Reactions between resonance stabilized radicals : propargyl + allyl

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. HardingChemical Sciences and Engineering DiVision, Argonne National Laboratory, 9700 South Cass AVenue,Argonne, Illinois 60439

Wesley D. Allen and Andrew C. SimmonettCenter 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 variousforms of transition-state theory (TST) to calculate microcanonical, J-resolved rate coefficients-conventionalTST 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 masterequation to determine phenomenological rate coefficients, k(T,p), for various product channels. The analysisindicates 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 aspreviously thought. A simplified description of the kinetics of the reaction is discussed, and correspondingrate coefficients are provided.

Introduction

The reactions of resonance-stabilized free radicals play acritical role in the formation of aromatic compounds in flamesof aliphatic fuels.1-6 They may also be important in the growthof polycyclic aromatic hydrocarbons (PAH).2,3,7 The resonance-stabilized propargyl radical (C3H3) is singularly important inthe formation of the first ring, and its reaction with itself

dominates the cyclization process in virtually all flames. Thereasons underlying the significance of propargyl are understoodand make an interesting story. However, we shall defer thatstory to another time and place.

Over the past 10 years, we have been developing a chemicalkinetic model to predict the formation of benzene and othercyclic compounds in low-pressure flames of light-hydrocarbonfuels. The experimental flames with which we constantlycompare our model predictions are fueled by acetylene, ethylene,allene, propyne, propene, and 1,3-butadiene.8-11 All of theflames are rich or stoichiometric, and at least one flame of eachfuel 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

which is short for a two-step process

where c-C6H7 is a hydrofulvenyl radical. In rich flames, fulveneis relatively easily converted to benzene by H-atom-assistedisomerization

Because of the importance of reaction R2 in our flamemodeling, we thought it prudent to investigate its kinetics insome 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 thathad no intrinsic energy barrier (the highest rearrangement barrierwas ∼30 kcal/mol below the reactants). They concluded thatthe rate coefficient for reaction R3 would be determined by theC3H3 + C3H5 association rate, and they approximated this ratecoefficient 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

† Part of the special section “30th Free Radical Symposium”.* To whom correspondence should be addressed. E-mail: jamille@

sandia.gov.

C3H3 + C3H3 f products (R1)

C3H3 + C3H5 f fulvene + H + H (R2)

C3H3 + C3H5 f c-C6H7 + H (R3)

c-C6H7 (+M) f fulvene + H (+M) (R4)

H + fulvene f H + benzene

J. Phys. Chem. A 2010, 114, 4881–4890 4881

10.1021/jp910604b 2010 American Chemical SocietyPublished on Web 02/03/2010

suggests that the rearrangement of the initial complex in C3H3

+ C3H5 may have a large impact on the rate coefficient forforming c-C6H7 + H. Clearly, a detailed theoretical analysis iswarranted.

Theoretical and Computational Approach

For this investigation, we employed the B3LYP hybrid DFT(density functional theory) method with Pople’s split-valence6-311++G(d,p) basis set to optimize geometries and computethe vibrational frequencies of all of the relevant stationary pointson the PES, both minima and saddle points. Additionally, wecalculated rotational potentials for molecular configurations withinternal rotations and intrinsic reaction coordinate (IRC) curvesusing this theoretical model. In order to obtain accurate ratecoefficients, we calculated single-point energies at all of thestationary points using Dunning’s basis sets, cc-pVxZ with x) T, Q, and the rQCISD(T) method (spin-restricted quadraticconfiguration interaction with perturbative estimate of the triplescontribution). Finally, the latter energies were extrapolated tothe infinite basis set limit using the formula15,16

where lmax is the maximum angular momentum in the basis setand B is a fitting constant. The final potential energy surface(PES) is depicted diagrammatically in Figure 2 for referenceand is discussed at length below; the structures correspondingto the wells and cyclic components of the bimolecular productsare shown in Figure 3.

