A theoretical study of β-hydroxybutenyl with O₂ on the HOC₄H₆OO· potential energy surface

Abstract

The reaction of β-hydroxybutenyl radicals with O₂ and subsequent reactions are regarded as crucial steps in the low-temperature oxidation of 1,3-butadiene. However, the reaction network is not complete and previous studies failed to include accurate rate constants. In this study, the reaction network of β-hydroxybutenyl radicals with O₂ is supplemented, as well as the temperature- and pressure-dependent rate constants which are investigated by high-level quantum chemical calculation, combining with the transition state theory (TST), the variational transition state theory (VTST) and Rice–Ramsperger–Kassel–Marcus/master equation method (RRKM/ME). The results of quantum chemical calculations indicate that Waddington reaction is the dominant channel, while the intramolecular addition reactions of the radical center atoms to the double bonds are found to be important non-alkyl-analogue reactions. Moreover, some of the reactions are investigated in the present study as well, whose pressure-dependent kinetics have not been reported so far. The calculations indicate that addition reaction of β-hydroxybutenyl with O₂ is pressure independent when the temperature is below 600 K, while above 600 K, the effect of pressure is increasingly significant. Furthermore, the formation of β-hydroxybutenylperoxy is a dominant channel when the pressure is above 1 atm, while below 0.1 atm, other channels begin to take place with the increase in temperature.

1 Introduction

1,3-Butadiene is not only an important intermediate during the pyrolysis and oxidation of olefins, but also an important precursor for the formation of soot and poly-aromatic hydrocarbons (PAHs) [1, 2]. The combustion of 1,3-butadiene will produce many toxic atmospheric pollutants, and the butadienyl radicals (i/nC₄H₅) and propargyl radicals (C₃H₃) generated by its oxidative pyrolysis are the important precursor of benzene [3,4,5]. Consequently, it is of great significance to study the oxidation mechanism of 1,3-butadiene for better improvement of detailed C₄ combustion model, formation of PAHs and soot and control of toxic exhaust emission.

In recent years, plenty of reports have been published for the oxidation of 1,3-butadiene. A recent extensive study by Zhou et al. [6] showed an oxidation mechanism of 1,3-butadiene over a wide temperature range, and a series of experiments, such as shock tube, rapid compression machine and flame speed, are conducted to validate this model. Zhou et al. [6] proposed that addition of ·OH radicals to the double bonds on 1,3-butadiene and its subsequent reactions are crucial in predicting fuel reactivity. But the rate constants for these reactions are taken by analogy with other similar reactions, such as ·OH radicals addition to the C=C double bonds, whose rate constants are analogous to that of ·OH radicals additions to propene from Zádor et al. [7]. And the rate constants of O₂ addition to C₂H₃CHOHCH₂· radicals were analogized with alkyl radicals addition to O₂ calculated by Miyoshi [8], respectively. For the reaction network of ·OH radicals added to alkene, Lizardo-Huerta et al. [9] proposed that the hydroxyalkylperoxy radicals (HOROO·) are center of reaction network for the low-temperature oxidation of alkenes (600–900 K), which is displayed in Fig. 1. Figure 1 shows that hydroxyalkyl radicals (·ROH) are formed via the ·OH radicals addition onto the double bond of alkene. Then, the ·ROH radicals react with O₂ to form hydroxyalkylperoxy (HOROO·) radicals. The HOROO· radicals are mainly consumed by intramolecular hydrogen transfer, including the generation of ·OROOH radicals and HOQ·OOH radicals through Waddington reaction and isomerization, respectively. And it can be also noted that HOQ·OOH radicals are consumed by isomerization and decomposition.

