Discussion of the Separation of Chemical and Relaxational Kinetics of

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Discussion of the Separation of Chemical and Relaxational Kinetics of Chemically Activated Intermediates in Master Equation Simulations Malte Döntgen, and Kai Leonhard J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b12927 • Publication Date (Web): 03 Feb 2017 Downloaded from http://pubs.acs.org on February 5, 2017

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The Journal of Physical Chemistry

Discussion of the Separation of Chemical and Relaxational Kinetics of Chemically Activated Intermediates in Master Equation Simulations Malte Döntgen†,‡ and Kai Leonhard∗,†,‡ †Chair of Technical Thermodynamics, RWTH Aachen University, Germany ‡AICES Graduate School, RWTH Aachen University, Germany E-mail: [email protected] Phone: +49 241 8098174

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Abstract Chemical activation of intermediates, such as hydrogen abstraction products, is emerging as basis for a fully new reaction type: Hot β-scission. While for thermally equilibrated intermediates chemical kinetics are typically orders of magnitude slower than relaxational kinetics, chemically activated intermediates raise the issue of inseparable chemical and relaxational kinetics. Here, this separation problem is discussed in the framework of Master Equation simulations, proposing three cases often encountered in chemistry: Insignificant chemical activation, predominant chemical activation, and the transition between these two limits. These three cases are illustrated via three .

.

example systems: Methoxy (CH3 O), diazenyl (NNH), and methyl formate radicals .

(CH3 OCO). For diazenyl, it is found that hot β-scission fully replaces the sequence of hydrogen abstraction and β-scission of thermally equilibrated diazenyl. Building on the example systems, a rule of thumb is proposed which can be used to intuitively judge the significance of hot β-scission: If the reverse hydrogen abstraction barrier height is comparable to or larger than the β-scission barrier height, hot β-scission should be considered in more detail.

Introduction Chemically activated intermediates enable reaction pathways which would be irrelevant for the thermally equilibrated intermediates and were shown to enhance rate constants of readily accessible reactions. 1–4 For unimolecular intermediates fed by Boltzmann-distributed reactants, the chemical activation is accurately described via the Master Equation (ME). 5,6 Although ME simulations give considerable insights into the temperature- and pressuredependent kinetics of a reaction network, they are based on the distinct separation of chemical and relaxational eigenvalues of the Master Equation. 7 In the course of studying chemically activated intermediates, however, the distinct separation of these eigenvalues was shown to be violated at high temperature. 8 This problem will be discussed based on the following


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general hydrogen abstraction reaction, in which .

∗

indicated chemical activation.

.

.∗

R1 + R2 H −−→ (HR1 + R2 )∗ −−→ HR∗1 + R2,A + R∗2,B

(1)

.

In this reaction, the multimolecular intermediates (HR1 + R2 )∗ are chemically activated .∗

and the chemically activated radical R2 participates in a further reaction prior to deactivation. The reactions of chemically activated multimolecular intermediates, such as hydrogen abstraction products, are not included in state-of-the-art kinetics predictions but were found to affect the global behavior of chemical processes largely under high-temperature and lowpressure conditions. 8–12 Recently, hot β-scission was proposed as a new reaction type, 8,11,12 in which chemically activated multimolecular intermediates participate in further reaction prior to deactivation. In this work, microcanonical aspects of hot β-scission will be discussed in order to facilitated a deeper understanding of this new reaction type. Three example sys.

.

.

tems, CH3 OH + H, N2 H2 + H, and CH3 OCH−O + H are used to discuss three different cases encountered in the context of hot β-scission: Insignificant chemical activation, predominant chemical activation, and the transition between these two limits, respectively. For the transition between the lower and upper limiting cases, an approximate separation between chemical and relaxational kinetics is achieved and will be discussed for the methyl formate (MF) radicals.

Hot β-scission Figure 1 shows the mechanism and potential energy surface (PES) of the recently proposed hot β-scission reaction type, 8 which describes the formation and ”prompt” dissociation of chemically activated radicals formed via hydrogen abstraction (multimolecular). Typically, the sequence of hydrogen abstraction and β-scission is treated as separate reaction steps, building on fully thermalized reactants. In case of hot β-scission in turn, the fact that the hydrogen abstraction products are formed in a rovibrationally excited state is taken into



account. As a consequence, the β-scission reaction is fed by non-Boltzmann-distributed reactants, enabling the prompt dissociation of the freshly formed radicals (cf. dashed line in Figure 1). Thus, hot β-scission is the direct reaction from the hydrogen abstraction reactants to the β-scission products.

