Shiqing Kang

Wanxiong Liao

Wenyu Sun

Keli Lin

Handong Liao

Kai Moshammer

Philippe Dagaut

Bin Yang
email: byang@tsinghua.edu.cn

Exploring low-temperature oxidation chemistry of 2-and 3-pentanone

Keywords: 2-pentanone, 3-pentanone, Jet-stirred reactor, Rapid compression machine, Photoionization mass spectrometry, Ignition delay time

The low-temperature oxidation chemistry of 2-and 3-pentanone was investigated over a wide range of conditions using a jet-stirred reactor (JSR) and a rapid compression machine (RCM). The JSR oxidation experiment was performed at the pressure of 93.3 kPa over a temperature range of 600-1000 K. Detailed speciation information was obtained using synchrotron vacuum ultraviolet photoionization mass spectrometry. Ignition delay times (IDTs) of 2-and 3-pentanone were measured in an RCM from 640 to 820 K at pressures of 15 and 25 bar and an equivalence ratio of 1.0.

The two C5 ketones showed NTC behavior and two-stage ignition phenomena. Mole fraction time histories of intermediate species during the two-stage ignition process of both ketones were obtained using a fast-sampling system coupled with gas chromatography. There are distinct differences between 2-and 3-pentanone in species concentration profiles and IDTs. A kinetic mechanism for the low-temperature oxidation of 2-and 3-pentanone was developed, which can satisfactorily predict all

Introduction

The growing scarcity of fossil hydrocarbons and environmental concerns, i.e., greenhouse gas emissions and air pollution, have stimulated the need to produce and consume sustainable and renewable biofuels. Ketones are one of the potential transportation biofuels that can be produced through fungal conversion of lignocellulosic biomass [START_REF] Li | Carbon-increasing catalytic strategies for upgrading biomass into energyintensive fuels and chemicals[END_REF][START_REF] Shen | Production of liquid 37 hydrocarbon fuels with 3-pentanone and platform molecules derived from lignocellulose[END_REF][START_REF] Singh | An endophytic Phomopsis sp. possessing bioactivity and fuel potential with its volatile organic compounds[END_REF] or can be synthesized using acetone-n-butanol-ethanol (ABE) fermentation [START_REF] Anbarasan | Integration of chemical catalysis with extractive fermentation to produce fuels[END_REF][START_REF] Zhang | Species diagnostics and modeling study of laminar premixed flames fueled by acetone-butanol-ethanol (ABE)[END_REF][START_REF] Xue | Bridging chemical-and bio-catalysis: high-value liquid transportation fuel production from renewable agricultural residues[END_REF]. The higher energy density [START_REF] Mccormick | Selection criteria and screening of potential biomass-derived streams as fuel blendstocks for advanced sparkignition engines[END_REF], the better anti-knock properties [START_REF] Hoppe | Tailor-Made Fuels from Biomass: Potentials of 2-butanone and 2-methylfuran in direct injection spark ignition engines[END_REF], and lower soot emissions [START_REF] Barak | Measuring the effectiveness of high-performance Co-Optima biofuels on suppressing soot formation at high temperature[END_REF][START_REF] Kwon | Sooting tendencies of 20 bio-derived fuels for advanced spark-ignition engines[END_REF] of ketones compared to alkanes facilitate their applications as transportation fuels or fuel additives. In addition, ketones are key intermediate species of low-temperature oxidation of alkanes [START_REF] Zador | Kinetics of elementary reactions in lowtemperature autoignition chemistry[END_REF][START_REF] Battin-Leclerc | Detailed chemical kinetic models for the low-temperature combustion of hydrocarbons with application to gasoline and diesel fuel surrogates[END_REF], and therefore, combustion kinetics studies of ketones can help to further understand the complex low-temperature oxidation reaction pathways. Pentanones (C5H10O) are the smallest ketones that exist in two different linear isomers, i.e., 2-pentanone and 3pentanone. Here, we present a comparative study presented of the combustion kinetics of these two ketones. Our results provide insights into the impact of the position of the carbonyl functional group in the fuel molecule on ignition delay times and intermediate identities and concentrations.

Previous combustion kinetics studies on 2-and 3-pentanone were focused on experimental work. Ignition delay times (IDTs) of 2-and 3-pentanone were mainly measured at high temperatures (> 1000 K) in shock tubes [START_REF] Serinyel | Experimental and chemical kinetic modeling study of 3-pentanone oxidation[END_REF][START_REF] Lam | Shock tube measurements of 3-pentanone pyrolysis and oxidation[END_REF][START_REF] Fikri | High-pressure shock-tube investigation of the impact of 3-pentanone on the ignition properties of primary reference fuels[END_REF][START_REF] Minwegen | An experimental and theoretical comparison of C3-C5 linear ketones[END_REF]. Serinyel et al. [START_REF] Serinyel | Experimental and chemical kinetic modeling study of 3-pentanone oxidation[END_REF] measured the laminar burning velocities (LBVs) of 3-pentanone in a combustion vessel.

Li et al. [START_REF] Li | Insight into fuel isomeric effects on laminar flame propagation of pentanones[END_REF] expanded LBV measurements of 2-and 3-pentanone to different initial temperature and pressure experimental conditions. To further explore the detailed combustion kinetics of 2-and 3-pentanone, several speciation experiments in shock tubes [START_REF] Lam | Shock tube measurements of 3-pentanone pyrolysis and oxidation[END_REF][START_REF] Lam | High-temperature measurements of the reactions of OH with a series of ketones: acetone, 2-butanone, 3-pentanone, and 2-pentanone[END_REF][START_REF] Dames | An improved kinetic mechanism for 3-pentanone pyrolysis and oxidation developed using multispecies time histories in shock-tubes[END_REF], laminar premixed flames [START_REF] Pieper | A high-temperature study of 2-pentanone oxidation: experiment and kinetic modeling[END_REF][START_REF] Sun | Probing fuel-specific reaction intermediates from laminar premixed flames fueled by two C5 ketones and model interpretations[END_REF], and flow tubes [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF] were performed, but also focused on high-temperature ranges. Kang et al. [START_REF] Kang | Isomer-specific influences on ignition and intermediates of two C5 ketones in an RCM[END_REF] discussed the influence of resonantly stabilized fuel radicals on the species pools based on the quantitative speciation information of 2-and 3-pentanone pyrolysis and ignition processes in a rapid compression machine (RCM). Experimental studies of these two C5 ketones at low temperatures are relatively limited. Scheer et al. [START_REF] Scheer | Low-temperature combustion chemistry of novel biofuels: resonance-stabilized QOOH in the oxidation of diethyl ketone[END_REF][START_REF] Scheer | Resonance stabilization effects on ketone autoxidation: isomer-specific cyclic ether and ketohydroperoxide formation in the low-temperature (400-625 K) oxidation of diethyl ketone[END_REF] studied the Cl initiated oxidation reactions of 3-pentanone at 8 Torr and temperatures between 550 K and 650 K. The potential reaction pathways of cyclic ether and keto-hydroperoxide formation were discussed based on the mass spectrometric measurements but quantitative measurements were not obtained. Fenard et al. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF] measured the IDTs of 2-and 3-pentanone in an RCM at pressures between 20 and 40 bar and temperature ranges between 650 K and 950 K. IDTs of 2-and 3-pentanone showed negative temperature coefficient (NTC) behavior and the low-temperature reactivity of both ketones were confirmed.