Fig. 1

Simplified scheme of the primary mechanism of alkene oxidation at low temperatures [9]

In the previous study, Heyberger et al. [10] used EXGAS to generate and analyze the detailed mechanism of propylene, indicating that the channels of propene added by ·OH to generate hydroxypropyl accounted for 55% of propylene consumption. And hydroxypropyl was mainly consumed by addition to oxygen. Subsequently, Zhang et al. [11] calculated reaction of 4,4-dimethyl-1-pentene with ·OH radicals in the presence of O₂ at CCSD(T)/6–31+G(d,p)//BH&HLYP/6–311++G(d,p) level. The result showed that additional channels for the reactions of 4,4-dimethyl-1-pentene with ·OH radicals were the most dominant pathways, while the H-abstraction channels could be neglected. Then, da Silva et al. [12] performed kinetics studies for the reactions of alpha- and beta-hydroxyethyl radicals with O₂ at the G3B3//B3LYP/6-31G(d) level. And the result showed that the well depth of the beta-hydroxyethylperoxy was lower than alpha-hydroxyethylperoxy. Such result is enough to illustrate the priority of the reaction network of the β-hydroxybutenylperoxy. Vivier-Bunge et al. [13] reported the study for addition reaction system of hydroxypropyl with O₂ employing ab initio treatment at the PMP2-DZ//UMP2/6–31++ level, and they found that this system reaction was dependent with temperature and pressure. And Zádor et al. [14] used the RQCISD(T)/cc-pV∞Z//B3LYP/6–311++G(d,p) method to study the reactions of hydroxyethyl with O₂, and their research showed that the reaction of β-hydroxyethyl plus O₂ displayed remarkable pressure dependence, perfectly illustrating the importance of pressure dependence. However, the system of β-hydroxybutenyl with O₂ was lack of pressure-dependent rate constants, which needed to be calculated in later researches. Whereafter, Sun et al. [15] studied the reactions of O₂ with products from ·OH addition to isobutene at the CBS-Q//B3LYP/6-31G(d,p) level, and the result showed that the influence of the hindered internal rotor was not negligible. Lizardo-Huerta et al. [16] investigated the intramolecular configuration effects on the energy barriers and the rate constants, involving β-HOROO· and HOQ·OOH radicals, and the result showed that the position of the ·OH group in the carbon atom had an effect on the barrier height and rate constants. Therefore, it is necessary to obtain accurate kinetic data for reactions of β-hydroxybutenyl radicals plus O₂, and a systematic approach is required as well.

It is well known that the integrity of the reaction network and the accuracy of the kinetic data are important criteria for evaluating whether the chemical reaction mechanism is reasonable. In this work, some isomerization and dissociation channels for the β-hydroxybutenyl radicals with O₂ are completed according to the primary mechanism of alkene oxidation at low temperatures (Fig. 1) proposed by Lizardo-Huerta et al. [9]. To obtain temperature- and pressure-dependent rate constants, systematic kinetics analyses are investigated by high-level ab initio calculation, the transition state theory (TST) and the variational transition state theory (VTST), combining with Rice–Ramsperger–Kassel–Marcus theory/master equation (RRKM/ME). The present study provides relatively complete reaction network and accurate kinetic data, which leads to a better understanding of the low-temperature oxidation characteristics of 1,3-butadiene.