Several of the stationary points on the potential (wells 4 and6 and transition states 4-6, 4-7, and 6-7) were found to havesufficiently high multireference character (as indicated by largeT1 (or Q1) diagnostics) that it was deemed necessary to re-evaluate their energies using multireference methods. For thesepoints, the geometries were reoptimized with (4e,4o)-CASPT2/cc-pVDZ. Here, the four active orbitals consist of the diradicalorbitals and the π and π* orbitals. At these geometries,multireference configuration interaction calculations includinga multireference Davidson correction, (4e,4o)-CAS+1+2+QC25,26

were done with both cc-pVTZ and cc-pVQZ basis sets, and theenergies were extrapolated to the infinite basis set limit asdescribed above. For each of these points, the multireferencecalculations were performed for both the singlet and thecorresponding triplet stationary points. The resulting multiref-erence singlet-triplet splittings were combined with rQCISD(T)energies for the triplet states (which do not have significantmultireference character) to yield the final energies of the singletstates. The overall corrections ranged from nearly 0 to about 7kcal/mol.

One possible source of error in the stationary point energiescomes from locating such points inaccurately using a lower-level method than is used for the energies themselves. To testthe magnitude of this effect, we calculated the structure ofTS1-4, the critical transition state in the present analysis (seeFigure 2), using CCSD(T)/cc-pVTZ in addition to the methoddescribed above. For practical purposes, this structure can beviewed as exact. The single-point energies determined by thesame rQCISD(T) method for the two different structures differedby ∼0.25 kcal/mol, an inconsequential difference.

Phenomenological (thermal) rate coefficients, k(T,p), weredetermined as a function of temperature and pressure using twodifferent forms of the master equation:17-21

(1) The time-dependent multiple-well ME. This approach islimited to a one-dimensional formulation in which E, thevibrational-rotational energy of the molecule or complex, isthe independent variable.

(2) The collisionless limit of the multiple-well ME. As thename implies, this equation is simply the p f 0 limit of theabove ME. However, in the limiting case, we can solve the prob-lem in both one and two dimensions. In the latter case, E andJ, where J is the total angular momentum quantum number,are the independent variables. In the present investigation, Jconservation was never a significant issue. Nevertheless, all ofthe results reported in this article are from two-dimensionalcalculations.

The methodology for obtaining phenomenological rate coef-ficients from the master equation would require too much spaceto describe here. However, extensive discussions can be foundin refs 13, 19-21, and particularly ref 17. There are two typesof transition probabilities in the master equation, collisional andreactive. For the present work, we employed Lennard-Jonescollision rates and approximated the energy-transfer functionusing the single-exponential-down model with ⟨∆Ed⟩ )400(T/300 K)0.85 cm-1. The collider in the present work wastaken to be molecular nitrogen. The energy-transfer functionwas also assumed to be independent of energy, angularmomentum, and the isomeric form of the complex. Probablynone of these approximations are accurate, not even theexponential form of the energy-transfer function.21 However,thermal rate coefficients are only weakly dependent on thesedetails, 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 theassociation/dissociation reactions and conventional transition-state theory for the isomerizations. For the one-dimensional MEcalculations, the k(E) functions are constructed directly fromthe k(E,J)’s in order to preserve the accuracy of the J-resolvedrate coefficients as much as possible. Tunneling and nonclassicalreflection (both of which are automatically included in thequantum transition probabilities) are included in the analysisone-dimensionally by assuming that the potential along theminimum-energy path can be approximated by an asymmetricEckart function. Tunneling plays little or no role in the C3H3 +C3H5 reaction because the important isomerization transitionstates lie so much lower in energy than the reactants. However,our prior experience suggests that, for thermal isomerizationstaking place on the same potential, the effect of tunneling willbe significant at temperatures up to about 1000 K.

The electronic structure calculations were done with Gaussian0322 and MOLPRO.23 The RRKM and master equation calcula-tions were performed with VARIFLEX.24

A number of channels in the reaction involve the recombina-tion or production of two radicals (i.e., C3H3 + C3H5, C2H3 +i-C4H5, and H + C6H7, where there are numerous isomers ofthe latter). For such processes, where the minimum-energy pathpotentials in the recombination direction are barrierless, bothvariational and anharmonic effects have a major impact on therate 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 toprovide an effective means for treating these complications.31,32

The calculations were performed as previously described forthe C3H3 + C3H5 system14 and in close analogy with thosedescribed for the recombination of H atoms with resonance-stabilized radicals.33 For completeness, we describe here someof the key features of the methodology.

