domain: chim spi

text:
Bingzhi Liu

Qimei Di

Maxence Lailliau

Nesrine Belhadj

Philippe Dagaut

Zhandong Wang
email: zhdwang@ustc.edu.cn

Experimental and kinetic modeling study of low-temperature oxidation of n-pentane

Keywords: kinetic modelling, pressure dependence, low-temperature oxidation, Korcek reaction, npentane, keto-hydroperoxides chemistry

Attractiveness in advanced low-temperature combustion engines drives a constantly updated understanding of low-temperature oxidation chemistry. In this work, the low-temperature oxidation chemistry of n-pentane in two jet-stirred reactors at atmospheric pressure and in the temperature range of 500-825 K was investigated using combined analysis methods of synchrotron vacuum ultraviolet photoionization mass spectrometry, gas chromatography, and Fourier transform infrared spectroscopy. Furthermore, the gaseous mixture from JSR was collected in acetonitrile for subsequent product characterization using flow injection analysis, high-pressure and ultra-highpressure liquid chromatography coupled to a Thermo Scientific™ Orbitrap® Q-Exactive highresolution mass spectrometry. Numerous intermediate species were identified by these analytical methods, which contributed to unravelling the low-temperature oxidation reaction network of npentane. A detailed n-pentane model was tentatively developed to reduce deviations between experimental measurements and model predictions by updating the rate constants of C5 ketohydroperoxide decomposition, C5 hydroperoxy cyclic ether decomposition, and Korcek reactions of C5 keto-hydroperoxide, and by introducing pressure-dependent rate constants for the reaction classes of Q ̇OOH + O2, Q ̇OOH decompositions, concerted HȮ2-elimination of RȮ2, C5 keto-hydroperoxide decomposition, C5 hydroperoxy cyclic ether decomposition, and Korcek reactions of C5 ketohydroperoxide, and by adding more detailed sub-mechanisms for C5 cyclic ethers and C5 keto-

Introduction

Concerns about the environmental degradation and depletion of energy resources drive the rapid development of advanced internal combustion engines (ICEs). Low-temperature combustion (LTC) is a cleaner and more efficient combustion strategy with promising applications for advanced ICE [START_REF] Krishnamoorthi | A review on low temperature combustion engines: Performance, combustion and emission characteristics[END_REF].

The design and evaluation of advanced LTC engines are dependent on accurate chemical kinetic models. Further elucidating the combustion intermediate species of model compounds and clarifying their reaction pathways is crucial to develop accurate reaction mechanisms and to improve model predictions. Experimental data derived from various diagnostic techniques play an essential role in chemical kinetic modelling. The advances in instrumental diagnostic and analytical methods enable the measurement of elusive intermediates, providing insights into the LTC chemistry of model compounds.

n-Pentane is an ideal model fuel for investigating the LTC chemistry of straight-chain alkanes, and its kinetic and thermochemical rules can be applied to larger fuel molecules [START_REF] Bugler | Revisiting the Kinetics and Thermodynamics of the Low-Temperature Oxidation Pathways of Alkanes: A Case Study of the Three Pentane Isomers[END_REF]. Several previous studies have contributed to the pioneering understanding of the oxidation chemistry and kinetic modelling of n-pentane, including jet-stirred reactors (JSR) [START_REF] Westbrook | A kinetic modeling study of n-pentane oxidation in a well-stirred reactor[END_REF], shock tubes (ST) [START_REF] Burcat | Shock-tube investigation of comparative ignition delay times for C1-C5 alkanes[END_REF][START_REF] Zhukov | Self-ignition of a lean mixture of n-pentane and air over a wide range of pressures[END_REF], and rapid compression machines (RCM) [START_REF] Minetti | The chemistry of pre-ignition of n-pentane and 1-pentene[END_REF][START_REF] Minetti | Autoignition Delays of a Series of Linear and Branched Chain Alkanes in the Intermediate Range of Temperature[END_REF]. Recently, Rodriguez et al. [START_REF] Rodriguez | Measuring hydroperoxide chain-branching agents during n-pentane low-temperature oxidation[END_REF] investigated the formation chemistry of chain-branching agents during n-pentane oxidation in JSRs using three advanced diagnostic methods, i.e., synchrotron vacuum ultraviolet photoionization mass spectrometry (SVUV-PIMS), laser single-photon-ionization mass spectrometry (SPI-MS), and cw-cavity ring-down spectroscopy (cw-CRDS). The intermediate evolution of hydrogen peroxide, alkyl hydroperoxides, alkenyl hydroperoxides, keto-hydroperoxides (KHPs), ketene, and diones was followed. Bugler et al. [START_REF] Bugler | Revisiting the Kinetics and Thermodynamics of the Low-Temperature Oxidation Pathways of Alkanes: A Case Study of the Three Pentane Isomers[END_REF][START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF] carried out ignition time studies of pentane isomers using one RCM and two STs, and developed a detailed kinetic model of pentane isomers by systematically updating the thermochemistry of important species and by evaluating the rate rules of crucial reaction classes. The proposed kinetic model predicts the ignition delay times (IDTs) well over a range of temperatures and equivalence ratio. Subsequently, Bugler et al. [START_REF] Bugler | An experimental and modelling study of n-pentane oxidation in two jet-stirred reactors: The importance of pressure-dependent kinetics and new reaction pathways[END_REF] used gas chromatography (GC) and the cw-CRDS method to study species distributions during n-pentane oxidation in two JSRs with a temperature range of 500-1100 K, detecting oxygenated species such as C5 aldehydes, ketones, and diones. The detailed kinetic model was further developed by including several important reaction classes proposed by Pelucchi et al. [START_REF] Pelucchi | Improved Kinetic Model of the Low-Temperature Oxidation of n-Heptane[END_REF] and Ranzi et al. [START_REF] Ranzi | New reaction classes in the kinetic modeling of low temperature oxidation of n-alkanes[END_REF] and by updating the rate constants of the reaction classes of C5 hydroperoxyl alkyl radicals to form C5 cyclic ether. This updated kinetic model mainly improved the prediction of JSRs data, but considerable deviations between experimental measurements and model predictions still remained for some final products and major intermediates, e.g., water (the difference is a factor of ca.

2 at 625 K), 2-pentene (the difference is a factor of >10 at 625 K), propanal (the difference is a factor of ca. 2-3 at 625 K), acetone (the difference is a factor of >10 at 625 K), formic acid (the difference is a factor of >10 at 625 K), acetic acid (the difference is a factor of >10 at 625 K), etc. Bourgalais et al. [START_REF] Bourgalais | Isomer-sensitive characterization of low temperature oxidation reaction products by coupling a jet-stirred reactor to an electron/ion coincidence spectrometer: case of n-pentane[END_REF] recently performed an analysis of intermediates during the low-temperature oxidation of npentane in a JSR, characterizing a variety of intermediate species using i²PEPICO (double imaging PhotoElectron PhotoIon COincidence). Belhadj et al. [START_REF] Belhadj | Experimental and kinetic modeling study of npentane oxidation at 10 atm, Detection of complex low-temperature products by Q-Exactive Orbitrap[END_REF] investigated the species distributions during the low-temperature oxidation of n-pentane in a JSR at a pressure of 10 atm. Many oxidation intermediates, such as cyclic ethers, pentanones, diones, alkylhydroperoxides, alkenylhydroperoxides, KHPs, and highly oxygenated molecules (HOMs), were identified using GC, Fourier transform infrared spectroscopy (FTIR), and liquid chromatography coupled to a Thermo Scientific™ Orbitrap® Q-Exactive high-resolution mass spectrometry (LC-HRMS) analysis. Discrepancies between the experimental results and model predictions for some important intermediates were also observed in the work of Belhadj et al. [START_REF] Belhadj | Experimental and kinetic modeling study of npentane oxidation at 10 atm, Detection of complex low-temperature products by Q-Exactive Orbitrap[END_REF], e.g., methanol (the difference is a factor of ca. 2-3 at 640 K), ethanol (the difference is a factor of >10 at 640 K), 1,3-pentadiene (the difference is a factor of >10 at 800 K), formic acid (the difference is a factor of >10 at 640 K), butanal (the difference is a factor of >10 at 640 K), and hydroperoxides (the difference in temperature distributions is ca. 25 -40K). Therefore, revisiting the low-temperature oxidation chemistry of n-pentane is still needed. Indeed, previous studies [2-10, 13, 14] contributed to valuable insights into n-pentane oxidation, but further elucidation of the reaction network and understanding of oxidation chemistry to reduce the discrepancies between measurements and model predictions are necessary.

