Easy Access To Allylic Sulfones Through Transition Metal-Free Hydrosulfonylation Of Allenes.

Lucas Pagès, † Sébastien Lemouzy, † Marc Taillefer, † * and Florian Monnier †, ‡ * † ICGM, Univ Montpellier, CNRS, ENSCM, Montpellier, France, 240 Avenue du Professeur Emile JEANBRAU -34296 Montpellier (France) ‡ Institut Universitaire de France (IUF), 1 Rue DESCARTES, Paris 75231 Cedex 5, France.

Supporting Information Placeholder

ABSTRACT: A Brønsted acid-mediated addition of (hetero)aryl and (cyclo)alkyl sodium sulfinates to N-allenyl derivatives, which proceeds in water, is described under very smooth conditions. This reaction provided a practical and efficient protocol for the regio-and stereoselective synthesis of allylic sulfones in an atom-and step-economic fashion.

Allylic sulfones are highly valuable structures commonly found in bioactive molecules for pharmaceutical or agrochemical use. [START_REF] Chen | Sulfonyl Group-Containing Compounds in the Design of Potential Drugs for the Treatment of Diabetes and Its Complications[END_REF][START_REF] Scott | Analysis of US FDA-Approved Drugs Containing Sulfur Atoms[END_REF][START_REF] El-Awa | Evolving Organic Synthesis Fostered by the Pluripotent Phenylsulfone Moiety[END_REF][START_REF] Alba | New Reagents in Organocatalysis[END_REF][START_REF] Back | Cycloadditions and Cyclizations of Acetylenic, Allenic, and Conjugated Dienyl Sulfones[END_REF][START_REF] Buynak | The Synthesis and Evaluation of 2-Substituted-7-(Alkylidene)Cephalosporin Sulfones as β-Lactamase Inhibitors[END_REF][START_REF] Reck | Identification of 4-Substituted 1,2,3-Triazoles as Novel Oxazolidinone Antibacterial Agents with Reduced Activity against Monoamine Oxidase A[END_REF] They can also be useful intermediates to variously functionalized compounds, either through the C(sp 2 )-C(sp 2 ) bond or the carbon atom adjacent to the sulfone group. [START_REF] Nielsen | Asymmetric Organocatalysis with Sulfones[END_REF][START_REF] Trost | Sulfones as Chemical Chameleons: Versatile Synthetic Equivalents of Small-Molecule Synthons[END_REF][START_REF] Clayden | Allylic Sulfones as Allyl Anion Equivalents: Homoallylic Alcohols from Metal Catalysed Reactions of Sulfones with Aldehydes and Ketones[END_REF] The SO2 moiety is classically introduced either through in situ oxidation of sulfides (S II ) or sulfoxides (S IV ) or the use of pre-oxidized sulfur-based reagents (S VI ) i.e. sulfinic acid derivatives and their salts, in substitution or cross-coupling reactions (Scheme 1, A).

Allenes are interesting precursors of these allylic structures via hydrofunctionalization [START_REF] Blieck | Metal-Catalyzed Intermolecular Hydrofunctionalization of Allenes: Easy Access to Allylic Structures via the Selective Formation of C-N, C-C, and C-O Bonds[END_REF] with hydrothiolation followed by in situ oxidation or with direct hydrosulfonylation. The first allene hydrosulfonylation strategies were based on the use of sulfonylhydrazines as a source of sulfone. They can be catalyzed by a palladium/diphosphine system (Yamamoto, [START_REF] Kamijo | Palladium Catalyzed Hydrosulfination of Allenes with Tosylhydrazine Leading to Allylsulfones[END_REF] Scheme 1, eq 1) or by a Rhodium/DPEPhos system (Breit, [START_REF] Khakyzadeh | Rhodium-Catalyzed Addition of Sulfonyl Hydrazides to Allenes: Regioselective Synthesis of Branched Allylic Sulfones[END_REF] Scheme 1, eq 2), occurring respectively on the distal and the proximal carbon of the allene. These methods require the presence of a Brønsted acid in at least stoichiometric quantity, and the use of an organic solvent under moderately high temperature (70°C-80°C). Breit's group also developed an asymmetric rhodium-catalyzed hydrothiolation of terminal allenes followed by oxidation to afford the chiral branched allylic sulfones in a two-step process (Scheme 1, eq 3). [START_REF] Pritzius | Asymmetric Rhodium-Catalyzed Addition of Thiols to Allenes: Synthesis of Branched Allylic Thioethers and Sulfones[END_REF] Over the last years, two transition metal-free allene hydrosulfonylation methods have been described. In 2016, Miao, Ren and coworkers reported a transition metal-free tunable hydrosulfonylation of 3cyclopropylideneprop-2-en-1-ones using sodium sulfinates in the presence of acetic acid affording the corresponding allylic sulfone (γ-adduct) in DMSO [START_REF] Miao | Solvent-Controlled, Tunable Hydrosulfonylation of 3-Cyclopropylideneprop-2-En-1-Ones[END_REF] (Scheme 1, eq 4). Very recently, the group of Loh reported a transition-metal catalyst-free regioselective Michael addition of sulfinic acids to α-ketoallenes in a mixture of water and ethanol as a solvent, affording the corresponding vinylic sulfone [START_REF] Goh | Synthesis of Vinylic Sulfones in Aqueous Media[END_REF] (Scheme 1, eq 5). This method requires the synthesis of sulfinic acids in one to two steps, either from sodium sulfinate or sulfonyl chloride. 17 In our course to build new original allylic structures, we started to investigate the C-S bond formation via coppercatalyzed hydrofunctionalization of allenes. After starting with usual conditions described in our previous work for the C-C, [START_REF] Blieck | Stereo-and Enantioselective α-Addition of Carbonyl Nucleophiles to Allenamides Catalyzed by a Synergistic Copper/Enamine System[END_REF][START_REF] Ali Abdine | Hydroarylation of N-Allenyl Derivatives Catalyzed by Copper[END_REF][START_REF] Blieck | Regio-and Stereoselective Copper-Catalyzed Allylation of 1,3-Dicarbonyl Compounds with Terminal Allenes[END_REF] C-N [START_REF] Blieck | Copper-Catalyzed Hydroamination of Terminal Allenes[END_REF][START_REF] Perego | Copper-Catalyzed Hydroamination of Allenes: From Mechanistic Understanding to Methodology Development[END_REF][START_REF] Perego | Copper-Catalyzed Hydroamination of N-Allenylazoles: Access to Amino-Substituted N-Vinylazoles[END_REF][START_REF] Blieck | Copper-Catalysed Hydroamination of N-Allenylsulfonamides: The Key Role of Ancillary Coordinating Groups[END_REF] and C-O [START_REF] Blieck | Copper-Catalyzed Hydrocarboxylation of N-Allenyl Derivatives[END_REF] bonds formation, we quickly realized that the reaction could occur without a transition metal, in the presence of a Brønsted acid and under very smooth and easy-to-handle conditions. We herein describe an efficient synthesis of allylic sulfones and 1,3-disulfones through regio-and stereoselective mono and double addition of various (hetero)aryl and (cyclo)alkyl sodium sulfinates to N-allenyl derivatives in aqueous conditions and at room temperature in the presence of trifluoroacetic acid (TFA).

Based on our previous works, we began to study the model reaction between N-allenyl-2-pyrrolidinone 1a and benzene sulfinic acid sodium salt 2a under our Cu(CH3CN)4PF6-catalyzed system (Table 1). With an acid source we showed that the presence of catalytic amount of copper allows the formation of 50% of 3aa (entry 2, Table 1). Different ligands were associated to copper but no real improvement was detected (entries 3-5, Table 1). In the next set of experiments we tested different sources of acid (entries 6-8, Table 1) and we defined trifluoroacetic acid (TFA) as the best one. However in the absence of the copper system, we surprisingly observed that 3aa could be obtained in total conversion with 90% yield (entry 9, Table 1). As above mentioned in table 1, best conditions were based on the use of only 1 equivalent of TFA in the absence of the copper catalyst. Hence, we opted to explore milder conditions (Table 2) and were finally able to perform the reaction in water, under open air at room temperature. Thanks to these improved system we could reach the target product 3aa in a better yield (from 90% up to quantitative yield). THF, 70°C, 1h 2) m-CPBA 2.5 equiv.

• R 1 R 1 S O O R 2 S-C bond R 2 SO 2 X X = H,
+ R 2 SO 2 NHNH 2 R 1 SO 2 R 2 • R 1 Breit (2015) + R 2 SH R 1 SO 2 R 2 • R 1 Breit (2017) + R 2 SO 2 NHNH 2 R 1 SO 2 R 2 [Rh(COD)Cl]2 2.

DCM, 0°C, 1h

Transition metal-free reaction

• Miao-Ren (2016) + R 3 SO 2 Na
AcOH 2 equiv.

DMSO (0.1 M), RT, 8-72h R 1 R 2 O R 1 R 2 O R 3 O 2 S • + R 3 SO 2 Na TFA 1 equiv. H2O (0.5 M), RT, 4-h C. This work N R 1 N R 1 SO 2 R 3 (1) (2)
(3) With these optimized conditions in hand, we first explored the scope of the method for the hydrosulfonylation of N-allenyl-2-pyrrolidinone 1a with various readily available (hetero)aryl and (cyclo)alkyl sodium sulfinates compounds 2a-o (Scheme 2). Different para-substituted benzene sodium sulfinates were tested, such as sodium ptoluenesulfinate 2b, 4-fluoro and 4-chlorobenzenesulfinic acid 2c and 2d, and afforded the corresponding products 3ab to 3ad in high yields (85-91%). Meta substituted aryl sulfinates partners, 3-chloro 4-methyl benzenesulfinic acid sodium salt 2e and 3-(trifluoromethyl)benzenesulfinic acid sodium salt 2f were also successfully engaged in the reaction and gave corresponding products in 80-89% yields. Finally, N-and Sheteroaromatics 2g and 2h provided the desired products 3ag and 3ah in good to high yields (60-83%). Alkyl sulfinates were then investigated, and both linear (2i, 2j) and cyclic (2k, 2l) derivatives yielded the hydrosulfonylation products, in good isolated yields (61-75%). Functionalized sulfinates such as sodium triflinate 2m and sodium 1methyl 3-sulfinopropanoate 2n were also able to provide the allylic sulfones in good yields (70-73%), showing the high tolerance of the process. To prove the robustness of the reaction, we were interested in performing it on a larger scale. The reaction between our model substrates 1a and 2a was then conducted on a 2 mmol scale, i.e. on a scale eight times larger than usual, and afforded the corresponding hydrosulfonylation product 3aa with a 97% isolated yield.