E(∞) ) E(lmax) - B/(lmax + 1)4

4882 J. Phys. Chem. A, Vol. 114, No. 14, 2010 Miller et al.

The direct VRC-TST approach requires a method for estimat-ing the interaction between the two reacting radicals for arbitraryorientation and separation. Here, we evaluate these interactionswith the CASPT2 method employing an eight-electron, eight-orbital (8e,8o) active space consisting of the two radical orbitalstogether with the full (6e,6o) π,π* space. These CASPT2calculations employ Dunning’s correlation- consistent, polarizedvalence, double-� basis set34 and were done using the formalismof Celani and Werner35 as implemented in the MOLPROprogram package.23,36

We include two corrections to these base CASPT2(8e,8o)/cc-pVDZ rigid body interaction energies. In particular, orienta-tion-independent corrections for limitations in the basis set areobtained from complete basis set estimates for the interactionenergies along a minimum-energy path. The present CBSestimates are obtained as the average of the separate extrapola-tions of cc-pVDZ, cc-pVTZ and aug-cc-pVDZ, aug-cc-pvtz pairsof calculations. These CBS extrapolations are the same as inthe stationary-point energy determinations described above. ThisCBS correction is included for all of the radical-radicalchannels except for the C3H3 + C3H5 reaction, where the basisset 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 radicalsof interest here, this correction is more significant than is typical.For the propargyl + allyl addition, the internal geometries ofthe fragments were optimized at the uB3LYP/cc-pVDZ levelfor each orientation of the fragments. Sample evaluationsindicated that the uB3LYP-optimized relaxation energies weresuitably close to those obtained from CASPT2/cc-pVDZgeometry optimizations.14 For the remaining reactions, one-dimensional geometry relaxation corrections were obtained fromuB3LYP/6-31G* calculations along a minimum-energy path.

The rate coefficient predictions are only weakly dependenton the equilibrium geometries of these radicals, and therefore,they were simply determined from uB3LYP/6-311++G** oruB3LYP/cc-pVDZ calculations using the GAUSSIAN98 pro-gram.37

A number of the linear C6H7 species have multiple torsionalisomers. Generally, barrierless addition rates are only weaklydependent on the torsional state of the reactants. In particular,changes in the torsional state generally affect the backside andfrontside additions in opposite ways, and therefore, the sum ischanged little. Thus, for simplicity, we have evaluated theaddition rates for only one of the isomers. However, we haveexplicitly considered both frontside and backside additions asappropriate.

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 em-ployed a (6e,6o) active space instead of the (8e,8o) active spacedescribed above. The (6e,6o) active space consists of the πsystem of the cyclic-C6H7 fragments and the H orbital. We alsoemployed the CASPT2(6e,6o)/cc-pVDZ approach in the deter-mination of the one-dimensional geometry relaxation correctionsfor these three reactions.

Wells 4 and 6 are cyclic-C6H8 species with open-shell singletbiradical wave functions containing two unpaired electrons inthe π system. The lowest-energy C-H bond fission from well6 yields product p5. For well 4, there are two distinct C-Hbond fissions from the C5 ring. One yields product p5 again,while the other yields the p6 fragments. The reverse additionreactions for these C-H bond fissions correspond to H additionacross a π bond. Such reactions generally involve modest

barriers and therefore can be treated with a rigid rotor, harmonicoscillator implementation of TST.

The production of singlet biradicals in these reverse additionsimplies the need to employ multireference electronic structuremethods in the analysis of the rovibrational properties of theirsaddle points. Here, we use the CASPT2 approach for thisanalysis and implement a (6e,6o) active space consisting of theπ system of the cyclic-C6H7 fragments and the H orbital. Thegeometries and rovibrational properties were first obtained forthe cc-pVDZ basis set. The energies were then extrapolated tothe CBS limit via calculations with the cc-pVTZ and cc-pVQZbasis sets at the cc-pVDZ geometries. The energies for thesetransition states relative to the corresponding products wereconverted to absolute energies (i.e., relative to reactants) viathe rQCISD(T)/CBS absolute energies for the products.