2 Computational details

2.1 Potential energy surface calculations

Geometry optimization, frequency analyses and hindered internal rotor potentials for all the stationary points are calculated at the M062X/6–311++G(d,p) level, while vibrational frequencies are scaled by a factor of 0.97 [17]. Moreover, the intrinsic reaction coordinate (IRC) [18] is used to verify the optimized transition states (TS) which are connected to the reactants and products. The Cartesian coordinates of the optimized structures and the corresponding frequencies are provided in Supporting Information. The single point energies are corrected via CCSD(T)/cc-pVTZ calculations. The high computational accuracy of the CCSD(T)/cc-pVTZ method for the calculation of the small molecules C/H/O system energies has been proved in previous literatures [19, 20]. The T1 diagnostic [21] of all the closed-shell species is lower than 0.020, while that of all the open-shell species is lower than 0.045. These results prove the rationality of CCSD(T)/cc-pVTZ method used in this system [22, 23]. The T1 diagnostic values for all species are shown in Supporting Information. For the barrierless reactions, the minimum energy paths (MEP) are scanned along the C–OO bond with a 0.05 Å interval at the B3LYP/6–311++G(d,p) level. Energy of each point on the MEP is further corrected with the CASPT2/cc-pVTZ level. The active space is chosen as (7e, 5o) for the C₂H₃CHOHCH₂· + O₂ with two pairs of O–O π and π ∗ orbitals and a radicals orbital. For the C₄H₆4,2-1OH + O₂ contains resonance-stabilized radicals, it needs the (3e,3o) active space. Thus, the active space is chosen as (9e,7o), including two pairs of O–O π and π∗, a pair of C=C π and π ∗ orbitals and a radicals orbital. The M062X, B3LYP, CCSD(T) calculations are performed by the Gaussian09 quantum chemistry package [24]. And all the CASPT2 calculations are performed by the MOLPRO-2019 program package [25].

2.2 Rate constants calculations

The transition state theory (TST) is used to calculate the high-pressure limit (HPL) rate constants with obvious barrier height reaction, which is calculated by [26]:

$$k_{\infty } \left( T \right) = \kappa \left( T \right)\sigma \frac{{k_{B} }}{h}\frac{{Q_{TS} \left( T \right)}}{{Q_{R} \left( T \right)}}exp\left[ { - \frac{{V^{ \ne } }}{RT}} \right]$$

(1)

where $\kappa \left( T \right)$,${\upsigma }$,$k_{B}$ and $h$ are the Eckartʼs tunneling correction factor, the reaction path degeneracy, Boltzmann’s constant and Planck’s constant, respectively. $Q_{TS} \left( T \right)$ and $Q_{R} \left( T \right)$ represent the partition function for the TS and the reactant (R), respectively. $V^{ \ne }$ is the barrier height of the reaction, T represents temperature, which ranges from 300 to 1500 K.

For the barrierless reactions, the HPL rate constants are calculated through the variational transition state theory (VTST). HPL rate constants of barrierless reactions $k_{vtst}$ are expressed as below [27, 28]:

$$k_{vtst} = mink_{tst} \left( {T,s} \right).$$

(2)

In Eq. (2), $k_{tst}$, which is calculated from TST, represents the rate constants of the reaction. $s$ is the reaction coordinate, and T represents the temperature from 300 to 1500 K.

The pressure-dependent rate constants, with the pressure varying from 0.001 atm to 100 atm, are obtained by the time-dependent RRKM/ME method. All the rate constants are calculated by MESS code [29] in which Klippenstein and Miller solved the one-dimensional (1-D) master equation using the eigenvalue–eigenvector method [30,31,32]. The master equation can be shown as:

$$\frac{d}{dt}\left| f \right. = - \hat{G}\left| f \right. + \mathop \sum \limits_{v} s_{v} \left( t \right)\left| {p^{\left( v \right)} } \right..$$

(3)

In Eq. (3), $\left| f \right.$ is the statistical state of the reactive complex, and $\hat{G}$ represents the kinetic relaxation operator, which is composed of the collisional energy relaxation operator and the chemical isomerization and dissociation operator. $\left| {p^{\left( v \right)} } \right.$ is the time-independent vector, which is given by the νth pair of bimolecular reactants and describes microscopic population. The quantity $s_{v} \left( t \right)$ is expressed as:

$$s_{v} \left( t \right) = n_{A}^{\left( v \right)} \left( t \right)n_{B}^{\left( v \right)} \left( t \right)/Q_{v}$$

