Kinetic and mechanistic investigations of the thermal decomposition of methyl-substituted cycloalkyl radicals

Abstract

A systematic theoretical study on the thermal decomposition of 2-Me-cyclobutyl, 2-Me-cyclopentyl and 2-Me-cyclohexyl radicals is performed using the high-level ab initio CBS-QB3 and CCSD(T) quantum chemical calculations. The calculation reveals that the detailed reaction mechanism of the thermal decomposition of these cyclic alkyl radicals incorporates ring opening, vinyl rearrangements (exocyclization), beta-site C–C bond cleavage and H-elimination processes. The standard reaction enthalpies (Δ_rH⁰₂₉₈) and Gibbs free energies (Δ_rG⁰₂₉₈) for each elementary reaction involved in the 2-Me-cyclohexyl radical reactive system are also determined with the composite CBS-QB3 method. All the investigated vinyl rearrangements reactions are exothermic and spontaneous, while the ring opening, C–C bond scission and H-elimination processes are endothermic and nonspontaneous. Among all the investigated elementary reactions, the exocyclization processes are kinetically accessible and readily proceed (due to their significantly lower barrier and they are highly exothermic). Compared with the barrier heights for the distinct vinyl rearrangement pathways in these cyclic alkyl radicals, it can be found that they decrease in the order of 1,3- > 1,2- > 1,4-vinyl transfer. The branching ratios are evaluated at different temperatures on the basis of the quasi-steady state approximation (QSSA). The calculated result shows that the 1,2-, 1,3- and 1,4-vinyl rearrangement reactions are advantaged at low temperature, while the formation of a cycloalkene is favoured at high temperature.

---

1. Introduction

Many studies have been performed on the reactions of straight and branched-chain alkanes in a wide range of temperatures and pressures,^1–3 while the chemistry of cyclic hydrocarbons has been investigated to a much lesser extent.^4,5 Cyclic hydrocarbons, particularly cycloalkanes, constitute an important source in practical fuels.⁶ Cycloalkyl radicals are key intermediate species in the thermal decomposition processes of cycloalkanes.⁴ They result from the initial steps of hydrocarbon pyrolysis through C–H bond fission or abstraction H-atom reactions by means of small chemical species (such as H, OH, CH₃ radicals etc.) attacking parent molecules. Herein, the mechanistic and kinetic properties of cycloalkyl radicals are important to further improve our understanding of the thermal decomposition processes of hydrocarbons.

To date, no experimental evidence on the thermal decomposition routes of cyclic alkyl radical has been reported. Such low stability, short lifetime and highly reactive radicals are excessively difficult to be determined and characterized in the gas phase experimentally. In 2006, Orme et al.⁵ investigated the oxidation and pyrolysis of methylcyclohexane (MCH) at 1200–2100 K and 1.0, 2.0, and 4.0 atm, by means of high temperature shock tube and flow reactor. The author found that the major pyrolysis products contain methane (CH₄), ethylene (C₂H₄), propene (C₃H₆), 1,3-butadiene (1,3-C₄H₆) and isoprene (C₅H₈). And they also proposed a detailed chemical kinetic mechanism on the pyrolysis of MCH. Similar product distributions were also drawn in Zeppieri et al. studies⁷ that the high temperature pyrolysis of pure MCH and MCH/toluene blends are performed in the princeton turbulent flow reactor. All of these works have provided insight into the thermal decomposition behavior of cycloalkanes and their radicals. However, the estimated rate coefficients of elementary reactions in Orme's study⁵ are not accurate adequately, because they adopted that the rate coefficients of chemical reactions of similar nature are equal to those reported by Curran et al. for n-heptane⁸ and isooctane⁹ oxidation. Moreover, the dominant reaction pathways have not been mentioned during the processes of MCH and its radicals pyrolysis.

To the best of our knowledge, the detailed reaction mechanisms of second reactions of cycloalkyl radicals have not been reported so far. Sirjean et al.⁴ studied the beta site C–C and C–H bonds breaking reactions for cyclic alkyl radicals from three-membered to seven-membered rings, with and without a lateral alkyl chain by means of quantum chemical calculations at the CBS-QB3 level of theory. It is concluded that the increase of the activation energy as the π bond is being formed in the ring in contrast to the cases in which the π bond is formed on the side chain. Sirjean et al.⁶ also investigated the gas phase unimolecular decomposition of cyclobutane, cyclopentane and cyclohexane molecules, and considered the formation of biradical species. The result showed that the main part of ring strain energies contained in the cyclic reactants is removed from the cycloalkanes to the transition states. Wang et al.¹⁰ studied the kinetics of a series of homoallylic/homobenzylic rearrangement reactions under combustion conditions. They mainly considered the 1,2-, 1,3- and 1,4-vinyl/phenyl migration for homoallylic and homobenzylic radicals, and compared theirs product yield. The calculation indicated that the 1,2-vinyl/phenyl migration is particularly important for the kinetics of unimolecular reactions of homoallylic radicals, whereas the 1,3- and 1,4-vinyl/phenyl migration channel play an insignificant role under combustion conditions. All of these works provide useful information for investigating the pyrolysis of cyclic alkyl radicals. Unfortunately, the vinyl migration process of cycloalkyl radicals is neglected in Sirjean's study,⁴ which is an very important reaction type in pyrolysis processes of hydrocarbon, especially at low temperature. Moreover, they merely considered the initial ring opening steps, not mentioned the second reactions of the radicals formed and not compared their relative importance.