Attempts to locate a saddle point for the C-H bond fission fromwell 4 to species p6 were unsuccessful and led instead to theisomerization transition state from well 4 to well 8. Constrainedoptimizations suggest that the reverse addition has a relatively largebarrier, while the C-H bond fission from well 8 to p6 is barrierless.Thus, it appears that the dynamical process for production of speciesp6 from well 4 involves first the isomerization to well 8 followedby 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. Thesestructures are shown in Figure 1. Again, the potential energysurface on which our rate coefficient calculations are basedis shown diagrammatically in Figure 2; the structures of thestable molecules corresponding to the wells and the cyclicproducts are shown in Figure 3. Propargyl and allyl cancombine in two different ways, with either the head (the CH2

end) or the tail (the CH end) of propargyl forming the bondwith allyl (the two ends of allyl are identical). The bond inthe former case is slightly weaker than that in the latter (byless than 1 kcal/mol). The barrier to isomerization is only34.8 kcal/mol in the exothermic direction, resulting in TS1-2lying 28.8 kcal/mol below the C3H3 + C3H5 asymptote. Thetwo adducts can dissociate in a number of ways by eliminat-ing various hydrogen atoms or by breaking a C-C bond toform C2H3 + i-C4H5, as shown in the diagram. The

Figure 1. Resonant Kekule structures for allyl and propargyl radicals.

Reactions between Resonance-Stabilized Radicals J. Phys. Chem. A, Vol. 114, No. 14, 2010 4883

product p1 represents CH2CCHCHCHCH2 + H, p2 denotesCHCCH2CHCHCH2 + H, p3 represents C2H3 + i-C4H5, and

p4 is short for CHCCHCH2CHCH2 + H. These products liebetween 11.8 and 24.0 kcal/mol above the reactants.

Figure 2. Potential energy diagram (including zero-point energies) for the C3H3 + C3H5 reaction. The numbers on the diagram are energies inkcal/mol relative to reactants.

Figure 3. Molecular structures corresponding to the wells and the cyclic fragments.


The most important feature of the potential for our purposesis TS1-4, which closes well 1 into a cyclic five-memberedsinglet biradical (well 4). This biradical can then eject ahydrogen atom, producing a hydrofulvenyl radical (c-C6H7).However, as discussed above, only the less stable of the twoc-C6H7 radicals, p5, is accessible from this well. Nevertheless,well 4 is the gateway not only to the two hydrofulvenyl radicalsbut 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 moretortuous path that can also lead to the hydrofulvenyl radicalsand the stable cyclic-C6H8 compounds through a biradicalintermediate (well 6). It starts by closing well 1 into a four-membered ring, reopening the ring at a different place, and thenclosing again into another cyclic five-membered singlet biradical(well 6). Because of the difference between the energies ofTS1-4 and TS1-3 (and TS3-5), the latter route makesnegligible contributions to the rate coefficients. Higher-energyroutes leading from wells 1 and 2 also make negligiblecontributions to the rate coefficients and are neglected in theanalysis.

Wells 5 and 6 are effectively isolated from the rest of thePES by the relatively high barriers at TS3-5, TS4-6, andTS6-7. We could eliminate these wells from our analysis andstill get the same results for the rate coefficients. We includedthem, but we allowed for only one dissociation channel for well5, C2H3 + i-C4H5, although several H-atom-producing channelsare possible. As discussed below, C2H3 + i-C4H5 is the dominantbimolecular product channel at high temperatures where wells5 and 6 would be most likely to become accessible.