(4) (6)
• Loh (2021) + R 2 SO 2 H H2O:EtOH 1:1 (0.15 M), RT, 1h R 1 O (5) R 1 O R 2 O 2 S R 3 = (Het)Ar, (Cyclo)Alk R 2 R 2 3
Interestingly, we also performed a one-step double hydrosulfonylation of 1a with 2a to obtain the corresponding 1,3-disulfone 4aa in high yield. It is noteworthy to mention that the same 1,3-disulfone 4aa can also be obtained through the hydrosulfonylation of 3aa, demonstrating that our reaction can be applied both to N-allenyl and N-alkenyl derivatives. Scheme 2. TFA-mediated hydrosulfonylation of allene 1a with various sodium sulfinates 2a-n.

Reaction conditions: 1a (0.25 mmol), 2a-n (2 equiv., 0.5 mmol), TFA (1 equiv., 0.25 mmol), H2O (0.5M, 0.5 mL), open air, 25°C, 4h. Isolated yields. a The reaction was performed on a 2 mmol scale. b The reaction was performed at 40°C over two days under open air with 4 equiv. of TFA, 4 equiv. of PhSO2Na and H2O/THF 1:1 (0.25M) as the solvent. c The reaction was performed with 2 equiv. of TFA.

We then explored the reactivity of different N-allenyl derivatives with several (hetero)aryl and (cyclo)alkyl sodium sulfinates (Scheme 3). The N-allenyl-2oxazolidinone 1b was successfully involved in the reaction, affording both aryl and alkyl allylic sulfones in medium to excellent yields (51-90%). The symmetrical diallene N,N-diallenyl-2-imidazolidinone 1c afforded the corresponding double hydrosulfonylation product 3ca in 60% yield when a mixture 1:1 of water and THF was used as solvent. We also investigated the reactivity of Nsulfonyl allenamides, and observed that these substrates and their corresponding hydrofunctionalization products are sensitive under our acidic conditions. Nevertheless, the sulfones 3da and 3db were obtained in moderate to good isolated yields (35-60%). Reaction conditions (unless otherwise mentionned): 1 (0.25 mmol), 2 (2 equiv., 0.5 mmol), TFA (1 equiv., 0.25 mmol), H2O (0.5M, 0.5 mL), open air, 25°C, 4h. Isolated yields. a H2O/THF 1:1 (0.5M) was used as the solvent.

The mechanism of the reaction could take place in two stages. The first step is the hydrocarboxylation of the allene with the TFA, leading to the formation of the corresponding allylic ester (observed in both 1 H and [START_REF] Ali Abdine | Hydroarylation of N-Allenyl Derivatives Catalyzed by Copper[END_REF] F NMR), which is in presence of water in equilibrium with the corresponding iminium/carbocation forms (observed in HRMS ESI+). The second step could be either a nucleophilic substitution on the ester by the sulfinate, with the trifluoroacetate as a leaving group, or either a conjugated addition of the sulfinate to the iminium species, which finally leads to the formation of the desired allylic sulfone. The nucleophilic addition occurred on the  position of the iminium salt, which is a classical pathway to obtain enamines. (Scheme 4, A.). [START_REF] Lakhdar | Electrophilic Reactivities of α,β-Unsaturated Iminium Ions[END_REF] When the reaction was performed using D2O as a solvent, while in the presence of TFA or deuterated TFA, 90% deuterium incorporation was observed at the central position of the allene, which is consistent with the mechanism proposed above. By using benzenesulfinic acid (PhSO2H) instead of the corresponding sodium salt (PhSO2Na), no product was formed, suggesting that the sulfinic acid is not strong enough to protonate the N-allenyl derivative. (Scheme 4, B). To conclude, we developed a Brønsted acid-mediated allylation of S(VI)-containing nucleophiles. This transition metal-free methodology occurs in mild conditions and allows the hydrosulfonylation of a wide range of Nallenyl derivatives substrates with total regio-and stereoselectivity in water. Unprecedented (E)-allylic sulfones and a 1,3-disulfone were obtained with moderate to excellent yields in a total atom economical fashion. Further investigations and mechanistic studies for hydrofunctionalization of allenes will be reported in due course.

ASSOCIATED CONTENT Supporting Information

Detailed experimental procedures and characterization data (NMR 1 H, [START_REF] Khakyzadeh | Rhodium-Catalyzed Addition of Sulfonyl Hydrazides to Allenes: Regioselective Synthesis of Branched Allylic Sulfones[END_REF] C, [START_REF] Ali Abdine | Hydroarylation of N-Allenyl Derivatives Catalyzed by Copper[END_REF] F, ESI-HRMS and m.p. if solid) for all new compounds are provided in the supporting information (PDF).

The Supporting Information is available free of charge on the ACS Publications website.

Scheme 1 .

 1 Scheme 1. Metal-catalyzed or metal-free hydrosulfonylation of allenes: known methods and novel pathway described herein with TFA.

  Na, Li, Cl, NHNH 2 , CN, N 3

Scheme 3 .

 3 Scheme 3. TFA-mediated hydrosulfonylation of various allenyl derivatives with different sodium sulfinates.

Scheme 4 .

 4 Scheme 4. Proposed mechanism and control experiments.

Table 1 . Copper-catalyzed hydrosulfonylation of 1a with 2a : selected data for reaction parametric study. a

 1

	Entry x	y	Ligand d	H + source	Conver-sion (%) b,c	Yield (%) c
	1	0	0	-	none		-	-
	2	10	0	-	p-toluic acid	63%	50%
	3	10	10	Bpy	p-toluic acid	86%	57%
	4	10	10	Phen	p-toluic acid	74%	33%
	5	10	10	TMEDA	p-toluic acid	95%	51%
	6	10	10	Bpy	TMBA	79%	39%
	7	10	10	Bpy	PTSA		76%	30%
	8	10	10	Bpy	TFA		>99%	58%
	9	0	0	-	TFA		>99%	90%
	Phen	=	1,10-phenantroline.	TMEDA	=	N,N,N',N'-
	tetramethylethylenediamine.

a Reaction conditions: 1a (0.25 mmol), 2a (2 equiv., 0.5 mmol), H + source (1 equiv.). Catalyst and ligand were placed in a Schlenk tube under argon in 0.5 mL of THF (0.5M) for 18 hours at 55°C. b Based on recovered starting material 1a. c Determined through 1 H NMR analysis using trichloroethylene (1 equiv., 0.25 mmol) as internal standard. d TMBA = 2,4,6-trimethylbenzoic acid. PTSA = p-toluenesulfonic acid. Bpy = 2,2'-bipyridine.

Table 2 . Acid-mediated hydrosulfonylation of 1a with 2a : Selected data for reaction parametric study. a

 2

	Entry	Variations from the standard conditions	Conver-sion (%) b,c	Yield (%) c
	1	THF, instead of H2O	>99%	80%
	2	under argon, instead of open air >99%	90%

AUTHOR INFORMATION

Author Contributions

The manuscript was written through contributions of all authors. / All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT

Financial support was provided by ENSCM (PhD fellowship for LP and postdoctoral fellowship for SL) and CNRS. IUF is warmly acknowledged with thanks.


Rémi Blieck

Sébastien Lemouzy

Arie Van Der Lee

Marc Taillefer

Florian Monnier

Synergistic Copper/Enamine Catalysis for the Regio-,Stereo-and Enantioselective Intermolecular α-Addition of Aldehydes to Allenamides

Keywords: Synergistic catalysis, α-Addition, Allenes, Copper, Enamine

We herein describe an intermolecular enantioselective α-addition of aldehydes to allenamides, using a dual copper / enamine catalytic system. Highly enantioselective addition of aldehydes were obtained thanks to secondary amine catalysts. The process was found to be highly regio-, stereo-and enantio-selective under mild conditions. a reaction perfomed with 0.25 mmol of 1a and 0.50 mmol of 2a. b Assessed by 1H NMR using 1,3,5-trimethoxybenzene as internal standard. c Assessed by chiral HPLC on isolated product.

Synergistic catalysis has recently emerged as a powerful tool in organic synthesis, by combining the divergent reactivity of two distinct catalytic systems in a single reaction mixture. 1 In the last decade, the combination of transition-metal catalysis with amine organocatalysts through imine or enamine species has been exploited to develop efficient methods for the preparation of synthetically relevant building blocks. [START_REF] Simonovich | A general approach to the enantioselective α-oxidation of aldehydes via synergistic catalysis[END_REF][START_REF] Mo | Selected examples on the generation of transient directing groups through imine or enamine catalysis in C-H functionalization[END_REF] Recently, the Carreira group has developed a strategy based on the combination of chiral amine catalyst and chiral iridium catalyst. This method has shown outstanding results in the context of enantioselective allylic substitution of aldehydes (scheme 1a), [START_REF] Krautwald | Enantio-and diasterodivergent dual catalysis: α-allylation of branched aldehydes[END_REF] affording high enantio-and diastereo-divergence. This reaction was successfully applied to the synthesis of several diastereomer of tetrahydrocannabinols, 4c showcasing the synthetic potential of synergistic catalytic processes. Scheme 1. Works of Carreira, Breit and our group However, such a strategy may suffer from the use of acidic promoter (carboxylic or phosphoric acids), which in turn leads to a loss of atom economy. In order to address the atom economy issues associated with classical "Tsuji-Trost-type" allylation processes, Breit and coworkers has developed a range of enantioselective branched-selective reactions based on rhodium-catalyzed additions of various nitrogen, oxygen, sulfur and carbon-based nucleophiles to terminal allenes (scheme 1b). [START_REF] Koschker | Rhodium-catalyzed diastereoselective cyclization of allenyl sulfonylcarbamates: a stereodivergent approach to 1,3aminoalcohol derivatives[END_REF] In parallel, our research group also reported the addition of nitrogen, oxygen and carbon nucleophiles to terminal allenamides using a copper catalyst (scheme 1c). Unlike Breit's work, exclusive regioselectivity for the linear product was observed in our conditions, owing to the addition of the nucleophile in the distal C=C bond. [START_REF] Blieck | Copper-catalyzed hydroamination of terminal allenes[END_REF] In our quest for the development of enantioselective coppercatalyzed transformations, we then set out to examine whether the use of carbonyl pronucleophiles (aldehydes) in the presence of a chiral amine catalyst afford the addition to terminal allenamides in a synergistic catalytic process (scheme 2).

