Master Equation Modeling of the Unimolecular Decompositions of

Article pubs.acs.org/JPCA

Master Equation Modeling of the Unimolecular Decompositions of Hydroxymethyl (CH2OH) and Methoxy (CH3O) Radicals to Formaldehyde (CH2O) + H Enoch E. Dames* and David M. Golden High Temperature Gasdynamics Laboratory, Department of Mechanical Engineering, Stanford University, Stanford, California 94305, United States ABSTRACT: α-Hydroxyalkyl radical intermediates (RCHOH, R = H, CH3, etc.) are common to the combustion of nearly all oxygenated fuels. Despite their importance in modeling the combustion phenomena of these compounds through detailed kinetic models, the unimolecular decomposition kinetics remains uncertain for even the simplest αhydroxyalkyl radical, hydroxymethyl (CH2OH). In this study, RRKM/ master equation simulations were carried out for CH2OH decomposition to formaldehyde + H between N2 pressures of 0.01−100 atm and temperatures ranging from 1000 to 1800 K. These simulations were guided by methoxy (CH3O) decomposition calculations between pressures of 0.01−100 atm and temperatures ranging from 600 to 1200 K, in both helium and nitrogen. Excellent agreement of the methoxy results was observed for all regions where experimental data exist. Rates were parametrized as a function of both density and temperature within the Troe formalism. Temperature- and pressure-dependent uncertainty estimates are provided, with the largest source of uncertainty being tunneling contributions at very low pressures and at the lowest temperatures. In the regimes relevant to combustion, uncertainties range from factors of 1.4−2 for CH3O decomposition, and from 1.5−2.6 for CH2OH decomposition. The results of this study are expected to have an impact on the high temperature combustion modeling of methanol, as formation rates to CH2O + H from CH2OH are notably different from previous estimates under some conditions. ature conditions (T > 1200 K), the resulting α-hydroxyalkyl radical undergoes a chain propagating unimolecular decomposition. Under lower temperatures, unimolecular decomposition is of course in competition with possible internal isomerizations and bimolecular reaction with O2. The unimolecular decomposition of oxygenated compounds can also directly lead to α-hydroxyalkyl radicals. For alcohols, C−C fission pathways are in direct competition with H2O elimination reactions. In the case of n-butanol, both pathways are important.12 In the high-pressure limit and below 1000 K, water elimination dominates n-butanol consumption, while above 1000 K, two C−C fission channels are predicted to equally dominate the product branching fractions: C2H5 + CH2CH2OH and C3H7 + CH2OH.12 Thus, hydroxymethyl radicals and their analogues are ubiquitous throughout the combustion of alcohol compounds, although no kinetic model accurately incorporates their pressure- and temperaturedependent rate coefficients, while many lack this pressure dependence altogether. α-Hydroxyalkyl intermediates are prevalent in the high temperature combustion of nearly all oxygenated fuels, perhaps

1. INTRODUCTION Accurate temperature- and pressure-dependent rates regarding the fate of α-hydroxyalkyl radicals (RCHOH, R = H, CH3, etc.) are important for describing the combustion kinetics of most oxygenated fuel compounds. The H-abstraction preference resulting in the smallest α-hydroxyalkyl radical is well documented in the context of the simplest alcohol, methanol, with regards to attack by H, OH, HO2, CH3, and halogens.1−3 Recent shock tube studies of methanol decomposition at high temperature (between 1261 and 1524 K) clearly illustrate the influence of the resulting CH2OH decomposition on the overall methanol decay rate.4 In this case, the rate of H-elimination in CH2OH is a direct source of hydrogen radicals via the reaction CH 2OH(+ M) → CH 2O + H( +M)

(R1)

which may then go on to attack the parent fuel compound, regenerating CH2OH. This behavior is ubiquitous to high temperature methanol combustion and has been observed in studies of premixed laminar flames, shock tubes, and flow reactors.5,6 The combustion of larger alcohols also results in αhydroxyalkyl radicals. Under many combustion conditions of the isomers of butanol, H-abstraction from the parent fuel by a radical species predominantly results in the formation of the corresponding α-hydroxyalkyl radical.7−11 Under high temper© 2013 American Chemical Society

