Jean-François Nierengarten
email: nierengarten@unistra.fr

Prof L Jonathan

Sessler

Density functional theory investigation of the coordination of Zn(II)-tetraphenylporphyrin with 1,2,3-triazole ligands

Keywords: density functional theory, metalloporphyrin, N-ligand, apical coordination, electrochemistry

In this paper, we report density functional theory (DFT) studies to fully understand the coordination interactions between 1,4-disubstituted 1,2,3-triazole derivatives and the metal center of Zn(II)-porphyrins. Coordination through the most electron-rich N3 atom of the triazole heterocycle is more favorable. However, the energy difference between the N2-and N3-coordinated isomers is moderate and the less stable is likely also present in the solution. The binding of 1,4-disubstituted 1,2,3-triazole ligands to oxidized and reduced Zn(II)-porphyrins has been also investigated. Upon reduction, the binding interactions are extremely weak and the formation of the complex is not favorable anymore. In contrast, the binding energy is considerably increased upon oxidation of the Zn(II)-porphyrin moiety thus leading to more stable complexes. The calculations fully support experimental findings and explain the differences observed for the redox potentials of complexed and uncomplexed Zn(II)-porphyrin residues.

INTRODUCTION

The copper(I)-catalyzed alkyne-azide cycloaddition (CuAAC) is without any doubt the most emblematic reaction in the field of click chemistry [1][START_REF] Chandrasekaran | Click Reactions in Organic Chemistry[END_REF]. Owing to its reliability, specificity, and efficiency, the CuAAC reaction has become an extremely useful synthetic tool. It has been used to prepare a rich variety of compounds for applications in all fields of science [START_REF] Chandrasekaran | Click Reactions in Organic Chemistry[END_REF]. As a result, 1,2,3-triazole rings are more and more often present in the molecular systems produced over the past two decades. For this reason, the chemical and physical properties of 1,2,3-triazole derivatives have been also intensively investigated [START_REF] Schulze | [END_REF]. In particular, it has been shown that 1,2,3-triazoles are potential ligands for transition metals due to the presence of two unsubstituted sp 2 -hybridized N atoms in their heterocyclic structure [START_REF] Crowley | Top. Heterocycl[END_REF]. In a monodentate coordination mode, 1,4-disubstituted-1,2,3-triazoles are typically coordinated through the N3 nitrogen which is the most electron-rich [START_REF] Crowley | Top. Heterocycl[END_REF]. Such ligands are also interacting with metalloporphyrins but the interaction is rather weak when compared to pyridine or imidazole ligands [START_REF] Nguyen | [END_REF]. This has been ascribed to the significantly lower basicity of the 1,2,3-triazole ring. The triazole-metalloporphyrin interaction alone is therefore not particularly attractive for the preparation of stable supramolecular ensembles. Nonetheless, it has been successfully used by Osuka and co-workers for the construction of welldefined macrocyclic supramolecular assemblies from triazole-appended Zn(II)-porphyrin derivatives [6][7]. They have for example shown that Zn(II)-porphyrin 1 can form a cofacial dimer by mutual coordination of the triazole substituents of one porphyrin to the Zn(II) center of the other one (Fig. 1). In this case, multiple interactions and cooperative effects ensure the stability of the macrocyclic ensemble. The X-ray crystal structure of the supramolecular dimer revealed coordination through the N3 nitrogen of the triazole ring [6]. Coordination of the N2 nitrogen would not allow for the concerted formation of a dimer in this particular case and to the best of our knowledge, the preferential intermolecular coordination of metalloporphyrins with triazole ligands through the N3 nitrogen in solution has not been demonstrated so far.

On the other hand, intramolecular triazole-metalloporphyrin interactions have been also observed in several systems [8][9][10][11][12][13]. This is for example the case of [START_REF] Chandrasekaran | Click Reactions in Organic Chemistry[END_REF] rotaxanes in which a triazole group of the axle is coordinating the metal center of a metalloporphyrin-containing macrocyclic component [8][9][10]. Beer and co-workers have shown that this intramolecular coordination preorganizes the rotaxane binding cavity and thus dramatically enhances its anion binding affinities [8]. In other examples, the intramolecular coordination interactions have been used to control the position of the metalloporphyrinic macrocycle onto the molecular axle of the rotaxane [9] or to modulate the excited state deactivation of porphyrin-fullerene conjugates [10]. As part of this research, our group has shown that intramolecular triazole-Zn(II)porphyrin interactions can be used to control the conformation of multiporphyrinic arrays 2 and 3 constructed on a pillar [START_REF] Nguyen | [END_REF]arene scaffold (Fig. 2) [12].

For both 2 and 3, intramolecular coordination interactions between some 1,2,3-triazole linkers and metal centers of neighboring Zn(II)-porphyrin moieties are responsible for their partial folding at room temperature. At high temperatures, the weak intramolecular coordination interactions are disrupted and both 2 and 3 adopt a fully open structure. In contrast, lowering the temperature favors the association thus leading to the folding of 2 and 3. Fig. 1. Zn(II)-porphyrin 1 and X-ray crystal structure of the cofacial dimer (the tBu substituents have been omitted for clarity) [6]. Moreover, we have also shown that the degree of folding can be fully controlled by an electrochemical input. Upon reduction of their ten peripheral Zn(II)-porphyrin subunits, total decoordination occurs thus leading to fully unfolded decaanions. In contrast, oxidation of the peripheral porphyrins generates folded decacations in which all the metal centers of the peripheral Zn(II)porphyrin radical cations are coordinated with a triazole unit. In this paper, we now report density functional theory (DFT) studies to fully understand the coordination interactions between 1,4-disubstituted 1,2,3-triazole derivatives and the metal center of Zn(II)-porphyrins. For this purpose, 1,4-dimethyl-1,2,3-triazole (L1) and Zinc(II) 5,10,15,20-(tetraphenyl)porphyrin (ZnTPP) have been selected for the theoretical calculations (Chart 1). For comparison purposes, calculations have been also performed with pyridine (L2) and 1-phenylimidazole (L3) as their coordination with metalloporphyrin is well documented and understood [14]. Additionally, reference ligands L4-6 have been also computed.

EXPERIMENTAL

All DFT calculations were performed with Spartan 20 [START_REF] Deppmeier | Spartan 20[END_REF][START_REF] Becke | [END_REF] on a Mac Pro with 8 cores and 3 GHz Intel Xeon E5 processors. For the pK A calculations of the ligands, geometry optimization as well as subsequent frequency analysis at 298.15 K were performed at the B3LYP/6-311+G(2df,2p) level using the CPCM model [17] with the dielectric constant of water (78.30). As revealed by the frequency analysis, imaginary frequencies were absent in all ground states. For ligands L1-6, the following equilibrium was considered to calculate the pK a (LH + ) values:

(LH + ) aq → (L) aq + (H + ) aq (1)
The pK a of LH + was calculated from

ΔG aq = G(L) aq + G(H + ) aq -G(LH + ) aq (2)
pK a (LH + ) = ΔG aq / 2.303RT

where R is the gas constant, T is the temperature, and ΔG aq is the free energy of the reaction. For the calculations, the Gibbs free energy of the solvated proton (G(H + ) aq = -1130.85 kJ/mol) reported by Coe and co-workers has been used [18]. An error of at least 8 kJ/mol is usually assumed to correspond to ca. 1.5 pK a units. To avoid the uncertainties related to the proton-free energy, the pK a values were also computed by using isodesmic reactions and the experimental pK a data of a reference system (pyridine, pK a (pyridine) exp = 5.23):

(LH + ) aq + (pyridine) aq → (L) aq + (pyridineH + ) aq ( 4)

ΔG exchange = G(L) aq + G(pyridineH + ) aq -G(LH + ) aq -G(pyridine) aq [START_REF] Nguyen | [END_REF] pK a (LH + ) = (ΔG exchange /2.303RT) + pK a (pyridine) exp (6) For the coordination of ZnTPP with the various N-ligands, geometry optimization of all the compounds was first performed in the gas phase with the B3LYP functional using the 6-31G(d) basis set. The structures were further optimized at the same level using the CPCM model in solvents of different polarities, namely CH 2 Cl 2 (dielectric constant of 8.82) and DMF (dielectric constant of 37.22). In all the cases, the frequencies were computed at the same level to confirm that each optimized structure is an energy minimum and to evaluate its zeropoint vibration energy (ZPVE) and thermal corrections at 298.15 K. The UV/vis spectra of ZnTPP and ZnTPP-L1-3 were also calculated in the gas phase at the TD-B3LYP/6-31G(d) level of theory. Single-point energy calculations were performed at the B3LYP/6-311+G** level to evaluate the distortion and interaction energies. For this purpose, the electronic energies of the ZnTPP and L1-3 fragments from the ZnTPP-L1-3 structures were also computed.

The Boltzmann distribution was calculated from:

N i /N = exp(-E i /kT)/∑ j exp(-E j /kT) (7)
where N i is the number of molecules in conformation i, N is the total number of molecules in the system, E i is the energy of state i, T is the equilibrium temperature of the system, and k is the Boltzmann constant.

The coordinates of all the B3LYP/6-31G(d) optimized structures are reported in the electronic supplementary information together with the corresponding DFT computed energies, the pK a calculations, the distortion/ interaction analysis, and the frontier molecular orbitals (FMO).

RESULTS AND DISCUSSION

As a preliminary step, the pK a values of ligands L1-3 were calculated in an aqueous solution. For this purpose, the neutral and the protonated form of L1-3 were calculated in water using the CPCM (conductor-like polarizable continuum model) approach [17]. CPCM does not account for explicit solvent-solute interactions such as H-bonds or hydrophobic effects. However, such simple continuum calculations typically provide good qualitative trends for the investigated systems [19][20]. To further evaluate the calculated pK a values, calculations were also performed on related systems for which experimental pK a values are available, namely imidazole (L4), 1-methyl-1,2,3-triazole (L5), and 1-methyl-4-phenyl-1,2,3-triazole (L6). Whereas only one protonable N atom is present in pyridines and imidazoles, the situation is more complex for 1-substituted 1,2,3-triazoles. Effectively, the 1,2,3-triazole ring contains two basic sp 2 -hybridized N atoms, namely N2 and N3, and two protonated forms are possible (Fig. 3). Close inspection of the atomic charge in the calculated triazole derivatives revealed more negative values for N3 when compared to N2 in all the cases. Protonation should be favored on the more electron-rich N atom and, effectively, in all the cases, the triazoliums protonated in the N3 positions were found more stable than the corresponding cations protonated in the N2 position (Fig. 3). This is in perfect agreement with related experimental and theoretical results reported in the literature [20a]. The difference in free energy between the two triazolium isomers of L1, L5, and L6 is rather large (32.5 to 36.9 kJ/mol, see Fig. 3), thus the Boltzman population of the less stable cation at 298.15 K is negligible (0.00020 to 0.00012%). Therefore, for the pK a calculation of the triazolium derivatives, only the most stable protonated forms were considered. As anticipated, the triazole ligands are all significantly less basic when compared to pyridines and imidazoles. Overall, direct calculations of the pK a values with Eq. ( 3) at the B3LYP/6-311+G(2df,2p) level in water gave a correct trend but comparison with the experimental values reveals errors up to almost 2 pK a units. In contrast, the difference between the calculated and the experimental pK a values are less important by using isodesmic reactions with pyridine as a reference system. The relative basicity of L1-6 can also be simply evaluated by the difference in the natural atomic charge From ref. [20].

of the N atom involved in the protonation. Increased basicity is systematically associated with an increased natural atomic charge of the N atom. This first series of calculations revealed that triazoles are only poorly basic and as such, they should be only weak ligands for ZnTPP. This has been already experimentally confirmed by the difference in the binding constant (K a ) values obtained for the association of 1H-1,2,3-triazole (480 M -1 ) and pyridine (5800 M -1 ) with ZnTPP in toluene at room temperature [START_REF] Nguyen | [END_REF].

