . Further simulations reveal that choosing an excessive values of c d tends to increase this error. For a tube with sharp edges at the open end, [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF] estimated a maximum value of c d = 2.8. For c d = 5, for instance, the mean relative error of linear regression is around 10 -5 . Although this error has increased, it remains very low. Moreover, Figure 3b shows that choosing an polynomial regression degree larger than 1 has no consequence on the error over z in between modal decomposition with fitted coefficients and the definition given by Eq. ( 9). This error is therefore inherent to the modal decomposition approximation (Eq. ( 2)) of the input impedance (Eq. ( 9)).

According to these previous results, in the rest of this article, the modal coefficients will both be fitted by a polynomial of order 1.

Application to the timeintegration simulation of a clarinet-like system

The cylindrical tube considered in section 3 is now treated as the resonator of a clarinetlike instrument. The physical model governing the self-sustained oscillations of the instrument is presented in section 4.1. It is then simulated by time integration, imposing a linear increase of the blowing pressure (crescendo) followed by a linear decrease (diminuendo). Details and results from the simulations are given in section 4.2.

Physical model for the selfsustained oscillations

The self-sustained system consists of three main parts: reed dynamics, reed channel and resonator. This last subsystem is enriched compared to the classical modal decomposition formalism [START_REF] Vincent Dubos | Theory of sound propagation in a duct with a branched tube using modal decomposition[END_REF][START_REF] Silva | Simulation of single reed instruments oscillations based on modal decomposition of bore and reed dynamics[END_REF][START_REF] Taillard | Modal analysis of the input impedance of wind instruments. application to the sound synthesis of a clarinet[END_REF] by the addition of nonlinear losses at the end of the pipe.

Reed dynamics

The dynamic behavior of the reed is modeled for its dimensionless displacement x(t) by the given by Eq. ( 9) and modal decomposition approximation z (modal) in defined by Eq. ( 2). This error is plotted with respect to the degree of the fitting polynomial of C n and s n . 3a), and its consequence on the discrepancy between modal decomposition and definition (9) on z in (Figure 3b). following equation :

1 ω 2 r ẍ(t) + q r ω r ẋ(t) + x(t) = p(t) -γ(t) + F c (x, ẋ), (15)
where ω r = 2π × 2200 rad/s is the reed resonance angular frequency [START_REF] Chatziioannou | Estimation of clarinet reed parameters by inverse modelling[END_REF] and q r = 0.4

is the reed damping [START_REF] Silva | Approximation formulae for the acoustic radiation impedance of a cylindrical pipe[END_REF]. γ(t) = P m (t)/p M is the dimensionless blowing parameter and p(t) = p (dim) (t)/p M is the dimensionless pressure at the input of the resonator. The beating pressure p M is the value of the blowing pressure P m for which the reed closes the reed channel, in quasistatic regime. F c (x, ẋ) is the contact force function. In the following, a model of "ghost reed" [START_REF] Colinot | Influence of the "ghost reed" simplification on the bifurcation diagram of a saxophone model[END_REF] is considered, i.e. F c = 0.

Reed channel

The characteristic of the flow u(t) through the reed channel is described by the following equation, according to [START_REF] Theodore | Operating modes of the clarinet[END_REF]:

u(t) = -λ ẋ(t) + ζ [x(t) + 1] + sgn γ(t) -p(t) |γ(t) -p(t)|, (16)
where λ = 5.5•10 -3 /c 0 is the reed flow parameter [START_REF] Dalmont | Nonlinear characteristics of single-reed instruments: Quasistatic volume flow and reed opening measurements[END_REF], ζ is the embouchure parameter, and γ(t) is the dimensionless blowing pressure.

The operator

[•] + = (• + | • |)/2
refers to the positive part function. For the simulations, the absolute value functions are regularized to smooth the irregularities of the quasi-static flow characteristic, without altering its behavior, according to [START_REF] Colinot | Numerical simulation of woodwind dynamics:investigating nonlinear sound production behavior in saxophone-like instruments[END_REF]. Thus, | • | → • 2 + η, with η = 0.001.

