Initiation Reactions in Acetylene Pyrolysis - The Journal of Physical

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Initiation Reactions in Acetylene Pyrolysis Judit Zádor, Madison D. Fellows, and James A. Miller J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 10 May 2017 Downloaded from http://pubs.acs.org on May 11, 2017

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The Journal of Physical Chemistry

Initiation Reactions in Acetylene Pyrolysis Judit Zádor,1,* Madison D. Fellows,1,† and James A. Miller2

1

Combustion Research Facility, Mail Stop 9055, Sandia National Laboratories, Livermore, CA

94551-0969, USA 2

Chemistry Division, Argonne National Laboratory, Argonne, IL 60439, USA

Submitted to the Journal of Physical Chemistry A

* †

Corresponding author: [email protected], 1 (925) 294-3603 Currently at Grinnell College, Grinnell, IA, USA

ACS Paragon Plus Environment

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Abstract In gas-phase combustion systems the interest in acetylene stems largely from its role in molecular weight growth processes. The consensus is that above 1500 K acetylene pyrolysis starts mainly with the homolytic fission of the C–H bond creating an ethynyl radical and an H atom. However, below ~1500 K this reaction is too slow to initiate the chain reaction. It has been hypothesized that instead of dissociation, self-reaction initiates this process. Nevertheless, rigorous theoretical or direct experimental evidence is lacking, to an extent that even the molecular mechanism is debated in the literature. In this work we use rigorous ab initio transition state theory master equation methods to calculate pressure- and temperature-dependent rate coefficients for the association of two acetylene molecules and related reactions. We establish the role of vinylidene, the high-energy isomer of acetylene in this process, compare our results with available experimental data, and assess the competition between the first-order and secondorder initiation steps. We also show the effect of the rapid isomerization among the participating wells, and highlight the need for timescale analysis when phenomenological rate coefficients are compared to observed timescales in certain experiments.


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1. Introduction Acetylene (C2H2) is one of the most well studied molecules in physical chemistry. In gas-phase combustion systems the interest in this molecule stems largely from its role in molecular weight growth processes: pure acetylene pyrolysis serves as a basis for the understanding of the pyrolysis of other hydrocarbons1-10 and to soot formation in general. Hurd11 proposed as early as in 1934 that below 1000 K the pyrolysis largely proceeds via a radical mechanism as a polymerization reaction forming C4H4 and larger unsaturated products, including benzene. By 1957 it was established12 that the reaction has an activation energy around 46 kcal mol-1, but the details of the mechanism with which the reaction proceeds remained unclear. In 1992 Benson1 summarized the main aspects of acetylene pyrolysis, which in his view clearly point to the direction of radical chain reactions both at low and high temperatures. The overall reaction appears to be second order in acetylene concentration over a wide temperature range, and the induction time becomes shorter with increasing temperature. The consensus is that above ~1500 K acetylene pyrolysis starts mainly with the homolytic fission of the C–H bond creating an ethynyl radical (C2H) and an H atom:

R1

C2H2 + M → C2H + H + M

where M is a third-body collider. The H atoms produced can in turn abstract another H atom from acetylene:

R2

C2H2 + H → C2H + H2


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forming another ethynyl radical. The chain proceeds by ethynyl adding to acetylene in a metathesis reaction:

R3

C2H2 + C2H → C4H2 + H

to form diacetylene, while recreating the chain-propagating H atom. In further steps, in analogy to reactions R2 and R3, longer and longer polyacetylene (C2nH2) molecules can be formed. However, mechanism R1-R3 cannot explain the weight growth taking place below ~1500 K, because reaction R1 is too slow here to initiate the chain reaction. The C–H bond enthalpy is ~133 kcal mol-1 in acetylene,13 and according to Tsang and Hampson,14 the rate coefficient is only 0.02 s-1 in the high-pressure limit at 1500 K. It has been hypothesized that instead of dissociation, self-reaction initiates this process. Nevertheless, rigorous theoretical or direct experimental evidence is lacking about this hypothetical self-reaction, to an extent that even its molecular mechanism is debated in the literature, except that it should include radicals and H atom production. Benson15 and many others proposed and argued for the formation of a short-lived biradical directly from acetylene dimerization as the key initiation step:

HC

CH + HC

CH

HC

C H

C H

CH

R4

One of the arguments for a radical playing a key role is that radical scavengers, such as NO, inhibit the formation of large molecular weight products. According to Benson the biradical in turn rearranges to vinyl acetylene (VA) with a rapid H-shift:


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HC

C H

C H

CH

H2 C

C H

C

CH

R5

Importantly, VA can lose an H atom to form a resonantly stabilized C4H3 radical:

H 2C

C H

C

CH

HC

C

C

CH 2 + H

R6

The H atom created in reaction R6 can add to acetylene to form a vinyl radical, C2H3, which in turn can add to acetylene as well to form again vinyl acetylene. The co-product of this last reaction is also an H atom, allowing the chain to propagate this way. For the full scheme see p. 226 of Benson’s 1992 paper.1 What concerns this paper is the very first step in this scheme, reaction R4, which involves two acetylene molecules. Colussi and co-workers5 and Kiefer and coworkers16-17 proposed an important alteration to reaction R4. They argued that the initiation begins with a reaction between acetylene and its high-energy isomer, vinylidene (:CCH2). As for the role of NO as a radical scavenger Kiefer and Von Drasek17 argued for the participation of methylene cyclopropene (MCP) and its equilibrium with an unstable CH2CCHCH radical, which is perhaps the direct adduct of vinylidene and acetylene, and that the observed inhibition effect of NO does not necessarily point to the importance of R4, it is rather a “strange and perplexing”3 phenomenon in view of many other important details. In his paper Benson1 argued, based on thermochemistry, that the participation of vinylidene can be excluded, but his arguments hinged on the heat of formation of :CCH2. Having no accurate thermochemical information at hand at that time he put a lower limit estimate of (120.5±6) kcal mol-1 on ∆fH0(:CCH2). In order for the HCCH + :CCH2 mechanism to


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be plausible, the ∆fH0 of vinyledene needs to be ≤ 96.2 kcal mol-1. However, the current value given in the Active Thermochemical Tables18-20 is (98.32±0.15) kcal mol-1, which, at least, does not support Benson’s estimate, and likely is in line with the required value for the mechanism to be active, at least on thermodynamic grounds. In summary, despite decades of experimental work and modeling, the low-temperature initiation mechanism is still unclear, primarily because it is derived from indirect arguments, which are often complicated by unavoidable experimental uncertainties. As is often the case with initiation reactions, they are very difficult to study in isolation, because they are slow and are quickly overtaken by fast chain propagation and branching reactions. Here, instead of reinvestigating these indirect arguments, we will provide a rigorous theoretical description of the acetylene + acetylene and acetylene + vinylidene reactions. The C4H4 potential energy surface (PES) is very complex and there are multiple pathways that can contribute to the self-reaction of acetylene, including the ones mentioned. The PES has been also very extensively studied in the literature because of its complexity, beauty,21 and fundamental nature. The latest two and perhaps most detailed investigations are from Cremer et al.22 and Mebel et al.23 Both papers give a thorough overview of the previous investigations, and more importantly, provide an exhaustive account of the stationary points. The Cremer et al.22 paper focuses more on the mechanistic and electronic structure aspects of the system, while the Mebel et al.23 paper – besides accurate electronic structure calculations – provides energy specific branching fractions and rate coefficients for the C2(X1Σg+) + C2H4(X1A1g) and C(1D) + C3H4 reactions. Despite the ocean of previous studies (Cremer et al. cite 72 papers in the second sentence of their article!), there is no rigorous theoretical work on the pressure- and temperaturedependent kinetics of the acetylene self-reaction.


