ARTICLE pubs.acs.org/JPCA
Kinetics and Mechanisms of the Allyl þ Allyl and Allyl þ Propargyl Recombination Reactions Akira Matsugi, Kohsuke Suma, and Akira Miyoshi* Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
bS Supporting Information ABSTRACT: The kinetics and mechanisms of the self-reaction of allyl radicals and the cross-reaction between allyl and propargyl radicals were studied both experimentally and theoretically. The experiments were carried out over the temperature range 295800 K and the pressure range 20200 Torr (maintained by He or N2). The allyl and propargyl radicals were generated by the pulsed laser photolysis of respective precursors, 1,5hexadiene and propargyl chloride, and were probed by using a cavity ring-down spectroscopy technique. The temperature-dependent absorption cross sections of the radicals were measured relative to that of the HCO radical. The rate constants have been determined to be k(C3H5 þ C3H5) = 1.40 108 T0.933 exp(225/T) cm3 molecule1 s1 (Δ log10 k = ( 0.088) and k(C3H5 þ C3H3) = 1.71 107 T1.182 exp(255/T) cm3 molecule1 s1 (Δ log10 k = ( 0.069) with 2σ uncertainty limits. The potential energy surfaces for both reactions were calculated with the CBS-QB3 and CASPT2 quantum chemical methods, and the product channels have been investigated by the steady-state master equation analyses based on the RiceRamspergerKasselMarcus theory. The results indicated that the reaction between allyl and propargyl radicals produces five-membered ring compounds in combustion conditions, while the formations of the cyclic species are unlikely in the self-reaction of allyl radicals. The temperature- and pressure-dependent rate constant expressions for the important reaction pathways are presented for kinetic modeling.
1. INTRODUCTION The aromatic ring formation processes through the recombination reactions of resonance-stabilized hydrocarbon radicals are possible rate-determining steps in the formation of polycyclic aromatic hydrocarbons (PAH) and hence soot in the combustion environment.16 Especially, the reactions between the resonance-stabilized C3 radicals, propargyl (C3H3) and allyl (C3H5) radicals, are considered to play critical roles in the first aromatic ring formation.3,68 A number of experimental918 and theoretical1921 works have been performed for the self-recombination reaction of propargyl radicals C3 H3 þ C3 H3 f products
ðR1Þ
and the mechanism leading to the formation of benzene has been well understood. In contrast, however, only a limited amount of information is available for the kinetics and mechanisms on the self-reaction of allyl radicals2130 and the cross-reaction between allyl and propargyl radicals.21,31 C3 H5 þ C3 H5 f products
ðR2Þ
C3 H5 þ C3 H3 f products
ðR3Þ
Most of the experimental studies on the self-reaction of allyl radicals, R2, have been performed by UV (∼220 nm) absorption spectroscopy.2629 Van den Bergh and Callear26 first reported the absorption cross section, σC3H5, at 223 nm measured relative r 2011 American Chemical Society
to the 216 nm absorption cross section of the methyl radical and estimated the rate constant for R2, k2, at room temperature. Tulloch et al.27 measured k2 over a temperature range 295691 K by applying 193 nm laser flash photolysis of 1,5-hexadiene to generate allyl radicals and using the σC3H5 reported by Van den Bergh and Callear. Jenkin et al.28 investigated the kinetics of R2 at room temperature by employing a similar technique but using the σC3H5 at 220 nm calibrated relative to the loss of allyl iodide. Boyd et al.29 also studied R2 in a temperature range 403540 K by assuming the temperature-independent σC3H5. Previous experimental values of k2 are in reasonable agreement, having roomtemperature values near 3 1011 cm3 molecule1 s1 and slightly negative temperature dependence. However, the assumption of the temperature-independent absorption cross section might cause significant error since the derived rate constants are directly affected by the accuracy of the absorption cross section. Recently, Selby et al. utilized the photoionization mass spectrometry method to the photolysis of 1,5-hexadiene and measured the rate constant of R2.30 Their value of k2 at 298 K, (2.7 ( 0.8) 1011 cm3 molecule1 s1, agreed well with those measured by the UV absorption spectroscopy. They also investigated the product of reaction R2 and found that 1,5-hexadiene Received: April 15, 2011 Revised: May 18, 2011 Published: May 20, 2011 7610
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Figure 1. Schematic of the PLP/CRDS apparatus.
was the sole product of the reaction under the conditions of 300600 K and 16 Torr of He. The recombination rate constants for R2 have also been studied theoretically. Georgievskii et al. applied the variable reaction coordinates transition state theory (VRC-TST) to compute the high-pressure limiting rate constant.21 The calculated value agreed well with the experimental determinations but showed stronger temperature dependence than experiments. They argued that this discrepancy may be due to the ignorance of the temperature dependence of the UV absorption cross section. However, no direct experimental determination of the temperature dependence of σC3H5 at ∼220 nm is available. Further, although the rate constants have been investigated experimentally and theoretically, the product channels of R2 have not been studied extensively, especially at high temperatures. For the cross recombination reactions between allyl and propargyl radicals, R3, no experimental rate constant has been reported. However, motivated by the potential importance of this reaction in the ring-formation chemistry, several computational studies have been performed to date. Marinov et al. first proposed the possible contribution of reaction R3 to the aromatic ring formation in combustion, based on the BAC-MP4 quantum chemical prediction.4 More recently, Miller and co-workers performed state-of-the-art electronic structure calculations of the potential energy surface of R3, analyzed the product channels by solving the time-dependent master equation,31 and suggested kinetic models involving the formation of five-membered ring compounds. In the present study, the absorption cross section of the allyl and propargyl radicals and the rate constants for reactions R2 and R3 have been measured in the temperature range 295800 K by using the pulsed-laser photolysis/cavity ring-down spectroscopy (PLP/CRDS) technique.32,33 The product channels of both reactions have been investigated by the RiceRamsperger KasselMarcus (RRKM)/master equation analysis based on the quantum chemical calculations involving several new reaction pathways which have not been identified previously. The temperature- and pressure-dependent, product-specific rate constants were evaluated for combustion modeling.