.

α β k H [R 1 ] .∗

R2

.

ˆβ k

.∗

R2,A + R∗2,B

k H [R 1 ] ˆth k .

R1 + R 2 H

.

k-H [R1 H]

kβ / k-β [R2,B ]

.

R2

.

αth kH [R1 ]

TSH Energy level

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-HR1

tmp

TSβ

Reaction coordinate Figure 1: Illustration of the two-step mechanism underlying the hot β-scission reaction type and how these steps relate to the respective potential energy surface. The solid and dashed lines correspond to the chemically complete interpretation and the modeling approach of Labbe et al., 11 respectively. The blue and red colors represent thermally equilibrated or active species, respectively. The two-step mechanism shown in Figure 1 illustrates how the hydrogen abstraction .

.∗

reactants R1 + R2 H form chemically activated radicals R2 and how these radicals either .

thermalize to thermally equilibrated radicals R2 or dissociate to the β-scission products .∗

.∗

R2,A + R∗2,B . Since the rate constants for thermalization and consumption of R2 are typically not available, Labbe et al. 11 proposed to implicitly account for chemical activation in the 4 ACS Paragon Plus Environment

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.

direct reaction to either the thermally equilibrated radicals R2 or the β-scission products .∗

R2,A + R∗2,B . .∗

Although the inclusion of R

2

would be physically exact, the modeling approach of .∗

Labbe et al. 11 is equally exact if thermalization and consumption of R2 are significantly faster compared to its production. In that case, the total rates of the two-step mechanism is .∗

primarily determined by the rate of R2 production and the branching between thermalization and consumption. Thus, the absolute rate constants for thermalization and consumption are not required. The systematic screening of saturated fuels performed by Döntgen et al. 8 showed which abstracting radicals produce fuel radicals which will likely undergo hot β-scission. Generally speaking, abstracting radicals amplify rovibrational excitation if: 1. The radicals are small, thus take only a small fraction of the energy released during hydrogen abstraction, leaving more rovibrational excitation for the fuel radicals. Here, the previously used energy equipartitioning approach 8 will be used to calculate the fraction of rovibrational excitation available for subsequent reactions of the fuel radicals. This approach is based on assuming that each degree of freedom of a molecule contains the exact same amount of energy. When considering hydrogen as abstracting radical, the newly produced molecular hydrogen (HR1 ) has only three rovibrational modes, thus contains only a small fraction of the excess energy. For larger abstracting radicals, however, a detailed analysis of the energy partitioning would be crucial to obtain accurate non-Boltzmann energy distributions, e.g. via trajectory simulations as proposed by Goldsmith et al. 10 2. The reverse hydrogen abstraction barrier height is sufficiently large to release a substantial amount of energy to enable ”prompt” dissociation of the newly formed fuel radicals. The role of reverse hydrogen abstraction and β-scission barrier heights will be discussed in the course of this study.



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3. The hydrogen abstraction rate constants are strongly temperature-dependent, meaning the n-exponent of the modified Arrhenius equation is large. In the hot β-scission formulation of Döntgen et al., 8 the n-exponent determines the shape of the non-Boltzmann energy distribution of the newly formed radicals. If n is large, the maximum of the non-Boltzmann energy distribution is shifted to higher energies, thus enables reactions on higher energy levels. This is in line with the observations by Warnatz, 13 who stated that the small and very .

.

.

reactive H, O, and OH abstracting radicals cause ”prompt” dissociation of newly formed .

CH3 C−O radicals. The work of Warnatz 13 is presumably the first time hot β-scission has been actively considered. Some fuel radicals for which hot β-scission might be important 8 are formate species with the radical site at the formate carbon, organic acid species with the radical site at the oxygen of the carboxyl group, ether species with the radical site adjacent to the ether oxygen (secondary and tertiary carbon), and alcohol species with the radical site at the alcohol oxygen. Fuel radicals for which hot β-scission is expected to be negligible 8 are cyclic species, ketyl species, alkyl species, and alkyl groups distant to oxygen-containing functional groups. Again, these statements are for saturated fuel molecules.