Scheme 2. Synergistic transition metal-enamine catalyzed anti-Markovnikov addition of aldehydes

Carreira [START_REF] Krautwald | Enantio-and diasterodivergent dual catalysis: α-allylation of branched aldehydes[END_REF] : Synergistic Ir / enamine catalysis Breit [START_REF] Koschker | Rhodium-catalyzed diastereoselective cyclization of allenyl sulfonylcarbamates: a stereodivergent approach to 1,3aminoalcohol derivatives[END_REF] : Rh-catalyzed enantioselective Markovnikov addition to allenes

Our group [START_REF] Blieck | Copper-catalyzed hydroamination of terminal allenes[END_REF] : Cu-catalyzed anti-markovnikov addition to allenes Such synergistic catalysis-based enantioselective functionalizations of allenes have been reported only using precious transition-metal catalysts such as rhodium, 7 palladium (scheme 2c) [START_REF] Zhou | Dual amine and palladium catalysis in diastereo-and enantioselective allene carbocyclization reactions[END_REF] or gold (scheme 2a-b) [START_REF] Ballesteros | I) operational in synergistic catalysis for the intermolecular α-addition reaction of aldehydes across allenamides[END_REF] and thus represents a challenging area of research. In particular, previous works on the enantioselective addition of aldehydes to allenamides catalyzed by a gold /enamine system was performed with limited efficiency, as the desired products were obtained in low yields and/or enantioselectivity (68-83% ee with yields ranging from 23-53%). 9b Thus, this reaction represents a significant synthetic challenge, especially in its enantioselective version. The functionalization of allenamide derivatives via distal addition is of high importance, as enamine products are chemical intermediates of high synthetic value. [START_REF]Enamines: Synthesis, Structure and Reactions[END_REF] In addition, the generation of an α-carbonyl quaternary stereocenter is of paramount importance in enantioselective catalysis. To test our hypothesis, we reacted allenamide 1a with aldehyde 2a in the presence of Cu(MeCN)4BF4 and aminecatalysts (table 1). We selected aldehyde 2a to avoid the formation of amine-catalyzed aldol oligomerization products. When 1a was heated at 70°C in THF in the presence of diphenylprolinol trimethylsilyl ether C1 as the chiral amine catalyst, the desired product 3aa was obtained in good yield (62%), complete regio-(exclusive addition of the carbon atom of the distal C=C bond) and stereo-selectivity (only the E alkene was obtained), [START_REF] Blieck | Copper-catalyzed hydroamination of terminal allenes[END_REF] but with very modest enantioselectivity (46:54) (table 1, entry 1).

O R 2 R 1 R 3 OH cat. [Ir(cod)Cl] 2 L* chiral amine acidic promoter O R 2 R 1 R 3 H R + NuH cat. [Rh(cod)Cl] 2 L* R

Nu

When the solvent was changed to toluene, the enantioselectivity could be increased (37:63), although the reactivity of the catalytic system was decreased (entry 2). At a slightly lower reaction temperature (60°C), the yield dropped down to 16%, but the enantioselectivity was largely improved (er = 21:79, entry 3). We then hypothesized that the use of a catalytic amount of weak acid would enhance the organocatalyst activity by accelerating the enamine formation, with limited risk of hydrocarboxylation of the allenamides at the working temperature in the absence of an inorganic base. 6f However, when 20 mol% of p-methylbenzoic acid A1 was added, the yield and enantioselectivity were only slightly improved (28% and 16:84 respectively, entry 4). Then, we thought that the addition of external nitrogen ligand would prevent the potential formation of (prolinol) copper (I) complex, [START_REF] Kirillova | Mixed-ligand aminoalcohol-dicarboxylate copper(II) coordination polymers as catalysts for the oxidative of cyclic alkanes and alkenes[END_REF] which could sequestrate the organocatalyst in such non-polar, weakly coordinating solvent as toluene. Indeed, with a combination of 20 mol% of A1 and 2,2'-bipyridine L1, the yield and enantioselectivity were further improved to 46% and 88:12 er, respectively (entry 5). With these conditions in hand, we screened a range of commercially available amine catalysts (entries 6-12). Although L-proline gave essentially racemic mixture (entry 6), prolinol based organocatalysts (C3, C5) gave the best enantioselectivities, with C5 giving an excellent enantiomeric ratio of 95.5:4.5, albeit affording 3aa in a somewhat lower yield (entry 9). Both McMillan-type catalysts (C6-C7) and cinchona alkaloids-derived amine catalyst C8 gave lower enantioselectivity (entry 10-12).

Using the conditions depicted in entry 9, we tested other parameters, such as the ligand, the solvent, and the acid additive (table 1). When, 4,4'-ditert-butyl-2,2'bipyridine L2 was used, the yield could be increased up to 65% (58% isolated yield) without affecting the enantioselectivity (entry 13). When other nitrogen ligands were screened (entries 14-16), similar high enantioselectivities were obtained, but phenanthroline and diamine ligands afforded lower amount of the desired product. With aromatic solvents product 3aa was obtained in similar high enantioselectivities (entries 17-19). In particular, α,α,α-trifluorotoluene afforded 3aa in high isolated yield (86%, entry 18), and with excellent enantiomeric ratio (95:5 er). Using THF as the solvent, the enantioselectivity was decreased to 87:13 er (entry 20). Further attempts to optimize the reaction by screening carboxylic acid co-catalysts (A1-5) at lower copper/ligand loadings were engaged (entries 21-26). Indeed, increasing the acidity of the proton resulted in the improvement of the yield of 3aa, but it was accompanied with a substantial loss of enantioselectivity in the case of stronger acids. Rate accceleration of the background enol addition to the unsaturated substrate under strong Brønsted acid conditions might account for this result. [START_REF] Dion | Asymmetric Bronsted acid catalysis enabling hydroaminations of dienes and allenes[END_REF] We then started to study the scope of this reaction by using the optimized conditions (table 1, entry 18) and screened a range of allenamides 1a-1g with the model aldehyde reagent 2a (scheme 3) in order to evaluate the influence of the substitution pattern on the nitrogen atom. When N-allenyl oxazolidinone 1b was selected as the substrate, similar enantioselectivity (91:9) was obtained, although the yield of the desired product (3ba) decreased when compared with 3aa. We also observed that other electron-withdrawing protecting groups on the nitrogen atom such as tosyl (1c, 1d-e), mesyl (1f) or tertbutulyoxycarbonyl (Boc) (1g) afforded high enantioselectivities (>80% ee) of the adduct regardless of the substitution on the nitrogen atom. However, it is worth noting that carbonylsubstituted allenamides (1a-b) seem to react with similar high enantioselectivity than sulfonyl-substituted ones. Indeed, while allyl substituted allenamide 1c yielded the adduct in moderate yield (3ca, 46%) aryl-substituted analogs afforded higher amounts of the desired product (3da-3ea, 82-63%). The preparation of 3ca was accompanied with the formation of unidentified side-product which resulting from reactions of the pending allyl moiety. When comparing the sulfonyl protecting groups, similar yields were obtained when switching from tosyl (3ea) to mesyl (3fa). Surprisingly, the use of Boc-protected allenyl carbamate also afforded the desired product (3ga) albeit in moderate yield (52%). The fact that the reaction conditions are compatible with such an acid sensitive nitrogen protecting group is remarkable, since Boc protecting group can be easily cleaved even under mild Lewis acidic copper(I) catalyzed conditions. 14

N N R R R = H L 1 R = H L 3 DMEDA L 5 tBu L 2 = Me L 4 N N R R NH HN

Bidentates N,N ligands

Carboxylic acids (RCOOH)

A 1 R = p-Me-C 6 H 4 A 2 R = n-pent A 3 R = o-Me-C 6 H 4 A 4 R = o-I-C 6 H 4 A 5 PTSA-H 2 O N H OR Ph Ph = TMS = TBS = H R C 2 C 3 C 4 N H N O Bn C6 N MeO H 2 N N N H CO 2 H C 1 N H N O Bn C7 C8 N H OTMS CF 3 CF 3 F 3 C CF 3 C5

Organocatalysts

Scheme 3. Scope of allenamide substrates and aldehydes nucleophiles (yield values are those of the pure isolated products)

Next, we turned our attention to the influence of aldehyde partner. We were pleased to observe that aryl(methyl) 2a-b, aryl(naphthyl) 2c and heteroaryl(methyl) 2d α-substituted aldehydes were found to be effective in this reaction. In nearly all cases, excellent enantioselectivities were obtained (≥89%ee). Quite remarkably, using p-methoxyphenyl substituted aldehyde 2b, good yields were obtained regardless of the nature of the allenamide partner (3ab-3cb). Using aldehyde 2b, the addition product 3cb was obtained in an excellent 98:2 enantiomeric ratio, highlighting once again the high enantioselectivity of the process.

To further demonstrate the synthetic potential of the process, we also set up a multi-millimole scale reaction. Indeed, the tenfold scale-up synthesis of 3aa afforded the product with nearly quantitative yield (98%, 630 mg) and essentially identical enantioselectivity (95:5 er). Furthermore, we performed post-functionalization reactions of the aldehyde moiety to show the synthetic applicability of the optically active aldehyde products. Surprisingly, such sterically hindered α-quaternary aldehydes could be easily functionalized under mild conditions. When performing the reductive amination of 3aa using morpholine under mild reducing conditions (with sodium tri(acetoxy)borohydride as the reducing agent at room temperature), the desired δ-amino alkene product 4aa featuring a stereogenic carbon center in the β-position was obtained in moderate yield (42%). The Henry reaction of 3aa with ni-tromethane under mildly basic conditions (TBAF-H2O) afforded the desired nitroaldol product 5aa in moderate yield and diastereo-selectivity (58%, 1.9:1 dr). To gain information on the stereochemistry of the allylation process, we have performed X-ray diffraction of 3fa. Indeed, needle-like crystals of 3fa could be obtained when crystallizing at 0°C in a DCM / CyMe mixture. X-ray diffraction analysis showed that 3fa was obtained with S configuration on the quaternary carbon center (scheme 5).

Scheme 5. X-ray structure of 3fa (ellipsoids are given with 50% probability) and proposed stereochemical model for the enantio-determining step According to the observed absolute configuration of 3aa, we propose a stereochemical model (scheme 6) in which the addition of the chiral enamine (resulting from the addition of the Hayashi-Jorgensen catalyst on the aldehyde) occurs from the Si face of the (E)-enamine (backside attack). Indeed, such a backside attack should be favored to minimize the steric repulsion between both the substituents on the nitrogen atom of the allenamide and the bulky aryl group on the prolinol catalyst, consistently with the proposed stereochemical models used in prolinol-based enamine catalysis [START_REF] Schmid | Formation and Stability of Prolinol and Prolinol Ether Enamines by NMR: Delicate Selectivity and Reactivity Balances and Parasitic Equilibria[END_REF] .

Scheme 6. Proposed stereochemical model for the enantiodetermining step

To conclude, we have developed a highly enantioselective method for the addition of aldehydes pronucleophiles to allenamides. To achieve this goal, a combination of two synergistic catalytic systems based on copper and chiral amine catalysts was found to be highly efficient. Using cheap, readily available, and environmentally benign catalysts (cationic copper (I) complex and Hayashi-Jorgensen prolinol), high regio-, stereo-and enantio-selectivities were obtained. The reaction could be scaled up to multi millimole scale without any loss of the catalyst activity and selectivity. We are currently investigating other enantioselective additions of various nucleophiles to allenes by means of copper catalysis.

