Impact of conformational structures on low-temperature oxidation chemistry for cyclohexyl radicals: a theoretical and kinetic modeling study on first oxygen addition

Abstract

This work aims to investigate the crucial role of the inherent conformations of cyclohexyl radical in low-temperature oxidation chemistry through theoretical calculations and kinetic modeling, which has not been explored previously. Potential energy surface for cyclohexyl + O₂ was precisely examined using high-level composite quantum methods, and temperature- and pressure-dependent rate coefficients were predicted via RRKM/master-equation analysis in the range of 200–2000 K and 0.001–100 atm, respectively. A detailed kinetic model for cyclohexane oxidation was constructed by incorporating Boltzmann-weighted rate coefficients based on the equilibrium of conformers. Results show that the addition of an O₂ molecule onto cyclohexyl in the chair and twist-boat forms yields chair-axial, twist-boat-axial and twist-boat-isoclinal adducts accordingly. Axial and isoclinal preferences in the three adducts facilitate the 1,5-H transfer, while only the twist-boat-isoclinal conformation proceeds with the 1,6-H transfer. The dissociations of cyclohexylperoxy and hydroperoxycyclohexyl species exhibit distinctive conformational-dependent features, and ring-opening reactions preferably occur in equatorial conformations with lower steric hindrance. Kinetic predictions reveal the importance of isomerization in cyclohexylperoxy in the order 1,5- > 1,6- > 1,4-H transfer, while that for OH eliminations follows the order 1,2- > 1,4- > 1,3-epoxycyclohexane cyclization at evaluated temperatures and pressures. Stabilization and HO₂ elimination in cyclohexylperoxy separately predominate the overall oxidation mechanism at correspondingly low and high temperatures, while OH elimination and hydroperoxycyclohexyl stabilization have minor contributions at high temperatures. The most rapid inversion-topomerization allows for equilibrium between various conformers in cyclohexylperoxy and hydroperoxycyclohexyl, thereby facilitating the inclusion of partition function contributions into kinetics. The new model reproduces cyclohexane oxidation measurements in jet-stirred reactors and laminar flame speeds for cyclohexane/air mixtures fairly well.

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1. Introduction

Cycloalkanes constitute approximately 40% of commercial diesel,¹ 20% of jet fuel,¹ and 10–30% of aviation and automotive gasoline,² and even higher concentrations (up to 99%) are found in alternative transport fuels refined from oil sands, oil shale, and coal-derived liquids.^3,4 For fuels with high cycloalkane concentrations,^3,5–7 a homogeneous-charge compression ignition (HCCI) engine has been designed and optimized using low-temperature (low-T) combustion strategies to achieve efficient and low-pollution combustion.⁸ This engine strongly relies on chemical kinetics to control the timing of compression ignition and exhibits rather high sensitivity to the molecular structure of the fuel.^5,6 Thus, a detailed molecular-level understanding of low-T oxidation chemistry is needed to probe fuel reactivity, autoignition process, and pollutant generation.

As one of the simplest cycloalkanes, cyclohexane exhibits multiple structural conformations and low torsional strain, making it suitable for investigating the ring conformation effect on low-T oxidation chemistry and providing insight into analogous ignition behavior for alkylated cycloalkanes.^2,4 The reaction between cyclohexyl and O₂ is of tremendous interest because of its crucial role in engine autoignition and atmospheric oxidation chemistry.^9,10 The kinetics, elementary reaction mechanism, and end products have been investigated using theoretical computation, kinetic modeling, and prototypical and combustion experiments.^{4,5,7,11–16} Gas-phase rate coefficients for O₂ addition to cyclohexyl were originally measured by laser flash photolysis at room temperature¹⁷ and further precisely determined by time-resolved UV-vis adsorption spectroscopy.⁹ Subsequently, the rate coefficients for the cyclohexane + HO₂ system¹⁰via 1,4-, 1,5- and 1,6-H transfers in the cyclohexylperoxy (ROO•) adduct were estimated at combustion-related temperatures of 673–773 K in aged Pyrex vessels coated with boric acid. Results primarily suggest that the chair and twist-boat conformations for cyclohexylperoxy have a strong influence on H transfers. Afterwards, the conformations and conformational changes of cyclohexyl and cyclohexylperoxy radicals were explicitly considered in barrierless oxygen addition and subsequent isomerization and dissociation, respectively, by employing G2(MP2)-like ab initio calculations and time-dependent master equation analysis.¹⁵ A kinetic model was constructed and validated against the production of OH and HO₂ in Cl-initiated low-T cyclohexane oxidation at 6.5–20.3 bar and 586–828 K, and simulations reveal the significant role of “formally direct” chemical activation routes in the generation of OH and large molecules.⁵

Meanwhile, chemical kinetic models for cyclohexane oxidation have been continuously developed to simulate higher temperatures based on data acquired from rapid compression machine (RCM),^7,18,19 jet-stirred reactor (JSR),^13,20 laminar burning velocities,²¹ and slow-flow quartz reactor.¹² Early simplified models were proposed by Granata²² and Buda et al.¹⁹ using empirically estimated rate rules and later refined by Cavallotti et al.^16,19 with theoretical predictions for H transfers in cyclohexylperoxy and the decomposition of hydroperoxycyclohexyl (QOOH) combined with a consideration of conformation. Moreover, Sirjean et al.¹⁴ performed high-level quantum calculations for cyclohexyl isomerization and cyclohexylhydroperoxy decomposition at the CBS-QB3 level, and high-pressure limit (HPL) rate coefficients were included in the mechanism by Buda et al.¹⁹ Serinyel et al.¹³ further improved this model with a reaction mechanism automatically generated using the EXGAS package. More recently, Zou et al.^4,11 investigated low-T oxidation mechanism for cyclohexane triggered by first and second oxygen additions using theoretical computations, JSR experiments with SVUV-PIMS and GC-MS detections. Only the chair-axial conformation was considered in kinetic predictions, however, its impact on reaction possibilities was indicated. These previous studies have been focused on elucidating low-T oxidation chemistry for cyclohexane and have attempted to uncover the conformational effect on primary oxidation reactions. However, given the flexibility of the cyclohexyl ring, the major issue of conformational effect on initial oxidation steps is still not well understood. In other words, barrierless entrance channels for oxygen addition on cyclohexyl and subsequent consumption for ROO• and QOOH require cautious treatment to evaluate their reaction possibilities, energetics, and kinetics regarding the specific orientations of the radical site and side chain in different conformations as well as conformational transitions.