Eigenvalues and Separation Problem In Figure 1, the rate constants of direct reaction to either the thermally equilibrated radicals .∗

or the β-scission products are based on the rate of R2 production, which is the rate of hydrogen abstraction. 11 The fractions of thermalization and consumption are computed as the branching ratios αth/β , respectively, which are in turn obtained from ME simulations for reactions on the radical PES. 11 In the ME framework, the chemical kinetics are calculated from the eigenvalues of the kinetic relaxation operator, which is the matrix of energy transfer probabilities with the microcanonical reaction probabilities on its diagonal. 7 The eigenvalues of this matrix correspond to chemical kinetics (low eigenvalues) and relaxational kinetics 6 ACS Paragon Plus Environment

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(high eigenvalues). Only if these eigenvalues are clearly separated, chemical and relaxational kinetics can be described distinctly.

λchem,1 /ω Eigenvalue Ratio λ/ω / -

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10−1 λrelax,1 /ω

λchem,2 /ω

10−2

500

1,000

1,500

2,000

Temperature T / K Figure 2: Eigenvalue ratio of the lowest relaxational eigenvalue λrelax,1 and the two highest chemical eigenvalues λchem,1/2 to the collision frequency ω for reactions on the MF radical PES at 0.1 atm. 8,14 The distinction between λchem,1 and λrelax,1 is based on ensuring differentiability at the intersection point. Figure 2 shows the violation of distinct separability between chemical and relaxational eigenvalues for reactions on the MF radical PES at 0.1 atm. In there, the lowest relaxational and the two highest chemical eigenvalues relative to the collision frequency are shown as function of temperature. While at low-temperature the lowest relaxational eigenvalue exceeds the highest chemical eigenvalue by at least one order of magnitude, at high-temperature, the highest chemical eigenvalue starts to exceed the lowest relaxational eigenvalue. The distinction between chemical and relaxational eigenvalues is not given mathematically when solving the eigenvalue problem. Here, the eigenvalue distinction is achieved for all temperatures via manual reconstruction of the eigenvalue curves in a way that the eigenvalues correspond to the same physical effects over the whole temperature range. The underlying approach is to ensure differentiability with respect to temperature, so that the eigenvalue curves are 7 ACS Paragon Plus Environment


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continuous. The same procedure could be used in state-of-the-art ME simulation software to obtain the curves for chemical and relaxational eigenvalues for all temperatures. This approach, however, would not avoid the separability problem, i.e. the merging of chemical and relaxational eigenvalues, which is inherent to the studied chemistry and will be described in the following. If the chemical and relaxational eigenvalues are clearly separated, different species are well-defined. In contrast, if the distinct separation of eigenvalues is violated, species become chemically ill-defined and must be treated as a ”super-species”. 7 For hot β-scission, the .∗

”super-species” comprises the unimolecular radical R2 and the bimolecular β-scission prod.∗

ucts R2,A + R∗2,B . Such a ”super-species” co-exists in both the unimolecular and bimolecular states, while each state contributes a certain fraction to the ”super-species” phase space. The fraction of the ”super-species” phase space associated with the unimolecular state is expected to be significantly smaller compared to the phase space fraction associated with the bimolecular state (anticipated from the partition function ratio Qbi /Quni = 2.31 × 105 –6.97 × 105 for T = 500–2000 K and p = 1 atm). As a consequence, the ”super-species” is expected to predominantly exist in the bimolecular state. In general, three different relations between chemical and relaxational eigenvalues can be distinguished: • If chemical eigenvalue relaxational eigenvalue (lower limiting case), then separate species are chemically well-defined, energy relaxation dominates over structural relaxation, and the chemical properties are described by the distinct species. For this lower limiting case, hot β-scission is of minor importance due to the fast rate of thermalization which comes with high relaxational eigenvalues. • If chemical eigenvalue relaxational eigenvalue (upper limiting case), then separate species are chemically ill-defined and a ”super-species” comprising the phase space of all ill-defined. For this upper limiting case, hot β-scission is expected to be the major pathway due to the presumably high probability of the ”super-species” being in the bimolecular state (see above) 8 ACS Paragon Plus Environment