ASSOCIATED CONTENT Supporting Information

The Supporting Information (containing all the NMR spectra, chiral HPLC chromatograms of all the described products, as well as the procedures to prepare both starting materials and products) is available free of charge on the ACS Publications website. FAIR Data is available as Supporting Information for Publication and includes the primary NMR FID files for all compounds.

Scheme 4 .

 4 Scheme 4. Scale-up synthesis of 3aa and post-functionalization reactions

Table 1 . Selected data for reaction parametric study. a

 1

			11
	entry	solvent	x mol %	A (y mol %) L (z mol %) C	tempera-	3aa Yield % b	e.r. c
							ture
	1	THF	20	A1 (0)	L1 (0)	C1	70	62	46:54
	2	toluene	20	A1 (0)	L1 (0)	C1	70	32	37:63
	3	toluene	20	A1 (0)	L1 (0)	C1	60	16	21:79
	4	toluene	20	A1 (20)	L1 (0)	C1	60	28	16:84
	5	toluene	20	A1 (20)	L1 (20)	C1	60	46	12:88
	6	toluene	20	A1 (20)	L1 (20)	C2	60	48	50:50
	7	toluene	20	A1 (20)	L1 (20)	C3	60	35	15:85
	8	toluene	20	A1 (20)	L1 (20)	C4	60	10	Nd
	9	toluene	20	A1 (20)	L1 (20)	C5	60	39	5:95
	10	toluene	20	A1 (20)	L1 (20)	C6	60	27	49:51
	11	toluene	20	A1 (20)	L1 (20)	C7	60	33	28:72
	12	toluene	20	A1 (20)	L1 (20)	C8	60	67	25:75
	13	toluene	20	A1 (20)	L2 (20)	C5	60	65(58)	4:96
	14	toluene	20	A1 (20)	L3 (20)	C5	60	12	Nd
	15	toluene	20	A1 (20)	L4 (20)	C5	60	22	8:92
	16	toluene	20	A1 (20)	L5 (20)	C5	60	41	4:96
	17	PhCl	20	A1 (20)	L2 (20)	C5	60	63	4:96
	18	PhCF3	20	A1 (20)	L2 (20)	C5	60	88(86)	5:95
	19	Mesitylene	20	A1 (20)	L2 (20)	C5	60	47	3:97
	20	THF	20	A1 (20)	L2 (20)	C5	60	87(81)	13:87
	21	PhCF3	10	A1 (20)	L2 (10)	C5	60	30	8:92
	22	PhCF3	10	A2 (20)	L2 (10)	C5	60	25	Nd
	23	PhCF3	10	A3 (20)	L2 (10)	C5	60	39	5:95
	24	PhCF3	10	A4 (20)	L2 (10)	C5	60	60	8:92
	25	PhCF3	10	A5 (20)	L2 (10)	C5	60	72	15:85
	26	PhCF3	20	A5 (20)	L2 (20)	C5	60	76	12:88

ACKNOWLEDGMENT

Financial support provided by Région Languedoc-Roussillon (PhD fellowship for R.B.), ENSCM (Postdoctoral fellowship for S.L.), IUF (Institut Universitaire de France to F.M.), and CNRS are warmly acknowledged. We also thank Aubane Malheiro for carrying out some experiments on the enantioselective addition of keto-esters as well as Pascale Guiffrey for helping with the chiral HPLC separations of racemic compounds.

Author Contributions † These authors equally contributed to this work. The manuscript was written through contributions of all authors. / All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interests.


Yakun Zhang

Marine Jeanson

Rémy Mével
email: mevel@mail.tsinghua.edu.cn

Zheng Chen

Nabiha Chaumeix

Keywords: Spherical expanding flame, Critical flame radius, Laminar flame speed, Uncertainty propagation

The uncertainty on the laminar flame speed extracted from spherically expanding flames can be minimized by using large flame radius data for the extrapolation to zero stretch-rate. However, at large radii, the hydrodynamic and thermo-diffusive instabilities would induce the wrinkling of the flame surface and limit the range of usable data. In the present study, we have employed the flame stability theory of Matalon to tailor the properties of the initial mixture so that onset of cellular flame would occur at a pre-determined, large radius. This approach was employed to measure the laminar flame speeds of H 2 /O 2 /N 2 /He mixtures with equivalence ratios from 0.6 to 2.0, at pressures of 50/80/100 kPa and initial temperature of 300 K. For most experiments we performed, the uncertainty related to the extrapolation to zero stretch-rate (performed with the linear curvature model) was below 2% as shown by the position of the data points in the (

where L b is the burned Markstein length; and R f,L and R f,U are the flame radii at the lower and upper bounds of the extrapolation range. Unsteady 1-D simulations using four chemical mechanisms were performed to show that unstretched laminar flame speed can be well-characterized with relative error below 10% for most conditions. The flame dynamical response to stretch rate could be captured by the mechanisms only

Tailored Mixture Properties for Accurate Laminar Flame Speed Measurement from Spherically Expanding Flames:

Application to H 2 /O 2 /N 2 /He Mixtures

Introduction

The laminar flame speed, S 0 u , is defined as the normal propagation velocity of fresh gas relative to a fixed, planar flame front [START_REF] Egolfopoulos | Advances and challenges in laminar flame experiments and implications for combustion chemistry[END_REF]. It contains the physicochemical information of the reactive mixture and helps in characterizing combustion phenomena, testing and validating reaction models [START_REF] Konnov | A comprehensive review of measurements and data analysis of laminar burning velocities for various fuel+air mixtures[END_REF][START_REF] Law | Combustion Physics[END_REF]. Furthermore, since turbulent combustion properties scale with the laminar ones [START_REF] Lapointe | Fuel and chemistry effects in high Karlovitz premixed turbulent flames[END_REF][START_REF] Boeck | Flame propagation across an obstacle: OH-PLIF and 2-D simulations with detailed chemistry[END_REF], accurate experimental data obtained under laminar conditions are needed to predict turbulent combustion which is relevant to the conditions found in real applications, such as internal combustion engine, gas turbine and accidental explosion [START_REF] Lapointe | Fuel and chemistry effects in high Karlovitz premixed turbulent flames[END_REF][START_REF] Boeck | Flame propagation across an obstacle: OH-PLIF and 2-D simulations with detailed chemistry[END_REF][START_REF] Boeck | Simultaneous OH-PLIF and schlieren imaging of flame acceleration in an obstacle-laden channel[END_REF]. Various methods have been developed to measure S 0 u including Bunsen flame, counterflow/stagnation flame, burner stabilized flat flame and spherically expanding flame (SEF). The advantages, limitations and technical updates of these methods have been reviewed in detail by Egolfopoulos et al. and Konnov et al. [START_REF] Egolfopoulos | Advances and challenges in laminar flame experiments and implications for combustion chemistry[END_REF][START_REF] Konnov | A comprehensive review of measurements and data analysis of laminar burning velocities for various fuel+air mixtures[END_REF]. Among these methods, the SEF method is widely employed because it is the most effective method for measuring S 0 u at engine-relevant conditions especially at elevated pressure [START_REF] Xiouris | Laminar flame speeds under engine-relevant conditions: Uncertainty quantification and minimization in spherically expanding flame experiments[END_REF]. Using the SEF method along with a high-speed visualization system, the laminar flame speed can be extrapolated to zero-stretch rate from the temporal evolution of the radius R f (t) of a centrally spark-ignited flame.

Substantial efforts have been made to reduce the uncertainty associated with laminar flame speed measurement using the SEF method [START_REF] Egolfopoulos | Advances and challenges in laminar flame experiments and implications for combustion chemistry[END_REF][START_REF] Chen | On the accuracy of laminar flame speeds measured from outwardly propagating spherical flames: Methane/air at normal temperature and pres-sure[END_REF][START_REF] Lipatnikov | Experimental assessment of various methods of determination of laminar flame speed in experiments with expanding spherical flames with positive Markstein lengths[END_REF][START_REF] Beeckmann | Flame propagation speed and Markstein length of spherically expanding flames: Assessment of extrapolation and measurement techniques[END_REF]. Possible perturbations for a laminar spherical flame propagating in a constant volume bomb have been summarized by Lipatnikov et al. [START_REF] Lipatnikov | Experimental assessment of various methods of determination of laminar flame speed in experiments with expanding spherical flames with positive Markstein lengths[END_REF]. These perturbations include (i) spark ignition effects, (ii) confinement effects, (iii) heat loss through radiation, (iv) nonuniformity of product density, (v) compression effects of unburned gas and products, (vi) flame instabilities and (vii) stretch effects. The uncertainties induced by these perturbations have been quantified and their contributions to S 0 u deviations have been investigated in [START_REF] Chen | Effects of Lewis number and ignition energy on the determination of laminar flame speed using propagating spherical flames[END_REF][START_REF] Chen | On the critical flame radius and minimum ignition energy for spherical flame initiation[END_REF][START_REF] Chen | Effects of radiation and compression on propagating spherical flames of methane/air mixtures near the lean flammability limit[END_REF][START_REF] Chen | Effects of radiation absorption on spherical flame propagation and radiation-induced uncertainty in laminar flame speed measurement[END_REF][START_REF] Zhou | Effect of ignition energy on the uncertainty in the determination of laminar flame speed using outwardly propagating spherical flames[END_REF][START_REF] Burke | Effect of cylindrical confinement on the determination of laminar flame speeds using outwardly propagating flames[END_REF]. In order to decrease the influence of the aforementioned perturbations, various approaches have been developed [START_REF] Egolfopoulos | Advances and challenges in laminar flame experiments and implications for combustion chemistry[END_REF][START_REF] Chen | On the accuracy of laminar flame speeds measured from outwardly propagating spherical flames: Methane/air at normal temperature and pres-sure[END_REF]. However, the approach of excluding data points that have been obviously influenced by the perturbations is simple and widely used for significantly mitigating the perturbations [START_REF] Lipatnikov | Experimental assessment of various methods of determination of laminar flame speed in experiments with expanding spherical flames with positive Markstein lengths[END_REF][START_REF] Grosseuvres | Combustion properties of H2/N2/O2/steam mixtures[END_REF]. Perturbations (i) and (iv) only affect spherical flames at early stages and consequently these small radius data can be removed from the R f (t) curve. Perturbations (ii) and (v) are related to large flame radii with respect to the characteristic length scale of the combustion vessel and the part of the R f (t) curve affected by these two perturbations can be readily excluded from the analysis. Perturbation (iii) is relevant to slow flames at very large radius, so that many conditions can be studied without considering this perturbation. For near flammability limit or decomposition flames [START_REF] Jerzembeck | Experimental investigation of very rich laminar spherical flames under microgravity conditions[END_REF], which demonstrate low propagation velocity, empirical correlations [START_REF] Yu | Radiationinduced uncertainty in laminar flame speed measured from propagating spherical flames[END_REF] are available to estimate the increased uncertainty on the flame speed. Perturbations (vi) and (vii) are by far the most challenging especially when targeting a wide range of pressure for the measurements.

