J. Phys. Chem. A 2010, 114, 4881–4890
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Reactions between Resonance-Stabilized Radicals: Propargyl + Allyl† James A. Miller* Combustion Research Facility, Sandia National Laboratories, LiVermore, California 94551-0969
Stephen J. Klippenstein, Yuri Georgievskii, and Lawrence B. Harding Chemical Sciences and Engineering DiVision, Argonne National Laboratory, 9700 South Cass AVenue, Argonne, Illinois 60439
Wesley D. Allen and Andrew C. Simmonett Center for Computational Chemistry and Department of Chemistry, UniVersity of Georgia, Athens, Georgia 30602 ReceiVed: NoVember 6, 2009; ReVised Manuscript ReceiVed: January 8, 2010
This article describes a detailed theoretical analysis of the reaction between allyl and propargyl. In this analysis, we employ high-level electronic structure calculations to characterize the potential energy surface and various forms of transition-state theory (TST) to calculate microcanonical, J-resolved rate coefficients-conventional TST for isomerizations, and the variable reaction coordinate form of variational TST for the “barrierless” association/dissociation processes. These rate coefficients are used in a time-dependent, multiple-well master equation to determine phenomenological rate coefficients, k(T,p), for various product channels. The analysis indicates that the formation of (cyclic) c-C6H7 and c-C6H8 species is suppressed by elevated pressure. Overall, the results suggest that the formation of these five-membered rings from the reaction is not as important as previously thought. A simplified description of the kinetics of the reaction is discussed, and corresponding rate coefficients are provided. C3H3 + C3H5 f fulvene + H + H
Introduction The reactions of resonance-stabilized free radicals play a critical role in the formation of aromatic compounds in flames of aliphatic fuels.1-6 They may also be important in the growth of polycyclic aromatic hydrocarbons (PAH).2,3,7 The resonancestabilized propargyl radical (C3H3) is singularly important in the formation of the first ring, and its reaction with itself
C3H3 + C3H3 f products
(R1)
dominates the cyclization process in virtually all flames. The reasons underlying the significance of propargyl are understood and make an interesting story. However, we shall defer that story to another time and place. Over the past 10 years, we have been developing a chemical kinetic model to predict the formation of benzene and other cyclic compounds in low-pressure flames of light-hydrocarbon fuels. The experimental flames with which we constantly compare our model predictions are fueled by acetylene, ethylene, allene, propyne, propene, and 1,3-butadiene.8-11 All of the flames are rich or stoichiometric, and at least one flame of each fuel has an equivalence ratio very close to the sooting limit. Overall, the second most important cyclization step, after C3H3 + C3H3, is the reaction between propargyl and allyl †
Part of the special section “30th Free Radical Symposium”. * To whom correspondence should be addressed. E-mail: jamille@ sandia.gov.
(R2)
which is short for a two-step process
C3H3 + C3H5 f c-C6H7 + H
(R3)
c-C6H7 (+M) f fulvene + H (+M)
(R4)
where c-C6H7 is a hydrofulvenyl radical. In rich flames, fulvene is relatively easily converted to benzene by H-atom-assisted isomerization
H + fulvene f H + benzene Because of the importance of reaction R2 in our flame modeling, we thought it prudent to investigate its kinetics in some detail. Reaction R2 was first proposed by Marinov et al.12 as a cyclization step (i.e., source of benzene) in studying a rich, atmospheric-pressure propane flame. They justified the c-C6H7 + H products using BAC-MP4 electronic structure calculations, which showed that c-C6H7 + H could be reached from C3H3 + C3H5 through one of the initial adducts by a rearrangement that had no intrinsic energy barrier (the highest rearrangement barrier was ∼30 kcal/mol below the reactants). They concluded that the rate coefficient for reaction R3 would be determined by the C3H3 + C3H5 association rate, and they approximated this rate coefficient by the relatively well-known rate coefficient for C3H5 + C3H5 in its high-pressure limit (the “capture” rate coefficient). Our subsequent experience with the C3H3 + C3H3 reaction13,14
10.1021/jp910604b 2010 American Chemical Society Published on Web 02/03/2010
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suggests that the rearrangement of the initial complex in C3H3 + C3H5 may have a large impact on the rate coefficient for forming c-C6H7 + H. Clearly, a detailed theoretical analysis is warranted. Theoretical and Computational Approach For this investigation, we employed the B3LYP hybrid DFT (density functional theory) method with Pople’s split-valence 6-311++G(d,p) basis set to optimize geometries and compute the vibrational frequencies of all of the relevant stationary points on the PES, both minima and saddle points. Additionally, we calculated rotational potentials for molecular configurations with internal rotations and intrinsic reaction coordinate (IRC) curves using this theoretical model. In order to obtain accurate rate coefficients, we calculated single-point energies at all of the stationary points using Dunning’s basis sets, cc-pVxZ with x ) T, Q, and the rQCISD(T) method (spin-restricted quadratic configuration interaction with perturbative estimate of the triples contribution). Finally, the latter energies were extrapolated to the infinite basis set limit using the formula15,16