Inspired by these insights, the present work aims to provide an in-depth understanding of the conformational transitions in cyclohexyl and its cyclohexylperoxy and hydroperoxycyclohexyl derivatives and revisit low-temperature oxidation chemistry to disclose comprehensive conformational effects on key initial oxidations through theoretical calculations and kinetic modeling simulations. Stationary point energies on the cyclohexyl + O₂ potential energy surface (PES) were characterized by quantum chemical computations, with temperature- and pressure-dependent rate coefficients predicted using RRKM/master equation simulation. Supported by conformational analysis, the contributions of distinct conformers towards kinetics were evaluated. Thus, a new model was constructed by incorporating Boltzmann-weighted rates based on the model by Zou et al.⁴ and was subsequently validated against JSR measurements for cyclohexane oxidation and laminar flame velocities for cyclohexane/air flames. This is the first kinetic study to fully account for the conformational dynamics of cyclohexane, which has not been explored in previous kinetic studies for O₂ addition to cyclohexyl.

2. Theoretical methodology

2.1 Electronic structure methods

Optimized geometries and ro-vibrational frequencies for stationary points on the potential energy surface were acquired at the B3LYP/6-311++G(d,p) level of theory.²³ All distinct cyclic conformers, including radicals and transition states (TSs), were generated by both synergistic ring motions and special orientations of the side chain and optimized using the same method. Intrinsic reaction coordinate (IRC) analysis was performed to confirm the correct connections of TSs to the corresponding minima, and vibrational mode animations were considered to verify the reactant conformer bound to the special TS. Along the barrierless reaction coordinate for cyclohexyl + O₂, optimized TSs were verified to have only one imaginary frequency as well as obvious transition state configurations on the potential energy surface. For low frequencies relevant to torsional motions, hindered rotor analysis was employed to compute the density of states for –CHOO• and –COOH moieties and long-chain end products as appropriate. The underlying hindrance potential was scanned along the assigned dihedral angle at the B3LYP/6-311++G(d,p) level with an interval of 10°.

Suggested by theoretical reliability for cycloalkyl and its derivatives,^2,12,24,25 high-level evaluations for energetics and ro-vibrational parameters were refined using model chemistries of CBS-QB3²⁶ and G4.²⁷ The CBS-QB3 procedure performs geometrical optimization and Hessian matrix analysis at the B3LYP/6-311G(2d,d,p) level, and a series of high-level single-point energies is determined systematically using the MP2, MP4, and CCSD(T) methods to further approach the complete basis-group extrapolation to refine energetics. The G4 procedure optimizes the geometry and calculates the Hessian matrix at the B3LYP/6-31G(2d,p) level and then calculates the single-point energy at the CCSD(T)/6-31G(d) level with basis set extrapolation to obtain the HF limit, followed by diffuse and higher polarization function corrections. For the two composite methods, zero-point vibrational energies were obtained at B3LYP/6-311G(2d,d,p) and B3LYP/6-31G(2d,p) levels, with scaling factors of 0.951 and 0.942, respectively,²⁸ and enthalpies for the studied species were obtained by both CBS-QB3 and G4 methods.

At the ground triplet state, molecular oxygen shows biradical characteristics,^4,24 and leads to a barrierless entrance channel or a flat saddle point at the transition state zone for cyclohexyl + O₂ addition. This, however, complicates the interaction potential between two moieties and the quantification of minimum reaction flux with varying temperatures and pressures. Notably, CBS-QB3 and G4 methods barely depict a smooth energy profile along the addition coordinate, as shown in Fig. S1 of ESI-I (ESI†). As reported in previous studies for isopentyl + O₂⁶ and cyclohexyl + O₂,⁴ the multi-reference chemical method, i.e. CASPT2(7e,5o)/CBS//CASSCF(7e,5o)/cc-pVDZ, yields reasonable results. Thus, this chemistry was also employed herein with the same active space (7e,5o), in which (6e,4o) is for two pairs of O=O π bonding and π* antibonding orbitals and (1e,1o) for the carbon-centered radical.^4–6 Accordingly, with respect to the chair and twist-boat conformers in cyclohexyl, the relaxed scans along the addition coordinates for O₂ were performed at the CASSCF(7e,5o)/cc-pVDZ level within 2.0–2.6 Å in intervals of 0.1 Å and 2.6–4.0 Å in intervals of 0.2 Å. Subsequently, high-level energies were extrapolated to the CBS limit using the CASPT2(7e,5o) method with both cc-pVDZ and cc-pVTZ basis sets, and its extrapolated equation is as follows:²⁹

E_CASPT2/CBS = E_{CASPT2/cc-pVTZ} + (E_{CASPT2/cc-pVTZ} − E_{CASPT2/cc-pVDZ}) × 0.4629 (1)

Gaussian 09³⁰ and Molpro³¹ quantum chemistry programs were employed in single and multi-reference chemical computations, respectively.

2.2 Master equation simulations

Kinetic predictions were performed with time-dependent RRKM-based master equations to obtain temperature- and pressure-dependent rate coefficients at 200–2000 K and 0.001–100 atm using the kinetic MESS code.³² The MESS code accurately predicts rates for complex reaction systems containing both “well-merging” and “formally direct” pathway schemes.^5,6 The well-merging scheme treats rapidly equilibrated species as a single reactive compound when their chemical conversion rate closely approaches the internal energy relaxation rate. This approach effectively reduces the number of species in multiple wells in rate calculations. However, the formally direct scheme assumes that the chemically activated species can traverse two or several transition states in one single elementary step without collisional stabilization into in-between wells. These two schemes are incorporated into the cyclohexyl + O₂ system with entrance addition and multi-step consumption channels.

Reactions with evident chemical barriers were handled using microcanonical transition state theory as implemented in the RRKM/master equation analysis. For barrierless addition, the optimal transition state significantly changes along the reaction coordinate with varying temperatures. Variable reaction coordinate transition state theory (VRC-TST) was employed to perform kinetic calculations, aiming at minimizing rate coefficients by dividing the surface. The interaction potentials and parameters between cyclohexyl and O₂ characterized at the CASPT2 (7e,5o)/CBS//CASSCF(7e,5o)/cc-pVDZ level of theory were utilized in VRC-TST kinetic predictions. For all computed species, the low frequencies in cyclic ring related to distortion motion were primarily dealt to search out all distinct conformational structures, and those in side chains and long-chain products related to internal torsions were simulated as one-dimensional hindered rotors. Other vibrational modes were regarded as harmonic oscillators with the implantation of rigid rotor harmonic oscillator approximation.