Received: May 16, 2013 Revised: July 13, 2013 Published: July 26, 2013 7686

dx.doi.org/10.1021/jp404836m | J. Phys. Chem. A 2013, 117, 7686−7696

The Journal of Physical Chemistry A

Article

only excluding ethers, for which other alkoxy radicals (e.g., methoxy, in the case of DME) are direct decomposition products.13−16 However, α-hydroxyethyl (CH3CHOH) is one of the major intermediates in diethyl ether pyrolysis.17 αHydroxyalkyl intermediates exist during the combustion of ester compounds as well. For instance, CH2OH is a significant intermediate in the combustion of methyl formate (MF). Under stoichiometric plug flow reactor conditions (0.005/0.01/ 0.985 MF/O2/N2, at 3.0 atm, and 900 K), CH3O and CH2OH decomposition are thought to account for one-half of the formaldehyde produced.18 Similar contributions can be expected from larger esters. Despite the importance of hydroxymethyl decomposition, not only in the context of small hydrocarbon combustion, but also as a foundation for larger analogous species, its temperature and pressure dependence remain highly uncertain.19 There are no direct experimental data for the kinetics of this reaction. Temperature- and pressure-dependent rates were first proposed by Tsang, who assumed the high-pressure limit rate constant of the reaction at 500 K had a preexponential factor of 2 × 1013 s−1. The H-elimination barrier height was determined using thermochemical data as well as an estimation of the reverse barrier (H-addition to the O π-bond) at 13 kJ (3.1 kcal).20 Pressure dependence was derived using an RRKM formulation. Although few details are provided, Tsang reported using a constant energy transfer step size of 500 cm−1. The pressure and temperature rate coefficients were tabulated, and an overall uncertainty factor of 4 was recommended for the rate coefficients. Held and Dryer later used the full range of tabulated temperature and molar concentration data and performed a least-squares analysis to fit the functions defined by Troe (see below for details).21,22 An approach similar to that of Tsang was adopted for the same rate in GRI Mech 3.0.23 The expression for the highpressure limit rate coefficient was estimated by analogy to

Figure 1. Zero Kelvin relative energy diagram (ZPE included) for the CH3O/CH2OH decomposition system. RCCSD(T)/CBS energies are listed in Table 1 and are taken from Kamarchik et al.24

data.26−28 The overall pressure- and temperature-dependent kinetics of this reaction are well-known.26 By performing and validating additional ME simulations for methoxy decomposition, confidence in the CH2OH portion of the PES24 was gained, as there is no reason to expect the adopted ab initio calculations to be less reliable here. The same is expected of the presently adopted collisional energy transfer models for helium and nitrogen. During the process of this investigation, controversy regarding the role of tunneling in methoxy decomposition has been laid to rest.26,28 In addition, the ME treatment of methoxy decomposition is shown to be in excellent agreement with available data in the falloff region. All major sources of uncertainty are highlighted. Uncertainties in the computed rates increase outside the conditional space where experimental data are available and our model can be verified.

2. METHODOLOGY Information pertaining to the RRKM/ME calculations is given in Table 1. Electronic energies, molecular geometries, and force constants were all adopted from recent computations at the RCCSD(T)/aug-cc-pVTZ level of theory.24 Electronic energies were extrapolated to the complete basis set limit through the 1/ n3 formula.24,29 Force constants were not scaled. For the conditions of this study, the contribution from scaling the force constants by the NIST recommended factor of 0.975 to the high-pressure limit rate coefficients for H elimination from both methoxy and hydroxymethyl was computed to be less than 7%. Multiwell 2011.131−34 was used to solve the master equation for the time-dependent rate of changes for methoxy and hydroxymethyl, enabling determination of their temperatureand pressure-dependent rate coefficients for hydrogen elimination (a test run with the 2013 version showed results within 1% of each other). The potential energy surface shown in Figure 1 was utilized for all simulations. For both methoxy and hydroxymethyl as reactants, isomerization followed by stabilization was negligible as compared to hydrogen elimination pathways, and therefore no rates are recommended for the isomerization between methoxy and hydroxymethyl. This is further elaborated upon below. Rate constants fitted to the formulations of Troe21,22 and suitable for combustion kinetic modeling are given in Table 2. In these formulations, the rate at a particular temperature and pressure, k(T,P), is represented by

H + C2H4(+ M) → C2H5( +M)

with an estimated 16.7 kJ (4.0 kcal) barrier.23 The most recent Baulch compilation recommends the re-evaluation of Held and Dryer while recognizing the large discrepancies that exist among available estimations at the time.19 However, with accurate potential energy surfaces of the CH2OH system available,24,25 it is now clear that the barrier of H-addition to the oxygen atom in the CO π-bond of formaldehyde estimated by Tsang is incorrect by over a factor of 2 (in fact, Haddition to this bond is over 10 kcal/mol). The net effect is of course an overestimation of the H-elimination barrier in CH2OH; this is evidenced by a comparison of the high-pressure limit rate constant for this reaction, depicted in Figure 5. Recent work by Aranda and co-workers25 for this system and for methoxy decomposition is recognized but hampered by several factors that will be elaborated upon below. In some cases, their rate recommendations fall outside the uncertainty estimates made in this work. For the reasons outlined above, and the additional reason that no reliable detailed rate theory calculation has thus far been performed, the hydroxymethyl system (shown in Figure 1) was reinvestigated using an accurate one-dimensional potential energy surface and by solution of the master equation (ME) for energy transfer. In an effort to ensure the accuracy of the calculations performed in this work with respect to CH2OH decomposition, analogous calculations were performed on the H-elimination in CH3O and compared to a wealth of available 7687



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Table 1. Summary of Zero Kelvin Electronic Energies (E0) Relative to CH2OH, Vibrational Frequencies, Rotational Data, Electronic Partition Functions (Qel), and Optical Isomers Used in RRKM/Master Equation Calculationsa internal rotorse

external rotors E0b

species

c

vibrational frequencies24 (cm−1)

(kcal/mol)