In the recent work, we perform systemically theoretical investigations about the thermal decomposition of 2-Me-cyclobutyl, 2-Me-cyclopentyl and 2-Me-cyclohexyl radicals at the high-level composite CBS-QB3 and the coupled-cluster CCSD(T) approaches. The calculations are laid out as follows: firstly, the pyrolysis mechanism of these radicals, including the ring opening, vinyl rearrangements (exocyclization), the beta site C–C bond cleavage and H-elimination reactions, are explored. Secondly, the standard reaction enthalpies (Δ_rH⁰₂₉₈) and Gibbs free energies (Δ_rG⁰₂₉₈) for every elementary reaction are calculated. Thirdly, the high-pressure limit (HPL) rate coefficients of conventional transition state theory for individual elementary reaction are determined at 500–2500 K. Finally, the branching ratios of thermal decomposition of these radicals are predicted at different temperatures. The computational results, along with detailed discussions, will be presented in Section 3 and main conclusions will be drawn in Section 4.

2. Computational approach

All the electronic structure calculations that are discussed in the present investigation are carried out using the Gaussian 09 quantum chemistry code.¹¹ The geometry optimizations for all species are performed with an unrestricted B3LYP functional, which has been successfully applied to the study of organic molecules.¹² Moreover, the effectiveness of B3LYP in modeling radical reactions has been proposed in previous studies.^13–16 The basis set 6-311G(2d,d,p), which is reasonably accurate and computationally affordable, is adopted for all stationary points calculations. The vibrational frequency calculations are performed to verify that the optimized structures are either real local minima (no imaginary frequencies) or first order saddle points (just one imaginary frequencies) and to estimate the thermodynamic quantities. Intrinsic reaction coordinate (IRC) calculations^17–20 are traced at the same level of theory to confirm that the located transition state structures indeed connect to the designated reactants and products. Then, to obtain reliable energies of each species on the potential energy surface (PES), the single point calculations are performed at the CBS-QB3 and CCSD(T) levels of theory. The composite CBS-QB3 methodology involves a five-step calculation: (i) a geometry optimization and a frequency calculation (scaled by 0.99 as recommended by Montgomery et al.²¹) at the B3LYP/6-311G(2d,d,p) level of theory;²² (ii) CCSD(T)/6-31+G(d′) energy corrections; (iii) MP4SDQ/CBSB4 (CBSB4 = 6-31+G(d(f),p)) energy; (iv) MP2/CBSB3 (CBSB3 = 6-311+G(3d2f,2df,2p)) energy; (v) a complete basis set (CBS) extrapolation to correct the total energy.^14,23,24 The CBS-QB3 approach is chosen because it gives adequately accurate energies for C/H/O system, with a standard deviation of about 1.5 kcal mol⁻¹, and it is less computationally cost than the more recent and accurate ones, as G4.¹⁵ The coupled-cluster approach CCSD(T), involving single and double substitutions including perturbative corrections for the triple excitations,²⁵ is used to obtain more reliable energies based on the B3LYP geometries. T₁ diagnostics in the CCSD(T) energy calculations are considered to evaluate the reliability of the calculations for all stationary points involved in the above mentioned reaction mechanisms. They are all less than critical value 0.02 for the singlet species (see Tables S1–S3†), revealing that the CCSD(T) method employed provides an adequate description of the wave function.²⁶ The theoretical rate coefficients of conventional transition state theory for every elementary reaction are estimated over the temperature range of 500–2500 K. Tunneling effects are contained on the base of an one-dimensional asymmetric Eckart transmission factor.^27–29where κ(T) is the asymmetric Eckart tunneling factor, σ is reaction symmetry number, k_B is the Boltzmann constant. h is the Planck constant, Q^≠(T) is the partition function for the transition state, Q_A(T) and Q_B(T) are the partition functions for the reactants and E_a is the activation energy barrier. The total molar partition function includes translation (Q_trans), vibration (Q_vib), rotation (Q_rot), electronic (Q_ele) and torsional (Q_tor) partition functions (Q = Q_transQ_vibQ_rotQ_eleQ_tor).³⁰ The one-dimensional hindered rotor (1D-HR) partition function Q_tor is calculated by the following eqn (2).³¹where σ′ is symmetry number associated with that rotation, ε_i is the energy. The internal rotations of both reactant and transition state are investigated using the 1-D hindered rotor treatment. The hindrance potential for an internal rotor is obtained by relaxed potential energy scan with the step of 12° at the B3LYP/6-311G(2d,d,p) level. The quasi-steady-state approximation (QSSA) is employed to induce the overall rate coefficients. The rate coefficients are fitted to the modified three parameters Arrhenius expression eqn (3):

k = A × Tⁿ × exp(−E_a/RT) (3)