One-dimensional corrections for tunneling were included in kinetic predictions using asymmetric Eckart functions to account for its prominent effect at low temperatures, especially on H transfers.³³ The collisional energy transfer function was represented with a single exponential-down model. As suggested in previous theoretical calculations for cyclohexyl + O₂,^4,15 a temperature-dependent form of 〈ΔE_down〉 = 250 × (T/300)^0.85 cm⁻¹ was considered for the average downward energy transfer per collision. The binary collision frequency between reactants and the Ar buffer gas was computed by using Lennard-Jones (L-J) potentials. The L-J parameters of σ = 4.5 Å and ε = 183 cm⁻¹ for the cyclohexylperoxy radical were from kinetic computations by Zou et al.,⁴ and σ = 3.47 Å and ε = 79.2 cm⁻¹ for Ar were recommended by Hippler et al.³⁴

2.3 Kinetic modeling

In the present work, the first oxygen addition steps in the sub-mechanism of low-temperature cyclohexane oxidation were revisited and examined through comprehensive conformational analysis on initial elementary reactions and conformational structures for all cyclic species. Phenomenological rate coefficients for cyclohexyl reacting with O₂ were used to update the previous model by Zou et al.,¹¹ with rigorous consideration of Boltzmann-weighted contributions from distinct conformers. In their model, the AramcoMech 2.0 model³⁵ served as the base mechanism, and theoretical reaction schemes and kinetic predictions for the first and second oxygen additions were subsequently adopted to construct a new model. Notably, the kinetics were calculated by considering only one specific chair-axial conformer for cyclohexylperoxy and one entrance oxygen addition channel.^4,11 That is, the contributions of other steric conformational structures for cyclohexyl, cyclohexylperoxy and hydroperoxycyclohexyl and conformer-specific reaction channels were excluded from kinetic predictions. Thermodynamic data for the studied species were also generally adopted from Silke et al.⁷ and Zou et al.¹¹ The newly acquired rate coefficients were employed in the Arrhenius equation model of k = ATⁿ exp(−E_a/RT), the kinetic mechanism of which is provided in ESI.†

3. Quantum chemical results and discussion

3.1 Conformational analysis on first oxygen addition

Stable chair and twist-boat conformers for cyclohexane allow for two and three distinct orientations for H atoms, respectively.^36,37 The addition of molecular oxygen to the cyclohexyl radical can be viewed as a substitute for those H atoms to yield five diacritic ring conformers for cyclohexylperoxy (W1), as depicted in Fig. 1. Other enantiomers were ignored as our computations suggest that the rotation of the –OO˙ group for each ring conformer converged to one conformer. Thus, five conformational structures for W1 were considered in our study, which is confirmed by performing quantum computations using the CREST software.³⁸ Note that cyclohexyl has two diacritical conformers with one H atom loss, and the quite flexible cyclic ring makes the radical site be fixed at axial and isoclinal positions in the chair (RCe) and twist-boat (RTi) in Fig. 1, between which the enthalpy deviation is 4.1 kcal mol⁻¹ at the G4 level (Table 1). Hence, they provide two favorable sites for the addition of molecular oxygen.

Fig. 1 Simplified reaction scheme from cyclohexyl + O₂ to cyclohexylperoxy (W1).

Table 1 Calculated energetics of reactants, intermediates, products, and conformational transition states (TSs) in inversion-topomerization processes relative to cyclohexyl in chair + O₂ at 0 K in kcal mol⁻¹

	CBS-QB3	G4	Knepp^a ^15	Zou^b ^4		CBS-QB3	G4	Knepp^a ^15	Zou^b ^4
a Calculated at the modified G2(MP2)//B3LYP level. b Calculated at the CBS/QB3//B3LYP level.
R + O 2					W3Ce	−23.8	−23.1	−23.7	−23.7
RCe + O2	0.0	0.0	0.0	0.0	W3Ta	−20.7	−19.8
RTi + O2	4.0	4.1			W3Te	−20.4	−19.6		−20.7
W1					W3Ti	−20.2	−19.3		−20.2
W1Ca	−37.1	−36.6	−37.1	−37.2	TSs
W1Ce	−37.4	−36.8	−37.3	−37.3	W3H1	−19.2	−18.4		−19.1
W1Ta	−31.8	−31.3	−31.5	−32.0	W3H2	−19.2	−18.4		−19.2
W1Te	−31.2	−30.6	−30.7	−31.1	W3H3	−19.3	−18.5		−19.3
W1Ti	−31.8	−31.2		−31.8	W3B4	−19.1	−18.4
TSs					W3B5	−19.3	−18.5
W1H1	−25.5	−25.7		−26.2	W3B6	−19.4	−18.7
W1H2	−26.9	−26.2		−26.9	W3B7	−19.2	−18.4		−18.9
W1H3	−26.3	−25.7		−26.9	W4
W1B4	−29.7	−29.0			W4Ca	−23.3	−22.6	−23.4	−23.6
W1B5	−30.3	−29.6			W4Ce	−23.4	−22.8	−23.4
W1B6	−30.3	−29.6		−30.2	W4Te	−20.0	−19.2
W1B7	−30.4	−29.8		−30.5	W4Ti	−20.4	−19.5
W2					TSs
W2Ca	−22.6	−22.1	−22.8	−23.0	W4H1	−19.0	−18.3
W2Ce	−22.1	−21.5	−22.0		W4H2	−19.0	−18.2
W2Ta	−19.5	−18.7			W4H3	−19.0	−18.2
W2Te	−22.5	−22.0			W4B4	−19.2	−18.4
W2Ti	−18.7	−18.1			W4B7	−19.4	−18.6
TSs					Products
W2H1	−18.6	−17.9			P1	−15.8	−14.4	−15.2	−15.8
W2H2	−17.7	−17.0			P2	−34.9	−32.8	−35.8	−34.9
W2H3	−18.1	−17.4			P3	−39.8	−38.0	−40.4	−39.8
W2B4	−17.8	−17.1			P4	−52.9	−50.6	−53.4	−52.9
W2B5	−16.5	−15.8			P5	−43.8	−44.8	−41.7	−40.2
W2B6	−18.6	−17.9			P6	−0.9	−1.2
W2B7	−18.1	−17.4			P7	−1.6	−1.9		−0.3
W3					P8	−64.4	−63.0	−65.8	−64.4
W3Ca	−23.2	−22.5	−23.2	−23.2

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Theoretical results demonstrate that the adjacent axial hydrogen atoms in the chair conformation generate the 1,3-syn repulsion, i.e., the anomeric effect,³⁶ resulting in molecular oxygen being attached to the chair form of cyclohexyl in an axial orientation (W1Ca) rather than yielding an equatorial adduct (W1Ce). As for cyclohexyl in its twist-boat form, molecular oxygen can flexibly settle at its isoclinal position for W1Ti. But, when the spatial orientation of the oxygen group is closer to the neighboring axial hydrogen during the addition process, the gauche effect among them will lead to the slight deformation of the twist-boat to position oxygen in a virtually axial position (i.e., W1Ta).³⁶ Consequently, there are three entrance channels to directly yield the chair-axial, twist-boat-axial, and twist-boat-isoclinal adducts, which fairly differs from the consideration of only one primarily W1Ca adduct by Zou et al.⁴ and Knepp et al.¹⁵ Thus, the first oxygen addition for cyclohexyl needs in-depth exploration from a stereochemical perspective. Note that initial adducts can convert to chair-equatorial W1Ce and twist-boat-equatorial W1Te conformers via inversion-topomerization, and the conformation mechanism will be investigated in the following sections to elucidate the structural-dependent properties for oxidation reactions.

