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J. Phys. Chem. A 2010, 114, 8240–8261
Experimental and Theoretical Investigation of the Self-Reaction of Phenyl Radicals Robert S. Tranter,*,† Stephen J. Klippenstein,*,† Lawrence B. Harding,† Binod R. Giri,†,§ Xueliang Yang,† and John H. Kiefer‡ Chemical Sciences and Engineering DiVision, Argonne National Laboratory, 9700 South Cass AVenue, Argonne, Illinois 60439, and Department of Chemical Engineering, UniVersity of Illinois at Chicago, 810 South Clinton Street, Chicago, Illinois 60607 ReceiVed: April 6, 2010; ReVised Manuscript ReceiVed: June 25, 2010
A combination of experiment and theory is applied to the self-reaction kinetics of phenyl radicals. The dissociation of phenyl iodide is observed with both time-of-flight mass spectrometry, TOF-MS, and laser schlieren, LS, diagnostics coupled to a diaphragmless shock tube for temperatures ranging from 1276 to 1853 K. The LS experiments were performed at pressures of 22 ( 2, 54 ( 7, and 122 ( 6 Torr, and the TOF-MS experiments were performed at pressures in the range 500-700 Torr. These observations are sensitive to both the dissociation of phenyl iodide and to the subsequent self-reaction of the phenyl radicals. The experimental observations indicate that both these reactions are more complicated than previously assumed. The phenyl iodide dissociation yields ∼6% C6H4 + HI in addition to the major and commonly assumed C6H5 + I channel. The self-reaction of phenyl radicals does not proceed solely by recombination, but also through disproportionation to benzene + o-/m-/p-benzynes, with comparable rate coefficients for both. The various channels in the self-reaction of phenyl radicals are studied with ab initio transition state theory based master equation calculations. These calculations elucidate the complex nature of the C6H5 self-reaction and are consistent with the experimental observations. The theoretical predictions are used as a guide in the development of a model for the phenyl iodide pyrolysis that accurately reproduces the observed laser schlieren profiles over the full range of the observations. 1. Introduction A detailed understanding of the mechanism for formation of polycyclic aromatic hydrocarbons (PAHs) is of central importance to the development of mechanisms for the formation of soot in flames. The phenyl radical is a key intermediate in the PAH formation process. Its addition reactions with various molecular and radical fragments have been postulated to provide the primary link between the first aromatic ring and multiple ring species. Notably, the addition of phenyl radical to other aromatics provides a direct pathway for growth in the number of rings.1–3 The self-recombination of two phenyl radicals to form biphenyl
C6H5 + C6H5 f C12H10
∆Hr,298 ) -117.6 kcal/mol (1)
is the prototypical example of an aromatic-aromatic radicalradical addition reaction. (The ∆Hr,298 value of -117.6 kcal/ mol in eq 1 is from refs 4 and 5.) This reaction has been identified by Richter et al.3 as an important step in the production of acenaphthylene, a key species in PAH formation. Reaction 1 has also been identified as an important sink for phenyl radicals,6,7 and it has been used as a reference reaction in comparative rate studies.8 Nevertheless, the self-reaction of phenyl radicals has not been studied in great detail. * To whom correspondence should be addressed. E-mail:
[email protected] (R.S.T.);
[email protected] (S.J.K.). † Argonne National Laboratory. ‡ University of Illinois at Chicago. § Current address: Alberta Sulphur Research Ltd., 3535 Research Road NW, Calgary, Alberta, T2L 2K8, Canada.
However, there are some previous studies of reaction 1, and these yielded estimates of the rate coefficient k1,6–10 that ranged from 3 × 1012 to 1 × 1014 cm3 mol-1 s-1. Three prior experimental studies6–8 have measured k1 at temperatures ranging from 300 to 1450 K, and pressures ranging from 0.1 to 2 atm. Park and Lin8 obtained k1 ) (1.39 ( 0.11) × 1013 exp[(-56 ( 36)/T] cm3 mol-1 s-1 at 6-7 Torr and 300-500 K in laser photolysis/mass spectrometry experiments, with C6H5NO as the precursor of C6H5. In these experiments, the rate coefficient was determined from C12H10 concentration/time profiles following photolysis of the precursor. Meanwhile, both Horn et al.6 and Heckmann et al.7 thermally dissociated C6H5NO behind shock waves to study reactions of C6H5 in high temperature experiments around 1-2 atm. Horn et al. measured consumption of C6H5NO by UV absorption at 275.6 nm and from modeling the absorption profiles obtained an estimate of k1 ) (2 ( 1) × 1013 cm3 mol-1 s-1 over 800-1000 K. Heckman et al. monitored biphenyl growth with UV absorption at 245 nm and obtained k1 ) 5.7 × 1012 cm3 mol-1 s-1 over 1050-1450 K. To the best of our knowledge there are no ab initio based theoretical treatments of reaction 1 in the literature. In their classic modeling study of aromatic formation, Wang and Frenklach11 suggested, on the basis of Rice-RamspergerKassel-Marcus (RRKM) calculations, that at 760 Torr k1 is given by 2 × 1019 T-2.05 exp(-1460/T) cm3 mol-1 s-1, which yields (1.6-2.3) × 1012 cm3 mol-1 s-1 over 1500-2000 K. However, these calculations simply presumed that the high pressure recombination rate was equal to the 3 × 1012 cm3 mol-1 s-1 estimate of Fahr et al.10 In this work we employ a combination of multiple experimental observations and high-level theoretical calculations to obtain an improved understanding of the C6H5 + C6H5 reaction
10.1021/jp1031064 2010 American Chemical Society Published on Web 07/22/2010
Self-Reaction of Phenyl Radicals kinetics. The experimental aspect of this work employs a diaphragmless shock tube (DFST) coupled to both a time-offlight mass spectrometer (TOF-MS) and laser schlieren (LS) densitometry diagnostics. The TOF-MS diagnostic permits identification of products and relative concentrations with minimal interference from the sampling method. The laser schlieren observations yield quantitative data on the progress of the reaction and, with an appropriate model, provide good estimates of the rate coefficients. Both of these experiments are performed for temperatures centered about 1500 K. These above sets of experimental observations each indicate that the self-reaction of phenyl radicals is more complicated than a simple addition. In particular, they suggest that abstraction reactions to produce benzene plus benzyne are a significant factor in the phenyl + phenyl kinetics. This finding has led us to the exploration of both the addition and abstraction mechanisms with high-level ab initio electronic structure calculations. The results of these electronic structure calculations are then incorporated in transition state theory based master equation simulations to predict the temperature- and pressure-dependent rate coefficients for both the addition and abstraction channels. These theoretical kinetics predictions then provide the framework for a model that accurately reproduces the observed laser schlieren density gradient profiles. Ultimately, the combination of experiment, theory, and modeling provides strong verification for the importance of both the addition and abstraction channels in the self-reaction of phenyl radicals and even show that the previously unconsidered abstraction channels are dominant at high temperatures. The prior experimental studies of the self-reaction of phenyl have employed pyrolysis of nitrosobenzene, C6H5NO, as a source of phenyl radicals. In this work, we instead pyrolyze phenyl iodide, C6H5I, because the vapor pressure of C6H5NO is too low to prepare sufficient quantities of the reagent mixtures, up to 10% C6H5I, needed for the present work. Such relatively large fractions are necessary for LS experiments and increase the importance of the secondary phenyl + phenyl reactions of interest here. In addition, large concentrations increase the signal for the TOF-MS observations. The DFST/LS method is wellsuited to the study of the primary and secondary chemistry in reaction systems where a rapid endothermic decomposition reaction, the C6H5I decomposition, is followed by a strongly exothermic recombination reaction (C6H5 + C6H5 f C12H10). These experiments also extend efforts to develop well-characterized radical sources for other DFST/LS and DFST/TOF-MS experiments.12,13 Previous shock tube studies of the decomposition rate of phenyl iodide have been conducted by Robaugh and Tsang,14 Rao and Skinner,15 and Kumaran et al.16 In order to reduce the importance of secondary bimolecular reactions, these experiments were carried out with much lower concentrations ( 40, which is the region of most interest in the current work, although ions with m/z < 40 were also acquired. The mass spectra in Figure 3 were obtained with 30 eV electron impact ionization (EI) and 2% C6H5I, whereas those presented in Figure 4 were obtained with 50 eV EI and 4% C6H5I. In the latter experiment, the increased concentration of C6H5I and the higher ionization energy improved the signal/noise ratio, but also resulted in somewhat broader mass peaks. The preshock mass spectra shown in Figures 3a and 4a are consistent with the 70 eV EI literature mass spectrum for C6H5I.25 The primary difference is that C6H5+ is the largest peak in the literature mass spectrum, whereas in this work the C6H5+
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Figure 5. Expansions of segments from the mass spectrum shown in Figure 4b.
peak is smaller than the C6H5I+ peak, which indicates that at the lower ionization energies used in the present DFST/TOFMS experiments there is less fragmentation of the parent molecule. The peak at m/z ) 40 in Figure 3a is from argon, which was added as an internal standard to the reagent mixture in Figure 3a and is thus not seen in Figure 4a or in the literature mass spectrum. For the experiments shown in Figure 3 the frozen temperature behind the reflected shock wave, T5, is just 1259 K and only a small fraction of the parent molecule has been consumed, indicating that the dissociation of C6H5I is relatively slow. In Figure 4, T5 ) 1511 K, and in Figure 4b the phenyl iodide is almost completely consumed. Several differences between the pre- and postshock spectra in Figures 3 and 4 are evident including the appearance of new peaks at m/z ) 62, 127, 128, 152, and 154 in Figures 3b and 4b. Furthermore, in Figure 4b the peaks at m/z ) 78 and 76 are larger relative to m/z ) 77 than in the preshock mass spectra shown in Figure 4a. This increase can be seen most clearly in Figure 5, which expands the peaks in Figure 4b around m/z ) 77, 127, and 154. The small peak at m/z ) 62 in Figures 3b and 4b most likely occurs from fragmentation of product species in the ion source. The remaining peaks, m/z < 62, in Figures 3b and 4b can be assigned to fragmentation in the ion source of the parent molecule or phenyl radical. The significance of several species observed in the postshock mass spectra and their relationships to the dissociation of phenyl iodide and the theoretical investigation of phenyl self-reaction are discussed below. m/z ) 127 and 128 (I and HI). The peaks at m/z ) 127 (I+) and 128 (HI+) are both found in the literature 70 eV mass spectrum of C6H5I. Comparing the heights of these peaks with the most abundant ion in the literature mass spectrum, C6H5+, gives peak height ratios of ∼0.07 for I+/C6H5+ and ∼0.003 for HI+/C6H5+. The only stable isotope of iodine is 127I, and the m/z ) 128 peak can thus be safely assigned to HI+. Neither m/z ) 127 nor m/z ) 128 is seen in Figure 3a, indicating that with 30 eV EI I+ and HI+ are either not formed by fragmentation of the parent ion or their concentrations are simply below the detection limit. In contrast, in Figure 4a the m/z ) 127 peak is observed, but m/z )128 still does not appear to be present. The mass spectrum shown in Figure 4 was obtained with 50 eV EI, and this experiment used not only a higher concentration of the parent molecule, but also a larger diameter skimmer, which increased the mass flux into the ion source of the TOF-MS. These differences allowed for the detection of lower concentration species, but at the expense of higher pressures in the TOFMS and slightly broader peaks.