Due to the limited amount of experimental studies on the low-temperature oxidation, previous kinetic mechanisms of 2-and 3-pentanone oxidation were also based on and validated in high-temperature experiments. Serinyel et al. [START_REF] Serinyel | Experimental and chemical kinetic modeling study of 3-pentanone oxidation[END_REF] developed the first 3-pentanone kinetic model through analogies to alkanes and 2-butanone reactions. This model was further modified by Dames et al. [START_REF] Dames | An improved kinetic mechanism for 3-pentanone pyrolysis and oxidation developed using multispecies time histories in shock-tubes[END_REF] based on the species histories measured in a shock tube using laser absorption diagnostics. Sun et al. [START_REF] Sun | Probing fuel-specific reaction intermediates from laminar premixed flames fueled by two C5 ketones and model interpretations[END_REF] calculated the reaction rates of isomerization and decompositions of 3-pentanone and proposed an updated mechanism which could satisfactorily predict their flame speciation measurements. Pieper et al. [START_REF] Pieper | A high-temperature study of 2-pentanone oxidation: experiment and kinetic modeling[END_REF] proposed a high-temperature submechanism for 2-pentanone by analogy to 2-butanone and n-pentane. Based on previous theoretical calculations [START_REF] Cheng | Kinetics study of H-atom abstraction from 3-pentanone by Ḣ and ĊH3 radicals and the subsequent isomerization reactions[END_REF] and kinetic modeling studies [START_REF] Pieper | A high-temperature study of 2-pentanone oxidation: experiment and kinetic modeling[END_REF][START_REF] Sun | Probing fuel-specific reaction intermediates from laminar premixed flames fueled by two C5 ketones and model interpretations[END_REF], Li et al. [START_REF] Li | Insight into fuel isomeric effects on laminar flame propagation of pentanones[END_REF] and Kang et al. [START_REF] Kang | Isomer-specific influences on ignition and intermediates of two C5 ketones in an RCM[END_REF] separately developed and validated a comprehensive hightemperature kinetic model for 2-and 3-pentanone, respectively. Compared to the hightemperature kinetics modeling studies, the low-temperature kinetics modeling studies of 2-and 3-pentanone are extremely insufficient due to the limited experimental data.

Fenard et al. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF] developed a low-temperature kinetics model to reproduce their measured IDTs showing NTC behavior of 2-and 3-pentanone. Based on the models of Pieper [START_REF] Pieper | A high-temperature study of 2-pentanone oxidation: experiment and kinetic modeling[END_REF] and Fenard [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF], an improved 2-pentanone kinetic model applicable from low to high-temperature was proposed using the Bayesian optimization algorithm [START_REF] Lin | An improved 2-pentanone low to high-temperature kinetic model using Bayesian Optimization algorithm[END_REF].

The fates of alkylperoxy (ROO•), hydroperoxyalkyl (•QOOH), and hydroperoxyalkylperoxy (•OOQOOH) radicals were determined by using rate constants of analogous reactions from 2-butanone. Kuzhanthaivelan et al. [START_REF] Kuzhanthaivelan | Computational investigations on the thermochemistry and kinetics for the autoignition of 2-pentanone[END_REF] calculated the reaction rates of intramolecular hydrogen migration reactions and decomposition of •OOQOOH radicals in 2-pentanone low-temperature oxidation process, but pressure-dependent reaction rate parameters were not given.

The high-temperature kinetics of 2-and 3-pentanones have been thoroughly studied, while discussion on the low-temperature kinetics of the isomers is scarce. Therefore, in this work, we focused on the reaction kinetics of the pentanone isomers over low-to-intermediate temperatures. The goal is to show that our general understanding of the low-temperature oxidation chemistry is sufficient to assemble a mechanism that explains the observed differences in the two isomer experiments. To this end, JSR oxidation and RCM ignition experiments have been carried out: species information in a JSR was reported at 93.3 kPa over a temperature range of 600-1000 K, with isomers resolved and quantitative profiles obtained using photoionization molecular-beam mass spectrometry (PIMBMS). IDTs and species profiles of both ketones showing NTC behavior were measured in the RCM under the condition of 15-25 bar, 640-820 K. A low-temperature kinetic mechanism for 2-and 3-pentanone was developed and validated. Experimental measurements together with modeling analysis were then used to illuminate the low-temperature oxidation of 2-and 3-pentanone and to explain isomer specific differences based on insights into the influence of the position of the carbonyl functional group in the two ketones.

Experimental setup

Jet-stirred reactor

The JSR experiments to study the low-temperature oxidation of 2-and 3pentanone were performed at the Advanced Light Source (ALS) of the Lawrence Berkeley National Laboratory (LBNL). The spherical fused-silica JSR was coupled with a time-of-flight molecular-beam mass spectrometer (TOF-MBMS), using the tunable synchrotron vacuum ultraviolet (SVUV) radiation as the photoionization (PI) source. Only a brief description of each component is given here as the details of the experimental setup are described elsewhere [START_REF] Moshammer | Detection and identification of the ketohydroperoxide (HOOCH2OCHO) and other intermediates during lowtemperature oxidation of dimethyl ether[END_REF][START_REF] Moshammer | Quantification of the keto-hydroperoxide (HOOCH2OCHO) and other elusive intermediates during lowtemperature oxidation of dimethyl ether[END_REF]. A JSR with a volume of 33.5 cm 3 was housed in an electrically heated oven. The reaction gases were sampled by a quartz nozzle with a 50 μm diameter orifice at the tip. The pressure difference between the reactor (93.3 kPa) and the first differential stage (<10 -4 kPa) can lead to the formation of a molecular beam, in which the chemical reactions were quenched immediately [START_REF] Qi | Combustion chemistry probed by synchrotron VUV photoionization mass spectrometry[END_REF].

The molecular beam entered the ionization area (<10 -6 kPa) via a skimmer and the molecular components were subsequently ionized and analyzed by a mass spectrometer with a mass resolution (m/Δm) of about 4000, allowing for distinguishing oxygencontaining species from pure hydrocarbons.

The experiments were performed at a constant pressure of 93.3 kPa (700 Torr), with a nominal residence time of 2000 ms. The argon-diluted experimental mixtures contained 1.0% fuel and 14% O2, resulting in an equivalence ratio of 0.5. The gas flows of oxygen and argon were controlled by calibrated mass flow controllers (MKS). The liquid fuels were supplied by a syringe pump (New Era) and quantitatively vaporized by the heating transfer line maintained at 120 °C before entering the JSR. The species concentration was measured at 16 different temperature points varying from 600 K to 1000 K. The data were evaluated following the same procedures as in previous works [START_REF] Moshammer | Quantification of the keto-hydroperoxide (HOOCH2OCHO) and other elusive intermediates during lowtemperature oxidation of dimethyl ether[END_REF][START_REF] Liao | Influence of ozone addition on the low-temperature oxidation of dimethyl ether in a jet-stirred reactor[END_REF][START_REF] Tao | A further experimental and modeling study of acetaldehyde combustion kinetics[END_REF] and the concentration uncertainties of major species were evaluated to be within 20%, while the uncertainties of the intermediate species can range from 30% to a factor of 3, which mainly come from the uncertainties of photoionization crosssections (PICSs) for each species. In addition, measurements of photoionization efficiency (PIE) spectra were performed from 8.8 eV to 11.5 eV at 900 K for species identification. A K-type thermocouple (Thermocoax) was placed near the tip of the probe at the exit of the JSR. Due to the influence of the cold probe, the measured temperature was typically lower than the actual temperature in the JSR. A factor of 1.1 was used for temperature correction, by following previous studies using the same experimental setup [START_REF] Moshammer | Quantification of the keto-hydroperoxide (HOOCH2OCHO) and other elusive intermediates during lowtemperature oxidation of dimethyl ether[END_REF][START_REF] Liao | Influence of ozone addition on the low-temperature oxidation of dimethyl ether in a jet-stirred reactor[END_REF][START_REF] Tao | Exploring the negative temperature coefficient behavior of acetaldehyde based on detailed intermediate measurements in a jetstirred reactor[END_REF].

Rapid compression machine

Measurements of IDTs and quantification of major intermediates concentrations were conducted on the Tsinghua University RCM (THU-RCM). A detailed description of the THU-RCM can be found in [START_REF] Di | Effects of buffer gas compositon on low temperature ignition of isooctane and n-heptane[END_REF]. Briefly, the configuration of pneumatically driven, hydraulic deceleration was adopted. A creviced piston was used to suppress the vortex generated during the compression. Experimental measurements [START_REF] Mittal | Aerodynamics inside a rapid compression machine[END_REF] and computational fluid dynamics (CFD) simulation studies [START_REF] Wurmel | CFD studies of a twin-piston rapid compression machine[END_REF] showed that the "adiabatic core hypothesis" is valid in the core region of an RCM with a carefully designed creviced piston. The mixture in the combustion chamber is compressed by the creviced piston to high pressure within 25-30 ms. The t50 time, which is the time from 50% of the peak pressure to the end of compression, is about 3 ms. The compression stroke is fixed at 500 mm, and the compression ratio is adjusted by changing the length of the combustion chamber. A piezoelectric pressure transducer (Kistler 6125C) coupled with a charge amplifier (Kistler 5018A) was used for the pressure measurements and the temperature at the end of compression is calculated based on the pressure history according to the isentropic adiabatic relationship.