Thermal Rate Coefficients

Figure 4 shows our predictions for the total rate coefficientat a series of pressures. By definition, the high-pressure limitingvalue of ktot, k∞, is the capture rate coefficient, that is, the ratecoefficient for complex formation. This represents an upper limiton the total rate coefficient. At the high-pressure limit (i.e., attruly infinite pressure), the only possible products are the initialadducts, well 1 and well 2. At the other end of the spectrum isthe collisionless (or p ) 0) limit, k0. At this limit, there is nostabilization in any of the wells, and the only possible productsare the bimolecular ones (p1-p6). This is the lower limit ofktot. At any pressure, including p ) 0, ktot starts out at lowtemperature very close to k∞ and ultimately coalesces with k0

at high temperature. This behavior is universal as long as there

is a bimolecular product channel whose threshold energy liesbelow the reactants. As the temperature increases at fixedpressure, unimolecular dissociation of the complexes (either toproducts or back to reactants) becomes increasingly dominantover collisional stabilization. The same behavior is observed ifall of the bimolecular products are endothermic, except that inthis 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 indicatesthat the endothermic product channels are beginning to comeinto play. The k(T) curves at all finite pressures have suchminima, but they appear at increasingly higher temperatures asthe pressure increases. Second, at T ) 1900 K (the minimumof the k0(T) curve), the ratio of k∞ to k0 is 71.6. This indicatesthat (in the absence of collisions) only one in approximately73 complexes formed goes on to produce products, even thoughthe subsequent transition states leading to p6, the lowest-energybimolecular channel, lie well below the reactants. This behavior,as well as the sharp decrease of k0(T) with increasing T attemperatures below 1900 K, is due to the tight, entropy-deficientcharacter of the rearrangement transition states, most notablyTS1-4.

In Figure 5, we examine the product distribution at thecollisionless limit. As one might expect, the lowest-energychannel (p6), a c-C6H7 radical + H, is dominant at lowtemperatures. Its rate coefficient is roughly equal to the capturerate coefficient up to T ≈ 400 K. For T g 400 K, kp6 drops offrapidly with temperature because of the tightness of therearrangement transition states. At T ≈ 1900 K, the higher-entropy endothermic product channels begin to take over.Interestingly, the dominant product channel at high temperaturesis p3, i-C4H5 + C2H3, even though two of the H-atom channelsformed from wells 1 and 2 are lower in energy. This is a“rotational entropy” effect, largely a result of two dissociatingpolyatomic fragments having more rotational degrees of freedomthan an atom and a polyatomic fragment. The endothermicH-producing channels (p1, p2, and p4) are never reallysignificant even at high temperature.

Figure 6 displays the individual rate coefficients as a functionof temperature for a pressure of 10 Torr. Figure 6a focuses onthe stabilized products and Figure 6b the bimolecular channels.In Figure 6b, for simplicity, the rate coefficients for all of thenoncyclic H-producing channels are added together, kp1 + kp2

Figure 4. Total rate coefficients for C3H3 + C3H5 f products as afunction of temperature at various pressures. Figure 5. Rate coefficients for various product channels of the C3H3

+ C3H5 reaction at the collisionless (p ) 0) limit.


+ kp4, and labeled “linear+H.” As expected, stabilization intowells 1 and 2 is dominant at the lower temperatures. Stabilizationinto wells 3, 4, 5, and 6 is never significant because these wellsare relatively inaccessible, too shallow, or both. Only attemperatures of T ≈ 1500 K and above can the bimolecularchannels compete. Interestingly, the rate coefficients for stabi-lization into well 1 (kw1) and well 2 (kw2) cross. At lowtemperature, the dominant factor in determining these ratecoefficients is the reactive flux through TSR-1 and TSR-2.The larger reactive flux through TSR-2 causes kw2 to be largerthan kw1. However, at high temperature, the rate-controllingfactor is collisional deactivation. Because the lifetimes of well1 complexes are longer than those of well 2 (a consequenceprimarily of the slightly stronger bond of well 1), kw1 is slightlylarger than kw2 at high T.

In addition to the c-C6H7 + H channels, five-membered ringscan also be formed by stabilization into wells 7 and 8. Theformation of these species is precluded at the lowest tempera-tures because collisional stabilization into wells 1 and 2 is sodominant. However, as the temperature rises, it is increasinglypossible for a complex in one of these early wells to isomerizeand become stabilized in one of the deeper cyclic wells. Thus,the rate coefficients kw7 and kw8 first increase with temperaturein Figure 6. They subsequently fall with increasing T as thebimolecular channels, particularly p6, begin to dominate. Ascan be seen from Figure 6, the reaction essentially produces

only bimolecular products above 2200 K. The rate coefficientsfor forming the c-C6H7 + H fragments, kp5 and kp6, also decayat high temperature, where they cannot compete with theproducts formed from fragmentation in the early wells, par-ticularly p3.