Tranter et al. As to the postshock mass spectra shown in Figures 3b and 4b, m/z ) 127 is one of the largest peaks and it is followed by a small peak at m/z ) 128. The strong I+ peak, which was absent or very weak in the preshock mass spectra, appears immediately following reflection of the shock wave and provides a useful indicator of the onset of reaction. In Figure 3b, the ratio of peak areas for (m/z ) 128)/(m/z ) 127) is ∼0.09, whereas in Figure 4b (see also Figure 5 for an enlarged view) the ratio is ∼0.06. These ratios are about 1.5-2 times greater than is seen in the literature mass spectrum for C6H5I. In Figure 3b, the extent of dissociation of C6H5I is small, but the ratio of HI+/C6H5+ is about a factor of 10 greater than in the literature C6H5I mass spectrum, where HI+ is formed by fragmentation in the ion source. These increases in HI+/I+ and HI+/C6H5+ indicate the presence of some source of HI other than fragmentation in the ion source in the experiments shown in Figures 3 and 4. The main source of H-atoms in this system is through the decomposition of phenyl, C6H5 f o-C6H4 + H, with the phenyl radicals being the primary product of C6H5I pyrolysis. However, the reaction temperature for the experiment shown in Figure 3 is too low for significant dissociation of phenyl radicals.26 Consequently, HI cannot be formed here by reaction between H and I atoms in either the shock tube or molecular beam. Of course, the lack of H-atoms also eliminates the reaction C6H5I + H ) C6H5 + HI as a source of HI. Furthermore, I-atoms have very low reactivities and are unlikely to participate in H-abstraction reactions. Thus, it appears that the HI+ ions must be formed from HI molecules that were thermally eliminated from C6H5I. The peak areas in the mass spectra are proportional to the concentration of a species through the ionization cross section. At 30 eV the theoretical electron impact ionization cross sections for HI and I27–29 are very similar, 5.5 × 10-16 cm2. Thus, the observed HI+/I+ ratios indicate that up to 10% of the dissociation of C6H5I must proceed through HI elimination, rather than C-I fission. As far as we are aware, this is the first direct evidence for HI elimination from C6H5I in the pyrolysis of phenyl iodide. Previously, Kumaran et al.16 concluded that C-I fission was the sole route for decomposition of C6H5I. In I-atom atomic resonance absorption spectrometry (I-ARAS) experiments they assumed that at long reaction times C6H5I would be completely consumed and then determined [I]∞/[C6H5I]0 ) 1.01 ( 0.1, where [C6H5I]0 was the initial phenyl iodide concentration and [I]∞ was the I-atom concentration at long reaction times. This ratio suggests that C-I fission is the only dissociation channel available to C6H5I. However, their error limits are also compatible with up to 10% elimination of HI. Rao and Skinner15 estimated that C6H5I dissociation would proceed by 73% elimination of HI, which is incompatible with both the current observations and the results of Kumaran and co-workers.16 m/z ) 76, 77, and 78 (C6H4, C6H5, and C6H6). Peaks at m/z ) 78 and m/z ) 76 are seen in both the pre- and postshock mass spectra and in the literature mass spectrum of C6H5I. In the preshock spectra the m/z ) 78 to m/z ) 77 ratio of ∼0.07 is consistent with the natural abundance of 13C in C6H5 and similar ratios are observed in the postshock mass spectrum in Figure 3b. Furthermore, in Figure 3b m/z ) 204 (C6H5I+) is still the dominant peak and the m/z ) 76 to m/z ) 77 ratio is only ∼0.04 in both the pre- and postshock mass spectra. These observations suggest that in Figure 3b the m/z ) 76, 77, and 78 peaks are mainly due to fragmentation of the parent molecule in the ion source, which is consistent with slow dissociation of C6H5I at the reaction conditions of this experiment. However,
Self-Reaction of Phenyl Radicals the presence of a strong I+ peak in Figure 3b does indicate some dissociation of C6H5I, as noted above. The postshock mass spectrum in Figure 4b is rather more informative as here almost all of the parent molecule has been consumed. In Figure 4b (see also Figure 5) the m/z ) 78 to m/z ) 77 ratio is ∼2.2 and the m/z ) 76 to m/z ) 77 ratio is ∼0.5, which shows that both C6H6 and C6H4 are now being formed during the pyrolysis of C6H5I. In principle, C6H6 could be formed by addition of H-atoms to C6H5; however, as mentioned earlier, the dissociation of C6H5 radicals to C6H4 + H is the main source of H-atoms in this system and even at 1511 K this reaction is quite slow.26 Thus, it is unlikely, especially at short reaction times, that either a significant amount of C6H6 is formed from H + C6H5 or that a significant fraction of the m/z ) 76 peak is due to C6H4 from dissociation of C6H5. The simultaneous appearance of C6H4+ and C6H6+ for small extents of reaction at low and high temperatures is consistent with the presence of abstraction channels in the self-reaction of phenyl radicals. The C6H4+/C6H6+ ratio from Figure 4b is ∼0.23, whereas the pure abstraction process should yield a C6H6/ C6H4 ratio of 1. However, C6H4 has significant diradical character and typically the electron impact ionization cross sections of radicals are much smaller than for stable species. Consequently, the actual concentrations of C6H4 and C6H6 may be closer than a simple comparison of the peak areas suggests. The possible contribution of C6H5 + C6H5 abstraction channels is explored again in detail later in the theory and LS modeling sections (sections 5 and 6). m/z ) 152 and 154 (C12H8 and C12H10). A peak at m/z ) 154 appears, apparently simultaneously, with a peak at m/z ) 152 in the postshock mass spectra immediately following reflection of the shock wave in both the T5 ) 1259 K and T5 ) 1511 K experiments; see Figures 3b, 4b, and 5, respectively. The observation of m/z ) 154 (C12H10+) is consistent with both previous experimental studies of phenyl recombination and with theoretical predictions for reaction 1a, the σ-bond recombination of phenyl radicals to biphenyl; see below. The m/z ) 152 peak (C12H8+) could be due to fragmentation of biphenyl in the ion source of the TOF-MS. However, the 70 eV literature mass spectrum25 for biphenyl gives ratios (m/z ) 153)/(m/z ) 154) ) 0.5 and (m/z ) 152)/ (m/z ) 154) ) 0.33, whereas in Figure 5 (50 eV), (m/z ) 152)/(m/z ) 154) ) 0.6 and the m/z ) 153 peak is barely discernible. If the m/z ) 152 peak in Figure 5 is an ionization fragment of m/z ) 154, then a stronger m/z ) 153 peak should be found in Figure 5 than is observed. Thus, the m/z ) 152 peak in these experiments actually represents a reaction product, C12H8, and is not an ionization fragment from biphenyl. Potential sources and identities of m/z ) 152 are discussed later in the section on modeling of the laser schlieren experiments (section 6). 3.2. Laser Schlieren. Thirty-nine experiments were performed behind incident shock waves over the range 1334 K < T2 < 1853 K and three reaction pressures, P2, of 22 ( 2, 54 ( 7, and 122 ( 6 Torr; here the subscript “2” refers to conditions behind the incident shock wave. In the 54 and 122 Torr experiments the driven gas consisted of 2% C6H5I dilute in krypton with helium as driver gas. In the 22 Torr experiments 4% and 10% C6H5I/Kr mixtures were used to enhance the signal. The loading and reaction conditions for each experiment are given in Table 1. In Figure 6 semilog plots of density gradients derived from Figure 1 are shown. In each of the subplots of Figure 6 the first few points are from the interaction of the shock front with the laser beam and the clear break in the signals around 1 µs represents the
J. Phys. Chem. A, Vol. 114, No. 32, 2010 8245 TABLE 1: Loading, Reaction Conditions, and Rate Coefficient k2a for Laser Schlieren Experimentsa T1 / K
P1/Torr
P2/Torr
T2/K
k2a/103 s-1
297.8 297.8 298.0 297.8 298.0 298.0 298.0 298.0 299.5 298.0 299.5 299.5 296.5 299.5 296.5 299.6 296.8 299.6 296.8 298.0 296.8 299.6 296.9 296.9 299.5 297.6 299.5 297.6 299.5 297.6 297.7 299.6 297.6
0.90 0.71 0.90 0.69 0.70 3.50 3.50 3.50 3.00 3.00 3.00 3.50 2.75 2.75 2.60 2.50 2.35 2.25 2.10 2.50 2.10 1.91 2.05 2.00 7.50 7.50 7.00 7.00 6.50 6.50 6.50 6.00 6.00
22 23 20 24 20 56 59 61 53 55 56 66 54 53 52 52 52 50 48 57 50 47 52 52 124 128 119 124 118 121 127 117 118
1221 1450 1464 1560 1762 1276 1334 1353 1390 1415 1434 1448 1482 1485 1512 1574 1624 1645 1675 1676 1738 1778 1818 1853 1312 1335 1344 1381 1405 1432 1484 1485 1494
1.04b 12.71b 12.83c 31.81b 94.35c 5.49 10.11 12.08 19.88 24.49 29.43 36.74 45.12 48.44 60.23 94.31 129.20 145.79 171.39 164.53 231.69 273.76 303.62 341.55 9.81 12.42 15.28 23.39 30.69 40.95 65.01 67.14 71.21
a Unless noted otherwise, all experiments were performed with 2% C6H5I/Kr. b Performed with 10% C6H5I/Kr. c Performed with 4% C6H5I/Kr.
appearance of signal due to chemical reaction. The pattern shown in Figure 6b,d, of positive density gradients followed by negative density gradients, is typical of a reaction system where a fast endothermic process precedes a strong exothermic reaction. In the following subsection, we initially discuss the results for the endothermic processes (i.e., C6H5I f C6H5 + I and C6H5I f C6H4 + HI). The later density gradients arising from the exothermic process (i.e., C6H5 + C6H5 f products) will be discussed further on in the modeling section (section 6), which focuses on the development of a realistic chemical mechanism for C6H5 self-reaction. Dissociation of C6H5I. The dissociation of phenyl iodide has two possible low energy channels: scission of the C-I bond (2a) and elimination of HI (2b).
C6H5I f C6H5 + I
∆Hr,298 ) 66.8 kcal/mol
(2a) f C6H4 + HI
∆Hr,298 ) 77.7 kcal/mol (2b)
(The ∆Hr,298 values of 66.8 kcal/mol in eq 2a and 77.7 kcal/ mol in eq 2b are from refs 4 and 5.) Previous work on these includes that of Kumaran et al.,16 who used I-ARAS to measure k2a for temperatures ranging from 1080 to 1420 K and pressures ranging from 103 to 731 Torr. Robaugh and Tsang14 used the comparative rate method in a single-pulse shock tube to measure
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Tranter et al.
Figure 6. Semilog density gradient plots derived from the laser schlieren profiles shown in Figure 1. Absolute values are plotted. Circles represent experimental data, while light and dark gray represent positive and negative density gradients, respectively. Solid lines are the results of simulations using the improved mechanism provided in Table 2; see text for details. In (a) and (b), the start of the simulation has been delayed by ti µs to include an estimate of incubation.