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Our goal in this paper, therefore, is to find out whether the acetylene + acetylene or the acetylene + vinylidene step is the main initiation process in acetylene pyrolysis below ~1500 K, and whether the calculated rate coefficients and branching fractions can account for the lowtemperature acetylene pyrolysis observations. Largely building on the PESs of Cremer et al.22 and Mebel et al.23, we identify the subset of stationary points on the C4H4 PES relevant for the description of the kinetics of this reaction under combustion conditions. On this surface we calculate temperature- and pressure-dependent rate coefficients using a time-dependent master equation. The results are provided also in CHEMKIN format to be used in combustion models. The association direction of the initiation reaction R1 was studied by Harding and coworkers at the CAS+1+2/aug-cc-pVTZ level,24 and we will use their results to establish the reverse rate coefficient and its importance for initiation relative to the association reactions as a function of temperature and initial acetylene concentration.

2. Theoretical Methods Based on the stationary point geometries of Cremer et al.22 and Mebel et al.23 we recalculated energies on the PES using CCSD(T)-F12b/cc-pVQZ-F12//M06-2X/MG3S level of theory25-27 with ultrafine grid and ultra tight convergence criteria in the DFT calculations. We always used the restricted ansatz in the calculations, but in certain cases we tested the unrestricted ansatz as well. Although the T1 diagnostic28 for the entrance channel is low, we confirmed some of the single-reference assumptions by reoptimizing some of the structures using multireference perturbation theory at the CASPT2(8,8) and CASPT2(10,10) levels29 with the cc-pVDZ, aug-ccpVDZ and aug-cc-pVTZ basis sets. We also investigated and, as will be shown, excluded the participation of certain excited states in the reaction. The coupled cluster and multireference


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calculations were done in Molpro 2012,30 while the DFT calculations were done in the Gaussian 09 program package.31 On the singlet ground state surface the single most important bimolecular product channel is the formation of HCCCCH2 + H. The HCCCCH2 radical is resonantly stabilized, however, there is no barrier for the H atom loss. Harding and coworkers32 previously investigated the reverse addition reactions for the two possible addition channels by variablereaction-coordinate transition state theory (VRC-TST):

HC

C

C

CH2 + H

HC

C

C H

CH2

HC

C

C

CH2 + H

H2C

C

C

CH2

R–6 R–7

The first adduct is VA, while the second one is 1,2,3-butatriene (123B). The related highpressure-limit rate coefficients over the 200–2000 K temperature range are 1.52×10-10 × (T/(298 K))0.125 × exp(52.9 K/T) cm3 molecule-1 s-1 and 9.98×10-12 × (T/(298 K))0.556 × exp(232.9 K/T) cm3 molecule-1 s-1, respectively. We used inverse Laplace transform33-34 to regenerate the related specific rate coefficients, k(E), to be used in the ME scheme. Harding and coworkers have shown that while the total capture rate coefficient can be predicted quite accurately using their methodology, the branching between the two channels is very sensitive to the electronic structure method that is used to optimize and describe the resonantly stabilized radical. Here, we will discuss the effect of this uncertainty. To solve the master equation, we used the Master Equation System Solver (MESS),35-37 a modern kinetics code that solves the 1-D time-dependent master equation for multiple-well systems and provides rate coefficients based on the eigenvectors and eigenvalues of the


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transition matrix, whose elements contain k(E) microcanonical rate coefficients and P(E',E) collisional energy transfer probabilities. The specific rate constants were calculated assuming RRKM theory within the rigid rotor and harmonic oscillator approximation. Hindered rotors were treated within the 1-D separable rotor approximation, and tunneling was included with asymmetric Eckart barriers. The properties of the Eckart barrier were simply determined from the imaginary frequency and the forward and reverse barrier heights. The Lennard-Jones parameters for all C4H4 complexes were estimated as Stearns et al.38 did, σ = 5 Å and ε = 280 K. We used He as a bath gas with σ = 2.58 Å and ε = 9.9 K. The rate coefficients were fit with the MESS_TPfit code of C. Franklin Goldsmith.37 Polino et al.39 recently investigated theoretically the addition kinetics of singlet methylene to a series of unsaturated hydrocarbons. They found that the reaction could proceed either by addition to the double or triple bonds, or by insertion into certain C–H bonds. More importantly, they found that both reaction channels are barrierless, which required the careful investigation of the PES and application of VRC-TST. Addition and insertion channels are also possible in the case of acetylene self-reaction, but in contrast, as will be shown, both channels have a saddle point. Therefore, no VRC-TST calculations were done for the entrance channel reaction. We also used inverse Laplace transform to invert the results of Harding et al.24 for the dissociation rate coefficient of acetylene into CCH + H, a reaction that represents the hightemperature initiation step for acetylene, to investigate the pressure- and temperature-dependent competition between the low- and high-temperature mechanisms. Their capture rate coefficient is 3.73×10-11 T0.32 cm3 molecule-1 s-1 over the 200–2000 K temperature range.


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Figure 1. ZPE-corrected CCSD(T)-F12b/cc-pVQZ-F12//M06-2X/MG3S potential energy surface for C4H4. Colors simply serve the purpose of readability.

3. Results As mentioned in the Introduction, the ground electronic state C4H4 PES is extremely complicated: it features about 40 wells and a dozen bimolecular product pairs. We have carefully studied the previously published surfaces, recalculated all relevant stationary points, and confirmed their identity. Here only the distilled information that is relevant for the title reaction is given. We included stationary points only up to ~100 kcal mol-1 energy above vinylacetylene, the lowest energy C4H4 isomer, which will also be regarded as the zero of energy throughout the paper, unless stated otherwise. We also excluded all shallow wells from the analysis that are high


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in energy and do not effectively connect wells to each other or wells and bimolecular products via barriers lower than 100 kcal mol-1. These cul-de-sac species do not play a role in the kinetics, and only slightly modify the partition functions of the species they are connected to. (Information about the omitted species is provided in the Supporting Information.) The resulting reduced potential energy surface is shown in Fig. 1, and the energies and some rovibrational properties of the important species are summarized in Table 1.