The effect of stretch rate on the flame propagation can be described using various linear or non-linear extrapolation equations [START_REF] Kelley | Nonlinear effects in the extraction of laminar flame speeds from expanding spherical flames[END_REF][START_REF] Kelley | Premixed flame propagation in a confining vessel with weak pressure rise[END_REF][START_REF] Chen | On the extraction of laminar flame speed and Markstein length from outwardly propagating spherical flames[END_REF] which enable to extract the unstretched laminar flame speed. Nevertheless, Wu et al. [START_REF] Wu | Uncertainty in stretch extrapolation of laminar flame speed from expanding spherical flames[END_REF] showed that the selection of extrapolation equations is one of the major causes to data discrepancy.

It was found that substantial errors resulting from extrapolation still exist for conditions where the Lewis number (Le) is away from unity, as for very lean or rich hydrogen/air flames [START_REF] Wu | Uncertainty in stretch extrapolation of laminar flame speed from expanding spherical flames[END_REF][START_REF] Liang | Extrapolation of laminar flame speeds from stretched flames: Role of finite flame thickness[END_REF]. Recent studies [START_REF] Wu | Uncertainty in stretch extrapolation of laminar flame speed from expanding spherical flames[END_REF][START_REF] Nativel | Laminar flame speeds of pentanol isomers: An experimental and modeling study[END_REF][START_REF] Huo | Uncertainty reduction in laminar flame speed extrapolation for expanding spherical flames[END_REF] have explored the effect of flame radius range on the uncertainty resulting from the extrapolation method and showed that extrapolation-induced uncertainty can be reduced using large flame radius data. In particular, Huo et al. [START_REF] Huo | Uncertainty reduction in laminar flame speed extrapolation for expanding spherical flames[END_REF] found that extrapolation uncertainty largely relates to the Markstein length (L b ) normalized by the flame radius (R f ) and the uncertainty can be limited to a maximum of 2% using flame radius from 3 cm to 4 cm. However, the use of large flame radii for measuring the flame speed increases the detrimental impact of perturbation (vi). The flame instability, which results in the formation of cellular structures on the flame surface, is induced, on the one hand, by the gas expansion across the flame front (hydrodynamic or Darrieus-Landau instability), and on the other hand, by the non-equidiffusion of mass and heat of the deficient reactant (thermo-diffusive instability).

The strategy to mitigate the development of cellular instabilities consists of using helium as diluant in the combustible mixture. With the addition of helium, hydrodynamic instability can be suppressed by the increasing flame thickness, especially at high pressures; thermo-diffusive instability can be mitigated by increasing the effective Lewis number (Le ef f ) of the mixture. This approach has been implemented by several groups [START_REF] Lamoureux | Laminar flame velocity determination for H 2 -air-He-CO 2 mixtures using the spherical bomb method[END_REF][START_REF] Tse | Morphology and burning rates of expanding spherical flames in H 2 /O 2 /inert mixtures up to 60 atmospheres[END_REF][START_REF] Shen | Experimental and kinetic studies of acetylene flames at elevated pressures[END_REF][START_REF] Zhang | Laminar flame speeds of lean high-hydrogen syngas at normal and elevated pressures[END_REF][START_REF] Han | Laminar flame speeds of H 2 /CO with CO 2 dilution at normal and elevated pressures and temperatures[END_REF][START_REF] Ai | Laminar flame speed and Markstein length of syngas at normal and elevated pressures and temperatures[END_REF] at normal and elevated pressures and temperatures. It is worth mentioning that few well-defined procedure has been proposed to minimize the uncertainty related to perturbations (vi) and (vii) through tailoring the mixture property.

In view of these considerations, the goal of the present study is to develop a procedure to tailor the mixture properties based on the flame stability theory of Matalon et al. [START_REF] Addabbo | Wrinkling of spherically expanding flames[END_REF], so that the theoretical onset of instability for SEF is delayed up to a predefined large flame radius, thus allowing to perform the flame speed measurement over a wide range of radii for which the uncertainty associated with the extrapolation method can be minimized. The procedure was applied to hydrogenair mixtures over a range of pressures from 50 to 100 kPa, equivalence ratios (Φ) from 0.6 to 2.0 and initial temperature of 300 K. Theoretical and experimental flame properties were compared to evaluate the performance of the flame stability theory and four chemical mechanisms. Then, theoretical analyses focused on critical flame radius (R c ) were implemented. The influence of changing flame property on the prediction of R c were explored through local sensitivity analysis. Uncertainty propagation for some fundamental parameters in the input kinetics, thermodynamics and transport data sets was investigated and quantified.

Materials and methods

Tailoring procedure

The onset of instability for SEF was first theoretically described by Istratov et al. [START_REF] Istratov | On the stability of gasdynamic discontinuities associated with chemical reactions. The case of a spherical flame[END_REF] considering only the hydrodynamic effect. Bechtold and Matalon [START_REF] Bechtold | Hydrodynamic and diffusion effects on the stability of spherically expanding flames[END_REF] extended the stability analysis accounting for both hydrodynamic and thermo-diffusive instabilities. Addabbo et al. [START_REF] Addabbo | Wrinkling of spherically expanding flames[END_REF] further improved the stability analysis using more realistic temperature-dependent transport coefficients and the theory was applied in [START_REF] Law | Cellular instabilities of expanding hydrogen/propane spherical flames at elevated pressures: theory and experiment[END_REF]. According to the theory of spherical flame stability of Matalon and coworkers [START_REF] Addabbo | Wrinkling of spherically expanding flames[END_REF][START_REF] Matalon | Flame dynamics[END_REF], the critical flame radius for cellularity onset (R c ) is given by

R c = δ ⋅ P e c = δ ⋅ Ω ω = δ ω Q 1 + β(Le ef f -1) σ e -1 Q 2 + P rQ 3 , (1)
where δ is the flame thickness; P e c is the critical Peclet number; ω corresponds to the contribution of the hydrodynamic effects to the flame instability; Ω corresponds to the contribution of thermo-diffusive effect to the flame instability; β is the Zeldovich number; Le ef f is the effective Lewis number; σ e = T ad T u is the expansion ratio where T ad is the adiabatic flame temperature and T u is the unburnt gas temperature; P r is the Prandtl number; Q 1 , Q 2 , and Q 3 are complex terms which depend on the expansion ratio, the spherical harmonic wavenumber (n), and the thermal conductivity (λ), see supplementary material for detailed equations. The flame thickness is calculated using the temperature gradient definition

δ = T ad -T u (dT dx) max , (2)
where (dT dx) max is the maximum temperature gradient in a steady 1-D planar flame. The Zeldovich number β is defined as

β = E a T ad -T u RT 2 ad , (3)
where E a is the activation energy and R is the perfect gas constant. E a is determined by

E a = -2R ∂[ln(m 0 )] ∂[ln(1 T ad )] , (4)
where m 0 is the mass burning rate. m 0 and T ad for mixtures with different mole fractions of argon (0%, 1%, 2%, 3%, 4%, 5%) were used to calculate the partial difference in Equation 4. To reduce the computational costs, the relationship between E a and helium percentage in the dilution was fitted using a quadratic polynomial prior to the tailoring procedure. The effective Lewis number Le ef f is obtained from

Le ef f = 1 + (Le E -1) + (Le D -1) ε 1 + ε ; (5)
where ε = 1 + β (Ξ -1) with Ξ = Φ for rich mixtures and Ξ = 1 Φ for lean mixtures.

All the required parameters were calculated using the thermo-chemical and transport utilities available in the Cantera package [START_REF] Goodwin | Cantera: An object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes[END_REF].

The tailoring procedure is described in Fig. 1. First, a target critical radius (R t c ) is defined. Then, the theoretical critical radius (R th c ) is calculated for H 2 -air mixture. If R th c is greater than or equal to R t c , the tailoring procedure is stopped. If R th c is lower than R t c , N 2 would be fully replaced by helium, while the mole fraction of other components remain unchanged. We define R th c (He) as the corresponding critical flame radius for the H 2 /O 2 /He mixture. If R th c (He) is greater than or equal to R t c , considering the initial mixture composition, N 2 would be partly replaced by helium.

If R th c (He) is less than R t c , the hydrogen to oxygen ratio would be maintained, and the mole fraction of the diluant (including only helium) is increased. Through iteration, the tailored composition with theoretical R c equal to the pre-defined radius can be obtained using a Matlab routine in order to measure the flame speed at large flame radii. This procedure was applied to H 2 -air mixture using Wang's model [START_REF] Davis | An optimized kinetic model of H 2 /CO combustion[END_REF] (14 species and 38 reactions). The initial temperature was set at 300 K. The pressure was considered as constant during flame propagation at 50/80/100 kPa. The optical window radius of the experimental combustion vessel, 10 cm, was selected as R t c . The evolution of R th c as a function of the proportion of helium is shown in Fig. 2 for mixtures initially at P=100 kPa, T=300 K, and Φ=0.6-2.0. The intersection between R th c =f(X He (X He +X N 2 )) and R t c =10 cm corresponds to the tailored mixture composition. The tailored compositions of the mixtures we studied experimentally are given in Table 1. eters, and some calculated parameters are also given. Conditions: P=100 kPa, T=300 K, Φ=0.6-2.0.

Φ X H2 X O2 X N2 X He P 1 (kP a) T 1 (K) R ex c (cm) R th c (cm) S 0 u (cm/s) L b (
equipped with 4 quartz windows with 200 mm optical diameter for visual access and 8 ports for mounting fans to create homogeneous environment or turbulent conditions.

A Z-type schlieren arrangement is utilized. The flame propagation is recorded using a high-speed camera (PHANTOM V1210 or V1610), with an acquisition rate up to 25,000 frames per second. Two tungsten electrodes linked to a high voltage power supply enable to ignite the mixture with a central electric spark. The spherical vessel is equipped with two Kistler piezo-electric pressure transducers (6001 and 601A models) synchronized with the imaging system to ensure that the pressure remains constant during the period employed to extract the flame speed. Prior to each test, the vessel is evacuated to a residual pressure on the order of 1 Pa. The reactive mixtures are prepared using the partial pressure approach. After introducing all the gases needed, the fans are turned on for 3 minutes. Then, the gas is let to settle for 3 minutes before ignition. Example schlieren images of SEF are shown in Fig. 3. By repeating the SEF experiments at similar conditions for two tailored cases (P=50 kPa, Φ=1.0 and P=100 kPa, Φ=0.8), the repeatability of experimental results could be characterized, see supplementary material.