k2a at temperatures of 1050-1200 K and pressures of 2-6 atm. They minimized secondary reactions by using low reagent concentrations in the presence of a large excess of cyclopentane, which resulted in quantitative conversion of the phenyl product to benzene. Robaugh and Tsang and Kumaran et al. concluded that the dissociation of phenyl iodide proceeds only via reaction 2a with no contribution from reaction 2b. Rao and Skinner15 observed the formation of H-atoms from benzene and halogenated benzenes by H-ARAS in very dilute (1-10 ppm) mixtures behind reflected shock waves, over 1400-1900 K and 0.4 atm. From their experiments with iodobenzene, and comparing the results of Robaugh and Tsang, they concluded that complete reaction through reaction 2a was not consistent with their H-ARAS results and they hypothesized that reaction 2b could be responsible for up to 73% of the loss of C6H5I, although no direct evidence for reaction 2b was obtained. The current TOF-MS results indicate that C6H5I decomposition does proceed to some extent by reaction 2b, but with a maximum consumption of about 10%. Thus, at t0, both C-I fission, (2a), and HI elimination, (2b), contribute to the net density gradient. Variation of the branching ratio k2b/(k2a + k2b) between 0 and 0.1 has a much less than 10% effect on the density gradient at t0 because of the small difference in ∆Hr,298. However, at high temperatures, inclusion of reaction 2b in the modeling does have a small effect on the density gradient after about 1 µs from subsequent reactions involving the product o-benzyne. In the lower temperature experiments at all reaction pressures the initial density gradient can be accurately estimated by simple eyeball extrapolation of the experimental data back to t0, with subsequent refinement of this estimate through
simulation of the whole profile with the mechanism of Table 2 (cf. Figure 6a,c). Typically, such a simulation-based refinement changes the initial rate estimates by no more than 10-15%. For the higher temperatures, where the initial slope of the density gradient profile is steeper, the density gradient at t0 is more difficult to estimate. Furthermore, the possibility of incubation delay must also be recognized. Nevertheless, simulations of the density gradient with the mechanism in Table 2 yield less than a 20% variation of the initial estimates (cf. modeling section (section 6) for more details). In Figure 6a,b the start of the simulation has been delayed by an incubation time of ti µs as shown by the vertical dotted lines. Without this delay, the simulations of the 22 and 54 Torr experiments run parallel and on the underside of the experimental data. This offset between the experimental results and the simulation is greater than can be attributed to uncertainty in the location of t0, and adjustment of rate coefficients within reasonable bounds cannot improve the fit of the simulations to the experimental results. In LS experiments, this situation is typically indicative of an incubation delay preceding dissociation; see, for instance, examples in refs 30 and 31. In this work we have not attempted to observe vibrational relaxation and incubation in C6H5I, and there are no literature data for these. Over the range of the current work, the incubation delay was estimated32 to be about 0.3-0.4 µs at 54 Torr and ∼0.6 µs at 22 Torr. In Figure 6a,b, the start of simulation is delayed accordingly. At 122 Torr the estimated incubation delay is less than the expected uncertainty in t0, and this effect is ignored. The rate coefficients for C-I fission, k2a, as determined from the LS experiments are presented in Table 1 and Figure 7, where they are compared with the experimental data from the
Self-Reaction of Phenyl Radicals
J. Phys. Chem. A, Vol. 114, No. 32, 2010 8247
TABLE 2: Reaction Mechanism Used To Simulate C6H5I/Kr Dissociationa reaction
C6H5 + C6H5 f C12H10 C6H5I f C6H5 + I C6H5I f C6H5 + I C6H5I f C6H5 + I C6H5I f o-C6H4 + HI C6H5 + C6H5 f o-C6H4 + C6H6 C6H5 + C6H5 f Z-C6H4 + C6H6 C6H5 f o-C6H4 + H C6H5I + C6H5 f C12H10 + I C6H5 + H f C6H6 C6H6 + H f C6H5 + H2 C6H6 + C6H5 f C12H10 + H C6H5I + H f C6H5 + HI H + HI f H2 + I o-C6H4 f C4H2 + C2H2 o-C6H4 f c-C6H3 + H c-C6H3 f C6H3 C6H3 f C6H2 + H C6H2 + M f C6H + H + M C4H2 f C4H + H C4H f C4 + H C2H2 + M f C2H + H + M o-C6H4 + C2H2 f C8H6 C6H5 + C2H2 f C8H6 + H C2H + H2 f H + C2H2 C4H + H2 f H + C4H2 C2H2 + C2H f C4H2 + H C4H2 + C2H f C6H2 + H C4H + C2H2 f C6H2 + H C4H + H + M f C4H2 + M H + H + M f H2 + M C6H5 + H f o-C6H4 + H2 o-C6H4 + o-C6H4 f C12H8
(1a) (2a) (2a) (2a) (2b) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30)
log(A)
n
22.508
-2.81
(Ea/R)/103
∆Hr,298
source
2.41
-117.6
this workb
72.783
-17.1
46.73
66.8
this workd
(54 Torr)
c
77.483
-18.6
47.63
66.8
this workd
(22 Torr)
c
83.151
-20.4
48.60
66.8
this workd
77.7
this worke
(122 Torr)c
e g
e
e
-2.768
4.57
-2.89
-30.9
this workf
-3.368
4.57
-2.89
-16.1
this workf
58.19
81.9
74.195
-17.2
26h
12.300
0
5.54
-50.8
58
49.495
-10.2
11.41
-112.8
26h
14.540
0
8.11
8.6
58
12.300
0
5.941
2.5
5.54
-4.8
26
-0.07
-4.1
66
13.600
0
0.00
-33.3
58
14.813
0
41.77
53.8
57
15.778
0
49.90
116.0
57
10.681
0
27.00
-11.2
57
10.362
0
18.14
57.1
57
16.699
0
40.00
132.1
64
14.342
0
58.72
135.7
65
14.130
0
58.72
106.0
65
16.556
0
53.59
133.4
65
13.301
0
10.06
-86.5
26
12.667
0
3.98
-4.6
26
5.690
2.5
0.28
-29.2
26
5.690
2.5
0.28
-31.5
26
13.982
0
0.00
-28.8
11
13.477
0
0.00
-25.4
26
13.477
0
0.00
-27.7
26
29.240
-5.45
0.78
-135.7
26
17.800
-1
0.00
-104.2
26
4.54
-22.3
3
-0.69
-123.4
-5.928 9.695
6.18 0.827
this work
a Rate constants are given as k ) ATn exp(-Ea/RT), with units of cm3, mol, s, and kcal. b Reaction 1 is treated as a composite of channels 1a, 1-C3s, and 1-C4s; i.e., k1 is estimated from k1a at 0.1 atm in Table 7 and k1-C3s and k1-C4s are estimated from Table 9. c Apart from reactions 2a, all pressure-dependent rate coefficients are given for 121 Torr. d k2a is obtained from the present modified Gorin model RRKM calculations (see text). e k2b ) (0.06 ( 0.01)k2a. f k3 and k4 are estimated from the total abstraction rate given in Table 8. See the text for details. g Z-C6H4 is (Z)-hexa-3-en-1,5-diyne. h Pressure-dependent rate coefficient. Calculated from the Troe parameters given in ref 26.
literature14,16–18 as well as with a restricted rotor Gorin model RRKM calculation. The present experimental k2a values show little scatter and a clear pressure dependence which is predicted quite well by the Gorin model calculations. The present RRKM calculations are based on the earlier Gorin model of Kumaran et al.,16 but employ revised values for the 0 K barrier, E0, and treat the energy transfer parameter, ∆Edown, and the hindrance parameter, η0,33 as fitting parameters. In their work, Kumaran et al.16 found that a ∆Edown of 447 ( 92 cm-1 and a temperatureindependent η0 of 0.04 allowed them to reproduce their data as well of those of the prior experiments. Their fitting procedure also yielded an estimated reaction energy of E0 ) 66.7 ( 0.7 kcal/mol. In this work, we instead employ E0 ) 67.1 ( 0.3 kcal mol-1, which was obtained from recent evaluations of the thermochemical data for C6H5I (∆Hr,0 ) 42.2 kcal/mol) and C6H5 (∆Hr,0 ) 83.8 kcal/mol) by Ruscic,5 and is within the error bars suggested by Kumaran et al. The fits to the present
experimental data, Figure 7, now require a larger ∆Edown ) 800 cm-1 and a temperature-dependent η0 ) 0.04T0.19, ∼0.16 over the range of the LS experiments. A consequence of the larger η0 is that the extrapolated high pressure limit rate coefficient, k∞ (Figure 7), is increased by a factor of 2-2.8 over 1000-1800 K relative to that of Kumaran et al. The experimental results of Robaugh et al.14 lie about a factor of 3.6 lower than k∞ and those of Kominar et al.17 are about 50% lower. It has not been possible to produce a Gorin model that provides a satisfactory fit to the literature data and the current work by adjustments of E0, ∆Edown, and/or η. However, the current data show little scatter, resolve the pressure dependence, cover a larger temperature range, and are reproduced by a model with reasonable fitted values for ∆Edown and η0, and with E0 determined independent of the LS experiments. The present 122 Torr, first order k2a values are about a factor of 1.6 higher than the recommendation of Kumaran et al. at
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Tranter et al. atures that are too low for H elimination from phenyl radicals. Furthermore, a peak appears at m/z ) 152, apparently simultaneously with m/z ) 154, which is not a fragment of biphenyl, and at the temperatures of these experiments, m/z ) 152 is unlikely to be formed by sequential H-atom eliminations from biphenyl. Also, as will be discussed in the modeling of the DFST/LS experiments, the laser schlieren profiles are inconsistent with a reaction mechanism that includes biphenyl as the sole product of phenyl recombination. In an attempt to rationalize these observations, we next apply high-level theoretical methods to a study of the mechanism and kinetics of the self-reaction of phenyl radicals. These theoretical methods and their results will be discussed in sections 4 and 5 prior to discussion of the modeling of the full density gradient profiles from the LS experiments. The DFST/TOF-MS experiments provide evidence that up to 10% of the C6H5I molecules dissociate by HI elimination rather than C-I fission. This evidence for the two dissociation channels was used to estimate the density gradient at t0 in calculating the rate coefficients for C-I fission from the DFST/ LS experiments. The C-I fission rate coefficients were simulated with a Gorin model RRKM calculation to obtain modified Arrhenius expressions that are then used later in modeling the complete density gradient profile for each experiment. 4. Theoretical Methods
Figure 7. Comparison of first order rate coefficients for the reaction C6H5I f C6H5 + I. Part a shows current results and the experimental points from Kumaran et al.16 Part b includes the lower temperature results of Robaugh and Tsang,14 Kominar et al.,17 and Butler and Polanyi.18 The 22, 54, and 122 Torr RRKM calculations and experimental results of Kumaran et al. have been removed for clarity. LS 2% C6H5I/Kr: red 2, 122 Torr; blue 9, 54 Torr. LS 4% and 10% C6H5I/ Kr: red b, 22 Torr. Results of Gorin model RRKM calculations with E0 ) 67.1 kcal mol-1, ∆Edown ) 800 cm-1, η ) 1 - 0.04T0.19, and molecular parameters from ref 16: red ---, 22 Torr; blue s, 54 Torr; red s, 122 Torr; s, k∞. Kumaran et al.16 experiments, O; first order fit, 1-1. Robaugh and Tsang,14 f-f. Kominar et al.,17 right-pointing triangles. Butler and Polanyi,18 9.
TABLE 3: Modified Arrhenius Parametersa for C-I Fission in C6H5Ib pressure/Torr
log(A)
n
E
∞ 20 60 130
19.022 72.783 77.483 83.151
-0.98 -17.1 -18.6 -20.4
34.28 46.73 47.63 48.60
a k ) ATn exp(-E/T) s-1, where T is in K and is valid over 1000-1900 K. b From fit of present Gorin model RRKM calculations to LS data.
1500 K, and this increases to 1.9 at 1300 K, although the present results do fall within the scatter of Kumaran et al.’s experimental points, Figure 7a. These differences in k2a, and the effect they have on simulation of the LS profiles, are considered later in the modeling section (section 6). Modified Arrhenius fits to the present Gorin model results are reported in Table 3. 3.3. Summary of Experimental Results. The results of the DFST/TOF-MS experiments indicate that the self-reaction of phenyl reactions does not simply lead to biphenyl. For instance, both benzene and benzyne are formed at temper-
Two phenyl radicals can undergo a variety of reactions including radical-radical recombination, abstraction, and radical π-bond addition. For each of these classes of reaction there are multiple possible products. Furthermore, in each case the reaction can occur on either a singlet or triplet electronic surface. Past studies have emphasized the radical-radical recombination to form biphenyl, reaction 1.7,8 However, the present TOF-MS and LS results indicate that the abstraction reaction is also of some significance. Here we provide detailed theoretical kinetics analyses for all three of these classes of reactions employing a variety of different methodologies, as discussed in the following subsections. Structures and vibrational frequencies for the various species examined theoretically are provided in the Supporting Information. At low temperatures, many of the transition states lie at large radical-radical distances and there is not yet a clear separation into the three distinct sets of reactions. However, our interest is specifically in predicting the rate coefficients at the high temperatures of relevance to soot formation as examined in the present experimental study. By such temperatures, the overall transition state dividing surface is at separations of 2-3 Å and appears to separate quite effectively into local ones for each of the three types of reactions. Thus, for simplicity, in the present analysis we consider only this picture of distinct transition states with the overall reactive flux given by the sum of the individual components and the branching given by the ratios of these fluxes. This restriction may cause some inaccuracies in the predicted rate coefficients and branching ratios near room temperature. 4.1. Radical-Radical σ-Recombination. Thisradical-radical recombination involves the formation of a chemically activated biphenyl adduct (C6H5-C6H5*) followed by either its stabilization or dissociation to bimolecular products. CH bond fission is expected to provide the lowest energy decomposition pathway and can occur at any one of the three distinct CH bonds in biphenyl. Thus, as illustrated schematically in Figure 8, the overall recombination mechanism considered here is
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Figure 8. Schematic plot of the potential energy surface for the recombination reaction. The numbers in parentheses denote the HL energies, EHL, in kcal/mol relative to two phenyl radicals. EHL is given by expression II in the text. Figure 10. Plot of phenyl-phenyl interaction energy along the minimum energy path for addition in both the singlet and triplet electronic states and for both cc-pvdz and cc-pvtz basis sets for the singlet state.
Figure 9. Structures for the stationary points in the recombination reaction.