2. EXPERIMENTAL SECTION The PLP/CRDS technique was employed for the present experimental kinetic studies. All the experiments were performed under slow flow conditions using He or N2 as bath gas. A schematic diagram of the apparatus is shown in Figure 1. The
reactant gas mixture of known composition was flowed into a quartz flow tube with an inner diameter of 25 mm and a length of 72 cm, which was held by a pair of stainless steel arms. The radicals were generated by the photolysis of the corresponding precursors RCl þ hν f R þ Cl ðR ¼ C3 H5 or C3 H3 Þ
ðR4Þ
C6 H10 ð1; 5-hexadieneÞ þ hν f 2C3 H5
ðR5Þ
at 193 nm by using an ArF excimer laser (Lambda Physik, Compex 102) with a typical pulse energy of 416 mJ pulse1. The concentrations of the radical precursors were (115) 1014 molecules cm3 (1,5-hexadiene) and (210) 1013 molecules cm3 (C3H3Cl) in the measurements of rate constants. The probe laser beam was introduced into the reaction cell through one of the two high-reflective mirrors mounted at both ends of the stainless steel arms separated 114 cm apart. The probe laser light was generated by a tunable pulsed dye laser (Lambda Physik, ScanMate, line width 97%); C3H5Cl (Wako Pure Chemical Industries, >98%); C3H3Cl (Tokyo Chemical Industry, >97%). The liquid samples were degassed by freezepumpthaw cycles, gasified, and stored in glass reservoirs. Formaldehyde (HCHO) was prepared by the thermal decomposition of paraformaldehyde44 (Sigma-Aldrich, 95%) at ∼400 K and purified by the trap-to-trap distillation. Four sets of cavity ring-down mirrors were used: the wavelength ranges and the reflectivity were 240260 nm (R > 99.5%), 320340 nm (R > 99.8%), 370410 nm (R > 99.97%), and 580660 nm (R > 99.97%).
3. EXPERIMENTAL RESULTS AND DISCUSSIONS 3.1. Absorption Spectrum and Cross Sections. The absorption spectra of the allyl and propargyl radicals observed after the photolysis of 1,5-hexadiene and C3H3Cl are shown in Figures S1 and S2 (Supporting Information), respectively. Both spectra exhibit diffuse vibronic structures associated with the predissociative nature of the excited states. The peak positions and shapes in the room-temperature spectra are essentially the same as those in the previously reported spectra of C3H535 and C3H311 radicals. As the temperature increases, the peaks become broader, and the spectra lose the structure almost completely at 800 K. The absorption cross sections were measured relative to that ~ 2A00 of the peak top of the P(8) rotational line of the A 2 0 ~ (0,9,0)X A (0,0,0) band of the HCO radical at 615.6 nm.40 The spectrum of the (0,9,0)(0,0,0) region consists of sharp rotational lines of the Σ band (Ka0 = 0 r Ka00 = 1) with linewidths less than 1 cm1 and a superimposed broad continuum assigned to the transitions involving Δ vibronic upper levels (Ka0 = 2) which are strongly predissociative and show the lifetime broadening of the order of ∼20 cm1.4547 Though the absorption cross sections of these bands were well studied at room temperature,40 there have been no direct measurements at elevated temperatures as high as 800 K. Therefore, in the present study, the temperature dependence of the peak cross section of the P(8) line, including the contribution of the continuum absorption, was estimated by the following equation
σ HCO, 615:6nm ðTÞ gðTÞ 1 f ðTref Þ ¼ σ HCO, 615:6nm ðTref Þ gðTref Þ 1 f ðTÞ
ð2Þ
where Tref is a reference temperature, 295 K; g(T) is the relative population of the N00 = 8, Ka00 = 1 levels; and f(T) is the fraction of the continuum part in the total cross section at the P(8) peak top. The relative population was calculated by using the ground state spectroscopic constants48 (in cm1), A = 24.329, B = 1.494, C = 1.399, ν1 = 2434.48, ν2 = 1080.76, and ν3 = 1868.17, while the fraction of continuum, f(T), was directly measured at each temperature. Here, the line shape of the P(8) transition was estimated to be independent of temperature and pressure since the reported line width40 (∼0.46 cm1 fwhm) is sufficiently larger than the estimated Doppler linewidths (0.037 and 0.061 cm1 fwhm at 300 and 800 K, respectively) and collisional broadening (1017 molecules cm3) was added to convert the allyl radicals to allylperoxy radicals via reaction R7. The ratio of the absorbance of the allyl radicals at 402.9 nm to that of the allylperoxy radicals at 245 nm was measured and multiplied by the literature value of the cross section of allylperoxy radicals,28 5.6 1018 ((15%) cm2 molecule1, to derive the absorption cross section of allyl radicals. Since the photolysis of 1,5-hexadiene produces various hydrocarbon radicals,27,30 their peroxy radicals may also contribute to the absorption at 245 nm, though the photolytic
Figure 3. Temperature-dependent absorption cross section of the propargyl radical at 332.5 nm. Dashed line is the result of the leastsquares fit to the exponential formula (see text). The values reported in the previous studies11,12 are also shown for comparison.