Table 1. ZPE-corrected energies and T1 diagnostic of the structures shown in Fig. 1 relative to that of vinylacetylene in kcal mol-1, their lowest (v0) and imaginary (v*) frequencies in cm-1, and symmetry number (σ). Wells and barriers not entering the ME calculations are ignored. This worka

Literatureb

σ d

E+ZPE (T1 diag)

v0, v*

E+ZPE

v0, v*

VA CH2CHCCH

0.0 (0.013)

226

0.0, 0.0

226

1

123B CH2CCCH2

7.8 (0.015)

227

7.7, 7.3

219

4

MCP [CHCH]CCH2

23.4 (0.013)

358

23.6, 23.4

361

2

CBD [CHCHCHCH]

33.4 (0.013)

539

33.4, 33.4

534

4

144

2i

c

Wells

B CHCHCHCH

72.2 (0.027), 70.2

CP [CHCH]CHCH

e

f

185

76.5, 70.0

89.4 (0.015)

126

90.6, 88.5

106

1

I HCCH + HCCH

39.2 (0.014)

713

40.3, 38.9

653

4

II HCCCCH + H2

39.3 (0.014)

238

41.2, 38.5

240

4

III HCCH + CCH2

81.8 (0.018)

297

82.8, 81.6

344

4

IV H2CCCC + H2

84.0 (0.020)

156

85.2, 82.3

162

4

V HCCCCH2 + H

99.4 (0.022)

77

102.3, 102.5

105

2

Bimolecular pairs

Entrance and early barriers I↔ttt-B

74.0 (0.021), 72.7e

85, 614i

81.5, 72.5f

101, 880i

1g

ttt-B↔ttc-B

-, 75.6e

136. 711i

-, -

-, -

1

ttc-B↔ctc-B

-, 78.0e

169, 776i

-, -

-, -

1

ctc-B↔CBD B ↔ VA

78.0 (0.031), 79.2

e

f

346, 119i

77.5 ,

332, 167i

.5

89.6 (0.047), 86.9

e

262, 1379i

88.8, -

276, 1806

1


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I ↔ CP

92.6 (0.022), 99.7e

190, 584i

94.0, 96.1

218, 482i

.5

CP ↔ CBD

89.9 (0.014)

690, 180i

89.8, 89.0

440, 186i

.5

CP ↔ MCP

101.8 (0.015)

356, 462i

101.1, 101.1

370, 700i

.5

III ↔ MCP

85.7 (0.016)

106, 238i

85.8, 85.5

95, 248i

.5

III ↔ VA

103.3 (0.017)

165, 1127i

103.0, -

133, 1244i

1

VA ↔ 123B

90.6 (0.016)

88, 622i

90.8, -

147, 780i

1

VA ↔ MCPh

66.6 (0.016)

380, 464i

65.8, 65.8

364, 580i

1

86.6 (0.039)

230, 932i

86.1, 84.7

227, 798i

.5

-

-

92.4, -

441, 2098i

1

80.5 (0.019)

229, 430i

80.5, 80.1

198, 452i

.5

85.4 (0.015)

337, 759i

85.3, 84.7

344, 933i

.5

90.7 (0.021)

343, 1179i

88.3, 87.4

300, 1283i

1

Isomerization barriers

123B ↔ MCPh

VA ↔ CBD

Barriers to bimolecular products 123B ↔ II

89.8 (0.015)

422, 1785i

90.1, -

410, 1478i

2

123B ↔ IV

97.5 (0.020)

200, 814i

97.3, 96.2

204, 748i

1

123B ↔ V

barrierless

-

barrierless

-

1

VA ↔ V

barrierless

-

barrierless

-

1

VA ↔ II syn

95.0 (0.016)

218, 541i

95.3, -

208, 389i

1

anti

97.2 (0.015)

204, 618i

95.4, -

-

1

a

CCSD(T)-F12b/cc-pVQZ//M06-2X/MG3S.

b

Cremer et al., CCSD(T)/cc-pVTZP, and Mebel et al., G2M(RCC,MP2), respectively.

c

[] means cyclic (sub)structure.

d

Cremer et al., mostly RDFT.

e

CASPT2/aug-cc-pVTZ//CASPT2/aug-cc-pVDZ using CBD as reference. The related CCSD(T)-F12b/cc-pVQZ

energies were calculated at the CASPT2 geometries. f

CASPT2/6-311+G(3df,2p)//CASSCF/6-311G(d,p) using CBD as reference.

g

Lowest frequency modeled as free rotor.

h

Belongs to a different geometry

h

There are two distinct channels.

i

For ttt-B. More details are in Fig. 2.


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3.1. Stable species As shown in Fig. 1a, there are four deep closed shell singlet wells on the C4H4 PES in the prescribed energy range, vinylacetylene (VA, 0.0 kcal mol-1), 1,2,3-butatriene (123B, 7.8 kcal mol-1), methylenecyclopropene (MCP, 23.4 kcal mol-1), and cyclobutadiene (CBD, 33.4 kcal mol-1), along with several weakly bound singlet biradicals (carbene or other). These shallow wells are only mechanistically important in order to explain the connectivity of the wells or bimolecular products they are connected to, and almost always the barrier on their one side is much higher than the one on their other, therefore, in the master equation calculations these species can be certainly excluded. In some cases, however, the barriers on both sides are comparable (e.g., in the case of the CHCHCHCH biradical B), and the inclusion of more than one barrier is necessary either in a two-transition-state scheme or variationally. The acetylene + acetylene (I) asymptote lies 39.2 kcal mol-1 above vinylacetylene (VA), while acetylene + vinylidene (III) is at 81.8 kcal mol-1. Their calculated 42.6 kcal mol-1 ZPEcorrected energy difference is in reasonably good accord with the difference of the recommended 0 K heat-of-formation value of acetylene and vinylidene obtained from ATcT,18-20 43.6 kcal mol1

. The ATcT uncertainty in the acetylene and vinylidene heat-of-formation at 0 K is 0.04 and

0.15 kcal mol-1, respectively. The ground state of vinylidene is singlet, the triplet lying 47.6 kcal mol-1 above it,40 making this excited state unimportant for the current investigation.

3.2. Entrance channel kinetics Initially we considered the following seven entrance channel pathways for the self-reaction of acetylene. Pathways 1–4 include the direct reaction of two acetylene molecules, while 5–7 are pathways where the vinylidene isomer of acetylene reacts with acetylene.


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Pathway 1. In a 2+2 cycloaddition reaction acetylene can dimerize to directly form the antiaromatic CBD molecule:

H C

H

H

H

C

C

C

C

C

C

C

H

H

+

H

H

R8

2+2 cycloadditions are typically not thermal processes and involve the excitation of one of the reactants to the S1 excited state,41 which lies higher in the case of acetylene than the energy range of interest. Also, the first triplet state of acetylene (13B2) is about 80 kcal mol-1 above the ground state (~120 kcal mol-1 above VA).42 On these grounds, we exclude the possibility of Pathway 1. Pathway 2. As hypothesized by Benson,1,

15

acetylene can directly add to another

acetylene forming a singlet biradical B (see reaction R4). We could locate neither the related minimum nor the saddle point on the PES with DFT. However, CASPT2(8e,8o)/aug-cc-pVDZ does find a saddle point for acetylene-to-acetylene addition with the two acetylene fragments being nearly perpendicular to each other, see Fig. 2. The active space in this calculation consists of all eight π electrons and the π, π* orbitals of the two acetylene molecules and the energies were calculated via the CASPT2 calculation for cyclobutadiene, for which the 8 electron, 8 orbital active space was also adequate. In principle, the adduct CHCHCHCH biradical can have six conformers with cis (c) or trans (t) conformations of the H atoms around each of the three C– C bonds. However, we found that the ccc-B conformer readily forms CBD because the radical orbitals overlap, while the cct-B and tct-B ones are rotational saddle points (i.e., only the


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conformers with trans conformation for the middle C–C bond are stable). The lowest energy conformer is the ttt-B. These structures and their CASPT2 energies are also shown in Fig. 2.