C6H5 + C6H5 f C6H5-C6H5* C6H5-C6H5* f C6H5-C6H5
(1a)
f o-C6H5-C6H4 + H
(1b)
f m-C6H5-C6H4 + H
(1c)
f p-C6H5-C6H4 + H
(1d)
where o, m, and p denote the radicals obtained via CH fission ortho, meta, and para to the CC bond between the two phenyl rings. The corresponding structures are illustrated in Figure 9. The rovibrational properties of the stationary points on this potential energy surface were studied with B3LYP (Becke-3 Lee-Yang-Parr)34 density functional theory employing the 6-311++G(d,p) basis set.35 Higher level estimates for the energies of the stationary points, EHL, are obtained from eq II, which includes a series of MP2 (second order Møller-Plesset perturbation theory)35 and QCISD(T) (quadratic configuration interaction with perturbative inclusion of triples)36 calculations at the B3LYP optimized geometries. These higher level estimates employ the 6-311++G(d,p) and 6-311++G(3df,2pd) basis sets to obtain an approximate QCISD(T)/6-311++ G(3df,2pd) estimate:
EHL ) EQCISD(T)/6-311++G(d,p) + EMP2/6-311++G(3df,2pd) EMP2/6-311++G(d,p) + E0
(II)
where E0 denotes the vibrational zero-point energy obtained from the B3LYP/6-311++G(d,p) evaluations. The second of the two MP2 calculations includes the core electrons as active, whereas the others do not. These higher level energy estimates, EHL, are employed in the master equation analyses. The minimum energy path potential for the initial recombination, reaction 1a, is barrierless. In this case, both variational and anharmonic effects have a major impact on the predicted kinetics within transition state theory (TST). Here, the direct variable reaction coordinate (VRC) transition state theory approach37–40 is implemented. This approach has been shown to provide an effective means for treating these complications.41,42 These calculations are performed as previously described for the case of two alkyl radicals.42 The VRC-TST approach requires some method for estimating the interaction between the two reacting radicals for arbitrary orientation and separation. Here, these interactions are evaluated directly with multireference second order perturbation theory (CASPT2) employing a two-electron two-orbital (2e,2o) complete active space (CAS) consisting of the radical orbitals on each of the phenyl radicals. These CASPT2 calculations employ Dunning’s correlation consistent polarized valence double-ζ basis set (cc-pvdz)43 and were done using the formalism of Celani and Werner44 as implemented in the Molpro program package.45–47 Sample calculations employing larger basis sets and incorporating some of the π-orbitals in the active space yield essentially identical reaction path energies. Thus, no basis set or active space corrections are incorporated here. The CASPT2(2e,2o)/cc-pvdz singlet and triplet energies along the singlet minimum energy path are illustrated in Figure 10. At short range the triplet state is repulsive for this geometry. Thus, the contribution from the triplet state to the addition kinetics is expected to be negligible, and only the singlet state is considered in the direct VRC-TST analysis. The presence of radical-radical σ-addition, radical π-bond addition, and abstraction channels all on the same potential energy surface complicates the analysis. Here the σ-addition to form biphenyl is separated from the remaining channels by placing pivot points on the C-atoms neighboring the radical C-atom as well as along the radical orbital of the radical C-atom. This placement allows the reactive flux on the face leading to biphenyl to be evaluated separately from that on the other faces. At low temperatures, the transition state lies at fairly large distances, and this separation is necessarily rather approximate.
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However, by about 500 K, the transition state has moved in to such short separations that the distinction between the different channels should be fairly accurate. In prior studies one-dimensional corrections for the effects of limitations in the basis set and the effect of geometric relaxation of the conserved vibrational modes have sometimes been included.41,42 However, the plots in Figure 10 for the phenyl recombination indicate that along the minimum energy path the interaction energies for the cc-pvtz basis are essentially identical to those for the cc-pvdz basis and so the basis set correction is expected to be negligible. Furthermore, the geometric relaxation is expected to be particularly small for this reaction since the formation of biphenyl involves relatively modest structural changes in the phenyl radical. Thus, these one-dimensional corrections are not included here. However, a standard dynamic correction factor of 0.85 was included, which improved the agreement between VRC-TST and trajectory simulations for CH3 radical selfrecombination reactions and between VRC-TST and experiment for related alkyl radical recombination reactions.42 This direct VRC-TST analysis should provide a quantitative estimate of the high pressure recombination rate coefficient. It is also of interest to predict the kinetics at lower pressures, particularly near 100 Torr as in the present experiments. The combination of a deep well and many vibrational modes implies that there should be little, if any, redissociation back to reactants for typical ranges of temperature and pressure. However, since the o-, m-, and p-C12H9 + H channels are moderately exothermic, the branching between stabilization and bimolecular product formation may be pressure dependent over the relevant pressure range. To examine this issue, master equation simulations were performed as described in some related studies.48,49 These master equation simulations employ an exponential down model for the collisional energy transfer process and Lennard-Jones collision rates. The average value of the downward energy transfer, 〈∆Edown〉, is taken to be 400(T/298)0.85 cm-1, where T is in kelvin. The Lennard-Jones parameters for biphenyl are taken from the related RRKM modified strong collider study of Wang and Frenklach,11 while those for Kr are taken from ref 50. For biphenyl and for each of the o-, m-, and p-C12H9 species, the stationary points on the torsional potential were explored for the relative rotation of the two C6 rings and hindered rotor treatments of this degree of freedom were employed. These master equation simulations also require some estimate of the reactive flux for the product channels in reactions 1b-1d. The potentials for these product channels are also barrierless. Thus, the direct CASPT2(2e,2o)/cc-pvdz based VRC-TST approach was also employed to predict the reactive flux for these channels. A one-dimensional basis set correction is included for these channels which is taken to be the same as that calculated earlier for C6H5 + H.41 A dynamic correction factor of 0.9 is also included, which improves the agreement between VRC-TST and trajectory simulations for a number of radical-Hatom recombinations.41 All of these VRC-TST predictions are only weakly dependent on the equilibrium geometries of the radicals, so they were simply determined from UB3LYP/6-311++G** or UB3LYP/ cc-pvdz calculations using the Gaussian 98 program.51 Previous direct VRC-TST calculations for related systems have agreed with experimental measurements to within about 10-20%.41,42 The more limited reaction coordinate optimizations together with the multiple reaction sites somewhat increases the uncertainties for the present calculations, particularly at low temperature. For the temperatures of interest here we crudely estimate the predictions for the high pressure recombination rates to be
Tranter et al. accurate to about 30%. Meanwhile, the pressure dependence of the branching ratios is considerably more uncertain, with error bars of perhaps a factor of 2 or 3. 4.2. Abstraction. The radical site on one phenyl radical can abstract any one of the three distinct H atoms on the other phenyl radical to yield benzene plus o-, m-, or p-benzyne. Furthermore, in each case, the abstraction may occur on either a singlet or triplet electronic surface to yield six independent abstraction channels:
C6H5 + C6H5 f C6H6 + o-C6H4
(3os)
C6H5 + C6H5 f C6H6 + m-C6H4
(3ms)
C6H5 + C6H5 f C6H6 + p-C6H4
(4ps)
C6H5 + C6H5 f C6H6 + 3o-C6H4
(3ot)
C6H5 + C6H5 f C6H6 + 3m-C6H4
(3mt)
C6H5 + C6H5 f C6H6 + 3p-C6H4
(4pt)
The rovibrational properties of the saddle points for each of these abstraction reactions were studied at the CASPT2(2e,2o)/cc-pvdz level, where the two active orbitals correlate with the two radical orbitals of the phenyl radical reactants. In each of these evaluations two distinct torsional geometries are considered: one where the two C6 rings are in the same plane, and one where they are in perpendicular planes. Further evaluations with the cc-pvtz basis at these CASPT2/cc-pvdz geometries allow for the determination of complete basis set (CBS) limit CASPT2 energy estimates as52,53
ECASPT2/CBS ∼ ECASPT2/cc-pvtz + 0.4629(ECASPT2/cc-pvtz ECASPT2/cc-pvdz) (III) Corresponding CASPT2 calculations were carried out for the phenyl radical. The results of these CASPT2 evaluations are summarized in Table 4, and the structures of the saddle points are illustrated in Figure 11. Notably, the zero-point corrected barriers are all lower than the electronic barriers. This reduction in the zeropoint energy is related to the decrease in the CH stretching frequency as the H is abstracted from one phenyl radical to the other. More importantly, the zero-point energy barriers for the ortho and meta singlet abstractions are both negative; i.e., they lie below the energy of the two phenyl radicals. Thus, the abstraction rate coefficient for these two channels should be quite large, perhaps comparable to those for the addition process. As expected, the lowest abstraction barrier is for that leading to singlet o-benzyne because this product channel has the greatest exothermicity. The barriers for the perpendicular orientation are generally lower than that for the parallel orientation. This preference for the perpendicular orientation is likely indicative of steric repulsions between the H-atoms on the different phenyl radicals in the planar orientation. Correspondingly, the lowest vibrational frequencies for the planar orientation are generally greater than those for the perpendicular orientation. This variation in vibrational frequencies implies that the usual assumption of the separability of torsional and other motions is problematic. Indeed, for the singlet ortho case, the variation in vibrational frequencies correlates with a factor of
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TABLE 4: CASPT2(2e,2o)/CBS//CASPT2(2e,2o)/cc-pvdz Stationary Point Properties for the Abstraction Saddle Points productsa
Eb/(kcal/mol)
E + E0c/(kcal/mol)
frequenciesd/cm-1
rotational constants/GHz
o-C6H4 + C6H6
-4.12 -1.52 0.56 -0.11 5.37 4.77 8.04 7.48 7.06 6.32 6.21 5.33
-5.33 -3.10 -2.07 -2.48 1.69 1.26 4.50 4.21 3.41 2.87 2.41 1.82
136i, 37, 49, 53, 101, 191 122i, 7, 17, 25, 104, 167 413i, 29i, 32, 38, 110, 126 352i, 11, 24, 26, 110, 134 1621i, 25i, 34, 35, 130, 147 1587i, 13, 28, 29, 130, 149 1787i, 22i, 31, 35, 129, 143 1754i, 13, 29, 30, 129, 142 1787i, 28i, 34, 37, 128, 146 1744i, 14, 28, 28, 130, 145 1688i, 29i, 35, 37, 132, 143 1626i, 15, 28, 28, 127, 152
2.845, 0.390, 0.343 2.882, 0.339, 0.338 2.849, 0.376, 0.332 2.813, 0.357, 0.356 2.772, 0.386, 0.339 2.771, 0.362, 0.362 2.870, 0.378, 0.334 2.875, 0.356, 0.355 2.885, 0.383, 0.338 2.883, 0.361, 0.360 2.761, 0.389, 0.341 2.758, 0.365, 0.364
m-C6H4 + C6H6 p-C6H4 + C6H6 3
o-C6H4 + C6H6
3
m-C6H4 + C6H6
3
p-C6H4 + C6H6
a For each set of products the first row of numbers is for the torsional state with both phenyls in the same plane, while the second set are for the two phenyls in orthogonal planes. b Electronic energy at the saddle point relative to C6H5 + C6H5. c Zero-point corrected energy at the saddle point relative to C6H5 + C6H5. d Vibrational frequencies for the six lowest frequency and/or imaginary modes.
by the fact that, even for the singlet o-benzyne channel (3os), the rate coefficient is already an order of magnitude lower than the long-range collision rate at 300 K. Overall, we expect that the theoretical predictions for the abstraction rate coefficients will be accurate to about a factor of 2 for temperatures in the 1000-2000 K range. These estimates are based on experiencebased estimates of the uncertainties in the barrier heights (∼1 kcal/mol) and state counts (∼50%). 4.3. Radical π-Bond Recombination. The radical site of one of the phenyl radicals can add across any of the CC π-bonds of the other phenyl radical. There are four distinct C atoms at which this addition can occur, and the process can involve either the singlet or triplet electronic state: Figure 11. Plot of the saddle point structures in C6H5 + C6H5 abstraction reactions.