quantum yields of radicals other than allyl are small. However, since the absorption cross section of allylperoxy radicals had also been measured following the 193 nm photolysis of 1,5hexadiene,28 the interference of the minor peroxy radicals is expected to be canceled out. The measured absorption cross sections are shown in Table 1 and plotted in Figures 2 and 3. The specified error limits are the 2σ levels consisting of the errors propagated from the cross sections of the reference radicals (HCO or C3H5O2) and those from the relative absorbance measurements. For allyl radicals, two different measurements showed excellent agreement at room temperature. The present room-temperature value also agrees well with the measurement by Tonokura and Koshi,35 who determined the value by fitting the absorption decay trace following the 193 nm photolysis of 1,5-hexadiene with the selfreaction rate expression of the allyl radicals, using the average of the rate constants determined by Jenkins et al.28 and Tulloch et al.,27 2.83 1011 cm3 molecule1 s1. Since this rate constant also agreed well with the present determination as 7613
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The Journal of Physical Chemistry A shown later, the present results are consistent with that of Tonokura and Koshi. The result reported by DeSain et al.,53 who measured the cross section at 404 nm relative to the 1315 nm cross section of the I atom following the 266 nm photolysis of allyl iodide, was higher than the present result. The discrepancy is in fact larger than that shown in Figure 2 because the cross section at 404 nm is smaller than the peak absorption at 402.9 nm as shown in Figure S1 (Supporting Information). The reason for this is not clear, but DeSain et al. attributed the discrepancy with Tonokura and Koshi to the difference in the linewidths of the probe laser; they employed a multimode diode laser with line width of 1 nm which was mush larger than that of pulsed dye lasers (∼0.01 nm) used by Tonokura and Koshi and in the present study. The absorption cross section of the allyl radical showed small positive temperature dependence at 402.9 nm. For the propargyl radical, the measured cross section was in good agreement with that reported by Atkinson and Hudgens,11 while those reported by Giri et al.12 were significantly smaller than the present results. The discrepancy may be attributed to the difference of the spectral bandwidths as discussed by Giri et al. The negative temperature dependence of the cross section observed in the present study can be attributed to the broadening of the rovibrational distribution in the electronic ground state at elevated temperatures. This seems to be consistent with the almost temperature-independent cross section reported by Giri et al. since, as mentioned by the authors, their cross section corresponds to the wider spectral range (332.5 ( 1.25 nm fwhm) especially inclined to the longer wavelength due to the strong emission at 334.1 nm of the HgXe arc lamp. The dashed lines in Figures 2 and 3 show the polynomial fits of the cross sections obtained in this study. The resultant expressions are shown below with the 2σ error limits for the temperature range 295800 K. σ C3H5, 402:9nm ðTÞ ¼ ð2:15 ( 0:39Þ 1019 expð 0:1582
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Figure 4. Typical concentration profiles of (a) the allyl radical following the photolysis of 1,5-hexadiene and (b) the propargyl and allyl (inset) radicals following the photolysis of the 1,5-hexadiene/C3H3Cl mixture at 295 K and 50 Torr (N2). Lines are the results of the fits with kinetic simulation (see text for details).
Table 2. Experimental Conditions and Measured Rate Constants, k2, for the C3H5 þ C3H5 Reaction
þ 6:544 104 T 2:225 107 T 2 Þ cm2 molecule1 σ C3H3, 332:5nm ðTÞ ¼ ð4:25 ( 0:77Þ 1018 expð 3:052 þ 2:855
102 T 9:056 105 T 2 þ 1:123 107 T 3 4:858 1011 T 4 Þ cm2 molecule1 3.2. Rate Constant Measurements. The rate constants for
reactions R2 and R3 were determined by observing the time profiles of the absorbance at 402.9 nm (allyl) and 332.5 nm (propargyl). The photolysis of 1,5-hexadiene was used as the source of the allyl radicals, while the propargyl radicals were produced by the photodissociation of C3H3Cl. The 193 nm photolysis of 1,5-hexadiene has been investigated by Tulloch et al.27 and Selby et al.30 The reported quantum yields for major product channels were in good agreement, 68%27 and 63.6%30 for C3H5 þ C3H5 and 27%27 and 26.1%30 for propene þ allene, except that Selby et al. observed a significant amount of propargyl radicals with the quantum yield of 25.4%. Since the propargyl radical cannot be expected as a primary photodissociation product, it was ascribed to the dissociation of hot allene. However, in the present study, no absorption of propargyl radicals was observed by the photolysis of 1,5-hexadiene. Considering the detection limit of the measurements, the yield of propargyl radicals was less than 1% of that of allyl radicals.
[C3H5]0c
k2 ( 2σd
5
2.42.9
3.28 ( 0.65
38
1.26.7
3.20 ( 0.61
2.35.0 2.26.0
3.00 ( 0.59 2.72 ( 0.61 2.48 ( 0.49
T/K
pa/Torr
nb
295
20e
295
20200f
400 500
50 50
11 11
600
50
9
2.17.8
700
50
8
2.05.4
2.19 ( 0.42
800
50200f
24
2.06.3
2.11 ( 0.41
a N2 was used as a buffer gas unless otherwise specified. b Number of measurements. c Nascent concentration of allyl radicals in units of 1013 molecules cm3. d In units of 1011 cm3 molecule1 s1. e Buffer gas is He. f The rate constants were pressure independent.
Therefore, the production of propargyl radicals from the 193 nm photodissociation of 1,5-hexadiene was concluded to be minor and was not taken into account in the following analyses. 3.2.1. Rate Constants for Self Reaction: C3H5 þ C3H5 (R2). The kinetic analysis of the self-reaction of allyl radicals (R2) was done by assuming that the decay of allyl radicals is dominated by the self-reaction after the photolysis of 1,5-hexadiene since the contribution of other radicals to the decay profile is considered to be negligible.27,35 A typical decay time profile is shown in Figure 4a. The solid line indicates the result of the least-squares fitting to the self-reaction rate law. The experimental conditions 7614
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Table 3. Experimental Conditions and Measured Rate Constants, k3, for the C3H5 þ C3H3 Reaction T/K
pa/Torr
nb
[C3H3]0c
[C3H5]0c
S(σC3H3)d
S(σC3H5)d
S(k1)d
S(k2)d
k3 ( 2σe
295
20f
2
0.200.26
2.42.7
0.04
0.69
0.08
0.34
8.61 ( 1.33
295 400
50 50
14 7
0.270.39 0.260.46
3.36.6 3.05.0
0.02 0.03
0.63 0.67
0.05 0.06
0.40 0.36
8.71 ( 1.26 7.56 ( 1.17
500
50
7
0.280.53
3.36.0
0.04
0.69
0.07
0.35
6.47 ( 1.17
600
50
7
0.290.44
3.27.8
0.03
0.65
0.06
0.39
5.82 ( 0.89
4
0.300.39
3.94.8
0.04
0.70
0.07
0.34
5.34 ( 0.82
17
0.270.55
2.96.3
0.06
0.69
0.08
0.34
4.47 ( 0.72
700
50
800
50200g
a
N2 was used as a buffer gas unless otherwise specified. b Number of measurements. c Nascent concentration of the radicals in units of 1013 molecules cm3. d Average normalized sensitivities of the parameters to the fitted rate constants, k3, in the kinetic simulations. e In units of 1011 cm3 molecule1 s1. f Buffer gas is He. g The rate constants were pressure-independent.