(4)

where $n_{A}^{\left( v \right)} \left( t \right)$ and $n_{B}^{\left( v \right)} \left( t \right)$ are the concentrations of the νth bimolecular reactants A and B, which are time dependence. $Q_{v}$ is the standard partition function of the relative and internal motions of the reactants, more detailed information can be found in [29]. Usually, rigid-rotor harmonic-oscillator (RRHO) is used to approximately treat low-frequency internal rotations, which will cause significant errors in the partition function. Consequently, in the calculation of the partition function, the low-frequency torsional modes are estimated by the one-dimensional (1-D) hindered internal rotor method [33, 34]. The symmetric or asymmetric torsional potentials computed at discrete torsional angles are represented by a Fourier series expansion with multiple terms which can express top rotations of almost any form [34]. The Lennard-Jones (L-J) parameters σ and ε are used to estimate the collision frequency between reactant and bath gas Ar, whose parameters are calculated with a new group contribution method proposed by Sun et al. [35]. For Ar, σ  = 3.47 Å, ε  = 79.2 cm⁻¹, for C₂H₃CHOHCH₂OO·, σ  = 5.83 Å, ε  = 469.15 cm⁻¹. In all calculations, the exponential down model [36] with $\Delta E_{down}^{o} = 300\left( {T/300} \right)^{0.8}$ is used to treat the collision energy transfer.

All of the rate constants are fitted with three parameters from the Arrhenius equation:

$$k\left( T \right) = AT^{n} exp\left( { - \frac{{E_{a} }}{RT}} \right)$$

(5)

where A, n, E_a are the pre-exponential factors, temperature exponent and activation energy, respectively.

2.3 Thermodynamic data calculations

This part refers to the calculation of thermodynamic data for the important species in this system. The composite methods G3 [37], G4 [38], CBS-QB3 [39] and CBS-APNO [40, 41] were used to calculate the enthalpies of formation for these species, and the average value of enthalpies of formation at 0 K was calculated based on the method of atomization enthalpy. Previously, Simmie and Somers [42] proved the high accuracy of such combination method, and the difference was within ~ 1 kcal/mol compared with the benchmark formation enthalpies in ATcT database, which indicated its reliability. Then the molecular partition functions were computed by the MESS code [29], and ThermP program [43, 44] was used to calculate the temperature-dependent enthalpy of formation, entropy and heat capacity. These data can be found in the Supporting Information.

3 Results and discussion

The β-hydroxybutenyl radicals are generated through OH radicals addition to the terminal and central carbon atom in 1,3-butadiene. Subsequently, the two radicals react with O₂ to form C₂H₃CHOHCH₂OO· (W1) radicals, C₂H₃CH(OO·)CH₂OH (W1’) radicals and HOCH₂C₂H₂CH₂OO (W5’). And they can produce more stable products by intramolecular hydrogen shift, concerted elimination, intramolecular addition or reverse reaction back to the β-hydroxybutenyl radicals and O₂, which are depicted in Schemes 1 and 2. Since the contents of the two schemes are similar, only Scheme 1 is introduced in detail. Firstly, the chemically activated addition reaction for the β-hydroxyethyl radicals reacts with O₂ to form the W1 radicals. Subsequently, the W1 radicals can generate C₂H₃C·OHCH₂OOH(W2), C₄H₅OH1-4OOH(W3) and C₄H₆O1-3OOH4(W4) by intramolecular hydrogen migration and can generate C₄H₅1,3OH2 + HO₂ (P1) and C₂H₃CHOHCHO + OH (P2) by elimination reaction. Moreover, CH₂yCOOCCOH (W5) can also be produced by intramolecular addition of peroxy radicals. As for the W2, W3, W4 radicals, they can convert to each other by hydrogen migration, and stable smaller molecules can be produced by dissociation. The CH₂yCOOCCOH (W5) is consumed by cracking reaction.