4 larger rate coefficient for the perpendicular orientation. For the other cases, this difference is generally much smaller, with a ratio of 1.35 being fairly typical. A proper treatment of the coupling between the torsion and low frequency vibrations should include a Boltzmann weighted average over the torsional angles. However, to do this averaging with high precision was deemed too computationally intensive for this study. Instead, an approximate scheme was used that employs the energy values for the planar and perpendicular torsional states to define a low order torsional potential via a Fourier series fit. A torsional average is then obtained by assuming that the remaining vibrational frequencies were equal to those for the planar geometry for torsional angles between 0 and 45° and were equal to those for the perpendicular geometry for torsional angles between 45 and 90°. The remaining degrees of freedom were treated with rigid rotor harmonic oscillator assumptions, and the rate constant was evaluated according to transition state theory. No treatment for tunneling was incorporated here because, at the high temperatures of interest here, tunneling should be insignificant. Furthermore, at low temperature, channels 3os and 3ms will dominate the abstraction process. The latter abstractions will have negligible tunneling contributions for all temperatures due to their low imaginary frequencies and negative activation energies. The effect of the long-range transition state for channels 3os and 3ms, where the saddle point is lower than the reactants, was also neglected. This long-range transition state is not expected to have a significant effect at the temperatures of interest here (1000 K and higher). This assumption is validated
C6H5 + C6H5 f -CHCHCHCHCHC(C6H5)- f ... (1-C1s) C6H5 + C6H5 f -CHCHCHCHCH(C6H5)C- f ... (1-C2s) C6H5 + C6H5 f -CHCHCHCH(C6H5)CHC- f ... (1-C3s) C6H5 + C6H5 f -CHCHCH(C6H5)CHCHC- f ... (1-C4s) C6H5 + C6H5 f 3-CHCHCHCHCHC(C6H5)- f ... (1-C1t) C6H5 + C6H5 f 3-CHCHCHCHCH(C6H5)C- f ... (1-C2t) C6H5 + C6H5 f 3-CHCHCHCH(C6H5)CHC- f ... (1-C3t) C6H5 + C6H5 f 3-CHCHCH(C6H5)CHCHC- f ... (1-C4t) CASPT2(2e,2o)/CBS estimates of the barrier heights are obtained according to eq III as described in the preceding subsection for the abstraction reactions (4.2). These energies are reported in Table 5, and the corresponding structures for the triplet saddle points are illustrated in Figure 12. Saddle points
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TABLE 5: CASPT2(2e,2o)/CBS//CASPT2(2e,2o)/cc-pvdz Stationary Point Properties for π-Addition Saddle Points addn site C1 C2 C3 C4 C4′ d c
electonic state
Ea/(kcal/mol)
E + E0b/(kcal/mol)
low frequenciesc/cm-1
rotational constants/GHz
3 3 1 3 1 3 3
0.65 -1.93 -1.65 -1.45 1.07 -1.98 -1.12
0.34 -1.86 -1.77 -1.44 0.72 -1.87 -1.05
453i, 6i, 48, 69, 123, 125 297i, 16, 55, 73, 120, 128 329i, 15, 55, 77, 122, 135 337i, 16, 56, 76, 125, 131 514i, 10, 60, 83, 132, 144 310i, 20, 57, 76, 126, 127 347i, 29i, 41, 104, 114, 140
2.070, 0.581, 0.544 2.114, 0.563, 0.524 2.105, 0.572, 0.532 2.112, 0.572, 0.532 2.098, 0.582, 0.546 2.089, 0.571, 0.536 2.009, 0.589, 0.520
a Electronic energy at the saddle point relative to C6H5 + C6H5. b Zero-point corrected energy at the saddle point relative to C6H5 + C6H5. Vibrational frequencies for the six lowest frequency and/or imaginary modes. d Alternative torsional state.
Figure 12. Plot of the structures for the saddle points in the C6H5 + C6H5 triplet π-addition reactions.
for the singlet addition to the first and second C atoms, channels 1-C1s and 1-C2s, could not be located. Attempts to find these saddle points lead instead to the σ-addition reaction to form biphenyl. The rate coefficients for these π-additions are calculated with conventional transition state theory employing rigid rotor harmonic oscillator assumptions for most of the modes. For the C4 addition on the triplet surface the saddle point properties were evaluated for two distinct torsional orientations. In contrast with the abstraction reactions, these two torsional states have fairly similar rovibrational properties and the usual separation of torsions and vibrations seems appropriate. Harmonic and hindered rotor treatments of this torsional mode for the C4 triplet π-addition have a maximum difference of 25% over the 200-2500 K range. Thus, with one exception, harmonic treatments of even this lowest frequency mode were deemed appropriate for the remaining π-additions. The one exception is for the C1 addition on the triplet state, for which the lowest frequency is 6i. The extremely small value of this frequency makes even its sign uncertain, and this mode was instead treated as a free rotor. Due to the absence of a saddle point for channels 1-C1s and 1-C2s, no predictions are made for these channels. It is not immediately clear what the ultimate products would be for these π-addition reactions. The initial complexes are bound by about 30 kcal/mol. For the singlet reactions, the carbene character of the complexes implies that the barriers for H-atom isomerization will be rather low, indeed likely below any dissociation thresholds. Furthermore, such H-atom transfers lead directly to biphenyl. For benzene there are analogous H-atom transfers that connect the benzene well to various carbene-like species. In benzene, the barriers for these H transfers are each at least 2 kcal/mol below the C6H5 + H asymptote.54 Thus, it seems likely that for the singlet additions the final product will be biphenyl. For the triplet additions, these same H-atom transfer barriers are expected to be significantly
higher. Then, CH fission at the addition site, which leads to the exothermic o-, m-, and p-C12H9 species, may be the dominant process. Explicit examination of these H transfer and CH fission processes was deemed beyond the scope of the present work. Correspondingly, master equation simulations of these addition processes cannot be performed. Nevertheless, it seems likely that their effective rate constants will be significantly reduced from the high pressure recombination rate constants that are presented below due to the relatively weak bonding of the initial adducts, which implies a substantial redissociation of them back to reactants in competition with the isomerization to biphenyl. Overall, we expect that the theoretical predictions for the radical π-bond high pressure recombination rate coefficients will be accurate to about a factor of 3 for temperatures in the 1000-2000 K range. Again, these estimates are based on crude, experience-based, estimates of the uncertainties in the barrier heights (∼1 kcal/mol) and state counts (factor of 2). 5. Theory Results 5.1. Radical-Radical Recombination. The direct CASPT2 VRC-TST predictions for the high pressure recombination rate coefficients for channel 1a and for the reverse of channels 1b-1d are illustrated in Figure 13a. The predictions for the H-atom addition to the m- and p-C12H9 radicals are essentially identical, showing a gradual rise from ∼8 × 1013 to ∼1.2 × 1014 cm3 mol-1 s-1 as the temperature rises from 200 to 2000 K. The rate coefficient for the addition to the o-C12H9 radical is about 0.7 times lower due to the greater steric repulsion arising from the closer proximity of the radical site to the other phenyl ring. The temperature dependence of the rate coefficient for the phenyl self-recombination is markedly different, decreasing by an order of magnitude from 5 × 1013 to 6 × 1012 cm3 mol-1 s-1 as the temperature increases from 200 to 2000 K. This change in behavior is related to the increased steric repulsion and greater number of rotational degrees of freedom. A similar contrast in behavior was found for alkyl + alkyl recombinations compared with alkyl + H-atom recombinations.41,42 Modified Arrhenius fits to these predicted high pressure recombination rate coefficients are provided in Table 6. The present direct CASPT2 VRC-TST predictions for the high pressure recombination to form biphenyl, reaction 1a, are compared with the available experimental data in Figure 13b. The predictions are in reasonably satisfactory agreement with the somewhat disparate experimental results, being about a factor of 1.5 higher than the experimental results of Heckman et al., and of Park and Lin, and about a factor of 3 lower than the results of Horn et al. Each of these experimental results is expected to be reasonably close to the high pressure limit and has been interpreted as relating to just the recombination channel (1a). However, it is not clear what role the abstraction and
Self-Reaction of Phenyl Radicals
Figure 13. (a) Plot of the direct CASPT2 VRC-TST predictions for the C6H5 + C6H5, o-, m-, and p-C12H9 + H high pressure recombination rates. (b) Comparison of the direct CASPT2 VRC-TST predictions for the C6H5 + C6H5 high pressure recombination rate with the available experimental data.
π-addition channels should have had on the interpretation of the experimental results. It is perhaps worth noting that the present estimates for the H + C12H9 recombination rates focused on the lowest energy twisted torsional geometry. The barriers to this torsional rotation are quite modest, with the planar and perpendicular geometries of o-C12H9, for example, being only 0.6 and 2.2 kcal/mol higher in energy, respectively. At combustion temperatures one expects significant contributions from the full range of torsional angles. Thus, a more accurate treatment should average the rate coefficient over a Boltzmann distribution of the ring-ring torsional angles. However, the steric repulsion for the twisted geometry, which has the lowest energy, should be intermediate between that for the planar and perpendicular geometries. Thus, the present emphasis on the twisted geometries should provide reasonable estimates. The present master equation predictions for the pressure dependence of the rate coefficients for channels 1a-1d are illustrated in Figure 14. For temperatures below 1500 K, the recombination process effectively yields only biphenyl. At 1750 K, channels 1b-1d are beginning to become significant for pressures of about 100 Torr or lower. By 2000 K the deviations from the high pressure limit are significant all the way up to 1000 Torr. The branching to p-C12H9 is lower than that to oand m-C12H9 because there are only two CH fissions that lead to this product, whereas there are four that lead to the ortho and meta products. The branching to o-C12H9 is lower than that to m-C12H9 because of the steric effects for this channel as mentioned in the discussion of the reverse high pressure
J. Phys. Chem. A, Vol. 114, No. 32, 2010 8253 recombination rate. Modified Arrhenius fits to the pressuredependent rate coefficients are provided in Table 7 for a range of pressures and 1000-2000 K. The only prior theoretical study of the pressure-dependent recombination appears to be an RRKM modified strong collider study of Wang and Frenklach.11 Their predicted branching between stabilization and C12H9 + H formation is remarkably similar to that of the present analysis. However, their assumed phenyl + phenyl high pressure recombination rate coefficient of 3 × 1012 cm3 mol-1 s-1 is substantially lower than the present a priori prediction. Meanwhile, their assumed H + C12H9 rate coefficient of 1.0 × 1014 cm3 mol-1 s-1 is fairly similar to the average of the present predictions for the three distinct H + C12H9 high pressure recombination rate coefficients. Nevertheless, the present predictions for the branching between stabilization and bimolecular products in the phenyl + phenyl recombination are quite similar to their earlier estimates, perhaps due to other differences such as the significantly lower CC bond energy of 113 kcal/mol in the work of Wang and Frenklach. The present higher level prediction of a greater CC bond energy, which implies an increased branching to C12H9 radicals arising from the phenyl radical self-reaction (since the CH bond strengths in the two studies are nearly equivalent), may indicate a more important role for these radicals in PAH growth. Notably, their reaction with acetylene then connects the single ring in phenyl to the conjugated three ring complex as discussed by Frenklach et al.,55 and as highlighted in recent pyrolysis studies of Koshi and co-workers.1,2 The possible importance of this pathway to PAH growth was also highlighted in the ab initio quantum chemical study of Mebel and co-workers.56 The study of Wang and Frenklach downplayed the importance of this pathway, apparently due to the shifting of the equilibrium between C6H5 + C6H5 and C12H9 + H to favor the two phenyl radicals at high temperature.11 The present prediction of an increased CC bond strength implies this shift will not happen until a higher temperature and the pathway may be of greater significance than predicted in their pioneering modeling study. 5.2. Abstraction. The present theoretical predictions for the phenyl + phenyl high pressure limit abstraction rate coefficients are plotted in Figure 15 and presented in Table 8. At low temperatures the ortho and meta abstractions on the singlet surface are dominant. At higher temperatures, the triplet abstractions increase in importance, with all three benzyne products being of roughly equal importance in the 1000-2000 K region. The overall abstraction rate coefficient is nearly constant at ∼3 × 1012 cm3 mol-1 s-1 in the 500-1000 K range and then gradually rises to ∼4 × 1013 cm3 mol-1 s-1 by ∼2500 K. By this point the total abstraction rate is almost an order of magnitude larger than the recombination rate to form biphenyl. The rise at higher temperature is related to the increasing importance of the triplet channels, which, in turn, is primarily the result of the 3-fold electronic degeneracy of the triplet channels. The relatively large magnitude of these abstraction rate coefficients suggests a more prominent role for benzynelike structures in PAH growth. Notably, o-benzyne is highly reactive due to its strong diradical character. For example, the addition of two o-benzyne radicals is barrierless and so should be quite rapid. Indeed, preliminary results of direct CASPT2 VRC-TST calculations suggest that the recombination rate is given by 4.96 × 109 T0.827 exp(690/T) cm3 mol-1 s-1, which reduces to ∼3 × 1012 in the 1000-2000 K range. Even the reaction of o-benzyne with a closed shell species like acetylene has a barrier of only ∼6 kcal/ mol. Thus, the addition reactions of benzyne-like structures
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TABLE 6: Modified Arrhenius Parametersa for High Pressure Recombination Reactions k∞b d
Keqc
reactants
A
n
E
A
n
E
C6H5 + C6H5 o-C12H9 + H m-C12H9 + H p-C12H9 + H
1.55 (14) 4.27 (13) 1.25 (13) 2.78 (13)
-0.446 0.338 0.284 0.185
-277 -79.6 -77.9 7.7
1.03 (-11) 8.19 (-3) 2.07 (-3) 4.44 (-3)
1.91 -0.083 -0.190 -0.197
-60 530 -56 070 -56 070 -56 240
a Valid for the temperature T in the range from 200 to 2500 K. b k∞ ) ATn exp(-E/T) cm3 mol-1 s-1, where T is in K. c Keq ) ATn exp(-E/ T) cm3 mol-1, where T is in K. d Numbers in parentheses denote powers of 10.