All kinetic calculations are evaluated by implementing VKLab program.³²

3. Results and discussion

The global flux diagram for the detailed reaction mechanism of 2-Me-cyclobutyl, 2-Me-cyclopentyl and 2-Me-cyclohexyl radicals is drawn in Scheme 1. As shown in Scheme 1, the detailed mechanism includes mainly the ring opening, vinyl rearrangements (exocyclization), beta C–C bond dissociation and H-elimination processes. The geometrical parameters for all stationary points involved in the 2-Me-cyclohexyl radical reaction system at the B3LYP/6-311G(2d,d,p) level together with the available experimental values, are depicted in Fig. 1. The expectation values of 〈S²〉 for all species are listed in Tables S1–S3,† after spin annihilation, the value for the open-shell systems is very close to the ideal value of 0.7500, indicating it can be negligible at the above depicted computation level. The standard reaction enthalpies (Δ_rH⁰₂₉₈) and Gibbs free energies (Δ_rG⁰₂₉₈) for every elementary reaction in Table 1 are estimated at the CBS-QB3 level, and are compared with the available literature values. The PESs for these cycloalkyl radicals reactions at the CBS-QB3 and CCSD(T) levels of theory, respectively, are constructed in Fig. 2–4. The full structural descriptions of all transition states are displayed in Fig. S1–S3 in the ESI.† The modified three parameter Arrhenius expressions for each elementary reaction rate coefficient in Table 2 are listed. The branching ratio of these radicals pyrolysis is calculated in Fig. 7 over the temperature range of 500–2500 K.

Scheme 1 The global flux diagram for the pyrolysis of 2-Me-cyclobutyl (a), 2-Me-cyclopentyl (b) and 2-Me-cyclohexyl (c) radicals (the prefix and postfix of the number represent the site of double bond and radical, respectively).

Fig. 1 Optimized geometries of all stationary points incorporated in 2-Me-cyclohexyl radical at the B3LYP/6-311G(2d,d,p) level along with the available experimental values (experimental values are indicated by a superscript a; bond lengths are in angstroms and bond angles are in degrees; red line represents the breaking bond).

Table 1 Thermodynamic data (kcal mol⁻¹) of 2-Me-cyclohexyl radical at the CBS-QB3 level^b

Channels	ΔrH^0298		ΔrG^0298
	Cal	Ref	Cal
a are the theoretical values taken from ref. 4 and 5.b The ring opening reactions include R3a, R3f and R3i; exocyclization reaction contain R3e and R3l; H-elimination reactions is R3h; the C–C bond scissions are remain reactions.
2-CH3-cyclohexyl → 4-CH3-5-C6H10-1 (R3a)	21.53		17.52
4-CH3-5-C6H10-1 → 2-CH3-3-C4H6-1 + C2H4 (R3b)	22.34		11.40
2-CH3-3-C4H6-1 → 1,3-C4H6 + CH3 (R3c)	18.61	17.52^a	8.02
2-CH3-3-C4H6-1 → C3H6 + C2H3 (R3d)	32.40	32.54^a	20.94
4-CH3-5-C6H10-1 → 2-CH3-c-C5H8-CH2 (R3e-cis)	−15.74		−13.14
4-CH3-5-C6H10-1 → 2-CH3-c-C5H8-CH2 (R3e-trans)	−15.80		−13.29
2-CH3-c-C5H8-CH2 → 6-C7H13-2 (R3f-cis)	13.85		10.76
2-CH3-c-C5H8-CH2 → 6-C7H13-2 (R3f-trans)	13.91		10.91
6-C7H13-2 → 3-C4H7-1 + C3H6 (R3g)	23.30		11.95
3-C4H7-1 → 1,3-C4H6 + H (R3h)	28.82	28.67^a	22.61
2-CH3-cyclohexyl → 5-C7H13-1 (R3i)	20.22		16.07
5-C7H13-1 → 3-C5H9-1 + C2H4 (R3j)	21.80	20.44^a	11.61
3-C5H9-1 → 1-C3H5-1 + C2H4 (R3k)	34.40		23.56
5-C7H13-1 → 2-C2H5-c-C5H8 (R3l)	−15.45		−13.76
2-CH3-cyclohexyl → c-C6H10 + CH3 (R3m)	23.21	24.47^a	11.16

---

Fig. 2 Potential energy profile of the thermal decomposition of 2-Me-cyclohexyl radical at the CBS-QB3 and CCSD(T) (italic) levels (the prefix and postfix of the number represent the site of double bond and radical, respectively; the ring opening reactions include R_3a, R_3f and R_3i; exocyclization reaction contain R_3e and R_3l; H-elimination reactions is R_3h; the remain reactions are C–C bond scission; relative energies are given in kcal mol⁻¹).