Resonator

The resonator is described in the frequency domain by its input impedance z in , under its modal decomposition form given by Eq. ( 2). The modal poles and residues s n and C n are chosen to be linearly dependent on v RM S , i.e.:

C n (v RM S ) = C (0) n + C (1) n v RM S , s n (v RM S ) = s (0) n + s (1) n v RM S .
The modal decomposition of z in allows to write the relation between the dimensionless pressure p and flow u in the temporal domain:

ṗn (t) -s n (v RM S )p n (t) = C n (v RM S )u(t),
where

p(t) = 2 N n ℜ(p n (t)). ( 17
)
The values of s n and C n must be updated at each time step by computing v RM S (L, t). The method for calculating the mean velocity at nonlinearity is detailed in the following section.

Computation of v RM S at each time step

To compute v RM S (L, t), the pressure at the termination L must be calculated. First, a linear problem is considered. The value of s n is therefore chosen for v RM S = 0 m/s. The modal components of the pressure are related to the total pressure at L through the following equation:

p(L, t) = 2ℜ N n p n (t)φ n (L) , where φ n (ξ) = cosh(Γ(s (0) n )ξ), (18)
and ξ ∈ [0, L] is the distance along the longitudinal axis of the resonator. For reading comfort, the expression p(t) will be reserved to denote p(0, t).

The dimensionless acoustic velocity at the open end is then calculated from its definition which is related to the pressure field through the dimensionless Euler equation:

∂p ∂ξ (ξ, t) = - 1 c 0 ∂v ∂t (ξ, t). (19)
The dimensionless acoustic velocity v(L, t) is thus obtained by numerical integration of the following equation :

v(L, t) = -2c 0 ℜ N n p n (t) dφ n dξ (L), where dφ n dξ (ξ) = Γ(s (0) n ) sinh Γ(s (0) n )ξ . ( 20
)
The RMS velocity is finally obtained by double time integration of Eq. ( 20), according to Eq. ( 6). Since the equations of the system characterizing the self-sustained oscillations are dimensionless, it is necessary to resize v RM S , because the modal coefficients C

(k) n and s

(k)
n presented in Eq. ( 3) have been regressed from the dimensioned velocity. To do so, v (adim) RM S is multiplied by p M /(ρ 0 c 0 ). In the following simulations, a value of p M = 8.5 kPa is set, according to experimental results from [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF].

Regarding the calculation of v(L, t), it is not numerically guaranteed that the mean value of v(L, t) remains zero. Consequently, the integral v(L, t) = t 0 v(L, t)dt may diverge. To avoid these divergence problems when integrating v(L, t) and v 2 (L, t), a short-memory term τ is added within each integral. Thus, v(L, t) is now defined by the following convolution product:

v(L, t) = t 0 v(L, u)e -t-u τ du, (21)
which is written, in the Laplace (L) domain, as:

L [v(L, t)] = L [ v(L, t)] 1 s + 1/τ . ( 22
)
By going back to the time domain, Eq. ( 21) becomes :

v(L, t) = -2c 0 ℜ N n p n (t) dφ n dξ (L) - 1 τ v(L, t). ( 23
)
The same method is applied for the computation of v RM S :

v 2 RM S = 1 τ t 0 e -t-u τ v 2 (L, u) du (24) ⇔ L v 2 RM S = 1 τ L v 2 (L, t) 1 s + 1/τ (25) ⇔ ∂(τ v 2 RM S ) ∂t = v 2 (L, t) -v 2 RM S . (26)

Complete system of equations

The complete self-sustained system is governed by the equations defined in sections 4.1.1, 4.1.2, 4.1.3 and 4.1.4. This article shows the resolution of this problem using an ode solver. The Cauchy problem Ẏ = F (Y, t) writes, according to Eq. ( 28), as:

F (Y, t) = F            x(t) ẋ(t) v(L, t) τ v 2 RM S p 1 (t) . . . p N (t)            (27) =               ẋ(t) -q r ω r ẋ(t) + ω 2 r [p(t) -γ(t) -x(t)] -2c 0 ℜ N n p n (t) dφ n dξ (L) -1 τ v(L, t) v 2 (L, t) -v 2 RM S C 1 (v RM S )u(t) + s 1 (v RM S )p 1 (t)
. . .