In this work, two JSRs were used to study the low-temperature oxidation of n-pentane.

Numerous intermediate species were identified by a combined analytical method of SVUV-PIMS, GC, FTIR and LC-HRMS, including C5H10O3 (KHPs), C5H10O (cyclic ether), C5H10 (pentene isomers), CH3CHO (acetaldehyde), C2H5CHO (propanal), CH3COCH3 (acetone), C2H5COCH3 (2-butanone), C1-C3 acids, etc. These intermediates are crucial for elucidating the oxidation reaction pathways of npentane and developing an accurate kinetic model. According to experimentally measured intermediates and previously available literature data, a detailed kinetic model of n-pentane was tentatively developed by updating the rate constants of hydroperoxide decomposition, by including pressure-dependent rate constants of key reaction classes, and by adding a more detailed submechanism for important intermediate species such as C5 cyclic ethers and C5 KHPs.

Experimental methods

The low-temperature oxidation of n-pentane in JSRs was studied using four analytical methods: SVUV-PIMS, GC, FTIR, and LC-HRMS. The experiments were carried out at the atomic and molecular physics beamline (BL09U) of National Synchrotron Radiation Laboratory (NSRL) and Centre National de la Recherche Scientifique (CNRS). The detailed experimental conditions are summarized in Table 1.

JSR coupled with SVUV-PIMS

This experiment was carried out at NSRL in Hefei, China, using the JSR-SVUV-PIMS experimental platform to study the low-temperature oxidation chemistry of n-pentane. The specific descriptions of the JSR-SVUV-PIMS experimental device can be found in a recent publication [START_REF] Xu | Comprehensive study of the low-temperature oxidation chemistry by synchrotron photoionization mass spectrometry and gas chromatography[END_REF]; only a brief description is given here.

The n-pentane (purity 99.9% from Shanghai Aladdin Bio-Chem Technology Co., Ltd) was vaporized and mixed with Ar (dilution gas) by a controlled evaporation and mixing (CEM) system.

The n-pentane/Ar mixture is then mixed with O2 (oxygen) and finally enters into the 78 cm 3 JSR.

There are four nozzles with a diameter of 0.3 mm in the JSR sphere, and the gas mixture forms jets through the four nozzles and is evenly mixed. The tapered quartz nozzle is welded to the JSR sphere for SVUV-PIMS sampling. The sampled molecular beam subsequently enters into the SVUV-PIMS to be ionized and analyzed. The mass resolution (m/Δm) of SVUV-PIMS is ca. 4000 at m/z 72 in this experiment. The JSR is heated to the specified temperature by a specially designed tubular heating furnace, and the actual reaction temperature is monitored online by a K-type thermocouple inserted into the outlet of the JSR. The uncertainty of the measured temperature is estimated to be within ± 10 K. The methods of species identification and quantification in SVUV-PIMS analysis can be found in a previous study [START_REF] Herbinet | Detailed product analysis during the low temperature oxidation of n-butane[END_REF]. The photoionization cross sections (PICSs) used in this work are from an online database [START_REF]Photonionization cross section database (Version 2.0[END_REF] and some available literature data [START_REF] Dodson | VUV photoionization cross sections of HO2, H2O2, and H2CO[END_REF][START_REF] Moshammer | Quantification of the Keto-Hydroperoxide (HOOCH2OCHO) and Other Elusive Intermediates during Low-Temperature Oxidation of Dimethyl Ether[END_REF][START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF][START_REF] Dong | Improving quantification of hydrogen peroxide by synchrotron vacuum ultraviolet photoionization mass spectrometry[END_REF][START_REF] Hu | Elucidating the photodissociation fingerprint and quantifying the determination of organic hydroperoxides in gas-phase autoxidation[END_REF][START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF]. Furthermore, the quantification of organic hydroperoxides is essential for understanding low-temperature oxidation chemistry. Battin-leclerc et al. [START_REF] Battin-Leclerc | Chemistry deriving from OOQOOH radicals in alkane lowtemperature oxidation: A first combined theoretical and electron-ion coincidence mass spectrometry study[END_REF] improved the quantification of the mole fraction of KHP by considering the contribution of fragmentation, but the lack of available PICS remains an important source of quantitative uncertainty.

Recently, Hu et al. [START_REF] Hu | Elucidating the photodissociation fingerprint and quantifying the determination of organic hydroperoxides in gas-phase autoxidation[END_REF] reported experimentally measured PICS for various hydroperoxides (alkyl hydroperoxides and 4-hydroperoxy-2-pentanone), enabling quantitative measurements of organic hydroperoxides. In this work, we quantified the mole fractions of C1, C2, and C5 alkyl hydroperoxides, and C5 KHP during low-temperature oxidation of n-pentane using PICSs measured by Hu et al. [START_REF] Hu | Elucidating the photodissociation fingerprint and quantifying the determination of organic hydroperoxides in gas-phase autoxidation[END_REF].

The uncertainties of the quantified mole fractions are estimated to be within ± 10% for n-pentane and O2, ± 30% for species with known PICSs, and a factor of two for species with estimated PICSs.

JSR coupled with FTIR/GC/LC-HRMS

This experiment was performed at CNRS in Orléans, France. JSR coupled with FTIR/GC/ LC-HRMS analysis was adopted to study the low-temperature oxidation of n-pentane. The details of this experimental platform are described in previous publications [START_REF] Belhadj | Experimental and kinetic modeling study of npentane oxidation at 10 atm, Detection of complex low-temperature products by Q-Exactive Orbitrap[END_REF][START_REF] Togbé | Experimental and detailed kinetic modeling study of 1-pentanol oxidation in a JSR and combustion in a bomb[END_REF][START_REF] Dagaut | Experimental and modeling study of the kinetics of oxidation of ethanol-gasoline surrogate mixtures (E85 surrogate) in a jet-stirred reactor[END_REF].

The experiment was carried out in a 35 cm 3 JSR, with N2 as the diluent gas. The gas mixture in the JSR is sampled by a low-pressure sonic probe, and the sampled mixture is transmitted to the analyzers (FTIR and GCs) via Teflon lines. The FTIR was used to detect and quantify H2O, CO, CO2, CH2O, etc. Two GCs were used for detecting and quantifying stable intermediate species. A GC equipped with a DB624 column and flame ionization detector (FID) was used for detecting some oxygenated species, while another GC equipped with a CP-Al2O3/KCl column and FID was used for detecting several hydrocarbons. Among them, the GC equipped with a CP-CarboPLOT P7 column and thermal conductivity detector (TCD) was also used to detect O2 and H2. In addition, GC/MS was used for species identification according to the NIST Mass Spectrometry database. The uncertainties of the measured reaction temperatures in this experiment were estimated to be within ± 10 K, and the uncertainties of the quantified mole fractions in FTIR/GC analysis were within ± 15%. For carboxylic acids, uncertainties are higher, ca. ± 30%

In order to perform LC-HRMS analyses, the gaseous samples exiting a sampling probe were collected for 75 min in a glass vial containing 25 mL of acetonitrile kept at 273 K. To limit the thermal degradation of dissolved samples, they were stored at 258 K in a freezer for future chemical analyzes.

Flow injection analysis (FIA) provided a global compound signal with no chromatographic separation.

The FIA, high-pressure and ultra-high-pressure liquid chromatography (HPLC, UHPLC) analyses were coupled to a Thermo Scientific™ Orbitrap® Q-Exactive HRMS. Atmospheric pressure chemical ionization (APCI) produced ions in positive (M + H) + and negative (M -H) -modes (see Table S1 in Supplementary Materials for details). MS/MS analyses were performed at 10 eV using higher energy collisional dissociation. This analytical method allowed the detection and separation of specific oxygenated intermediates and products, providing additional qualitative information. In the current work, dozens of oxygenated intermediates from the low-temperature oxidation of n-pentane were characterized by this analytical method, and these intermediates are recorded in detail in Table S1 of the Supplementary Materials.