E(∞) ) E(lmax) - B/(lmax + 1)4 where lmax is the maximum angular momentum in the basis set and B is a fitting constant. The final potential energy surface (PES) is depicted diagrammatically in Figure 2 for reference and is discussed at length below; the structures corresponding to the wells and cyclic components of the bimolecular products are shown in Figure 3. Several of the stationary points on the potential (wells 4 and 6 and transition states 4-6, 4-7, and 6-7) were found to have sufficiently high multireference character (as indicated by large T1 (or Q1) diagnostics) that it was deemed necessary to reevaluate their energies using multireference methods. For these points, the geometries were reoptimized with (4e,4o)-CASPT2/ cc-pVDZ. Here, the four active orbitals consist of the diradical orbitals and the π and π* orbitals. At these geometries, multireference configuration interaction calculations including a multireference Davidson correction, (4e,4o)-CAS+1+2+QC25,26 were done with both cc-pVTZ and cc-pVQZ basis sets, and the energies were extrapolated to the infinite basis set limit as described above. For each of these points, the multireference calculations were performed for both the singlet and the corresponding triplet stationary points. The resulting multireference singlet-triplet splittings were combined with rQCISD(T) energies for the triplet states (which do not have significant multireference character) to yield the final energies of the singlet states. The overall corrections ranged from nearly 0 to about 7 kcal/mol. One possible source of error in the stationary point energies comes from locating such points inaccurately using a lowerlevel method than is used for the energies themselves. To test the magnitude of this effect, we calculated the structure of TS1-4, the critical transition state in the present analysis (see Figure 2), using CCSD(T)/cc-pVTZ in addition to the method described above. For practical purposes, this structure can be viewed as exact. The single-point energies determined by the same rQCISD(T) method for the two different structures differed by ∼0.25 kcal/mol, an inconsequential difference. Phenomenological (thermal) rate coefficients, k(T,p), were determined as a function of temperature and pressure using two different forms of the master equation:17-21
Miller et al. (1) The time-dependent multiple-well ME. This approach is limited to a one-dimensional formulation in which E, the vibrational-rotational energy of the molecule or complex, is the independent variable. (2) The collisionless limit of the multiple-well ME. As the name implies, this equation is simply the p f 0 limit of the above ME. However, in the limiting case, we can solve the problem in both one and two dimensions. In the latter case, E and J, where J is the total angular momentum quantum number, are the independent variables. In the present investigation, J conservation was never a significant issue. Nevertheless, all of the results reported in this article are from two-dimensional calculations. The methodology for obtaining phenomenological rate coefficients from the master equation would require too much space to describe here. However, extensive discussions can be found in refs 13, 19-21, and particularly ref 17. There are two types of transition probabilities in the master equation, collisional and reactive. For the present work, we employed Lennard-Jones collision rates and approximated the energy-transfer function using the single-exponential-down model with 〈∆Ed〉 ) 400(T/300 K)0.85 cm-1. The collider in the present work was taken to be molecular nitrogen. The energy-transfer function was also assumed to be independent of energy, angular momentum, and the isomeric form of the complex. Probably none of these approximations are accurate, not even the exponential form of the energy-transfer function.21 However, thermal rate coefficients are only weakly dependent on these details, and knowledge of them is severely lacking in general. Microcanonical, J-resolved RRKM rate coefficients, k(E,J), were calculated using variational transition-state theory for the association/dissociation reactions and conventional transitionstate theory for the isomerizations. For the one-dimensional ME calculations, the k(E) functions are constructed directly from the k(E,J)’s in order to preserve the accuracy of the J-resolved rate coefficients as much as possible. Tunneling and nonclassical reflection (both of which are automatically included in the quantum transition probabilities) are included in the analysis one-dimensionally by assuming that the potential along the minimum-energy path can be approximated by an asymmetric Eckart function. Tunneling plays little or no role in the C3H3 + C3H5 reaction because the important isomerization transition states lie so much lower in energy than the reactants. However, our prior experience suggests that, for thermal isomerizations taking place on the same potential, the effect of tunneling will be significant at temperatures up to about 1000 K. The electronic structure calculations were done with Gaussian 0322 and MOLPRO.23 The RRKM and master equation calculations were performed with VARIFLEX.24 A number of channels in the reaction involve the recombination or production of two radicals (i.e., C3H3 + C3H5, C2H3 + i-C4H5, and H + C6H7, where there are numerous isomers of the latter). For such processes, where the minimum-energy path potentials in the recombination direction are barrierless, both variational and anharmonic effects have a major impact on the rate coefficients predicted by transition-state theory (TST). Here, we implement the direct variable reaction coordinate (VRC) transition-state theory approach,27-30 which has been shown to provide an effective means for treating these complications.31,32 The calculations were performed as previously described for the C3H3 + C3H5 system14 and in close analogy with those described for the recombination of H atoms with resonancestabilized radicals.33 For completeness, we describe here some of the key features of the methodology.