Data reduction

To obtain the evolution of the flame radius as a function of time, the raw Schlieren images were processed using an in-house Matlab program that detects the flame edge thus allowing to extract the flame radius profile with time. Further details about this laminar flame analysis program can be found in [START_REF] Nativel | Laminar flame speeds of pentanol isomers: An experimental and modeling study[END_REF][START_REF] Mevel | Spherical expanding flames in H 2 -N 2 O-Ar mixtures: flame speed measurements and kinetic modeling[END_REF]. The unstretched laminar flame speed was extracted using the linear curvature (LC) equation which considers a linear relationship between the flame speed and curvature

S b = S 0 b -2S 0 b L b R f (6)
where S b is the stretched flame speed with respect to the burnt gas; S 0 b is the unstretched flame speed with respect to the burnt gas; and L b is the Markstein length.

The choice of the LC equation was motivated by the results in [START_REF] Chen | On the accuracy of laminar flame speeds measured from outwardly propagating spherical flames: Methane/air at normal temperature and pres-sure[END_REF][START_REF] Chen | On the extraction of laminar flame speed and Markstein length from outwardly propagating spherical flames[END_REF][START_REF] Wu | Uncertainty in stretch extrapolation of laminar flame speed from expanding spherical flames[END_REF] which showed that this equation is the most accurate to extract the flame speed for mixtures with Lewis number larger than unity. To avoid differentiating the experimental data, numerical integration of Equation 6 is used for extracting the flame properties.

Due to the effect of ignition energy and cellular flame formation, the range of data used for extrapolating the unstretched flame speed should be appropriately selected. The effect of minimum and maximum flame radius (R f,L and R f,U ) on the flame properties was investigated. The extrapolation surfaces of S 0 u using different R f,L and R f,U are shown in Fig. 4. The shapes of the L b surfaces are similar to the corresponding S 0 u surfaces, see supplementary material. From Fig. 4 (a), we can see the surface folding along the R f,U axis induced by cellular onset resulting from the dynamics of the cells which demonstrate alternative phases of growth and breakup [START_REF] Huo | Self-acceleration and global pulsation in hydrodynamically unstable expanding laminar flames[END_REF][START_REF] Bradley | The development and structure of flame instabilities and cellularity at low Markstein numbers in explosions[END_REF]; in Fig. 4 (b), we can observe the crest along the R f,L axis induced by ignition effect; combination of the two aforementioned effects is shown in Fig. 4 (c); in Fig. 4 (d), the surface is smoother than the first three surfaces. In addition to the obvious oscillations, wavy surfaces with small oscillations are present for all the cases due to noise. Probability density profile was obtained using the data points in the surfaces. Then the profile was fitted to a normal distribution. A one-standard deviation range ([µ s -σ s , µ s +σ s ], we call it 1-σ s range) in S 0 u was considered to guide the choice of flame radius range, see Fig. 5 (a). Three rules were employed in the R f,L and R f,U determination:

• R f,L and R f,U were fixed at the radius not affected by the obvious oscillations if folding or crest existed in the surface.

• If the surface is wavy with small oscillations, R f,L and R f,U were selected within the 1-σ s range. R f,L and R f,U correspond to the points starting to deviate from the 1-σ s range.

• The minimum range of radius was selected to be 20 mm.

An example of R f,L and R f,U selections is shown in Fig. 5. After removing data outside 1-σ s range, the mean value and standard deviation decreased. For Fig. 5 (b), colored area includes data points in 1-σ s range, other areas are not considered due to the limitations of minimum radius range and 1-σ s range. Then, the data points in part of the extrapolation surface we selected were fitted to a normal distribution to extract laminar flame properties and their uncertainties.

Numerical methods

Spherical flame simulation

The simulations of SEF were conducted using the A-SURF code, which has been validated and successfully used in previous studies [START_REF] Chen | Effects of Lewis number and ignition energy on the determination of laminar flame speed using propagating spherical flames[END_REF][START_REF] Chen | Effects of radiation absorption on spherical flame propagation and radiation-induced uncertainty in laminar flame speed measurement[END_REF]. The initial velocity and temperature at each grip point before ignition were set as 0 cm/s and 300 K. To avoid perturbation due to confinement effects, the radius of the spherical numerical domain was set at 120 cm. The flame front was defined as the position of maximum heat release rate. The flame radius range used for determining the laminar flame speed and the Markstein length was consistent with the corresponding experiments. The radiation effect was neglected. Mixture-averaged transport and Soret effects were included. In order to explore the effect of chemical mechanisms on the calculation of S 0 b and L b , besides Wang's model, three other mechanisms were also used: Konnov's model [START_REF] Konnov | Yet another kinetic mechanism for hydrogen combustion[END_REF] (15 species and 75 reactions); Burke's model [START_REF] Burke | Comprehensive H 2 /O 2 kinetic model for high-pressure combustion[END_REF] (13 species and 27 reactions); and Curran's model [START_REF] Conaire | A comprehensive modeling study of hydrogen oxidation[END_REF] (11 species and 21 reactions).

Local sensitivity analysis

Local sensitivity analysis can be used to determine the influence of changes in model input parameters on the output parameter. This method is the most frequently applied method in combustion research for identifying key reaction pathways and reactants which control model outputs or product distribution [START_REF] Turányi | Analysis of kinetic reaction mechanisms[END_REF]. Laminar flame speed is a commonly used model output in mechanism construction and reduction [START_REF] Zador | Local and global uncertainty analyses of a methane flame model[END_REF][START_REF] Turányi | Effect of the uncertainty of kinetic and thermodynamic data on methane flame simulation results[END_REF]. In the present study, we performed a local sensitivity analysis on R c . R c depends on the following parameters

R c = f (n, ρ, c p , T ad , E a , (dT dx) max , λ, µ, D H 2 , D O 2 ), (7)
where n is wavenumber; ρ, c p , λ, µ are respectively the density, heat capacity at constant pressure, thermal conductivity and dynamic viscosity of the reactive mixture;

S i = ln(R ′ c ) -ln(R ′′ c ) ln(z ′ i ) -ln(z ′′ i ) , (8)
where z An in-house Matlab code was developed to execute the calculation with the thermokinetics and transport properties obtained through Cantera. The mechanism used for this analysis was Wang's model [START_REF] Davis | An optimized kinetic model of H 2 /CO combustion[END_REF]. Then, all the sensitivity coefficients obtained for the same initial condition were normalized by the maximum absolute value among them to clearly compare the relative contributions of the parameters on predicted R c .

Uncertainty quantification

On the basis of local sensitivity analysis, uncertainty propagation resulting from input model parameter can be explored if its uncertainty range is known. However, we noticed that most properties in Equation 7 cannot be obtained directly from the input chemistry set (containing species and reactions descriptions, thermodynamics, and transport data). To explore the uncertainty on the predicted R c , we have chosen to study the uncertainty propagation of the kinetics, thermodynamics, and transport properties of H atom. The reason of targeting H atom is that it is one of the most studied reactive species and the most efficient chain carrier in hydrocarbon and hydrogen oxidation [START_REF] Law | Combustion Physics[END_REF][START_REF] Zador | Local and global uncertainty analyses of a methane flame model[END_REF]. Note that given the large amount of calculations required to determine the uncertainty related to H atom only, i.e. approximately 100,000, the uncertainty propagation for the parameters of the other species of the H 2 -O 2 chemical system were not investigated.

To be consistent with the tailoring procedure and the local sensitivity analysis, Wang's mechanism was used as the basic model [START_REF] Davis | An optimized kinetic model of H 2 /CO combustion[END_REF]. For the kinetics input, we chose the rate coefficients of H + O 2 → OH + O reaction. This reaction is a chain branching reaction controlling the level of the radical pool during the combustion of hydrogenated fuels in O 2 or air [START_REF] Turányi | Effect of the uncertainty of kinetic and thermodynamic data on methane flame simulation results[END_REF]. Many experimental data about this rate constant are available [START_REF] Yu | Reexamination of shock-tube measurements of the rate coefficient of H+ O 2 → OH+ O[END_REF][START_REF] Hong | A new shock tube study of the H+ O 2 → OH+ O reaction rate using tunable diode laser absorption of H 2 O near 2.5 µm[END_REF]. From the definition of uncertainty factor in Wang's model, it is assigned to the pre-exponential factor A. For the thermodynamics input, the enthalpy of formation for H atom was selected. The effect of the uncertainty on entropy was not investigated because it would also modify the kinetics through the calculation of the backward reaction rate. The standard enthalpy of the k th species (H 0 k ) is given by

H 0 k = R(a 1k T + a 2k T 2 2 + a 3k T 3 3 + a 4k T 4 4 + a 5k T 5 5 + a 6k ), (9)
where T is the temperature in Kelvin; a 1k to a 6k are the NASA polynomial fitting coefficients. Previous works have determined H 0 k at 298.15 K theoretically and experimentally [START_REF] Saeys | Ab initio calculations for hydrocarbons: Enthalpy of formation, transition state geometry, and activation energy for radical reactions[END_REF][START_REF] Curtiss | Assessment of Gaussian-2 and density functional theories for the computation of enthalpies of formation[END_REF][START_REF] Ruscic | IUPAC Critical evaluation of thermochemical properties of selected radicals. Part I[END_REF]. Since uncertainty over a wide range of temperature is not available, the deviation is presumed to be the same as the deviation at 298.15 K.

The effect of the uncertainty on the thermodynamics was investigated by perturbing a 6k . For the transport input, the Lennard-Jones potential well depth ε in cm -1 and collision diameter σ t in Å of H atom were selected.

The uncertainty of the studied parameters are assumed to be characterized by log-normal distributions. Each uncertain parameter can be normalized into a nondimensional variable x i , which follows the normal distribution,

x i = ln(p i p i,0 ) lnf i , (10)
where p i is the value of the i th parameter we selected; p i,0 is its nominal value; and f i is the uncertainty factor. This uncertainty characterization method has been widely employed in evaluating the kinetics-induced uncertainty and in optimizing kinetics models [START_REF] Wang | Combustion kinetic model uncertainty quantification, propagation and minimization[END_REF][START_REF] Urzay | Uncertaintyquantification analysis of the effects of residual impurities on hydrogen-oxygen ignition in shock tubes[END_REF]. After determining the distribution of x i , Monte Carlo sampling was used based on Equation 10 to generate new inputs. Monte Carlo analysis required 10,000 runs for each tailored composition. An in-house Matlab routine was developed to rewrite chemical models, applying the altered input parameters, and then calculate the corresponding R c .