and derived rate constants are summarized in Table 2. The specified error limits are 2σ level including the deviations of the measurements and the propagated error from the absorption cross sections. No apparent pressure dependence was observed at 800 K, the highest temperature studied, in the range of 50200 Torr (N2), indicating that the rate constants are already in (or, at least, close to) the high-pressure limit at 50 Torr. 3.2.2. Rate Constants for Cross Reaction: C3H5 þ C3H3 (R3). The mixture of 1,5-hexadiene and C3H3Cl was introduced into the cell and exposed to the photolysis laser to produce allyl and propargyl radicals simultaneously. The relative concentration of the precursors was controlled so that the resultant concentration of propargyl radicals was less than one tenth of that of allyl radicals. Under this condition, the decay of allyl radicals was found to be unaffected by the presence of C3H3Cl within the experimental uncertainty, and the decay of propargyl radicals is governed by their reactions with allyl radicals. The time profiles of propargyl radicals were fitted to kinetic simulations including three reactions: C3H5 þ C3H5 (R2), C3H5 þ C3H3 (R3) and C3H3 þ C3H3 (R1). The rate constant of the cross reaction R3 was optimized as a parameter in the simulations, while the other two were fixed. The rate constants determined in the present study were used for reaction R2. For the self-reactions of propargyl radicals (R1), the high-pressure limiting rate constants calculated by Georgievskii et al.,21 k = 8.47 1012T0.101 exp(295.3/T) cm3 molecule1 s1, which satisfactory reproduced the available experimental data,18 were used. A typical concentrationtime profile of propargyl radicals is shown in Figure 4b. The line indicates the result of nonlinear least-squares fitting to the numerical kinetic simulation. The subsidiary figure embedded in Figure 7 shows the C3H5 time profile observed under the same condition. Table 3 summarizes the rate constants for R3, k3, with experimental conditions. In each measurement, the sensitivity coefficients on k3 were calculated with respect to the absorption cross sections of allyl and propargyl radicals and the rate constants for R1 and R2, and the normalized sensitivity coefficients averaged over n experiments are also shown in each row. Large sensitivity coefficients were found for the absorption cross section of the allyl radical, σC3H5, and the rate constants for R2, k2. The propagated uncertainties to derived k3 were estimated by Monte Carlo simulations with typically 3000 trials, by assuming the normally distributed errors. For the standard deviations of k2, only the scattering of the measurement was included because the inclusion of the errors propagated from σC3H5 causes double
Figure 5. Arrhenius plot of the measured rate constants for C3H5 þ C3H5 (R2) and C3H5 þ C3H3 (R3) reactions (solid circles). Solid lines are the modified Arrhenius fits of the rate constants. Dashed line represents the rate constants for R3 estimated using the geometric mean rule (see text). The previous experimental2730 and computational21 results are also shown.
counting. For k1, the standard deviations were assumed to be 30% of the calculated values of Georgievskii et al.21 The resultant error limits of k3 mainly reflected those of σC3H5, while the relative uncertainties were somewhat smaller than σC3H5 itself because the absolute sensitivity coefficients, |S(σC3H5)|, were less than unity. Here, possible interference from the photolytic subproduct, Cl atoms, is discussed briefly. Since 1,5-hexadiene was introduced in great excess over C3H3Cl, Cl atoms are expected to react dominantly with 1,5-hexadiene to form the adduct radical (C6H10Cl) or allylic radical (C6H9) via H-abstraction. To estimate the contribution of these radicals, the following reactions were included in the kinetic simulations: C6X þ C6X, C6X þ C3H5, and C6X þ C3H3 where C6X represents C6H10Cl and/ or C6H9. Since no kinetic data are available for these radicals, the rate constant of the self-reaction of C6X was assumed to be the same as k2, and those of the latter two were estimated by the geometric mean rule.5456 The incorporation of C6X reactions slightly decreased the resultant k3, typically 3% smaller. Since the deviation is small compared to the uncertainty limits of k3, the influence from Cl atoms was concluded to be negligible under the conditions of the present experiments. 7615
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The pressure dependence of k3 was also investigated and was found to be negligible over the pressure range 50200 Torr (N2) at 800 K. Thus, the measured rate constants are in, or near, the high-pressure limit, as the case of R2. 3.3. Comparison with Previous Data. Figure 5 shows the Arrhenius plots of the present and literature21,2730 rate constants for reactions R2 and R3. For R2, in the past, there exist several experimental determinations utilizing the flush photolysis or pyrolysis methods. Since those by Throssell,24 Golden et al.,23 and Rossi et al.,25 employing pyrolysis to generate allyl radicals, were performed under high-temperature and low-pressure conditions where the falloff effect is significant, they are discussed later along with the results of the RRKM/master equation calculations. The rate constants reported by Tulloch et al.,27 Jenkin et al.,28 Boyd et al.,29 and Selby et al.,30 all of which employed 193 nm laser photolysis of 1,5-hexadiene for the allyl radical source, showed good agreement with the present study. The first three2729 monitored UV (223 or 220 nm) absorption of allyl radicals, and the temperature dependence of the rate constants reported by Tulloch et al.27 and Boyd et al.29 showed excellent agreement with the present study, indicating the validity of their assumption of the temperature-independent absorption cross section. Selby et al.30 reported the rate constant for R2 as (2.7 ( 0.8) 1011 cm3 molecule1 s1 after a correction for the reaction with propargyl radical R3 based on their observation of propargyl radicals in the photolysis of 1,5-hexadiene. It should be noted that their uncorrected result, (3.2 ( 1.0) 1011 cm3 molecule1 s1, agrees better with the present result, in which reaction R3 was ignored since no trace of the production of propargyl radicals was observed. The dashed lines in Figure 5 are the theoretical high-pressure limiting rate constants by Georgievskii et al.21 based on the quantum chemical and the VRC-TST calculations. Considering the accuracy of the potential energy calculations for large systems involving six carbon atoms, the agreement with experimental rate constants is remarkable, except that the predicted temperature dependence is larger than the experimental results. The dotted line shows the cross recombination rate constant, k3, estimated by the geometric mean rule5456 using the present results for k2 and the literature values21 of k1. The present result and past studies on the cross recombination reactions of the hydrocarbon radicals57,58 imply that the geometric mean rule can be used for the estimations of cross recombination rate constants including the resonance-stabilized hydrocarbon radicals. The rate constants measured in the present study in the temperature range of 295800 K and the pressure range of 50200 Torr were expressed in the modified Arrhenius form (with 2σ uncertainty limits), the results of which are also shown in Figure 5 by solid lines, as follows