Reactions between Resonance-Stabilized Radicals The direct VRC-TST approach requires a method for estimating the interaction between the two reacting radicals for arbitrary orientation and separation. Here, we evaluate these interactions with the CASPT2 method employing an eight-electron, eightorbital (8e,8o) active space consisting of the two radical orbitals together with the full (6e,6o) π,π* space. These CASPT2 calculations employ Dunning’s correlation- consistent, polarized valence, double-ζ basis set34 and were done using the formalism of Celani and Werner35 as implemented in the MOLPRO program package.23,36 We include two corrections to these base CASPT2(8e,8o)/ cc-pVDZ rigid body interaction energies. In particular, orientation-independent corrections for limitations in the basis set are obtained from complete basis set estimates for the interaction energies along a minimum-energy path. The present CBS estimates are obtained as the average of the separate extrapolations of cc-pVDZ, cc-pVTZ and aug-cc-pVDZ, aug-cc-pvtz pairs of calculations. These CBS extrapolations are the same as in the stationary-point energy determinations described above. This CBS correction is included for all of the radical-radical channels except for the C3H3 + C3H5 reaction, where the basis set correction was previously shown to be insignificant.14 The second correction is for the effect of geometry relaxation. Due to the resonance in the electronic structure of the radicals of interest here, this correction is more significant than is typical. For the propargyl + allyl addition, the internal geometries of the fragments were optimized at the uB3LYP/cc-pVDZ level for each orientation of the fragments. Sample evaluations indicated that the uB3LYP-optimized relaxation energies were suitably close to those obtained from CASPT2/cc-pVDZ geometry optimizations.14 For the remaining reactions, onedimensional geometry relaxation corrections were obtained from uB3LYP/6-31G* calculations along a minimum-energy path. The rate coefficient predictions are only weakly dependent on the equilibrium geometries of these radicals, and therefore, they were simply determined from uB3LYP/6-311++G** or uB3LYP/cc-pVDZ calculations using the GAUSSIAN98 program.37 A number of the linear C6H7 species have multiple torsional isomers. Generally, barrierless addition rates are only weakly dependent on the torsional state of the reactants. In particular, changes in the torsional state generally affect the backside and frontside additions in opposite ways, and therefore, the sum is changed little. Thus, for simplicity, we have evaluated the addition rates for only one of the isomers. However, we have explicitly considered both frontside and backside additions as appropriate. For the barrierless C-H bond fissions of the cyclic C6H8 species (well 7 f p5, well 8 f p5, and well 8 f p6), we employed a (6e,6o) active space instead of the (8e,8o) active space described above. The (6e,6o) active space consists of the π system of the cyclic-C6H7 fragments and the H orbital. We also employed the CASPT2(6e,6o)/cc-pVDZ approach in the determination of the one-dimensional geometry relaxation corrections for these three reactions. Wells 4 and 6 are cyclic-C6H8 species with open-shell singlet biradical wave functions containing two unpaired electrons in the π system. The lowest-energy C-H bond fission from well 6 yields product p5. For well 4, there are two distinct C-H bond fissions from the C5 ring. One yields product p5 again, while the other yields the p6 fragments. The reverse addition reactions for these C-H bond fissions correspond to H addition across a π bond. Such reactions generally involve modest
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Figure 1. Resonant Kekule´ structures for allyl and propargyl radicals.