I↔B

ttt-B

ttc-B

ctc-B

tct-B*

tcc-B*

B↔VA

72.7

70.2

74.0

74.0

75.6

76.9

86.9

Figure 2. Structures constituting Pathway 2: the I↔B saddle point, the various conformers of the CHCHCHCH singlet biradical B, and the B↔VA saddle point. The numbers indicate the ZPEcorrected energies calculated at the CASPT2(8,8)/aug-cc-pVTZ//CASPT2(8,8)/aug-cc-pVDZ level of theory in kcal mol-1. The letters refer to the configuration of the H atoms (cis or trans) around each C–C bond, in order from top to bottom. Note that out of the six possible conformers ccc is missing, because it readily closes to CBD, while the structures marked with * are conformational saddle points.

The barrier height for the Pathway 2 addition channel I↔B is 74.0 kcal mol-1 at the CCSD(T)-F12 level, and 72.7 kcal mol-1 at the CASPT2 level (see Table 1). The addition can lead either to ttt-B or tct-B*, but the latter is a saddle point. To create the ccc-B conformer, the one that spontaneously makes CBD, the terminal H atom positions need to be flipped and the central C–C bond need to be rotated by π in the ttt-B conformer. The lowest energy path for this is the ttt-B → ttc-B → ctc-B → CBD sequence, with the highest barrier being the last step at the


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Page 16 of 72

CASPT2 level (9.0 kcal mol-1 relative to ttt-B). Flipping the terminal H atoms has a barrier of 5.4 kcal mol-1 and 7.8 kcal mol-1 at this level, while the calculations cannot be completed with CCSD(T)-F12 due to convergence errors likely caused by the open shell singlet nature of these structures. In summary, to get from I to CBD, four saddle points need to be passed, which are increasing in energy but vary in tightness. The real behavior can be more complicated, but the full exploration of this part of the potential is difficult because of the multireference character. In order to partially take into account the complexity, we included all four saddle points along the I → ttt-B → ttc-B → ctc-B → CBD path. In the ME calculations we used the CASPT2 energies for this portion of the potential. Another fate for biradical B is to rearrange into VA by an internal H atom transfer in a 4membered transition state via a significantly higher energy saddle point. This channel was also included as part of Pathway 2, since it also includes the ephemeral intermediate B. Pathway 3. Two acetylene molecules can also directly form the cyclic biradical marked CP in Fig. 1. This cyclic carbene can either rearrange and form CBD via an intermediate well, or isomerize into MCP via a simple 1,2 H atom transfer. The first reaction is controlled by the entrance barrier, while the second one by the isomerization one. Note the large discrepancy between the CCSD(T)-F12 (92.6 kcal mol-1) and the CASPT2 (99.7 kcal mol-1) energies for the I↔CP energy. In the ME calculations the latter was used.

Figure 3. The two conformers of the I↔CP saddle point, which constitute Pathway 3.


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Pathway 4. The C atom of one acetylene can directly insert between the C–H bond of the other acetylene, forming initially a CHCCH2CH carbene. Based on the work of Cremer et al.22 this barrier is significantly above our cutoff (~113-120 kcal mol-1). Therefore, we did not investigate this channel further. Pathway 5. Vinylidene is separated from acetylene by a 43.3 kcal mol-1 barrier at the CCSD(T)-F12b/cc-pVQZ//M06-2X/MG3S level of theory. Vinylidene could add to the triple bond, forming either a biradical or MCP directly:

HC

CH +

C

CH2

HC

C H

C

CH2

R9

CH2 HC

CH +

C

CH2

C HC

CH

R10

The saddle point for the addition is 46.5 kcal mol-1 above the HCCH + HCCH (I) entrance channel, and 3.8 kcal mol-1 above the HCCH + :CCH2 (III) asymptote (85.7 kcal mol-1 relative to VA). The C–C distance at the saddle point is 2.08 Å. With IRC calculations we confirmed that this saddle point is connected to a very weak van der Waals complex (-0.3 kcal mol-1 below the HCCH + :CCH2 asymptote) on the reactant side, and to MCP on the product side (i.e., R9 is not operational, and even if it were, it would rapidly form MCP), in line with the previous studies.2223

The geometries of the saddle point and the van der Waals complex are shown in Fig. 4. Note

that this channel and the related changes in electronic structure were investigated in detail by Kraka et al.43


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van der Waals of III

Page 18 of 72

III↔MCP

Figure 4. Critical geometries along Pathway 5.

Pathway 6. Vinylidene can also insert between the C–H bond of acetylene forming VA directly:

HC

C

H +

C

CH2

HC

C

C H

CH2

R11

The saddle point for this pathway, shown in Fig. 5, is 64.1 kcal mol-1 above the HCCH + HCCH (I), and 21.4 kcal mol-1 above the HCCH + :CCH (III) asymptotes. Again, on the reactant side 2

the IRC leads into the van der Waals well, while in the product direction VA is formed. Even though this is a thermally allowed pathway, reaction R10 will heavily dominate over this channel. Such dominance of the addition and the presence of the barrier clearly differentiates this reaction from the addition of singlet methylene to unsaturates.39

Figure 5. The structure of the III↔VA saddle point that constitutes Pathway 6.


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Pathway 7. Finally, vinylidene could abstract an H atom from acetylene:

HC

CH +

C

CH2

C

CH + HC

CH2

R12

However, we were not able to locate such a barrier: the trials either did not converge, or converged to the insertion saddle point. Based on the above considerations it is clear that at the very first step the reactions starting in I (acetylene + acetylene) compete with the ones starting in III (acetylene + vinylidene) as summarized in Fig. 6. It is interesting to note that in both cases the most favorable pathways do not lead directly to VA, the global minimum on our PES and the commonly observed C4H4 isomer in acetylene pyrolysis. Instead, CBD appears to be the most important primary product followed by MCP. Therefore, VA formation must include further processes.

Figure 6. Summary of the important proposed initial steps of acetylene self-reaction. Number are the barriers in kcal mol-1 at the CCSD(T)-F12b/cc-pVQZ-F12//M06-2X/MG3S except the starred ones, which are at the CASPT2/aug-cc-pVTZ//CASPT2/aug-cc-pVDZ level. Thicker arrows indicate the more important pathways.