initio electronic structure theory. The geometry optimizations, vibrational frequency calculations, and single point energy calculations of the reactants, products, intermediates, and transition states were carried out according to the procedure of the complete basis-set model chemistry, CBS-QB3 method,59,60 developed by Petersson and co-workers. This composite method employs the B3LYP/6-311G(2d,d,p) level of theory61,62 for geometry optimization and frequency analysis, followed by single-point calculations with a sequence of the post-Hartree Fock methods. The total energy is obtained by the additive corrections to the MP2 energy including CBS extrapolation,63 MP4 and CCSD(T) correlation corrections, the empirical higher-order correlation,64,65 spin-contamination correction,59,65 and the vibrational zero-point energy (ZPE) correction calculated at the B3LYP/6-311G(2d,d,p) level and scaled59 by 0.99. The locations of the potential minima and the transition states were verified by the results of the vibrational analysis. The intrinsic reaction coordinate (IRC) calculations at the B3LYP/6-311G(d,p) level were also performed to identify the reaction path in some cases. Although there are some issues on the spin correction term of the CBS-QB3 composite method since this term can sometimes introduce overcorrection,6668 recent studies showed that the CBS-QB3 method accurately reproduces the bond dissociation enthalpies for a number of hydrocarbon radicals including the resonance-stabilized radicals.6971 Some parts of the potential energy surface were found to have strong open-singlet character. Since the single-determinant basing methods may be inaccurate for such diradicals, the properties and energies were calculated by the following composite procedure similar to that employed by Miller et al.31 The geometries and vibrational properties of the singlet diradicals were calculated at the UB3LYP/6-311G(d,p) level. The corresponding triplet geometries were also calculated at the same level, and the triplet states’ energies were evaluated by the CBSQB3 method. Then, the singlettriplet (ST) energy gaps were calculated by the multireference perturbation theory, CASPT2,72,73 with Dunning’s correlation-consistent polarized valence triple-ζ (cc-pVTZ) basis set.74 The active spaces employed were six-electrons in six-orbitals (6e,6o) for the C6H10 (allyl þ allyl) system and eight-electrons in eight-orbitals (8e,8o) for the C6H8 (allyl þ propargyl) system. The orbitals in the active space correspond to the π, π*, and singly occupied orbitals of the C3H5 and C3H3 fragments. The zero-point energy corrected energies of the singlet diradicals, E0(S), were estimated as follows.
k2 ¼ 1:40 108 T 0:933 expð 225=TÞ cm3 molecule1 s1
where Epot,CBS-QB3(T) denotes the CBS-QB3 potential energy for the triplet states; ΔEpot,CASPT2(ST) is the ST potential energy gaps calculated at the CASPT2/cc-pVTZ//UB3LYP/6311G(d,p) level; and ZPE(S) is the UB3LYP/6-311G(d,p) ZPE of the singlet states scaled by 0.99. This method is rationalized since the electronic wave function of the triplet diradical can be well approximated by a single (degenerate) Slater determinant. For the transition states connecting the diradicals, the geometries and frequencies were calculated at the UB3LYP/6-311G(d,p) level, and the energies were calculated relative to the corresponding diradical wells at the CASPT2/cc-pVTZ level. The Gaussian 0375 and MOLPRO 2008.176 programs were used for the ab initio and density functional theory calculations.
ðΔ log10 k ¼ ( 0:088Þ k3 ¼ 1:71 107 T 1:182 expð 255=TÞ cm3 molecule1 s1 ðΔ log10 k ¼ ( 0:069Þ
4. COMPUTATIONAL METHODS 4.1. Quantum Chemical Calculation. The potential energy surfaces of the reactions R2 and R3 were characterized by ab
E0 ðSÞ ¼ Epot, CBS-QB3 ðTÞ þ ΔEpot, CASPT2 ðS TÞ þ ZPEðSÞ ð3Þ
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Figure 6. Energy diagram for the self-reaction of allyl radicals.
Computed rotational constants and vibrational frequencies of the reactant, products, intermediates, and transition states are listed in Table S1 in the Supporting Information. Corresponding thermodynamic functions (standard enthalpies of formation, entropies, and heat capacities) of the reactants, important intermediates, and products are also shown in Tables S2 and S3 (Supporting Information). 4.2. TST Calculation. The transition state theory (TST) and RRKM/master equation calculations were carried out to evaluate the rate constants and to analyze the product channels. The partition functions and the density/sum of states were calculated by using the (U)B3LYP/6-311G(d,p) harmonic vibrational frequencies scaled77,78 by 0.97. The internal rotations about the single rotatable CC bonds of the initial adducts of reactions R2 and R3 were treated as hindered rotors by using the Pitzer Gwinn approximation.79,80 The hindrance parameters were estimated from the rotational potential energy curves calculated at the B3LYP/6-311G(d,p) levels. One-dimensional semiclassical tunneling corrections were included in the rate coefficients used in the RRKM calculation by assuming the asymmetric Eckart potential.81,82 High-pressure limiting rate constants for the barrierless recombination and dissociation reactions were calculated by the microcanonical variational transition state theory. The minimum energy paths along the CC or CH internuclear distances were followed at the UB3LYP/6-311G(d,p) level, and frequency analyses were performed every 0.1 Å. Then, the single point energy calculations were performed by the CASPT2/ cc-pVTZ method with (6e,6o) and (8e,8o) active spaces for the C3H5 þ C3H5 and C3H5 þ C3H3 systems, respectively. The internal rotations around the newly forming CC bonds of the C3H5 þ C3H5 and C3H5 þ C3H3 reactions were assumed to be free rotors. The microscopic rate coefficients at each point on the reaction paths were calculated with the energy grain size of 10 cm1 and variationally minimized for each grain at the same energy.