C N (v RM S )u(t) + s N (v RM S )p N (t)               , (28)
where u(t) is computed by using Eq. ( 16), C k (v RM S ) are computed with Eq. ( 14), and s k (v RM S ) are computed with Eq. ( 12). The initial conditions Y 0 will be set to 0 in the following simulations.

Simulation including nonlinear losses at the open end

The model including nonlinear losses at the open end as represented by the system of Eq. ( 28) is now simulated by time integration, using the solver ode45 from Matlab. The input data of the problem are based on experimental results from [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF].

Simulation parameters

The parameters related to the resonator, the reed and the reed channel are presented in table 1. Six values of c d have been taken from [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF], which are between c d = 0 and c d = 2.8. The value of ζ = 0.28 was calculated using data corresponding to a "loose embouchure" configuration. For the calculation of v RM S , the short-memory term was set to the period of the first impedance peak, i.e. τ = 2π/ℑ(s 1 ). The evolution of γ is first linearly ascending, from γ = 0 to γ = 3 in 8 s, then linearly descending, from γ = 3 to γ = 0 in 8 s. The total simulation time is therefore 16 s. Simulations were performed for N = 4 modes. A convergence study showed that the dynamic behavior of the system did not change for a higher number of modes.

Table 1: Main constants used for the simulations.

L (cm) R (mm) N modes ω r /2π (Hz) 64 8 4 2200 q r λ (s) ζ p M (kPa) 0.4
1.6 • 10 -5 0.28 8.5

Results

The bifurcation diagram of the input pressure p (dim) = p•p M over the blowing pressure P m = γ • p M is represented on Figure 4, with the parameters detailed in section 4.2.1. 4a), the oscillation threshold is located near 5.7 kPa (γ th ≈ 0.67). This threshold remains the same, whether nonlinear losses are taken into account (c d = 0) or not (c d = 0). However, nonlinear losses have a significant influence on the extinction threshold γ extup , i.e. the blowing pressure from which the reed stops oscillating and is pressed completely against the mouthpiece. This influence is shown in Table 2. When c d is increased (i.e. nonlinear losses increase), γ extup diminishes. Similarly to the experiments conducted by [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF], nonlinear losses have an important influence on the dynamic playing range of the musician. Figure 4 shows that when nonlinear losses are low, the range of stable oscillation amplitude that can be obtained is larger.

In the diminuendo phase, the inverse threshold is the same (γ inv ≈ 0.89) for each geometry at the open end of the pipe. Around this threshold, the amplitude of the input pressure decreases slightly as the losses increase. This behavior is also observed in the experimental curves of [2, Figure 12 a, c ande].

This comparaison with the experimental results of Atig et al. is limited to a qualitative study. Parameters such as the reed resonance angular frequency ω r and reed damping q r have a strong influence on the dynamics of the system, as demonstrated by [START_REF] Silva | Interaction of reed and acoustic resonator in clarinetlike systems[END_REF] for the oscillation threshold. The complete recalibration of the model on experimental results is beyond the scope of this article.

Comparison with experimental results from Dalmont and Frappé (2007)

An attempt to quantitatively validate the model of nonlinear losses is carried out. To do so, the values of the saturation threshold γ sat measured by [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF] are employed (see Figure 5). This threshold corresponds to the blowing pressure γ for which p is maximum during a crescendo. The authors measured this threshold for different reed openings, which are transcribed here in different values of ζ. The dimensioning pressure values p M used here correspond to the beating pressure estimated by the authors for a diminuendo. Furthermore, the measurements are performed on a cylindrical tube of dimensions L = 50 cm and R = 8 mm. The geometry at the opening has been estimated by the authors at c d = 2.8.