For clarity, cyclohexylperoxy (ROO•, W1) and 2-, 3-, 4-hydroperoxycyclohexyl (QOOH, W2–W4) species are labeled as W1, W2, W3 and W4 individually, followed by the uppercase letter (C for “chair” vs. T for “twist-boat ring form”), indicating the position of the radical site relative to the side chain, and then lowercase (a, e, or i) to distinguish axial, equatorial vs. isoclinal side chain, respectively. For transition states, half-chair and boat configurations are indicated by the uppercase (H and B), followed by numbers, like B1 in Fig. 3. Table 1 tabulates the energetics for reactants, transition states, intermediates and end products computed by the two energy schemes, G4 and CBS-QB3, and Table 2 compares the present theoretical energetics for key stationary points on PES to those by CBS-QB3⁴ and G2.¹⁵ Note that energies at the CBS-QB3 level are generally higher than those at the G4 and G2 levels, and the energy gap is roughly less than 2 kcal mol⁻¹ for all studied species. Given the good performance in alkylcycloalkyl radicals,²⁵ the G4 values are used as the reference in kinetic calculations and discussions that follow. Cartesian coordinates and ro-vibrational frequencies in optimized structures for all species are provided in ESI-II (ESI†).

Table 2 Calculated energies of transition states in subsequent isomerization and dissociation for ROOH species relative to cyclohexyl in chair + O₂ at 0 K (kcal mol⁻¹)

	CBS-QB3	G4	Knepp^a ^15	Zou^b ^4		CBS-QB3	G4	Knepp^a ^15	Zou^b ^4
a Calculated at the modified G2(MP2)//B3LYP level. b Calculated at the CBS-QB3//B3LYP level.
R + O 2					TS10Ce	−6.6	−6.2	−4.3
RCe + O2	0.0	0.0	0.0	0.0	TS10Ta	−6.6	−4.6
RTi + O2	4.0	4.1	4.1		TS10Te	−7.2	−5.0
1,2-H transfer					TS10Ti	−6.6	−4.6
TS2Ce	19.3	20.0			1,2-Epoxycyclohexane
TS2Te	17.5	18.3			TS7Ca	−12.9	−9.6	−9.3	−12.9
TS3Ca	18.6	19.4			TS7Ce	−12.9	−9.0	−6.6
TS3Ce	18.1	18.8			TS7Ta	−12.9	−9.6
TS3Te	18.1	18.8			TS7Te	−12.9	−9.6
1,4-H transfer					TS7Ti	−10.9	−7.6
TS1Ca	−5.2	−3.1	−3.7	−5.2	1,3-Epoxycyclohexane
TS1Ce	−3.9	−1.7	−2.5		TS9Ca	−3.8	0.0	0.1	−3.8
TS1Ta	−1.7	0.5			TS9Ta	−3.4	0.4
TS1Te	0.2	2.5			TS9Ti	−3.4	0.4
TS1Ti	−0.7	1.6			1,4-Epoxycyclohexane
1,5-H transfer					TS14Ca	−6.6	−2.9	−2.7	−6.6
TS4Ca	−7.1	−4.7	−10.1	−11.6	TS14Ce	−6.6	−2.9
TS4Ta	−9.7	−6.9			TS14Ti	−6.6	−2.9
TS4Ti	−9.7	−6.9			Cyclohexanone
1,6-H transfer					TS8Ce	1.9	4.5
TS5Ti	−6.5	−4.0			W1Ca → P8			3.6	2.1
W1Ca → W4Ca				−7.6	β-C–C scission
W1Ce → W4Ce			−5.2		TS11Ce	6.2	8.0
Concerted elimination					TS11Te	8.5	10.6
TS6Ca	−5.1	−4.0	−5.5	−5.1	TS13Te	9.0	11.0
TS6Ce	−4.8	−3.9	−5.5		W4Ca → P7				6.7
TS6Ta	−5.1	−4.0			5-Hexenal
TS6Te	−5.1	−4.0			TS12Ca			−1.0	1.3
TS6Ti	−5.1	−4.0			TS12Ce	−1.4	3.3
β-C–O scission					TS12Te	4.5	9.2
TS10Ca	−7.6	−5.6	−4.5	−7.6

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3.2 Inversion-topomerization mechanisms for ROO• and QOOH

This work presents a comprehensive conformational analysis for ROO• and QOOH radicals. Zero-point inclusive energetics for all stationary points on inversion-topomerization PESs are listed in Table 1. For simplicity, quantum chemical results for ROO• are briefly discussed. As depicted in Fig. 2, the conversions for the three twist-boat conformers, i.e., W1Te, W1Ti and W1Ta, proceed through one topomerization circle following the sequence Te–B7–Ti–B6–Ta–B5–Ta–B6–Ti–B7–Te–B4–Te. This process causes the –OO• moiety to switch from the pseudo-equatorial to isoclinal and then to pseudo-axial positions. Inversion of Te can transform to Ce and Ca, and Ta merely connects to Ca, following the sequence Ce–H1–Te–B4–Te–H2–Ca–H3–Ta–B5–Ta. Note that half-chair transition states in inversion have high energies of 10.6–11.2 kcal mol⁻¹, of which the larger steric repulsion arises from five –CH₂– moieties in a quasi-plane. The boat transition states in topomerization have lower energies in the range of 5.5–6.3 kcal mol⁻¹. Thus, inversion dominates the whole conformational mechanism for cyclohexylperoxy. Moreover, energies for Ce and Ca chair conformers are clearly lower than those for Ta, Te, and Ti twist-boat ones with 5.5–6.3 kcal mol⁻¹. This phenomenon correlates with theoretical results for C₆H₁₁O₂ complexes at both the G2¹⁵ and CBS-QBS⁴ levels.

Fig. 2 Conformational inversion-topomerization process of W1. Energies (0 K, kcal mol⁻¹) for various conformers and transition states relative to W1Ce at the G4 level.