Table 2 The theoretical rate coefficients expression of 2-Me-cyclohexyl radical pyrolysis^a

Reactions	log A	n	Ea/R	Reactions	log A	n	Ea/R
a The ring opening reactions include R3a, R3f and R3i; exocyclization reactions contain R3e and R3l; H-elimination reactions is R3h; the C–C bond scissions are remain reactions; the unit of s^−1.
2-CH3-cyclohexyl → 4-CH3-5-C6H10-1 (R3a)	12.86	0.33	13 232	4-CH3-5-C6H10-1 → 2-CH3-3-C4H6-1 + C2H4 (R3b)	12.80	0.23	13 719
2-CH3-3-C4H6-1 → 1,3-C4H6 + CH3 (R3c)	11.91	0.29	12 478	2-CH3-3-C4H6-1 → C3H6 + C2H3 (R3d)	13.76	0.23	17 203
4-CH3-5-C6H10-1 → 2-CH3-c-C5H8-CH2 (R3e-cis)	10.64	0.02	3917	4-CH3-5-C6H10-1 → 2-CH3-c-C5H8-CH2 (R-3e-cis)	12.69	0.19	11 070
4-CH3-5-C6H10-1 → 2-CH3-c-C5H8-CH2 (R3e-trans)	10.80	0.02	3763	4-CH3-5-C6H10-1 → 2-CH3-c-C5H8-CH2 (R-3e-trans)	12.85	0.18	10 999
2-CH3-c-C5H8-CH2 → 6-C7H13-2 (R3f-cis)	12.61	0.14	10 037	2-CH3-c-C5H8-CH2 → 6-C7H13-2 (R3f-trans)	12.70	0.13	9969
6-C7H13-2 → 3-C4H7-1 + C3H6 (R3g)	13.16	0.26	13 755	3-C4H7-1 → 1,3-C4H6 + H (R3h)	11.04	0.64	13 910
2-CH3-cyclohexyl → 5-C7H13-1 (R3i)	13.07	0.29	12 951	5-C7H13-1 → 3-C5H9-1 + C2H4 (R3j)	13.22	0.25	13 551
3-C5H9-1 → 1-C3H5-1 + C2H4 (R3k)	13.88	0.19	17 732	5-C7H13-1 → 2-C2H5-c-C5H8 (R3l)	11.02	0.03	3574
2-CH3-cyclohexyl → c-C6H10 + CH3 (R3m)	13.50	0.33	14 294

---

3.1 Geometrical parameters and thermodynamic properties

Fig. 1 details the optimized geometries of all stationary points involved in 2-Me-cyclohexyl radical at the B3LYP/6-311G(2d,d,p) level of theory, as well as available experimental values. The NIST Standard Reference Database (http://cccbdb.nist.gov) values are chosen as a reference to assess the accuracy of the computational methodology employed through comparing the deviation of the bond lengths and angles.

As shown in Fig. 1, the calculated values of the bond lengths and angles are in good agreement with available experimental ones. The mean average deviations of bond lengths and bond angles between the calculated and experimental values are 0.01 Å and 0.87°, respectively. The largest deviations of bond lengths and bond angles is 0.02 Å for C=C bond in C₃H₆ molecule and 1.43° for ∠C–C–C angle in 1,3-C₄H₆ molecule. These calculated results reveal that the method employed is suitable to describe the geometries in the reaction mechanisms of the title reaction system.

The standard reaction enthalpies (Δ_rH⁰₂₉₈) and Gibbs free energies (Δ_rG⁰₂₉₈) are evaluated under the condition of 298 K and 1 atm.^16,33 The enthalpies of formations for partial species with available experimental values come from NIST Chemistry Webbook (http://webbook.nist.gov/chemistry) or (ref. 4 and 5). The Δ_rH⁰₂₉₈ and Δ_rG⁰₂₉₈ for each elementary reaction involved in 2-Me-cyclohexyl radical are calculated at the CBS-QB3 model chemistry, and the results are listed in Table 1.

As is readily apparent from Table 1, the calculated reaction enthalpies are in good agreement with available literature ones for reaction R_3c, R_3d, R_3h, R_3j and R_3m. The largest deviation is equal to 1.36 kcal mol⁻¹ (R_3j), suggesting the present CB3-QB3 approach is reasonable to discuss the thermodynamic property of the title reaction system. The conclusion is also supported by the pervious literature reported²⁴ that CBS-QB3 reproduces the experimental results and recommends as a reference where experimental values are not available. The exocyclization reactions (R_3e-cis, R_3e-trans and R_3l) are exothermic and spontaneous with releasing heat ∼16 kcal mol⁻¹, whereas the ring opening, H-elimination and C–C bond scission processes are endothermic and nonspontaneous with absorbing heat 14–34 kcal mol⁻¹. The reaction exothermicities for the distinct exocyclization channels are almost equivalent in our system studied.

In summary, the CBS-QB3 approach used provides adequately accurate geometrical parameters and thermodynamic values in the title reaction system. The exocyclization reactions are exothermic and spontaneous, while the ring opening, C–C bond scission and H-elimination processes are endothermic and nonspontaneous.