Kinetic model

The The rate constants for the reaction class of Q ̇OOH + O2, Q ̇OOH decompositions, and concerted HȮ2-eliminations of RȮ2 are critical for the low-temperature oxidation reactivity and autoignition behavior of hydrocarbons, as they directly determine the branching ratios of chain propagation and chain branching pathways. Previous publications [START_REF] Yao | High-Pressure-Limit and Pressure-Dependent Rate Rules for Unimolecular Reactions Related to Hydroperoxy Alkyl Radicals in Normal Alkyl Cyclohexane Combustion. 1. Concerted HO2 Elimination Reaction Class and β-Scission Reaction Class[END_REF][START_REF] Xing | Theoretical kinetic studies for low temperature oxidation of two typical methylcyclohexyl radicals[END_REF][START_REF] Goldsmith | Role of O2 + QOOH in low-temperature ignition of propane. 1. Temperature and pressure dependent rate coefficients[END_REF][START_REF] Duan | A theoretical kinetics study on low-temperature oxidation of n-C4H9 radicals[END_REF][START_REF] Liu | Experimental and Updated Kinetic Modeling Study of Neopentane Low Temperature Oxidation[END_REF] highlighted the pressure dependence of the rate constants of these reaction classes, but high-pressure limit rate constants were used for these reaction classes in NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF]. In this updated model, the rate constants at 1 atm for the reaction classes of Q ̇OOH + O2, Q ̇OOH decompositions, and concerted HȮ2-eliminations of RȮ2 are tentatively added, with an estimated factor of ca. 2-3 slower than their high-pressure limits.

KHPs are critical chain branching agents that directly control autoignition chemistry. Although numerous experimental studies of KHPs during the low-temperature oxidation of many hydrocarbons and oxygenated fuels have been reported [START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF][START_REF] Battin-Leclerc | Products from the oxidation of linear isomers of hexene[END_REF][START_REF] Bourgalais | Jet-Stirred Reactor Study of Low-Temperature Neopentane Oxidation: A Combined Theoretical, Chromatographic, Mass Spectrometric, and PEPICO Analysis[END_REF][START_REF] Zhang | New insights into propanal oxidation at low temperatures: An experimental and kinetic modeling study[END_REF], studies on the kinetic parameters of KHPs in the literature are scarce, leading to considerable uncertainty in kinetic models. In the case of n-pentane, the rate constants of C5 KHP decomposition from the Polimi model [START_REF] Ranzi | New reaction classes in the kinetic modeling of low temperature oxidation of n-alkanes[END_REF], LLNL model [START_REF] Sarathy | Comprehensive chemical kinetic modeling of the oxidation of 2-methylalkanes from C7 to C20[END_REF],

NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF], and the presently updated model are compared as a function of temperature in A similar approach was recently adopted in the evaluation of the rate constants of C7 KHP decomposition in the n-heptane model [START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF]. In NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF], the rate constants of the reaction class of C5 KHP decomposition adopt the O-OH bond scission of heptyl hydroperoxides (C7H15O2H)

experimentally measured by Sahetchian et al. [START_REF] Sahetchian | Determination of the gas-phase decomposition rate constants of heptyl-1 and heptyl-2 hydroperoxides C7H15OOH[END_REF], but these rate constants are too fast for predicting the C5 KHP distributions; the prediction is ca. 30 K earlier than the experimental measurements.

Moreover, it was noted that several published rate constants of O-OH bond scission of hydroperoxides are a factor of ca. 4 slower than heptyl hydroperoxides measured by Sahetchian et al. [START_REF] Sahetchian | Determination of the gas-phase decomposition rate constants of heptyl-1 and heptyl-2 hydroperoxides C7H15OOH[END_REF] at 550-750 K, e.g., tert-butyl hydroperoxide [START_REF] Kirk | The pyrolysis of alkyl hydroperoxides in the gas phase[END_REF] and ethyl hydroperoxide [START_REF] Baulch | Evaluated kinetic data for combustion modelling[END_REF]. Therefore, the rate constants of the reaction class of C5 KHP decomposition were decreased by a factor of ca. 4 in the updated model to better predict the measured C5 KHP distribution at 1 atm. Recently, Goldsmith et al. [START_REF] Goldsmith | Effect of non-thermal product energy distributions on ketohydroperoxide decomposition kinetics[END_REF] computed temperature-and pressure-dependent rate constants for the decomposition channels of a C3 KHP using RRKM/ME methods, including the O-OH bond scission and Korcek reactions. The computed results indicated that the rate constants of the O-OH bond scission for KHP at 1 atm are a factor of ca. 2 slower than at 10 atm and for T > 575 K, and the rate constants of the Korcek reactions exhibited considerable pressure dependence. In the updated model, the pressure-dependent rate constants of the O-OH bond scission for C5 KHP are included, and the rate constant at pressures of 10 atm and higher is estimated to be a factor of ca. 2 faster than that at 1 atm, i.e., the rate constants at 10 atm and higher are a factor of ca. 2 slower than for heptyl hydroperoxides as measured by Sahetchian et al. [START_REF] Sahetchian | Determination of the gas-phase decomposition rate constants of heptyl-1 and heptyl-2 hydroperoxides C7H15OOH[END_REF]. The pressure-dependent rate constants of the Korcek reactions are updated following the theoretical calculations of Goldsmith et al. [START_REF] Goldsmith | Effect of non-thermal product energy distributions on ketohydroperoxide decomposition kinetics[END_REF], with a rate constant of 0.17 s -1 at 600 K. This is consistent with the calculations (0.16 s -1 at 600 K) of Ranzi et al. [START_REF] Ranzi | New reaction classes in the kinetic modeling of low temperature oxidation of n-alkanes[END_REF]. Recently, Popolan-Vaida et al. [START_REF] Popolan-Vaida | Formation of Organic Acids and Carbonyl Compounds in n-Butane Oxidation via γ-Ketohydroperoxide Decomposition[END_REF] provided an estimated upper limit of 1.9 s -1 at 600 K for the rate constant of Korcek decomposition. Therefore, the rate constant used in the updated model is approximately 10 times smaller than the upper limit estimated by Popolan-Vaida et al. [START_REF] Popolan-Vaida | Formation of Organic Acids and Carbonyl Compounds in n-Butane Oxidation via γ-Ketohydroperoxide Decomposition[END_REF].

Furthermore, the H-atom abstraction reactions on KHP can form several intermediates, e.g., diones, keto cyclic ethers, and olefinic ketones. Xing et al. [START_REF] Xing | Degradation of Carbonyl Hydroperoxides in the Atmosphere and in Combustion[END_REF] carried out theoretical calculations for the H-atom abstraction reactions of 4-hydroperoxy-2-pentanone in an n-pentane oxidation system.

Recently, the H-atom abstraction reactions of C7 KHP and its subsequent reactions were added to the n-heptane model by Xie et al. [START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF] based on the calculations of Xing et al. [START_REF] Xing | Degradation of Carbonyl Hydroperoxides in the Atmosphere and in Combustion[END_REF], to consider the contribution of diones, keto cyclic ethers and olefinic ketones to C7H12O and C7H12O2. In this updated model, the H-atom abstraction reactions on C5 KHPs by ȮH, Ö, Ḣ, HȮ2, CH3Ȯ, ĊH3, Ċ2H5, Ċ2H3, CH3Ȯ2 and O2 were also included according to these previous works [START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF][START_REF] Xing | Degradation of Carbonyl Hydroperoxides in the Atmosphere and in Combustion[END_REF].

Cyclization reactions of Q ̇OOH radicals can lead to cyclic ethers and ȮH radicals. A recent publication [START_REF] Tran | Chemical kinetics of cyclic ethers in combustion[END_REF] reviews the chemical kinetics of cyclic ethers, emphasizing their importance in lowtemperature oxidation. Previous studies [START_REF] Doner | Stereoisomer-dependent unimolecular kinetics of 2,4dimethyloxetanyl peroxy radicals[END_REF][START_REF] Doner | Isomer-dependent reaction mechanisms of cyclic ether intermediates:cis-2,3-dimethyloxirane andtrans-2,3-dimethyloxirane[END_REF][START_REF] Christianson | Reaction mechanisms of a cyclic ether intermediate: Ethyloxirane[END_REF] revealed that cyclic ether chemistry enables the potential formation of a variety of intermediate species, such as bicyclic ethers, olefinic cyclic ethers, and diones. In the recent work of Xie et al. [START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF], the predictions of C7H12O and C7H12O2 in the temperature range of 625-750 K were significantly improved by adding the detailed C7 cyclic ether sub-mechanism to the n-heptane model. However, the sub-mechanism of C5 cyclic ethers was not fully detailed in NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF]. 2-Methyltetrahydrofuran and 2,4-dimethyloxetane are the two major cyclic ether intermediates in the low-temperature oxidation of n-pentane. In this work, we added the detailed sub-mechanism of these two cyclic ethers to better understand cyclic ether oxidation kinetics.