4.3. RRKM/Master Equation Calculation. The pressuredependent rate constants were calculated by solving the steady-state master equation. The energy grained, multiple-well master equation can be expressed as83,84
d n ¼ Jn dt
ð4Þ
where n is the internal energy distribution vector and J is a transition matrix containing the rate coefficients for collisional energy transfer and reaction. The reaction of allyl þ allyl (R2) or allyl þ propargyl (R3) can be described as a chemically activated complex-forming reaction. The steady-state of the chemically activated reaction can be written as85 d n ¼ Jn þ kin r ¼ 0 dt
ð5Þ
where kin is the rate constant for complex formation and r is the chemically activated energy distribution. The steady-state energy distribution, nss, can be obtained by solving the linear eq 5. Then, the rate constants for each channel can be calculated from nss and the microscopic rate coefficients. The steady-state thermal decomposition rate constants of the major intermediates were also computed. For the multiple-well systems, the isomer-specific steady-state dissociation process is no longer represented by a single eigenpair. Instead, the steadystate dissociation of one certain well (one intermediate) can be expressed as84 Jnss ¼ kss nss 0
ð6Þ
where kss donates the steady-state total rate constant and nss0 is the steady-state distribution for the well of interest. A solution for eq 6 can be obtained by the iterative procedure.84 The stabilization to an intermediate was treated by truncating the transition matrix, J, at a certain threshold.8688 The stabilization rate constants were calculated from nss as the collisional 7617
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Figure 7. Microscopic isomerization/dissociation rate coefficients of 1,5-hexadiene (shown as wa1 in Figure 6).
energy transfer rate to the energy below the threshold. In the present study, after some examinations, the truncation threshold was set to 20% of the barrier height of the lowest-energy channel for each well. It was confirmed that changing the truncation threshold does not significantly affect the branching fraction for the stabilization channel at least at the temperatures where the lifetimes of the activated complex and stabilized intermediate differ clearly. The density of states and microscopic rate coefficients were calculated by using a modified version of the UNIMOL program89and GPOP program suit.90 The steady-state unimolecular master equations were solved by using SSUMES program.91 The linear equations were solved by LAPACK routines92 after symmetrizing the transition matrix.93 The energy grain sizes of 10 and 100 cm1 were adopted for reactions R2 and R3, respectively. The coarser grain for the latter was due to the computational limitation. To check the convergence of the grain size for R3, the 10 cm1 grained system was also solved for some conditions by the preconditioned conjugate gradient method94 by utilizing the sparse structure of the linear system. The difference between the solution with 100 and 10 cm1 grains was at most 10% for the rate constants for the main reaction pathways. The exponential-down model93 was used to estimate collisional energy transfer probability, with the average downward energy transferred per collision, ÆΔEdownæ = 400(T/ 300)0.7 cm1, with which Miller and co-workers20,21 satisfactorily reproduced the pressure-dependent rate constants for the selfrecombination reaction of propargyl radicals at high temperature.16,18 The collision frequency was estimated by assuming the Lennard-Jones (LJ) potential. The LJ parameters for C6H10 and C6H8 isomers were assumed to be the same as those of benzene95 (σ = 5.46 Å and ε/kB = 401 K). The buffer gas was assumed to be N2 (σ = 3.798 Å and ε/kB = 71.4 K96).
5. THEORETICAL RESULTS AND DISCUSSIONS 5.1. Energy Diagram and Computed Rate Constant for C3H5 þ C3H5. Figure 6 shows the energy diagram for the self-
reaction of allyl radicals. Here, prefixes “wa” and “pa” are used to denote wells and products, respectively. The first well, wa1
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Figure 8. Computed high-pressure limiting and overall rate constant for the self-reaction of allyl radicals (R2), in comparison with the present and previous2325,2730 experimental data: This work (20200 Torr in He or N2); Selby et al. (16 Torr in He); Boyd et al. (760 Torr in N2); Jenkin et al. (760 Torr in N2); Tulloch et al. (0250 Torr in Ar); Throssell (10 Torr in toluene); Golden et al. (VLPP of diallyl oxalate); Rossi et al. (VLPP of diallyl oxalate or 3,30 -azo-1-propene).