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3.3. Further isomerization and dissociation reactions on the C4H4 PES There are many isomerization reactions among the four main wells, VA, 123B, MCP, and CBD. VA is directly connected to 123B and CBD, while indirectly, via an unstable carbene structure to MCP, one of the latter channels having the lowest overall barrier on the PES (66.6 kcal mol-1). 123B is also connected to MCP indirectly via two weakly bound carbene structures, and is not connected to CBD, at least not in the investigated energy range. The lowest energy bimolecular product pair is the resonance stabilized HCCCCH + H2 (II), which can be formed either from VA or from 123B. This channel is likely not to promote chain propagation, similar to the higher lying H2CCCC + H2 products connected to 123B. The only H-atom producing channel is the one forming the resonance-stabilized HCCCCH2 radical as a co-product (V), and is connected to the VA (R6) and 123B (R7) wells. This radical is formed in barrierless processes. The energy of this resonantly stabilized radical is difficult to calculate due to its multireference character, but we are in reasonable accord (lower by 1.2 kcal mol-1) with very accurate heat-of-formation calculations.44-45 The ME calculation result, described in the next section, will uncover which of the possible competing channels are the main contributors to H atom formation.

3.4. Initiation kinetics The results of the ME calculations reveal the competition between the important entrance channels that were presented in Fig. 6. First, the high-pressure limit rate coefficients starting from I and III are shown Fig. 7a. Note that rigorously the high-pressure limit rate coefficients starting from I are I→BB and I→CP CP, because I is connected directly to B and CP only. However,

these are very shallow wells, so they are never stabilized under any conditions of practical relevance. The three relevant high-pressure limit rate coefficients for I to VA, I to CBD, and I to


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MCP were instead calculated by either just considering the dominant (significantly higher) barrier, or, when the barriers are of comparable height, using a simple two-transition-state model.46 Fig. 7b demonstrates that hindered rotor treatment have moderate effects on the

-1 calculated III→MCP III MCP value, and decreasing the barrier by 1 kcal mol increases the rate coefficient

only modestly at higher temperatures.

As shown in Fig. 7a, the rate coefficients originating from III (III→ III ) are orders of magnitudes faster than the ones starting in I, which would lead to the (false) conclusion that the latter pathways are irrelevant. However, in a real system the concentration of vinylidene is not only minute, but unlike a regular bimolecular reactant, its concentration is strongly linked to the concentration of the other bimolecular reactant, acetylene, via the equilibrium constant, therefore: III→ III III = :CCH / HCCH III→

(1)

The molecular parameters used in the master equation calculations suggest that at 300 K the mole fraction of vinylidene is only 8.9×10-31, at 1000 K its 7.3×10-9, reaching 0.0027 at 2500 K. To look at the competition of the five channels in light of this, the rate coefficients starting in III are multiplied by this internally consistent equilibrium constant and are shown in Fig. 7c. In this 47 figure we also show III→ as a reference scaled by the III from the model of Laskin and Wang

equilibrium constant as well. We chose the value of Laskin and Wang47 as an orientation point

here because it is often used in comprehensive mechanisms.


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(a)

Page 22 of 72

(b)

(c) Figure 7. (a) The calculated high-pressure limit rate coefficients for the entrance channel reactions shown in Fig. 6. (b) Effects of the hindered rotor (HR) treatments, and of the barrier height on the III→MCP rate coefficient (HO is harmonic oscillator). (c) The same rate constants but the ones starting in III and that of Laskin and Wang47 are multiplied by the acetylene ⇌ vinylidene equilibrium constant. The right axis shows the ratio of our total rate coefficient to that of Laskin and Wang.


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These results suggest that formation of MCP from III and the formation of CBD from I at ~800 K are of equal importance, with more important I initiation at lower, while more important III initiation at higher temperatures. It can be seen that the agreement between the total calculated high-pressure limit rate coefficient and that of Laskin and Wang is good when all channels and the effect of the fast acetylene ⇌ vinylidene equilibrium are considered. An important question is what the Laskin and Wang (and other similar values) really mean. The reaction as it stands in the mechanism is an association reaction forming C4H4, without any reference to the conformer. If we assume that C4H4 stands for VA (indirectly supported by further reactions of C4H4 in the mechanism) then the question arises: do the initially formed MCP and CBD eventually form VA, or something else, in the high-pressure limit. In fact, our calculations show that CBD largely dissociates back to I, below 1000 K by about more than 1000, between 1000 and 1500 K by more than 100 times faster than its forward isomerization to VA. It means that even though CBD is the dominant primary channel, it is in fact in a rapid equilibrium with acetylene, and does not significantly contribute to VA, and eventually to the H atom formation (channel V). The other two channels starting in I, as was

/ seen, are too slow also to compete with the reactions starting in III. (Note that the ratio III→ III

I→ is about 15 below 1500 K. If we assume that the first barrier is too low, while the other

one is too high by 1 kcal mol-1 this ratio decreases to ~5.5 at 1000 K and to ~3.3 at 1500 K. Further uncertainties in the calculations could in principle bring these values even closer to each other.) Conversely, MCP, the direct product of acetylene and vinylidene, isomerizes mostly to VA (and to a smaller extent to 123B) under these conditions, by about 1000 times faster below 1000 K, and 100 times faster between 1000 and 1500 K, than its dissociation to III. Overall, it is then the case that the acetylene + acetylene reaction largely just creates an equilibrium


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Page 24 of 72

concentration of CBD, while acetylene + vinylidene allows the reaction to explore much more of the PES and eventually exit via the H atom producing channel V. Besides the estimate of Laskin and Wang, there are many experimental results for the initiation reaction. Importantly, however, none of the experimental results are direct, and all of them include extensive modeling with a chemical mechanism of a few dozen reactions, many of which themselves are poorly known. Moreover, often these experimental studies only report a second-order overall decay constant for acetylene, which is not equivalent with a rigorously defined rate constant. Nevertheless, in Fig. 8a we compare experimental results with our calculations for the initiation reaction at 1 bar, which is the most typical experimental pressure employed in these studies. One can see that theory predicts about an order of magnitude slower reaction than what is observed experimentally, even as we consider the I→ →CBD reaction part of the total rate coefficient in this comparison. It is important, however, to emphasize the large scatter in the experimental data, at both ends of the temperature range. Without details of the experimental studies and modeling the whole system it is not easy to recover what the apparent second order acetylene loss rate coefficient is, therefore, we do not attempt to tune our calculations. Several studies have tried to also estimate the acetylene + acetylene rate coefficient leading to H atom formation (bimolecular product V). At low temperatures branching into V is very small (at 1 bar), at the same time, this is the only channel that is available for chain propagation. There is no experimental determination of this rate coefficient, only estimates from modeling studies. Fig. 8b shows the comparison between theory and these values. The estimates are inconclusive and spread over many orders of magnitudes, although show a similar apparent activation energy. Our calculation, although is within their large range, has a somewhat larger


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activation energy for the III→ →V reaction at 1 bar. The I→ →V reaction is slow in comparison, meaning that initiation largely happens due to the vinylidene + acetylene entrance channel. It appears that the H-atom producing rate coefficient estimates have a similar activation energy to the total rate coefficient shown in Fig. 8a, which is clearly unphysical: branching into the H-atom producing well-skipping channel ought to increase with temperature. One likely cause for these large discrepancies is that not just the self-reaction rate coefficient, but also the dissociation rate coefficient in R1, especially its pressure dependence was not known, therefore, the bulk experiments were not able to constrain these two initiation reactions independently.