For 2-methyltetrahydrofuran, its sub-mechanism is mainly derived from the published 2methyltetrahydrofuran model [START_REF] Fenard | Experimental and kinetic modeling study of 2-methyltetrahydrofuran oxidation under engine-relevant conditions[END_REF], and some reactions and their rate constants were updated in this work according to some available models [START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF][START_REF] Fenard | Experimental and kinetic modeling study of 2-methyltetrahydrofuran oxidation under engine-relevant conditions[END_REF][START_REF] Fenard | 2,5-Dimethyltetrahydrofuran combustion: Ignition delay times at high and low temperatures, speciation measurements and detailed kinetic modeling[END_REF][START_REF] Vanhove | Experimental Study of Tetrahydrofuran Oxidation and Ignition in Low-Temperature Conditions[END_REF]. The 2-methyltetrahydrofuran sub-mechanism mainly includes various pathways of successive additions of radicals to O2 and subsequent reactions, which leads to olefinic cyclic ethers, bicyclic ethers, cyclic KHPs, etc. Especially, Hansen et al. [START_REF] Hansen | Isomer-Selective Detection of Keto-Hydroperoxides in the Low-Temperature Oxidation of Tetrahydrofuran[END_REF] recently reported isomer-resolved detection and quantification of cyclic KHPs in the low-temperature oxidation of tetrahydrofuran. For 2,4-dimethyloxetane, its sub-mechanism was added by reference to a previous publication [START_REF] Doner | Stereoisomer-dependent unimolecular kinetics of 2,4dimethyloxetanyl peroxy radicals[END_REF] and by analogy with the existing mechanism of tetrahydrofuran and its derivatives [START_REF] Fenard | Experimental and kinetic modeling study of 2-methyltetrahydrofuran oxidation under engine-relevant conditions[END_REF][START_REF] Fenard | 2,5-Dimethyltetrahydrofuran combustion: Ignition delay times at high and low temperatures, speciation measurements and detailed kinetic modeling[END_REF][START_REF] Vanhove | Experimental Study of Tetrahydrofuran Oxidation and Ignition in Low-Temperature Conditions[END_REF]. The thermochemistry of related species was estimated using the THERM [START_REF] Ritter | THERM: Thermodynamic property estimation for gas phase radicals and molecules[END_REF].

The 2,4-dimethyloxetane sub-mechanism mainly includes the first and second O2 addition and subsequent reaction pathways, which leads to olefinic cyclic ethers, bicyclic ethers, diones, cyclic KHPs, etc.

ȮH radical addition to acetaldehyde can lead to the formation of formic acid. The rate constant is estimated to be the same as the ȮH radical addition to formaldehyde due to their similar bond dissociation energies. Furthermore, the rate constant of the H-atom abstraction reaction of acetone by the CH3COCH2Ȯ2 radical is estimated to be the same as the H-atom abstraction reaction of acetone by the CH3Ȯ2 radical to better predict the formation of acetone.

Results and discussion

The detailed pool of species formed during the low-temperature oxidation of n-pentane was determined by SVUV-PIMS, GC, and FTIR analysis. It includes hydroperoxides, aldehydes, carboxylic acids, ethers, alkanes, alkenes, and some final products. Furthermore, dozens of specific oxygenated intermediates (see Table S1 in Supplementary Materials for details) were characterized by LC-HRMS: C3-C5 KHPs, C5 hydroperoxides, unsaturated C5 hydroperoxides, 2-butanone, cyclic ethers, diones, HOMs, etc. CHEMKIN PRO software [START_REF]CHEMKIN-PRO 15092[END_REF] was adopted to simulate the experimental results. In low-temperature oxidation, successive O2 additions of radicals and intramolecular H-atom migration reactions drive the oxidation process. Especially, the exploration of the third O2 addition reactions contributes to the insights of more complex oxygenated intermediates [START_REF] Wang | Unraveling the structure and chemical mechanisms of highly oxygenated intermediates in oxidation of organic compounds[END_REF][START_REF] Wang | Exploring hydroperoxides in combustion: History, recent advances and perspectives[END_REF][START_REF] Wang | Additional chain-branching pathways in the low-temperature oxidation of branched alkanes[END_REF]. A previous study [START_REF] Wang | Third O2 addition reactions promote the low-temperature auto-ignition of n-alkanes[END_REF] demonstrated that the third O2 addition reaction pathway can accelerate the autoignition of n-hexane, but the effect on n-pentane is negligible. Therefore, the third O2 addition reaction pathways were not considered in this work, although the products of the third O2 addition reactions (e.g., keto-

The first and second additions to O 2

As shown in Scheme 1, n-pentane forms n-pentyl radicals (Ṙ) initially through H-atom abstraction reactions by O2 and subsequently by radical pools (e.g., ȮH radicals). Ṙ radical addition to O2 leads to n-pentyl peroxy radicals (RȮ2), i.e., the first addition to O2. RȮ2 radicals dominantly form hydroperoxy pentyl radicals ( Q ̇ OOH) via intramolecular H-atom migration reactions, accompanied by a competing channel for the formation of the C5 alkene via HȮ2-elimination reactions.

Subsequently, the O2 addition to Q ̇OOH radicals lead to ȮOQOOH radicals, i.e., the second addition to O2. The ȮH-elimination reactions of ȮOQOOH radicals are responsible for the key intermediate KHP.

Figure 2 shows the measured and model-predicted mole fraction profiles of n-pentane, O2, H2O, CO, CO2, 1-pentene (C5H10-1), 2-pentene (C5H10-2), and 2-methyltetrahydrofuran (C5H10O1-4) during the low-temperature oxidation of n-pentane. Under the present experimental conditions, n-pentane showed significant oxidation, with a ~ 50% conversion at 650 K. NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF] overestimates the n-pentane reactivity at 550-650 K, which can also be observed in the predictions of some to the formation of a ketoalkoxyl radical and an ȮH radical. This is the dominant consumption channel for KHP. Additionally, the Korcek reaction is considered a competitive pathway for the chain branching step. Jalan et al. [START_REF] Jalan | New pathways for formation of acids and carbonyl products in low-temperature oxidation: the Korcek decomposition of gamma-ketohydroperoxides[END_REF] determined the γ-KHP decomposition reaction pathways that form cyclic peroxide isomers using ab initio calculations in a propane oxidation system. They indicated that γ-KHP can decompose to carbonyl compounds and carboxylic acids via the Korcek decomposition channel. Popolan-Vaida et al. [START_REF] Popolan-Vaida | Formation of Organic Acids and Carbonyl Compounds in n-Butane Oxidation via γ-Ketohydroperoxide Decomposition[END_REF] recently provided experimental evidence for carbonyl compounds and carboxylic acid formation through Korcek decomposition of γ-KHP in an n-butane oxidation system using the SVUV-PIMS method.

For the H-atom abstraction reactions of KHP, Xing et al. The subsequent reaction of the CH3COĊH2 radical forms acetone via a H-atom abstraction reaction.

Furthermore, the β-C-C bond scission and subsequent reactions of ketoalkoxyl radicals deriving from other C5 KHPs is one of the formation routes for propanal, 2-butanone, formaldehyde, and methanol.

The decomposition reactions of formaldehyde, acetaldehyde, and propanal mainly lead to HȮ2/ĊH3/Ċ2H5 radicals, and the subsequent reactions of these radicals contribute to H2O2, CH3O2H, and C2H5O2H production. The experimental measurements and model predictions for these intermediate species are presented in Fig. 5. The updated model provides a better prediction of these species but overestimates the mole fraction of C2H5O2H by a factor of ca. 2-3. and these reactions can lead to the formation of three carbonyl acids (formic acid, acetic acid, and propionic acid) and four aldehydes/ketones (acetaldehyde, propanal, acetone, and 2-butanone). 4-Hydroperoxy-2-pentanone is the most dominant C5 γ-KHP, and it leads to the formation of acetic acid and acetone via the Korcek reaction (R3). The measured and model-predicted mole fractions of formic acid, acetic acid, and propionic acid are presented in Fig. 5. Formic acid and acetic acid were detected and quantified by both FTIR and SVUV-PIMS, but FTIR measurements are a factor of ca. 2 higher than SVUV-PIMS measurements. Additionally, acetic acid was also analyzed by the LC-HRMS method, and the LC-HRMS signal profiles were scaled to FTIR data for comparison.

NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF] overpredicted the mole fractions of formic acid by a factor of ca. 10 and underpredicted the formation of acetic acid and propionic acid. For propionic acid, the presence of the allylhydroperoxide isomer is one of the potential reasons for the inconsistencies between model prediction and experimental measurements. The updated model improved the predictions of formic acid mainly by updating the rate constants of ȮH radical addition to acetaldehyde. Additionally, the rate constant reduction of C5 KHP decomposition and the rate constant increase of the Korcek reaction lead to more carbon flux to these three carboxylic acids and improve their predictions. As shown in Fig. 6, this branching ratio deviated very large from the calculation of Goldsmith et al. [START_REF] Goldsmith | Effect of non-thermal product energy distributions on ketohydroperoxide decomposition kinetics[END_REF]. Therefore, a thorough evaluation of the branching ratio of the Korcek reaction, both from experiments and/or theoretical calculations, is believed to help understanding the formation chemistry of carbonyl acids and evaluate the prediction of carboxylic acid.

Scheme 2

The Korcek reaction pathways of C5 γ-KHPs in the n-pentane oxidation system. Since no PICSs are available for these isomers, the total mole fractions of C5H8O and C5H8O2 are estimated at 11.5 eV using the PICSs of 3-hepten-2-one and 2,3-pentanedione measured by Xie et al. [START_REF] Xie | Revisiting low temperature oxidation chemistry of n-heptane[END_REF], respectively. The measured (symbols) and model-predicted (lines) mole fraction profiles of C5H8O and C5H8O2 are presented in Fig. 7; large deviations between the measurements and simulations were observed if only the H-atom abstraction reaction pathways of C5 KHP were considered. For the H-atom abstraction reactions of C5 KHPs, olefinic ketones contribute to the formation of C5H8O, while diones and keto cyclic ethers contribute to the formation of C5H8O2. The temperature range over which these species are formed is ca. 550-650 K, because they are directly produced from KHP, which is

The oxidation of pentenes and C 5 cyclic ethers

Alkenes are one of the ubiquitous intermediates in combustion systems, and alkene chemistry can form numerous key oxidation species. ȮH radical addition to the double bond leads to hydroxyalkyl radical formation, which is an important reaction class for alkene chemistry. The O2 addition reactions of hydroxyalkyl and subsequent β-scission reactions lead to aldehydes production, i.e., the Waddington mechanism. Furthermore, large alkenes (including pentenes) can also undergo successive radical additions to O2 and intramolecular H-atom migration reactions. In the case of npentane, 1-pentene and 2-pentene are formed. In NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF], the pentene isomer submechanism has been well established based on the work of Dong et al. [START_REF] Dong | A comprehensive experimental and kinetic modeling study of 1-and 2-pentene[END_REF]. The pentene isomers undergo H-atom abstraction, addition to O2, intramolecular isomerization and subsequent cyclization reactions, leading to the formation of olefinic cyclic ethers, which are the major components of C5H8O in Fig. 7a.

The cyclization reaction of the Q ̇OOH radical is responsible for the formation of cyclic ethers, and this reaction class has a significant effect on autoignition chemistry. In NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF], the C5 cyclic ether sub-mechanism has not been well established. predicts the formation of the C5 unsaturated ROOH by ca. 100 K. This could be due to the fact we could not differentiate between unsaturated C5 hydroperoxides and hydroxy-carbonyls. Additionally, the distributions of the three HOM species, i.e., diketohydroperoxides, unsaturated dihydroperoxides and dihydroperoxides, mainly cover a wider temperature range of ca. 575-750 K. Both NUIGMech1.1 [START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF] and the present model lack formation channels for HOMs and cannot predict their formation.

Further development of more detailed kinetic models including the introduction of more complex reaction networks is expected to contribute to the understanding of the formation of these C5 HOMs.

Summary and conclusion

The low-temperature oxidation of n-pentane in two JSRs at atmospheric pressure and over the temperature range of 500-825 K was investigated using a combined analytical method of SVUV-PIMS, GC, FTIR and LC-HRMS. Numerous intermediate species were identified, including KHPs, cyclic ethers, H2O2, alkyl hydroperoxides, alkenes, aldehydes, ketones, carboxylic acids, etc. In particular, LC-HRMS analysis characterized some specific oxygenated intermediates, such as C3-C5 KHPs, C2-C5 unsaturated ROOHs and several HOMs.

  detailed chemical kinetic model NUIGMech1.1[START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF] served as the starting point of the npentane model updates in this work. The n-pentane sub-mechanism in NUIGMech1.1[START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF] is derived from the previously published n-pentane model[START_REF] Bugler | Revisiting the Kinetics and Thermodynamics of the Low-Temperature Oxidation Pathways of Alkanes: A Case Study of the Three Pentane Isomers[END_REF][START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF][START_REF] Bugler | An experimental and modelling study of n-pentane oxidation in two jet-stirred reactors: The importance of pressure-dependent kinetics and new reaction pathways[END_REF], and it was developed by systematically updating the thermochemistry of important species and rate rules of key reaction classes. Although the n-pentane model was validated against various experimental data, some inconsistencies between model predictions and measurements drive further model development, e.g., the reactivity of n-pentane overestimated at 550-650 K, the considerable deviations between model predictions and experimental measurements for C5 KHP, CH3COCH3, formic acid, etc. Details of the model update in this work are given in the next paragraphs.

Fig. 1a .

 1a Fig.1a. The model-predicted mole fractions of C5 KHP obtained with NUIGMech1.1[START_REF] El-Sabor Mohamed | An experimental and kinetic modeling study of the auto-ignition of natural gas blends containing C1-C7 alkanes[END_REF] using the

Fig. 1 .

 1 Fig. 1. (a) The rate constants of C5 KHP decomposition from the Polimi model [12], LLNL model [36], NUIGMech1.1 [27], and the updated model in this work. (b) The model-predicted mole fractions of C5 KHP derived by NUIGMech1.1 [27] using these rate constants, with the measured C5 KHP signal

Scheme 1

 1 Scheme 1 Reaction pathway analysis during the low-temperature oxidation of n-pentane at 600 K. The numbers represent the percentages of the corresponding reaction fluxes. Only the dominant reaction pathways are shown, with the model update highlighted.

intermediates shown in Figs. 1 - 3 ,

 13 Fig. 5, and Fig. S1. The rate constants of the reaction class of Q ̇OOH + O2 directly determine O2 consumption and control the carbon flux to the chain branching pathways; subsequently, the KHP chemistry dominantly drives chain branching chemistry. As shown in Fig. S2, the reaction classes of Q ̇OOH + O2 and KHP decomposition have large sensitivity values for the npentane oxidation reactivity in the temperature range of 550-650 K. The model predictions are improved by updating the rate constants of these reaction classes, i.e., the predictions of n-pentane, O2, CO, H2O, and other intermediates mole fractions.The concerted HȮ2-elimination reactions of RȮ2 radicals are the dominant source of pentene isomers, and they have the potential to inhibit ignition by forming relatively unreactive HȮ2 radicals and impact negative temperature coefficient (NTC) chemistry. Two pentene isomers (1-pentene and 2pentene) were detected and quantified by both SVUV-PIMS and GC; the SVUV-PIMS and GC data are in good agreement. Both models can predict their mole fraction distributions well. The experimental measurements of SVUV-PIMS and GC for 2-pentene in Fig.2are the sum of the two isomers (cis-2-pentene and trans-2-pentene). Neither NUIGMech1.1 nor the updated model in this work distinguishes them. Additionally, the cyclization reactions of Q ̇ OOH radicals are mainly responsible for the formation of the C5 cyclic ethers, with ȮH radicals as a byproduct. This reaction is competitive with the second O2 addition reaction. As shown in Fig.2h, one dominant C5 cyclic ether (2-methyltetrahydrofuran) was well separated and quantified by the GC method, and it was also characterized by the LC-HRMS analytical method. Both models can well capture its mole fraction profile.