Scheme 1

The reaction channels of C₂H₃CHOHCH₂· + O₂ during the low-intermediate-temperature oxidation

Scheme 2

The reaction channels of C₄H₆4,2-1OH + O₂ during the low-intermediate-temperature oxidation

3.1 Potential energy surfaces

3.1.1 C₂H₃CHOHCH₂· + O₂

The potential energy surface for the system of the C₂H₃CHOHCH₂· + O₂ is shown in Fig. 2. The reaction of C₂H₃CHOHCH₂· radicals with O₂ generates W1 radicals with a well depth of 28.4 kcal/mol. And W1 can be consumed with six pathways including intramolecular hydrogen migration, intramolecular addition, OH elimination and HO₂ elimination. Among them, the channel that generates W5 through the addition of internal peroxy radicals has the lowest energy barrier of 15.2 kcal/mol, and its subsequent channels also have relatively low energy barriers. In previous studies, the intramolecular addition reactions of the radical center atoms were found to be important non-alkyl-analogue reactions, which is consistent with the results obtained in this work [45]. Moreover, the energy barrier for the 1,5 H-migration reaction to generate W4 is 22.5 kcal/mol, which is 4.1 kcal/mol lower than 1,4 H-shift reaction producing W2. The reaction of ‘W1 → W3,’ as a 1,5 H-transfer reaction, has a very high energy barrier of 33.0 kcal/mol. Generally, the energy barriers for hydrogen shift reactions should be presented as 1,5 H-shift < 1,4 H-shift. For the reaction that generates W3, the hydrogen atom it transferred is located on the double bond carbon, which gives a higher dissociation energy to the C–H bond and results in a higher energy barrier. Through further analyzing the reaction of 1,4 H-shift, it generates allyl group and forms resonance stability, which makes its energy barrier lower. Moreover, Lizardo-Huerta et al. [16] reported that the hydroxyl group is bonded to the carbon atom which bears the transferred hydrogen atom (α-carbon position), it will contribute to the decrease in the barrier height, to a large extent. As a result, the energy barrier of 1,5 H-shift reaction for the ‘W1 → W3’ is higher than reaction of 1,4 H-shift. The remaining two concerted elimination reactions have higher energy barriers, 32.4 kcal/mol and 44.4 kcal/mol, corresponding to HO₂ elimination reaction and OH elimination reaction, respectively. The W2, W3, W4 radicals, therefore, produce more stable and smaller products, including enols and cyclic ethers. For example, W4 can be consumed through five pathways, and the most competitive channel is the second step of Waddington reaction with a barrier of 4.2 kcal/mol to produce C₂H₃CHO, CH₂O and OH (P7). Secondly, W4 isomerizes to produce W2 and decomposes to produce C₂H₃ + HO₂CH₂CHO (P6) when the energy barriers are 23.2 kcal/mol and 23.0 kcal/mol, respectively. The remaining two paths both carry higher energy barriers. One is to generate C₂H₃y(COC) + HO₂ (P8) by β-scission, and the other is to produce C₂H₃y(COOC) + OH (P9) through a four-membered cyclic transition state. The corresponding energy barriers are 36.3 kcal/mol and 41.4 kcal/mol, respectively. The other reaction channels are similar to the channels mentioned above.

Fig. 2

Potential energy profiles of C₂H₃CHOHCH₂· + O₂ reaction system calculated at the CCSD(T)//CASPT2/cc-pVTZ level

3.1.2 C₄H₆4,2-1OH + O₂

The potential energy surface for the system of the C₄H₆4,2-1OH radicals with O₂ is shown in Fig. 3. Since C₄H₆4,2-1OH has a delocalized π bond, its addition with O₂ can generate C₂H₃CH(OO)CH₂OH (W1’) and HOCH₂CHCHCH₂OO (W5’), and the well depth is 10.4 kcal/mol and 6.1 kcal/mol, respectively. They are much lower than the well depth of C₂H₃CHOHCH₂· + O₂ addition. Similar to W1, W1’ and W5’ are consumed by intramolecular hydrogen shift, intramolecular addition, OH elimination and HO₂ elimination. Whether W1’ or W5’, the energy barrier of each consumption pathway is much higher than the reverse reaction backing to the C₄H₆4,2-1OH + O₂. The C₄H₆4,2-1OH radicals have p-π conjugate that resonance stabilization [46] is generated, which causes the reverse reaction of C₄H₆4,2-1OH + O₂ → W1’ or C₄H₆4,2-1OH + O₂ → W5’ to be more dominant. Therefore, the kinetic calculation for this potential energy surface is not performed.