CH3O

9.44

CH2OH

0.00

TS3 [CH3O → CH2O + H] TS2 [CH3O → CH2OH] TS1 [CH2OH → CH2O + H] species

34.10 38.90 39.95

758, 960, 1106, 1393, 1403, 1518, 2940, 3019, 3065 418,h 595, 1055, 1198, 1368, 1488, 3138, 3279, 3840 967i, 466, 581, 1169, 1242, 1499, 1659, 2933, 3000 1934i, 792, 987, 1136, 1142, 1482, 2441, 3096, 3183 1756i, 186, 626, 1068, 1234, 1474, 1617, 2994, 3087 σ (Å) ε/kB (K)

N230 26

He

3.621 2.55

d

optical isomers

3 + exp(−89/T)

1

2

2

0

2

1

5.807

0

2

1

5.253

0

2

1

active (cm−1)

no. rotors

0.919

5.251

0

0.929

6.419

1

0.999

3.717

0.945 0.976

97.53 10.2

V0 (kcal)

Qelf

inactive (cm−1)

4.3

⟨ΔE⟩d,300Kg

n

120 80

0.76 0.95

A maximum energy of 85 000 cm−1 was assigned with energy grains of 25 cm−1. The number of necessary stochastic trials for rate constant extrapolation ranged from 100 000 to 50 million depending on the temperature and pressure. bRCCSD(T)/CBS energies, RCCSD(T)/aug-ccpVTZ harmonic frequencies, and geometries taken from ref 24. cTwo-dimensional (all symmetry number σ = 1). dOne-dimensional (symmetry number σ = 3 for CH3O; σ = 1 otherwise). eSee text for description of assumed symmetric internal rotor for CH2OH. fSee text. gSee description in text and eq R1. hNot used in RRKM/ME simulations, replaced with hindered internal rotor. a

Table 2. Troe Format Fitted Rate Recommendations for Methoxy and Hydroxymethyl Decompositiona Troe12 fitting results to RRKM/ME simulations reaction channel

bath gas, M

CH2OH (+M) → CH2O + H (+M)

N2

CH3O (+M) → CH2O + H (+M)

He

N2

k∞ ko Fc: 0.844, 900, 1, 3315 k∞ ko Fc: 0.372, 44, 510, 1979 ko Fc: 0.341, 28, 1000, 2339

A

n

E/R

7.37 × 1010 5.81 × 10−03

0.811 −1.99

19 920 12 077

1.13 × 1010 4.00 × 10−10

1.210 0.121

12 120 8610

1.00 × 10−07

−0.547

9070

a

Fitted for RRKM/ME results in the rate of 600−1200 K (CH3O) and 1000−1800 K (CH2OH) and for pressures of 0.01−100 atm. Units in cm3, molecule, s. In the same order of appearance, Fc parameters correspond to α, T***, T*, and T**. With reference to N2, collision enhancement factors are: He/0.67 ± 0.05, Ar/0.85 ± 0.05, O2/1, H2/2, CO/1.5, CO2/2, H2O/6, CH4/2, CH2O/2.5, C2H6/3, CH3OH/3 (see text). See text for uncertainty assignments.

k(T , P) k∞

=

Pr F 1 + Pr

In this work, the temperature and pressure dependence of methoxy and hydroxymethyl decay were fitted to the functions defined by Troe via a least-squares fit of all Multiwell results for a given reaction to the four constants (α, T***, T*, and T**) and the low-pressure limit, ko. The low-pressure limit rate defined in this manner is thus not the true low-pressure limit rate, but merely a fitting parameter. For both elementary rate constants determined this way, the fitting error was less than 30% between the results of each ME simulation and the resulting Troe fit. The exponential down model for collisional energy transfer was adopted in this work. A temperature-dependent formulation was used for the average downward energy transferred per collision:

(1a)

where k∞ is the high-pressure limit rate constant, Pr is the reduced pressure (Pr = ko/k∞), and F is defined as −1 ⎡ ⎡ ⎤2 ⎤ log + P c r log F = ⎢1 + ⎢ ⎥ ⎥ log Fcent ⎢⎣ ⎣ n − d(log Pr + c) ⎦ ⎥⎦

(1b)

where Fcent, known as the central broadening factor, is defined as Fcent = (1 − α) exp( −T /T ***) + α exp(−T /T *) + exp( −T **/T )

(1c)

⟨ΔEd⟩ = ⟨ΔEd⟩300 (T /300)n cm−1

and is dependent on temperature. The four constants α, T***, T*, and T** are fitted and reaction-specific. The constants c, n, and d are defined as:

with ⟨ΔEd⟩300 = 80 and 120 cm−1, and n = 0.95 and 0.76 for He and N2, respectively. The parameters for He were chosen to agree with previous work on ethyl radical decomposition35 due to its similarities with methoxy and hydroxymethyl in molecular weight and electronic structure. The temperature coefficient for He, n, agrees with that computed by Jasper and Miller for

c = −0.4 − 0.67 log Fcent n = 0.75 − 1.27 log Fcent d = 0.14

(2)