Fig. 2 .

 2 Fig. 2. Experimentally measured (symbols) and model-predicted (lines) mole fraction profiles of npentane (nC5H12), oxygen (O2), water (H2O), carbon monoxide (CO), carbon dioxide (CO2), 1-pentene (C5H10-1), 2-pentene (C5H10-2), and 2-methyltetrahydrofuran (C5H10O1-4) during the low-temperature oxidation of n-pentane. The error bars of SVUV-PIMS, GC, and FTIR data are given. The LC-HRMS measurement were only qualitative; signal profiles were scaled to GC data for comparison.

Fig. 3 .

 3 Fig. 3. Model-predicted mole fraction distributions of C5 ROOH, C5 KHP, and C5 HPCE, and the experimental measured mole fraction distributions of C5 ROOH and C5 KHP/HPCE by SVUV-PIMS during low-temperature oxidation of n-pentane. Since C5 ROOH and C5 KHP/HPCE could not be quantified by LC-HRMS analysis, the signal profiles of LC-HRMS were scaled to SVUV-PIMS data for comparison.

[ 42 ]

 42 performed theoretical calculations for C5 KHP (4-hydroperoxy-2-pentanone) in an n-pentane autoxidation system, showing that this reaction class can lead to crucial oxidation intermediates, such as diones, keto cyclic ethers, and olefinic ketones. To further improve the predictions of the chemical kinetic model, it is necessary to revisit KHPs chemistry. In the case of n-pentane, the rate of production (ROP) analysis for C5 KHP consumption is presented in Fig. 4, highlighting the three aforementioned reaction classes. The C5 KHP consumption occurs mainly at a temperature of 575-725 K, dominantly through unimolecular O-O dissociation. The other two reaction classes are competing channels of Reaction class 1, and their contribution to KHP consumption becomes negligible with increasing temperature. Even so, Reaction classes 1 and 2 contribute ca. 20% C5 KHP consumption flux at 600 K.

Fig. 4 .

 4 Fig. 4. The ROP analysis for C5 KHP consumption during low-temperature oxidation of n-pentane. The results are based on the simulation of the updated kinetic model in this work. The left coordinate corresponds to the black lines and represents the ROP of these reaction classes. The right coordinate corresponds to the stacked area and represents the percentage contribution of these reaction classes to the C5 KHP consumption flux. Reaction class 1 represents the homolytic O-O dissociation reaction pathways of C5 KHP. Reaction class 2 represents the Korcek reaction pathways. Reaction class 3 represents the H-atom abstraction reaction pathways of C5 KHP.

Fig. 5 .

 5 Fig. 5. Experimentally measured (symbols) and model-predicted (lines) mole fraction profiles of acetaldehyde (CH3CHO), acetone (CH3COCH3), propanal (C2H5CHO), 2-butanone (C2H5COCH3), formaldehyde (CH2O), methanol (CH3OH), hydrogen peroxide (H2O2), methyl hydroperoxide (CH3OOH), ethyl hydroperoxide (C2H5O2H), formic acid (HOCHO), acetic acid (CH3COOH), and propionic acid (C2H5COOH) during low-temperature oxidation of n-pentane. The error bars of SVUV-PIMS, GC, and FTIR data are given. Since LC-HRMS analysis are only qualitative, LC-HRMS signal profiles were scaled to GC or FTIR data for comparison.

Figure 6 .

 6 Figure 6. Branching ratio for the rate constants of Reaction class 1 and Reaction class 2 at 1 atm. These rate constants are derived from the calculation of Goldsmith et al. [40] and the estimates in this work , respectively. Reaction class 1 represents the homolytic O-O dissociation reaction pathways of C5 KHP. Reaction class 2 represents the Korcek reaction pathways.

Figure 7 .

 7 Figure 7. Measured (symbols) and model-predicted (lines) mole fraction profiles of C5H8O and C5H8O2. Short dot, dash dot and dash lines represent the contributions of the oxidation/decomposition of C5 KHP, C5 cyclic ether, and pentene to C5H8O and C5H8O2, respectively. Solid lines represent their total contributions to C5H8O and C5H8O2 mole fractions. The error bar for the experimental data is given in the figure.

2 - 7 . 4

 274 Scheme 4 Reaction pathways of 2-methyltetrahydrofuran to form olefinic cyclic ethers, C5 bicyclic

4. 4

 4 Hydroperoxides characterized by LC-HRMSThe LC-HRMS analysis characterized numerous elusive oxygenated intermediate species. In this work, in addition to the aforementioned C5H10O1-4, C5 ROOH, C5 KHP, C2H5COCH3 and CH3COOH, several hydroperoxides including C2-C5 unsaturated ROOHs, C3-C4 KHPs and three C5 HOMs (diketohydroperoxides, unsaturated dihydroperoxides and dihydroperoxides) were also well characterized by the LC-HRMS analytical method. Hydroperoxides are ubiquitous during the atmospheric oxidation of volatile organic compounds and are responsible for the two-stage ignition processes of various fuels. The elucidation of the distribution of hydroperoxide species is helpful for understanding autoignition chemistry.

Figure 8

 8 Figure 8 presents the LC-HRMS signal profiles of these nine hydroperoxide species and the

Figure 8 .

 8 Figure 8. Model-predicted mole fraction distributions and the LC-HRMS signal profiles of C2-C5 unsaturated ROOHs, C3-C4 KHPs and three C5 HOMs (diketohydroperoxides, unsaturated dihydroperoxides and dihydroperoxides) during low-temperature oxidation of n-pentane. Since LC-HRMS analysis are only qualitative, the signal profiles of C2-C5 unsaturated ROOHs and C3-C4 KHPs are scaled to the updated model for comparison. The temperature-dependent signal profiles of the three HOMs are also given.

  These intermediates are used to elucidate the reaction network and update the reaction mechanisms of n-pentane oxidation. In the model development, the rate constants of C5 KHP decomposition, C5 HPCE decomposition, and Korcek reactions of C5 KHP are updated. The pressure-dependent rate constants for the reaction classes of Q ̇OOH + O2, Q ̇OOH decompositions, concerted HȮ2-elimination of RȮ2, C5 KHP decomposition, C5 HPCE decomposition, and Korcek reactions of C5 KHP are tentatively included. Additionally, more detailed sub-mechanisms for C5 cyclic ethers and C5 KHP are added. The updated model is validated against measurements in this work and experimental data available from the literature, including JSR species data and IDT data from both RCMs and STs. These exploratory updates of the n-pentane combustion model mainly highlight the governing influence of the rate constants of hydroperoxide decomposition and the influence of the pressuredependent rate constants of the aforementioned reaction classes on the model predictions. The results in this work indicate that further high-level quantum chemistry calculations for the pressure-dependent kinetics of these reaction classes are necessary to deeply understand the combustion chemistry of hydrocarbon fuels. The introduction of the pressure dependence of important reaction classes and more detailed sub-mechanisms of important combustion intermediates is essential to reduce mechanism uncertainties and to develop accurate chemical kinetic models.

Table 1 .

 1 The JSR experimental conditions used in this work.

	Location	T (K)	p (bar)	 (s)		n-pentane
	NSRL	500-825	1.03	0.75 0.25	0.0075
	CNRS	520-760	1.03	0.75 0.25	0.0075

Acknowledgements

This work was supported by National Natural Science Foundation of China (51976208) and the Hefei Science Center, CAS (2020HSC-KPRD001, 2021HSC-UE005). Support from the CAPRYSSES project (ANR-11-LABX-006-01) funded by ANR through the PIA (Programme d'Investissement d'Avenir) is gratefully acknowledged.

inconsistent with the SVUV-PIMS measurements of C5H8O and C5H8O2.