barriers and therefore can be treated with a rigid rotor, harmonic oscillator implementation of TST. The production of singlet biradicals in these reverse additions implies the need to employ multireference electronic structure methods in the analysis of the rovibrational properties of their saddle points. Here, we use the CASPT2 approach for this analysis and implement a (6e,6o) active space consisting of the π system of the cyclic-C6H7 fragments and the H orbital. The geometries and rovibrational properties were first obtained for the cc-pVDZ basis set. The energies were then extrapolated to the CBS limit via calculations with the cc-pVTZ and cc-pVQZ basis sets at the cc-pVDZ geometries. The energies for these transition states relative to the corresponding products were converted to absolute energies (i.e., relative to reactants) via the rQCISD(T)/CBS absolute energies for the products. Attempts to locate a saddle point for the C-H bond fission from well 4 to species p6 were unsuccessful and led instead to the isomerization transition state from well 4 to well 8. Constrained optimizations suggest that the reverse addition has a relatively large barrier, while the C-H bond fission from well 8 to p6 is barrierless. Thus, it appears that the dynamical process for production of species p6 from well 4 involves first the isomerization to well 8 followed by C-H bond fission in well 8. Potential Energy Surface Both propargyl and allyl are resonance-stabilized radicals. Each has two Kekule´ structures of comparable importance, although the two are chemically equivalent for allyl. These structures are shown in Figure 1. Again, the potential energy surface on which our rate coefficient calculations are based is shown diagrammatically in Figure 2; the structures of the stable molecules corresponding to the wells and the cyclic products are shown in Figure 3. Propargyl and allyl can combine in two different ways, with either the head (the CH2 end) or the tail (the CH end) of propargyl forming the bond with allyl (the two ends of allyl are identical). The bond in the former case is slightly weaker than that in the latter (by less than 1 kcal/mol). The barrier to isomerization is only 34.8 kcal/mol in the exothermic direction, resulting in TS1-2 lying 28.8 kcal/mol below the C3H3 + C3H5 asymptote. The two adducts can dissociate in a number of ways by eliminating various hydrogen atoms or by breaking a C-C bond to form C2H3 + i-C4H5, as shown in the diagram. The
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Figure 2. Potential energy diagram (including zero-point energies) for the C3H3 + C3H5 reaction. The numbers on the diagram are energies in kcal/mol relative to reactants.
Figure 3. Molecular structures corresponding to the wells and the cyclic fragments.
product p1 represents CH2CCHCHCHCH2 + H, p2 denotes CHCCH2CHCHCH2 + H, p3 represents C2H3 + i-C4H5, and
p4 is short for CHCCHCH2CHCH2 + H. These products lie between 11.8 and 24.0 kcal/mol above the reactants.
Reactions between Resonance-Stabilized Radicals
Figure 4. Total rate coefficients for C3H3 + C3H5 f products as a function of temperature at various pressures.
The most important feature of the potential for our purposes is TS1-4, which closes well 1 into a cyclic five-membered singlet biradical (well 4). This biradical can then eject a hydrogen atom, producing a hydrofulvenyl radical (c-C6H7). However, as discussed above, only the less stable of the two c-C6H7 radicals, p5, is accessible from this well. Nevertheless, well 4 is the gateway not only to the two hydrofulvenyl radicals but also to the two most stable c-C6H8 species, wells 7 and 8. The route from well 1 through TS1-3 and TS3-5 is a more tortuous path that can also lead to the hydrofulvenyl radicals and the stable cyclic-C6H8 compounds through a biradical intermediate (well 6). It starts by closing well 1 into a fourmembered ring, reopening the ring at a different place, and then closing again into another cyclic five-membered singlet biradical (well 6). Because of the difference between the energies of TS1-4 and TS1-3 (and TS3-5), the latter route makes negligible contributions to the rate coefficients. Higher-energy routes leading from wells 1 and 2 also make negligible contributions to the rate coefficients and are neglected in the analysis. Wells 5 and 6 are effectively isolated from the rest of the PES by the relatively high barriers at TS3-5, TS4-6, and TS6-7. We could eliminate these wells from our analysis and still get the same results for the rate coefficients. We included them, but we allowed for only one dissociation channel for well 5, C2H3 + i-C4H5, although several H-atom-producing channels are possible. As discussed below, C2H3 + i-C4H5 is the dominant bimolecular product channel at high temperatures where wells 5 and 6 would be most likely to become accessible. Thermal Rate Coefficients Figure 4 shows our predictions for the total rate coefficient at a series of pressures. By definition, the high-pressure limiting value of ktot, k∞, is the capture rate coefficient, that is, the rate coefficient for complex formation. This represents an upper limit on the total rate coefficient. At the high-pressure limit (i.e., at truly infinite pressure), the only possible products are the initial adducts, well 1 and well 2. At the other end of the spectrum is the collisionless (or p ) 0) limit, k0. At this limit, there is no stabilization in any of the wells, and the only possible products are the bimolecular ones (p1-p6). This is the lower limit of ktot. At any pressure, including p ) 0, ktot starts out at low temperature very close to k∞ and ultimately coalesces with k0 at high temperature. This behavior is universal as long as there
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Figure 5. Rate coefficients for various product channels of the C3H3 + C3H5 reaction at the collisionless (p ) 0) limit.