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• If chemical eigenvalue ≈ relaxational eigenvalue (inseparable case), then separate species are chemically ill-defined, but conversion between these species is still dynamically limited by, e.g., the collision frequency of bimolecular fragments. Here, this case is predominantly encountered for the reactions of the MF radicals, as the ratio of chemical to relaxational eigenvalues range from 0.1–2 roughly (cf. Figure 2). As for the upper limiting case, i.e. significantly dominating chemical eigenvalues, hot β-scission is expected to be the an important pathway in the inseparable case. The branching ratio for hot β-scission, however, is smaller than for the upper limiting case, leading to an overall reduced flux to the bimolecular state. For the lower and upper limiting cases, the hydrogen abstraction reactants would solely .

.∗

produce thermally equilibrated radicals R2 or bimolecular β-scission products R2,A + R∗2,B , respectively. For the inseparable case (λchem ≈ λrelax ), the hot β-scission branching ratio varies from the latest distinguishably separable case to unity (upper limiting case). The latest distinguishably separable case is defined by a kinetic relaxation operator with relaxational eigenvalues being larger than its chemical eigenvalues. The three cases given above will be further discussed in the following, using reactions of methoxy (lower limiting case), diazenyl (upper limiting case), and MF radicals (inseparable case) as examples.

Results and Discussion Lower limiting case: λchem λrelax A case in which the chemical eigenvalues are significantly smaller than the relaxational ones can be found for any system when considering the high-pressure limit (infinite rate of thermalization). For technically relevant pressures, however, the above relation of chemical and relaxational eigenvalues is met if the chemical activation is below the smallest barrier .

height of the respective species. Here, the β-scission of methoxy (CH3 O) is used as an example. The hot reaction involving chemically activated methoxy is given as follows. 9 ACS Paragon Plus Environment


.

.

.

CH3 OH + H −−→ (CH3 O + H2 )∗ −−→ CH2 −O + H2 + H

(2)

Figure 3 shows the ”glued” PES of hydrogen abstraction from methanol (CH3 OH) via hydrogen radicals (on the left) and the β-scission and isomerization reaction network of methoxy (on the right). The ”gluing” is done by adding the electronic energy of H2 to the β-scission PES. The potential energies for hydrogen abstraction and β-scission are taken from Meana-Pa˜ neda et al. 15 and Kamarchik et al. 16 at the MC3BB and CCSD(T)/aug-ccpV(T,Q)Z // CCSD(T)/aug-cc-pVTZ levels of theory, respectively. The ”gluing” points of the two PESs are shown as a horizontal cut, similar to the illustration in Figure 1. The chemical activation of methoxy radicals is illustrated by the non-Boltzmann energy distribution at the hydrogen-abstraction transition state (red). Only the reactions on the solid line PES are considered for discussion. E

40

TSiso TSβ

30 E / kcal/mol

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.

CH2 −O + H + H2

20

Pˆ (E)

TSH

10

.

.

CH3 OH + H

0

CH3 O + H2

.

CH2 OH + H2

Figure 3: Potential energy surface for hot β-scission of the methoxy radical. The isomerization transition state is shown in the top middle and connects the two unimolecular species . . CH2 OH and CH3 O. Chemical activation of the hydrogen abstraction products is illustrated by the non-Boltzmann energy distribution (red) at the transition state (T = 1000 K). .

.