(a)

(b)

Figure 8. (a) Total association rate constant from a wide range of experimental and modeling studies, compared to the theoretical value plotted at 1 bar. The experimental values1-2, 4-7, 9-10, 12, 47-64

are marked with symbols, and are referred to with the first letters of the authors’ last names.

(b) The experimentally derived H-atom formation rate coefficients2, 6-7, 10, 54, 63-65 are marked with symbols, and are referred to with the first letters of the authors’ last names. We show the


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Page 26 of 72

calculated I→ →V and III→ →V rate constants at two pressures. For reference, the total association rate coefficient is also shown at 1 bar.

3.5. Dissociation from VA and other wells Under finite pressures all five (I through V) bimolecular channels are accessible when VA dissociates, as summarized in Fig. 9. The dominant channels – according to theory – are VA→ III and VA→IV. There are a number of direct experimental studies measuring the unimolecular dissociation rate coefficients starting in VA. Braun-Unkhoff et al.66 found that both of these main channels are at or close to the high-pressure limit under ~1900 K and above ~1 bar. Our calculations agree with their measured values (Fig. 9a), and the stated pressure-independence for VA→III. Hidaka et al.67 in their 1986 paper thought to have measured VA → V (H atom channel), but the nature of the reaction was only hypothetical, as they simply followed the decay of VA, therefore, we attribute their results approximately to a total VA decomposition rate coefficient, which is in good accord with the Braun-Unkhoff et al.66 values for the major channel. Kiefer et al.16 measured VA decomposition up to ~400 Torr, and derived a second-order decomposition rate coefficient expression for the low-pressure limit. Evaluated at 1 bar, it agrees well both with the results of the other experiments and our calculations, even at 2400 K. By incorporating theoretical parameters, they also derived the high-pressure limit rate coefficient for this reaction, which, however, is about an order of magnitude higher than the other experimental values and our calculations. As the authors note, their experiments were probably too far into the falloff for accurate extrapolations, which is likely the reason for the discrepancy. It is worth noting a peculiarity about the VA→III reaction. Under finite pressures III can be formed either directly via the III ↔ VA saddle point, or via multiple steps (well-skipping68)


26

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involving isomerizations and finally crossing either the III ↔ MCP or the III ↔ VA barriers. The well-skipping rate coefficients always increase with decreasing pressure, and go to zero as the pressure goes to infinity. In this case the lower barrier for III ↔ MCP tells us that wellskipping is the main eventual escape route to III. However, the rigorously defined high-pressure rate coefficient (i.e., the one that is only defined for reactions that traverse exactly one transition

state) VA→III VA III is equal to the relatively slow III ↔ VA reaction’s high-pressure limit, because this

limit is only possible for adjoining configurations. Therefore, the VA→III VA III rate coefficient,

contradicting simplistic views of pressure-dependent reactions, first increases and then decreases as a function of pressure. The rigorous high-pressure limit rate coefficient, VA→III VA III , is more than

an order of magnitude slower than the 1-bar one (see Fig. 9a). Very importantly, however, in a VA dissociation experiment one will never observe this phenomenon! The reason for it is that VA still – for instance – isomerizes into MCP at a faster rate as the pressure is increased, and concomitantly, MCP also at an increasing rate dissociates into III, therefore, the effective III formation rate coefficient will appear to increase and plateau with pressure. By diagonalizing the underlying kinetic equations it is possible to show that the slowest eigenvalue corresponds to an overall VA → bimolecular reaction even at the high-pressure limit, which means that all isomerization modes will be exhausted on a relatively short timescale, and the formation of III will appear to be governed by a single, relatively slow, timescale. This timescale is the one that is measured in the experiments and is identified with the rate coefficient. Because the ME gives access to all rate coefficients, and thus to all kinetic eigenmodes, it allows us to calculate an effective VA→ →III formation rate coefficient, akin to the experimental one. This effective rate coefficient shows the expected behavior, i.e., reaches a maximum and plateaus, and also agrees


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Page 28 of 72

with the experimental findings, see Fig. 9a. The same applies for VA→I VA I . For the details of the

derivation see the section 4.

The hydrogen-molecule-forming channels, VA→ →II and VA→ →IV, are shown in Fig. 9b. The dominant channel is II, the 1,4-elimination from VA, and our calculations are slightly lower than the measurements, moreover, we observe pressure-dependence in the 1–10 bar range, not in full accord with the experiments. In the experimental temperature range this channel is, however, minor, therefore, larger experimental uncertainties are expected.

a)

b)

c)


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Figure 9. Comparison of our calculated VA dissociation rate coefficients to experimental values. The experimentally determined expressions16,

66-67, 69

are plotted on a 100 K grid in the

temperature range of validity at representative pressures. (a) VA → I and VA → III rate coefficients, (b) VA→ →II and VA→ →IV rate coefficients, (c) VA→ →V rate coefficients.

Finally, the lowest yield, but perhaps the most important channel for VA from a combustion perspective is its dissociation to form an H atom (V). Braun-Unkhoff et al.66 has observed a pressure-dependence for this reaction below 8 bar, and their rate coefficients are in reasonable agreement with those of Hidaka et al.69 However, the calculated rate coefficients show less pronounced pressure dependence, and also predict a lower H-atom formation rate coefficient especially at the lower end of the temperature range (Fig. 9c). Although the H-atom channel is directly connected to VA, it is one of the highest exit channels on our C4H4 PES, well above many isomerization pathways. Again, experimentally, these relatively fast isomerization channels take place on a short timescale, and the effective (observed) H-atom formation rate coefficient is an overall H-atom formation form VA and from the other wells. Because these timescales are very different, the experimentalists, again, observe an effective rate coefficient, a timescale corresponding to the lowest eigenvalue in the kinetic system. Fig. 9c also shows these effective rate coefficients, and one can see that the agreement with the experiments is very good. As mentioned earlier, the capture rate coefficients for HCCCCH2 + H → calculated by Harding et al. that were used in the inverse Laplace transform for this channel, have an uncertainty with regard to the V → VA vs. V → 123B branching ratio. Our results for VA dissociation mean that the exact value of the branching ratio is not important for VA→ → V,


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Page 30 of 72

because, under relevant conditions, the H atom formation timescale is slower than the equilibration of the wells. At 1.5 bar and 1500 K the C2H2 : C4H3 : C4H2 ratio from VA in the Hidaka et. al67 experiments is 92 : 3 : 5, in the experiments of Braun-Unkhoff et al.66 one it is 98.8 : 0.2 : 1, while our calculations yield (for 1 bar) 94 : 4 : 2, in general good accord with the experiments. Stearns et al.38 propagated the ME for VA decomposition using a stochastic ME solver and the PES of Cremer et al.,22 and got a 61 : 38 : 1 ratio, which is likely due to moderate differences in the energetics between their and our PES.