3.2 Reaction mechanisms

The radical chain mechanism is nowadays accepted for the pyrolysis of hydrocarbons.³⁴ According to this reactive mechanism, the 2-Me-cyclohexyl, 2-Me-cyclopentyl and 2-Me-cyclobutyl radicals undergoes the ring opening, exocyclization, beta C–C bond scission and H-elimination processes (Scheme 1). Fig. 2–4 present the PESs of the thermal decomposition of these radicals at the CBS-QB3 and CCSD(T) levels of theory. The full structural descriptions for all transition states are presented in Fig. S1–S3.† The exhaustive descriptions for the thermal decomposition processes of these radicals are discussed as follows.

As shown from Fig. 2–4, the calculated barriers by using the CCSD(T) (italic) method are in qualitative agreement with those from the CBS-QB3 results, although some stationary points have small apart in energy. Thus, in this work, unless otherwise mentioned, the energetic description obtained by CBS-QB3 model chemistry is applied to discuss in the subsequent analysis. As seen from Fig. 2, the barriers for exocyclization processes (R_3e-cis, R_3e-trans, and R_3l) are much lower than that of other pathways, in which the R_3e-cis and R_3e-trans have near identical energies (the difference is 0.25 kcal mol⁻¹). The barrier heights are attributed to the influence of low strain energy in the cyclic transition states of 1,4-vinyl migration reactions, which will be discussed detailedly in the following paragraph.

The initial steps of 2-Me-cyclohexyl radical pyrolysis include three pathways: the ring opening R_3a (C5–C6 bonds cleavage produces 4-Me-5-C₆H₁₀-1), R_3i (C2–C3 bonds scission forms 5-C₇H₁₃-1), and CH₃-elimination R_3m (leads to c-C₆H₁₀ + CH₃). These processes are accompanied by the barrier heights lies 27.77, 27.13 and 29.84 kcal mol⁻¹ above the total energy of the reactant. The result shows that the C–C bond cleavage on the ring is more advantage than that of the side chain. In the viewpoint of geometrical structures TS3a, TS3i and TS3m, as showed in Fig. S3,† the breaking C–C bond is elongated by 50.9, 48.0 and 46.2%, whereas the forming C=C bond lengths are 1.330, 1.332 and 1.333 Å, respectively, compared to the equilibrium structure calculated for 2-Me-cyclohexyl radical. Therefore, these three transition states are late and product-like, and these reactions with the high energy barrier, strong endothermic (∼22 kcal mol⁻¹) and nonspontaneous (11–18 kcal mol⁻¹), which are coincide with the Hammond's postulate.³⁵

The 4-Me-5-C₆H₁₀-1 radical formed by channel R_3a, not only produces 2-Me-3-C₄H₆-1 + C₂H₄ (via TS3b) by the beta C–C bond scission with a barrier of 28.20 kcal mol⁻¹, but also forms 2-Me-c-C₅H₈-CH₂-cis/trans radicals (via 1,4-vinyl migration TS3e-cis/trans) through a five-membered transition state with the barriers of 8.09 and 7.84 kcal mol⁻¹. The energies of these two transition states are almost equivalent, meaning that these two reactions play an equal importance in the title reactions. Moreover, the channels R_3e-cis and R_3e-trans are strongly exothermic and spontaneous. The result shows that the 1,4-vinyl migration reactions are thermodynamically and kinetically favored. Then the 2-Me-3-C₄H₆-1 radical in turn dissociates to 1,3-C₄H₆ + CH₃ (via TS3c) or C₃H₆ + C₂H₃ (via TS3d). These two reactive processes accompany with the barriers of 26.04 and 35.39 kcal mol⁻¹, respectively. Channel R_3c is kinetically more favorable than the R_3d, which is attributed to the effect of the breaking C–C bond is in conjunction with a C=C double bond.

The ultimate products of 2-Me-c-C₅H₈-CH₂-cis/trans radical are 1,3-C₄H₆ + C₃H₆ + H through a series of reactions (2-Me-c-C₅H₈-CH₂-cis/trans → R_3f → R_3g → R_3h → 1,3-C₄H₆ + C₃H₆ + H). These processes accompany with the barriers of 20.74, 20.58, 28.20, 31.66 kcal mol⁻¹, respectively. The most barrier height is H-elimination process by the cleavage of the strong C–H bond (R_3h), whereas the lowest barrier is the ring opening (R_3f-cis or R_3f-trans) reaction by C1–C2 bond breaking. It is implied that R_3h is the rate limiting step and is thus expected to be a minor decomposition channel. The H-elimination exhibits larger activation energies than the beta C–C bond cleavage (owing to the high bond dissociation energy (BDE) of C–H bond breaking and the stability of products formed). The conclusion is supported by the previous studies.^4,36 Equivalent to the 4-Me-5-C₆H₁₀-1 radical decomposition, the 5-C₇H₁₃-1 radical formed by channel R_3i, also has two reactive channels. One is to product 1-C₃H₅-1 + 2C₂H₄ by two consecutive C–C bond scission reactions (R_3j and R_3k) with the barriers of 27.93 and 35.95 kcal mol⁻¹, respectively. Another is to form 2-Me-c-C₅H₈ through 1,4-vinyl transfer rearrangement (R_3l) with a barrier of 7.52 kcal mol⁻¹. The result confirms the above conclusion again that 1,4-vinyl transfer is kinetically favored.