(1,5-hexadiene), has several isomerization and dissociation channels including a 1,3-hydrogen shift to wa2, concerted 1,4-hydrogen shift with three-membered ring formation to wa3, CH bond fission to pa1, and intramolecular addition to form a cyclic diradical (wa4). The diradical further isomerizes via a bond formation (to wa5) or H-atom transfer (to wa6 and wa7). The transition state connecting wa4 and wa5 was found to lie lower than wa4, mostly due to the difference in the methods employed for the geometry optimization (B3LYP) and the energy calculation (CASPT2). No further pursuit was made for this anomaly since this reaction was found to be minor. For C3H5 þ C3H5 recombination, there are two additional routes as previously reported by Georgievskii et al.,21 which correspond to the two rotational conformers of 1,5-hexadiene around the CC bonds adjacent to the double bonds. The potential energy curves for the recombination channels as a function of the length of the newly formed CC bond calculated at the CASPT2/cc-pVTZ//UB3LYP/6-311G(d,p) level are shown in Figure S3 in the Supporting Information. Since the isomerization barrier between two conformers is significantly high (1770 cm1 at CC bond length of 3 Å) at large CC separations due to the delocalized π orbitals of the allyl radicals, the two channels were treated separately, and the recombination rate coefficient was calculated by summing those for two channels. Figure 7 shows the microscopic rate coefficients from 1,5hexadiene (wa1). Clearly, the back-decomposition to the reactants (2C3H5) dominates over other channels. The master equation analyses indicated that the reaction pathways leading to wa27 or pa13 are negligible under the conditions of interest (300 2000 K and 176 000 Torr) in the self-recombination of allyl radicals or in the thermal decomposition of 1,5-hexadiene, with branching fractions of at most 3 104. Therefore, the reaction was concluded to be dominated by the recombination to 1,5-hexadiene, which is consistent with the previous photoionization mass spectrometry experiment where the 1,5-hexadiene was the only isomer of C6H10 detected as a product of R2.30 The temperature- and pressure-dependent rate constants for the self-recombination reaction of allyl radicals were computed 7618
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defined as the ratio of the rate constant at pressure, p, to that of the high pressure limit, k¥, was 0.97 at 50 Torr and 0.98 at 200 Torr at 800 K. At the conditions of higher temperatures and lower pressures, the falloff effect becomes significant. The rate constants reported by Throssell24 were a reanalysis of the experimental result of Akers and Throssell,22 who pyrolyzed 1,5-hexadiene in toluene buffer (∼10 Torr) and performed end-product analysis. Both of the works of Golden et al.23 and Rossi et al.25 were performed in a very low-pressure pyrolysis (VLPP) apparatus. As noted by Rossi et al., the rate constants reported by Golden et al. might be overestimated due to the heterogeneous reactions at the chamber wall and were omitted from the comparison. The VLPP study of Rossi et al. was performed by pyrolizing diallyl oxalate or 3,30 azo-1-propene at 102103 Torr. Considering the difference of the third body and the experimental and theoretical uncertainty, the calculated rate constants reasonably agreed with the experiment by Throssell and Rossi et al. Experimental pressure dependence at high temperature is necessary for further validation of the calculated falloff behavior. The computed high-pressure limiting rate constants over the temperature range 3002000 K, as well as the pressure dependence for g1 Torr, N2, were expressed by the Troe formula97 as follows k2 ¥ ¼ 5:28 1011 T 0:236 expð296=TÞ cm3 molecule1 s1 k2 0 ¼ 1:11 1022 T 16:050 expð 1431=TÞ cm6 molecule2 s1 Fcent ¼ expð T=200:5Þ þ expð 689=TÞ
Figure 9. Energy diagrams for the recombination reaction of the allyl radical with the propargyl radical: (a) entrance/cyclization paths, (b) five-membered ring channels; and (c) six-membered ring channels. The suffixes “(T)” after the energies represent the triplet states.
by the detailed balance from the steady-state thermal decomposition rate constants of 1,5-hexadiene and are shown in Figure 8 with the experimental values.2325,2729 To obtain the best fit to the present experimental rate constants, the potential energy curves were shifted by þ1.5 kJ/mol within the uncertainty limit of the computational method employed. The computed high-pressure limiting rate constants were slightly overestimating the negative temperature dependence. The source of the discrepancy is unclear at this point. However, considering that the VRC-TST calculation of Georgievskii et al.21 also exhibited the same tendency as seen in Figure 5, the same problem is likely to be the cause of the discrepancy. The error in the potential energy curve calculations may also be the origin, but more detailed analysis is required to resolve the disagreement. The pressure independence of the rate constants measured in the present study was also reproduced by the master equation calculations. The calculated falloff factor, k(p)/k¥, which is
5.2. Energy Diagram for C3H5 þ C3H3. Figure 9 shows the energy diagram for the cross-recombination reaction of allyl and propargyl radicals, R3. The major features are similar to those recently reported by Miller et al.31 The recombination takes place at either the CH2 or CH end of the propargyl radical, producing the intermediate having a propargyl (CH2CCH) or an allenyl (CHdCdCH2) group, respectively. The latter intermediate, w2, can form a cyclic diradical, w8, via intramolecular addition of the methylenic carbon to the allenic carbon site. As direct β CH fission channels from the diradical w8, two reaction pathways producing isomers of methylenecyclopentenyl radicals (p6 and p7; c-C6H7) were found. The pathway connecting w8 directly to p7 þ H had not been identified by Miller et al. The present UB3LYP/6-311G(d,p) level of calculation clearly showed the first-order saddle point between w8 and p7 þ H, which was further confirmed by the IRC calculation. Therefore, this channel was included in the master equation analysis, though its contribution was found to be minor, typically 67 times slower than the w8 f p6 þ H channel. The H-elimination channels from the triplet diradical, w8(T), were also identified. Since the energy gap between the singlet and triplet states is sufficiently small to allow rapid intersystem crossing (ISC), these channels may contribute to the formation of p6 and p7 products. Other pathways found from w8 are the hydrogen transfer reactions forming closed-shell intermediates, methylenecyclopentenes (w13 and w14; c-C6H8), which then decompose to produce c-C6H7 species or fulvene (p8). The formation of fulvene þ H2 from w14 is one of the important pathways which were not included in the previous calculations by Miller et al. Although this H2 elimination involves a six-centered transition 7619
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Figure 10. Computed high-pressure limiting and overall rate constant for the reaction of the allyl radical with propargyl radical R3, in comparison with the present experimental results.
state and is entropically less favored than the simple CH fission channels, the branching fractions were comparable, as will be described later, because of the much lower barrier height for the H2-elimination to form fulvene. There are other cyclization pathways involving diradicals w7, w6, and w9. However, these pathways are energetically unfavorable and were concluded to be negligible. The lower energy for the transition state between w1 and w7 than that of w7 is a small artifact of employing different methods for geometry optimization and energy calculation. The six-membered ring compound formation pathways from w13 were also identified, proceeding through the intermediates, w15 or w18, both of which have diradical carbon centers and thus lie at higher energies than the closed-shell intermediates. The barrier heights to w15 and w18 are comparable to those of CH fission channels from w13. Nevertheless, the master equation analysis described below indicated that these channels are negligible due to the small pre-exponential factors. Compared to the energy diagram for allyl þ allyl, the allyl þ propargyl reaction involves a much smaller barrier for the cyclization step. This noticeable difference can be attributed to the resonant stabilization of the diradical, w8. Although the one localized structure is illustrated in Figure 9, the unpaired electron indicated at the ortho-position is delocalized by the resonance with another structure with the unpaired electron on the methylene carbon. The analogical discussions have also been done for the “Cope reaction family”, for which Schreiner and coworkers concluded that the cyclization reactions of the polyunsaturated hydrocarbon take place when diradical intermediates are stabilized either by allylic resonance or gain of aromaticities.98,99 This also implies the role of the recombination reactions involving the radicals with propargylic resonance2,100 in the chemistry of the PAH formation because their recombination reactions with hydrocarbon radicals commonly generate allene moieties and products of insertion reactions to the allenic carbon have allylic resonance. 5.3. Computed Rate Constants for C3H5 þ C3H3. The potential energy curves for the recombination channels are shown in Figure S4 (Supporting Information). At shorter CC length, the potential energy for the w1 channel is more attractive since the radical orbital is mainly localized on the CH2 side of the propargyl radical, as discussed by Georgievskii et al.21 In contrast, at larger CC separation, the potential energy curve for the w2 channel shows attractive interaction between the
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Figure 11. Product-specific rate constants for the chemically activated C3H5 þ C3H3 reaction at the total pressure of 1 atm (N2).