The reverse hydrogen abstraction barrier of both products, CH3 O+H2 and CH2 OH+H2 , is significantly lower than the β-scission barrier heights of these products (roughly 14 kcal/mol 10 ACS Paragon Plus Environment

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at least). As can be seen from the non-Boltzmann energy distribution shown in Figure 3 (at TSH , for T = 1000 K), chemical activation is unlikely to exceed the β-scission barrier heights and the lifetime of the chemically activated radicals is long enough to allow for deactivation .

via collisions with the bath gas. Deactivation of rovibrationally excited CH2 OH radicals is .

expected to be even more pronounced compared to CH3 O radicals, due to the first being less excited by roughly 10 kcal/mol. With increasing temperature, however, the chemical activation would be shifted to higher energies and would enable ”prompt” dissociation of the chemically activated methoxy radicals. Using the RRKM/ME-based approach of Döntgen et al., 8 hot β-scission branching ratios for ”prompt” dissociation of chemically activated methoxy radicals are calculated. The RRKM/ME simulations are conducted using the MESS software package, 7 which is used throughout the present work. Geometries and energies are taken from Kamarchik et al., 16 Lennard-Jones parameters for collision frequency calculations are taken from Hippler et al., 17 and energy transfer is modeled as ΔEdown = 200 cm−1 ·(T /300 K)0.85 throughout the present work. The exact parameters for this and all following ME simulations can be found in the supporting information. For unimolecular kinetics of the thermally equilibrated methoxy, please refer to Dames and Golden, 18 in which the QM data of Kamarchik et al. 16 has been used in detailed RRKM/ME simulations. The major outcome of the present RRKM/ME investigation is that methoxy hot β-scission branching ratios at 0.1 atm are below 5 % for temperatures below 1200 K and do not exceed 25 % up to 2000 K. With increasing pressure, however, the hot β-scission branching ratios decrease and reach zero eventually. At 1 atm, for instance, hot β-scission branching ratios do not exceed 10 % below 1600 K and reach roughly 17 % at 2000 K. In conclusion, hot β-scission of methoxy radicals converges to the lower limiting case with decreasing temperature and increasing pressure. Generally speaking, any hot β-scission reaction is expected to converge to the lower limiting case as thermal energy decreases (temperature) and the rate of thermalization increases (pressure). Another example system illus-



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trating the lower limiting case and its temperature- and pressure-dependence is 2-butanone, as studied by Kopp et al. 19

Upper limiting case: λchem λrelax As mentioned previously, a case in which the chemical eigenvalues are significantly larger than the relaxational ones can be found for any system when considering the high-temperature limit. For technically relevant temperatures, however, the above relation of chemical and relaxational eigenvalues is met if the chemical activation is above the smallest barrier height .

of the respective species. Here, the β-scission of diazenyl (NNH) is used as an example. The hot reaction involving chemically activated diazenyl is given as follows. .

.

.

N2 H2 + H −−→ (NNH + H2 )∗ −−→ N2 + H2 + H

(3)

Figure 4 shows the ”glued” PES of hydrogen abstraction from diazene (N2 H2 ) via hydrogen radicals (on the left) and the β-scission reaction network of diazenyl (on the right). The potential energies for hydrogen abstraction and β-scission are taken from Linder et al. 20 and Bozkaya et al. 21 at the MRCI(5,5)/cc-pVTZ and CCSD(T)/cc-pV(D-6)Z // CCSD(T)/ccpCVQZ levels of theory. The ”gluing” point of the two PESs is shown as a horizontal cut, similar to the illustration in Figure 1. The chemical activation of the diazenyl radicals is illustrated by the non-Boltzmann energy distribution at the hydrogen abstraction transition state (red). In contrast to the methoxy example, the reverse hydrogen abstraction barrier height for .

NNH + H2 largely exceeds the β-scission barrier height (about 35 kcal/mol). As a conse.

quence, the chemical activation of NNH enables ”prompt” dissociation in the framework of hot β-scission. Again, RRKM/ME simulations via the MESS software package 7 are used to describe hot β-scission branching ratios, as described by Döntgen et al. 8 Here, the geometries and potential energies of the β-scission reaction network are taken from Bozkaya et al. 21 and


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E

60

E / kcal/mol

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Pˆ (E)

40 .

TSH

N2 H2 + H

20

0

TSβ .

NNH + H2 .