The energetics of the five stable conformers were employed to calculate their temperature-dependent populations using the Boltzmann distribution.² As displayed in Fig. 3, two chair conformers, i.e., W1Ce and W1Ca, are the most abundant from 200–2000 K. They contribute to 99.5% at 500 K, then decrease to 92% at 1000 K, and 73.6% at 2000 K. At higher temperatures above 2000 K, W1Ca and W1Ce have a similar population. However, the rate-limiting inversion between them must proceed through twist-boat W1Te via topomerization, implying the coexistence of all conformers at the low-T oxidation zone. For W2–W4, the presence of a radical site in the cyclic ring could alter the molecular symmetry and significantly reduce steric repulsion, leading to the cyclic ring being more easily deformed and associated with lower conversion barriers. Note that the W4Ta conformer is not stable enough to maintain the –COOH group in the pseudo-axial position.

Fig. 3 Equilibrium populations for W1 conformers over 200–2000 K obtained by Boltzmann distributions.

3.3 Potential energy surface for cyclohexyl + O₂

Fig. 4 displays the multi-step PESs for cyclohexyl + O₂ at the G4 level. The conformer-specific channels were scrutinized and explored for initial oxidation reactions according to the distinct conformers identified for W1–W4. As mentioned earlier, there are three barrierless paths for O₂ additions, i.e., one from RCe to W1Ca and two other paths from RTi to W1Ti and W1Ta. Afterwards, their axial and isoclinal conformations favor 1,5-H transfers to trigger chain branching reactions via TS4. Only W1Ti can undergo a 1,6-H transfer reaction via a slightly higher barrier TS5 of −4.0 kcal mol⁻¹. Notably, the equatorial –OO• groups in W1Ce and W1Te cannot relocate the radical site to the ring via 1,5- and 1,6-H transfers, which is consistent with previous findings for cyclohexyl + O₂^3,4,15 and alkyl-cyclohexane.¹ However, 1,4-H transfers for W1 conformers fail to exhibit structural dependence via TS1 with the highest barriers of −3.1 to 2.5 kcal mol⁻¹. The same situation occurs with concerted elimination through five diacritical routes to yield cyclohexene (P1) and HO₂. Cyclohexanone (P8) + OH is produced from W1Ce via TS8 with the highest barrier of 41.4 kcal mol⁻¹ referenced to W1Ce.

Fig. 4 Potential energy surface of cyclohexyl + O₂ through ROO• and QOOH wells at the G4 level. The cyclohexyl in chair + O₂ is set to have zero energy. Red, green, blue, pink and yellow colors represent Ca, Ce, Ta, Te and Ti conformers, respectively.

Notably, as also pointed out by Knepp et al.,¹⁵ all the above discussion assumes that the rapid inversion-topomerization maintains equilibrium populations among all conformers since conformational barriers are significantly lower than chemical reaction barriers for W1 to W4, as shown in Fig. 3 and 4. In this work, this assumption will be verified by the following kinetic predictions. Furthermore, theoretical results also suggest that the global minimum of the cyclic species should initially shift to a specific configuration to facilitate a conformer-specific reaction. That is, the coexistence of each conformer in W1–W4 enables all strong structural-dependent channels to become accessible. Therefore, omitting certain conformers not only introduces errors in calculating partition functions but also excludes important conformer-specific reaction pathways from the overall mechanism.

All five conformers of 2-hydroperoxycyclohexyl (W2) can decompose into cyclohexene (P1) + HO₂ and 1,2-epoxycyclohexane (P3) + OH via TS10 and TS7, with −6.2 to −4.6 kcal mol⁻¹ and −9.6 to −7.6 kcal mol⁻¹ lower energies than those of the reactants, respectively. In comparison, 1,2-H transfer from W2 to W3 is only available in the equatorial –COOH group with fairly high barriers of 18.3 and 20 kcal mol⁻¹. Similarly, 1,2-H transfers for 3-hydroperoxycyclohexyl (W3) have the largest barriers to 4-hydroperoxycyclohexyl (W4). Its next lower barriers are for ring-opening reactions via TS11 and TS12 to yield P6 and P5 + OH, respectively. It is noted that ring-opening reactions can proceed with W3Ce, W3Ca and W3Te conformers via TS12, among which the axial –COOH group in the chair conformation greatly reduces steric repulsion raised by C–C fissions.^4,15 OH elimination for W3 is most favored by W3Ca, W3Ta and W3Ti via TS9 of 0–0.4 kcal mol⁻¹ because the axial and isoclinal –COOH groups are more conducive to forming a four-membered ring in P2. Additionally, OH elimination overwhelmingly dominates W4 consumption to produce a five-membered ring in P4 with the lower energy barrier TS14, whereas a ring-opening reaction channel from W4Te proceeds via TS13 with an energy barrier of 11 kcal mol⁻¹. Consequently, H-transfers for W1 facilitate chain branching by shifting the radical site to W2, W3 and W4, especially for 1,5-H transfer. Also, the eliminations for W1 to W4 are more likely to control the autoignition process by releasing OH and HO₂ active radicals. The barriers for 1,2-H transfers and ring-opening reactions are roughly 20 and 10 kcal mol⁻¹ above the reactants of R1Ce + O₂, and therefore, negligible in the currently evaluated conditions.

4. Kinetic predictions

4.1 Kinetic analysis for initial addition

As mentioned above, three entrance addition pathways for O₂ onto cyclohexyl are barrierless, with the PESs divided by transition states that vary significantly as a function of temperature.¹⁵ Thus, a variational approach to transition state theory is introduced to calculate the optimal transition state at different temperatures.⁶Fig. 5(a) illustrates smooth minimum energy pathways for O₂ addition over R_CO distances of 2.0–3.2 Å, evaluated by the CASPT2(7e,5o)/CBS//CASSCF(7e,5o)/cc-pVDZ method. The predicted energy curves correlate well with the active space of separations. Notably, the orientations of optimal transition states, employed to minimize high-pressure limit rate coefficients, fall within the range of 2.2–3.0 Å. As the temperature increases, the weaker enthalpic effect shifts the transition state to a configuration with a shorter R_CO separation between two reactants. This trend aligns with the previous results for phenyl + O₂³⁹ and iso-pentane + O₂.⁶ Additionally, compared to the preferred chair-axial conformation, steric repulsion in the twist-boat results in a more distant orientation for molecular oxygen in an optimal transition state. In addition, RTi arranges transition state configurations with separations of 3.0 Å and 2.5 Å at 200 K and 800 K, while the corresponding separations for RCe are located at 2.8 Å and 2.4 Å, respectively.