is a bimolecular product channel whose threshold energy lies below the reactants. As the temperature increases at fixed pressure, unimolecular dissociation of the complexes (either to products or back to reactants) becomes increasingly dominant over collisional stabilization. The same behavior is observed if all of the bimolecular products are endothermic, except that in this case, k∞ and k0 do not coalesce at low T. Two other features of Figure 4 are worthy of mention. First, k0(T) decreases with temperature up to T ≈ 1900 K, and then, it starts to increase again. This increase in k0(T) with T indicates that the endothermic product channels are beginning to come into play. The k(T) curves at all finite pressures have such minima, but they appear at increasingly higher temperatures as the pressure increases. Second, at T ) 1900 K (the minimum of the k0(T) curve), the ratio of k∞ to k0 is 71.6. This indicates that (in the absence of collisions) only one in approximately 73 complexes formed goes on to produce products, even though the subsequent transition states leading to p6, the lowest-energy bimolecular channel, lie well below the reactants. This behavior, as well as the sharp decrease of k0(T) with increasing T at temperatures below 1900 K, is due to the tight, entropy-deficient character of the rearrangement transition states, most notably TS1-4. In Figure 5, we examine the product distribution at the collisionless limit. As one might expect, the lowest-energy channel (p6), a c-C6H7 radical + H, is dominant at low temperatures. Its rate coefficient is roughly equal to the capture rate coefficient up to T ≈ 400 K. For T g 400 K, kp6 drops off rapidly with temperature because of the tightness of the rearrangement transition states. At T ≈ 1900 K, the higherentropy endothermic product channels begin to take over. Interestingly, the dominant product channel at high temperatures is p3, i-C4H5 + C2H3, even though two of the H-atom channels formed from wells 1 and 2 are lower in energy. This is a “rotational entropy” effect, largely a result of two dissociating polyatomic fragments having more rotational degrees of freedom than an atom and a polyatomic fragment. The endothermic H-producing channels (p1, p2, and p4) are never really significant even at high temperature. Figure 6 displays the individual rate coefficients as a function of temperature for a pressure of 10 Torr. Figure 6a focuses on the stabilized products and Figure 6b the bimolecular channels. In Figure 6b, for simplicity, the rate coefficients for all of the noncyclic H-producing channels are added together, kp1 + kp2
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Miller et al. only bimolecular products above 2200 K. The rate coefficients for forming the c-C6H7 + H fragments, kp5 and kp6, also decay at high temperature, where they cannot compete with the products formed from fragmentation in the early wells, particularly p3. The reason that wells 7 and 8, as well as fragments p5 and p6, are so difficult to reach at high temperature is a classic competition between loose and tight transition states. All of the isomerization transition states, most notably TS1-4, are tight; all fragmentation transition states are loose. Tight transition states have compact structures, resulting in relatively small moments of inertia (large rotational constants) and high vibrational frequencies; loose transition states are just the opposite. Because of these differences, tight transition states are lower in entropy than loose ones. More to the point, if both have the same ground-state energy, a loose transition state will generally have a sum-of-states that is larger than a similar tight one for a given vibrational-rotational energy, that is, there is more flux through the loose transition state. This difference can be compensated for if the tight transition state lies lower on the potential energy surface than the loose one. For larger molecules (with more degrees of freedom), the compensating energy must be larger than that for smaller molecules because the loose/ tight sum-of-states ratio is larger for a given vibrational-
Figure 6. Rate coefficients of the C3H3 + C3H5 reaction for various product channels at a pressure of 10 Torr. The label “bimolecular products” indicates the sum of the rate coefficients for the p1-p6 product channels. “Linear+H” is the sum of the rate coefficients for the p1, p2, and p4 product channels.