(1d) 7688



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methane,36 although there is little reason to suggest the temperature dependence of this parameter should be the same for methane and methoxy. Nevertheless, the temperature dependence computed for methane decomposition in N2 by Jasper and Miller has also been adopted here.36 However, ⟨ΔEd⟩300 for N2 is more appropriately taken from the work of Feng et al.35 who fitted this value to the low-pressure ethyl radical decomposition data of Michael et al.37 The ME simulations results are somewhat sensitive to ⟨ΔEd⟩, and this can be seen in the comparisons of Figure 9 for methoxy decomposition at 808 K. Although ⟨ΔEd⟩ is a fitting parameter to an empirically derived model, it does have theoretical basis,38 and capturing its temperature dependence as accurately as possible helps to ensure more accurate extrapolations of the computed rates to regions where no experimental data exist. Other information necessary for the rate theory calculations can be found in Table 1. The electronic partition function of the Jahn−Teller distorted methoxy radical (2E3/2, 2E1/2) was adopted from recent work,25 and is of the form

q=

∑ g j e −β / T

experimental data are available. A discussion of the primary sources of uncertainties is presented below.

3. RESULTS AND DISCUSSION Methoxy Decomposition in Helium. Figure 2 compares present master equation (ME) simulation results with

j

j

(3) Figure 2. Comparison of selected experimental data in the falloff region for the unimolecular H-elimination in methoxy, with helium as a bath gas. Large red filled symbols: individual RRKM/ME simulation results at 808 K (▲), 748 K (◆), 713 K (●), and 678 K (■). Thick red lines: Troe fits to corresponding RRKM/ME results. Dashed red lines: uncertainty bands, see text. Filled black symbols: Hippler and coworkers at 808 K (▲), 748 K (◆), 713 K (●), and 678 K(■), reported 30% uncertainties shown for 808 and 678 K data;26 open symbols from Oguchi and co-workers at 740 K (◇), 710 K (○), and 684 K (□).28

All species in this study were treated as symmetric tops, with one degree of rotational freedom designated as active and allowed to exchange energy with internal vibrational degrees of freedom. The only assigned internal hindered rotor is that for the CH2−OH rotation in hydroxymethyl, the potential of which was assumed to be symmetric about the axis of rotation and a barrier height determined from the relation: V=

1 ⎛⎜ ν ⎞⎟ B⎝n⎠

2

experimental data for methoxy decomposition in the range of 678−808 K and helium pressures ranging from 0.1−100 atm. As previously mentioned, this exercise was conducted for the purpose of increasing the confidence in hydroxymethyl decomposition kinetics. As illustrated in Figure 2, the present results are in excellent agreement with the data. For this reason, Troe fits were performed on the current ME simulation results and new rate recommendations made, valid in the temperature range of 600−1200 K, and from 0.01−100 atm. These rate parameters are presented in Table 2. Above 1200 K, methoxy decomposition is so fast that accurate determination of its rate has little value except at very low pressures. At and below 600 K, methoxy decomposition competes with O2 addition and subsequent reaction to produce HO2 + CH2O.42 Tunneling and Comparison with Direct Rate Constant Measurements. Oguchi and co-workers found that the inclusion of Eckart tunneling in their RRKM calculations resulted in better agreement with their measurements of methoxy decomposition.28 Furthermore, Dertinger et al. argue that tunneling plays an important role, and must be considered to reconcile their RRKM model with their direct measurements of the state-resolved microcanonical rate, some of which are nonzero below the reaction threshold.27 Figure 3 compares direct microcanonical rate constant measurements27 for methoxy decay with those computed here. The data were taken from Dertinger et al., who utilized stimulated emission pumping spectroscopy (SEP).27 The SEP is operated within a molecular beam environment, allowing for decay measurements to be performed under near-collisionless conditions and with no measurable diffusional loss effects. Whereas Dertinger et al. fit both the imaginary frequency of the transition state (830

(4)

where B is the rotational constant of the rotor in question, in this case adopted from recent work.25 ν is the harmonic frequency associated with the hindered rotation, 418 cm−1, and n is the number of potential wells, two in this case, thus yielding a value of V = 1517 cm−1 (4.3 kcal/mol) for the present work. This barrier height is in excellent agreement with past experimental and theoretical work.39,40 Although not symmetric in nature, a more sophisticated incorporation of this internal degree of freedom was not necessary because Aranda and coworkers25 showed this approximation results in only modest errors of entropies and formation enthalpies between 298 and 1200 K. Thus, under the temperature ranges of the present work, there are negligible differences between the thermodynamic contributions from a hindered rotor treatment and that of a harmonic oscillator treatment. Last, asymmetric Eckart41 tunneling corrections have been applied to all rate calculations in this work, as implemented in Multiwell 2011.1.31−34 Further discussion on the effects of tunneling on the rates will follow below. A rigorous estimation of the uncertainty in the computed rate coefficients was not attempted due to the availability of previous measurements in the case of methoxy decomposition. However, such an exercise would necessarily include a great number of factors, including contributions from aspects not considered in the current simulations (e.g., higher dimensional tunneling and consequence of using a 2D ME), and is beyond the scope of the present work. Rather, uncertainty estimates are based on comparison with experimental data, with liberally increasing uncertainties outside the T and P domains where 7689