Fig. 3

Potential energy profiles of C₄H₆4,2-1OH + O₂ reaction system calculated at the CCSD(T)//CASPT2/cc-pVTZ level

3.2 Kinetic calculations

3.2.1 β-Hydroxybutenyl + O₂

The β-hydroxybutenyl with O₂ addition is the initial step in this system and it will directly affect the subsequent reaction channels. The potential energy profiles of dissociation of the C–OO bond are shown in Fig. 4. Distance ranges of C–OO bound breaking vary from 1.45 Å to 3.70 Å at the CASPT2/cc-pVTZ//B3LYP/6–311++G(d,p) level. Since there are few experimental or theoretical studies on the rate constants of β-hydroxybutenyl with O₂, Zhou et al. [6] studied the β-hydroxybutenylperoxy system by analogy with alkyl radicals adding O₂. Figure 5 shows the comparison of present calculations with previous studies of Zhou et al. [6], Sun et al. [15] and Heyberger et al. [16] about the HPL rate constants. It shows that calculations of this work possess strongly negative temperature dependence, and results of this study are significantly lower than the analogical data of Zhou et al. [6] by 3 orders of magnitude in the temperature range of 600–900 K. The results show the competitiveness of this reaction and subsequent reaction channels are overestimated for the 1,3-butadiene mechanism in previous studies [6].

Fig. 4

Potential energy profiles about dissociation of the C–OO bond at the CASPT2/cc-pVTZ//B3LYP/6–311++G(d,p) level

Fig. 5

HPL rate constants of C₂H₃CHOHCH₂ + O₂ in the present work, Zhou et al. (C=CCOHC· + O₂) [6], Sun et al. (C₂COHC· + O₂) [15] and Heyberger et al. (C·(C)COH + O₂) [10]

Figure 6 shows the pressure dependence for the reaction of C₂H₃CHOHCH₂· + O₂ obtained in the present work. It shows that the pressure dependence becomes more obvious with the increase in temperature. Comparing the rate constants of 0.1 atm with HPL, there is a gap approximating one order of magnitude at 1000 K. It reminds that the pressure dependence of β-hydroxybutenyl + O₂ recombination at the temperature over 600 K cannot be ignored.

Fig. 6

The pressure-dependent rate constants for the reaction of C₂H₃CHOHCH₂· + O₂

3.2.2 Waddington reaction

The Waddington reaction is a very important channel for the low-temperature oxidation of hydrocarbon, which is proposed by Ray et al. [47]. Scheme 3 shows the two steps of this mechanism. The first step is H-shift reaction, with the hydrogen atom on the hydroxyl group transfers to the position of the peroxy group. The second step of the reaction is to form aldehydes or ketones and OH radicals by decomposition. As can be seen in Fig. 2, the Waddington reaction has a very low energy barrier for this reaction network, which is a dominant channel.

Scheme 3

An example of the two-stage Waddington mechanism for the decomposition of C₂H₃CHOHCH₂OO·

Figure 7 compares results of this work with those of previously similar studies [6, 16] about the HPL rate constants of Waddington reaction for the C₂H₃CHOHCH₂· + O₂ system. Figure 7a is the first step of Waddington reaction and Fig. 7b is the second one. There are few experimental or theoretical studies on the rate constants of Waddington reaction of this system. The Zhou et al. [6] obtained the data by analogizing with that of alkyl system. Liardo-Huerta et al. [16] calculated the rate constants for the Waddington reaction of C₂C(OH)COO· radicals with CBS-Q method, and Sun et al. [15] also calculated hydroxyisobutyl + O₂ system with the CBS-QB3 method.