terminal H atom on the CH side of the propargyl radical with the π orbital of the allyl radical. The computed high-pressure limiting and pressure-dependent rate constants for R3 are shown in Figure 10. As in the case of R2, the potential energy curves were shifted by þ1.5 kJ mol1 for the best fit to the experimental rate constants. The temperature dependence of the experimental rate constants was well reproduced by the calculation. This was in contrast to the case of R2 where the calculation showed stronger temperature dependence than the experiment. The difference of these two reactions may be attributed to a problem in the treatment of the transitional vibrational mode which correlates to the torsion motion between C4 and C2 parts since there is no such bond in the propargyl and allenyl moieties, and thus the effect of the problem is expected to be smaller than the allyl þ allyl case. The predicted pressure falloff factor, k(p)/k¥, at 800 K was 0.94 and 0.98, respectively, at 50 and 200 Torr, which supports the pressure independence of the experimental rate constants. The high-pressure limiting rate constant was fitted to the modified Arrhenius expression in the temperature range of 3002000 K as follows k3 ¥ ¼ 7:98 108 T 1:094 expð153=TÞ cm3 molecule1 s1 The product-specific rate constants obtained by the steadystate master equation analysis for the chemically activated reaction at total pressure of 1 atm (N2) are shown in Figure 11. The pressure dependence of these rate constants are also shown in Figure S5 of the Supporting Information. Unless otherwise noted, the reactions on the triplet surface involving the ISC of diradicals were not included in the calculations. The stabilizations to the initial adducts are dominant at low temperature. At temperatures above 1200 K, productions of the c-C6H8 isomers (w13 and w14) through the diradical (w8) were found to be dominant. As the temperature increases further, the stabilization to the wells w13 and w14 decreases, and the formation of the decomposition products, c-C6H7 isomers (p6, p7) and fulvene (p8), becomes dominant. The formation of fulvene occurs dominantly from w14, while for p6 and p7, dissociations from w8, w13, and w14 are competing. At higher temperature, the formation of the dissociation products from the initial adducts, especially p5 (i-C4H5) þ C2H3, also becomes significant because these channels are entropically favored. The contribution of the reactions involving the triplet diradical was qualitatively estimated. The maximum possible contribution 7620
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Figure 12. Product-specific rate constants for the steady-state isomerization/dissociation reactions of w1 (a) and w2 (b) at the total pressure of 1 atm (N2).
was estimated by assuming the rate constant for the singlet to triplet ISC, kISC, of 1012 s1, with which the singlet and triplet diradicals are nearly equilibrated. The reverse rate constants, kISC, were calculated by detailed balance. With this value, the rate constants for the formation of five-membered ring compounds at the pressure of 1 atm were found to be enhanced by factors of 1.04, 1.09, and 1.13 at 1400, 1700, and 2000 K, respectively, compared to the rate constants calculated without triplet channels. Therefore, there is a contribution from the H-elimination channels of the triplet diradical if rapid ISC occurs, but the effect was predicted to be minor. When compared with the results of Miller et al.,31 there was a minor difference with regard to the stabilization process to the first wells at high temperature. In the analysis of Miller et al., the collisional stabilization to w1 and w2 takes place even at the elevated temperature, whereas in the present analysis, the stabilization becomes negligible at temperature above 1500 K. This comes from the difference of the ways to separate reactive and stabilization processes (the eigenpair separation101,102 and the matrix truncation), which become hard to be separated at high temperatures due to the nearly degenerate time constants. However, the difference is minor in practice since the overall phenomenon is unchanged whether it is modeled as a one-step chemical activation reaction or as a sequence of stabilization and dissociation reactions when it occurs with the same time constant. Miller et al. also estimated the “effective” rate constants for the cyclic C6H7 and C6H8 formation which include the secondary decomposition/isomerization of the initially formed wells.31
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Figure 13. Product-specific rate constants for the steady-state isomerization/dissociation reactions of w13 (a) and w14 (b) at the total pressure of 1 atm (N2).
Their value of the effective rate constants was 7.19 1013 cm3 molecule1 s1 at 1500 K and 10 atm, which is comparable to the value of the present study, 1.07 1012 cm3 molecule1s1. In either case, the rate constants for the cyclization were predicted to be slower than that used in the previous modeling studies which employed the estimated value of Marinov et al., 4.65 1012 cm3 molecule1 s1, at 1500 K.4 5.4. Computed Rate Constants for Thermal Isomerization/ Decomposition of C6H8 Isomers. The steady-state thermal isomerization and decomposition rate constants for the initial adducts (w1, w2) and the cyclic intermediates (w13, w14) were also computed. The results at total pressure of 1 atm are plotted in Figures 12 and 13. The results for other pressures, 10 Torr and 10 atm, are also shown in Figures S6 and S7 (Supporting Information). For simplicity, only the major product channels are shown. For reactions of w1 and w2, the major product channels are similar to those of the chemically activated reactions. At temperature below 1200 K, dominant reactions are isomerization to another initial adduct. The rate coefficients for other pathways are more than an order of magnitude slower, and w1 and w2 are in partial equilibrium. The back-dissociation to the C3H5 þ C3H3 becomes significant as the temperature increases, and the formation of the five-membered ring compounds (w13 and w14) is also important, at relatively low temperature (