Fig. 5 (a) Interaction potential for O₂ addition onto cyclohexyl along the minimum energy pathway. The energy of cyclohexyl in chair + O₂ is set to zero, indicated by a light dashed line in pink; and (b) high-pressure limit rate coefficients for three entrance pathways computed in this work, along with former theoretical calculations^4,5,15 and experimental determinations.^9,17

Fig. 5(b) illustrates the CASPT2 VRC-TST predictions for conformer-specific addition rate coefficients at the high-pressure limit (HPL) within 200–2000 K, showing negative temperature coefficient (NTC) behavior as mentioned in the previous studies.^4,5,15 The predicted rate at 300 K for RCe + O₂ to W1Ca, i.e. 1.27 × 10⁻¹¹ cm³ molecule⁻¹ s⁻¹, aligns well with (1.4 ± 0.2) × 10⁻¹¹ cm³ molecule⁻¹ s⁻¹ at 298 ± 2 K and 0.0046 atm¹⁷ and (1.3 ± 0.2) × 10⁻¹¹ cm³ molecule⁻¹ s⁻¹ at 296 ± 2 K and 0.99 atm⁹ acquired by laser scintillation photolysis and pulse radiolysis experiments, respectively. Our observation is also consistent with the viewpoint by Zou et al.⁴ that the kinetics for cyclohexyl + O₂ addition are insensitive to ambient pressure at the low-temperature end, as clearly demonstrated in Fig. 6(b). However, as seen from Fig. 5(b), the present rate coefficients generated for W1Ca are 1.37–5.48 times larger than those computed by Zou et al.⁴ and 0.76–1.70 times as those by Knepp et al.,¹⁵ whereas the kinetics for W1Ta generation from RTi + O₂ are coincident with the kinetics by Knepp et al.,¹⁵ except the slight divergences below 350 K and above 1300 K. Therefore, for barrierless addition reactions, the kinetic prediction strongly relies on energetic and structural parameters for optimal transition state via variational minimization along with interacting potential, and the discrepancies in kinetics may result from different best transitional structures selected in computations.³⁹ However, the underlying reason for this cannot be explicitly ascertained because the minimum interaction energies for RCe + O₂ to W1Ca have not been disclosed in detail in previous studies.^4,5,15

Fig. 6 (a) Boltzmann-weighted rate coefficients at the high-pressure limit for three entrance channels, and (b) temperature-dependent rate coefficients for RCe + O₂ to W1Ca addition channel at 0.001–1000 atm.

The HPL rate coefficients for W1Ti are 1.04 to 1.75 times higher than those for W1Ta above 300 K due to the lower barriers for variational transition states, as shown in Fig. 5(a). This pathway to W1Ti is significant within the range of 300–850 K, while W1Ca formation dominates at high temperatures, facilitated by the increasingly significant entropic effect in computing the density of states. This finding contradicts the former assumption that the twist-boat form in cyclohexyl would increase the rate coefficients by a factor of 1.3 compared to the chair form over the entire temperature range.¹⁵ Thus, the contributions of the two cyclohexyl conformers to the overall addition kinetics are evaluated according to their equilibrium populations. Fig. 6(a) presents the Boltzmann-weighted HPL rate coefficients for three addition pathways that align with those for the formation of W1Ca, similar trends are observed at varying pressures, as shown in Fig. S2 of the ESI-I (ESI†). Thus, these kinetic results could be readily used in modeling simulations. Additionally, when the pressure exceeds 1 atm (Fig. 6(b)), the rate coefficients for the formation of W1Ca converge to those of HPL, and they coincidentally cover the NTC region where the core low-T oxidation reactions proceed.

4.2 Kinetic analysis for ROO• and QOOH consumption

Fig. 7 plots the HPL rate coefficients for the evolution of ROO• (a) and ROOH originating from diverse conformers in W1 to W4. Kinetics reveal that the rapid equilibrium between conformers dominates over chemical isomerization and dissociation. This gives the direct theoretical evidence for the coexistence of distinct conformers within the range of 200–2000 K and requires reexamination of the previous assumption that only chair conformer existed,^4,15 ignoring the contributions of conformer-specific pathways to one reaction type, e.g., 1,5-H transfers favored by W1Ta and W1Ti shown in Fig. 7(a). Notably, 1,5-H transfers from W1Ca, W1Ta and W1Ti play a leading role in the consumption of W1 converting to W3 below 600 K due to their lower barriers. As the temperature increases, the more influential entropy effect makes the concerted elimination (P1 + HO₂) from three twist-boats significant. However, W1Ca and W1Ce are not favorable for concerted eliminations mainly due to their relatively lower energies, as shown in Fig. 4. The 1,6-H transfer from W1Ti becomes competitive at the NTC zone between 500 and 1000 K. Moreover, the 1,2- (P3), 1,3- (P2) and 1,4-epoxycyclohexane (P4) cyclization reactions individually are the major consumption pathways for W2, W3 and W4. Compared to 1,2-epoxycyclohexane cyclization, the β-C–O bond cleavage in W2 is less competitive but has larger rate coefficients than those for 1,3- (P2) and 1,4-epoxycyclohexane (P4) generated in Fig. 7(b). Therefore, the competition among these reactions should be further examined at evaluated temperatures and pressures.

Fig. 7 Temperature-dependent rate coefficients at the high-pressure limit of exit consumption channels of ROO• (a) and ROOH (b) for cyclohexyl + O₂ reaction system.

Fig. 8 illustrates the Boltzmann-weighted rate coefficients at the high-pressure limit for the main consumption pathways considering all diverse conformers for W1–W4 compared with the previous kinetics.^{4,7,10,13,14,16,19} The predicted results agree well with the former studies for exit reactions in W1. However, a small deviation in kinetics is found for 1,5-H and 1,6-H transfers at the low-temperature end for the difference in energy barriers. In Fig. 8(d)–(f), the HPL rates for cyclization in W2 to W4 exhibit strong discrepancy, where the rates computed by Sirjean et al.¹⁴ with 10.1, 19.0 and 16.7 kcal mol⁻¹ barriers are two or three orders of magnitude higher than those by Cavallotti et al.¹⁶ with 15.4, 23.4, and 20.7 kcal mol⁻¹ barriers, respectively. Our calculated activation energies for the three cyclization reactions are 9.5–12.5, 19.7–22.6, and 16.4–19.9 kcal mol⁻¹, resulting in kinetics lying between the above-mentioned results. The barriers for 1,2-epoxycyclohexane cyclization with TSs in the chair-axial and chair-equatorial forms by Knepp et al.¹⁵ are higher (i.e., 12.79 and 14.95 kcal mol⁻¹) than those with 9.2–12.5 kcal mol⁻¹ in Table 2, resulting in their lower rates than the current kinetics in Fig. 8e. Also, for the same reason, the rates of Buda et al.¹⁹ (at 1 atm) and Silke et al.⁷ are about three orders of magnitude lower than that in our calculations and by Zou et al.⁴ In general, the current kinetics for major channels are slightly smaller but still in good agreement with those by Serinyel et al.¹³ and Zou et al.⁴ Note that the rates for QOOH dissociations in Serinyel's model were computed by reducing those reported by Sirjean et al.¹⁴ by a factor of 10.