The geometries of the ZnTPP-L1-3 complexes, ZnTPP and L1-3 were optimized at the B3LYP/6-31G (d) level in both the gas phase and in solvents of different polarities (CH 2 Cl 2 and DMF) using the CPCM model. The B3LYP functional combined with 6-31G* or higher basis set has proven its ability to display and predict the physicochemical properties of porphyrin derivatives and large metalloporphyrin supramolecular ensembles [21][22][23]. Relevant thermodynamic and structural parameters of the optimized ZnTPP-L1-3 complexes are summarized in Table . 2.

In the particular case of ZnTPP-L1, the two possible isomers resulting from the coordination of the metal center through N2 or N3 of the 1,2,3-triazole ligand have been calculated. As shown in Fig. 4, the coordination of L1 with the more electronegative N atom is favored. However, the free energy difference between the N2-and N3-coordinated isomers is less important when compared to the one calculated for the two isomeric protonated forms of L1. In the case of ZnTPP-L1, the Boltzmann population of the less stable N2-coordinated isomer is not negligible anymore. In the gas phase, the N3/N2 isomer distribution was found to be 96/4. In CH 2 Cl 2 and DMF, the difference in Gibbs free energy was slightly increased when compared to the gas phase. Accordingly, the Boltzmann population of the N2 isomer was estimated lower in solution than in the gas phase (2.3-2.4% vs. 4%). Solvation effects on the coordination preference through the N3 atom are however limited as the Boltzmann distribution of the N3-and N2-coordinated isomers of ZnTPP-L1 is similar in CH 2 Cl 2 and DMF despite the large difference in dielectric constants. Analysis of the enthalpy difference (ΔH 0 ) for the coordination of ligand L1 through N2 or N3 to the Zn(II) center of ZnTPP is also consistent for the preferential interaction of the metal with the more electronegative N atom of the triazole heterocycle. Comparison of the ΔH 0 values for the binding of L1-3 to ZnTPP reflects well the affinity of the different ligands for ZnTPP. The strongest ligand is imidazole L3 followed by pyridine L2 and the weakest one is triazole L1. These relative affinities follow perfectly experimental trends reported for the binding of imidazole, pyridine, and 1,2,3-triazole ligands to ZnTPP [START_REF] Nguyen | [END_REF]23]. The red-shifted absorption of both the Soret and the Q bands of the porphyrin chromophore upon binding of N-ligands to ZnTPP is a diagnostic spectroscopic signature for the complexation [24]. The lower HOMO-LUMO gap found for ZnTPP-Ln (n = 1, 2 or 3) when compared to ZnTPP is thus in perfect agreement with the experimental data. The absorption spectra of ZnTPP and ZnTPP-Ln (n = 1, 2 or 3) were also calculated. Whatever the ligand, the calculated UV/vis allowed transitions were all significantly red-shifted in the complexes when compared to ZnTPP in good agreement with experimental observations.

The factors controlling the coordination trends of L1-3 with ZnTPP were further elucidated using the interaction energy decomposition method [25]. This analysis decomposes the electronic energy of the ZnTPP-L complexes into two terms: the distortion energy (ΔE dist ) which corresponds to the energy required for the deformation of the individual components (ZnTPP and L1-3) to form the complex and the interaction energy (ΔE int ) which corresponds to the difference between the electronic energies of the two distorted components and the complex. The energy gain for the complexation of the N-ligand to the metal center of ZnTPP was then evaluated by adding the positive distortion energy to the negative interaction energy. The calculated energy values are summarized in Table 3.

The interaction energy and the energy gain are increased when the basicity of the ligand is increased. In the gas phase, the Zn-N distances in the DFT-optimized structures of ZnTPP-Ln (n = 1, 2, or 3) are shortened when the binding interaction of the N-ligand to the metal center is increased. This is also the case in CH 2 Cl 2 and DMF. Interestingly, the Zn-N distances are however significantly affected by the solvation. They are effectively shorter when going from the gas phase to the solution. At the same time, their enthalpy differences and their interaction energies are decreased suggesting weaker binding interactions in solution. This observation is counterintuitive. However, for dative bonds, the widely accepted assumption that stronger bonds are shorter than weaker ones is not generally correct as the bond distances are very sensitive to the environment [24]. In coordination compounds, the strength of dative bonds in which the two electrons shared in the bond are provided by the ligand donor to the metal acceptor is typically destabilized by the solvent [26].

Finally, we were also interested in fully rationalizing the electrochemical properties of Zn(II)porphyrins coordinated to 1,2,3-trazole ligands. As mentioned in the introduction, electrochemistry has been used to control the folding/unfolding of multi-Zn(II)-porphyrinic arrays 2 and 3. As a typical example, the experimental observations done for compound 3 are summarized in Fig. 5.

In the neutral state, compound 3 is partially folded. The coordinated and uncoordinated Zn(II)-porphyrin moieties of 3 gave rise to different oxidation and reduction waves (E 1 ox and E 2 ox -E 1 Red and E 2

Red

) [12]. In both cases, the dynamic interconversion process between coordinated and uncoordinated redox centers does not occur within the voltammetric timescale. Upon complete oxidation of the ten peripheral Zn(II)-porphyrin subunits of 3, a single wave (E 3 °×) has been observed for the ten peripheral radical cations suggesting that they are all coordinated in 3 10+ . When compared to a model Zn(II) porphyrin, this third oxidation is effectively more difficult. In agreement with literature data [22,24], this is a diagnostic signature for a coordinated Zn(II)-porphyrin radical cation. Therefore, the complete oxidation of

ΔE dist = (E Frag Zn + E Frag L ) -(E Zn + E L )
, where E Frag Zn is the electronic energy of the Zn(II)porphyrin fragment, E Frag L is the electronic energy of the ligand fragment, E Zn is the electronic energies of ZnTPP and E L is the electronic energy of the ligand. b

ΔE int = E -(E Frag

Zn + E Frag L ), where E is the electronic energy of the complex. the ten peripheral Zn(II)porphyrins triggers a complete folding of the compound. In contrast, total decoordination occurs upon complete reduction of the ten peripheral Zn(II)-porphyrins thus leading to a fully unfolded deca-anion. A single wave (E 3

Red

) is effectively observed for the reduction of the ten peripheral anionic residues at a redox potential typical for a second reduction of an uncomplexed Zn(II)-porphyrin system. All these experimental observations revealed dramatic changes in the binding ability of the 1,2,3-triazole ligand to the metal center depending on the redox state of the Zn(II)-porphyrin moiety. On the other hand, the redox properties of the Zn(II)-porphyrin unit are also modulated by the coordination of the triazole ligand. This prompted us to evaluate the binding interactions and the electronic properties of L1 with oxidized and reduced ZnTPP. For this purpose, the geometries of ZnTPP and ZnTPP-L1 were optimized at the B3LYP/6-31G* level for the different redox states. Both N2-and N3-coordinated isomers of ZnTPP-L1 were analyzed. Further calculations were also performed with L3 (see ESI). For the sake of clarity, only the most stable N3-coordinated isomer of ZnTPP-L1 is discussed in the present section, the general trend is the same for all the systems. The calculated electron affinity (EA) and ionization potential (IP) of ZnTPP and N3-coordinated ZnTPP-L1 are listed in Table 4. Relevant energetic and structural parameters of the complexes obtained by association of L1 with ZnTPP 2+ , ZnTPP +• , ZnTPP -• , and ZnTPP 2-are reported in Table 5.

Table 5. Calculated Zn-N distance in the B3LYP/6-31G(d) optimized N3-coordinated ZnTPP-L1 complexes at different redox states (gas phase), ΔH 0 for the binding of L1 to the metal center of the Zn(II)-porphyrin, and distortion/interaction analysis from the energies calculated at the B3LYP/6-311+G** level. An ΔE dist and ΔE int were calculated assuming that the charge is localized on the porphyrin moiety.

When triazole ligand L1 is interacting with ZnTPP, the Zn(II)-porphyrin unit is more electron-rich when compared to uncoordinated ZnTPP. This can be conveniently visualized by comparing the electrostatic potential map of ZnTPP and ZnTPP-L1 (Fig. 6). Intuitively, the first reduction is expected more difficult in the complex. This was confirmed by the difference in the first electron affinity (EA 1 ) calculated for ZnTPP and ZnTPP-L1. Based on the ΔH 0 values, the complexation of L1 to radical-anion ZnTPP -• is less favorable when compared to ZnTPP. The interaction energy is also considerably reduced and considering the distortion energy, the overall energy gain for the complexation is almost non-existent. This is in full agreement with the observed decoordination upon the first reduction of the ten peripheral Zn(II)-porphyrin moieties of compound 3. Upon the second reduction, the enthalpy difference for the association of L1 with ZnTPP 2-is positive. The interaction energy between ZnTPP 2-and L1 is also extremely low and unable to compensate for the distortion energy. The formation of ZnTPP 2--L1 is thus thermodynamically unfavorable.

Upon oxidation of ZnTPP, an opposite trend is observed. The binding of L1 to radical-cation ZnTPP +• and dication ZnTPP 2+ is enthalpically more favorable when compared to neutral ZnTPP. The binding interactions of L1 with the oxidized forms of ZnTPP are also considerably increased. This is fully consistent with the complete folding observed for compound 3 upon oxidation of the ten peripheral Zn(II)-porphyrin moieties.