Just like in the case of the thermal decomposition of 2-Me-cyclohexyl radical, the detailed mechanisms of 2-Me-cyclopentyl and 2-Me-cyclobutyl radicals also include the ring opening, exocyclization and the beta C–C bond scission processes. As can be seen from Fig. 3, the most favored channels are 1,3-vinyl transfer rearrangement (R_2c-cis, R_2c-trans and R_2h) through a four-membered ring transition state. Similar conclusion is also drawn in the processes of 2-Me-cyclobutyl radical pyrolysis (see Fig. 4) that 1,2-vinyl migrate rearrangement (R_1d-cis, R_1d-trans and R_1i) reactions are dominated through a three-membered ring transition state. To avoid redundancy, we will not be discussed in detail for these two radicals decomposition. In addition, we also compare the barrier heights for 1,4-, 1,3- and 1,2-vinyl migrate rearrangement reactions, which proceed by five-, four- and three-membered ring transition state structures. The calculation shows that the barrier heights decrease in the order of 1,3- > 1,2- > 1,4-vinyl transfer, and the largest difference among them is amount to 7.79 kcal mol⁻¹. Our viewpoint is also supported by recent literature reports.¹⁰

Fig. 3 Potential energy profile of the thermal decomposition of 2-Me-cyclopentyl radical at the CBS-QB3 and CCSD(T) (italic) levels (the prefix and postfix of the number represent the site of double bond and radical, respectively; the ring opening reactions include R_2a, R_2d and R_2f; exocyclization reaction contain R_2c and R_2h; the remain reactions are C–C bond scission; relative energies are given in kcal mol⁻¹).

Fig. 4 Potential energy profile of the thermal decomposition of 2-Me-cyclobutyl radical at the CBS-QB3 and CCSD(T) (italic) levels (the prefix and postfix of the number represent the site of double bond and radical, respectively; the ring opening reactions include R_1a, R_1e and R_1g; exocyclization reaction contain R_1d and R_1i; the remain reactions are C–C bond scission; relative energies are given in kcal mol⁻¹).

As a result, the cis and trans transition states with energies very close to each other. Same conclusion is also drawn in cis and trans isomers. The exocyclization process is the most favored channel among all of elementary reactions due to lower barrier and high exothermic. The reaction barrier heights for the distinct reaction channels decrease in the order of 1,3- > 1,2- > 1,4-vinyl rearrangements.

3.3 Rate coefficients and branching ratios

Table 2 summarizes the modified three parameters Arrhenius expressions of rate coefficients of every elementary reaction involved in the processes of 2-Me-cyclohexyl radical pyrolysis. Other rate coefficients Arrhenius expressions incorporated in 2-Me-cyclobutyl and 2-Me-cyclopentyl radicals are presented in Tables S4 and S5,† respectively. The computations are done by employing conventional transition state theory together with an asymmetric Eckart tunneling correction based on the energies derived from the CBS-QB3 level of theory, in the temperature range from 500 to 2500 K. Each dihedral angle of both reactant and transition state is investigated using the 1-D hindered rotor treatment. Fig. 5 shows an example of the hindrance potential for an internal rotor, obtained by relaxed potential energy scan with the step of 12° at the B3LYP/6-311G(2d,d,p) level. Fig. 6 presents the Arrhenius plots of rate coefficients for reactions of R_3c and R_3h, compared with available theoretical results.

Fig. 5 Potential energy diagram for the internal rotation of the CH₃-cCH-˙CH(CH₂)₄ dihedral angle in 2-Me-cyclohexyl radical.

Fig. 6 Arrhenius plots of rate coefficients of the reactions of R_3c and R_3h are calculated at the CBS-QB3 level of theory along with the literature data from ref. 37 and 38.

From Fig. 6(a) we can see the rate coefficients of the beta C–C bond scission reaction R_3c linearly increase with rising temperature, and they satisfy Arrhenius behavior in the whole temperature range. The rate coefficients, both corrected (TST/Eckart) and uncorrected (TST) one, are compared over the temperature range of 500–2500 K. The result shows that the rate coefficients are nearly independent on the tunneling effects. The calculated rate coefficients are within one order of magnitude greater than Tsang's theoretical results,³⁸ which were determined through the solution of the master equation in the processes of n-pentenyl radical decomposition. For example, at 1800 K, the calculated rate coefficients are 6.82 × 10⁹ (TST) and 7.14 × 10⁹ s⁻¹ (TST/Eckart), which are higher than the corresponding theoretical value (1.91 × 10⁹ s⁻¹) by 3.57 and 3.74 times, respectively. Such discrepancy between the computational values and the corresponding literature ones is acceptable.