Figure 14. Plot of master equation predictions for pressure-dependent rate coefficients at three temperatures for reactions 1a-1d in C6H5 + C6H5 recombination.
TABLE 7: Modified Arrhenius Parametersa for Reactions 1a-1d at Various Pressures channel
pressure/atm
Ab
n
E
1a
0.01 0.1 1 10 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100
1.66 (64) 6.14 (37) 7.34 (20) 1.11 (14) 3.09 (12) 1.51 (76) 1.20 (48) 2.44 (13) 2.76 (-12) 7.65 (-20) 2.44 (76) 2.81 (48) 3.44 (13) 6.92 (-12) 1.60 (-19) 1.25 (76) 1.10 (48) 2.13 (35) 3.34 (-12) 8.61 (-20)
-14.68 -7.140 -2.335 -0.405 0.036 -16.8 -8.82 0.885 7.72 9.58 -16.86 -8.90 0.868 7.63 9.52 -16.88 -8.89 0.826 7.62 9.49
16740 7903 2076 -307 -857 39390 32270 21730 13750 11520 39440 32390 21750 13870 11620 39520 32410 21870 13940 11710
1b
1c
1d
a k ) ATn exp(-E/T) cm3 mol-1 s-1 with T in K. Valid for T in the range from 1000 to 2000 K. For lower T the high pressure expressions are appropriate over this range of pressure. b Numbers in parentheses denote powers of 10.
provide an interesting alternative pathway for hydrocarbon growth in flames that should be more fully explored. 5.3. Radical π-Bond Recombination. The present TST predictions for the temperature dependence of the high pressure π-addition rate coefficients are illustrated in Figure 16 and given in Table 9. The additions on the triplet surface are significantly more rapid than those on the singlet surface. For the triplet
Figure 15. Plot of TST predictions for temperature-dependent abstraction rate coefficients for C6H5 + C6H5. The solid and dashed lines denote rate coefficients for the singlet and triplet electronic states, respectively. The dash-dot-dot-dot line denotes the total abstraction rate coefficient for all singlet and triplet channels.
TABLE 8: Modified Arrhenius Parametersa for C6H5 + C6H5 Abstraction Rate Coefficients Ab
reaction
n
E
C6H5 + C6H5 f C6H6 + o-C6H4 1.35 (3) C6H5 + C6H5 f C6H6 + m-C6H4 488 C6H5 + C6H5 f C6H6 + p-C6H4 78.3
2.70
-2216
2.90
-1402
3.13
494
C6H5 + C6H5 f C6H6 + o-C6H4
849
3.07
2006
C6H5 + C6H5 f C6H6 + 3m-C6H4
526
3.12
1327
C6H5 + C6H5 f C6H6 + p-C6H4
243
3.13
805
4.57
-2888
3
3
total abstraction, kabs,Tot
4.26 (-3)
a k ) ATn exp(-E/T) cm3 mol-1 s-1 with T in K. The fits are valid over the 300-2500 K temperature range. b Numbers in parentheses denote powers of 10.
additions, the C2 and C3 addition rate coefficients are roughly equivalent, while those for the C1 and C4 additions are reduced by about a factor of 3. The overall triplet π-addition rate coefficient increases from ∼7 × 1011 cm3 mol-1 s-1 near room temperature to ∼1 × 1013 cm3 mol-1 s-1 near 2000 K. Although master equation simulations of the pressure dependence have not been performed, we expect there to be a significant reduction in the effective rate coefficients for these π-addition channels. The wells that are formed are only bound by ∼30 kcal/mol, so most of the entrance flux is expected to result in simple redissociation back to phenyl + phenyl. The remaining flux should perhaps yield biphenyl or C12H9 + H, with the former favored for the singlet reactions and the latter perhaps favored for the triplet reactions. In any case, due to the relatively low rates for these additions coupled with the expected
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Figure 16. Plot of TST predictions for temperature-dependent π-addition rate coefficients for C6H5 + C6H5.
TABLE 9: Modified Arrhenius Parametersa for C6H5 + C6H5 π-Addition Rate Coefficients channel 1-C1t 1-C2t 1-C3t 1-C3s 1-C4t 1-C4s
addition site
electronic state
Ab
n
E
C1 C2 C3 C3 C4 C4 all all
3 3 3 1 3 1 3 1
2.21 (3) 372 206 81.3 3.18 (3) 21.2 704 16.4
2.66 2.97 3.04 3.05 2.53 3.18 3.01 3.30
-35.1 -1100 -935 -1060 -978 137 -1050 -1110
a k ) ATn exp(-E/T) cm3 mol-1 s-1 with T in K. The fits are valid over the 300-2500 K temperature range. b Numbers in parentheses denote powers of 10.
dominance of the redissociation back to reactants, we have chosen to simply ignore the contributions from these channels in the development of the final improved mechanism. 5.4. Summary of Theory Results. The present theoretical predictions for the high pressure radical-radical σ-addition to form biphenyl, the abstractions to form benzene + benzynes, and the singlet and triplet π-additions are contrasted in Figure 17. At lower temperatures the σ-addition is most important. At higher temperatures the abstractions are dominant, where the triplet abstractions cause the change in relative importance and are most significant at the highest temperatures due to their higher electronic degeneracy. The π-addition rate coefficients are always lower than the abstractions. Furthermore, it is expected that the effective π-addition rate coefficients are greatly reduced by the effect of back-dissociation to the reactants from the relatively weak π-entrance complexes. These theoretical results and the experimental data were used to develop a model for the self-reaction of phenyl radicals. 6. Modeling of LS Profiles Simulations were performed using a computer integrator designed to model reactive flows in shock waves and also account for temperature changes due to nonisothermal reaction. All reactions were treated as reversible with the rate coefficients of the reverse reactions being calculated from the equilibrium constants and detailed balance. The thermochemical parameters were taken from Burcat and Ruscic4 with the exception of those
Figure 17. Plot of TST predictions for temperature-dependent high pressure rate coefficients for forming biphenyl (solid), for abstraction (dashed), and for triplet (dotted), and singlet (dash-dot-dot-dot) π-additions.
for C6H5I and C6H5, which were taken from a recent evaluation by Ruscic.5 Incubation delays were estimated in the 22 and 54 Torr simulations as described above. Simulation of the LS profiles provides a solid test for a reaction mechanism, and the ability of different schemes to reproduce the experimental density gradients has been examined with two models. The initial “literature” model contains only reactions from the literature. The TOF-MS work and the current theoretical study suggest that some reaction paths for phenyl self-reaction are absent from the literature. The final “improved” model extends the literature model by incorporating these new reaction paths. The sensitivity of the simulated density gradients to specific reactions in both models is tested by adjusting the appropriate rate coefficients. The literature and improved models share a core set of reactions, (5)-(29), whose rates are fixed throughout the modeling at those given in Table 2. These reactions encapsulate the extant literature for the pyrolysis of phenyl radicals,26 dissociation of o-benzyne,57 the reaction of C6H5I with H and C6H5,59 and the reactions of C6H5 with H.3,26 For those reactions where Troe parameters were available, the appropriate pressuredependent rate coefficients were calculated. The majority of the Arrhenius expressions were used without adjustment; however, k13, the rate for dissociation of o-benzyne, was increased by a factor of 2 from the 100 Torr expression calculated by Moskaleva et al.59 This increase produced a small improvement in simulation of the highest temperature experiments and was retained throughout the modeling, and is included in Table 2. Reactions 2a and 2b, dissociation of C6H5I by C-I fission and HI elimination, respectively, are also common to the literature and improved models. The temperature dependence of reaction 2a, dissociation of C6H5I to C6H5 + I, is described by the modified Arrhenius fits to the results of the RRKM calculations discussed in the subsection Dissociation of C6H5I of section 3.2. The optimal values of k2a at T2 for each experiment (Table 1) are introduced by slight adjustments of the A factor. The expressions for k2a are then fixed, apart from testing the sensitivity of the simulation to k2a. A branching ratio of k2b/(k2a + k2b) of ∼0.06 was used in all the simulations. Reaction 1a, the self-reaction of phenyl radicals to form biphenyl, is treated differently in the literature model and in our improved model and is discussed in the later sections for each model. Reactions 3 and 4, which represent the abstraction
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channels discussed in the theory section, do not appear in the literature and are discussed under the improved model where they appear. 6.1. Literature Model. The literature model was developed to test the ability of a mechanism based on the current literature to simulate the density gradient profiles from the current work. It consists of reactions 1a, 2a, 2b, and 5-29 from Table 2, with k2a and k2b set as described above. The literature experimental rate coefficients for reaction 1a lie in the range 5 × 1012-2 × 1013 cm3 mol-1 s-1. The higher temperature results of Heckmann et al.7 overlap the lower temperatures of the current experimental work, and their rate coefficient of 5.7 × 1012 cm3 mol-1 s-1 was used for k1a, with the assumption that the pressure dependence of this reaction is negligible. Results from the literature model are shown by the red dashed lines in parts b and c of Figure 18, which show high and low temperature, 122 Torr experiments, respectively. The higher pressure, 122 Torr, experiments are more sensitive to the effects of bimolecular reactions than the lower pressure experiments and have thus been chosen to illustrate the modeling. The simulation of the low temperature experiment, Figure 18c, is rather good except for a lack of upturn after about 6 µs. However, at higher temperatures, Figure 18b, the literature model causes the density gradient to drop sharply in the first microsecond, yielding a negative net density gradient much too early. Furthermore, the simulated profile remains too negative throughout. Analysis of the simulation results shows that the primary contributors to the calculated density gradients are reactions 1a and 2a. Of course, the influence of the strongly exothermic formation of biphenyl, (1a) (∆Hr,298 ) -117.6 kcal mol-1), is also controlled by reaction 2a, which determines the formation of phenyl radicals. For instance, at T2 ) 1335 K, Figure 18c, the literature model predicts that in 1 µs only 3.5% of the reagent is consumed and consequently the concentration of phenyl radicals is sufficiently low that reaction 1a makes only a minor contribution to the net density gradient. In fact, not until about 5 µs does the magnitude of the negative density gradient generated by reaction 1a become comparable to the magnitude of the positive gradient generated by the sum of reactions 2a and 2b. The simulation results also indicate that at 1335 K all the other reactions in the literature model make no contribution to the calculated density gradients due to their low rates of reaction. Thus, in the lower temperature experiments, the LS profiles are almost entirely due to dissociation of C6H5I and the density gradients at t0 give an accurate measure of k2a + k2b. As T2 increases, the rate of dissociation of C6H5I also increases and the influence of reaction 1a on the simulated density gradient becomes more significant. When T2 ) 1727 K (Figure 18a,b), approximately 55% of the C6H5I is consumed within 1 µs and the magnitude of the density gradient from reaction 1a is about 67% of that from dissociation of C6H5I by reactions 2a and 2b. From 1.5 µs on, the self-reaction of C6H5 dominates the net calculated density gradient. At high reaction temperatures several reactions other than reaction 1a also contribute to the calculated density gradient. However, in comparison to reaction 1a they are all minor. In the literature model all secondary chemistry except reactions 1a and 6, C6H5 + C6H5I, is initiated by the decomposition of phenyl radicals to o-benzyne and atomic hydrogen, reaction 5, or, by dissociation of o-benzyne, reactions 12 and 13. However, even for the experiment in Figure 18a,b, one of the highest temperature experiments, k5 is still only ∼6 × 103 s-1. Thus, in
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Figure 18. Comparison of experimental density gradients and simulations using the literature model (see text for details) and effects of varying k1a and k2a. Points represent experimental data and lines represent simulations. Absolute values are plotted with light gray and dark gray representing positive and negative density gradients, respectively. (a) Black dashed line, literature model and k2a from ref 16; black solid line, as black dashed line but 1.07k1a and 1.20k2a. (b) Red dashed line, literature model; blue dash-dot line, as red dashed line and k1a/2. (c) Red dashed line, literature model; blue dash-dot line, as red dashed line with k1a/2; black solid line, as black solid line in (a).