A fast-sampling system was applied to realize online species sampling and quantification [START_REF] Ji | Intermediate species measurement during iso-butanol auto-ignition[END_REF][START_REF] Zhang | Measurement of reaction rate constants using RCM: A case study of decomposition of dimethyl carbonate to dimethyl ether[END_REF]. A capillary tube connected with a fast-acting solenoid valve (Parker Series 9) was used for sampling. A controller (IOTA ONE) can precisely adjust the opening time and duration of the solenoid valve so that gas mixtures at different times can be sampled. A pressure sensor (Kistler 4045A) mounted on the sampling tank can record the equilibrium pressure after samples from the combustion chamber expand into the sampling tank. The samples were then sent to a GC (Agilent 7890B) for qualitative and quantitative analysis. Only one sampling procedure was operated during a single shot. A series of experiments need to be conducted under the same condition to sample at different times, yielding a time-resolved species concentration profile. The uncertainty of concentration measurements is estimated to be about 30% which mainly comes from the dilution of the unreacted gases trapped in the dead volume of the sampling probe and the leak of the sampling system.

A typical pressure history of the two-stage ignition process in the RCM is shown in Fig. 1. Effective pressure (peff) and effective temperature (Teff) are used to represent the experimental conditions which are similarly determined as in our previous works [START_REF] Kang | Isomer-specific influences on ignition and intermediates of two C5 ketones in an RCM[END_REF][START_REF] Di | Effects of buffer gas compositon on low temperature ignition of isooctane and n-heptane[END_REF][START_REF] Wang | Hydrogen formation from methane rich combustion under high pressure and high temperature conditions[END_REF]. The total ignition delay time (total ID) is defined as the time interval between the end of compression (EOC) and the maximum pressure rise rate, while the first-stage ignition delay time (1 st ID) is the time interval between the EOC and the maximum pressure rise rate during the first heat release resulting from the lowtemperature oxidation. The uncertainty of the IDT measurement comes mainly from the uncertainties of the measured pressure, the difference between computed temperature (Teff) and the real temperature, and the potential reactions during the compression process. The uncertainty of pressure originates from the accuracy of the pressure transducer and charge amplifier, while the uncertainty of temperature is estimated to be 5% based on the experimental measurements [START_REF] Uddi | Temperature measurements in a rapid compression machine using mid-infrared H2O absorption spectroscopy near 7.6 μm[END_REF][START_REF] Nasir | Time-resolved temperature measurements in a rapid compression machine using quantum cascade laser absorption in the intrapulse mode[END_REF] and calculations [START_REF] Weber | On the uncertainty of temperature estimation in a rapid compression machine[END_REF]. The potential reactions that occurred during the compression process will only decrease the IDT, this partial uncertainty is calculated based on the Livengood-Wu integral [START_REF] Livengood | Correlation of autoignition phenomena in internal combustion engines and rapid compression machines[END_REF] from the start to the end of compression. The combined uncertainty of the IDTs is calculated based on the error propagation formula. All the experimental data and corresponding uncertainty are provided in the Supplementary Material.

Kinetic modeling and simulation method

The high-temperature mechanism of the current model was adopted from our previous work [START_REF] Kang | Isomer-specific influences on ignition and intermediates of two C5 ketones in an RCM[END_REF], which developed a high-temperature kinetic mechanism for 2-and 3-pentanone to explain the influence of the carbonyl functional group and resonantlystabilized fuel radicals on ignition delay time and species pools. In that model, H-atom abstractions by H, CH3, and HO2 radicals were updated from theoretical calculations by Cheng et al. [START_REF] Cheng | Kinetics study of H-atom abstraction from 3-pentanone by Ḣ and ĊH3 radicals and the subsequent isomerization reactions[END_REF] and Mendes et al. [START_REF] Mendes | Theoretical and kinetic study of the reactions of ketones with HO2 radicals. Part I: abstraction reaction channels[END_REF], respectively. The fuel radical isomerization and decomposition reactions of 2-pentanone were supplemented by analogies to 2butanone, while those for 3-pentanone were from the calculations of Sun et al. [START_REF] Sun | Probing fuel-specific reaction intermediates from laminar premixed flames fueled by two C5 ketones and model interpretations[END_REF].

The kinetic model can well reproduce relevant high-temperature measurements, such as ignition delay time [START_REF] Serinyel | Experimental and chemical kinetic modeling study of 3-pentanone oxidation[END_REF][START_REF] Lam | Shock tube measurements of 3-pentanone pyrolysis and oxidation[END_REF][START_REF] Minwegen | An experimental and theoretical comparison of C3-C5 linear ketones[END_REF], laminar burning velocities [START_REF] Li | Insight into fuel isomeric effects on laminar flame propagation of pentanones[END_REF], and laminar premixed flame structures [START_REF] Pieper | A high-temperature study of 2-pentanone oxidation: experiment and kinetic modeling[END_REF][START_REF] Sun | Probing fuel-specific reaction intermediates from laminar premixed flames fueled by two C5 ketones and model interpretations[END_REF] for both ketones, and the relevant validation results are provided in the Supplementary Material. As a comparison, the rate parameters of unimolecular decomposition and β-scission reactions in the high-temperature mechanism of Fenard et al. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF] were given in analogy to the relevant reactions of 2butanone and n-pentane from Thion et al. [START_REF] Thion | An experimental study in a jet-stirred reactor and a comprehensive kinetic mechanism for the oxidation of methyl ethyl ketone[END_REF] and Bugler et al. [START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF].

In this paper, we focus on the low-temperature oxidation kinetics of both ketones.

Compared to high-temperature kinetics, reaction pathways at low temperatures are more complicated. Fenard et al. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF] developed a detailed kinetic model containing the low-temperature oxidation mechanism to explain the NTC behavior of 2-and 3pentanone through analogy to 2-butanone and n-pentane. In this work, the reaction pathways and reaction rates were basically obtained by analogy to 2-butanone and npentane following the same strategy of Fenard et al. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF] (see Fig. 2) due to their similar molecular structure and bond dissociation energy (BDE) of the C-H bonds. On the basis of that, reaction rates of some key elementary reactions were updated according to the newly reported theoretical calculation from the literature and experimental results in this work, which will be discussed in detail in the following. The reaction classes at low temperatures considered in this mechanism can be divided into 10 classes, which are listed in Table S1 in the Supplementary Material. The nomenclature adopted in this mechanism for species in the C5 ketone sub-mechanism is briefly described here.

Species prefixed with C5KET2 and C5KET3 are derived species of 2-pentanone and 3pentanone, respectively. The important structural information, including the positions of radicals, double bonds, and other functional groups is stated behind the hyphen. For example, R stands for a radical position, D for a double bond, KET for a carbonyl functional group, and ALD for an aldehyde functional group. The numbers following the abbreviation provide the position of the functional groups in the carbon chain. For example, the species C5KET2-OOH3-R1 (see detailed structure in Fig. 3) denotes a derived molecule of 2-pentanone with a hydroperoxyl group on the third carbon and a radical site on the first carbon. The detailed structures of the species in the C5 ketone sub-mechanism in this work can be found in the Supplementary Material.