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a)

b)

c) Figure 10. Unimolecular dissociation rate coefficients to bimolecular products at 1 bar from (a) 123B, (b) MCP, and (c) CBD.

There are no measurements for the dissociation of CBD, 123B, or MCP. The important bimolecular dissociation channels for these species at 1 bar are summarized in Fig. 10. The two deeper wells, 123B and MCP, preferably dissociate via the vinylidene + acetylene (III) channel, while CBD mostly forms acetylene + acetylene (I) directly. Unlike VA, H-atom formation (V) is more important than H2 elimination (II and IV) for these species, although it needs to be noted


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Page 32 of 72

that we only show here the rate coefficients to bimolecular channels. Isomer scrambling still is a fast process at least for MCP and 123B; therefore, the full kinetic mechanism of the C4H4 PES needs to be included in order to determine effective (under a given condition) outcomes of these reactions.

3.6. HCCCH2 + H reaction Product V, HCCCCH2 + H, includes two species that are important in combustion: a highly reactive H atom and a resonantly stabilized radical, therefore, it is interesting to see what products are formed in this reaction. Fig. 11 shows the calculated bimolecular rate coefficients at 1 bar as a function of temperature. As expected, stabilization into the wells dominates at low temperatures; the temperature at which bimolecular products become more important is around 1200 K. Also, at low temperatures 123B is the dominant well, as the reaction under these conditions is controlled by the entrance channel. However, at higher temperatures the ratio of 123B and VA shifts to VA due to well-skipping processes contributing to the rate coefficient significantly. We know this, because in the high-pressure limit (not shown) 123B stays dominant by about a factor of 10 in the entire temperature range. A related consequence is the nonArrhenius behavior of the VA formation rate coefficient. From 300 to ~1000 K it increases due to the increased importance of well-skipping, and after going through a maximum it decreases, because overall, bimolecular channels start to dominate. Such strong non-Arrhenius behavior often requires a double-Arrhenius (sum of two extended Arrhenius expressions) fit to the rate coefficients.


32

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Figure 11. Calculated, channel-specific bimolecular rate coefficients for the HCCCCH2 + H reaction at 1 bar. The V→CBD rate coefficient is very small, and is therefore, omitted in this figure.

The most important bimolecular channel is III (acetylene + vinylidene), which can either form somewhat directly via VA, but through a high barrier, or, indirectly, including several isomerization steps, before reaching MCP followed by dissociation. In either case, the V→III reaction is overall a well-skipping reaction.

4. Determination of effective dissociation rate coefficients for rapidly equilibrating multiwell systems. Imagine a system of N interconnected wells and M bimolecular products, where the bimolecular products are all treated as infinite sinks. The system’s time evolution is captured by a phenomenological kinetic model formulated as the following ODE: =

(2)


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Page 34 of 72

where c is the concentration vector of the wells, K is the rate coefficient matrix describing isomerization and dissociation reactions of these wells (importantly, however, the bimolecular products are not explicitly represented), and the dot is the time derivative operator. Let us further assume that the reaction is initiated experimentally in well W1, which can isomerize into other wells, W1→Wi (i = 2, 3, …, N), or form bimolecular products, W1→(B1, B2, …, BM). The other wells can of course isomerize back to the starting well, Wi→W1, or to other wells, Wi→Wj, or themselves form bimolecular products, Wi→(B1, B2, …, BM). In a general case if there are N wells, for any of the B1, B2, …, BM product formations one observes N timescales characterized

by the < 0 eigenvalues of the K matrix. It means that assuming an infinitely fine

experimental time resolution and no noise, the concentration–time profiles of B1, B2, …, BM

products is a linear combination of ! functions plus a constant, where " is the lth eigenvalue of Eq. 2. (Note that these " eigenvalues of K are the same as the chemically significant

eigenvalues of the much larger transition matrix of the ME.)

However, real experiments do not have an infinite time resolution and/or zero noise. Moreover, it can easily happen that some of the timescales of the kinetic system are short compared to the time-response or resolution of the experiments. The experiment is blind to these fast processes and sees a simpler concentration-time profile for product formation effectively driven by fewer timescales.

Let us take the extreme case when the # , … , & timescales of K are faster than the

experimental time resolution, but of course still significantly slower than collisional relaxation time. It means that the reason these timescales cannot be observed and resolved is an experimental difficulty rather than a fundamental one. In the experiments only the slowest

eigenvalue ' is observed, and, therefore, the experimentalist sees a pure exponential formation


34

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kinetics for all bimolecular products within the noise of the data. Having no information about the complexity of the underlying processes, and in some cases not even knowing that wells W2, W3, …, WN exist, the reported result is that this observed exponential behavior corresponds to a -./ (!1)

reaction W1→Bj, and that ()* = ' +, , where +, = -

34 (!1)

is the branching fraction of W1 into

Bj. Using theory, of course one can reconstruct the complete network of wells and can calculate timescales without the restriction of time resolution and noise, and in turn calculate true phenomenological rate coefficients.35 Note that the relevant aspect of the definition of a phenomenological reaction is that “its rate constant be time-independent and not depend on initial conditions over a very wide range of thermodynamic conditions (temperature, pressure, and possibly compositions).”35,

68

However, ()* is not a phenomenological rate coefficient,

because if for instance another experimentalist looks at the reaction at a significantly better time resolution (but still with one that is slower than collisional energy relaxation), he or she will see

that this new ()* changes with time. We can argue, therefore, that in the original experiment

()* is the slowest eigenvalue, and as a consequence, no one 56(789 matches ()* , and comparison is not possible. However, one can simply reconstruct the eigenvalues and

eigenvectors from the rate coefficients (or directly from the ME), and compare whether ()* =

',56(789 +,,56(789 holds, and also, whether it is really the case that |' | ≪ |# |, … , |< |, and,

finally, whether the fast timescales are really so fast that the experiment should be blind to them. If all requirements are met, then the comparison between theory and experiment is meaningful. As an example, let us look at the VA→III reaction depicted in Fig. 9a and use 2000 K and the high-pressure limit. The kinetic ODE using the rate coefficients from the solution of the ME is:


35


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> −4.57 × 10M R > = 4?@. E = = 6.37 × 10M >ABC 3.89 × 10 >B.D 1.47 × 10R

1.47 × 10M −4.94 × 10T 1.50 × 10T 0.

7.08 × 10O 1.19 × 10O −1.13 × 10U 0.

> 1.41 × 10O > 0. E V 4?@. W >ABC 0. U >B.D −.5.73 × 10

Page 36 of 72

(3)

The eigenvalues are as follow: −2.01 × 10M , −4.8 × 10T , −1.13 × 10U , −5.73 × 10U in s-1 units.