the current work, reaction 5, and reactions initiated by it, makes only minor contributions to the density gradient. Furthermore, for the higher pressure experiments, only when T2 > 1800 K does the dissociation of o-benzyne, primarily by the reverse Diels-Alder reaction, (12),59,60 begin to contribute to the calculated density gradients. Thus, the thermal stability of C6H5 and o-C6H4 effectively minimizes the importance of the secondary reactions in the literature model, apart from the recombination of phenyl radicals via reaction 1a. Though only a small fraction of the C6H5 and o-C6H4 radicals dissociate, those that do generate hydrogen atoms, which are
Self-Reaction of Phenyl Radicals mainly consumed in the literature model by reaction with phenyl radicals to form benzene,26 (7), and consumed to a lesser extent by reaction with C6H5I, (10).58 Reaction 7 is exothermic by 112.8 kcal/mol, whereas reaction 10 is only 4.1 kcal/mol exothermic. Their combined contribution to the net density gradient at T2 ) 1727 K, Figure 18a,b, is still less than 5% that of reaction 1a. Richter et al.3 point out that, as well as recombining to give benzene via reaction 7, H + C6H5 can also form o-C6H4 + H2, reaction 29 (∆Hr,298 ) -22.3 kcal/mol). However, at 1727 K k29/k7 ∼ 0.2 and so reaction 29 is not competitive with reaction 7. Thus, the contribution from reaction 29 to the net density gradient is negligible and it is included in Table 2 simply for completeness. The simulation shown in Figure 18b with the literature model shows that at high reaction temperatures, where the dissociation of C6H5I is fast and the concentration of C6H5 radicals is relatively large, the rate of reaction 1a is too high, leading to excessive negative density gradients. The influence of reaction 1a on the density gradient profile could be reduced by lowering k1a, or indirectly by reducing k2a, which would reduce the C6H5 concentration. Lower concentrations of C6H5 could also be achieved by increasing k5, which would result in faster dissociation of C6H5. However, k5 is so low that increasing it by a factor of 2 in the literature model has a negligible effect on the simulated density gradients and a much larger increase is not justified given the existing literature.26 Furthermore, a large increase in k5 would lead to excessive H-atom production which in turn would simply increase the importance of reaction 7 with ∆Hr,298 ) -112.8 kcal mol-1, and this would again force the simulated density gradients negative too early. Reducing k1a by a factor of 2 in the literature model from the value of Heckmann et al.7 improves prediction of the location of the switch from net positive to net negative density gradients in the high temperature experiment, as seen in the blue dash-dot line of Figure 18b. However, the predicted density gradient prior to the cusp is now much too positive and after the cusp it is still too negative. Also, halving k1a has only a modest effect on the simulation of the lower temperature experiment, Figure 18c, due to the relative unimportance of reaction 1a at low temperatures, as was discussed above. We conclude that only the initial dissociation of phenyl iodide remains as a means of improving the simulations with the literature model. It was noted in the discussion of the dissociation of phenyl iodide that the values for k2a determined here are between a factor of 1.6 and 1.9 higher than the “best fit” to the first order rate coefficients for reaction 2a determined by Kumaran et al.16 Smaller k2a would reduce both the initial density gradient and the contribution of reaction 1a to the net density gradient through a lower concentration of phenyl radicals, and could thereby improve simulation of the higher temperature experiments. The effect of reducing k2a in the literature model to that obtained from the expression of Kumaran et al., keeping the branching ratios the same as the original simulations, and retaining k1a from Heckmann et al., is shown in Figure 18a, T2 ) 1727 K, by the black dashed line. The reduction in the rate of production of C6H5, resulting from the smaller k2a, reduces the contribution of biphenyl formation, reaction 1a, to the calculated density gradient moving the positive to negative cusp in the simulation, Figure 18a, to around 1.5 µs compared to the 1.1 µs predicted by the literature model, Figure 18b. However, the calculated density gradient is now too positive prior to the cusp. The solid line in Figure 18a represents the literature model where both k1a and k2a have been optimized to reproduce the location of the cusp and the density gradient prior to the cusp.
J. Phys. Chem. A, Vol. 114, No. 32, 2010 8257 The best fit was obtained with 1.07k1a and 1.20k2a, with k2a taken from Kumaran et al.16 This “optimized” literature model reproduces these data rather well, but the simulations are still much too negative after the cusp. Furthermore, the “optimized” literature model fails to reproduce the lower temperature experiments, solid line Figure 18c. The literature model encapsulates the existing literature on the decomposition and self-reaction of phenyl radicals. Without modification it does not accurately simulate the range of experimental data from the LS experiments presented here. Even with the adjustment of k2a and k1a within reasonable bounds, it cannot produce a model that simulates all the LS data. Furthermore, since the remaining reactions in the literature mechanism are of secondary importance, simple adjustment of their rate coefficients also cannot resolve the discrepancies. It is clear that the literature model is inadequate for describing the reactions of phenyl radicals that dominate the high temperature LS experiments and that additional reactions must be considered. 6.2. Improved Model. The literature model has reaction 1a as the sole route for C6H5 self-reaction. However, the experimental and theoretical work presented here indicates that this assumption is too simplistic and that the self-reaction of phenyl radicals is actually a complex process with multiple channels. These channels were classified in the theoretical part of this work as σ-recombination, abstraction, and π-recombination. We now discuss the incorporation of this information into the improved reaction mechanism provided in Table 2. As part of this discussion, we consider the formation of the various products seen in the mass spectra. The improved model provides a reasonably satisfactory fit to all of the laser schlieren experiments, as illustrated with the solid lines in each panel of Figure 6. Formation of Biphenyl. Biphenyl is apparently formed immediately in the TOF-MS experiments and, from the discussion of the theoretical results, reaction 1a, σ-bond recombination, is the dominant route leading to C12H10. However, the theoretical work indicates that C12H10 could also be formed from the π-bond addition channels. With respect to the LS experiments, the singlet π-bond addition and σ-bond recombination channels are indistinguishable as they share the same reagents and stable products and therefore have the same heat of reaction. The triplet π-bond additions may also make some contribution to the biphenyl formation, but in this instance the final products are less clear. As indicated in the discussion of the theoretical work, the contributions of the singlet and triplet π-bond addition channels are difficult to estimate, but are likely to be small due to falloff in the rate coefficients. For simplicity, in the improved model these reactions are not considered separately, but are instead combined in Table 2 as reaction 1a. We then treat the overall k1a as a variable parameter and compare the fitted values with the theoretical predictions for the σ-bond recombination. The best simulations for the combination of the 22, 54, and 122 Torr experiments were obtained with the following expression for the overall k1a:
k1a ) 3.22 × 1022 T-2.81 exp(-2410/T)
cm3 mol-1 s-1 (IV)
where T is in kelvin. This expression is about a factor of 1.4 larger at 1800 K than the 0.1 atm expression for σ-bond recombination (reaction 1a) in Table 7, but is only a factor of 1.1 larger at 1300 K. These differences are well within the expected accuracy of the
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theoretical predictions and the modeling simulations. They may simply indicate that the π-bond additions do make a minor contribution at higher temperatures, or that the falloff in reaction 1a has been overestimated by the present master equation simulations. At 1450 K, k1a from Table 2 is about a factor of 1.4 larger than the temperature-independent expression of Heckmann et al.7 that was used in the literature model. Formation of C12H8. C12H8, m/z ) 152, is observed as a primary product in mass spectra at both low and high temperatures. It was not possible to determine the structure of C12H8 from the DFST/TOF-MS results; however, potential candidates include biphenylene and acenaphthylene, both of which have been detected in flames.3 Richter et al.3 suggested that acenaphthylene could be formed by isomerization of C12H9, biphenyl radical, followed by H-atom elimination. The current theoretical work suggests that C12H9 can be formed from the self-reaction of phenyl radicals by the σ-bond addition reactions (1b), (1c), and (1d) and by the π-bond addition reactions that occur on the triplet surface, (1-C1t)-(1-C4t). However, the sum of the rate coefficients from Table 7 for channels 1b, 1c, and 1d at 0.1 atm and 1800 K is a factor of 6 lower than k1a, recombination of phenyl to form biphenyl, from Table 2, and this becomes a factor of 11 at 1700 K and 500 at 1300 K. Consequently, over the temperature range of the current work, reactions 1b-1d make only negligible contributions to the net density gradient and are not likely sources of C12H8. Pressure-dependent rate coefficients were not calculated for the second potential route to C12H9, the π-bond addition channels (1-C1t)-(1-C4t). However, the sum of the high pressure rate coefficients for these channels (cf. Table 9) are factors of 2 and 13 lower than k1a at 1800 and 1300 K, respectively. Furthermore, as was mentioned in the theoretical discussion of these channels, falloff in these is expected to be severe. Thus, their contributions to the net density gradients and C12H9 concentration are also likely to be negligible. To estimate the maximum effect on the calculated density gradients, reactions 1-C1t-1-C4t were initially included in the improved model using the high pressure limit rate coefficient expression in Table 9 and assuming rapid conversion of C12H9 to C12H8 + H. However, even then the effect was small and reactions 1-C1t-1-C4t were finally removed from the improved model and are not included in the mechanism presented in Table 2. It seems a better route to products with m/z ) 152 is the self-reaction of o-benzyne radicals. The formation of biphenylene (C12H8) from two o-benzyne radicals, reaction 30, has been studied by Schaefer and Berry61 with UV absorption and mass spectrometry at room temperature and by Porter and Steinfeld62 using absorption spectroscopy at 363-473 K. The work of Schaefer and Berry gave k30 ) 9 × 1012 cm3 mol-1 s-1, while that of Porter and Steinfeld yielded k30 ) 4.2 × 1012 cm3 mol-1 s-1. A preliminary direct CASPT2 VRC-TST theoretical study of the o-benzyne self-reaction, performed by analogy to the present analysis for the phenyl radical recombination, yields the following expression for the o-benzyne high pressure recombination rate over the 200-2000 K temperature range, which gives k30 ) 3.3 × 1012 cm3 mol-1 s-1 at 1500 K.