Fig. 8 Comparison of Boltzmann-weighted HPL rate coefficients for the main consumption channels in this work, non-pressure rate coefficients calculated by Sirjean et al.¹⁴ and Cavallotti et al.,¹⁶ HPL rate coefficients calculated by Zou et al.,⁴ rate coefficients calculated at a total density of 8.5 × 10¹⁷ cm⁻³ by Knepp et al.,¹⁵ rate coefficients in Silke model⁷ and Serinyel model,¹³ rate coefficients at 1 atm in Buda model,¹⁹ and rate coefficients at 0.02 atm fitted from experimental data by Handford-Styring et al.¹⁰

Fig. 9 displays the temperature-dependent rate coefficients and branching ratios for the major entrance and exit consumption channels in the cyclohexyl + O₂ system at 0.1 and 10 atm pressure. Kinetics at 1 and 100 atm are provided in Fig. S3 of ESI-I (ESI†). At evaluated pressures, the reactants via entrance addition prefer to flow into deep well W1 via collisional stabilization to form three more stable conformers, i.e. W1Ce, W1Ca and W1Ta. This is owing to the rapid conformational change of initial adducts (e.g., W1Ti) into those with lower energies. Furthermore, the preference for axial- and isoclinal conformations significantly promotes conformer-specific exit reaction channels, eventually resulting in W1Ca and W1Ta with slightly smaller rate coefficients and W1Ti only at higher temperatures. Note that W1Ta and W1Ca are separately formed below 350 K and 550 K, while the generation rates for W1Ce extend to 1350 K in Fig. 9(a). As the pressure increases, unreactive collision becomes more influential on ROO• stabilization. It gives rise to higher rate coefficients and a larger temperature range to form W1 conformers. However, as mentioned by Zou et al.,⁴ NTC behavior for entrance channels is elusive, and the well-merging behavior tends to occur at higher temperatures accordingly.

Fig. 9 Temperature-dependent rate constants (a) and (b) and branching ratios (c) and (d) for the main consumption channels in the cyclohexyl + O₂ system at 0.1 and 10 atm with Ar bath gas.

As depicted in Fig. 9(c) and (d), ROO• stabilization dominates the overall mechanism below 900 K at 0.1 atm and below 1600 K at 10 atm, which is in line with the former findings.^{4,7,10,13,14,16,19} Above 900 K, HO₂ elimination (P1) plays a leading role and competes with ROO• stabilization at 1100 K and 1 atm, as shown in Fig. S4 of ESI-I (ESI†). Among all subsequent consumption pathways, energetically excited ROO• species is most likely to undergo HO₂ elimination with the largest rate coefficients above 300 K, of which the barrier for TS6 lies at −4.0 to −3.9 kcal mol⁻¹ below that of the reactants. Also, competition exists between QOOH (i.e., W2 to W4) stabilization and OH elimination to form its relevant bimolecular products (OH + P2 to P4). It is noted that the rate coefficients for P3 are mainly three orders of magnitude higher than those for W2Te formed at 0.1 atm, but their maximum deviation is within a factor of 20, and the kinetics for W2Te become larger below 300 K at 10 atm. It also results from the fact that the energetically excited QOOH species prefer to stabilize in the well rather than decompose at high pressures and low temperatures. According to energy barriers characterized in G4, the present kinetics indicate that the relative importance for OH eliminations is 1,2- > 1,4- > 1,3-epoxycyclohexane cyclization and that for isomerization follows the order 1,5- > 1,6- > 1,4-H transfer, as demonstrated in Fig. S4 and S5 of ESI-I (ESI†).

5. Kinetic simulation

As stated in the Kinetic modeling section, Boltzmann-weighted rate coefficients for major entrance and subsequent consumption channels in the cyclohexyl + O₂ system are incorporated into the previous model by Zou et al.¹¹ to establish a new model. It is used to simulate the low-temperature oxidation of cyclohexane in JSR experiments at 1.04 bar, ϕ = 0.25 and τ = 4 s by SVUV-PIMS and GC-MS detections,¹¹ and laminar burning velocities of cyclohexane/air flames at 1 atm with unburned mixture temperatures of 298, 353, 358 and 398 K.^13,40–42 Experimental and simulated concentration profiles for reactants and C₀–C₆ major products detected by GC-MS are given in Fig. 10 and those by SVUV-PIMS are plotted in Fig. 11 and Fig. S6 of ESI-I (ESI†). The present model reasonably predicts NTC behavior and low temperature oxidative reactivity for cyclohexane detected by GC-MS in Fig. 10(a). Also, our simulated results perfectly reproduce the molar fraction distributions for cyclohexene (P1) and cyclohexanone (P8) by comparing with those predicted by the models of Zou et al.¹¹ and Serinyel et al.¹³ in Fig. 10(e) and (f).

Fig. 10 Mole fraction profiles of cyclohexane, four C₆H₁₀O isomers, and cyclohexene during low-temperature oxidation of cyclohexane at 1.04 bar, ϕ = 0.25, and τ = 4s. Empty triangle symbols are JSR GC-MS data by Zou et al.,¹¹ in comparison to the simulation results using the new model (solid) and former kinetic models by Zou et al.¹¹ and Serinyel et al.¹³

Fig. 11 Mole fraction profiles of the C₀–C₆ products in the low-temperature oxidation of cyclohexane at 1.04 bar, ϕ = 0.25, and τ = 4s. Empty triangle symbols are JSR SVUV-PIMS data by Zou et al.,¹¹ in comparison to the results using the new model (solid) and former kinetic models by Zou et al. (dotted)¹¹ and Serinyel et al. (dashed).¹³

As seen from Fig. 12, the formation of cyclohexene (P1) consumes 69.2% cyclohexylperoxy (ROO•), favored by three entrance additions and 0.06% cyclohexyl via the formally direct pathway at 775 K, while it only consumes 50% cyclohexylperoxy and 0.4% cyclohexyl, as reported by Zou et al.¹¹ The present results for cyclohexanone (P8), consuming 6.56% cyclohexylperoxy at 575 K, are roughly two times smaller than those by Zou et al.¹¹ with 13.4% consumption of cyclohexylperoxy. Moreover, the concentration profiles predicted for 5-hexanal (P2 and P5), 1,2- (P3) and 1,4-epoxycyclohexanes (P4) are closer to experimental data in comparison with the previous simulations in Fig. 10(b)–(d). Note that these three products are primarily generated through controlled 1.5-, 1,4- and 1,6-H transfers, followed by HO eliminations, respectively. As mentioned before, the rate coefficients for the 1,5-H transfer are lower than those reported by Zou et al.¹¹ and Serinyel et al.,¹³ resulting in the smaller mole fractions for 5-hexanal (P2 and P5) in Fig. 10(b). Furthermore, this model also exhibits good performance in predicting concentrations and trends for small end products (CO and C₂H₄), ketones (CH₂O and C₃H₄O), and hydroperoxides (HCOOH, C₂H₅OOH and C₆H₁₂O₂) in Fig. 11, as well as other major intermediates and products in Fig. S6 of ESI-I (ESI†).