As far as the redox potential of the first oxidation is concerned, the complexation of L1 to ZnTPP leads to a smaller first ionization potential (IP 1 ) value meaning that less energy is required to oxidize the Zn(II)-porphyrin subunit in the complex. This is fully consistent with electrochemical data and DFT calculations reported for related imidazole-Zn(II)porphyrin complexes [22] and supports the assignment for E 1 ox and E 2 ox in the voltammograms obtained for compound 3. However, the second ionization potential (IP 2 ) is not in agreement with the experimental findings which revealed that the second oxidation is more difficult for complexed Zn(II)-porphyrins. The complexation of L1 to dication ZnTPP 2+ is associated with a particularly large distortion energy. Indeed, the dicationic porphyrin moiety of ZnTPP 2+ -L1 is unable to adopt the saddle-shaped conformation observed for the uncomplexed ZnTPP 2+ dication because of the steric hindrance resulting from the presence of the N-ligand bound to the Zn(II) center (Fig. 7). Moreover, the dihedral angles between the porphyrin and the meso-phenyl substituents are significantly larger in ZnTPP 2+ -L1 when compared to ZnTPP 2+ thus reducing possible π-conjugation that may play a role in the stabilization of the dicationic Zn(II)-porphyrin moiety [22]. The positive shift observed for the second oxidation of the complexed Zn(II)porphyrin subunit must therefore result from a destabilization of the dication and not from a substantial stabilization of the intermediate radical cation.

CONCLUSION

DFT studies of the binding interactions between 1,4-disubstituted 1,2,3-triazole ligands and the metal center of Zn(II)-porphyrins revealed preferential coordination through the most electron-rich N3 atom. However, the energy difference between the N2-and N3-coordinated isomers is moderate and the less stable is likely also present in the solution. When compared to imidazole and pyridine ligands, the binding energy of 1,4-disubstituted 1,2,3-triazole derivatives to the metal center of Zn(II)-porphyrins is substantially weaker. This is related to a reduced atomic charge of the unsubstituted sp 2 -hybridized N atom in the triazole heterocycle leading to a weaker basicity when compared to imidazoles and pyridines. Finally, the binding of 1,4-disubstituted 1,2,3-triazole ligands to oxidized and reduced Zn(II)porphyrins has been also investigated. Upon reduction, the binding interactions are extremely weak and the formation of the complex is not favorable anymore. In contrast, the binding energy is considerably increased upon oxidation of the Zn(II)-porphyrin moiety thus leading to more stable complexes. The calculations fully support the experimental observations reported for multiporphyrin arrays 2 and 3 and support the proposed conformational changes upon oxidation or reduction of their peripheral Zn(II)-porphyrin subunits. They also explain the differences in redox potentials observed for the coordinated and uncoordinated Zn(II)-porphyrin residues in compounds 2 and 3.

Fig. 2 .

 2 Fig. 2. Multi-Zn(II)-porphyrin arrays 2 and 3. The inset shows a schematic representation of the dynamic conformational equilibrium resulting from the intramolecular coordination of a 1,2,3-triazole moiety to a neighboring Zn(II)-porphyrin subunit in 2.

Chart 1 .

 1 Structure of the compounds used for the theoretical investigations.

Fig. 3 .

 3 Fig. 3. (a) Relative Gibbs free energy at 298.15 K (ΔG 0 ) calculated at the B3LYP/6-311+G(2df,2p) level for the isomeric protonated forms of L1, L5, and L6. (b) Electrostatic potential map of L1-6 and natural atomic charge of their unsubstituted sp 2 -hybridized N atoms.

Fig. 4 .

 4 Fig. 4. DFT optimized structures and relative Gibbs free energy at 298.15 K (ΔG 0 ) for the two isomeric ZnTPP-L1 complexes. Boltzmann distribution at 298.15 K for the N3-/N2-coordinated ZnTPP-L1 complexes: 96/4 (gas phase), 97.6/2.4 (CH 2 Cl 2 ) and 97.7/2.3 (DMF).

Fig. 5 .

 5 Fig. 5. Top: schematic representation of the electrochemically triggered conformational changes of compound 3. Bottom: Osteryoung square wave voltammograms (anodic and cathodic scans) recorded for compound 3 on a Pt electrode in CH 2 Cl 2 + 0.1 M nBu 4 N.BF 4 at room temperature [12].

Fig. 7 .

 7 Fig. 7. DFT optimized structures of dications ZnTPP 2+ and ZnTPP 2+ -L1.

Table 1 .

 1 Experimental and calculated pK a (LH + ) values of

	1,4-dimethyl-1,2,3-triazole	(L1),	pyridine	(L2),
	1-phenylimidazole (L3) imidazole (L4), 1-methyl-1,2,3-
	triazole (L5) and 1-methyl-4-phenyl-1,2,3-triazole (L6). The
	Gibbs free energy values calculated at the B3LYP/6-
	311+G(2df,2p) level in water have been used for the pK a (LH + )
	calculations.
	Compound	pK a (LH + )	pK a (LH + )		pK a (LH + )
		Calculated	Calculated from	Experimental
		from Eq. (3)	Eq. (6) with L2	value
			as reference
	L1	1.14	2.05
	L2	4.32			5.23 a
	L3	5.02	5.93
	L4	5.97	6.88		6.95 a
	L5	-0.61	0.30		1.25 a
	L6	-0.86	0.05		0.05 a
	a

Table 2 .

 2 Calculated Zn-N distance in ZnTPP-L1-3, ΔH 0 for the binding of L1-3 to ZnTPP and HOMO-LUMO gap.

	Compound	Zn-N (Å)	ΔH 0	HOMO-LUMO
			(kJ/mol)	gap (eV)
	B3LYP/6-31G* (Gas phase)
	ZnTPP			2.82
	ZnTPP-L1
	(N2 coordination)	2.256	-42.8	2.71
	(N3 coordination)	2.215	-50.9	2.69
	ZnTPP-L2	2.201	-57.5	2.68
	ZnTPP-L3	2.152	-63.7	2.67
	B3LYP/6-31G* (CPCM model in CH 2 Cl 2 )
	ZnTPP			2.86
	ZnTPP-L1-3
	(N2 coordination)	2.234	-30.1	2.71
	(N3 coordination)	2.182	-39.6	2.68
	ZnTPP-L2	2.164	-44.8	2.73
	ZnTPP-L3	2.131	-51.2	2.67
	B3LYP/6-31G* (CPCM model in DMF)
	ZnTPP			2.86
	ZnTPP-L1-3
	(N2 coordination)	2.227	-29.4	2.70
	(N3 coordination)	2.182	-38.9	2.68
	ZnTPP-L2	2.161	-44.2	2.74
	ZnTPP-L3	2.131	-50.8	2.67

Table 3 .

 3 Distortion/interaction analysis.

	Compound	ΔE dist a	ΔE int b	ΔE dist + ΔE int
		(kJ/mol)	(kJ/mol)	(kJ/mol)
	B3LYP/6-311+G** (Gas phase)
	ZnTPP-L1
	(N2 coordination)	27.2	-47.1	-19.9
	(N3 coordination)	29.4	-57.8	-28.4
	ZnTPP-L2	26.2	-64.3	-38.1
	ZnTPP-L3	31.7	-71.6	-39.9
	B3LYP/6-311+G** (CPCM model in CH 2 Cl 2 )
	ZnTPP-L1-3
	(N2 coordination)	25.9	-30.3	-4.4
	(N3 coordination)	29.7	-43.9	-14.2
	ZnTPP-L2	24.5	-46.9	-22.4
	ZnTPP-L3	30.6	-55.2	-24.6
	B3LYP/6-311+G** (CPCM model in DMF)
	ZnTPP-L1-3
	(N2 coordination)	25.9	-29.4	-3.5
	(N3 coordination)	25.9	-39.2	-13.3
	ZnTPP-L2	23.9	-45.5	-21.6
	ZnTPP-L3	29.0	-52.6	-23.6
	a

Table 4 .

 4 Calculated first and second electron affinities (EA 1 and AE 2 ), and first and second ionization potentials (IP 1 and IP 2 ).

	Compound	EA 1 (eV)	EA 2 (eV)	IP 1 (eV)	IP 2 (eV)
	ZnTPP	1.1	1.7	5.9	8.8
	ZnTPP-L1	0.8	2.1	5.4	8.5

Fig.

6

. Electrostatic potential map of ZnTPP and ZnTPP-L1.

Acknowledgments

Financial support by the ANR (projects FastGiant ANR-17-CE07-0012-01 and Pillar ANR-19-CE06-0032), the Fondation Jean-Marie Lehn, and the LabEx "Chimie des Systèmes Complexes" is gratefully acknowledged.

Supporting information

Additional data are given in the supplementary material. This material is available free of charge via the Internet at https://www.worldscientific.com/doi/suppl/ 10.1142/S1088424623500803


Alexis Plaquet

Hervé Bredin

Powerset multi-class cross entropy loss for neural speaker diarization

Keywords: speaker diarization, loss function, powerset classification

Introduction

Speaker diarization is the task of partitioning an audio stream into homogeneous temporal segments according to the identity of the speaker. Most dependable diarization approaches consist of a cascade of several steps [START_REF] Landini | Bayesian HMM Clustering of x-vector Sequences (VBx) in Speaker Diarization: Theory, Implementation and Analysis on Standard Tasks[END_REF]: voice activity detection to discard non-speech regions, speaker embedding to obtain discriminative speaker representations, and clustering to group speech segments by speaker identity. This family of approaches has two main drawbacks: • errors made by each step are propagated to the subsequent steps, possibly escalating into even larger errors; • they are not designed to detect (let alone assign to the right speakers) overlapping speech regions -an additional postprocessing step is needed to handle them [START_REF] Bullock | Overlapaware diarization: Resegmentation using neural end-to-end overlapped speech detection[END_REF].

To circumvent these limitations, a new family of approaches have recently emerged, rethinking speaker diarization completely [START_REF] Fujita | End-to-End Neural Speaker Diarization with Permutationfree Objectives[END_REF][START_REF] Fujita | End-to-end neural speaker diarization with selfattention[END_REF][START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF]. Dubbed end-to-end diarization (EEND), the main idea is to train a single neural network that ingests the audio recording and directly outputs the diarization, hence addressing the error propagation issue. They formulate the problem as a multi-label classification task, allowing multiple overlapping speakers to be active simultaneously. Furthermore, permutation-invariant training [START_REF] Fujita | End-to-End Neural Speaker Diarization with Permutationfree Objectives[END_REF] is the critical ingredient that turns clustering -an intrinsically unsupervised task -into a supervised classification task. EEND approaches do have a few significant limitations: • they struggle to predict the correct number of speakers, espe-cially when it is larger in test data than during training -additional mechanisms such as encoder-decoder attractors were proposed to partially solve this issue [START_REF] Horiguchi | End-to-End Speaker Diarization for an Unknown Number of Speakers with Encoder-Decoder Based Attractors[END_REF];

• training such approaches is extremely data hungry as each conversation in the training set constitutes a single training sample -one has to resort to synthetic (hence unrealistic) conversations instead [START_REF] Landini | From Simulated Mixtures to Simulated Conversations as Training Data for End-to-End Neural Diarization[END_REF]; • because of the internal self-attention mechanism, they do not scale well to long conversations. Therefore, a few recent works took a step back to borrow the best of both worlds (BoBW, i.e. multi-stage and end-to-end approaches) [START_REF] Kinoshita | Integrating End-to-End Neural and Clustering-Based Diarization: Getting the Best of Both Worlds[END_REF][START_REF]Advances in Integration of End-to-End Neural and Clustering-Based Diarization for Real Conversational Speech[END_REF]. The general principle is based on three steps: 1. split long conversations into shorter chunks; 2. apply end-to-end speaker diarization on each of them; 3. stitch them back together using speaker embeddings and unsupervised clustering.