+ kp4, and labeled “linear+H.” As expected, stabilization into wells 1 and 2 is dominant at the lower temperatures. Stabilization into wells 3, 4, 5, and 6 is never significant because these wells are relatively inaccessible, too shallow, or both. Only at temperatures of T ≈ 1500 K and above can the bimolecular channels compete. Interestingly, the rate coefficients for stabilization into well 1 (kw1) and well 2 (kw2) cross. At low temperature, the dominant factor in determining these rate coefficients is the reactive flux through TSR-1 and TSR-2. The larger reactive flux through TSR-2 causes kw2 to be larger than kw1. However, at high temperature, the rate-controlling factor is collisional deactivation. Because the lifetimes of well 1 complexes are longer than those of well 2 (a consequence primarily of the slightly stronger bond of well 1), kw1 is slightly larger than kw2 at high T. In addition to the c-C6H7 + H channels, five-membered rings can also be formed by stabilization into wells 7 and 8. The formation of these species is precluded at the lowest temperatures because collisional stabilization into wells 1 and 2 is so dominant. However, as the temperature rises, it is increasingly possible for a complex in one of these early wells to isomerize and become stabilized in one of the deeper cyclic wells. Thus, the rate coefficients kw7 and kw8 first increase with temperature in Figure 6. They subsequently fall with increasing T as the bimolecular channels, particularly p6, begin to dominate. As can be seen from Figure 6, the reaction essentially produces
Figure 7. Rate coefficients of the C3H3 + C3H5 reaction for various product channels at a pressure of 10 atm. The label “bimolecular products” indicates the sum of the rate coefficients for the p1-p6 product channels. “Linear+H” is the sum of the rate coefficients for the p1, p2, and p4 product channels.
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Figure 8. Rate coefficient for the formation of cyclic species (well 7, well 8, p5, and p6) in the C3H3 + C3H5 reaction as a function of temperature at various pressures.
Figure 10. Dissociation/isomerization rate coefficients for CHCCH2CH2CHCH2 (well 2) at (a) 10 Torr and (b) 10 atm. The label “bimolecular products” indicates the sum of the rate coefficients for the p1-p6 product channels.
Figure 9. Dissociation/isomerization rate coefficients for CH2CCHCH2CHCH2 (well 1) at (a) 10 Torr and (b) 10 atm. The label “bimolecular products” indicates the sum of the rate coefficients for the p1-p6 product channels.
rotational energy for larger molecules. For example, if we were dealing with a system of four or five atoms, the tightness of TS1-4 would be inconsequential because it lies so low on the potential. However, for a 14-atom system, the tightness of TS1-4 is the dominant factor determining access to the cyclic
species from C3H3 + C3H5. Of course, higher temperatures reflect higher energies, and any advantage from energy considerations that a tight transition state might have disappears at high T. It is interesting and of practical importance to investigate the effect of elevated pressure on the formation of the cyclic products (wells 7 and 8 and bimolecular products p5 and p6). To this end, we have plotted our results for a pressure of 10 atm in Figure 7. The rate coefficients at this pressure display many of the same qualitative features as those at low pressure. However, the most striking effect of increased pressure is the suppression of the cyclic product channels. The rate coefficients for these channels rise at higher temperatures and attain peak values that are more than an order of magnitude smaller at 10 atm than that at 10 Torr. This reduction in the size of the rate coefficients is due to the increased likelihood of complexes being stabilized in the early wells because of the increased collision rates. The values of kw1 and kw2 increase by about an order of magnitude at 1500 K in going from 10 Torr to 10 atm. Because their lifetimes are shorter, higher-energy complexes are less likely than the lower-energy ones to suffer collisions. Consequently, the rate coefficients for the high-energy bimolecular channels, particularly p3, are considerably less affected by increased pressure than those for the cyclic products (which necessarily involve TS1-4).