N2 + H + H 2

Figure 4: Potential energy surface for hot β-scission of the diazenyl radical. Chemical activation of the hydrogen abstraction products is illustrated by the non-Boltzmann energy distribution (red) at the transition state (T = 1000 K). the collision frequency and energy transfer probability are modeled as described for methoxy (see above). While the methoxy-based hot β-scission exhibits some small none-zero fraction of ”prompt” dissociation at most, the chemically activated diazenyl radicals solely dissociate prior to thermalization for temperatures from 300–2000 K and pressures from 0.1–100 atm. Thus, diazenyl radicals formed via hydrogen abstraction would dissociate to molecular nitrogen and hydrogen radicals for any technically relevant condition. In current reaction models, however, hydrogen abstraction from diazene and β-scission of diazenyl are modeled as separate steps, e.g. in the hydrazine combustion model of Konnov and co-workers. 22,23 Future kinetic modeling studies involving hydrogen abstraction from diazene should consider the hot β-scission pathway to avoid incompleteness of the reaction mechanisms. As mentioned in the introductory discussion on the separation problem, the hot β-scission rate constants for the upper limiting case are equal to the hydrogen abstraction rate constants and the hot β-scission reaction would fully replace the diazenyl forming reaction.



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Inseparable case: λchem ≈ λrelax For the lower and upper limiting cases, the fraction of chemically activated intermediates being deactivated can be anticipated from the PES via chemical intuition. If the energy release from hydrogen abstraction and the barrier height for β-scission are on the same level, however, a detailed hot β-scission kinetics study is essential for qualitative and quantitative .

insights. Here, the β-scission of MF radicals (CH3 OC−O) is used as an example, which were shown to be especially prone to hot β-scission previously. 8 The hot reaction involving chemically activated MF radicals is given as follows. .

.

.

CH3 OCH−O + H −−→ (CH3 OC−O + H2 )∗ −−→ CO2 + H2 + CH3

(4)

Figure 5 shows the ”glued” PES of hydrogen abstraction from methyl formate (CH3 OC−O) via hdyrogen radicals (on the left) and the β-scission reaction network of MF radicals (on the right). The potential energies for hydrogen abstraction and β-scission are taken from Tan et al. 24 and Tan et al. 14 at the MRACPF/cc-pV(D,T)Z // B3LYP/cc-pVTZ and CCSD(T)/cc-pV(T,Q)Z // CCSD(T)/cc-pVTZ levels of theory, respectively. The ”gluing” points of the two PESs are shown as a horizontal cut, similar to the illustration in Figure 1. The chemical activation of the MF radicals is illustrated by the non-Boltzmann energy distribution at the hydrogen abstraction transition state (red). Only the reactions on the solid line PES are considered for discussion. As can be seen from the non-Boltzmann energy distribution, chemical activation exceed the barrier height for β-scisison. The actual non-Boltzmann energy distribution of the radicals, however, is shifted to lower energies and compressed, as only a fraction of the hydrogen abstraction energy release is distributed to the radicals. 8,11,12 For this particular example, the fraction of chemically activated radicals dissociating ”promptly” rises from zero to unity over the temperature regime ranging from 500–2000 K. In that temperature regime, the ME formulation encounters the inseparability problem; i.e. chemical and relaxational eigenval-


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40

TSiso

30 20 E / kcal/mol

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.

CH−O + CH2 −O + H2

Pˆ (E) TSH

10 0

TSβ

.

CH3 O + CO + H2

.

CH3 OCH−O + H

−10

.

−O + H 2 / CH2 OCH . CH3 OC−O + H2

−20

.

CH3 + CO2 + H2

−30 Figure 5: Potential energy surface for hot β-scission of the MF radicals. The isomerization transition state is shown in the top middle and connects the two unimolecular . . species CH2 OCH−O and CH3 OC−O. Chemical activation of the hydrogen abstraction products is illustrated by the non-Boltzmann energy distribution (red) at the transition state (T = 1000 K). ues cannot be separated clearly. In the following, an approximate description is applied to enable description of hot β-scission over the whole temperature regime. The microcanonical balancing of the reactions on the PES illustrated in Figure 5 is conducted via the MESS software package, as described above. In there, the geometries and potential energies are taken from Tan et al. 14 The collision frequency and the energy transfer are modeled as described for methoxy, but with collision parameters for MF. Figure 6 shows the energy-dependent branching ratios for β-scission at 500 K (solid lines) and at the highest accessible temperature, for pressures ranging from 0.1–100 atm. In this context, the highest accessible temperature is the highest temperature for which chemical and relaxational eigenvalues of the Master Equation can be separated distinctly. Referring to the kinetic relaxation operator, the highest accessible temperature comes with the latest distinguishably separable case. The low-temperature (solid lines) and high-temperature (dashed lines) results are shown