The experimental data are compared to simulations using the same parameters L and R as in the experiment. Simulations are performed for two cases: the first one does not take into account nonlinear losses (c d = 0), the other one takes them into account (c d = 2.8). The simulations are performed for ascending ramps of blowing pressure varying from γ = 0 to γ = 3 in 10 s. The other parameters are the same as those given in table 1.

The model presented in this paper is also compared to the Raman model including nonlinear losses proposed by [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF]. In this model, viscothermal losses are simplified by a frequency-independent coefficient α which replaces ℜ(Γ). The simplicity of this model allows the authors to obtain an analytical expression of the saturation threshold γ sat as a function of the parameter c d .

In the present work, the value of α was first adjusted at ℜ(Γ(jω 1 )), where ω 1 is the angular frequency of the impedance peak supporting the oscillation. However, this value produced excessively high estimates of γ sat compared to experimental results, especially for high values of ζ. To better match the experimental data, α = 2.7ℜ(Γ(jω 1 )) ≈ 0.147 was chosen. It should be noted that the analytical expression of γ sat exhibits a linear dependence in 1/α, for low values of α. Thus, γ sat is extremely sensitive to the value of α in Raman's model. It appears first that the saturation threshold is overestimated when nonlinear losses are not taken into account (+). In comparison, the two models taking into account nonlinear losses produce results much closer to the experiment. This shows the importance of taking this phenomenon into account in wind instrument simulations.

Furthermore, for the analytical solution given by the Raman model ( * ), the evolution of γ sat follows a straight line whose slope depends on 1/α. Although the results are close to the experimental ones, the high sensitivity of γ sat to α as well as the linear evolution of γ sat reflect the excessive simplicity of the Raman model to describe the dynamics of a wind instrument.

Finally, the model presented in this paper (×) has a good agreement with the experimental data, except for very low values of the mouthpiece parameter (ζ < 0.12). For these low values of ζ, the model overestimates γ sat compared to the experimental results. This overestimation may be related to the transient behavior at extinction caused by the temporal evolution of the control parameter γ. A characterization of the system by continuation could give results independent of the evolution rate of γ(t).

In conclusion, the correspondence between the results from the model presented in this article and the experimental data from [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF] highlights its ability to describe the dynamic behavior of a simplified wind instrument at saturation.

Conclusion

This article develops the design of a sound synthesis model of a reed instrument by modal decomposition of the input impedance, taking into account viscothermal losses as well as nonlinear losses at the end of the resonator. The input impedance now depends on the RMS acoustic velocity at a geometric discontinuity (here, the open termination). Poles and residues resulting from the modal decomposition are fitted as a function of this velocity. In a physical model of wind instrument, the pressure-flow relation defined by the resonator is then completed by new equations which account for this dependence with the velocity at the end of the pipe.

To evaluate the ability of the model to reproduce a real phenomenon, comparisons with the experimental results of [START_REF] Atig | Saturation mechanism in clarinet-like instruments, the effect of the localised non-linear losses[END_REF] and [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF] have been made. In the first case, simulations show a qualitatively similar behavior regarding the evolution of the extinction threshold depending on the geometry at the open end. In the second case, the model gives a good correspondence with the experimental results, in particular compared to a model without non-linear losses.

This formalism could be promising in the sound synthesis of wind instruments in real time, which employs the modal decomposition formalism in order to keep a minimum of memory. The inclusion of nonlinear losses in a complete clarinet model could more accurately translate dynamic phenomena essential to the experience of the musician, as suggested by [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF] : "any realistic model of the clarinet should include nonlinear losses in the side holes".

  (a) Real and imaginary parts of poles s n (v RM S ), calculated by solving Eq. (12). Detailed view on s 1 .(b) Real and imaginary parts of residues C n (v RM S ), calculated with Eq. (14). Detailed views on C 1 and C 4 .

Figure 2 :

 2 Figure 2: Graphical representation of the N = 8 first poles and residues. The values of v RM S are linearly chosen between 0 m/s and 24 m/s, according to [1]. As in Figure 1, c d = 13/9.