Other reaction pathways can also contribute C5H8O and C5H8O2, e.g., the oxidation chemistry of C5 cyclic ethers and pentenes, which will be discussed in the next section. Furthermore, 2,4pentanedione is separated from the dione isomers and quantified by the GC method, but its mole fraction quantified by GC is higher than the total mole fraction of C5H8O2 estimated using SVUV-PIMS data, as shown in Fig. 7b. Interestingly, the mole fraction profile of 2,4-pentanedione is different from the C5H8O2 mole fraction profile measured by SVUV-PIMS. Instead, its temperature distribution is the same as that of C5 KHP (ca. 550-650 K), which indicates that 2,4-pentanedione is mainly derived from C5 KHP. A similar observation is observed in n-heptane low-temperature oxidation, where the authors stated that the diones may come from KHP decomposition in the transfer line from the JSR to the GC analysis [START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF]. Scheme 3 Reaction pathways of 4-hydroperoxy-2-pentanone leading to diones, keto cyclic ethers, and olefinic ketones via H-atom abstraction reactions.


Anthony Gauvan

Perron's capacity of random sets

come

Perron's capacity of random sets

Introduction

Given any set of directions Ω ⊂ S 1 , one can define the directional family of rectangle R Ω as R Ω := {R ∈ R : ω R ∈ Ω} and then consider the directional maximal operator M Ω defined for f : R 2 → R locally integrable and x ∈ R 2 as

M Ω f (x) := sup x∈R∈RΩ 1 |R| R |f | .
The boundedness property of the operator M Ω is deeply related to the geometric structure of the set of directions Ω. For example, in the case where Ω = S 1 , the following obstruction of the Euclidean plane (which is also true in higher dimension) allows us to completely describe the boundedness property of the operator M S 1 .

Theorem 1 (Kakeya blow with R). Given any large constant A 1, there exists a finite family of rectangles {R i : i ∈ I} ⊂ R such that we have

i∈I T R i ≥ A i∈I R i .
Here, we have denoted by T R the rectangle R translated along its longest side by its own length

The reader can find a proof of this Theorem in [START_REF] Fefferman | The multiplier problem for the ball[END_REF]: it follows that given any large constant A 1, there exists a bounded set E satisfying the following estimate

M S 1 1 E ≥ 1 2 ≥ A |E| .
It suffices to set E = ∪ i∈I R i and to observe that we have the following inclusion

i∈I T R i ⊂ M S 1 1 E ≥ 1 2 .
The previous estimate easily implies the following.

Theorem 2. The operator M S 1 is not bounded on L p for any p < ∞.

Far from being exotic, Theorem 1 has deep implications in harmonic analysis: for example, it is a central part of Fefferman's work in [START_REF] Fefferman | The multiplier problem for the ball[END_REF] where he disproves the famous Ball multiplier conjecture. A natural question is the following: given a set of directions Ω, is it possible to make a Kakeya blow only with the directional family R Ω ? This question have been investigated by different analyst among which [START_REF] Moonens | Differentiating Orlicz spaces with rect-angles having fixed shapes in a set of directions[END_REF], [START_REF] Gauvan | Application of Perron trees to geometric maximal operators[END_REF], [START_REF] Katz | A counterexample for maximal operators over a Cantor set of directions[END_REF], [START_REF] Nagel | Differentiation in lacunary directions[END_REF] or [START_REF] Hare | Applications of generalized Perron trees to maximal functions and density bases[END_REF] to cite a few. In [START_REF] Bateman | Kakeya sets and directional maximal operators in the plane[END_REF], Bateman answered this question as he classified the L p (R 2 ) behavior of those operators according to the geometry of the set Ω. Precisely, he proved that the notion of finitely lacunary for a set of directions were the correct one to consider. Theorem 3 (Bateman). We have the following alternative:

• if Ω is finitely lacunary then M Ω is bounded on L p for any p > 1.
• if Ω is not finitely lacunary then it is possible to make a Kakeya blow with the family R Ω . In particular, the operator M Ω is not bounded on L p for any p < ∞.

Let us define the notion of finite lacunarity following a nice presentation made by Kroc and Pramanik [START_REF] Kroc | Lacunarity, Kakeya-type sets and directional maximal operators[END_REF]: we start by defining the notion of lacunary sequence and then the notion of lacunary set of finite order. We say that a sequence of real numbers L = { k : k ≥ 1} is a lacunary sequence converging to ∈ R when there exists 0 < λ < 1 such that

| -k+1 | ≤ λ| -k |
for any k. For example the sequences 1 2 k : k ≥ 2 and 1 k! : k ≥ 4 are lacunary. We define now by induction the notion of lacunary set of finite order. Definition 4 (Lacunary set of finite order). A lacunary set of order 0 in R is a set which is either empty or a singleton. Recursively, for N ∈ N * , we say that a set Ω included in R is a lacunary set of order at most N + 1 -and write Ω ∈ Λ(N + 1) -wether there exists a lacunary sequence L with the following properties : for any a, b ∈ L such that a < b and (a, b) ∩ L = ∅, the set Ω ∩ (a, b) is a lacunary set of order at most N i.e. Ω ∩ (a, b) ∈ Λ(N ).

For example the set

Ω := π 2 k + π 4 l : k, l ∈ N, l ≤ k
is a lacunary set of order 2. In this case, observe that the set Ω cannot be re-written as a monotone sequence, since it has several points of accumulation.

We can finally give a definition of a finitely lacunary set.

Definition 5 (Finitely lacunary set). A set Ω in [0, π) is said to be finitely lacunary if there exists a finite number of set Ω 1 , . . . , Ω M which are lacunary of finite order such that

Ω ⊂ k≤M Ω k .
2 Can we apply Bateman's Theorem ?

A classic example of set which is known to be not finitely lacunary is the set

Ω lin = π n : n ∈ N * .
Indeed, the classic construction of Perron trees shows that it is possible to make a Kakeya blow with the family R Ω lin : hence an application of Bateman's Theorem implies that Ω lin is not finitely lacunary. The second classic example of set which is known to be finitely lacunary is the set

Ω lac = π 2 n : n ∈ N * .
One can see that the set Ω lac is finitely lacunary by definition and it was in [START_REF] Nagel | Differentiation in lacunary directions[END_REF] that Nagel, Stein and Waigner proved that the maximal operator M Ω lac is bounded on L p (R 2 ) for any 1 < p < ∞ ; their proof relies on Fourier analysis. Also, let us say that the comprehension of the sets Ω lin and Ω lac is important because they are the most simple (and smallest) cases of infinite sets which yield maximal operator having different boundedness properties. However, even if Bateman's Theorem is extremely satisfying, it appears to be difficult to decide if a given set Ω is finitely lacunary or not. The most striking example was raised by Stokolos : at the present time, it is not known if the set

Ω sin,lac := π sin(n) 2 n : n ∈ N *
is finitely lacunary or not. The main problem of this set of directions is that we have a very poor control on the deterministic sequence {sin(n) : n ≥ 1} and that initially, the set Ω lac is finitely lacunary: hence, the perturbations are quite difficult to handle. In [START_REF] D'aniello | Un)boundedness of directional maximal operators through a notion of "Perron capacity" and an application[END_REF], with D'Aniello and Moonens, we were able to show that the following set

Ω sin,lin := π sin(n) n : n ∈ N *
is not finitely lacunary (this was also a set considered by Stokolos). More precisely, we studied the maximal operator M Ω sin,lin associated and, improving on concrete techniques, we proved that this operator is not bounded on L p (R 2 ) for any 1 < p < ∞: the heart of the method relied on the introduction of the Perron's capacity of a set of directions. We need some notations to recall our results: given an infinite set of directions Ω ⊂ S 1 whose only point of accumulation is 0 and we denote, for notational convenience, by Ω -1 the set π u : u ∈ Ω , that is Ω -1 := π Ω and we order Ω -1 asastrictlyincreasingsequence{u k : k ∈ N * }. With those notations, we define the Perron's factor of Ω as

G(Ω) := sup k≥1 l≤k u k+2l -u k+l u k+l -u k + u k+l -u k u k+2l -u k+l .
In [START_REF] Hare | Applications of generalized Perron trees to maximal functions and density bases[END_REF], Hare and Ronning proved the following Theorem.

Theorem 6 (Hare and Ronning). If we have G(Ω) < ∞ then it is possible to make a Kakeya blow with the family R Ω .