To both start with a strong baseline and ensure reproducibility, we build on top of the speaker diarization pipeline available in version 2.1.1 of pyannote.audio open source toolkit [START_REF] Bredin | pyannote.audio 2.1 speaker diarization pipeline: principle, benchmark, and recipe[END_REF] that embraces this aforementioned three-steps principle. More precisely, we focus on the loss function used for training neural networks towards end-to-end speaker diarization of short audio chunks. [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF] calls this task speaker segmentation and states that working with short audio chunks has the following advantages: • it circumvents the scalability issue because the neural network only ever ingests short fixed-duration (5s in our case) audio chunks; • it makes it easier to train (as each conversation can now provide multiple training samples); • it allows to set a small upper bound on the (local) number of speakers Kmax -for instance, there is a 99% chance that any 5s chunk contains less than Kmax = 3 speakers in the 9 benchmarking datasets used in this paper. Since its introduction a few years ago, the EEND line of work has been processing whole conversations with up to Kmax ≈ 20 (partially overlapping) speakers. Therefore, addressing it as a frame-wise multi-label classification problem was a brilliant (and efficient!) idea that we summarize in Section 2. Working with a much smaller number of (Kmax = 3) speakers opens up a new set of opportunities which have not yet been investigated by recent BoBW approaches. Inspired by [START_REF] Du | Speaker overlapaware neural diarization for multi-party meeting analysis[END_REF], we propose to address speaker segmentation as a framewise powerset multi-class classification problem. Section 3 provides a detailed description of the approach but one can understand the main idea by looking at Figure 1. The advantages of this approach are two-fold: it completely removes the need for the (sensitive) detection threshold hyperparameter, and it leads to significant performance improvement of the overall 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 y t y 1 y T y t y 1 y T Figure 1: From multi-label to powerset multi-class encoding. While powerset classes are mutually exclusive, several multilabel classes must be active at the same time to encode overlapping speakers.

speaker diarization pipeline (as shown by a thorough evaluation on 9 benchmarking datasets in Sections 4 and 5). In particular, we claim state-of-the-art performance on AISHELL-4 [START_REF] Fu | AISHELL-4: An Open Source Dataset for Speech Enhancement, Separation, Recognition and Speaker Diarization in Conference Scenario[END_REF], AliMeeting [START_REF] Yu | M2MeT: The ICASSP 2022 Multi-Channel Multi-Party Meeting Transcription Challenge[END_REF], AMI [START_REF] Carletta | The AMI Meetings Corpus[END_REF], Ego4D [START_REF] Grauman | Ego4D: Around the World in 3,000 Hours of Egocentric Video[END_REF], MSDWild [START_REF] Liu | MSDWild: Multimodal Speaker Diarization Dataset in the Wild[END_REF], and REPERE [START_REF] Kahn | A Presentation of the REPERE Challenge[END_REF] benchmarks, make the approach available in the pyannote.audio library, and release pretrained models publicly.

2. From multi-label classification...

As depicted in the upper part of Figure 1, the reference segmentation of an audio chunk X can be encoded into a sequence of Kmax-dimensional binary frames y = {y1, . . . , y T } where yt ∈ {0, 1} Kmax and y k t = 1 if speaker k is active at frame t and y k t = 0 otherwise. We may arbitrarily sort speakers by chronological order of their first activity but any permutation of the Kmax dimensions is a valid representation of the reference segmentation. Therefore, the binary cross entropy loss function LBCE (classically used for such multi-label classification problems) has to be turned into a permutation-invariant loss function L by running over all possible permutations perm(y) of y over its Kmax dimensions:

L (y, ŷ) = min perm(y) LBCE (perm(y), ŷ) (1)
with ŷ = f (X) where f is the segmentation model whose architecture is described below. In practice, for efficiency, we first compute the Kmax × Kmax binary cross entropy losses between all pairs of y and ŷ dimensions, and rely on the Hungarian algorithm to find the permutation that minimizes the overall binary cross entropy loss.

For the segmentation model f , we use the exact same architecture as the one introduced by [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF]. It consists in a SincNet convolutional block [START_REF] Ravanelli | Speaker recognition from raw waveform with sincnet[END_REF] (that performs frame-wise feature extraction), four bi-directional LSTMs (that contains most of the learnable weights), two fully-connected layers, a classification layer with Kmax outputs, and a final sigmoid activation function to squash the frame-wise speaker activities ŷ between 0 and 1.

At test time, a subsequent binarization step is needed to decide whether each speaker s is active in each frame t. This is achieved by comparing ŷst to a detection threshold θ ∈ [0, 1] whose value needs to be tuned carefully.

... to powerset multi-class classification

While the standard multi-label loss relies on Kmax = 3 classes (one for each speaker s1, s2, s3) that may be active simultaneously to encode overlapping speech, the powerset multi-class encoding considers a grand total of Kpowerset = 7 mutually exclusive classes: • ∅ for non-speech frames; • {s1}, {s2}, {s3} for frames with one active speaker; • {s1, s2}, {s1, s3}, {s2, s3} for two overlapping speakers 1 .

Switching to powerset encoding entails no radical change to the rest of the approach. It only amounts to changing the output size of the classification layer from Kmax = 3 to Kpowerset = 7, the final activation function from sigmoid to softmax, and the training loss function from binary to regular cross-entropy. Furthermore, a nice by-product is that the critical detection threshold θ is removed in favor of a simple argmax at test time.

Yet, powerset permutation space is slightly more complex than in the multi-label setting. For example, swapping speakers s1 and s2 in the multi-label case obviously implies swapping classes {s1} and {s2}, but also overlapping speakers classes {s1, s3} and {s2, s3}. Therefore, to find the optimal permutation in the powerset space, we use the following process: • convert target (y) from powerset to multi-label encoding; • convert binarized prediction (arg max k ŷ) from powerset to multi-label encoding; • find the optimal permutation as described in Section 2;

• permutate the target accordingly in the multi-label space, and convert it back to powerset encoding; • compute the cross-entropy loss in the powerset space.

While [START_REF] Du | Speaker overlapaware neural diarization for multi-party meeting analysis[END_REF] introduced the use of powerset encoding for speaker diarization, it has two main limitations. First, powerset encoding does not play well with a variable number of speakers: adding an extra speaker to a pool of K speakers results in K + 1 new powerset classes (one for the actual speaker, and one for each possible overlapping speakers classes). Therefore, [START_REF] Du | Speaker overlapaware neural diarization for multi-party meeting analysis[END_REF] had to resort to use a fixed and conservatively high number of speakers (Kmax = 16). Second, this leads to a number of powerset classes which is prohibitively high. For instance, using Kmax = 16 results in Kpowerset = 0 K + 1 K + 2 K = 137 classes, most of which are almost never seen during training. For reasons already explained in the introduction, our approach overcomes both issues: we can stick to a small upper bound on the number of speakers (Kmax = 3) thanks to our processing of short audio chunks, and the unsupervised clustering step takes care of estimating the number of speakers specific to each conversation later in the pipeline.

Experiments

To ensure reproducibility, the research described in this paper builds on top of the speaker diarization pipeline from version three-steps principle of BoBW approaches. More details on the pipeline internals are available in [START_REF] Bredin | pyannote.audio 2.1 speaker diarization pipeline: principle, benchmark, and recipe[END_REF] but we describe the gist of it in the following section.

Baseline

The first step consists in applying the pretrained end-to-end neural speaker segmentation model pyannote/segmentation introduced in [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF] using a sliding window of 5s with a step of 500ms. A binarization step is further applied using the detection threshold θ ∈ [0, 1] which constitutes the first hyperparameter of the approach. The second step consists in extracting one speaker embedding per active speaker in each 5s window. More precisely, speaker embeddings are only extracted from audio samples with exactly one active speaker: overlapping speech regions inferred automatically from the first step are discarded before computing the embeddings. All experiments reported in this paper rely on the pretrained ECAPA-TDNN model [START_REF] Desplanques | ECAPA-TDNN: Emphasized Channel Attention, Propagation and Aggregation in TDNN Based Speaker Verification[END_REF] from SpeechBrain [START_REF] Ravanelli | SpeechBrain: A General-Purpose Speech Toolkit[END_REF] available at hf.co/speechbrain/spkrec-ecapa-voxceleb. The third and final step consists in applying agglomerative clustering on the aforementioned embeddings, using a second hyperparameter δ as the maximum allowed distance between centroids of two clusters for them to be merged. Once each local (i.e. from first step) active speaker is assigned to a global cluster, the global speaker diarization output can be constructed.

Datasets

We report performance on the official test sets of 9 different datasets: AISHELL-4 [START_REF] Fu | AISHELL-4: An Open Source Dataset for Speech Enhancement, Separation, Recognition and Speaker Diarization in Conference Scenario[END_REF], AliMeeting [START_REF] Yu | M2MeT: The ICASSP 2022 Multi-Channel Multi-Party Meeting Transcription Challenge[END_REF], AMI [START_REF] Carletta | The AMI Meetings Corpus[END_REF] (with two variants), DIHARD III [START_REF] Ryant | The Third DI-HARD Diarization Challenge[END_REF], Ego4D2 [START_REF] Grauman | Ego4D: Around the World in 3,000 Hours of Egocentric Video[END_REF], MSDWild [START_REF] Liu | MSDWild: Multimodal Speaker Diarization Dataset in the Wild[END_REF], REPERE [START_REF] Kahn | A Presentation of the REPERE Challenge[END_REF], and VoxConverse [START_REF] Chung | Spot the Conversation: Speaker Diarisation in the Wild[END_REF]. AMI variants are headset mix (or IHM, merged audio of the participants' headset microphones) and array 1 channel 1 (or SDM, first channel from the first far-field microphone array).