(a) p = 0.1 atm

(b) p = 1 atm

(c) p = 10 atm

5 15 25 E / kcal/mol


(d) p = 100 atm

1 Branching ratio α / -

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0.8 0.6 0.4 0.2 5 15 25 E / kcal/mol


Figure 6: Energy-dependent branching ratios for thermalization of MF radicals as function of pressure (0.1 - 100 atm) and temperature. The temperature-dependence is given by the solid and the dashed lines, which correspond to 500 K and the highest accessible temperature, respectively. Note that the highest accessible temperature decreases with pressure due to the increasing rate of thermalization and amounts to 750, 800, 900, and 1060 K for 0.1, 1, 10, and 100 atm, respectively. for different pressures to illustrate the branching ratios’ low sensitivity to temperature for all studied pressures. Although the sensitivity increases with increasing pressure, the lowenergy and high-energy limits are similar for all conditions: All MF radicals thermalize at low-energy and all MF radicals dissociate at high-energy. For 0.1 atm, the temperaturedependent part is fairly below the dissociation limit of the MF radicals, which amounts to 15 kcal/mol roughly. With increasing pressure, however, the temperature-dependent part shifts to higher energies, since the rate of thermalization increases with pressure. The low- and high-energy limits and the exponential transition between these limits is approximated with a logistic function in the present study:

α(E) = 1/(1 + exp(−c · (E − ε)),

(1)

with the curvature parameter c and the transition energy parameter ε. This function is fitted to the energy-dependent thermalization branching ratios to understand how the branching 16 ACS Paragon Plus Environment

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ratios change as temperature approaches the highest accessible temperature. Note that the fitting procedure is exclusively used to analyze the trend of the energy-dependent branching ratios over temperature and will not be used for further kinetics predictions. (a) Curvature parameter c

(b) Transition energy parameter ε

p = 0.1 atm p = 1 atm p = 10 atm p = 100 atm

3 2 1 0

400

600

800

1E4 Trans. energy ε / 1/cm

4 Curvature c / cm

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8E3 7E3 6E3 5E3 4E3

1,000

p = 0.1 atm p = 1 atm p = 10 atm p = 100 atm

9E3

400

Temperature T / K

600

800

1,000

Temperature T / K

Figure 7: Temperature-dependence of the curvature and transition energy parameters of the logistic function used to fit the energy-dependent MF radical thermalization branching ratios. Pressure ranges from 0.1 to 100 atm. The temperature- and pressure-dependence of the curvature and transition energy parameters of the logistic function, c and ε respectively, is shown in Figure 7. Some points at low-temperature are excluded from the plots, since the respective fits failed. Fitting issues also caused the oscillations in the curvature parameter at low-temperature. The general curvature trend, however, is clearly visible in Figure 7. For all pressures, the curvature decreases with increasing temperature and flattens out as it approaches the highest accessible temperature. Thus, it appears that the curvature parameter is almost converged to a constant value for the highest accessible temperature. A similar trend is observed for the energy transition parameter, although the pressure dependence is inverted. Most importantly, both parameters flatten out with increasing temperature, and the energy-dependent branching ratios at the highest accessible temperature can therefore be used as high-temperature approximations. Note that the parameters of the logistic function 17 ACS Paragon Plus Environment


are not used any further, but the energy-dependent branching ratios above the highest accessible temperature are described by the dashed lines in Figure 6, i.e. the energy-dependent branching ratios at the highest accessible temperature. 1