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TABLE 1: Modified Arrhenius Representation of Rate Coefficients for Modeling (see text)a pressure (Torr)
A1 (cm3/molecule-s)
10 30 760 7600 76000
0.35 × 1045 0.773 × 1042 0.543 × 1032 -0.248 × 108 -2.45
C3H3 + C3H5 f CH2CCHCH2CHCH2 (well 1) -16.68 25753 0.655 × 109 -15.75 26038 0.586 × 107 -12.42 24912 26.56 -5.578 5166 1832 -3.346 3103 0.799 × 10-2
-6.688 -5.981 -4.153 -4.253 -2.545
3853 3458 2364 3790 2226
10 30 760 7600 76000
0.325 × 1042 0.235 × 1041 0.602 × 1033 -0.162 × 1011 -728
C3H3 + C3H5 f CHCCH2CH2CHCH2 (well 2) -15.88 22677 0.242 × 1012 -15.39 23547 0.142 × 1010 -12.78 23846 3.82 × 103 -6.404 5767 0.930 × 106 -4.054 3660 1.87
-7.475 -6.693 -4.775 -5.043 -3.221
4277 3878 2762 4364 2743
10 30 760 7600 76000
1.89 × 1021 3.80 × 1019 3.13 × 1014 0.737 1.54 × 1017
n2
E02 (cal/mol)
pressure (Torr)
a
A1 (1/s)
n1
-9.656 -9.123 -7.526 -3.364 1.291 n1
E01 (cal/mol)
A2 (cm3/molecule-s)
n2
E02 (cal/mol)
C3H3 + C3H5 f cyclics 19652 20504 23878 18249 10523 E01 (cal/mol)
A2 (1/s)
CHCCH2CH2CHCH2 f CH2CCHCH2CHCH2 -9.678 49848 -8.821 49074 -6.242 45805 -4.021 42105 -1.696 37722
10 30 760 7600 76000
1.60 × 1043 3.04 × 1040 1.25 × 1032 4.22 × 1024 4.68 × 1016
10 30 760 7600 76000
3.41 × 1037 3.19 × 1036 3.20 × 1032 2.20 × 1026 5.30 × 1021
CH2CCHCH2CHCH2 f cyclics -8.054 50960 -7.676 51281 -6.314 51585 -4.448 49352 -3.010 49033
10 30 760 7600 76000
3.72 × 1075 5.22 × 1084 4.34 × 1083 9.08 × 1070 3.13 × 1022
CHCCH2CH2CHCH2 f cyclics -19.70 83466 -21.16 93165 -20.57 102400 -16.89 99876 -3.514 64992
Nominally, the fits for kw1 and kw2 are applicable from 250 to 2500 K, while the others are from 700 to 2500 K.
In order to illustrate the effect of pressure on the formation of the cyclic products in the C3H3 + C3H5 reaction more clearly, we have added the four rate coefficients, kc ) kw7 + kw8 + kp5 + kp6, and plotted kc as a function of temperature for various pressures in Figure 8. Clearly, increased pressure suppresses the formation of the c-C6H7 and c-C6H8 species at all temperatures. At low temperatures, where kc at zero pressure is essentially equal to the capture rate coefficient (k∞), the effect is dramatic. It becomes smaller as the temperature goes up. As T f ∞, kc approaches its collisionless value at all pressures. Also shown in Figure 8 is the rate coefficient for reaction R2 most commonly used in modeling, originally due to Marinov, et al,12 and discussed in the Introduction. At 1500 K, a temperature typical of the reaction zones in premixed flames, our prediction of kc varies from being a factor of 6.5 smaller than Marinov’s estimate at zero pressure to a factor of 673 at 100 atm. The considerations of the last paragraph suggest that we (and others) have overestimated the importance of C3H3 + C3H5 as a cyclization step in our flame modeling. This is probably true, but so far, we have overlooked the possibility that molecules stabilized in wells 1 and 2 could subsequently dissociate/ isomerize to form cyclic products. Let us examine this point now.