Fig. 7

HPL rate constants of Waddington reactions for the decomposition of C₂H₃CHOHCH₂OO· in the present work, Zhou et al. [6], Liardo-Huerta et al. [16] and Sun et al. [15]

Figure 7a shows that rate constants of the first step of the Waddington reaction, and the results of this study are very close to those of Liardo-Huerta et al. [16]. The results of this work are higher than the data of Sun et al. [15], and the differences about an order of magnitude. Figure 7b shows that rate constants of the second step of the Waddington reaction, and there are significant differences when comparing the data from literatures. During the 600–900 K, the results of the present work are higher than those of Zhou et al. [6] over five orders of magnitude. The results indicate that the rate constants of the Waddington reaction have been seriously underestimated. Due to the importance of Waddington reaction in low-temperature oxidation process of hydrocarbon [48], overestimating or underestimating will definitely cause great deviation.

3.2.3 Isomerization

Intramolecular H-shift reactions are usually known as the critical channels at the low-temperature oxidation of hydrocarbon. Figure 8 shows the hydrogen transfer reactions of this work, and their H-shift types and energy barrier height are shown in Table 1. It can be seen that the two 1,5 H-shift reactions have the well depth with a gap of 9.3 kcal/mol, and the deviation of rate constants exceeds 3 order of magnitudes. Since the C–H dissociation energy of the two reactions is different, their well depth is different. Zhou et al. [49] studied the C–H bond dissociation energy for the isobutene, which indicates that the C–H bond dissociation energy is significantly higher when H is located on the C=C than C–C. This means that the formation or breaking of the C–H bond is more difficult for the C=C bond, which leads to higher well depth.

Fig. 8

HPL rate constants of H-shift reactions for the C₂H₃CHOHCOO· potential energy surfaces

Table 1 Energy barriers of H-shift reactions for the C₂H₃CHOHCOO·

Figure 8 shows that the rate coefficients of 1,4 H-migration are close to reaction of 1,5 H-migration (C₂H₃CHOHCH₂OO·=C₄H₆O1–3OOH4) and faster with the increase in temperature. The reason is that the entropy effect plays a major role at a high temperature, while 1,4 H-migration reaction has a higher entropy feature to counteract the gap between energy barriers [50]. For another reason, Liardo-Huerta et al. [16] believe that OH group of 1,4 H-shift located on the carbon atom bearing the hydrogen transferred, it will change the partial charge on the carbon atom, which leads to the disappearance of the attractive interaction with the hydrogen atom and promotes the reaction. For the 1,3 H-shift reaction, a high energy barrier is required for the formation of a 4-membered ring transition state, so that the slowest rate coefficients are owned by 1,3 H-shift reaction, which corresponds with the intramolecular hydrogen migration rate rule [51, 52]. As for the 1,2 H-shift reaction, the highest energy barrier possesses a 3-membered ring transition state. The C₂H₃C·OHCH₂OOH radicals possess a p-π conjugate that produces resonance stabilization [46], and this hydrogen migration breaks its resonance stability, resulting in the generation of a very high energy barrier.

3.2.4 Chemical activations

The effects of temperature and pressure are mainly reflected in chemical activations, such as C₂H₃CHOHCH₂· + O₂ ←  → [C₂H₃CHOHCH₂OO·]* → products, including formation of bimolecular products and formation of stabilized unimolecular. Since the shallow well of HO·QOOH species (for example, W2 in Fig. 9), a very fast chemical equilibrating between HO·QOOH species and HOROO· species, the chemical significant eigenvalue which describes the equilibration between HO·QOOH species and HOROO· species is merged to the quasicontinuum of internal energy relaxation eigenvalues (IEREs) [27, 33, 53], which leads to difficulty in obtaining phenomenological rate constants. One way to solve the problem is to extrapolate the rate constants under reasonable conditions, and the other is to change in the number of species with conditions [53].