The reason that wells 7 and 8, as well as fragments p5 andp6, are so difficult to reach at high temperature is a classiccompetition between loose and tight transition states. All of theisomerization transition states, most notably TS1-4, are tight;all fragmentation transition states are loose. Tight transitionstates have compact structures, resulting in relatively smallmoments of inertia (large rotational constants) and highvibrational frequencies; loose transition states are just theopposite. Because of these differences, tight transition statesare lower in entropy than loose ones. More to the point, if bothhave the same ground-state energy, a loose transition state willgenerally have a sum-of-states that is larger than a similar tightone for a given vibrational-rotational energy, that is, there ismore flux through the loose transition state. This difference canbe compensated for if the tight transition state lies lower on thepotential energy surface than the loose one. For larger molecules(with more degrees of freedom), the compensating energy mustbe larger than that for smaller molecules because the loose/tight sum-of-states ratio is larger for a given vibrational-

Figure 7. Rate coefficients of the C3H3 + C3H5 reaction for variousproduct channels at a pressure of 10 atm. The label “bimolecularproducts” indicates the sum of the rate coefficients for the p1-p6product channels. “Linear+H” is the sum of the rate coefficients forthe p1, p2, and p4 product channels.

Figure 6. Rate coefficients of the C3H3 + C3H5 reaction for variousproduct 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 productchannels.


rotational energy for larger molecules. For example, if we weredealing with a system of four or five atoms, the tightness ofTS1-4 would be inconsequential because it lies so low on thepotential. However, for a 14-atom system, the tightness ofTS1-4 is the dominant factor determining access to the cyclic

species from C3H3 + C3H5. Of course, higher temperaturesreflect higher energies, and any advantage from energy con-siderations that a tight transition state might have disappears athigh T.

It is interesting and of practical importance to investigate theeffect of elevated pressure on the formation of the cyclicproducts (wells 7 and 8 and bimolecular products p5 and p6).To this end, we have plotted our results for a pressure of 10atm in Figure 7. The rate coefficients at this pressure displaymany of the same qualitative features as those at low pressure.However, the most striking effect of increased pressure is thesuppression of the cyclic product channels. The rate coefficientsfor these channels rise at higher temperatures and attain peakvalues that are more than an order of magnitude smaller at 10atm than that at 10 Torr. This reduction in the size of the ratecoefficients is due to the increased likelihood of complexes beingstabilized in the early wells because of the increased collisionrates. The values of kw1 and kw2 increase by about an order ofmagnitude at 1500 K in going from 10 Torr to 10 atm. Becausetheir lifetimes are shorter, higher-energy complexes are lesslikely than the lower-energy ones to suffer collisions. Conse-quently, the rate coefficients for the high-energy bimolecularchannels, particularly p3, are considerably less affected byincreased pressure than those for the cyclic products (whichnecessarily involve TS1-4).

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 oftemperature at various pressures.

Figure 9. Dissociation/isomerization rate coefficients forCH2CCHCH2CHCH2 (well 1) at (a) 10 Torr and (b) 10 atm. The label“bimolecular products” indicates the sum of the rate coefficients forthe p1-p6 product channels.

Figure 10. Dissociation/isomerization rate coefficients forCHCCH2CH2CHCH2 (well 2) at (a) 10 Torr and (b) 10 atm. The label“bimolecular products” indicates the sum of the rate coefficients forthe p1-p6 product channels.


In order to illustrate the effect of pressure on the formationof 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 variouspressures in Figure 8. Clearly, increased pressure suppressesthe formation of the c-C6H7 and c-C6H8 species at all temper-atures. At low temperatures, where kc at zero pressure isessentially equal to the capture rate coefficient (k∞), the effectis dramatic. It becomes smaller as the temperature goes up. AsT f ∞, kc approaches its collisionless value at all pressures.Also shown in Figure 8 is the rate coefficient for reaction R2most commonly used in modeling, originally due to Marinov,et al,12 and discussed in the Introduction. At 1500 K, atemperature typical of the reaction zones in premixed flames,our prediction of kc varies from being a factor of 6.5 smallerthan Marinov’s estimate at zero pressure to a factor of 673 at100 atm.