Branching ratio α / -

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0.8

p=

0.6

p= p=

0.4

p=

0.2

0

500

0. 1

10

10

1a

0a

tm

at m

tm

at m

1000 1500 Temperature T / K

2000

Figure 8: MF radical thermalization branching ratios for the temperature regime of chemically well-defined MF radicals (dense thick lines) and for the temperature regime of chemically ill-defined MF radicals (loose thin lines). The hot β-scission branching ratios are given by the difference between the thermalization branching ratios and unity. Figure 8 shows the temperature- and pressure-dependent thermalization branching ratios of rovibrationally excited MF radicals. While the dense thick lines are adopted from the previous work of Döntgen et al., 8 the loose thin lines are calculated using the presently proposed high-temperature approximation for the separation of chemical and relaxational kinetics. This high-temperature approximation is expected to give more realistic branching ratios compared to keeping the branching ratio constant (very pessimistic) or setting it to zero (very optimistic), but is still somewhat pessimistic. This is due to the non-zero slope of the curvature parameter shown in Figure 7, indicating a minor decrease of thermalization as the temperature exceeds the highest accessible temperature. As a consequence, thermalization is expected to be slightly over-predicted for temperatures above the highest accessible 18 ACS Paragon Plus Environment

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temperature. Physically speaking, the proposed high-temperature approximation is keeping the energy-dependent branching ratios constant, leading to hot reaction kinetics fully governed by the reactants’ non-Boltzmann energy distribution above the highest accessible temperature.

Conclusion The present discussion of hot β-scission illustrated how the underlying two-step mechanism relates to the PES of the ”glued” hydrogen abstraction and β-scission reaction network. For the chemically activated hydrogen abstraction products, three cases were proposed: Insignificant chemical activation (lower limit), predominant chemical activation (upper limit), and the transition between these two limits (inseparable case). Each case was illustrated by an example system: Hot β-scission of methoxy radicals (lower limiting case), of diazenyl radicals (upper limiting case), and of MF radicals (inseparable case). For diazenyl, hot β-scission is found to dominate for technically relevant temperatures from 500–2000 K and pressures from 0.1–100 atm. Therefore, direct reaction from diazene and a hydrogen-abstracting radical to the β-scission products would fully replace the separate hydrogen abstraction and β-scission reaction steps. For future kinetic modeling, this direct abstraction should be considered, with the kinetics being described via hydrogen abstraction rate constants. For the inseparable case, an approximate distinction between chemical and relaxational kinetics has been proposed and used to describe the kinetics of hot β-scission for the whole temperature range of interest. In principle, this approximation can be applied to any inseparable case. However, a more fundamental description would be available if the chemical and relaxational eigenvalues of the ME would be distinguished mathematically. A mathematical distinction could be based on ensuring differentiability of the eigenvalue curves with respect to temperature, as described in the introductory part of this work.



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From the present discussions, rules of thumb can be derived. Hot β-scission can be ignored if the reverse hydrogen abstraction barrier is much smaller than the barrier for βscission. Hot β-scission has to be considered if the barriers are comparably large, and hot β-scission replaces the separate reaction steps if the reverse hydrogen abstraction barrier is much larger than the barrier for β-scission. The lower or upper limiting cases, however, can be encountered for any system irrespective of the barrier heights, if the high-pressure or high-temperature limits are reached, respectively.

Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111 is gratefully acknowledged. KL is thankful for financial support to the Cluster of Excellence ”Tailor-Made Fuels from Biomass” (EXC 236), which is funded by the Excellence Initiative by the German federal and state governments to promote science and research at German universities.

Supporting Information Available MESS software input files for methoxy, diazenyl, and the methyl formate radicals are attached. This material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Dean, A. M. Predictions of pressure and temperature effects upon radical addition and recombination reactions. J. Phys. Chem. 1985, 89, 4600–4608. (2) Westmoreland, P. R.; Howard, J. B.; Longwell, J. P.; Dean, A. M. Prediction of rate constants for combustion and pyrolysis reactions by bimolecular QRRK. AIChE J. 1986, 32, 1971–1979. 20 ACS Paragon Plus Environment

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of formyl radicals and its effect on laminar flame speeds. J. Phys. Chem. Lett. 2016, 7, 85–89. (12) Labbe, N. J.; Sivaramakrishnan, R.; Goldsmith, C. F.; Georgievskii, Y.; Miller, J. A.; Klippenstein, S. J. Ramifications of including non-equilibrium effects for HCO in flame chemistry. Proc. Combust. Inst. 2016, 36 . (13) Warnatz, J. Combustion chemistry, Ch. 5 ; Springer-Verlag, NY, 1984. .

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Discussion of the Separation of Chemical and Relaxational Kinetics of

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