Figures 9 and 10 show the dissociation/isomerization rate coefficients for the molecules that are represented by wells 1 and 2, respectively. The (a) panels are for 10 Torr, and the (b) panels are for 10 atm. In all cases, the reaction is dominated by isomerization to the other of the two wells at low temperature (the energetically favored channel) and dissociation to C3H3 + C3H5 at high temperature (entropically favored over the isomerization). Higher pressures produce higher collision rates and allow higher energies to be accessed in the wells, thus promoting the higher-energy C3H3 + C3H5 channel over the isomerization. Consequently, the crossing point of the two rate coefficient curves moves to lower temperatures as the pressure increases. The cyclic products must be reached through TS1-4 regardless of which of the linear isomers, well 1 or well 2, is the reactant. This transition state is higher in energy than TS1-2 and tighter than TSR-1 or TSR-2, making it difficult for the cyclic product channels to compete with the isomerizations at low temperature and with dissociation to C3H3 + C3H5 at high T. Note that formation of the cyclic species from well 2 is a “well-skipping” reaction that forces a complex to pass through two tight transition states (TS1-2 and TS1-4) to access the cyclic part of the potential, whereas formation of cyclics from well 1 requires passage only through one tight transition state, TS1-4. This is the reason why cyclic product formation is so
Reactions between Resonance-Stabilized Radicals much larger for well 1 than that for well 2 under all conditions. The rate coefficients for formation of the cyclic products generally increase with pressure, at least at high temperature, but not as fast as that for the C3H3 + C3H5 product channel. Consequently, the fraction of the total rate coefficient that produces c-C6H7 and c-C6H8 molecules drops off with increased pressure. To get an estimate of how important secondary reactions could be to the formation of c-C6H7 and c-C6H8, we calculated the fraction of the well 1 and well 2 total dissociation/ isomerization rate coefficients that are due to these products and multiplied the results by kw1 and kw2, as discussed above. Comparing these “effective rate coefficients” with kc gives us an idea of how much “secondary” c-C6H7 and c-C6H8 might be produced, which in turn gives us a better idea of how much cyclization might occur as a result of the C3H3 + C3H5 reaction, both directly and indirectly. At 1500 K, these effective rate coefficients are 2.58 × 10-13 cm3/molecule-s at 30 Torr and 7.19 × 10-13 at 10 atm. These compare with 1500 K values of kc of 4.60 × 10-13 at 30 Torr and 4.50 × 10-14 at 10 atm. In both cases, the additional contributions are significant, and at high pressure, the secondary contribution is dominant. However, in neither case does the sum of the direct and secondary contributions come close to the Marinov estimate of 4.65 × 10-12 cm3/molecule-s. It should be noted that our method of estimating the secondary contribution to the formation of c-C6H7 and c-C6H8 is flawed in at least two respects: it neglects cyclic product formation preceded by one or more isomerizations between well 1 and well 2, and it neglects reactions of the well 1 and well 2 species with other molecules in a more complicated reacting environment. These two effects act in opposite directions and should be included in a phenomenological model. In any event, it appears that the Marinov estimate significantly overpredicts the formation rates of c-C6H7 and c-C6H8 for the C3H3 + C3H5 reaction.
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CH2CCHCH2CHCH2 a fulvene + H + H CHCCH2CH2CHCH2 a fulvene + H + H If desired, one can substitute fulvene + H2 for fulvene + H + H in these reactions since they will contribute negligibly to H or H2 formation in any ordinary combustion environment. In either case, the appropriate rate coefficient to use is the sum of those for all of the cyclic products (well 7, well 8, p5, and p6). Rate coefficients for these reactions are provided in Table 1; they are given in the form 2
k(T, p) )
∑ Aj(p)Tn (p) exp[-E(j)0 (p)/RT] j
j)1
that is, the rate coefficient at any pressure is represented as a sum of two modified Arrhenius functions (sometimes one function is sufficient). These results can be used directly in CHEMKIN 4.138 or higher, which interpolates log k linearly as a function of log p at any temperature. Rate coefficients for reverse reactions can be obtained from detailed balance. Acknowledgment. This work was supported by the United States Department of Energy, Office of Basic Energy Science, Division of Chemical Sciences, Geosciences and Biosciences. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94A185000. The work at Argonne was supported under Contract No. DE-AC02-06CH11357, and the work at the University of Georgia was supported under Contract No. DE-FG02-97ER14748. References and Notes
Concluding Remarks The work described above is a relatively complete analysis of the rate coefficient and product distribution of the reaction between propargyl and allyl, with a focus on the temperature and pressure dependence of the c-C6H7 (p5 and p6) and c-C6H8 (well 7 and well 8) product channels. Unfortunately, there are no experimental rate coefficients with which to compare our predictions. Nevertheless, the theoretical results indicate that the rate coefficient for reaction R2 commonly used in modeling is probably too large, although our simple analysis of secondary formation of c-C6H7 and c-C6H8 from the stabilized CH2CCHCH2CHCH2 (well 1) and CHCCH2CH2CHCH2 (well 2) adducts does not account for the production of these cyclic species following one or more CH2CCHCH2CHCH2 a CHCCH2CH2CHCH2 isomerizations. A satisfactory, but highly simplified, approximate model should consist of the following reactions:
C3H3 + C3H5 a fulvene + H + H C3H3 + C3H5 a CH2CCHCH2CHCH2 (well 1) C3H3 + C3H5 a CHCCH2CH2CHCH2 (well 2) CH2CCHCH2CHCH2 a CHCCH2CH2CHCH2
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