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though a slight thermodynamic driving force favors CH2OH, the rate of isomerization was suspected to be too slow as compared to H-elimination and was therefore not included in a large detailed model for methanol combustion.6 Indeed, for all of the conditions studied here, the CH3O ME results show negligible stabilization of the CH2OH isomer. In addition, a negligible amount of methoxy decomposes through isomerization followed by chemically activated H-elimination. On the other hand, such a process does occur with CH2OH as the reactant species and can be represented as CH 2OH( +M) ↔ CH3O*( +M) → CH 2O + H( +M) (R2)

where CH3O* denotes the chemically activated methoxy radical. Thus, CH2OH decay takes place via two pathways, the latter pathway, and direct H-elimination from the oxygen atom of CH2OH (R1). For example, isomerization to CH3O followed by chemically activated H-elimination accounts for almost 50% of the total CH2OH consumption at 1000 K and ∼1 Torr. At 1000 K and 10 atm, this channel accounts for roughly 20% of CH2OH consumption. Because there is still no stabilization of CH3O, and because both products are chemically equivalent, the total rate of CH2OH decomposition is best represented through one elementary rate coefficient. Thus, the rate of CH2OH decomposition recommended in Table 2 includes the net decay through both possible channels. Last, for the purpose of atmospheric and combustion modeling, there is no need to consider the isomerization directly; thus no rates are recommended for this reaction. Echoing the work of Held and Dryer, such an elementary reaction need not be included in detailed kinetic models. High-Pressure Limit and Falloff Rates. In comparing the high-pressure limit (k∞) rate constants for H-elimination in methoxy (see Figure 5), those computed here are uniformly larger (between a factor of 4 and 5) than those determined by Hippler et al.,26 and nearly the same as those by Aranda and coworkers.25 The agreement with Aranda et al. is expected because many of the input parameters are nearly equivalent and verified to be so. Hippler et al. recognized that previous theoretically derived rate constants for this reaction tended to be larger, yet chose to fit the high-pressure limit rate constant to maintain good agreement with their falloff data. As described in their manuscript, Hippler et al. performed a fit to ko and k∞ at a particular temperature by assuming an initial broadening factor (Fc) of 0.5 and backing out ko and k∞. These values were then used to recalculate Fc, and this process was repeated until a fully consistent set of parameters within the Troe formalism accurately captured their experimental data. Because the k∞ calculated in this work is higher than that of Hippler et al., fitting of the Troe parameters in the falloff region simply resulted in lower Fc values. The present authors note, however, that in the context of methoxy decomposition, the highpressure limit hardly represents an accessible kinetic region as falloff extends well beyond 100 atm even for the lowest temperature considered (600 K). The same holds true for calculations performed with N2 as the bath gas. Figure 4 shows N2 falloff plots computed herein for selected temperatures. High-pressure limit rate recommendations are given in Table 2 in modified Arrhenius form. Conventional Arrhenius fits resulted in errors of 30% or more at the lowest temperatures, but because they help to rationalize the nature of the transition state, they are still reported here. For CH3O = CH2O + H, k∞

Figure 3. Comparison of direct rovibrationally resolved rate measurements (○) of the rate for CH3O → CH2O + H27 with the microcanonical rates, k(E), computed in this work (solid line). Also shown is the k(E) computed without considering the contribution from tunneling (dashed line).

cm−1) and the barrier height corresponding to H-elimination in methoxy (8500 cm−1) to their SEP data, thereby achieving excellent agreement between their RRKM model and their experimental data, in this work no such fitting was performed. The overall agreement remains satisfactory, albeit the k(E) calculated here is uniformly smaller than the state-specific decay constants observed and computed by Dertinger et al.27 One reason for this discrepancy in the computed k(E) could be that the vibrational frequencies used by Dertinger et al. for methoxy and the transition state are, respectively, from experimental observations and ab initio calculations.43 As noted by Dertinger et al. and repeated here, the energy specific rate constants computed are for a microcanonical ensemble and not for single molecular quantum states.27 Nevertheless, the influence of tunneling is clear and its inclusion necessary to reproduce the SEP data below the RCCSD(T)/CBS reaction barrier of 8630 cm−1 (24.66 kcal/mol). More recently, Hippler et al. conducted a theoretical study in addition to obtaining a wealth of experimental data between 2 and 90 atm,26 depicted in Figure 2. Hippler et al. show that their RRKM parameters (without tunneling) successfully reproduce the state resolved specific rate constants of Dertinger et al., but only above the reaction threshold. Because tunneling was not included, they by definition do not reproduce the Dertinger et al. data below the reaction threshold where methoxy decay was still observed. Thus, Eckart tunneling contributions were included in this work and concluded to be necessary on the basis of comparison and evaluation of the aforementioned work. Another reason tunneling was included is because it simply represents a real phenomenon with a very large impact on the calculated rates at low pressures and low temperatures, to be discussed below. CH 2 OH Isomerization and Decomposition. The CH2OH radical was once thought to be important in tropospheric methane oxidation due to the prevalence of CH3O radicals and the expected thermodynamically driven isomerization to CH2OH.39,44,45 This was quickly ruled out,46 although some speculation remained regarding the importance of this isomerization under temperatures common to combustion. As was later noted by Held and Dryer, even 7690