For datasets distributed with an official training/development split, we use the training subset as is. For those only providing a development set, we split it into two parts which we call training and development subsets in the rest of the paper. Models were trained on a compound multi-domain training set made of the concatenation of the training sets of all aforementioned datasets, but DIHARD. DIHARD was kept aside to measure how robust our models are to domain mismatch: it is composed of 11 sub-domains, all tested individually in this paper. Figure 2 showcases the amount of data from each dataset in the compound training set as well as their source and language. We do expect that including DIHARD in the compound training set would significantly improve the results found in this paper.

While Figure 2 highlights the relative imbalance between each dataset in the compound training set, we do make sure that the compound development set is balanced, in two variants: • dev duration is built by picking random 5s chunks from the development subset of each dataset, for a grand total of 8 hours (one hour per dataset); • dev files is built by randomly picking 25 files from the development subset of each dataset3 , for a grand total of 164 files.

Experimental protocol

Speaker segmentation models are trained on the compound training set for at most one hundred hours, using Adam optimizer, an initial learning rate of 10 -3 , and a scheduler that divides the learning rate by 2 after 30 epochs with no improvement. The validation metric used by the scheduler and for model selection is the local (i.e. computed on 5s chunks) diarization error rate on the dev duration development set. hyperparameters tuning is performed with pyannote.pipeline in order to minimize the average diarization error rate on the dev files compound development subset. More specifically, in the case of an internal multi-label speaker segmentation model, two hyperparameters need to be optimized: the detection threshold θ and the clustering threshold δ. In the case of powerset segmentation models, the detection threshold θ is not needed (because it is replaced by a parameter-less argmax): only the clustering threshold δ needs to be optimized.

Domain adaptation experiments reported in Section 5 are done by fine-tuning speaker segmentation models further on one specific domain. In those cases, for each dataset (or domain in case of DIHARD), model fine-tuning relies on the corresponding training set, model selection relies on the whole development set, and so does the tuning of pipeline hyperparameters.

Results and discussions

While the first line of Table 1 reports the performance of the speaker diarization pipeline based on the pretrained multi-label model available at hf.co/pyannote/segmentation, the second line is our (successful) attempt at retraining it from scratch with the proposed compound training set.

Powerset compound training consistently gets better results than its multi-label counterpart -with an average relative improvement of 8% for in domain datasets (from 25.6% to 23.5% diarization error rate). Further adapting the powerset model to each dataset gives an additional 8% relative improvement, reaching state of the art performance for most of them.

Both approaches are tested for robustness to domain mismatch using the DIHARD III dataset, which is composed of 11 domains that were never seen during (compound) train-Table 1: Performance on 8 different datasets. We report diarization error rates on the official test sets (with the exception of Ego4D validation set because test labels are not available). Best result for each dataset are reported in bold (as well as those less than 5% worse relatively). No forgiveness collar is used for evaluation, except for numbers in italics with a grey background computed with a 250ms forgiveness collar (to allow comparison with some results reported in the literature).

indicates the winning (ensemble) submission to the VoxSRC or DIHARD III challenges. ing. Powerset training shows a 11% relative improvement over multi-label (from 33.8% to 29.9%), even larger than for the in domain case. This suggests that powerset training is more robust to domain mismatch than its multi-label variant -a behavior that could be explained by the removal of the otherwise sensitive detection threshold θ.

AISHELL-

Figure 3 provides a detailed analysis of the errors committed by the various approaches. We find that most of the improvement comes from a consistent reduction in missed overlapping speech. This also holds for in-domain data (with missed detection rate reduced from 13.1% to 9.9% on average), supporting the idea that powerset explicit modeling of overlapping speakers classes has a huge impact on the overall performance.

Fine-tuning the powerset segmentation model on each DI-HARD domain ('with domain adaptation' bars in Figure 3) tends to converge towards an overall better compromise between false alarm and missed detection rates, with little to no impact on speaker confusion. This suggests that this additional domain adaptation step also plays the role of the detection threshold θ (which is a separate hyperparameter in the multilabel case); thus making the whole approach even more robust.

Conclusion

In this paper, we study the impact of framing the speaker diarization task as a powerset multi-class classification problem. We extensively test this approach on 8 in domain datasets, as well as the 11 domains of DIHARD used as out-of-domain data. Compared to the classic multi-label approach, using a powerset multi-class encoding results in significant diarization error rate improvement (mostly due to better predictions on overlapping speech) and better robustness to domain mismatch. We obtain state of the art performance on AISHELL-4, AliMeeting, AMI, Ego4D, MSDWild, and REPERE. In the spirit of reproducible research, the powerset multi-class segmentation code is available in the open-source pyannote.audio library. The models trained on the compound training dataset, as well as their precomputed output on each separate dataset are available at github.com/FrenchKrab/IS2023-powerset-diarization.

Acknowledgements

This work was granted access to the HPC resources of IDRIS under the allocations AD011013477 and AD011012177R2 made by GENCI.

2 . 1 . 1 Figure 2 :

 2112 Figure 2: Compound multi-domain training set.

Figure 3 :

 3 Figure 3: Decomposition of diarization error rate on DIHARD III in terms of its false alarm, missed detection, and speaker confusion components. Errors happening in conjunction with overlapping speakers are hatched.

We do not consider the case of three or more overlapping speakers because the number of frames for which this happens in our benchmarking datasets is marginal: 1.6% in the compound dataset, 0.73% in DIHARD III.

At the time of writing this paper, test labels of Ego4D were not available. Hence, Ego4D results were obtained on its development subset and should therefore be taken with a grain of salt.

Except for AISHELL-4, AliMeeting, and AMI which have only 20, 8, and 18 files in their respective development subsets.


Frédéric Lechenault
email: frederic.lechenault@phys.ens.fr

Iyad Ramdane

Sébastien Moulinet

Martin Roman-Faure

Matteo Ciccotti
email: matteo.ciccotti@espci.fr

Soft coring: how to get a clarinet out of a flute?

Cutting mozzarella with a dull blade results in poorly shaped slices: the process occurs in a configuration so deformed as to yield unexpectedly curved surfaces. We study the rich morphogenetics arising from such process through the example of coring: when a thin cylindrical hollow punch is pushed into a soft elastomer, the extracted core is "clarinet-shaped", reaching diameters far smaller than those of the tool. With contributions from fracture mechanics and large strain theory, we build a simple yet quantitative understanding of the observed shapes, revealing the crucial role of friction.

When we use a cookie cutter on a thin layer of dough we obtain the shape we meant, e.g. a gingerbread man! Yet when coring soft elastic materials like rubber with long cylindrical punches, we are surprised to obtain thin clarinet shapes like those shown in Fig. 1. As a matter of fact, while both ends of the core have the same radius R as the coring blade, the central part of the core reaches a steady-state radius R core which can be significantly smaller than the former if the cored substrate is thick enough. The overall shape is surprisingly symmetric despite the fact that the coring process has a well defined direction involving large asymmetric strains.

In most cutting operations, we impose the path of the blade (at least we try to). However, as illustrated by our soft coring experiments, the final shape of the cut parts is selected by the deformation of the sample induced by the blade. In practical situations, in order to enforce a close match between the blade path and the final shape, different strategies are used to minimize this deformation, or to reduce the applied forces. Several lines of work have investigated classical ways to cut efficiently, such as sharpening the blades [1,2], introducing slicing motion [3][4][5][6], or minimizing friction [7][8][9]. Notably, a plethora of tools have been developped historically, specialized for each material, cutting speed, and target chip size [START_REF] Atkins | The science and engineering of cutting: the mechanics and processes of separating, scratching and puncturing biomaterials, metals and non-metals[END_REF]. However, when addressing soft materials such as food, gels or living tissues, these strategies can fail and shape control is lost.

The soft coring is investigated here as a robust model experiment for rationalizing the geometrical and material nonlinearities induced by the coupling of large strains and cutting. In this situation, the cutting edge of the blade moves into the material after the latter has been compressed by the applied force required for cutting, resulting into a biaxial extension in a plane perpendicular to the direction of cutting. This is the key geometrical nonlinearity of the problem: the final shape of the core results from its elastic recovery once the coring is completed.

A preliminary insight into the scaling of this phe-

mm

FIG. 1. Some examples of clarinet-shaped cores obtained by cutting rubber with a cylindrical punch. In this case, the punch had a radius of 2mm, as witnessed by the similar size of the core extremities, but the core radius changes as the samples differ in stiffness.

nomenon can be obtained by considering the brittle limit, where deformations are small, achieved when using very sharp tools. We consider a cylindrical coring tool of radius R pushed with a force F orthogonally into a soft material with Young's modulus E, as illustrated in Fig. 2. For small strain, the cut length δ cut can be identified with the vertical blade displacement δ. Invoking Saint-Venant's principle we can expect a steady-state cutting regime when the distance of the cutting edge from the external surfaces is larger than R, resulting in a core of constant final radius R core < R. While this provides a rationale for the clarinet shape of the two edges of the core, the focus here will be on the modeling of the ratio λ R = R/R core for steady-state cutting. When initially neglecting friction on the blade and bulk energy dissipation, we can describe the cutting process with fracture mechanics tools such as the Griffith energy release rate. The work performed when the blade is advanced by a distance dδ is:

dW = F dδ ≃ F dδ cut FIG.
2. Sketch of the coring mechanism. The compressed region below the blade of radius R undergoes a contraction of λz < 1 in the vertical direction and an equibiaxial expansion of λR in the horizontal plane. While the external portion of the cut sample remains stretched around the blade, the inner part relaxes into the final core shape. The vertical displacement δ of the blade can be split into the cut length δcut and an elastic component δ el .

The average compressive stress at the scale of the blade radius R is σ ∼ F/R 2 . The elastic energy U el stored in the compressed sample can be written as

U el ∼ σ 2 E R 3 ∼ F 2 ER
which does not depend on the crack length δ cut in the considered steady-state regime. The cutting criterion thus reads

G = dW dA - ∂U el ∂A = F 2πR = Γ cut (1)
F cut = 2πRΓ cut (2)
where the cutting energy Γ cut represents the cost for cutting a new unit area of material dA = 2πRdδ cut . The average vertical contraction at the scale R below the blade is thus:

ε cut ∼ - F cut R 2 E ∼ - Γ cut ER
Due to incompressibility of rubber, this results into a local equibiaxial extension in the horizontal directions ε R = -ε cut /2, which sets the scaling for the final radius R core of the cored sample:

ε R = R R core -1 ∼ Γ cut ER (3)
The outcome of this preliminary scaling analysis is that the shape of the core region is ruled by the ratio between the size R of the cylindrical cutting blade and a characteristic length scale of the material ℓ T E ≡ Γ cut /E, that we will name "tomo-elastic" length scale (from greek τ oµóς = cut). The morphogenetic effect should only be substantial when R ≤ ℓ T E . A rough estimation of ℓ T E can be obtained for rubbers by considering a typical tearing energy of Γ ∼ 1000 J/m 2 [7], and modulus E ∼ 1 MPa, which yields millimetric scales.