The considerations of the last paragraph suggest that we (andothers) have overestimated the importance of C3H3 + C3H5 asa cyclization step in our flame modeling. This is probably true,but so far, we have overlooked the possibility that moleculesstabilized in wells 1 and 2 could subsequently dissociate/isomerize to form cyclic products. Let us examine this pointnow.

Figures 9 and 10 show the dissociation/isomerization ratecoefficients for the molecules that are represented by wells 1and 2, respectively. The (a) panels are for 10 Torr, and the (b)panels are for 10 atm. In all cases, the reaction is dominated byisomerization 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 isomer-ization). Higher pressures produce higher collision rates andallow higher energies to be accessed in the wells, thus promotingthe higher-energy C3H3 + C3H5 channel over the isomerization.Consequently, the crossing point of the two rate coefficientcurves moves to lower temperatures as the pressure increases.

The cyclic products must be reached through TS1-4 regard-less of which of the linear isomers, well 1 or well 2, is thereactant. This transition state is higher in energy than TS1-2and tighter than TSR-1 or TSR-2, making it difficult for thecyclic product channels to compete with the isomerizations atlow temperature and with dissociation to C3H3 + C3H5 at highT. Note that formation of the cyclic species from well 2 is a“well-skipping” reaction that forces a complex to pass throughtwo tight transition states (TS1-2 and TS1-4) to access thecyclic part of the potential, whereas formation of cyclics fromwell 1 requires passage only through one tight transition state,TS1-4. This is the reason why cyclic product formation is so

TABLE 1: Modified Arrhenius Representation of Rate Coefficients for Modeling (see text)a

pressure (Torr) A1 (cm3/molecule-s) n1 E01(cal/mol) A2 (cm3/molecule-s) n2 E02

(cal/mol)

C3H3 + C3H5 f CH2CCHCH2CHCH2 (well 1)10 0.35 × 1045 -16.68 25753 0.655 × 109 -6.688 385330 0.773 × 1042 -15.75 26038 0.586 × 107 -5.981 3458760 0.543 × 1032 -12.42 24912 26.56 -4.153 23647600 -0.248 × 108 -5.578 5166 1832 -4.253 379076000 -2.45 -3.346 3103 0.799 × 10-2 -2.545 2226

C3H3 + C3H5f CHCCH2CH2CHCH2 (well 2)10 0.325 × 1042 -15.88 22677 0.242 × 1012 -7.475 427730 0.235 × 1041 -15.39 23547 0.142 × 1010 -6.693 3878760 0.602 × 1033 -12.78 23846 3.82 × 103 -4.775 27627600 -0.162 × 1011 -6.404 5767 0.930 × 106 -5.043 436476000 -728 -4.054 3660 1.87 -3.221 2743

C3H3 + C3H5f cyclics10 1.89 × 1021 -9.656 1965230 3.80 × 1019 -9.123 20504760 3.13 × 1014 -7.526 238787600 0.737 -3.364 1824976000 1.54 × 1017 1.291 10523

pressure (Torr) A1 (1/s) n1 E01(cal/mol) A2 (1/s) n2 E02

(cal/mol)

CHCCH2CH2CHCH2f CH2CCHCH2CHCH2

10 1.60 × 1043 -9.678 4984830 3.04 × 1040 -8.821 49074760 1.25 × 1032 -6.242 458057600 4.22 × 1024 -4.021 4210576000 4.68 × 1016 -1.696 37722

CH2CCHCH2CHCH2f cyclics10 3.41 × 1037 -8.054 5096030 3.19 × 1036 -7.676 51281760 3.20 × 1032 -6.314 515857600 2.20 × 1026 -4.448 4935276000 5.30 × 1021 -3.010 49033

CHCCH2CH2CHCH2f cyclics10 3.72 × 1075 -19.70 8346630 5.22 × 1084 -21.16 93165760 4.34 × 1083 -20.57 1024007600 9.08 × 1070 -16.89 9987676000 3.13 × 1022 -3.514 64992

a Nominally, the fits for kw1 and kw2 are applicable from 250 to 2500 K, while the others are from 700 to 2500 K.


much larger for well 1 than that for well 2 under all conditions.The rate coefficients for formation of the cyclic productsgenerally 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 thatproduces c-C6H7 and c-C6H8 molecules drops off with increasedpressure.