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CCSD/6-311G(d,p) level of theory.25 Additionally, the vibrational frequencies here were not scaled, while those use by Aranda were scaled by the NIST recommended scaling factor of 0.954, thus further increasing the discrepancy between the ratio of vibrational partition functions. The resulting effect introduces another factor of 1.6 into k∞ between 1000 and 1500 K. Taken together, the total difference is significant and results in discrepancies that are between factors of 3−4 for the high-pressure limit rate coefficient. As previously mentioned, the vibrational frequencies utilized in this work were not scaled because the contribution to the difference in the computed k∞ was calculated to be no greater than 7% under all T. Finally, as was also mentioned before, the k∞ utilized in most earlier studies clearly exhibits a lower activation energy that is a direct result of an incorrect barrier height for the reverse reaction.6,20,23 Figure 6 shows the falloff behavior for the decomposition of hydroxymethyl in N2 at 1000, 1200, and 1800 K (1500 K simulation results not depicted). Depicted therein are also the falloff estimates/calculations of Held and Dryer, Aranda et al., and GRI 3.0.6,25,47 Immediately evident from the plots is the considerable discrepancy between the rates computed here and those of Held and Dryer and GRI 3.0. Again, this unsurprising result is primarily due to the incorrect barrier height to Helimination. At 1000 K and 1 atm, the rate estimates of Held and Dryer and GRI 3.0 are a factor of 4 faster than the rates computed in this work. At 1200 K and 1 atm, the rate estimates of Held and Dryer and GRI 3.0 are a factor of 2−3 faster than those computed here, while at 1500 K, agreement is within a factor of 2. In some cases, such discrepancies are larger than the uncertainties estimated here, which are discussed below. In general, the predictions of Aranda et al. agree with those computed here for lower pressures, but begin to fall outside the predicted uncertainty at elevated pressures for all but the highest temperature. The pressure and temperature dependence of both decomposition channels is illustrated in Figure 7. At high temperatures and between just 1 and 5 atm, the rate of CH2OH decomposition varies by almost a factor of 5. With respect to

Figure 4. Falloff plots for methoxy decomposition in nitrogen. Symbols: RRKM/ME results, ▼, 1200 K; ■, 1000 K; ●, 800 K; ◆, 700 K; ▲, 600 K. Solid black lines: Troe fits to RRKM/ME results. Dashed lines: uncertainty bands, those for 1000 K not included to avoid confusion. See Table 2 for bath gas specific energy transfer parameters.

= 1.86(1014) exp(−13 480/T) s−1. For CH2OH = CH2O + H, k∞ = 7.36(1013) exp(−21 400/T) s−1. Comparison of the computed k∞ for CH2OH with that of Aranda et al.25 reveals a large discrepancy between 1000 and 2000 K, with the rate predicted by Aranda and co-workers roughly a factor of 3.6 higher. There are two reasons for this discrepancy. First, Aranda et al. did not account for the fact that the hydroxymethyl radical has two optical isomers. This results in their rates being too fast by a factor of 2 on this account. The remaining factor of 1.6 comes directly from the large differences in vibrational partition functions. In comparing vibrational frequencies used here and those of Aranda et al., those used here are larger than those computed by Aranda et al. for the transition state, yet smaller than theirs for CH2OH. Those utilized here were computed with a higher level of theory (CCSD(T)/aug-cc-pVTZ, NIST recommended scaling factor of 0.975) than those used by Aranda et al., who utilized the

Figure 5. Comparison of available high-pressure limit rate constants for the unimolecular H-elimination from methoxy (CH3O) and hydroxymethyl (CH2OH). Black line: this work. Red long dashed line: Hippler et al.26 (for CH3O only). Red dash−dotted line: Held and Dryer6 (for CH2OH only). Green dashed line: Aranda et al.25 (line for CH3O overlaps that of this work). Blue dotted line: GRI 3.0.12 7691



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Figure 8. Arrhenius plot of the low-pressure limit rate constant, ko, for CH3O + M → CH2O + H + M. Solid line: this work, M = He. Dashed line: this work, M = N2. Filled circles (●): M = He.28 Diamonds (◇): M = N2.28 Filled squares (■): M = He.47 See text for discussion of uncertainties at low P limit.

data. The fitted ko for helium essentially splits the data between Oguchi et al.28 and Choudhury et al.,47 with preference leaning toward the rates of Choudhury et al. The fitted ko values here are a factor of 2 higher than those derived by Oguchi and coworkers.28 Oguchi et al. derived ko by performing linear regressions to observed first-order decay rates without the constraint that the rate approaches zero at null pressure. ko for helium is in better agreement with the data obtained by Choudhury et al., albeit lower.30 The large scatter in data is not surprising due to the nature of the low-pressure limit. Not only is it difficult to experimentally interrogate the rate at zero pressure, but accurate modeling of CH3O decomposition in this limit is made difficult by its sensitivity to the effects of tunneling

Figure 6. Falloff plots for CH2OH decomposition in N2 at 1200 and 1800 K. Solid black lines: Troe fits to current RRKM/ME results. Black circles (●): individual RRKM/ME simulation results. Red lines and open circles (○): Held and Dryer.6 Blue lines and open diamonds (◇): GRI 3.0.12 Green lines and open squares (□): Aranda et al.25

the combustion kinetics of hydroxymethyl and other αhydroxyalkyl radicals, this clearly illustrates why accurately capturing the pressure dependence of their decomposition is important to the modeling of combustion phenomena. Methoxy Low-Pressure Limit Rate. Figure 8 illustrates the second-order rate constants for methoxy decomposition fitted for both He and N2, in addition to available experimental