It should be emphasized that the reaction rates of intramolecular hydrogen migration reactions were obtained by analogy to n-pentane in the model proposed by Fenard et al. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF], and the activation energy was reduced by 3 kcal/mol to account for the influence of the carbonyl functional group on the BDEs of C-H in the α-position of the carbonyl group. However, different from n-pentane, the carbonyl groups of 2-and 3-pentanone participate in the intramolecular hydrogen migration reactions, and the influence of the carbonyl group itself on the intramolecular hydrogen migration reactions was not considered in their model. Recently, Kuzhanthaivelan et al. [START_REF] Kuzhanthaivelan | Computational investigations on the thermochemistry and kinetics for the autoignition of 2-pentanone[END_REF] provided reasonable reaction rates of intramolecular hydrogen migration reactions and decomposition of •OOQOOH radicals of 2-pentanone based on theoretical calculations.

As shown in Fig. 3, the barriers of the intramolecular hydrogen migration reactions of the C5KET2-OO3 radical producing C5KET2-OOH3-R1 and C5KET2-OOH3-R5 calculated by Kuzhanthaivelan et al. [START_REF] Kuzhanthaivelan | Computational investigations on the thermochemistry and kinetics for the autoignition of 2-pentanone[END_REF] are 26.0 and 20.7 kcal/mol, respectively, while the BDEs of the corresponding C-H bonds are 95.6 and 101.1 kcal/mol. The transition states of both intramolecular hydrogen migration reactions are 6-membered rings. Although the formation of C5KET2-OOH3-R1 breaks a C-H with a lower BDE, the participation of the carbonyl group in the transition state makes the barrier higher.

The reaction rates of the intramolecular hydrogen migration and decomposition of •OOQOOH radicals for 2-pentanone were adopted from the calculations of Kuzhanthaivelan et al. [START_REF] Kuzhanthaivelan | Computational investigations on the thermochemistry and kinetics for the autoignition of 2-pentanone[END_REF] in our current mechanism. The corresponding reaction rates for 3-pentanone are obtained by analogy to 2-pentanone according to the ring members of the transition state and the position of the site relative to the carbonyl group. As illustrated in Fig. 2, the position of the carbon site relative to the carbonyl group can affect the dissociation energy of the C-H bond. The intramolecular hydrogen immigration from the carbon site which is closer to the carbonyl group was given a larger rate constant due to the lower C-H bond energy. The specific values of rate constants were determined based on our experimental results. The current kinetic mechanism for 2-and 3-pentanone combustion includes 639 species and 3326 reactions and is provided in the Supplementary Material. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF]). The numbers in parenthesis are the dissociation energy of the C-H bond in 2-pentanone [START_REF] Li | Insight into fuel isomeric effects on laminar flame propagation of pentanones[END_REF], 3-pentanone [START_REF] Hudzik | Thermochemistry and bond dissociation energies of ketones[END_REF], and 2-butanone [START_REF] Thion | An experimental study in a jet-stirred reactor and a comprehensive kinetic mechanism for the oxidation of methyl ethyl ketone[END_REF] molecules. Unit: kcal/mol. [START_REF] Kuzhanthaivelan | Computational investigations on the thermochemistry and kinetics for the autoignition of 2-pentanone[END_REF], and the numbers in parenthesis are the BDE of C-H in 2-pentanone [START_REF] Li | Insight into fuel isomeric effects on laminar flame propagation of pentanones[END_REF]. Unit: kcal/mol.

The module of the Perfectly Stirred Reactor of the CHEMKIN-PRO software [START_REF] Kee | CHEMKIN-PRO Reaction Design[END_REF] was employed to simulate the measured 2-and 3-pentanone low-temperature oxidation intermediate species concentration in the JSR. The simulations for IDTs of both ketones were conducted with the Cantera software [START_REF] Goodwin | Cantera: An object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes[END_REF] using the multi-zone model [START_REF] Zhang | First-stage ignition delay in the negative temperature coefficient behavior: Experiment and simulation[END_REF]. Due to the heat release during the first-stage ignition process, the pressure and temperature of the core regime are increased, and mass transfer between the boundary layer and crevice of the piston can occur [START_REF] Di | Effects of buffer gas compositon on low temperature ignition of isooctane and n-heptane[END_REF][START_REF] Park | Rapid compression machine measurements of ignition delays for primary reference fuels[END_REF][START_REF] Lee | Rapid compression machines: Heat transfer and suppression of corner vortex[END_REF][START_REF] Mittal | Effect of crevice mass transfer in a rapid compression machine[END_REF]. Since the zero-dimensional approach with adiabatic volume expansion cannot take the mass transfer into account, it is more reasonable to use the multi-zone model when a two-stage ignition process is observed.

The multi-zone model for simulating heat loss and mass transfer during the compression and ignition process was reviewed in [START_REF] Goldsborough | A computationally efficient, physicsbased model for simulating heat loss during compression and the delay period in RCM experiments[END_REF]. Considering the calculation costs of combining the multi-zone model and the detailed mechanism, a three-zone model was used in the study on first-stage ignition delay in NTC behavior [START_REF] Zhang | First-stage ignition delay in the negative temperature coefficient behavior: Experiment and simulation[END_REF]. In this work, this three-zone model is used for the two-stage ignition study. Three 0-D homogeneous zones, representing the core zone, boundary layer zone, and crevice zone, respectively, are combined to simulate the mass transfer from the combustion chamber to the crevice volume during compression and first-stage ignition. The boundary layer zone and crevice zone can produce heat transfer between environments, while the core zone keeps adiabatic assumptions. The piston velocity history is recorded by a highspeed camera and is used as an input parameter to simulate the compression process.

The detailed three-zone Cantera code is provided in the Supplementary Material.

Results and discussion

Influence of carbonyl functional group on the species pool under JSR condition

A total of 34 species were detected during the oxidation process of both ketones in JSR, and those with known PICSs were quantified with the MBMS technique. The measured species, ionization energies, PICSs used in the quantification, and the measurement uncertainty factors (UFs) for separate species are provided in Table S2 in the Supplementary Material. As shown in Fig. 4, the main hydrocarbon intermediates of 2-pentanone and 3-pentanone were generated over the temperature range of 800-1000 K, and the concentration of these intermediates showed an obvious difference during the oxidation of the two fuels. The concentration of CH4, C2H6, C3H6, C4H6, and C4H8 were higher in the low-temperature oxidation process of 2-pentanone, while the concentration of C2H4 in 3-pentanone was higher. The predictivity of the 2-and 3pentanone low-temperature kinetic model was validated against the experimental measurements. The rate of production (ROP) analyses for relevant species were performed to further illustrate the effects of the carbonyl functional group and resonance-stabilized radicals on the low-temperature oxidation process of 2-pentanone and 3-pentanone. Solid lines: the modified model in this work; dashed lines: the original model from Fenard et al. [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF].