It turns out that it is difficult to compute the Perron factor of the set

Ω sin,lin = π sin(n) n : n ≥ 1
since the oscillation of the cosinus prevent us to obtain a good description of the increasing sequence {u k : k ∈ N * } associated to Ω sin,lin . Based on a careful read of the proof of Theorem 6, for an arbitrary set of directions Ω included in S 1 , we define its Perron's capacity as

P (Ω) := lim inf N →∞ inf U ⊂Ω -1 #U =2 N G(Ω) ∈ [2, ∞]
where as before

G(Ω) = sup k,l≥1 k+2l≤2 N u k+2l -u k+l u k+l -u k + u k+l -u k u k+2l -u k+l if U = {u 1 < • • • < u 2 N }.
In [START_REF] D'aniello | Un)boundedness of directional maximal operators through a notion of "Perron capacity" and an application[END_REF], we proved the following (in contrast with Hare and Ronning Theorem, we do not assume that the set Ω is ordered): Theorem 7 (D'Aniello, G. and Moonens). For any set of directions Ω, if we have P (Ω) < ∞ then it is possible to make a Kakeya blow with the family R Ω . In particular, for any p < ∞, one has M Ω p = ∞.

Loosely speaking, if P (Ω) < ∞ then the set Ω contains arbitrary large set which are (more or less) uniformly distributed and this geometric pattern prevents the set Ω to be finitely lacunary. The advantage of Theorem 7 is that it allows us to make effective computation. However, as mentioned earlier, the following case is still unsettled. Question 1. Is the following set of direction

Ω sin,lac := π sin(n) 2 n : n ∈ N *
finitely lacunary or not ?

Results

Our result concerns random sets of directions which are meant to give a generic comprehension of the two classic examples Ω lin and Ω lac when they are randomly perturbated. Precisely we consider the following random sets of slopes

Ω rand,lin := πX k k : k ≥ 1
and

Ω rand,lac := πX k 2 k : k ≥ 1
where {X k : k ≥ 1} are random variables uniformly distributed in (0, 1) and independent. To begin with, we prove the following Theorem.

Theorem 8. The Perron's capacities of Ω rand,lin is finite almost surely i.e. we have almoste surely

P (Ω rand,lin ) < ∞.
In some sense, Theorem 8 means that if a set Ω presents structured patterns -like large uniformly distributed sequence -then a small perturbation of Ω will still exhibit those patterns. The second result reads as follow.

Theorem 9. The Perron's capacities of Ω rand,lac is finite almost surely i.e. we have almoste surely

P (Ω rand,lac ) < ∞.
The proof of Theorems 8 and 9 relies on the possibility to compute effectively the Perron's capacity of the random sets Ω rand,lin and Ω rand,lac .

Proof of Theorem 8

We wish to prove that the Perron's capacity of Ω rand,lin is finite almost surely. We are simply going to prove that the set Ω -1 rand,lin contains small perturbation of arbitrarily long homogeneous sets. We say that a set H of the form

H := H a,N = {ka : 1 ≤ k ≤ 2 N }
for some integer a ∈ N * is an homogeneous set. The following claim is easy.

Claim 1. For any a, N ∈ N, one has G(H a,N ) = 2.
We wish to perturb a little an homogeneous set H into a set H such that the Perron's factor of H is still controlled. Precisely, fix any a, N ∈ N * and let be an arbitrary function

: H a,N → (0, ∞).
Define then the set H a,N ( ) as

H := H a,N ( ) := {(1 + (l))l : l ∈ H a,N } .
If the perturbation is small enough compared to the integer N , one can control uniformly G(H ).

Proposition 1. With the previous notations, if we have

2 N ∞ ≤ 1 2
then we have G(H a,N ( )) < 6.

We are now ready to prove Theorem 8. We fix a large integer N ∈ N and consider the following set of indices

E N := k ∈ N : |X k -1| ≤ 2 -N .
In other words, an integer k belongs to E N when X k is close to 1 with precision 2 -N . We claim that this random set E N contains almost surely large (with at least 2 N points) homogeneous sequences.

Claim 2. For any N ≥ 1, the set E N contains an homogeneous set of cardinal 2 N almost surely.

Proof. Observe that for any a ∈ N * , the following probability P(H a,N ⊂ E N ) is independent of a: indeed since the random variables {X k : k ≥ 1} are independent and uniformly distributed, we have

P(H a,N ⊂ E N ) = k∈H a,N P |X k -1| ≤ 2 -N = 1 2 N 2 N . H a i ,N H a j ,N a j a i Figure 1: We choose the sequence {a i : i ≥ 1} such that H ai,N ∩ H aj ,N = ∅ for any i = j.
Hence we fix a sequence {a i } i≥1 satisfying for any i = j H ai,N ∩ H aj ,N = ∅.

For example, setting a i = 2 2N (i+1) works since we have a i 2 N < a i+1 for any i ≥ 1. In particular this means that the events

{(H ai,N ⊂ E N ) : i ≥ 1}
are independent and since we have We can now conclude the proof : we define a perturbation for any n ≥ 1 as 1 + (n) = X -1 n . We fix a large integer N 1 and we know that almost surely there exists an integer a ∈ N * such that H a,N ⊂ E N . Observe now that by definition one has the following inclusion H a,N ( ) ⊂ Ω -1 rand,lin . However since H a,N ⊂ E N and that it is not difficult to see that we have

|H a,N ∞ 2 -N . Indeed, for k ∈ H a,N ⊂ E N , we have |1 + (k)| = 1 1 + (X k -1) 1 + 2 |X k -1| 1 + 2 -N .
It follows that we have P (Ω rand,lin ) < 6 almost surely applying Proposition 1.

Proof of Theorem 9

We wish to prove that the Perron's capacity of Ω rand,lac is finite almost surely.

Observe that if U is a set in R who is well distributed then one can control its Perron's capacity.

Claim 3. If we have δ > 0 and a set

U = {u 1 < • • • < u 2 N } such that for any 1 ≤ i ≤ 2 N -1 we have δ ≤ u i+1 -u i ≤ 3δ
then one has G(U ) 1.

Proof. For any i, j such that i + j ≤ 2 N -1, one has u i+j -u i jδ. Hence i, j such that i + 2j ≤ 2 N -1 , we have

u i+2j -u i+j u i+j -u i + u i+j -u i u i+2j -u i+j 2jδ jδ + jδ 2jδ 1.
Hence we obtain G(U ) 1 as claimed

We are going to prove the following proposition.

Proposition 2. For any N ≥ 1, there exists almost surely a scale δ > 0 and a set U ⊂ Ω -1 rand,lac such that

U := {u 1 < • • • < u 2 N -1 }
and for any i ≤ 2 N -1 one has

δ < u i+1 -u i < 3δ.
Theorem 9 is a consequence of Claim 3 and Proposition 2: for any N , we can exhibit almost surely a set U ⊂ Ω -1 rand,lac of cardinal 2 N such that G(U ) 1 and so we obtain P (Ω rand,lac ) < ∞ as expected. The rest of the section is devoted to the proof of Proposition 2.

Proof of Proposition 2

We consider the following dyadic intervals for d ∈ N

I d := 2 d , 2 d+1 .
We wish to obtain information on the distribution of the points of the set Ω rand,lac that may be in the interval I d . We fix a large integer N 1 and we divide each dyadic interval I d into 2 N intervals of same length i.e. for any 1 ≤ l ≤ 2 N we set

I d,l = 2 d 1 + l -1 2 N , 2 d 1 + l 2 N .
We claim that the union of those events

B N := d≥1 A d,N
occurs almost surely.

i≥1P(

  H ai,N ⊂ E N ) = ∞an application of Borel-Cantelli lemma yields the conclusion.

Claim 5 . 1 .Figure 3 :

 513 Figure 3: The random set Ω rand,lac := Ω r2 contains almost surely uniformly distributed subset of arbitrarily large cardinal.We can now prove Proposition 2 since we have

is independent of d.

Proof. One has 2 d-l X d-l ∈ I d,l if and only if

Since the random variable X d-l is uniformly in (0, 1), the probability

We fix an extraction {d s : s ≥ 1} satisfying the following

for any s ≥ 1 ; this growth condition will assure that the events we will consider are independent and we will be able to apply Borel-Cantelli lemma when needed. Thanks to Claim 4 and independence, one can see that for any s ≥ 1, the following probability is independent of s

Now for any d ≥ 1, we consider the following event A d,N defined as

In other words, the event A d,N occurs when the random set Ω -1 rand,lac fills each sub-intervals I d,l : 1 ≤ l ≤ 2 N with at least one point. In particular, observe that we have