Following this estimation, we performed systematic experiments by changing the elastic properties of the rubbers and the radius of the coring tools in this regime. As model soft elastic rubbers, we selected a family of four commercial silicon-based elastomers (Elite Double ® , Zhermack). Their mechanical properties were measured in a standard uniaxial tensile test and reported in Fig. 3. The corresponding elastic moduli are reported in Table I. The samples were cast into cylinders 100 mm in diameter and variable heights up to 30 mm in order to be at least four times larger than the blade radius. The cylindrical blades were made from a set of steel tubes and syringes, with 15 different diameters ranging from 0.5 mm to 8 mm. The blade thickness h remains in approximately constant proportion R/5. The tubes were sharpened using a conical milling tool providing variable internal blade angles θ from 10 • (sharp) to 90 • (flat). The coring protocol is as follows. A tube is mounted vertically in the jaws of a mandrel attached to the moving part of a testing machine Instron ® (model 5965) with a 100 N load cell. An elastomer sample is laid on a flat wooden sheet sitting horizontally underneath the tool. The sample is then cored by moving the tube into the elastomer at a velocity of 10 mm/min until it touches the wooden plate, resulting in a sharp increase in the measured force. The steady-state diameter of the cut 3 with Eq. ( 11). Γcut and ℓT E are the values of the cutting energy and tomo-elastic length obtained by fitting the coring data in Fig. 4 with version α of Eq. ( 8).

part is then measured by taking a high resolution picture with a D800 Nikon camera. The experimental results for the core radius R core and cutting force F cut measured in the steady-state regime are reported in Fig. 4 and Fig. 5 as a function of the tool radius R. The values of h and θ are not reported here since they had no appreciable influence on the results. As predicted by the preliminary linear scaling model, the coring experiments confirm that the samples undergo large strains when the cylindrical blade radius R becomes comparable or smaller than the millimetric scale estimated for the tomo-elastic length ℓ T E . Unsurprisingly, the linear model fails to capture the functional dependency of the core size on the blade radius. Since the observed strains can become larger than 300%, we developed a more advanced model, based on hyperelasicity, in order to account for both finite strains and nonlinear material behavior.

From a geometrical point of view, the elastic indentation δ el becomes larger than the blade radius R, and it can no longer be neglected in front of the length δ cut of the cut material, so that the blade displacement should be written as δ = δ el + δ cut (see Fig. 2). The final core radius R core is selected by the value of the in-plane equibiaxial stretch λ R ≥ 1 of the elastomer in the compressed region of size R just below the cylindrical blade tip. The compressive force F cut = πR 2 σ T applied by the blade on the elastomer should be related to the local average true stress σ T by considering the homogeneous uniaxial compression (λ z ≤ 1) of an incompressible hyperelastic material with energy density W(I 1 ), where I 1 is the first invariant of the Cauchy strain [START_REF] Ogden | Nonlinear elastic deforma-tions[END_REF]:

I 1 = λ 2 z + 2 λ z λ z = 1 λ 2 R σ N = dW(I 1 ) dλ z = 2 dW dI 1 λ z - 1 λ 2 z ( 4
)
σ T = λ z σ N = 2 ∂W ∂I 1 1 λ 4 R -λ 2 R ( 5
)
The value of the local compressive force at the scale R of the blade front during steady-state cutting can also be related to the cutting energy Γ cut by a Griffith-like energy balance similar to Eq. ( 1). During steady-state cutting all the local parameters such as F cut and λ R are constant. In the absence of friction, the elastic component of the displacement δ el is also constant, and the displacement of the loading point is thus simply related to the variation of the cut length δ cut as measured in the reference frame: dδ = dδ cut . However, the evaluation of the relevant area for the expression of the surface energy term is delicate under large strain: while the blade front effectively advances along a cylindrical surface of radius R, the surface cut in the unstrained material is a cylinder of smaller radius R core . We chose to initially preserve both choices in the formulation of the model, before discussing their significance and consistency with our measurements:

dA = 2π {R; R core } dδ cut = 2π R {1; λ R } dδ cut
where the term in brackets {α; β} means the alternative between the two different model hypothesis. The Griffith-like cutting condition becomes then:

G = F cut {1; λ R } 2πR - {1; λ R } 2πR dU el dδ cut = Γ cut (6)
Although the local strain field has a complex shape, when considering the steady-state cutting regime away from external surfaces, the variation of the total elastic energy for a unit cut advance can be estimated by the residual stretch energy in the outer portion of the cut cylinder (the inner part being relaxed as sketched in Fig. 2). For the plane-strain stretch (λ z = 1) of a cylindrical cavity of radius R core in an incompressible hyperelastic material, the stretch field is uniquely determined by the stretch λ R imposed by the punch as [START_REF] Cheewaruangroj | [END_REF]:

λ r (r) = 1 λ θ = 1 1 + R 2 core r 2 (λ 2 R -1) I 1 (r) = 1 + R 2 core r 2 (λ 2 R -1) + 1 1 + R 2 core r 2 (λ 2 R -1) + 1 dU el dδ cut (λ R ) = ∞ Rcore W(r)2πrdr
According to Eq. ( 6) the steady-state cutting force F cut can be expressed as:

F cut = Γ cut 2πR {1; λ R } + dU el dδ cut (λ R ) (7)
The value of the radial stretch λ R can thus be determined as a function of the cutting energy by equating the two expressions for the true stress σ T = -F cut /πR 2 between Eq. ( 5) and ( 7):

2 E dW dI 1 1 λ 4 R + λ 2 R + 1 πR 2 E dU el dδ cut (λ R )+ 2Γ cut ER {1; λ R } = 0
(8) where we chose to divide every term by the elastic modulus E in order to obtain a dimensionless equation for λ R . This constitutes the main relationship of our finite strain modeling for the shape of the central steady-state portion of the core.

When comparing with the linear model, we remark that the large strain model contains four different sources of non-linearity, which can be attributed either to large geometry changes or to the large strain material response. The equibiaxial expansion of the compressed region underneath the blade of size R has three different consequences: (1) the hyperelastic uniaxial compression law becomes non-linear due the stretch of the application area (left term); (2) the external part of the cut material remains stretched around the blade and stores elastic energy (middle term); (3) the area of the unstrained cut surface is much smaller than the area swept by the blade (right term). Moreover, the left and middle terms are also affected by the specific nature of the large strain material response encoded in W(I 1 ). The competition between these different terms selects a single value of the stretch λ R as a function of the cutting energy Γ cut , the elastic modulus E and the tool radius R. Since the left and middle terms scale as W, which scales with E, dimensional analysis confirms that the dimensionless equilibrium stretch λ R must be written as a function of the dimensionless combination Γ cut /ER = ℓ T E /R.

In order to gain more insight into the relative impact of the four nonlinearities, let us first focus on the geometry by restricting our model to a Neo-Hookean energy density functional [START_REF] Cheewaruangroj | [END_REF]:

W = E 6 (I 1 -3) dU el dδ cut (λ R ) = π E 3 R 2 core (λ 2 R -1) log λ R (9) 1 λ 4 R -λ 2 R + 1 - 1 λ 2 R log λ R + 6 Γ cut ER {1; λ R } = 0 (10)
The solution of this simplified model does not correctly predict the dependency of our experimental data for λ R on the radius of the cutting tool, which can only be captured by taking into consideration the full nonlinear material behavior. However, this Neo-Hookean expression allows to appreciate that the middle term (residual cylindrical stretch) becomes very small in front of the left term (uniaxial compression) when the radial stretch λ R is increased for small values of R. Moreover, it is easily shown that when expanding any elastic energy potential W(I 1 ) into a series of power law terms (I 1 -3) n in order to capture material strain hardening, the contribution to the left term will always remain nλ 2 R times larger than the middle term. We can thus comfortably approximate Eq. ( 8) by ignoring the middle term when evaluating our model for an arbitrary nonlinear constitutive law.

In particular, in order to capture the strain hardening behavior of our elastomers, displayed in Fig. 3, which present an initial softening behavior followed by a linear strain hardening of the σ N (λ) curve, we use Eq. ( 4) with the following custom expression:

dW dI 1 = E 6(1 + a) 1 + a 1 + b(I 1 -3) c (11)
The fitted parameters for each elastomer are reported in Table I. The corresponding solutions of equation ( 8) (where we dropped the middle term) are plotted in Fig. 4. Both versions α and β of the model can adjust the experimental dataset very well for each elastomer over a wide range of cutting blade radii. However, the values of the tomo-elastic length ℓ α,β T E = Γ α,β cut /E that are extracted by the fitting procedure depend on the choice of the model α or β. The corresponding values of Γ α,β cut are reported in the inset of Fig. 5.

Let us now discuss more extensively the important role of friction in the coring process. When pushing a blade into a soft material we can soundly separate two main (8) .

contributions of friction, to be respectively associated to the contact of the cut material with the side blade walls and the tip region. The first contribution F f increases with the progression of the cut length δ cut . In contrast, the second one is localized at the scale of the blade tapered region, which in steady-state cutting conditions is expected to be invariant during the advance of the blade. We remark that only the latter contributes to the local compressive force F cut , which is responsible for the selection of the lateral stretch λ R at the moment of cutting, according to the model defined above. This explains why the observed core radius is almost constant during the cutting process, while the total applied force F = F cut + F f is slowly increasing due to increase of the sliding surface. Following the approach of Irwin [13], the local dissipation energy due to friction on the tapered region can be added to the fracture energy required to break the polymer network resulting into a global energy cost for cutting a unit area of material Γ cut . However, when considering soft materials, the fracture energy term should be evaluated in the unstrained reference (radius R core ), while the local friction energy term should be determined by the actual reference (radius R), as described respectively by models β and α introduced above. Although the two contributions to Γ cut were estimated to have the same order of magnitude [1,2,7], the relevance of the friction contribution to Eq. ( 8) will be dominant for increasing values of λ R . This explains why only model α provides values of F cut which are physically consistent, i.e. lower than the forces F ss measured independently during the initial stages of steady-state cutting, as reported in Fig. 5 along with the corresponding values of Γ cut in the inset. This interpretation is also consistent with the relatively weak influence of the blade sharpness on the measured radius R core of the cored regions. Future investigations might allow independant identification of these two contributions.