To get an estimate of how important secondary reactionscould be to the formation of c-C6H7 and c-C6H8, we calculatedthe fraction of the well 1 and well 2 total dissociation/isomerization rate coefficients that are due to these productsand multiplied the results by kw1 and kw2, as discussed above.Comparing these “effective rate coefficients” with kc gives usan idea of how much “secondary” c-C6H7 and c-C6H8 might beproduced, which in turn gives us a better idea of how muchcyclization might occur as a result of the C3H3 + C3H5 reaction,both directly and indirectly. At 1500 K, these effective ratecoefficients are 2.58 × 10-13 cm3/molecule-s at 30 Torr and7.19 × 10-13 at 10 atm. These compare with 1500 K values ofkc of 4.60 × 10-13 at 30 Torr and 4.50 × 10-14 at 10 atm. Inboth cases, the additional contributions are significant, and athigh pressure, the secondary contribution is dominant. However,in neither case does the sum of the direct and secondarycontributions come close to the Marinov estimate of 4.65 ×10-12 cm3/molecule-s. It should be noted that our method ofestimating the secondary contribution to the formation of c-C6H7

and c-C6H8 is flawed in at least two respects: it neglects cyclicproduct formation preceded by one or more isomerizationsbetween well 1 and well 2, and it neglects reactions of the well1 and well 2 species with other molecules in a more complicatedreacting environment. These two effects act in opposite direc-tions and should be included in a phenomenological model. Inany event, it appears that the Marinov estimate significantlyoverpredicts the formation rates of c-C6H7 and c-C6H8 for theC3H3 + C3H5 reaction.

Concluding Remarks

The work described above is a relatively complete analysisof the rate coefficient and product distribution of the reactionbetween propargyl and allyl, with a focus on the temperatureand pressure dependence of the c-C6H7 (p5 and p6) and c-C6H8

(well 7 and well 8) product channels. Unfortunately, there areno experimental rate coefficients with which to compare ourpredictions. Nevertheless, the theoretical results indicate thatthe rate coefficient for reaction R2 commonly used in modelingis probably too large, although our simple analysis of secondaryformation of c-C6H7 and c-C6H8 from the stabilizedCH2CCHCH2CHCH2 (well 1) and CHCCH2CH2CHCH2 (well2) adducts does not account for the production of these cyclicspecies following one or more CH2CCHCH2CHCH2 aCHCCH2CH2CHCH2 isomerizations. A satisfactory, but highlysimplified, approximate model should consist of the followingreactions:

If desired, one can substitute fulvene + H2 for fulvene + H+ H in these reactions since they will contribute negligibly toH or H2 formation in any ordinary combustion environment. Ineither case, the appropriate rate coefficient to use is the sum ofthose 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

that is, the rate coefficient at any pressure is represented as asum of two modified Arrhenius functions (sometimes onefunction is sufficient). These results can be used directly inCHEMKIN 4.138 or higher, which interpolates log k linearly asa function of log p at any temperature. Rate coefficients forreverse reactions can be obtained from detailed balance.

Acknowledgment. This work was supported by the UnitedStates Department of Energy, Office of Basic Energy Science,Division of Chemical Sciences, Geosciences and Biosciences.Sandia is a multiprogram laboratory operated by SandiaCorporation, a Lockheed Martin Company, for the United StatesDepartment of Energy’s National Nuclear Security Administra-tion under Contract DE-AC04-94A185000. The work at Ar-gonne was supported under Contract No. DE-AC02-06CH11357,and the work at the University of Georgia was supported underContract No. DE-FG02-97ER14748.

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C3H3 + C3H5 a CH2CCHCH2CHCH2 (well 1)

C3H3 + C3H5 a CHCCH2CH2CHCH2 (well 2)

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Reactions between resonance stabilized radicals : propargyl + allyl

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