The ROP analysis of both ketones was conducted at 840 K at which the concentrations of the cyclic ethers are highest. The results are shown in Fig. 5 together with a comparison of n-pentane under the same condition. The reaction pathways of the three fuels are similar, H-abstraction reactions lead to different fuel radicals following the consumption through β-scission reactions and oxygen addition reactions. Due to the existence of the carbonyl functional group, the bond energy of carbon-hydrogen α to the carbonyl carbon atom is weakened, and the formation of resonance-stabilized fuel radicals like C5KET2-R3 and C5KET3-R2 are favored through H-abstraction reactions in 2-and 3-pentanone oxidation than those in n-pentane, which leads to a high fuel radical concentration. The different lengths of the carbon chain on both sides of the carbonyl group in 2-and 3-pentanone resulted in the difference in the species distribution. As shown in Fig. 5, the formation of C2H4 and C3H6 was mainly from the β-scission reactions of the fuel radicals. Fuel radicals of 2-pentanone tended to yield C3H6, while the production of C2H4 was more favored for fuel radicals of 3-pentanone due to its symmetric structure. According to the analysis of the reaction pathway, the formation of butene primarily came from the reaction: C3H5-A+CH3 (+M) = C4H8-1(+M), and the yielded butene can further lead to the production of C4H6. The production of CH4 and C2H6 originated from CH3 radicals. The carbonyl functional group separated the carbon chain of 2-pentanone with one and three carbons on both sides, which made it tend to generate C3H5 and CH3 radicals. Therefore, larger productions of CH4, C2H6, C3H6, C4H8, and C4H6 were observed in 2-pentanone oxidation. Besides hydrocarbon species, many oxygenated intermediate components were also measured, and the analysis of their production is more helpful to understand the consumption reaction pathways of the carbonyl functional group and the oxygenaddition related reaction pathways. A total of 9 types of oxygenated intermediates were quantified and the results are shown in Fig. 6. C5H8O is the product of ROO radicals through the HO2-elimination reaction. Comparing the concentration of C5H8O in the oxidation process of 2-pentanone and 3-pentanone, it can be found that the concentration of C5H8O generated in 3-pentanone is slightly larger. Due to the symmetric structure of the 3-pentanone molecule, there is only one structure of C5H8O (C5KET3-D1) during 3-pentanone oxidation, while two structures (C5KET2-D3 and C5KET2-D4) in 2-pentanone oxidation. Since C5KET2-R4 is consumed mainly through β-scission reactions (88.6%), the formation of C5KET2-OO4 and the following C5KET2-D4 is less important, which leads to the difference in concentration of C5H8O between the two fuels. According to the experimental study of Scheer et al. [START_REF] Scheer | Low-temperature combustion chemistry of novel biofuels: resonance-stabilized QOOH in the oxidation of diethyl ketone[END_REF], the signal of C5H8O2 is primarily from the cyclic ether formed through the further reaction of QOOH radicals, the contribution of diones was thus considered to be not important in 2-and 3-pentanone oxidation and not considered in our mechanism. There are six potential structures of the cyclic ether in 2-pentanone oxidation and four structures in 3-pentanone oxidation, and the reaction pathways of producing all cyclic ethers were included in our mechanism. Based on the reaction pathway analysis, the cyclic ether of C5KET2-3CYEEt (4-member ring) is favored in 2-pentanone oxidation, while 20 C5KET3-3CYE4 (4-member ring) is favored in 3-pentanone oxidation. However, due to the lack of PICSs, the concentration of C5H8O2 cannot be quantified. Only the relative signal intensity is compared with the model prediction. C5H8O and C5H8O2 are the characteristic products corresponding to the two competition reaction channels of ROO radicals, so the concentration profiles of these two species play an important role in constraining the reaction rates of these two competition reaction pathways of ROO.

In future work, the theoretical calculation and experimental measurements of PICSs of C5H8O2 are of great significance. Solid lines: the modified model in this work; dashed lines: the original model from [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF]. Ketene (CH2CO) and methyl ketene (CH3CHCO) are the products of C5KET2-R1 and C5KET3-R2 through β-scission reactions (Fig. 5) respectively. It can be seen from Fig. 6 that the concentration of CH2CO is larger than that of CH3CHCO. Although the formation of C5KET3-R2 through the H-abstraction reaction of 3-pentanone is more preferred than the formation of C5KET3-R1, due to its resonance-stabilized structure it can first reach the energy barrier of radical isomerization reaction before the β-scission reaction occurs, resulting in 32% of C5KET3-R2 isomerizing to C5KET3-R1 [START_REF] Sun | Probing fuel-specific reaction intermediates from laminar premixed flames fueled by two C5 ketones and model interpretations[END_REF]. C5KET2-R1 can also undergo isomerization reactions, but the proportion of β-scission reaction (21%) is still dominant. The isomerization reaction of the fuel radicals changes the distribution of the species pools. The concentration of acetaldehyde (CH3CHO) in the oxidation process of 3-pentanone is significantly higher than that in 2-pentanone. The C5KET3-RO2 radicals in 3-pentanone can directly produce CH3CHO through the β-scission reaction, while CH3CHO in 2-pentanone oxidation cannot be produced through fuel-related radicals directly. C3H6O is the sum of the mole fractions of propionaldehyde (C2H5CHO) and acetone (CH3COCH3).

However, since the ionization energies of these two species are very close, they cannot be quantified separately in this work. Since the PICSs of these two species are similar, the average PICS of these two species is used for quantification. Based on the ROP analysis, the C3H6O in 3-pentanone and 2-pentanone are mainly C2H5CHO and CH3COCH3, which are produced through the addition-dissociation reactions of 3pentanone with H radicals and 2-pentanone with CH3 radicals, respectively [START_REF] Kang | Isomer-specific influences on ignition and intermediates of two C5 ketones in an RCM[END_REF]. CH3 radicals can be largely produced in 2-pentanone oxidation through the dissociation of 2-pentanone fuel radicals, while the H radicals in 3-pentanone oxidation are only produced from small-molecule species. Therefore, the concentration of CH3 radicals in 2-pentanone oxidation is much higher than the concentration of H radicals in 3pentanone oxidation, which leads to the difference in the concentration of C2H5CHO and CH3COCH3 during the oxidation of these two fuels.

As shown in Fig. 7, the presence of acetic acid (CH3COOH) and propionic acid (C2H5COOH) can be proven by comparing the measured PIE curves with literaturereported PICSs [START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF] respectively. CH3COOH was only detected in 2-pentanone oxidation, while C2H5COOH was only detected in 3-pentanone oxidation. In the previous studies on the oxidation of alkanes [START_REF] Battin-Leclerc | To better understand the formation of short-chain acids in combustion systems[END_REF][START_REF] Ranzi | New reaction classes in the kinetic modeling of low temperature oxidation of n-alkanes[END_REF] and aldehydes [START_REF] Rodriguez | A study of the low-temperature oxidation of a long chain aldehyde: n-hexanal[END_REF], it was proposed that the formation of acids was mainly produced from the consumption of aldehydes. Taking acetic acid as an example, it can be produced from the acetyl group through CH3CO+O2 → CH3CO3 → CH3CO3H → CH3CO2 → CH3COOH. Since acetaldehyde is a main oxygenated intermediate in 3-pentanone oxidation, acetic acid was anticipated to be among the products, but only the signal of propionic acid was detected. In the investigation of the potential energy surface of butanone reacting with HO2 radicals, Zhou et al. [START_REF] Zhou | Theoretical and kinetic study of the reaction of ethyl methyl ketone with HO2 for T = 600-1600 K. Part II: addition reaction channels[END_REF] found that in addition to H-abstraction reactions, HO2 can also add to the carbonyl functional group to form ROO radicals of 2-butanol, which further go through the intramolecular hydrogen migration reaction producing QOOH radicals and finally produce acetic acid. In analogy to this reaction, as shown in Fig. 8, the HO2 addition reactions of 2-pentanone and 3-pentanone can produce acetic acid and propionic acid, respectively. eV. The PICSs of CH 3 COOH and C 2 H 5 COOH are obtained from [START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF]. It can be seen from Fig. 5 that at the temperature of 840 K, a part of the fuel radicals is still consumed by the oxygen addition reaction, especially to the resonance-stabilized fuel radicals C5KET2-R3 and C5KET3-R2, respectively. To analyze the subsequent reaction of the formed ROO radicals and the influence of the carbonyl functional group, the comparison of reaction pathways of the ROO radicals of 2-pentanone, 3-pentanone, and n-pentane are shown in Fig. 9. It is observed by modeling that 83.9% of the ROO radical of 3-pentanone (C5KET3-OO2) was converted to C5KET3-OOH2-R4 by intramolecular hydrogen migration reaction, while the proportion of intramolecular hydrogen migration for ROO radial of n-pentane at the same site was 40.5%. Similarly, 59.3% of ROO radicals of 2-pentanone (C5KET2-OO3) were converted to C5KET2-OOH3-R1, while the corresponding branching ratio converting to QOOH radicals in npentane was 7.1%. It should be noted that both the produced C5KET3-OOH2-R4 and C5KET2-OOH3-R1 have resonance-stabilized structures. To explain the reason why the ROO radicals of 2-and 3-pentanone are more likely to produce resonance-stabilized QOOH radicals, the forward and reverse reaction rates of ROO radicals producing QOOH radicals through intramolecular hydrogen migration reactions are compared. Figure 10 shows that the forward reaction rate of ROO radicals producing non-resonance-stabilized QOOH radicals is larger than that producing resonance-stabilized QOOH. However, Fig. 9 shows that the branching ratio of ROO radicals producing resonance-stabilized QOOH radicals is larger, which is caused by the difference in the reverse reaction rate. It can be seen from Fig. 10 that the reverse reaction rate of ROO to produce resonance-stabilized QOOH radicals is 1 or 2 orders of magnitude smaller than the reverse reaction rate of non-resonance-stabilized QOOH radicals. This is because the produced resonance-stabilized QOOH radicals have lower energy in the reaction potential energy surface, and the energy barrier that needs to be crossed to return to ROO radicals through reverse reaction is also higher [START_REF] Scheer | Low-temperature combustion chemistry of novel biofuels: resonance-stabilized QOOH in the oxidation of diethyl ketone[END_REF]. Since the branching ratio reflects the reaction equilibrium, the difference in the reverse reaction rates causes the difference in the branching ratio. In summary, the forward reaction rate of the intramolecular hydrogen migration reaction is decreased when the carbonyl functional group participates in the ring formation of the transition state, while the produced resonance-stabilized QOOH radicals need to overcome a much higher energy barrier to isomerize back to ROO than undergoing other competing channels. Therefore, the ROO radicals are more likely to produce resonance-stabilized QOOH radicals.