As seen from Fig. 6(b), for C–H bond cleavage reaction R_3h, the rate coefficients increase linearly as the temperature increases, and they also obey positive temperature dependence. The calculated rate coefficient, the agreement with the theoretical data of Weissman et al.³⁷ at 1260–1310 K is quite satisfactory. For example, at 1300 K, the calculated values, 2.25 × 10⁸ s⁻¹, is quantitatively comparable with the corresponding literature value (4.50 × 10⁷ s⁻¹). In the following discussion, the theoretical rate coefficients with tunneling effect corrections are applied to discuss the thermal decomposition of 2-Me-cyclohexyl radical.

According to our computations, the rate coefficients of the ring closure reactions R_3e-cis and R_3e-trans are dramatically higher than that of the ring opening pathways R_3f-cis and R_3f-trans. Thus, we assume the intermediate 2-Me-C₅H₈-CH₂ radical quickly equilibrates with the free reactants. According to the quasi-steady state approximation (QSSA), the rate coefficient of the formation of 6-C₇H₁₃-2 radical leads to the following expression as eqn (4).

where k_3e and k_−3e are forward and reverse rate coefficients from reactant 4-Me-5-C₆H₁₀-1 to intermediate 2-Me-C₅H₈-CH₂, respectively. k_3f is the forward rate coefficient from 2-Me-C₅H₈-CH₂ to 6-C₇H₁₃-2 radical. Similar methodology is adopted to calculate the total rate coefficients of the formation of final products, followed by the branching ratios being estimated at different temperatures. Same computational approach is employed to predict the thermal decomposition of 2-Me-cyclopentyl and 2-Me-cyclobutyl radicals. Fig. 7 displays a graph of the correction between the branching ratios for these radicals against temperatures.

Fig. 7 The branching rations of 2-Me-cyclohexyl (a), 2-Me-cyclopentyl (b) and 2-Me-cyclobutyl (c) radicals decomposition as a function of temperatures.

As shown in Fig. 7, the temperature changes have a significant influence on the branching ratio variations. From Fig. 7(a) we can see the branching ratio of 2-C₂H₅-c-C₅H₈ radical reduces rapidly at 500–1800 K (from 69.36% to 7.98%), whereas the c-C₆H₁₀ + CH₃ exceeds gradually it with the temperature rising (>1050 K), and they amount to as much as 79.13% at 2500 K. It is concluded that this reaction channel could be overwhelmingly competitive comparing with other pathways at elevated temperatures. The branching ratio of 3-C₄H₇-1 + C₃H₆ passes through a turning point with an increase in temperature, and the maximum value is 10.79% (at 750 K). The character is in good agreement with the feature of consecutive reactions. The branching ratios of other routes not exceed 9.0% throughout the entire temperature range, meaning that these pathways can be negligible under normal pyrolysis conditions.

As can be seen from Fig. 7(b) and (c), the similar conclusions can be drawn in the thermal decomposition of 2-Me-cyclopentyl and 2-Me-cyclobutyl radicals. The branching ratios of c-C₅H₈ + CH₃ (see Fig. 7(b)) and c-C₄H₆ + CH₃ (see Fig. 7(c)) exceed gradually 2-C₂H₅-c-C₄H₆ (>800 K) and 2-C₂H₅-c-C₃H₄ (>1500 K), and these channels are significantly favoured at high temperature. As above discussion, it is found that the vinyl rearrangement reactions have a significant superiority at low temperature, whereas the formation of cycloalkenes is favoured at high temperature.

To summarize, firstly, the tunneling effect for the calculation of rate coefficients in all of consideration reactive types is almost no influence in the entire temperature range. Secondly, the 1,2-, 1,3- and 1,4-vinyl rearrangement reactions are more advantaged at low temperature, while the formations of cycloalkene are favored at high temperature. Thirdly, the main products of the thermal decomposition of 2-Me-cyclohexyl, 2-Me-cyclopentyl and 2-Me-cyclobutyl radicals are c-C₆H₁₀, c-C₅H₈ and c-C₄H₆ under normal pyrolysis conditions.

4. Conclusions

In the present works, the thermal decomposition of 2-Me-cyclohexyl, 2-Me-cyclopentyl and 2-Me-cyclobutyl radicals have been investigated thoroughly from the geometries, thermodynamic and kinetic points of view. The following conclusions may be drawn.

(1) The reaction mechanism of the pyrolysis of cyclic alkyl radicals mainly incorporates the ring opening, vinyl rearrangements (exocyclization), beta site C–C bond cleavage and H-elimination processes.

(2) All investigated exocyclization reactions are exothermic and spontaneous, while the ring opening, C–C bond scission and H-elimination processes are endothermic and nonspontaneous.

(3) The reaction barrier heights for the distinct reaction channels decrease in the order of 1,3-vinyl > 1,2-vinyl > 1,4-vinyl rearrangements.