Figure 7. Arrhenius plots showing temperature and pressure parametrized RRKM/ME simulation results for the unimolecular decomposition of methoxy and hydroxymethyl in nitrogen (M = N2). 7692



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and angular momentum conservation.31 It is expected that solution of a two-dimensional ME may result in different (lower) low P limit rate determinations.31 Additionally, the contribution of tunneling is very large at lower pressures and temperatures. For example, the predicted rate at 808 K and 10−5 atm (0.0076 Torr) is a factor of 25 smaller without tunneling included, decreasing to factors of 3.2 and 1.3 at 0.01 and 10 atm, respectively. Thus, tunneling corrections of higher order than Eckart could also affect the computed ko, especially at lower temperatures where the Eckart correction can have large deviations from its higher order counterparts.32 Because tunneling plays an important role at low pressures (at the lowest temperatures studied here), the lack of it in the model of Hippler et al. explains why their predicted ko is significantly lower than that found here. In the context of this work, ko is a fitting parameter within the formalism introduced by Troe.21,22 Once again consider that the second-order rate coefficient (and attending k∞, Fc values) fitted here results in excellent agreement with the combined data of Oguchi (740 K) and Hippler (748 K), shown in Figure 2. The absolute value of ko is less important than the

on the surface of dust grains before nonthermal desorption.48,49 Under such low P and T conditions, the resulting methoxy radical will very likely experience photoinduced dissociation as opposed to thermal dissociation, negating the need for an accurate theoretical low P limit. Interestingly, laboratory attempts to reproduce this interstellar process have resulted in the thermodynamically preferred CH2OH being the preferred desorbed species.34 Third-Body Enhancement Factors. The third-body enhancement factors of He relative to N2 can be derived from the RRKM/ME results for methoxy decomposition by taking the ratio of the computed rates at corresponding temperatures and pressures. Between 600 and 1200 K and for all P considered, the collision efficiency of He relative to N2 was computed to be 0.67 with a single standard deviation of 0.05. We note that negligible temperature dependency was observed in the derived collision efficiency between 600 and 1200 K, although a slight negative pressure dependence was observed. For the purposes of detailed combustion and atmospheric chemistry modeling, the third-body collision efficiency of argon relative to N2 was similarly derived to be 0.85 with a single standard deviation also of 0.05. The Lennard-Jones parameters necessary for the Ar derivations are obtained from the literature.30 On the basis of the calculations of Jasper and Miller,36 ⟨ΔE d⟩ for Ar was scaled at 300 K by the corresponding value used here for N2, giving an only slightly different ⟨ΔEd⟩300,Ar of 106 cm−1, as compared to that computed by Jasper and Miller (115 cm−1). The temperature coefficient, n, was left unchanged at 0.75. The values for He and Ar relative to N2 are in fair agreement with the generic GRI 3.0 efficiencies,23 which recommend efficiencies for other bath gases: O2/1, CO/1.5, H2/2, CO2/2, H2O/6, CH4/2, CH2O/ 2.5, C2H6/3, CH3OH/3. It is with caution that these same parameters are recommended, noting that the uncertainties in these values for the reactions here may be a factor of 2 or greater. Uncertainty Estimates. Uncertainties in the RRKM/ME simulations and resulting Troe fits are smallest for the reaction of CH3O (+He) = CH2O + H (+He) in the region between 600−808 K and 0.01−100 atm due to the availability of kinetic data in this parameter space. Here, the resulting Troe fits are estimated to have absolute 2σ uncertainties between 40% and 60%. This estimate is based on the claimed 30% uncertainty in the measurements of Hippler et al., with additional consideration from the Troe fitting errors to individual RRKM/ME simulations (less than 30% for all cases) and uncertainty associated with energy transfer. Above 100 atm, although irrelevant for most practical purposes, uncertainties are estimated to be higher than 60%. Below 0.01 atm, uncertainty rapidly increases due to a number of factors, as explained above (tunneling, energy transfer, and 1D vs 2D ME treatments). An uncertainty factor of 4 was assigned to methoxy decomposition at 808 K and 0.001 atm. At even lower pressures, the uncertainty exceeds an order of magnitude. Extending the uncertainty estimates to methoxy decomposition in N2 is simply a matter of considering the different collisional energy transfer parameters. As explained above, a model with both a theoretical basis in addition to one with empirically supporting evidence was chosen.35,36 The additional uncertainty is expected to be negligible. Thus, for methoxy decomposition at 808 K in both He and N2, the uncertainties for pressures of 0.001, 0.01, 0.1, 1, 10, and 100 atm are 4, 1.6, 1.5, 1.4, 1.4, and 1.6, respectively. An additional 10%