Two-stage ignition kinetics under RCM condition

During the low-temperature oxidation of 2-and 3-pentanone under JSR conditions, the cool flame phenomenon was not observed. To have an in-depth understanding of the low-temperature oxidation process, the concentration of fuel and experimental pressure were increased, and an RCM was selected to study the two-stage ignition process of 2-and 3-pentanone. IDTs of both ketones were measured in the RCM under the condition of 15-25 bar, 640-820 K, and showed NTC behavior. In addition, the concentration profiles of nine major intermediates during the ignition process were quantified using a fast sampling system equipped with GC at 15 bar. The detailed experimental condition and results can be found in the Supplementary Material.

The overall IDTs can reflect the reactivity of the fuel, while the first-stage ignition can further reflect the low-temperature reaction process. The experimental measurements and modeling predictions on IDTs are compared in Fig. 11. Both the Fenard model [START_REF] Fenard | Experimental and modeling study of the low to high temperature oxidation of the linear pentanone isomers: 2-pentanone and 3pentanone[END_REF] and the model of this work can give a reasonable prediction of the overall IDTs, while the Fenard model is not capable to well predict the first-stage IDTs.

To facilitate relevant discussions, brute-force sensitivity analyses concerning the firststage IDTs, total IDTs, and their ratio were performed. The most sensitive 9 reactions are shown in Fig. 12. The secondary oxygen addition reaction of QOOH radicals, the decomposition reaction of KHP, and the hydrogen-abstraction reaction of both fuels promote the first-stage and total ignition process, while the HO2 elimination reaction of ROO radicals has a negative effect. In addition, it can be found that some reactions have different effects on the first-stage and total ignition process. The HO2 elimination reaction of C5KET3-OO2 radicals in 3-pentanone has a stronger influence on the firststage ignition process, while the secondary oxygen addition reaction of C5KET3-OOH2-R4 promotes the total ignition more obviously. The difference in reaction rates and branching ratios between oxygen-addition reactions and HO2 elimination reactions of ROO radicals, and between cyclic ether formation reactions and secondary oxygen addition reactions of QOOH radicals, leads to the difference in modeling prediction on first-stage IDTs. The total IDTs can reflect the reactivity of 2-and 3-pentanone. As shown in Fig. 13, 2-and 3-pentanone exhibit different reactivity at different temperature ranges.

Below the NTC region temperature, the total IDTs of 3-pentanone are shorter than that of 2-pentanone. When the temperature reaches the NTC region (approximately 715-770 K), the total IDTs of 2-and 3-pentanonoe are almost the same, and when the temperature crosses the NTC region (T > 770 K), the total IDTs of 2-pentanone are shorter, which is also consistent with the experimental measurement of a shorter IDTs 29 for 2-pentanone at temperatures less than 910 K in ref. [START_REF] Kang | Isomer-specific influences on ignition and intermediates of two C5 ketones in an RCM[END_REF]. The above discussion can be simplified to the reaction pathways in Fig. 16. When the ignition process of both fuels is dominated by oxygen addition reactions, the concentration of fuel radicals C5KET2-R3 and C5KET3-R2 affects the IDTs. Due to the symmetry of the molecular structure of 3-pentanone, there are two sites for the hydrogen abstraction reaction to produce C5KET3-R2, while in 2-pentanone, only one site can produce C5KET2-R3. As can be seen in Fig. 14 and Fig. 15, the ROP of C5KET3-R2 is higher than that of C5KET2-R3, so the total IDTs of 3-pentanonoe are shorter at temperatures below the NTC region. When the temperature rises and enters the NTC region, the HO2 elimination reaction of the ROO radicals becomes more important. Since C5KET3-OO2 produced in 3-pentanone oxidation has a higher concentration than that of C5KET2-OO3 in 2-pentanone oxidation, more HO2 is produced in 3-pentanone. In the NTC region, the temperature does not reach the H2O2 decomposition temperature, so the reactivity of 3-pentanone decreases as the chainterminating reaction of C5KET3-OO2 = C5KET3-D1 + HO2. Therefore, a similar reactivity of 2-and 3-pentanone was observed. The difference in total IDTs between 2and 3-pentanone as the temperature crosses the NTC region has been discussed in [START_REF] Kang | Isomer-specific influences on ignition and intermediates of two C5 ketones in an RCM[END_REF].

Conclusion

In this study, the low-temperature oxidation chemistry of 2-and 3-pentanone was investigated experimentally in a JSR and an RCM. JSR oxidation experiments combined with the PI-MBMS technique were performed under the equivalence ratio of 0.5 at 93.

Fig. 1 .

 1 Fig. 1. Typical pressure profile in the RCM. Red solid line: combustion chamber pressure; blue

Fig. 2 .

 2 Fig. 2. The strategy of obtaining reaction pathways and reaction rates by analogy to 2-butanone and

Fig. 3 .

 3 Fig. 3. The barriers of intramolecular hydrogen migration reactions of C5KET2-OO3 radical. The

Fig. 4 .

 4 Fig. 4. Experimental mole fraction profiles (symbols) and modeling results (lines) of hydrocarbon

Fig. 5 .

 5 Fig. 5. Initial fuel consumption pathways for 2-pentanone (a), 3-pentanone (b), and n-pentane (c)

Fig. 6 .

 6 Fig. 6. Experimental mole fraction profiles (symbols) and modeling results (lines) of oxygenated

Fig. 7 .

 7 Fig. 7. The PIE curve of C 2 H 4 O 2 and C 3 H 6 O 2 at 840 K, with photon energy ranging from 9.8 to 11.6

Fig. 8 .

 8 Fig. 8. The reaction pathways of HO 2 radicals with 2-and 3-pentanone producing acetic acid and

Fig. 9 .

 9 Fig. 9. Comparison of the branching ratios of ROO radicals of 2-pentanone (blue), 3-pentanone

Fig. 10 .

 10 Fig. 10. The forward and reverse reaction rates of ROO radicals of 2-pentanone (a) and 3-pentanone

Fig. 11 .

 11 Fig. 11. Experimental measurements (dots) and modeling prediction (red: Fenard model, blue: the

Fig. 12 .

 12 Fig. 12. Sensitivity analysis of first-stage IDTs (green), total IDTs (blue) and their ratios (purple) of

Fig. 13 .

 13 Fig. 13. The total IDTs of 2-and 3-pentanone over the temperature range 640-820 K and at 15 and

Fig. 14 .

 14 Fig. 14. (a) Rate of production of OH radicals in 2-pentanone ignition process at 719 K and 25 bar

Fig. 15 .

 15 Fig. 15. (a) Rate of production of OH radicals in 3-pentanone ignition process at 698 K and 25 bar

Fig. 16 .

 16 Fig. 16. Main low-temperature reaction pathways and corresponding components of 2-and 3-

17

 17

.