(4) The vinyl rearrangement reactions are advantaged at low temperature, while the formations of cycloalkene are favored at high temperature.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (no. 21173139, 21473108), the Fundamental Research Funds for the Central Universities (GK: 201101004, 201303004) and Shaanxi Innovative Team of Key Science and Technology (2013KCT-17)

References

1. C. K. Westbrook, W. J. Pitz, O. Herbinet, H. J. Curran and E. J. Silke, Combust. Flame, 2009, 156, 181–199.
2. J. X. Ding, L. Zhang, Y. Zhang and K. L. Han, J. Phys. Chem. A, 2013, 117, 3266–3278.
3. M. R. Zeng, W. H. Yuan, Y. Z. Wang, W. X. Zhou, L. D. Zhang, F. Qi and Y. Y. Li, Combust. Flame, 2014, 161, 1701–1715.
4. B. Sirjean, P. A. Glaude, M. F. Ruiz-Lopèz and R. Fournet, J. Phys. Chem. A, 2008, 112, 11598–11610.
5. J. P. Orme, H. J. Curran and J. M. Simmie, J. Phys. Chem. A, 2006, 110, 114–131.
6. B. Sirjean, P. A. Glaude, M. F. Ruiz-Lopèz and R. Fournet, J. Phys. Chem. A, 2006, 110, 12693–12704.
7. S. Zeppieri, K. Brezinsky and I. Glassman, Combust. Flame, 1997, 108, 266–286.
8. H. J. Curran, P. Gaffuri, W. J. Pitz and C. K. Westbrook, Combust. Flame, 1998, 114, 149–177.
9. H. J. Curran, P. Gaffuri, W. J. Pitz and C. K. Westbrook, Combust. Flame, 2002, 129, 253–280.
10. Z. H. Wang, L. D. Zhang and F. Zhang, J. Phys. Chem. A, 2014, 118, 6741–6748.
11. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb and J. R. Cheeseman, et al., Gaussian 09, revision D.01, Gaussian, Inc., Wallingford, 2004.
12. J. Tirado-Rives and W. L. Jorgensen, J. Chem. Theory Comput., 2008, 4, 297–306.
13. L. L Ye, F. Zhang, L. D. Zhang and F. Qi, J. Phys. Chem. A, 2012, 116, 4457–4465.
14. B. Sirjean, E. Dames, H. Wang and W. Tsang, J. Phys. Chem. A, 2012, 116, 319–332.
15. B. Sirjean and R. Fournet, J. Phys. Chem. A, 2012, 116, 6675–6684.
16. F. Wang, D. B. Cao, G. Liu, J. Ren and Y. W. Li, Theor. Chem. Acc., 2010, 126, 87–98.
17. C. Gonzalez and H. B. Schlegel, J. Chem. Phys., 1989, 90, 2154–2161.
18. C. Gonzalez and H. B. Schlegel, J. Phys. Chem., 1990, 94, 5523–5527.
19. M. Page, J. Chem. Phys., 1988, 88, 922–935.
20. K. Fukui, Acc. Chem. Res., 1981, 14, 363–368.
21. J. A. Montgomery, M. J. Frisch, J. W. Ochterski and G. A. Petersson, J. Chem. Phys., 1999, 110, 2822–2827.
22. A. D. Becke, J. Chem. Phys., 1993, 98, 5648–5652.
23. G. P. Wood, L. Radom, G. A. Petersson, E. C. Barnes and M. J. Frisch, J. Chem. Phys., 2006, 125, 094106–094116.
24. S. Snitsiriwat and J. W. Bozzelli, J. Phys. Chem. A, 2013, 117, 421–429.
25. J. Mendes, C. W. Zhou and H. J. Curran, J. Phys. Chem. A, 2013, 117, 4515–4525.
26. J. Mendes, C. W. Zhou and H. J. Curran, J. Phys. Chem. A, 2013, 117, 14006–14018.
27. B. C. Garrett and D. G. Truhlar, J. Phys. Chem., 1979, 83, 2921–2926.
28. C. Eckart, Phys. Rev., 1930, 35, 1303–1309.
29. H. S. Johnston and J. Heicklen, J. Phys. Chem., 1962, 66, 532–533.
30. J. Mendes, C. W. Zhou and H. J. Curran, J. Phys. Chem. A, 2014, 118, 12089–12104.
31. C. Y. Lin, E. I. Izgorodina and M. L. Coote, J. Phys. Chem. A, 2008, 112, 1956–1964.
32. W. T. Duncan, R. L. Bell and T. N. Truong, J. Comput. Chem., 1998, 19, 1039–1052.
33. A. Shenghur, K. H. Weber, N. D. Nguyen, W. Sontising and F. M. Tao, J. Phys. Chem. A, 2014, 118, 11002–11014.
34. A. Kossiakoff and F. O. Rice, J. Am. Chem. Soc., 1943, 65, 590–595.
35. G. S. Hammond, J. Am. Chem. Soc., 1955, 77, 334–338.
36. J. Z. Ding, L. Zhang and K. L. Han, Combust. Flame, 2011, 158, 2314–2324.
37. M. Weissman and S. W. Benson, Int. J. Chem. Kinet., 1984, 16, 307–333.
38. W. Tsang, J. Phys. Chem. A, 2006, 110, 8501–8509.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra02525g

---