Figure 9. Sensitivity of ME simulation results to downward energy transferred upon collision. Black circles (●): individual ME simulation results with black line drawn as a guide. Dashed lines: ME simulation results with ⟨ΔEd⟩ of 150 cm−1 (lower dotted line) and 250 cm−1 (upper dotted line). Open circles (○): data of Hippler and coworkers.26

actual rate of decomposition at finite pressure. In other words, the present ME simulations and resulting density and temperature parametrization result in excellent agreement with experimental data in the falloff region, where high accuracy is desired. For this reason, a more accurate determination of ko was not attempted, and large uncertainty factors have been assigned to all rates in this region (P < 0.01 atm). These uncertainties are dominated by tunneling. It is also unlikely that any terrestrial conditions exist where only an accurate second-order rate constant for methoxy or hydroxymethyl decomposition is needed. First, methane destruction primarily occurs in the troposphere, where the atmospheric pressure does not decrease further than a couple tenths of an atmosphere.33 Methoxy’s nearly exclusive fate under these conditions is through reaction with O2. For combustion applications, the low-pressure limit CH3 O/CH 2OH of decomposition is never achieved, leaving interstellar space as the only relevant domain for CH3O/CH2OH decomposition at or near the low-pressure limit. To little surprise, methoxy has been recently observed in cold dark clouds and thought to form 7693



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different bath gases. With N2 as a reference, collision enhancement factors for He and Ar are 0.67 and 0.85 with single standard deviations of 0.05 for both. Collision enhancement factors are cautiously recommended for other bath gases, based on values in GRI 3.0:23 O2/1, CO/1.5, CO2/2, H2O/6, CH4/2, CH2O/2.5, C2H6/3, CH3OH/3, valid for both methoxy and hydroxymethyl decomposition, although the uncertainties in these may be a factor of 2 or higher. Tunneling was found to be important, with more influence at lower pressures and temperatures. Although this behavior is well-known, controversy regarding its role in methoxy decomposition has been laid to rest as its inclusion is necessary to reproduce the state resolved direct rate measurements of Dertinger et al.27 Furthermore, an elementary rate describing the direct isomerization of CH3O ↔ CH2OH is not necessary for the detailed modeling of combustion phenomenon related to these species. Under the conditions of this study, CH3O exclusively undergoes direct H-elimination, while CH2OH decays via both direct H-elimination and isomerization followed by chemically activated H-elimination, with no stabilization of CH3O taking place. The results of this study are expected to have an impact on the high temperature combustion modeling of methanol and other oxygenated compounds, as formation rates to CH2O + H from CH2OH are notably different than previous estimates. Last, the CH2OH decomposition rates determined here should provide some guidance to analogous H-elimination pathways in larger αhydroxyalkyl radicals, as the pressure dependence can be expected to be similar.

uncertainty in rate is added per 200 K increase beyond 800 K and up to 1200 K; this is an attempt to account for extrapolation outside the temperature range for which experimental methoxy decomposition data are available. An additional 10% of uncertainty in rate is also added to the lower 600 K results from the 808 K baseline uncertainties. The resulting uncertainty isotherms are depicted in Figure 10 as a function of pressure. Uncertainty bands are illustrated in Figures 2 and 4.

Figure 10. Uncertainty factor isotherms for methoxy decomposition as a function of H2 and He pressure: red line with ●, 1200 K; blue line with ◆, 600 and 1000 K; black line with ■, 700−808 K.

The uncertainties in the hydroxymethyl rates are higher due to the lack of experimental data for this reaction. Thus, an additional uncertainty of 10% was assigned to the rates computed here. The T- and P-dependent uncertainties were estimated by first locating the corresponding temperatures that result in similar unimolecular decomposition rates for CH3O and CH2OH at 1 atm. Methoxy decomposition at 808 K and 1 atm He occurs with kuni ≈ 7 × 104 s−1. This is matched to hydroxymethyl decomposition at 1200 K and 1 atm N2, which occurs with kuni ≈ 6 × 104 s−1, and is thus assigned an uncertainty of 1.54 (1.4 with an additional 10%). This point served as an anchor from which the remaining uncertainties were mapped, again with additional 10% increases per 200 K increase, up to 1800 K. The authors note that uncertainties reported here are atypical because, in the absence of experimentally supporting data, it is common for rates in the falloff region to be least certain (assuming an accurate highpressure limit rate coefficient is used).

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS

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REFERENCES

This work was supported by the Combustion Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Basic Energy Sciences under Award Number DESC0001198.

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4. CONCLUSIONS Detailed master equation simulations have been performed for the decomposition of the hydroxymethyl radical (CH2OH) in N2, with corresponding elementary reaction rates fitted to the formalisms of Troe21,22 for pressures of 0.01−100 atm and between 1000−1800 K. Attendant uncertainty estimates are provided, primarily guided by the CH3O decomposition experiments of Hippler et al.26 and Oguchi et al.,28 with increasing uncertainty outside the temperature and pressure ranges of their work. Thus, ME simulations were also carried out for CH3O decomposition in the range of 0.01−100 atm for 600−1200 K, with attendant uncertainty estimates. The results are in excellent agreement with available data. Elementary rate recommendations have been made for methoxy decomposition as well, in both He and N2 bath gases. The collision enhancement factors of He and Ar were additionally computed by directly comparing ME simulation results between the 7694



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Master Equation Modeling of the Unimolecular Decompositions of

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