Fig. 12 ROP analysis for low-temperature cyclohexane oxidation at 1.04 bar, 575 K (green), 675 K (blue), 775 K (orange), φ = 0.25, and τ = 4s.

Fig. 12 displays reaction pathway analysis for cyclohexyl + O₂ at 675, 575, and 775 K, illustrating reactive flux in typical oxidation regions of NTC at low and high temperatures, respectively. H-Abstractions by the OH radical dominates the conversion of cyclohexane to cyclohexyl with 96.5% contribution at 585 K, while HO₂ and O abstractions become more significant at 685 K and 785 K. Entrance O₂ addition to ROO• entirely controls cyclohexyl consumption, and the flux via the formally direct path to cyclohexene (P1) and HO₂ is within 0.1% over 575–775 K. Subsequently, 14.2–69.2% ROO• decomposes to cyclohexene (P1) and HO₂via concerted elimination, which accurately predicts the highest molar fraction of P1 in Fig. 10(f). Regarding the importance of H-transfers, ROO• separately transforms to W2, W3 and W4 with the ratios of 0.06%, 26.6% and 3.56% at 775 K, and thus the molar fractions of their subsequent products are smaller than those for P1.

Furthermore, Fig. 13 depicts the laminar burning speeds for cyclohexane/air mixtures measured at 1 atm with variational equivalence ratios,^13,40–42 along with numerical results using the new model and Serinyel's model.¹³ Compared to the results by Serinyel et al.,¹³ the present simulations are in closer agreement with experimental data, especially in the fuel-lean zone. Although this model slightly over- and under-predicts the fuel-rich and peak burning velocities, respectively, the largest discrepancy of 4.3 cm s⁻¹ is less than the deviation from entire over-predictions using Serinyel's model. Consequently, the overall agreement between model simulations and measurements indicates the reliable performance of this new model in kinetic simulations.

Fig. 13 Laminar flame velocities of cyclohexane/air flame with unburned gas temperature of (a) 298 K, (b) 353 and 358 K, and (c) 398 K at 1 atm. Symbols represent measurements by Ji et al. (circle),⁴⁰ Davis and Law (diamond),⁴² Wu et al. (star)⁴¹ and Serinyel et al. (triangle).¹³ Lines are simulation results using current (solid) and former model by Serinyel et al. (dashed).¹³

6. Conclusions

Low-temperature oxidation chemistry for cyclohexane was thoroughly examined with full consideration of the conformational effect on oxygen addition reactions, and a detailed kinetic model was subsequently proposed by incorporating Boltzmann-weighted rate coefficients. Energetics and kinetics for the addition of O₂ to cyclohexyl were investigated by combining quantum chemical computations and RRKM/master equation analysis. Kinetics for barrierless entrance addition were explored using direct CASPT2-based VRC-TST theory. Phenomenological rate coefficients and branching ratios for primary oxidation reactions were determined in the range of 200–2000 K and 0.001–100 atm. Indeed, there are three entrance channels for O₂ addition to yield the chair-axial, twist-boat-axial, and twist-boat-isoclinal adducts. Their configurations are favorable for 1,5-H transfers, and only twist-boat-isoclinal preference can proceed with 1,6-H transfer. Conformation-dependent properties are observed for exit dissociation channels, among which ring-opening reactions are more likely to occur in equatorial configurations. Kinetics demonstrate the relative significances for isomerization and OH eliminations following 1,5- > 1,6- > 1,4-H transfer and 1,2- > 1,4- > 1,3-epoxycyclohexane cyclization, respectively. ROO• stabilization and HO₂ elimination separately predominate the overall consumption mechanism at correspondingly low and high temperatures. Nevertheless, inversion-topomerization conformations among all distinct conformers in each ring radical are exceedingly rapid, with rate coefficients several orders of magnitude larger. Such a conformational process establishes the equilibration of conformers and facilitates taking into account their partition function contributions to kinetics in terms of thermodynamics. Hence, Boltzmann-weighted rate coefficients for each reaction type were obtained based on the equilibrium populations of all conformers in the ROO• and QOOH species using the Boltzmann distribution. Then, they were utilized to update the former kinetic model for low-temperature cyclohexane oxidation. A new model can accurately predict the literature measurements for JSR and laminar speed experiments, especially for mole fraction distributions of primary oxidation species, such as 1,4-epoxycyclohexane, cyclohexene and cyclohexanone.

Author contributions

Hui-Ting Bian: supervision, writing – review & editing, methodology, formal analysis, and funding acquisition. Yang Wang: writing – original draft, investigation, methodology, and formal analysis. Shi-Hao Feng: conceptualization. Long Zhao: formal analysis and methodology. Wen-Chao Lu: formal analysis, methodology, and writing – review & editing. Hui-Ling Jiang: funding acquisition and methodology. Kai-Yuan Li: writing – review & editing and formal analysis.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The data supporting this article have been included as part of the ESI.†

Acknowledgements

This work was supported by the National Natural Science Foundation of China (no. 52476100), China Postdoctoral Science Foundation (no. 2024M750185), the Opening project of Key Laboratory of Microgravity, Institute of Mechanics, Chinese Academy of Sciences (no. NML202406), the Key technologies for safe and high-efficient exploitation and comprehensive utilization of deep carbonate geothermal resources in Xiongan New Area (no. 2023XAGG0061), the Interdisciplinary Research Project for Young Teachers of USTB (no. FRF-IDRY-24-013), the Fundamental Research Funds for the Central Universities (no. QNXM20250007). The authors appreciate Prof. Chi-Min Shu and Dr. Lili Ye for their help.

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Footnote

† Electronic supplementary information (ESI) available: ESI-I for temperature- and pressure-dependent rate coefficients for major channels in low-temperature cyclohexane oxidation, and experimental and simulated mole fractions for C₀–C₆ products in JSR SVUV-PIMS results are provided. ESI-II for geometries and frequencies for all studied species are provided. Rate calculation input file, chemical kinetic mechanism, thermodynamic and transport data are provided. See DOI: https://doi.org/10.1039/d5cp01550b

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