  The first-stage ignition of 3-pentanone occurs earlier than that of 2-pentanone. The model developed in this study, compared to the Fenard model, can better predict the species concentrations which exhibit two-stage formation behaviors. Discrepancies still exist between the experimental and modeling concentration profiles of CO2, CH4, CH3CHO, and C2H5CHO. As discussed in Section 3, the first-stage ignition might increase the gradience of temperature and species concentration in the RCM, which results in larger uncertainties of the sampled species concentration after the first-stage ignition. Still, since the experimental results of species concentration profiles for 2-and 3-pentanone ignition were measured at similar experimental conditions, the relative comparison between 2-and 3-pentanone can provide useful information on reaction path analysis.

Fig. 17 .

 17 Fig. 17. Experimental (symbols) and simulated (dashed lines: Fenard model, solid lines: the

  3 kPa and temperatures from 600 to 1000 K. Numerous species were identified and quantified, including C1-C4 hydrocarbons and C1-C5 oxygenated intermediate species, which are closely related to fuel consumption. IDTs of 2-and 3-pentanone were measured from 640 to 820 K under 15 and 25 bar at equivalence ratios of 1.0 in an RCM. IDTs of both ketones showed NTC behavior under these conditions, and the relative magnitude of total IDTs of 2-and 3-pentanone changed at different temperatures. The mole fraction time histories of intermediate species during the twostage ignition process of both ketones were obtained. A new kinetic model was constructed, which showed satisfactory predictive performances and therefore used to interpret speciation behaviors. Model analyses indicate that ROO radicals tend to yield QOOH radicals with resonance-stabilized structure through intramolecular hydrogen migration reaction during the oxidation of 2-and 3-pentanone. The secondary oxygen addition reaction of resonance-stabilized QOOH radicals is the most important source of OH radicals in the low-temperature oxidation of ketone fuels. The carbonyl functional group inhibits the intramolecular hydrogen migration when it participates in the ring formation, which makes the low-temperature reactivity of 2-and 3-pentanone lower than that of n-pentane with the same carbon chain length. The first-stage ignition process of 2-and 3-pentanone is initiated by the OH radicals produced by the decomposition of the KHP, while there is competition between the HO2 elimination reaction and the intramolecular hydrogen migration reaction of the ROO radicals. The position of the carbonyl functional group has a great influence on the component and concentration of the species pools because the length of the carbon chain on both sides of the carbonyl group is different in the two ketones. Higher production of CH4, C2H6, C3H6, CH3COCH3, and C2H5CHO were observed in 2-pentanone oxidation compared with that of 3-pentanone oxidation. Future model development requires more accurate rate coefficients of crucial reactions such as intermolecular hydrogen immigration reactions C5KET2-OO3 = C5KET2-OOH3-R1 and C5KET3-OO2 = C5KET3-OO2-

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 91841301 and 52076116) and Seed Fund of Shanxi Research Institute for Clean Energy, Tsinghua University. NH is supported by the Low Temperature Plasma Research Facility at Sandia National Laboratories, which is a collaborative research facility supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under contract number DE-NA0003525. Sandia National Laboratories is a multimission laboratory managed and operated by the National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE's National Nuclear Security Administration under contract DENA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. DOE or the U.S. Government.


Steffen Bohn

Marcelo O Magnasco

Structure, Scaling and Phase Transition in the Optimal Transport Network

Keywords: numbers: 89.75.Da, 89.75.Fb, 89.75.Hc, 89.75.Kd

We minimize the dissipation rate of an electrical network under a global constraint on the sum of powers of the conductances. We construct the explicit scaling relation between currents and conductances, and show equivalence to a a previous model [J. R. Banavar et al Phys. Rev. Lett. 84, 004745 (2000)] optimizing a power-law cost function in an abstract network. We show the currents derive from a potential, and the scaling of the conductances depends only locally on the currents. A numerical study reveals that the transition in the topology of the optimal network corresponds to a discontinuity in the slope of the power dissipation.

The optimal distribution of valuables such as electricity or telephone signals has been a subject of much study since Westinghouse and Edison's War of the Currents in the late 19 t h century [15]; more recently, natural systems such as river networks and vascular systems have been fruitfully interpreted in this light [START_REF] Rodtiguez-Iturbe | Fractal River Basins[END_REF][START_REF] Rinaldo | [END_REF]3]. Hence formal models of optimal transport networks have attracted attention over many years [4,5]. However, different studies use different definitions of network and optimize different functionals. For example, Durand [6,7] considers hydraulic networks whose currents derive from a potential, explicitly analogous to electrical networks; the networks are embedded in an ambient space, and he studies the optimal geometry and the relation between the local geometry and local topology. On the other side, Banavar et al. [3] propose a more abstract model where the graph is not assumed to be embedded in a target space, nor are the currents through the nodes explicitly constrained to derive from a potential. This allows them to furnish a strict proof that the topology of the optimized flow pattern [3] depends on the convexity of their cost function, but makes a direct physical interpretation of the model more elusive. In the following, we shall introduce a third model of an optimal transport network from whom both of these previous models can be derived, so all formulations are, in fact, equivalent. Consider an electrical transport network on a graph composed of nodes k interconnected by links (k, l). There is a given current source i k at each node and the total current input must add to zero:

k i k = 0. There are variable currents I kl flowing through the links; the sum of all currents impinging on a given node k must equal the given current sources: i k = l I kl (Kirchhoff's current law). We associate a resistor R kl ≥ 0 to each link (k, l) and decompose its value as R kl = (d kl κ kl ) -1 , where d kl > 0 is a given weight and the conductances κ kl are variable; considering κ kl as a conductivity per unit length, d kl can be thought of as the length of the link. The dissipation rate J is then a function of the currents I kl through the links and the conductances κ kl :

J = links (k,l) I 2 kl (d kl κ kl ) (1)
We shall minimize this dissipation rate J over the currents I kl and the conductances κ kl with the local constraint given by Kirchhoff's current law, and a supplementary global constraint that the sum over the conductances raised to a given power γ > 0 is kept constant:

K γ = κ γ kl
One may interpret this constant as an amount of resources we have at our disposal to build the network.

[16] Since we allow κ kl and I kl to vary independently, the currents are not explicitly constrained to derive from a potential at the nodes U k and Kirchhoff's voltage law (the sum of the potential differences on a loop vanishes) need not apply. Using a Lagrange multiplier λ, we define the function Ξ({κ kl }, {I kl }) as

Ξ({κ kl }, {I kl }) = (k,l) I 2 kl (d kl κ kl ) -λ (k,l) κ kl γ (2)
The necessary conditions for a minima of J with constant K are then:

∂Ξ ∂I kl = 0, ∂Ξ ∂κ kl = 0 (3)
Let us first consider the derivatives with respect to I kl . Let { Ĩkl }, {κ kl } minimize J. Adding a circular current X α on a loop α to the currents (fig. 1) does not violate the constraints. We (re)define the directions of the currents Ĩkl on the loop to be parallel to the loop current X α . Then

0 = ∂Ξ ∂X α Xα=0 = loop α Rkl Ĩkl (4)
Thus Kirchhoff's voltage law holds at the minimum of J, so the currents though the links derive from potential differences between the nodes: Ĩkl = Rkl (U l -U k ). Note that this is not the case for every arbitrary current distribution. For instance, if all currents on the loop in fig.

1 are positive Ĩkl > 0, then there exists no set of Rkl ≥ 0 to fulfill this relation.

Let us now consider the derivatives of Ξ with respect to κ kl (eq. 3). With the constraint of a constant K, we obtain an explicit scaling relation between the currents and the conductivity in the minimal configuration:

κ kl = (I 2 kl /d kl ) 1 1+γ mn (I 2 mn /d mn ) γ 1+γ 1/γ K (5)
We can now write the total dissipation (eq. 1) in terms of the currents alone as

J({I kl }) = 1 K kl (I 2 kl /d kl ) γ 1+γ 1+ 1 γ (6)
Since for γ > 0, the function x 1+ 1 γ is monotonically increasing, the original minimization problem is reduced to the minimization of

C({I kl }) = kl (I 2 kl /d kl ) γ 1+γ (7) By setting Γ = 2γ γ + 1 (8)
and rescaling the weights as

w kl = d -γ 1+γ kl
, the quantity to be minimized is now

C({I kl }) = kl w kl |I kl | Γ (9)
which is exactly the model used by Banavar et al. [3].