It can be seen that that the slowest eigenvalue is ~24× faster than the next one, or in other words,

the slowest eigenvalue is on the ~5 ms timescale, while the following one on the ~200 ns one, which can be significant experimentally. We show in Fig. 12 the calculated decay of VA at 2000 K in the high-pressure limit using both the solution of the full kinetic model Eq. 3 and assuming that VA is only driven by the slowest eigenvalue. There is no visible difference between the two curves on the 10 ms timescale, enough to consume >90% of VA, and there is only negligible difference ( −8.50 × 10Y' YR > V 4?@. W = =−8.77 × 10YZ E >ABC −3.05 × 10 >B.D −2.18 × 10Y[

(4)

What is important here is that the sign of all four vector components is the same, therefore, this eigenmode clearly does not equilibrate two or more wells with each other. It is, rather, their collective loss (or production, but this is unphysical when bimolecular products are sinks). The eigenvector also shows that this process is largely a loss of VA, because it has the largest amplitude in this vector. We do not provide the full matrix here, but if the bimolecular products are also included in the kinetic equations (note that they do not change the results shown so far) then this slowest


36

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eigenmode shows to produce products I : II : III : IV : V in a 0.08 : 0.10 : 0.54 : 0.01 : 0.28 ratio. It means that to get the effective rate coefficient for III formation from VA, the quantity shown in Fig. 9a, we need to multiply 2.01 × 10M s-1, the absolute value of the slowest

eigenvalue by 0.54, which yields 1.08 × 10M s-1, plotted as the 2000 K value of the dashed line in Fig. 9a. This agrees well with the experimentally assigned formation of III. In contrast, the

true phenomenological rate coefficient calculated from the ME for VA→III is 1.45 × 10R s-1 in

the high-pressure limit at 2000 K. In Fig. 9c, where the VA→V rate coefficient is shown, the

effective rate coefficient at 2000 K and in the high-pressure limit is 2.01 × 10M s-1 multiplied by

0.27, which is 5.35 × 10Z s-1, contrasted with the ME value of 2.20 × 10Z s-1.

a)

b)

Figure 12. Calculated concentration profile of VA at 2000 K in the high-pressure limit when the reaction is started in VA. The two panels show the same information on a) longer and b) shorter timescales. The solid line corresponds to the full solution of the kinetic equations (Eq. 3), while the dashed line accounts for the slowest timescale only. In panel a) the two lines are indistinguishable.


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Page 38 of 72

The phenomenon described in this section is not unique to the acetylene + acetylene system. Miller and Klippenstein described it in the case of ethyl + O2, where at high temperatures the exponential decay constant of the ethyl radical in an excess of O2 corresponds to an eigenmode of the ME rather than a single rate coefficient.70 They also found that isomerization of 1,2-dimethylenecyclobutene to fulvene and benzene between ~700 and ~800 K is also not a single elementary reaction as appears to be experimentally, rather an eigenmode-driven observable.71 Comparison of the eigenvalues with the experiment yielded excellent agreement. It should be pointed out that such effective rate coefficients are only useful when no experimental traces, just experimentally evaluated rate coefficients are available, and a mixture of fast and slow chemical timescales is suspected. When the concentration–time profiles are available, it is also possible to directly model those curves, but it is essentially the same as our analysis when timescale separation among chemical timescales is large. Finally, effective rate coefficients should not be used in detailed kinetic models, as they disguise important chemical details.

4. Conclusions We provided a very detailed description of the C4H4 PES that is relevant for the self-reaction of acetylene, largely building on the work of Cremer et al. and Mebel et al. We refined the previously calculated quantum mechanical results, and constrained the full ME for this system, which contains 4 deep wells, 5 bimolecular products, 19 transition states, and 3 shallow wells. We calculated pressure- and temperature-dependent rate coefficients in the 0.001–100 bar and 300–2500 K range, which we present in the form of double Arrhenius fits as a Supporting


38

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Information. The comparison of the theoretical results to the available experimental results was found to be satisfactory. We are now in the position to answer the decades-old question: What is the most important initiation reaction for acetylene pyrolysis, and what is the mechanism? We found that acetylene reacts with itself via the vinylidene form, formally creating methylene cyclopropene as the initial adduct, which rapidly interconverts to 1,2,3-butatriene and the lowest energy isomer, vinylacetylene. Both of these species can directly connected to the H-atom-forming channel necessary for the chain reaction. Moreover, via the well-skipping HCCH + :CCH2 → HCCCCH2 + H reaction prompt H atom formation is also possible at finite pressures as shown in Fig. 8b. Our calculations also show that the direct dimerization of acetylene is also important, especially below ~1200 K, where it is faster than the acetylene + vinylidene initiation. However, this reaction largely just produces cyclobutadiene, with no energetically favorable channels for further isomerization or dissociation. Unless cyclobutadiene contributes to chain propagation in some yet unknown way, this is not a chain propagating reaction. The well-skipping HCCH + HCCH → HCCCCH2 + H reaction is negligible (at most 5 few percent) under the conditions we investigated, see Fig. 8b. Therefore, we can firmly establish that an H atom driven acetylene pyrolysis mechanism is largely initiated by the acetylene + vinylidene entrance channel at low temperatures. We can also now see under what conditions the self-reaction dominates over simple bond scission (R1) to produce H atoms. At the beginning (t = 0) of the pyrolysis the uni- and bimolecular reactions exactly compete if ∗ ∗ C2H2→C2H+H C2 H2 ∗ = (I→V C2 H2 ∗ + III→ III→V :CCH2 ) C2 H2


(5a)

39


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C2 H2 ∗ = ^

^C2H2 →C2 H+H

I→V _ ^III→ III→V :CCH / HCCH

Page 40 of 72

(5b)

When Eq. 5 holds, the H-atom formation rate from the unimolecular and the bimolecular reactions are equal. If the acetylene concentration is higher than C2 H2 ∗ , the critical value, then

the bimolecular, if lower, then the unimolecular process dominates in the H-atom production.

From the previous paragraph we also know that the I→V term is only a small correction in the

denominator. Fig. 13 shows this critical initial acetylene concentration as a function of temperature at 1, 10, and 100 bar pressures expressed in mole fraction.

Figure 13. Critical initial acetylene concentration (see Eq. 5) at various pressures expressed as a mole fraction. At conditions above a given line second-, while below the first-order initiation process dominates at that pressure.

Our calculations show that for dilute mixtures of acetylene (few %, typical of shock tube experiments) the unimolecular initiation dominates H-atom production more and more with


40

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increasing temperatures. However, for more concentrated acetylene mixtures the bimolecular reactions dominate even at high temperatures, especially considering that the formed vinylacetylene, not included in this comparison, will also add to the H atom balance on a slightly longer timescale, diminishing the importance of the first order initiation even further for these scenarios.

Supporting Information Geometries and frequencies of all species included in the kinetics calculations, double Arrhenius fits to the rate coefficients formatted for use with CHEMKIN, and the description of the omitted species.

Acknowledgement This work was funded by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES). JZ was funded under DOE BES, the Division of Chemical Sciences, Geosciences, and Biosciences. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The work of MDF was supported by DOE’s Science Undergraduate Laboratory Internships (SULI) program. JAM was supported under contract number DE-AC02-2006-CH11357 as part of the ASC-HPCC (ANL FWP #59044).

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Initiation Reactions in Acetylene Pyrolysis - The Journal of Physical

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