The presented model captures the main features of the steady-state core radius, the selection of which is mainly dominated by the cylindrical blade radius R, the cutting energy Γ cut and the elastic modulus E, in the form of the ratio between R and the tomo-elastic length ℓ T E = Γ cut /E. The core shape is thus determined by a combination between geometrical and material nonlinearities. Although the physical phenomenon can be qualitatively explained by the geometrical nonlinearity associated to cutting the soft material in a stretched condition, the quantitative relation between the final core radius and the cylindrical blade radius is mainly determined by the details of the nonlinear -strain hardening -material behavior. We remark that if the material were Neo-Hookean, the geometrical effect would be considerably larger than observed in our experiments, since strain hardening has the effect of limiting the stretch in the compressed region. Although the role of friction on the side walls of the blade does not affect the shape of the core, local friction on the blade edge has a dominant effect on the value of the cutting energy, and thus on the tomo-elastic length, which cannot be identified with the elasto-cohesive length Γ f ract /E defined for fracture phenomena [14]. As a final remark, friction is also very important for the origin of the two clarinet shapes in the edges, since it prevents the lateral stretching and thus enforces the core and tool radius to be the same. For the first (incoming) clarinet the lateral stretch is suppressed by static friction of the blade before the initiation of cutting. For the second (outcoming) clarinet the lateral stretch is suppressed by static friction between the sample and the bottom support plate. In both cases, the transient regions shaping the clarinet bells extend over a length R from Saint-Venant's principle. Our study opens perspectives into complex programmable surface shaping, in a mundane, accessible setting fitted for an array of design applications.

FIG. 3 .

 3 FIG.3. Uniaxial extension curves (nominal stress against stretch) for the four elastomers (•), and the corresponding fit with Eq. (11) (lines), as well as with Neo-Hookean law (dash-dot).

FIG. 4 .

 4 FIG.4. Measured ( ) coring ratios against the radius of the cylindrical blade. The full and dashed lines represent the predictions of models α and β respectively (cf. Eq. (8)).

2 FIG. 5 .

 25 FIG.5. Scaling of the measured cutting force Fss in the initial stage of steady-state cutting against the blade radius R. The inset reports the cutting energy Γcut estimated by force measurements (•) and by fitting the core shape data from Fig.4with model α (⋄) and β ( ) defined by Eq.(8) .


Lucas Mele

Rachida Babouri

Jean-Luc Pirat

] Eder Tomás-Mendivil

] David Mar

Tahar Ayad

David Virieux

Beyond the Limits of Atropochirality: Design of Highly Conforma:onally Restrained Biaryls with Bridgehead Phosphine Oxide

In the last three decades, reac/ng sterically congested ortho-subs/tuted arenes to form atropochiral biaryls is an appealing venture and a challenging subject that has garnered significant a=en/on. Therefore, there is interest in developing methods to prepare these compounds. In this study, an efficient approach to produce a new class 2,2' disubs/tuted biaryls bridgehead phosphine oxides with an unusual topology and excep/onal conforma/onal stability is presented. Our methodology demonstrates that depending on the subs/tu/on pa=ern on the aryl moie/es, the methanophosphocine backbone could be rigid enough to observe a double atropochirality, resul/ng in an under covered class of molecules. Notably, our studies revealed that replacing only one hydrogen at the ortho posi/on by a fluorine atom led to sufficiently restricted rota/on at temperatures below 80 °C, extending "far away" the limits of atropostability. Finally, our inves/ga/ons, which employed variable-temperature NMR spectroscopy and DFT calcula/ons, yielded unique insights into the isomerisa/on mechanism, indica/ng that the two biaryl mo/fs are fully independent in spite of their proximity

Introduc-on

 The literature is also replete with naturally occurring bioac/ve compounds or materials that exhibit axis of chirality. 3 Although ini/ally seen as a source of complexity by medicinal chemists, modern asymmetric methods 4 have allowed the development of conforma/onally stable atropisomers that are now marketed drugs or drug candidates in clinical phases. 5 For all these reasons, the biaryl mo/f is universally recognized as a privileged scaffold that is increasingly used as a chiral inducer in numerous asymmetric catalysis. 6 Atropisomerism generally occurs when the rota/on of a single σ-bond connec/ng two molecular fragments is restricted. This hindered rota/on is usually caused by steric constraints, which can be enhanced by electronic effects, resul/ng in slow interconversion of the two enan/omers. Biaryl compounds are undoubtedly fundamental atropochiral building-blocks, where the size and number of ortho-ortho'-subs/tuents strongly determine the conforma/onal stability and hinder the rota/on of the pivotal bond connec/ng the biaryl mo/f. 7 The energy barrier of atropisomerisa/on is then directly related to the bulkiness of the subs/tuents. However, the reac/on of sterically congested ortho-subs/tuted arenes to form atropochiral biaryls is s/ll a challenging task. Therefore, there is s/ll need for efficient synthesis of highly conforma/onally constrained biaryl atropisomers. 8 2,3 : 6,7-Dibenzobicyclo[3.3.1]nona-2,6-diene-4,8-dione 1, 9 Kagan ether 10 2 and Tröger bases 11 3 are representa/ve examples of conforma/onally rigid Λ-shaped molecules. Among these, Tröger bases have suited significant interest in supramolecular chemistry and in ligand design. 12 Following our recently published synthesis of methanophosphocines 4, a new class of P-bridgehead bicyclic molecules adop/ng a similar shape, 13 we hypothesized that aryl disubs/tuted methanophosphocines 5 may present an

unusual topology. Specifically, we an/cipated that methanophosphocine backbone could be rigid enough to observe a double atropochirality when the biaryl mo/f was polysubs/tuted resul/ng that have received li=le a=en/on but were recently highlighted by Bao and Bonne (Figure 1). 14 To the best of our knowledge, only very few papers embarked bridgeheaded structural mo/fs closely fused to a biaryl and none of them have discussed the asymmetric behavior of such systems. 15 Even the recent and elegant work of Wencel-Delord et al., who reported an efficient and tunable method to unsymmetrical bisbiaryl via a pericyclic reac/on, did not highlight this feature. 16 In this paper, we describe how the topology of constrained biaryls bridgehead phosphine oxides, namely disubs/tuted methanodibenzophosphocines 5, pushes the boundaries of axial chirality, thus paving the way for designing and developing poten/ally new chiral ligands.

Results and Discussion

Synthesis of bis(diaryl)methanodibenzophosphocines 5

Palladium catalyzed reactions are known to be sensitive to the steric bulk of the partners, and typically require high temperatures, highly reactive reagents or carefully designed catalytic systems for effective C-C bond formation. 17 In keeping with this knowledge, we explored the Suzuki-Miyaura coupling of 4,8dibromo-5,7,12-trihydro-6,12-methanodibenzo[c,f]-phosphocine oxide 6, which has been previously obtained by gold-catalyzed intramolecular double hydroarylation of alkynes. 13 To this end, bromo substituted methanophosphocine 6 was reacted with an excess of phenyl boronic acid 7 a (Ar=Ph, 8 equivalents) in the presence of 10 mol % of palladium tetrakis(triphenylphosphine) in a mixture of dioxane/water (96:4) and potassium carbonate as a base (Table 1). The mixture was heated at 110 °C for 1 h under microwave irradiation. After purification, the expected methanophosphocine 5 a was isolated in excellent 96 % yield (entry 1). Notably, attempts to repeat the aforementioned Suzuki-Miyaura reaction led to irreproducible results, possibly due to the high sensitivity of palladium tetrakis(triphenylphosphine) complex to oxygen. Gratifyingly, after intensive optimization, we found that this issue could be resolved by conducing the reaction with 10 mol % of the less sensitive Pd2dba3 as a palladium source in the presence of 20 mol % of the bidentate Xantphos ligand, at 140 °C for 2 h under microwave irradiation, providing 5 a in similar high yield. With the optimal conditions established, we explored the scope of the reaction using dibromo-substituted methanophosphocine 6 and various functionalized arylboronic acids. As shown in Table 1, the reactions generally proceeded smoothly, providing the corresponding bis(diaryl)methanodibenzophosphocines 5 a-m in yields ranging from 26 % to 96 % (entries 1-13).

Table 1. Synthesis of bis(biaryl)methanophosphocines 5 a-m. [a]

Atropisomery in bis(diaryl) methanodibenzophosphocines 5

To investigate the presence of different diastereomers, 31 P{ 1 H} NMR analyses were performed. Results showed that only one 31 P NMR signal was observed for bis(aryl)methanophosphocine derivatives 5 a-g which were unsubstituted in ortho-positions. It is generally accepted that most of tri-and tetra-substituted biphenyls exhibit atropochirality. Interestingly, bis(aryl)methano-dibenzophosphocines 5 h-m that only possessed ortho substituents on the aryl moieties displayed the presence of three signals in 31 P NMR in a ratio 1:2:1 (Figure 2). This result indicated the presence of three diastereomers and clearly demonstrated that ortho substituted derivatives 5 h-m were constrained enough to adopt non-planar ground states. The nature of these diastereomers depended solely on the relative orientation of the ortho-substitution of biarylsubstituents. Considering the methylene CH2-bridge, two meso-diastereomers (meso-down/down and meso-up/up) could be formed whereas the latter was the racemic chiral derivative (chiral-up/down or chiraldown/up) (Figure 2a). The existence of four different stereoisomers was confirmed by the presence of the corresponding four signals when the 31 P NMR was recorded in the presence of L-dibenzoyl tartaric acid as a chiral resolver (Figure 2b). The two extreme signals were assigned to the meso derivatives while the signal of chiral enantiomers was split into two singlets. Notably, this splitting was also confirmed by chiral HPLC analysis of the racemic 1-naphthyl derivative 5 l as illustrated in Figure 2c. Indeed, for compound 5 l, we were able to separate the two meso compounds from the racemic chiral derivative by flash column chromatography thus opening the way to access enantiopure chiral ligands with this unusual topology. From a stereochemical perspective, this system shares some similarities with 2,3,4-trihydroxyglutaric acid. 18 By analogy, the two meso stereoisomers are achiral and the central phosphorus atom is stereogenic but not chirotopic. For the two enantiomers, although chirotopic, the phosphorus atom is not a stereogenic center as it bears two identical ligands. This situation is not common for molecules bearing chiral centers, and to our knowledge, has never been described for axially chiral systems. Remarkably, beside the hindered 1-naphthyl derivative 5 l, stable stereoisomers were also observed from mono-substituted biaryl methanophosphocines 5 i-5 k where the ortho-substituents were respectively a methyl (5 h and 5 i), a methoxy (5 j and 5 k). Even more interestingly, the observation of diastereomers at room temperature for the bis(2,4-difluorophenyl)methanophosphocine 5 m is noteworthy given that conformationally stable 2,2' disubstituted biphenyls remains extremely rare in literature, and only those with bulky ortho and ortho'-groups do not racemize at room temperature. 19 The apparent size of orthosubstituent (steric hindrance) determines the atropostability. With a Van der Waals effective radius of 1.47 Å, fluorine as individual atom is slightly larger than hydrogen (1.20 Å) and quasi isosteric to the "naked" oxygen (1.52 Å). For these reasons, fluorine is considered as a good isostere of a hydrogen atom while being smaller than a hydroxyl group. 20