Fig. 9

Rate coefficients for C₂H₃CHOHCH₂· + O₂ → Products as a function of temperature at 0.01, 0.1, 1, 10 atm

Pressure- and temperature-dependent rate constants for the ‘C₂H₃CHOHCH₂· + O₂ → products’ are shown in Fig. 9. It can be seen that in the temperature range of 300–1500 K, C₂H₃CHOHCH₂OO· (W1) is the dominant generation channel when the pressure is higher than 1 atm, while the other channels of bimolecular products and stabilized unimolecular (such as W2) can hardly compete with it. Under pressure less than 1 atm, the rate constants of W1 begin to decrease with the increase in temperature, namely, the formation of W1 gradually loses its competitive advantage, while the channels of bimolecular products were becoming more significant, especially, the formation channel of P3. In the low-temperature oxidation, the HO₂ elimination reaction acts as a chain termination channel, while the dissociation pathways forming OH radicals which are chain propagation channels [50], which leads to competition between the formation of P3 and P1. At the same time, it can also be seen that the P11 generation channel is also a very competitive channel.

Figure 10 shows that the branching ratios for ‘C₂H₃CHOHCH₂· + O₂ → products’ to further analyze the competitive relationship among channels. Here the ‘QOOH stabilization’ includes all QOOH species formation, the ‘Intra O–O’ includes all channels after intramolecular addition, while ‘OH channel’ and ‘HO₂ channel’ include all channels producing OH radicals and HO₂ radicals, respectively. Figure 10 indicates that ‘ROO stabilization’ has a negative dependence on temperature, which is in competition with the chain growth channel, the ‘OH channel.’ At low pressure, the ‘OH channel’ becomes dominant as the temperature rises. As the pressure increases, ‘ROO stabilization’ becomes more competitive. And at 0.01 atm, the ‘Intra O–O’ is also highly competitive, while ‘QOOH stabilization’ and ‘HO₂ channel’ depend little on temperature, especially ‘QOOH stabilization,’ and are less competitive than other channels. All pressure-dependent rate constants in this study are fitted to the Arrhenius forms in the supplemental material.

Fig. 10

Branching ratios for C₂H₃CHOHCH₂· + O₂ → Products as a function of temperature at 0.01, 0.1, 1, 10 atm

4 Conclusions

The reaction kinetics of β-hydroxybutenyl with O₂ and consequent reactions are investigated by high-level quantum chemical calculation combining with RRKM/ME. The results show that β-hydroxybutenyl can be consumed with hydrogen migration, intramolecular addition, OH elimination, HO₂ elimination and Waddington reaction, in which the Waddington reaction and the intramolecular addition channel are highly advantage channel. Moreover, the hydrogen transfer reactions are not only related to the structure of the transition state, but also affected by the molecular configuration and entropy effect. The reaction of β-hydroxybutenyl with O₂ addition is pressure independent at lower temperatures below 600 K. However, while the temperature is higher than 600 K, the pressure dependence becomes increasingly obvious as temperature rises. The competitive relationship, between the ‘ROO stabilization,’ ‘QOOH stabilization,’ ‘Intra O–O,’ ‘OH channel’ and ‘HO₂ channel,’ is revealed by their branch ratios. When the pressure is higher than 1 atm, ‘ROO stabilization’ is the main consumption channel with middle-low temperature range, while it is below 1 atm, ‘OH channel’ and ‘Intra O–O’ begin to dominate with the rising temperature and even surpass the ‘ROO stabilization’ at higher temperature. The accurate kinetic data presented in this work for the potential energy surfaces of β-hydroxybutenyl radicals with O₂ are valuable for improving the kinetic model of 1,3-butadiene low-temperature oxidation.