The nominal value and uncertainty factor of A were obtained from Wang's model [START_REF] Davis | An optimized kinetic model of H 2 /CO combustion[END_REF]. A two-standard deviation uncertainty in A was used. Wang et al. [START_REF] Wang | Combustion kinetic model uncertainty quantification, propagation and minimization[END_REF] pointed out that the choice for the distribution for rate coefficient does not influence the forward uncertainty propagation projections significantly. The f i of a 6k was determined based on [START_REF] Ruscic | IUPAC Critical evaluation of thermochemical properties of selected radicals. Part I[END_REF] which showed the experimentally measured enthalpy of formation for H atom is 217.998±0.006 kJ/mol at 298.15 K. The deviation between theoretical and experimental H 0 k of H atom can be up to 1.4 kJ/mol which is much larger than the experimental uncertainty [START_REF] Saeys | Ab initio calculations for hydrocarbons: Enthalpy of formation, transition state geometry, and activation energy for radical reactions[END_REF]. A one-standard deviation uncertainty in a 6k was considered. The uncertainty range of transport parameters was determined based on the calculations done by Jasper et al. [START_REF] Jasper | Lennard-Jones parameters for combustion and chemical kinetics modeling from full-dimensional intermolecular potentials[END_REF]. In [START_REF] Jasper | Lennard-Jones parameters for combustion and chemical kinetics modeling from full-dimensional intermolecular potentials[END_REF], calculated Lennard-Jones parameters for H atom in a He bath obtained using different intermolecular potentials and for several normal hydrocarbons in several bath gases were compared with the ones from CHEMKIN transport database [START_REF] Kee | CHEMKIN-PRO, Release 15105[END_REF]. The effect of different baths on H atom was assumed to be the same as the one observed for small hydrocarbons.

Considering different intermolecular potential, the maximum uncertainty factors for ε and σ t for H-He are 1.5 and 10, respectively. To account for different bath gases, we used the average value obtained for He and N 2 ; the maximum uncertainty factors for ε and σ t are 1.1 and 2, respectively. After comparing the effects of intermolecular potential and bath gas, we selected 1.5 and 10 to be the uncertainty factor of ε and σ t , respectively. A two-standard deviation uncertainty in σ t was used. A threestandard deviation uncertainty in ε was used due to its large uncertainty factor.

The nominal value, uncertainty factor and distribution for selected parameters are listed in Table 2.

Results and discussion

Experimental results and model performance

Critical radius for cellularity onset

As the SEF propagates, a cellular structure would develop possibly from the initial perturbation induced by the electrodes and progressively cover the flame surface.

The development of the cellular structure is accompanied by apparent oscillations of the flame speed as the cells undergo successive phases of growth and break-up. Because determining the critical radius for cellularity onset through a visual inspection can lead to a subjective interpretation, we have chosen not to rely on such an approach. We defined the critical radius as the last flame radius (R f,U ) that we employed in our flame speed analysis. As a consequence, the R c values that we are reporting correspond to the maximum flame radius for which the flame propagation is not affected by the development of the cellular flame front. These values are reported in Table 1.

Fig. 6 shows the evolution of the ratio R ex c /R th c with equivalence ratio for the three investigated pressures. The uncertainty on R ex c is ±0.3 mm. This ratio ranges between 0.29 and 0.84, which indicates that the theory of Matalon over-estimates the critical radius for cellularity onset. Helium addition in the mixture has little effect on the deviation ratio. It is also noteworthy that the deviation is lower at 80/100 kPa as compared to the one observed at 50 kPa. It is noted that, at low pressure, R th c was predicted to be very large. For example, for rich H 2 -air mixture at 50 kPa, R th c was predicted to be greater than 20 cm. The large discrepancy observed at low pressure indicate that either the destabilizing effect of the hydrodynamic instability is under-estimated and/or that the stabilizing action of the thermo-diffusive effects are over-estimated. It also noted that, at low pressure, the accuracy of the stability theory may be compromised since it assumes that the flame thickness is negligible whereas experimentally, the flame thickness is significant under such conditions. between the flame thickness and the accuracy of the theory. For example, at P=50 kPa, the smallest and largest discrepancies were respectively observed at Φ=0.6 and 1.2, whereas the flame thickness is larger at Φ=0.6 than at Φ=1.2. The change of activation energy with equivalence ratio might also decrease locally the accuracy of the prediction since the high activation energy assumption might not be completely respected under some conditions. The experimental critical radius was related to the initial schlieren image, see Fig. 3. The surface of the flame has been covered with wrinkles at early stage of propagation due to the ignition effect. During the initial growth of the instability, the flame propagation velocity is not significantly affected which indicates that the flame surface area has not been significantly modified by the cellular structure. The data extracted during this period may still be usable to extract the unstretched flame speed. Although under certain conditions, the theory of Matalon predicts with good accuracy the critical radius (or Peclet number) for cellularity onset, discrepancies of up to a factor of 4 between the predicted and measured critical radius were previously reported by Law et al. [START_REF] Law | Cellular instabilities of expanding hydrogen/propane spherical flames at elevated pressures: theory and experiment[END_REF], consistent with the present results.

Nevertheless, it is not clear from the experimental results what is the relationship

Laminar flame speed and reaction model performance

The position of the experimental data points in the diagram of extrapolationinduced uncertainty established by Huo et al. [START_REF] Huo | Uncertainty reduction in laminar flame speed extrapolation for expanding spherical flames[END_REF] is shown in Fig. 7. The uncertainty related to the extrapolation to zero stretch-rate is below 2% (black area) which reveals the relevance of the mixture properties tailoring and data reduction procedures we applied. Since R f,U in the present study is relatively larger than the upper bound used in Huo et al. [START_REF] Huo | Uncertainty reduction in laminar flame speed extrapolation for expanding spherical flames[END_REF], most points are on the left side of the black area. Nevertheless, the uncertainty due to extrapolation is still expected to be low in this region. The overall uncertainty on the flame speed measured (∆S 0 u ) can be estimated using the formula employed by Nativel et al. [START_REF] Nativel | Laminar flame speeds of pentanol isomers: An experimental and modeling study[END_REF] ∆S

0 u = (∆S u,ext ) 2 + (∆S u,T ) 2 + (∆S u,rf ) 2 + (∆S u,rad ) 2 (11)
where ∆S u,ext is the uncertainty due to the extrapolation to zero-stretch rate, taken equal to 2%; ∆S u,T is the uncertainty due to the uncertainty on the temperature, taken equal to 0.4%; ∆S u,rf is the uncertainty due to the uncertainty on the flame radii, taken equal to 0.9%; and ∆S u,rad is the uncertainty due to radiation, taken equal to 0.7%. This analysis leads to a maximum uncertainty of 2.3%.

After evaluating the performance of the flame stability theory, the experimental S 0 u and L b were used to assess the performance of four chemical models. The relative error on S 0 u (E r (S 0 u )) was calculated using

E r S 0 u = ∆S 0 u S 0 u(Exp) = S 0 u(Cal) -S 0 u(Exp) S 0 u(Exp) , (12)
where S 0 u(Exp) and S 0 u(Cal) are the experimental and calculated unstretched laminar speed, respectively. Fig. 8 shows the evolution of the flame speed as a function of equivalence ratio for the three investigated pressures. At all pressures, the flame speed increases in the range Φ=0.6 to Φ=1.5. In the range Φ=1.5 to Φ=2.0, the flame speed plateaus at P=50 and 80 kPa and decreases at P=100 kPa. The trend can be explained by the respective changes of the mixture diffusivity and of the chemical energy release rate with equivalence ratio. All four models reproduce qualitatively the trends observed experimentally. Quantitatively, the deviation between curve observed experimentally. It is also noted that Konnov's model fails to predict a negative Markstein length in the case for which Φ=0.6 and P=50 kPa. Quantitatively, the mean absolute error on the Markstein length, calculated in the same way as the error on the flame speed, see Equation 12, for the models of Burke, Curran, Konnov and Wang are, 30.2%, 30.0%, 30.4%, 26.3%, respectively. The uncertainty on L b measurement is about one order of magnitude larger than that on S 0 u . Consequently, the errors in the prediction of the detailed models are on the same order as the uncertainty, which indicates that the predictions may actually be considered as satisfactory overall. Further efforts need to be devoted to improving the accuracy of L b measurement. After successfully reproducing S 0 u , the correct characterization of L b should be paid close attention to in the future reaction model development.

Local sensitivity analysis for R c

Local sensitivity coefficients for critical flame radius predicted by the flame stability theory of Matalon are shown in Fig. 10. Among these parameters, R c is most sensitive to T ad . It can be explained by the appearance of T ad in the calculation of expansion ratio and Zeldovich number. Furthermore, the expansion ratio is a basic variable that characterizes the effect of hydrodynamic and thermo-diffusive instability. R c is least sensitive to mixture viscosity µ. This is because Q 3 is much It is noted that the uncertainty range for the parameters of Equation 7 are very different. For example, the adiabatic temperature usually has an uncertainty below 0.2% [START_REF] Turányi | Effect of the uncertainty of kinetic and thermodynamic data on methane flame simulation results[END_REF], while for E a the uncertainty range is relatively large as will be discussed later. Results in Fig. 10 are based on the assumption that the uncertainty range of these parameters are identical due to lack of error limits of some parameters. Actual parameter sets with reliable uncertainty values are needed in order to quantify the uncertainty of the predicted critical radius.

Uncertainty propagation analysis for R c

Using Monte Carlo sampling, the uncertainty propagation was investigated assuming log-normal distributed input parameters. The results obtained for the tailored composition at P=100 kPa are shown in Fig. 11. The relative frequency histograms and fitted probability distributions of R c for the three equivalence ratios are shown in Fig. 12. The fitted mathematical expectations and variances are listed in Table 3. As seen in Fig. 11, the uncertainties on transport properties ε and σ t lead to larger uncertainty on R c than the uncertainties on the kinetic and thermodynamic parameters. The variance induced by the uncertainty on a 6k is the smallest among these four studied parameters. The present results are consistent with the previous study by Zador et al. [START_REF] Zador | Local and global uncertainty analyses of a methane flame model[END_REF] which showed that the kinetic contributions to the overall uncertainty on laminar flame speed and the maximum concentration of H, O and OH radicals are more significant than the thermodynamic contributions. This is due to the higher accuracy of thermodynamic data as compared to the kinetics parameters. Connecting the uncertainty factor in Table 2 to the standard deviation results, we noticed that although ε demonstrates the largest f , its uncertainty does not lead to the widest R c range. The widest R c range comes from the uncertainty on σ t which indicates that the collision diameter has a more significant impact on the calculation of the transport parameters, i.e. thermal conductivity and mass dif-fusivity and thus alter the Lewis and Prandtl numbers. With the mixture changing from lean to rich, the standard deviations resulting from the uncertainties of A, ε and σ t are increasing while for a 6k the maximum standard deviation is located at Φ=1.5 and the minimum standard deviation is located at Φ=1.2. Previous studies by Wang et al. [START_REF] Wang | Combustion kinetic model uncertainty quantification, propagation and minimization[END_REF][START_REF] Sheen | The method of uncertainty quantification and minimization using polynomial chaos expansions[END_REF] show the great potential of uncertainty quantification based on log-normal distributions of rate constants. From Fig. 12, log-normal distribution is also an appropriate method for uncertainty characterization for transport parameters.