Molecular structure, conformation and DFT calculations

Bicyclo[3,3,1]nonane and related compounds generally exhibit conformational flexibility. As mentioned by Issleib 22 and later discussed, 23 these compounds coexist under three main conformations: boat-boat, boat-chair and chair-chair conformations (Figure 4a). However, due to the presence of fused phenyl rings, dibenzomethanophosphocines 4 exhibit a unique conformation with a cleft-like shape (Figure 4b and4c). This strained system resulted in a specific orientation of methylene PCH2Ar where the hydrogen atoms are positioned in pseudo-axial and pseudo-equatorial positions (Figure 4d). Ab initio methods (B3LYP/6-311G**) were used to calculate the different conformations of ground states and transition states. 24 First, optimized calculations were performed on the fluorine-derivative 5 m to estimate the energy difference for the smallest ortho-substituent (R = F). The calculations showed that the ortho-substituted biaryl motifs adopt an angular shape in their ground states (Figure 5a). Little energy differences were observed among all stereoisomers (0.4-0.5 kcal.mol -1 ), indicating that there is no preferred conformation. To determine the energy barrier for rotation of the C-C bond, DFT calculations were performed using bis(2,4-fluorophenyl)methanophosphocine 5 m as a model substrate (Figure 5b). The system consists of four transition states where one biaryl system is planar (in either anti-or syn-route), while the second one is an angular and stable conformation (either up or down). There are two anti-TS with the second fluoroaryl group oriented up or down and two syn-transition states. The preferred path for atropisomerisations occurs through a transition state with the minimum of energy. With a barrier of 16.4 kcal.mol -1 , the anti-routes are systematically preferred to the syn-paths which are 4.1 kcal.mol -1 higher in energy. As observed for the ground states, the conformation of the second aryl group does not affect the energy, meaning that the two biaryl-systems are fully independent, and their respective conformations do not affect each other. The exchange rate between two stereoisomers was calculated at the coalescence temperature and compared with the one extrapolated from NMR analysis. As expected, the results were consistent with a mean error of less than 10 % (see Supporting Information for details).

We hypothesized that atropisomerization in methano-phosphocines 5 could be compared to the same phenomenon in 2,2'-disubstituted biaryls 7 (Table 2). The lowest energy transition states for substituted methanodibenzophosphocines 5 h-m (anti vs syn-route) showed that the two phenyl rings of a biphenyl motif are almost in the same plane. The ortho-hydrogen (blue H) bisects the hydrogen atoms of PCHaHe group, and the system is rigidified by the bicyclic core. The phosphoryl is then rejected in the anti-position and it does not interact with the biaryl motif. On the other hand, 2-methyl-biphenyls 7 with variable Rgroup in position 2' have a methyl substituent that is free to move and has no fixed conformation. The difference of transition state energy between methanophosphocines 5 h-m and biaryls 7 is directly linked to the rigidification induced by the bicyclic core. DFT calculations were extended to the bis(diaryl)methanophosphocines 5 h-m and their rotational barriers were compared with reported data for the 2,2'-disubstituted biaryls. 25 The structural strain resulted in an increase in energy of around 5 kcal.mol -1 . This difference is observed even for substituents as small as fluorine and is significant enough to observe an atropostability for Me-and naphthyl-substituted methanophosphocines 5 h-i and 5-l.

Table 2. Compared atropisomeriza@on barriers of 2,2'-disubs@tuted biaryls 7 vs methanophosphocines 5. [c] t1/2 biaryl 7 [a] 5 [b] H 7. In order to investigate the electronic properties of methanodibenzophosphocines, complementary experiments were investigated. First, we examined the proton-decoupled 31 P NMR spectrum of the phosphine 8 which was synthesized by reducing dinaphthyl methanophosphocine oxide 5 l with DIBAL-H (scheme 1). 26 The spectrum showed three singlets at very high field at -77.8, -77.9 and -78.0 ppm (CDCl3), respectively. The chemical shifts of resulting phosphine 8 felt well upfield from those of simple trialkylphosphines (δPMe3=-62 ppm, δPnBu3=-32.3 ppm, δPiPr3=-19.4 ppm) 27 as well as for strained phosphatrypticenes (δ =-64.8 ppm) 28 and the simplest methanophosphocine (phospha bicyclo[3,3,1]nonane, δ =-62.0 ppm). 29 The rhodium(I) complex 9 [ClRh(CO)2( 8)] was also prepared by reaction of 8 with [RhCl(CO)2]2. 30 The 31 P NMR spectrum of complex 9 displayed three doublets corresponding to the three diastereoisomers at δ 3.3, 3.0 and 2.8 ppm, with an identical 1 JPRh coupling constant of 170 Hz. The infrared stretching frequencies, which reflects the π-acceptor character, were found to be 2024.8 and 1999.0 cm -1 . Next, we examined the reaction between methanophosphocines 8 with powdered elemental selenium, using a protocol compatible with strongly donating phosphine. 31 Following the reaction by 31 P{ 1 H}NMR, we observed the disappearance of the signals at -77.8 to -78.0 ppm and the rise of 2 new signals centered at 2.7 and 2.6 ppm. The signal at 2.7 ppm was the superposition of two diastereoisomers. The 31 P-77 Se coupling constant were measured and found to be 725 and 723 Hz, respectively. They were close to those observed by Stammler for phosphabicyclo[3,3,1]nonane (721.4 Hz). 29

R [kcal.mol -1 ] T °C

Scheme 1 : Reduc@on of 5 l into phosphine 8; rhodium complex 9 and phosphine selenide 10 synthesis.

The percent buried volume (%Vbur) was calculated to gauge the steric properties of such new phosphine 8.

It was found to be 26.1 %. 32 Surprisingly, the accessibility of the phosphorus center appeared to be similar to trimethylphosphine with a %Vbur of 26.1 for a metal-phosphorus bond length of 2.28 Å.

Conclusion

In summary, we have developed an efficient entry into a new class of highly conformationally rigid [3.3.1] biaryls bridgehead phosphine oxides. The procedure involves a double Suzuki-Miyaura coupling reaction of readily available dibromo-substituted methanodibenzophosphocine 6 with various arylboronic acids, leading to the formation of diversely substituted bis(diaryl) methano-dibenzophosphocines 5 that possess an unusual topology. Depending on the substitution pattern on the aryl moieties, the methanophosphocine backbone could be rigid enough to observe a double atropochirality, resulting in an under covered class of compounds. Our study revealed that even small ortho substituents, such as methyl, methoxy, and fusedphenyl groups induced hindrance large enough to observe stable configurations. Surprisingly, replacing just one hydrogen at the ortho position with a fluorine atom led to sufficiently restricted rotation at temperature below 80 °C pushing "far far away" the limits of atropostability as illustrated in Figure 6. DFT computations (B3LYP/6-311G**) were used to probe the mechanism of isomerisation, which showed that the two biaryl motifs are fully independent despite their proximity and that the isomerization proceeds through the antipathway which exhibits the lowest energy. We are continuing our experimental work to explore the potential of these compounds as novel classes of ligands and are also investigating further modifications to improve their atropostability. We will report on these findings in due course.

Experimental Section

All experimental and DFT-calculation data associated with this work are available in the Supporting Information. Were included synthetic procedures, structural analyses, variable temperature NMR and DFT computational results as well as the spectral data of the new compounds.

Figure 1 :

 1 Figure 1 : Λ-shaped molecules 1-3 and methano(dibenzo)phosphocine deriva@ves 4-5.

  Reac(on condi(ons: 6 (1 mmol), ArB(OH)2 ⋅ 7 (8 equiv.), Pd2dba3 (10 mol %), Xantphos (20 mol %), K2CO3 (2 equiv.), dioxane/H2O (1.98:0.02 mL), 140 °C (MW), 2 h.[b] Isolated yields.

Figure 2 :

 2 Figure 2 : a) Structure of the stereomers (R≠H), b) 31 P{ 1 H} NMR of stereoisomers and spliTng of enan@omers in the presence of L-dibenzoyl tartaric acid, c) Chiral HPLC (IA, 85/15: cyclohexane/isopropanol 1.0 mL/min, 25 °C) of racemic 5 l.

  Variable-temperature NMR spectroscopy is considered as the most convenient and sensitive technique to detect and measure the rotational barrier of biaryl compounds. 21 Due to the presence of phosphorus atom, investigation of the aryl-aryl rotational barrier was accomplished directly by 31 P{ 1 H} NMR experiments.To study the dynamic behavior of methanophosphocines, the ortho-methoxy-substituted compound 5 k was used as a model substrate in deuterated DMSO as solvent. The sample was heated from 25 °C to 148 °C and the barrier of rotation was calculated by measuring changes in the shape of NMR spectra as a function of temperature (Figure3). At room temperature, the line shape is constituted by the three NMR signals, indicating that no equilibration occurred at NMR timescale. Broadening of the signals was observed when the temperature rose to 90 °C. At 110 °C, only two broad singlets were visible in the NMR spectra. The complete coalescence was achieved at 130 °C and rising the temperature to 148 °C resulted in a sharper singlet signal. Cooling down to room temperature, clearly showed the reversibility of the phenomenon and resulted in the recovery of the three singlets confirming that they were related to the same molecule with different configuration/conformation.

Figure 3 :

 3 Figure 3 : Variable temperature 31 P{ 1 H} NMR spectra of methanophosphocine 5 k in [D6]DMSO.

Figure 4

 4 Figure 4 : a) Conforma@ons of phospha bicyclo[3,3,1] nonane vs dibenzomethanophosphocine 4, b) and c) clec view, d) Orienta@on of hydrogens of PCH2 group.

Figure 5 :

 5 Figure5: Ab ini@o calcula@on (B3LYP/6-311G**), a) of the different conforma@ons of ground states and transi@on states for dibenzomethanophosphocine 5 m, b) Energy barrier for rota@on for 5 m. (DFT-calcula@ons were accomplished at 25 °C).

  Experimental barrier of disubs(tuted-biphenyls at 25 °C.[b] Calculated rota(onal barrier.[c] Temperature of atropimeriza(on for 5.

Figure 6 :

 6 Figure 6 : Compared DG + rot and half-life of rota@onal stereoisomers according to their structures at 25 °C.

Acknowledgments

L. M. thanks the Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation for a Ph.D. grant. The authors greatly acknowledge C. Totée (ICGM, Univ. Montpellier, CNRS, ENSCM) for his NMR analytical contribution to this work.