Figure 3 :

 3 Figure 3: Error due to the regression of modal coefficients C n and s n (Figure3a), and its consequence on the discrepancy between modal decomposition and definition (9) on z in (Figure3b).

Figure 4 :

 4 Figure 4: Bifurcation diagram of the L 2 norm of p (dim) with respect to P m , for different values of c d , in an ascending blowing pressure configuration (4a) and a descending one (4b).

Figure 5 :

 5 Figure 5: Experimental results from [11] for the saturation threshold (•), compared with three methods: modal decomposition without nonlinear losses (+); modal decomposition including nonlinear losses (×); analytical solution provided by Raman model including nonlinear losses ( * ).

Figure 5

 5 Figure5illustrates the evolution of the saturation threshold as a function of ζ in the experimental case, through modal decomposition simulation, and through the analytical expression based on the Raman model. It appears first that the saturation threshold is overestimated when nonlinear losses are not taken into account (+). In comparison, the two models taking into account nonlinear losses produce results much closer to the experiment. This shows the importance of taking this phenomenon into account in wind instrument simulations.Furthermore, for the analytical solution given by the Raman model ( * ), the evolution of γ sat follows a straight line whose slope depends on 1/α. Although the results are close to the experimental ones, the high sensitivity of γ sat to α as well as the linear evolution of γ sat reflect the excessive simplicity of the Raman model to describe the dynamics of a

  This shift remains almost the same regardless of the index of the pole. However, in terms of relative deviation, the ratio |ℜ(s n -s

	(lin) n

) give s n , for different input values of v RM S . They are plotted on Figure

2a

. The consideration of nonlinear losses at the end of a cylindrical pipe has almost no influence on ℑ(s n ), which is the resonance angular frequency of peak n. However, a larger v RM S increases |ℜ(s n )|.

Table 2 :

 2 Values of the extinction threshold P extup = p M • γ extup (kPa), for the simulation of Figure4a.

	c d	0	0.15 0.9 1.4 1.7 2.8
	P extup 17.7 17.3 15.3 14.5 14.2 13.2
	During a crescendo (Figure

Acknowledgments

This study has been supported by the French ANR LabCom LIAMFI (ANR-16-LCV2-007-01). The authors thank S. Maugeais and J.-P. Dalmont for their valuable comments.

List of Figures

Input impedance of a cylinder z in (computed through Eq. ( 9) and Eq. ( 10)) of length L = 64 c 2 Graphical representation of the N = 8 first poles and residues. The values of v RM S are linearly 3

Error due to the regression of modal coefficients C n and s n (Figure 3a), and its consequence on 4

Bifurcation diagram of the L 2 norm of p (dim) with respect to P m , for different values of c d , in an 5

Experimental results from [START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments[END_REF] for the saturation threshold (•), compared with three methods:

List of Tables 1 Main constants used for the simulations. 9 2

Values of the extinction threshold P extup = p M • γ extup (kPa), for the simulation of Figure 4a.


Lucía Ormaechea

Pierrette Bouillon

Maximin Coavoux

Johanna Gerlach

Jérôme Goulian

Benjamin Lecouteux

Cécile Macaire

Jonathan Mutal

Magali Norré

Adrien Pupier

Emmanuelle Esperanc ¸a-Rodier

Jerôme Goulian

Didier Schwab

PROPICTO: Developing Speech-to-Pictograph Translation Systems to Enhance Communication Accessibility

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Introduction

Alternative and augmentative communication (AAC) devices have taken an increasingly important role among people with disabilities and their relatives. However, usage of these technologies (i.e., communication boards or electronic media) may be cumbersome (Vaschalde et al., 2018). To surmount this problem, we argue that Speech-to-Pictograph (S2P) translation systems can be helpful for AAC users. In addition, we believe that they can improve the accessibility of health services for patients not speaking the local language. Developing such tools requires in-depth research on several areas of natural language processing (NLP). In this paper, we present a research project aimed at creating systems that automatically translate spoken French into pictographs.