The combined uncertainty stemming from the rate coefficients of H + O 2 → OH + O reaction, thermodynamics and transport parameters for H atom can be calculated using the equation employed by Xiouris et al. [START_REF] Xiouris | Laminar flame speeds under engine-relevant conditions: Uncertainty quantification and minimization in spherically expanding flame experiments[END_REF] σ

Rc = (σ A ) 2 + (σ a 6k ) 2 + (σ ε ) 2 + (σ σt ) 2 (13)
where σ Rc is the total uncertainty on R c ; σ A , σ a 6k , σ ε , σ σt are the fitted standard deviation of four selected parameters. Errors of different parameters are assumed to be independent. The calculated σ Rc are listed in Table 3.

As seen in Table 3, the final uncertainty on R c prediction is no better than 10% with 95% confidence interval. Only the uncertainties on H atom were considered.

In addition, the above discussion is based on the assumption that the calculated activation energy data is accurate. However, the numerical and experimental global E a for H 2 -air mixtures have been compared in [START_REF] Grosseuvres | Combustion properties of H2/N2/O2/steam mixtures[END_REF] and relative errors of up to 40% have been demonstrated. Further experimental determination of E a would be needed to improve the predictions of R c . The results of Grosseuvres et al. have been reproduced in the supplemental material of the present paper. The present results demonstrate that (i) the critical radius for cellularity onset predicted by the stability theory of Matalon is affected by a significant uncertainty due to the uncertainty on the input parameters; and (ii) the accuracy of the theoretical critical radius may be improved as more accurate input parameters become available in the literature.

Conclusion

Using large flame radius is an efficient way to minimize the extrapolation-induced uncertainty in laminar flame speed measurement. In order to obtain more data by limiting flame instability, a numerical implementation of the flame stability theory of Matalon et al. has been employed to tailor the mixture properties. This method was applied to H 2 -air mixtures with equivalence ratios from 0.6 to 2.0, at T=300 K and P=50/80/100 kPa. Comparing the theoretical and experimental results, we can see that over-estimation for R c is obvious at all pressures and equivalence ratios.

Especially at 50 kPa, the ratio between experimental R c to the theoretical one can be as low as 0.27 at Φ=1.2. Although the experimental critical radius was found to be significantly lower than the theoretical one, SEFs without cellular structure at large radius were obtained with a total uncertainty on S 0 u below 2.3%.

Fig. 3 .

 3 Fig. 3. Evolution of flame radius as a function of time. Conditions: H 2 /O 2 /N 2 /He mixture, Φ=1.5, P=80 kPa, T=296 K (same experiment as in Fig. 4 (d))

Fig. 4 .

 4 Fig. 4. Extrapolation surfaces of S 0 u using different R f,L and R f,U . Conditions: (a) tailored H 2 /O 2 /N 2 /He mixture, Φ=1.2, P=100 kPa, T=294 K; (b) H 2 /air mixture, Φ=1.5, P=80 kPa, T=295 K; (c) tailored H 2 /O 2 /N 2 /He mixture, Φ=1.5, P=100 kPa, T=296 K; (d) tailored H 2 /O 2 /N 2 /He mixture, Φ=0.8, P=80 kPa, T=296 K.

Fig. 5 .

 5 Fig. 5. (a) Probability density and fitted Gaussian distribution for S 0 u surface before and after removal of data outside [µ s -σ s , µ s + σ s ] range. (b) Flame radius range determination for S 0 u wavy surface. Conditions: H 2 /O 2 /N 2 /He mixture, Φ=1.5, P=80 kPa, T=296 K (same experiment as in Fig. 4 (d))

D H 2

 2 , D O 2 are the mass diffusion coefficients relating the diffusive fluxes to gradients in the H 2 and O 2 mole fraction fields. From Equation 7, it is seen that the prediction of R c depends on thermodynamics (ρ, c p , T ad ), kinetics (E a , (dT dx) max ) and transport (λ, µ, D H 2 , D O 2 ) parameters. The commonly used logarithmic sensitivity factor which indicates the effect of change in i th parameter on computed R c was employed [50]:

′ i and z ′′i

 z are the perturbed values of the i th parameter shown in Equation7; R ′ c and R ′′ c are the corresponding predicted output with the altered i th parameter. z ′ i and z ′′ i were set at 1.1 and 0.9 times of their nominal value. S is therefore showing the linear approximation of the effects of parameter changes on the model solutions.

Fig. 6 .

 6 Fig. 6. Deviation of the experimental critical radius R ex c from the theoretical one R th c for H 2 /O 2 /N 2 /He mixtures at T=300 K and P=50/80/100 kPa.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Experimental data points in the map of extrapolation-induced uncertainty. The color bar presents the level of error expressed as ∆S 0u S 0 u . Conditions: H 2 /O 2 /N 2 /He mixtures; Φ=0.6-2; P=50-100 kPa; T=293-300 K. Further details about this uncertainty map can be found in[START_REF] Huo | Uncertainty reduction in laminar flame speed extrapolation for expanding spherical flames[END_REF].

Fig. 9 .

 9 Fig. 9. Comparison between the experimental and calculated Markstein length for H 2 /O 2 /N 2 /He mixtures at T=300 K and P=50/80/100 kPa.

Fig. 10 .

 10 Fig. 10.

Fig. 11 .

 11 Fig. 11. Scatter plot showing the uncertainty propagation from input parameters to R c for tailored H 2 /O 2 /N 2 /He mixtures at P=100 kPa.

Four

  models were assessed with respect to the unstretched laminar flame speed and the Markstein length using unsteady numerical simulation. The error on both quantities were close to the experimental uncertainties although some of the qualitative trends could not be reproduced for the Markstein length.Further investigations on R c predictions were performed through local sensitivity analysis and uncertainty quantification using Monte Carlo sampling. Important flame properties from the flame stability theory were identified. The largest and smallest effects come from the change of adiabatic temperature and mixture viscosity, respectively. Uncertainty quantification was focused on the effect of the uncertainty on the kinetics, thermodynamics, and transport parameters of the H atom. The variances of the calculated R c are arranged in descending order: collision diameter σ t , Lennard-Jones potential well depth ε, pre-factor A of H + O 2 → OH + O reaction, thermodynamic polynomial coefficient a 6k of H atom. Transport-induced uncertainty is obvious which reveals the significance of accurate transport data. We believe that more precise critical radius predictions could be achieved if some of the parameters, in particular the activation energy, would be measured experimentally instead of being calculated. The present study was founded by the Natural Science Foundation of China under grant number 51750110502 and the Programme Investissement d'Avenir-MITHYGENE grant agreement no. ANR-11-RSNR-0015. RM was also supported by a start-up fund of the Center for Combustion Energy of Tsinghua University. RM is grateful to the French Labex CAPRYSSES program, grant agreement no. ANR-11-LABX-0006-01, for supporting his visit at ICARE-CNRS Orléans as a visiting professor.

Fig. 12 .

 12 Fig. 12. Relative frequency histograms and fitted normally distributed probability density function of R c at Φ=0.6/1.0/2.0 and P=100 kPa.

Table 1 :

 1 Composition of the H 2 /O 2 /N 2 /He mixtures. The initial conditions, the measured param-

		Define a target radius, 𝑅 𝑐 𝑡
	Calculate the theoretical critical radius,
		𝑅 𝑐 𝑡ℎ , for H2/O2/N2 mixture
	Yes	𝑅 𝑐 𝑡ℎ ≥ 𝑅 𝑐 𝑡	No
		Calculate the theoretical critical
		radius, 𝑅 𝑐 𝑡ℎ (𝐻𝑒), for H2/O2/He
		mixture (replace N2 by He)
		Yes	𝑅 𝑐 𝑡ℎ (𝐻𝑒) ≥ 𝑅 𝑐 𝑡
		Replace part of N2		Maintain the H2 to O2 ratio, increase the
		by He		dilution (only include He)
		Calculate the theoretical critical	Calculate the theoretical critical radius,
		radius, 𝑅 𝑐 𝑡ℎ (𝑚𝑜𝑑)		𝑅 𝑐 𝑡ℎ (𝐻𝑒/𝑚𝑜𝑑)
	No	𝑅 𝑐 𝑡ℎ 𝑚𝑜𝑑 = 𝑅 𝑐 𝑡		𝑅 𝑐 𝑡ℎ 𝐻𝑒/𝑚𝑜𝑑 = 𝑅 𝑐 𝑡	No
		Yes		Yes
		Stop the procedure, save the composition

No

Fig.

1

. The flow chart of tailoring procedure for H 2 /O 2 /N 2 /He mixture.

2.2. Experimental methodology 2.2.1. Experimental set-up

Experiments of SEF were carried out in a constant volume vessel. Detailed description of the apparatus has been given in

[START_REF] Goulier | Experimental study of laminar and turbulent flame speed of a spherical flame in a fan-stirred closed vessel for hydrogen safety application[END_REF]

so only a brief introduction is given here. It is a stainless steel spherical bomb with an inner diameter of 563 mm.

It can sustain a maximum pressure of 20 MPa and can be heated up to 573 K. It is

Table 2 :

 2 Nominal value, uncertainty factor and deviation for rate coefficients of H + O 2 → OH + O reaction, the enthalpy of formation, the Lennard-Jones potential well depth and collision diameter of H atom.

	Parameter, p i	p i,0	f i	Deviation
	A	2.644E+016	1.15	2-σ
	a 6k	2.547365990E+004 1.00023554	1-σ
	ε	145.00	10.00	3-σ
	σ t	2.05	1.50	2-σ

Table 3 :

 3 Fitted normally distributed probability density function of R c for tailored H 2 /O 2 /N 2 /He mixtures at P=100 kPa.

		Kinetics A	Thermodynamics a 6k	Transport ε	Transport σ t Total
	Φ	µ	σ	µ	σ	µ	σ	µ	σ	σ Rc
	0.6 10.205 0.051 10.208	7.78E-04	10.179 0.079 10.227 0.212 0.232
	0.8 10.223 0.065 10.220	5.38E-04	10.232 0.148 10.240 0.293 0.335
	1.0 10.320 0.056 10.339	1.11E-03	10.305 0.154 10.356 0.363 0.398
	1.2 10.228 0.112 10.226	9.34E-05	10.245 0.202 10.225 0.361 0.428
	1.5 10.064 0.162 10.103	5.58E-03	10.171 0.220 10.098 0.384 0.471
	2.0 10.182 0.189 10.192	1.26E-03	10.252 0.228 10.178 0.376 0.478

Acknowledgments