They give a strict proof that for Γ < 1, the resulting structure may not have any loop, and each spanning tree is a local minimum. For Γ > 1, there are in general loops and a unique minimum. Due to the correspondence between γ and Γ, this result must apply also to our original model where γ < 1 (γ > 1) corresponds to a Γ < 1 (Γ > 1). On the other hand, the correspondence between the different models allows an important conclusion about the model of Banavar et al.. Since in both formulations, the is obtained by the same set of currents, and since in our model these currents must derive from potential differences between the nodes, this must be true for the minimum of the Banavar et al. model, too. We can furthermore write down directly the values of the corresponding resistors as

R kl = (d kl κ kl ) -1 = A w kl |I kl | Γ-2 (10)
with an arbitrary positive constant A. R kl thus scales explicitly with the local currents for Γ = 2.

Since positive γ corresponds to 0 < Γ < 2, the equivalence of the two models is restricted to this parameter range. Γ > 2 corresponds to values γ < -1, for which our model collapses into infinitely many degenerate minima. The relations 3 correspond instead to a saddle node of J: a minimum with respect to the I kl and a maximum with respect to the κ kl . Nevertheless direct inspection shows that the current flow in the Banavar et al. model is potential with the set of resistors given by eq. 10 even for Γ > 2.

[17]

In order to get a deeper insight into the transition at γ = 1, we search numerically for the minimal dissipation configuration of an example network, a triangular network of conductivities with a hexagonal border, with equal weights d kl ≡ 1. The total number of nodes N nodes scales roughly as the square of the linear dimension of the network, given by the diameter of the graph N dia . Except for those on the border, each node is linked by conductivities to six neighboring nodes. We place a current source at a corner of the hexagon (i 0 ), the remaining (N nodes -1) nodes present homogeneous distributed sinks; each node absorbs i k = -i 0 /(N nodes -1).

As an order parameter, we will consider the normalized dissipation rate J min /J homo , where J homo is the total dissipation with a constant conductivity distribution κ kl ≡ const., and J min is the dissipation for the optimized distribution of the conductivities. Note that J homo corresponds also to γ → ∞.

The previous discussion allows us to simplify the minimization problem enormously: using the scaling relation between κ kl and I kl , one can restrict the search of the minimum to the space of the currents or the space of conductivities. Furthermore, we can use the fact that the optimized current distribution derives from a potential U k to construct a simple relaxation algorithm. Starting with random distribution of κ kl , we calculate first the values of the potential at the nodes by solving the system of linear equations i k = l R kl (U k -U l ), then the currents through the links I kl are determined. We use these currents to determine a first approximation of the optimal conductivities on the basis of the scaling relation. Then, the currents are recalculated with this set of conductivities, and the scaling relation is reused for the next approximation. These steps are repeated until the the values have converged. We check by perturbing the solution that it actually is a minimum of the dissipation, which was always the case. For all γ > 1, independently of the initial conditions, the same conductivity distribution is obtained, which conforms to the analytical result of [3]: there exists a unique minimum which is therefore global. Furthermore, the distribution of κ kl is "smooth", varying only on a "macroscopic scale", as show in Fig. 2(a)). No formation of any particular structure occurs. However, the conductivity distribution is not isotropic. We can interpret the conductivity distribution as a discrete approximation of a continuous, macroscopic conductivity tensor (see also [6]). The smooth aspect of the distribution is conserved while approaching γ → 1 while the local anisotropy increases, while the values of all κ kl remain finite, even if they get very small. For γ = 1.5 and N dia = 15, the conductivity distribution spreads already over eight decades and becomes still broader as γ → 1 + , in which limit the number of iteration steps diverges as the minima becomes less and less steep. γ = 1 presents a marginal case. The results of the simulation suggest that the minimum is highly degenerate, i.e., there are a large number of conductivity distributions yielding the same minimal dissipation. For γ < 1, the output of the relaxation algorithm is qualitatively different (fig. 2(b)). The currents are canalized in a hierarchical manner: a large number of conductivities rapidly converge to zero and thus vanish transforming the topology from a highly redundant network to a spanning tree. This, too, is predicted by the analytical results [3]. In contrast to γ > 1, the conductivity distribution can not be interpreted as a discrete approximation of a conductivity tensor: for N dia → ∞, the structure becomes fractal. For different initial conditions, the relaxation algorithm yields trees with different topologies: each local minima in the high-dimensional and continuous space of conductivities {κ kl } correspond to a different tree topology. Given a tree topology, the currents through the links are given directly by the topology and do not depend on the values of the κ kl , and so using the scaling relation, one may directly write down the dissipation rate for a given tree. For γ < 1, we do thus not need to apply the relaxation algorithm, but we should search for the global minima in the (exponentially large) space of tree topologies using a Monte-Carlo algorithm. This regime has been widely explored in the context of river networks [START_REF] Rodtiguez-Iturbe | Fractal River Basins[END_REF][START_REF] Rinaldo | [END_REF]8], mainly for a set of parameters that corresponds, in our case, to γ = 0.5. An example of a resulting minimal dissipation tree structure is in fig. 2(c). Note also, that the scaling relations can be seen as some kind of erosion model: the more currents flows trough a link, the better the link conducts [START_REF] Rodtiguez-Iturbe | Fractal River Basins[END_REF]. The qualitative transition is reflected also quantitatively in the value of the minimal dissipation (fig 3(a)). The points for γ > 1 were obtained with the relaxation algorithm, the points γ < 1 by optimizing the tree topologies with a Monte-Carlo algorithm. For γ → ∞, J min /J homo → 1 by definition; for γ → 0, J min /J homo → 0, because the vanishing κ kl allow the the remaining κ kl → ∞. Figure 3(b) shows the behavior of minimal dissipation rate close to γ = 1. For γ smaller than one, the relaxation method spends a long time only to furnish a local minimum, while the Monte-Carlo algorithm searching for the optimal tree topologies gives lower dissipation values. The different values corresponding to different realization indicate that the employed Monte-Carlo method does not find the exact global minima. For γ > 1, the relaxation algorithm gives the lower J min because the global minima does not have a tree topology. While the curve is continuous, the crossover at γ = 1 shows a clear change in the slope of J min (γ). One could interpret this behavior as a second order phase transition. (The change in slope is of course preserved in the function C(I kl ) used by [3].) As an intriguing practical application of these models, one may for instance cite the venation of plant leaves. Experimental evidence [9] shows that the water transport through the veins derives from a pressure gradient. The venation pattern however shows a enormous redundancy of loops [START_REF] Esau | Plant anatomy[END_REF][START_REF] Roth-Nebelsick | [END_REF]12]. On the basis of some examples, it has been proposed [START_REF] Roth-Nebelsick | [END_REF]13] that the loops are actually meaningful to optimize the water transport in the leaf. The results presented in this paper however shows that this is not the case: optimization either leads to a tree topology, or to no structure at all. If the venation pattern is really based a optimization principe, it cannot simply be optimization of a steady state water transport, even if Murray's law seems to hold at the nodes of the venation [14].

FIG. 1 :

 1 FIG.1: Sketch of a loop α indicating the direction of the currents. Every perturbation of the I kl satisfying the constraints can be written as a weighted sum of such loops.

FIG. 2 :

 2 FIG. 2: Examples of the conductivity distributions. Results of the relaxation algorithm with different initial conditions: (a) γ = 2.0 and (b) γ = 0.5. (c) Result with optimized topology with γ = 0.5. the arrow indicated the localized inlet, the remaining nodes are outlets with constant i k .

  FIG. 3: (a) The normalized minimum dissipation rate Jmin/J homo as a function of γ for a network with N dia = 15 (462 links) and a network with N dia = 31 (2070 links, in red). Note the discontinuity of the slopes at γ = 1. (b) A detailed view of the crossover at γ = 1 for N dia = 15. Cross symbols show data points obtained by optimizing a tree topology, circles show the output of the relaxation algorithm. The continuous lines indicate the actual minimum.