Launched in early 2021, PROPICTO 1 (the acronym stands for PROjecting spoken language into PICTOgraphs) is a French-Swiss four-year project, funded by both the French National Re-© 2023 The authors. This article is licensed under a Creative Commons 4.0 licence, no derivative works, attribution, CC-BY-ND. 1 https://www.propicto.unige.ch/ search Agency2 and the Swiss National Science Foundation. 3 It is conducted as a collaboration between the Department of Translation Technology at the University of Geneva and the Study Group for Machine Translation and Automated Processing of Languages and Speech, affiliated to the Grenoble Informatics Laboratory.

For the purpose of this project, we will examine several NLP-related areas; namely, speech recognition, syntactic parsing, sentence simplification as well as pictograph generation. By integrating this series of tasks into different workflows (depending on the target scenario), we propose novel cross-modal machine translation systems that convert spoken language into pictographic units. Using this approach, we intend to tackle societal and communicative needs in the fields of: (1) disability, where an individual seeks to communicate with a person having a cognitive disorder, and (2) medical settings, where a language barrier exists between patient and practitioner.

Architectural Overview

PROPICTO aims at improving the usability of AAC devices by leveraging NLP-based solutions for greater accessibility.

We will design new methods and corpora so as to enable spoken utterances to be transcribed directly into sequences of pictographs, either generalpurpose like ARASAAC,4 or specific-purpose (i.e., SantéBD5 for health-related concepts). The project will face two major challenges:

• The scarcity of parallel speech-pictographs corpora, which constitutes a high hurdle to the implementation of state-of-the-art machine learning (especially end-to-end-based); • The need for extensive human and automatic evaluation to assess the comprehensibility of the output sequences with diverse target groups.

To better address them, we will adopt a cascade approach for our S2P processing workflow, which will be adapted according to the target setting. Thus, a first approach will favor a concept-based strategy to address pictographic generation, and will be integrated within a medical-purpose S2P architecture, 6 consisting of an automatic speech recognition (ASR) system and a neural text-to-UMLS 7 module that will define the pictographs to be produced and the syntax. An alternative pictograph generation strategy will rely on a wordbased approach, and will be preceded by the next stages (as shown in Figure 1): ASR, Dependency parsing (DP) and sentence simplification.

Using a cascade approach is motivated by the expected benefit of one phase over the next. Additionally, it helps to ensure greater model explainability. Our second proposed cross-modal architecture will start from an ASR module, relying on state-of-the-art Wav2Vec2.0 models. The DP task will be addressed with an end-to-end parser whose input is the raw signal for a given utterance. Using the raw signal instead of transcriptions enables us to use prosodic information to better predict syntactic boundaries (Pupier et al., 2022). Extracting a syntactic-based representation can in turn provide key information for a more effective sentence-level simplification. Reducing the linguistic complexity of the input transcript is likely to help the subsequent step, where the translation into pictographs will also be governed by expert grammar rules. 8 3 Contributions PROPICTO will make available to the scientific community methods and resources enabling a translation from spoken French into pictographs. The licenses will be as permissive as possible and conform to those of the pictographic sets being used. Furthermore, several prototypes for different target audiences will be put into production at the end of the project: (1) in emergency settings at the 6 For further details on this architecture, see (Mutal et al., 2022). 7 This acronym refers to Unified Medical Language System (UMLS) concepts. 8 Like multi-word expressions, verb tenses or proper names. Geneva University Hospitals,9 and (2) in institutions for children and adults with multiple disabilities. These will be tested in real conditions and evaluated using human and automatic methods.

Figure 1 :

 1 Figure 1: An overview of the Speech-to-Pictograph cascade architecture using a word-based approach.

https://anr.fr/en

https://www.snf.ch/en

https://arasaac.org/

https://santebd.org/

A demo is available on: https://propicto.demos. unige.ch/pictoDrClient/translate/

Acknowledgements

This work is part of the PROPICTO project, funded by the Swiss National Science Foundation (N°197864) and the French National Research Agency (ANR-20-CE93-0005).

PROPICTO is a project funded by the French National Research Agency and the Swiss National Science Foundation, that aims at creating Speech-to-Pictograph translation