k30 ) 4.96 × 109 T0.827 exp(690/T)
cm3 mol-1 s-1
(V) where T is in kelvin. This prediction for the high pressure limit rate coefficient for reaction 30 is in good agreement with the measurement of Porter and Steinfeld. In the o-benzyne recombination there will
be a competition between stabilization of the initial C12H8 adduct and decomposition to C12H7 (aceanaphthyl) + H, just as with phenyl radical recombination. However, in the current work m/z ) 151 was not observed in the TOF-MS experiments; of course it could simply be below the detection limit. Nonetheless, formation of C12H7 + H from o-benzyne self-reaction was excluded from the improved model. Initial simulations of the experimental density gradient profiles with the improved model including expression V for k30 indicated that the self-reaction of o-benzyne affects mainly the late gradient of the high temperature, high pressure (122 Torr) experiments. However, the calculated density gradient profiles are not particularly sensitive to k30. Thus, it was considered unnecessary and beyond the scope of the current work to investigate the self-reaction of benzyne radicals in any more detail, although this may be a focus of future studies. Formation and Reaction of o-/m-/p-Benzynes. The apparently immediate appearance of m/z ) 76 and 78 in the mass spectra from low and high temperature experiments is consistent with reactions 3os-4pt, where the self-reaction of two phenyl radicals results in a disproportionation to o-/m-/p-benzyne + benzene. For the purposes of modeling, the assumption has been made that the triplet benzynes will convert rapidly to the singlet state. Summation of the rate coefficients for formation of singlet and triplet states of each benzyne from Table 8 results in a branching ratio for o-/m-/p-benzyne that varies from 0.44/0.40/ 0.16 at 1300 K to 0.38/0.41/0.21 at 1800 K. This small variation in branching ratio is not likely to be significant in the simulation of the LS experiments. Consequently, the o-/m-/p-benzyne branching ratio was fixed at 0.40/0.40/0.20 and the rate coefficients for formation of each isomer were initially estimated by partitioning the total abstraction rate coefficient, kabs,Tot, from Table 8 in this ratio. Subsequently, the fates of each benzyne isomer were considered and the refinements discussed below were incorporated into the final improved mechanism in Table 2. In a theoretical study of o-benzyne Moskaleva et al.59 calculated pressure-dependent expressions for the isomerization of o-benzyne to m-benzyne and p-benzyne. For the reaction conditions in the present work, the equilibrium conditions for these isomerization reactions strongly favor formation of o-benzyne. Initially, the improved model was constructed to allow isomerization of all three benzyne isomers with rate coefficients taken from Moskaleva et al. However, p-benzyne can also react rapidly by ring opening in a Bergman decyclization reaction59,63 to yield (Z)-hexa-3-en-1,5-diyne:
p-C6H4 f HC≡C-CH)CH-C≡CH
(31)
To the best of our knowledge the rate of p-benzyne decyclization has not been measured or calculated for the current reaction conditions, although the original gas-phase experiments at 473 K of Jones and Bergman,63 where the existence of p-benzyne was postulated, do indicate it is fast. From their experimental results Jones and Bergman63 estimated that ∆Hr ∼ -14 kcal/mol for reaction 31, and the ab initio calculations of Moskaleva et al.59 agree with this value. Moskaleva et al. also estimated E0 ) 9.3 kcal/mol for reaction 31, whereas they estimated the barrier for isomerization between p-benzyne and m-benzyne to be 49.7 kcal/mol. Consequently, conversion of p-benzyne to (Z)-hexa-3-en-1,5-diyne is likely to be efficient and p-benzyne will be removed preferentially by reaction 31 rather than by isomerization to o-/m-benzyne. The formation of p-benzyne and subsequent decyclization are incorporated in
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the improved model as reaction 4, which assumes that p-benzyne formation is rate determining. In the current modeling (Z)-hexa3-en-1,5-diyne is considered a stable product and, as it is only a minor product, the effect of this assumption should be negligible. Initially, the formation of m-benzyne by disproportionation of two phenyl radicals and the subsequent isomerization to o-benzyne was included in the improved model as two separate steps.
C6H5 + C6H5 f m-C6H4 + C6H6
m-C6H4 f o-C6H4 The rate coefficient for the first step was estimated from Table 8 assuming that 40% of the disproportionation reaction would form m-benzyne. The second step, isomerization to o-benzyne, was entered as the reverse reaction with Arrhenius parameters for 100 Torr taken from Moskaleva et al.59 Simulations that included these two reaction steps indicated that, for the conditions of the current work, isomerization of m-benzyne to o-benzyne was sufficiently rapid that the two steps could be combined, effectively converting two phenyl radicals into o-benzyne + benzene. Thus, in Table 2, reaction 3 (2C6H5 f o-C6H4 + C6H6) represents the combination of the above two steps as well as the disproportionation reactions, (3os) and (3ot) from the theory section (section 4.2), that directly form o-benzyne. Little difference was noted between simulations that used reaction 3 from Table 2 or combinations of individual reactions. The rate coefficient for reaction 3 was initially estimated as 80% of kabs,Tot from Table 8. However, as discussed later, kabs,Tot is reduced by a factor of 2 in the final modeling and the lower value for k3 now appears in Table 2. The immediate result of including the disproportionation reactions in the improved model is to increase the concentration of o-benzyne relative to the literature model, where dissociation of C6H5 was the main source of o-benzyne with a minor contribution from reaction 2b. Reactions 3 and 4 are much less exothermic than reaction 1a, and consequently their inclusion in the improved model does have the desired result of generating more positive density gradients than those predicted by the literature model. The various changes made to the literature model to create the improved model are summarized here. (a) k1a is now expressed in a modified Arrhenius form unlike the temperature-independent estimate of Heckmann et al.7 The new k1a is about 1.4 times greater than that suggested by theory at 1800 K and 0.1 atm, but the difference decreases at lower temperatures, reaching a factor of 1.1 at 1300 K. Comparing the highest temperature of the Heckmann et al. study, k1a from Table 2 is a factor of 1.4 higher than their value. (b) Reaction 3 is the formation of o-benzyne by the abstraction reactions (3os), (3ot), (3ms), and (3mt), and this assumes both rapid conversion of triplet o-/m-benzyne to the singlet state and rapid isomerization of m-benzyne to o-benzyne. k3 was initially set to 80% of kabs,Tot from Table 8, although it was finally adjusted to half this initial value, as it appears in Table 2. The sensitivity of the improved model to reactions 3 and 4 is discussed below. (c) Reaction 4 is the formation of p-benzyne and benzene by disproportionation between two C6H5 radicals, here followed by the rapid Bergman decyclization of p-benzyne. k4 was
Figure 19. Comparison of experimental density gradients with simulations using the improved model showing the effect of varying k1a and k3 + k4 in the improved model. Points represent experimental data and lines represent simulations. Absolute values are plotted with light gray and dark gray representing positive and negative density gradients, respectively. For clarity only every second experimental data point is plotted. Black solid line is the improved model with rate coefficients from Table 2. Red dash-dot line, improved model with k3 + k4 equal to the total theoretical abstraction rate coefficient expression, Table 8, and k3/(k3 + k4) ) 0.8. Blue dash-dot-dot line as red dash line but with 3.2k1a to accurately predict the positive to negative cusp in (a).
initially set to 20% of kabs,Tot from Table 8. In analogy with k3, the value for k4 in Table 2 is half the initial estimate. (d) Reaction 30 represents the formation of biphenylene from recombination of two o-benzyne radicals. The value of k30 in Table 2 is that given in expression V. In all the simulations with the improved model k2a, k2b, and k5-k29 were left as they are in the literature model. Simulation with ImproWed Model and SensitiWity to k1, k3 + k4, and k30. In Figure 19 the same experimental data are shown as in Figure 18. The red dash-dot lines in Figure 19 show the results of simulations with the improved model, but with the rate coefficients for reactions 3 and 4 set to the initial values as summarized in points (b) and (c) above. For the lower temperature experiment, Figure 19b, the simulation results are satisfactory and comparable to those from the literature model. This agreement is expected, as in both models dissociation of C6H5I dominates the low temperature density gradients. However, from Figure 19a it is apparent that, with these initial values of k3 and k4, the simulation fails to capture the essential features of the high temperature experimental data. After 1 µs the simulation results are too positive throughout and the location of the cusp is 0.5 µs later than its experimental location. This is in contrast to the literature model, which predicted density
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gradients that were too negative throughout at high temperatures, as in Figure 18b. In this improved model k1a is 1.1-1.6 times greater than k1a in the literature model, which, if this were the only modification, would create even more negative density gradients as reaction 1a is exothermic by -117.6 kcal/mol. However, in the improved model k1a/(k3 + k4) ∼ 1.4 at 1300 K and this ratio reduces to ∼0.6 at 1700 K, which indicates that at high temperatures the disproportionation reactions (3) and (4), which are much less exothermic than the recombination reaction (1a), are now dominating the calculated density gradients. Furthermore, the simulations also show that at high temperatures these three reactions account for almost all of the density gradient with all other reactions making only minor contributions. Consequently, to improve the simulation of the high temperature experiments with the improved model, the relative importance of reaction 1a to reactions 3 and 4 should be increased. This can be achieved by either reducing the total disproportion rate, k3 + k4, or by increasing the recombination rate, k1a. If k1a is increased, the location of the cusp in Figure 19a can be reproduced, dash-dot-dot blue line. However, the predicted early density gradient is lower than the observed data. Additionally, the calculated density gradient at low temperatures is much too small throughout as in Figure 19b. Smaller increases in k1a cannot produce satisfactory fits to both Figure 19a and Figure 19b. Furthermore, to reproduce the cusp in Figure 19a required an increase of a factor of 3.2 in k1a and such a large value of k1a is considerably greater than the uncertainty of the present theoretical calculations for the high pressure recombination rate. On the other hand, simply reducing k3 + k4 by a factor of 2 from the theoretical high pressure values, Table 8, and keeping k3/(k3 + k4) ) 0.8 results in a very good simulation of the high and low temperature experimental data as shown by the solid black lines in Figure 19. Such a reduction is just within the estimated uncertainties in the theoretical calculations. The reduced values for k3 and k4 are those given in Table 2. With this improved model the main deficiency is that the late density gradient after the minimum remains slightly too negative relative to the experimental profile at high temperatures, Figure 19a. However, with respect to the focus of this work on the reactions of phenyl radicals, this is not a serious discrepancy. Analysis of the simulations shows that the negative late gradient is mainly due to the self-reaction of o-benzyne radicals through reaction 30. A large number of simulations have been conducted to explore the sensitivity of the predicted density gradients to k12 (dissociation of o-benzyne), to k30, and to different reaction channels for o-benzyne recombination (i.e., o-C6H4 + o-C6H4 f biphenylene or o-C6H4 + o-C6H4 f aceanapthyl + H). The changes observed in the simulated profiles are small and affect only the tail of the high temperature, high pressure experiments with negligible effect on the low pressure experiments. The final reaction mechanism arrived at and used is presented in Table 2, and simulations that cover the complete range of experimental conditions in the LS work are shown in Figure 6 by the solid black lines. These results are representative of the complete data set, and the largest discrepancies between the simulation and experimental results are seen in the higher pressure, higher temperature experiments such as shown in Figure 6d, where the predicted late density gradient is too negative. 7. Conclusions The combination of experimental observations employing time-of-flight mass spectrometer and laser schlieren diagnostics
Tranter et al. coupled to a diaphragmless shock tube and high level theoretical predictions have yielded a greatly improved understanding of the self-reaction kinetics of phenyl radicals. The DFST/LS experiments provide convincing evidence for the existence of reactions 3 and 4, which is supported by the product analyses obtained from the DFST/TOF-MS experiments. The high-level theory provided a detailed exploration of the self-reaction of phenyl radicals and elucidated many important features that are not accessible experimentally. Finally, the theoretical results were incorporated into a model that is able to quantitatively describe the laser schlieren observations for decomposition of phenyl iodide at temperatures ranging from 1276 to 1853 K and at a pressure of 22, 54, or 122 Torr. This model is based on literature values for most reactions coupled with reasonable adjustments to the theoretical predictions for the phenyl radical self-reactions. This multipronged experimental and theoretical approach provides a powerful means to enumerating the kinetics of such complex reactions. Notably, the analysis demonstrates that the total rate coefficient for the phenyl + phenyl abstraction reactions to produce o-/m-/p-benzyne + benzene is comparable to the corresponding high pressure recombination rate coefficient over this same temperature range. At 1400 K, the theoretical predictions for the two rate coefficients are identical. The recombinations of o-benzyne with itself and other closed shell species, such as acetylene, are also expected to be reasonably rapid. Thus, the observation of o-benzyne as a primary product in phenyl + phenyl reactions suggests the possible importance of o-benzyne, and other o-benzyne-like structures, in polyaromatic hydrocarbon growth. Rate coefficients for the decomposition and further reaction of o-benzyne play a significant role in the laser schlieren density gradients at longer times. Further experimental and theoretical studies of the kinetics of o-benzyne reactions are planned in future work. The theoretical calculations also suggest some role for π-bond addition reactions at the higher temperatures. This role depends on the isomerization rates of the initial triplet π-adducts. Furthermore, even in the high pressure limit, the predicted π-addition rate coefficients are never greater than the predicted total abstraction rate. Thus, the pressure dependence and product branching of these π-addition channels were not thoroughly investigated. As part of the analysis, we have also obtained rate coefficients and branching ratios for the decomposition of phenyl iodide that extend the available data to temperatures where pyrolysis of C6H5I is used as an “instantaneous” source of phenyl radicals. In contrast to prior work, we find that roughly 6% of the thermal decomposition leads to o-benzyne + HI. The contribution from this secondary decomposition channel should be included in future studies that employ phenyl iodide as a source for phenyl radicals. Acknowledgment. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, U.S. Department of Energy, under Contract No. DE-AC02-06CH11357. This article has been created by the University of Chicago as Operator of Argonne National Laboratory (“Argonne”) under Contract No. W-31-109-ENG-38 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.
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