A chemically consistent rate constant for the reaction of nitrogen dioxide with the oxygen atom

Abstract

In the present study, a chemically consistent rate constant for the reaction between nitrogen dioxide and the oxygen atom has been obtained by combining low-temperature experimental data from the literature and new high-temperature quantum chemical calculations. The expression for our rate constant is k_NO2+O=NO+O2= 2.589 × 10¹⁵T^−1.035 exp(−226/RT) + 4.242 × 10¹⁶T^−0.861exp(−50 917/RT) cm³ mol⁻¹ s⁻¹, where R = 8.314 J mol⁻¹ K⁻¹, and is valid over the temperature range T = 221 to 3000 K. The effect of the inclusion of the new rate constant on the prediction of three detailed reaction models from the literature has been studied using (i) new experimental oxygen atom profiles obtained in a shock tube during nitrogen dioxide pyrolysis, and (ii) published shock tube and jet-stirred reactor data for H₂–NO_x mixtures with and without dioxygen. The impact of the new rate constant on the sensitivity coefficients and reaction pathways has also been analyzed under some conditions. Overall, the predictive capability of the reaction models were improved. The present study suggests that our chemically consistent rate constant should be included in detailed reaction models for combustion applications.

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1 Introduction

Nitrogen dioxide (NO₂) is a well known pollutant originating from combustion processes. It participates in ozone formation in urban and suburban areas.¹ The kinetics of nitrogen dioxide is also relevant to the combustion of solid propellants and energetic materials which are characterized by chemical structures with one or several nitro groups.² Nitrogen dioxide could also be attractive as an auto-ignition promoter of hydrogen–oxygen mixtures since its addition induces a strong reduction of the auto-ignition delay time in the second explosion limit region.^3–5 The strong impact of NO₂ addition in reducing the ignition delay-time was also demonstrated for methane–oxygen mixtures based on the study of Mathieu et al..⁶ In addition to practical propulsion applications, the chemistry of nitrogen dioxide is relevant to more fundamental subjects since a number of mixtures such as CH₃NO₂(–O₂), H₂–NO₂/N₂O₄, CH₄–NO₂/N₂O₄ and C₂H₆–NO₂/N₂O₄ exhibit a double cellular structure while detonating.^7–10 The thermal decomposition of nitrogen dioxide leads to the formation of NO and O¹¹ and numerous studies are available for the high-temperature rate constant of this reaction.^11–16 On the contrary, the reaction of atomic oxygen with NO₂ has been much less investigated at high temperatures^11,17 and most studies have been performed in the low-^18–23 and intermediate-temperature²⁴ ranges. It has been shown previously that the reactions of the oxygen atom with nitrous oxide (N₂O) need to be accurately known to describe the dynamics of nitrous oxide based mixtures.²⁵ A similar impact may be expected for the reactions between NO₂ and the oxygen atom.

The present work aims at (i) obtaining a chemically consistent rate constant for the reaction between nitrogen dioxide and oxygen atom over a wide range of temperatures, and (ii) understanding the impact of the rate constant update on the prediction of recently established detailed reaction models for combustion applications.

2 Methods and materials

2.1 Rate constant calculation

We performed dynamics calculations for the NO₂ + O = NO + O₂ reaction using the canonical variational transition-state theory (CVT)^26–29 to obtain accurate high-temperature (3000 K > T > 1000 K) rate constants in the high-pressure limit, based on the potential energy surface obtained by the TPSSh^30,31 density functional combined with the MG3S³² basis set, which is the same as the 6-311+G(2df,2p) basis set for N and O.

We chose the TPSSh/MG3S method for geometry, electronic energy, and frequency calculations because it has the best performance on barrier prediction for this reaction in a variety of Kohn–Sham (KS) model chemistries as compared to high-level benchmark calculations using the composite method W3X-L.³³ The tested KS model chemistries are TPSSh/MG3S and the combinations of M08-HX³⁴ and MN15^35,36 exchange–correlation functionals with the MG3S and aug-cc-pVTZ^37,38 basis sets. The T₁ diagnostic³⁹ calculated using the CCSD^40,41/MG3S method based on TPSSh/MG3S geometry shows that the transition state of this reaction (T₁ = 0.049) has an inherently moderate multi-configurational character, and hence a multi-reference method is needed for accurate barrier prediction. The W3X-L method offers an accurate approximation to the all-electron scalar-relativistic CCSDT(Q)/CBS energy by a combination of valence-electron nonrelativistic CCSD(T)/CBS, core-valence correlation contribution, scalar relativistic effects, and a series of post-CCSD(T) treatments. These post-CCSD(T) terms include the [CCSDT-CCSD(T)] and the [CCSDT(Q)-CCSDT] components to account for higher-order excitation effects, and hence, the W3X-L method can provide accurate electronic energies for the system with a moderate multi-reference character. Using the W3X-L method based on the TPSSh/MG3S geometries, we obtained the best estimation of the forward barrier for the NO₂ plus O reaction, which is a negative barrier, −9.83 kJ mol⁻¹. As shown in Table 1, only the TPSSh/MG3S method predicts a negative barrier as the W3X-L method does, with a deviation of only 3.42 kJ mol⁻¹ from the best estimation, which is accurate enough for high-temperature rate constant calculations. The other tested KS model chemistries significantly overestimate the barrier by 19.3–36.5 kJ mol⁻¹. Note that all the barriers discussed here are classical barriers with zero-point-energies (ZPEs) excluded. The calculated high-temperature rate constants are listed in Table S1 of the ESI.†

Table 1 The calculated forward barriers V^≠_f with a variety of KS model chemistries and their unsigned deviations (UD) from the best estimation obtained using the W3X-L//TPSSh/MG3S method (in kJ mol⁻¹)

	V ^≠f	UD
M08-HX/MG3S	26.71	36.54
M08-HX/aug-cc-pVTZ	26.67	36.50
MN15/MG3S	9.89	19.72
MN15/aug-cc-pVTZ	9.43	19.26
TPSSh/MG3S	−6.41	3.42
W3X-L//TPSSh/MG3S	−9.83	0

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We carried out all the density functional calculations using the Gaussian 09 package⁴² with the MN-GFM⁴³ module and the W3X-L calculations using the Molpro 2015 software package.⁴⁴ The CVT dynamics calculations were performed using the Polyrate 2016⁴⁵ and Gaussrate 17 programs.⁴⁶ In the dynamics calculations, ZPEs and partition functions were calculated with the vibrational frequencies scaled by a scaling factor of 0.984.

2.2 Experimental technique

The experimental setup and the method used to obtain new experimental data to study the effect of the inclusion of the new rate constant are presented in this section.

2.2.1 Mixture preparation. Experimentally, both nitrous oxide–argon and nitrogen dioxide–argon mixtures were employed. The N₂O–Ar mixtures were used to calibrate the Atomic Resonance Absorption Spectroscopy (ARAS) set-up (see next paragraph), whereas the NO₂–Ar mixtures were used to obtain new experimental absorption profiles of the oxygen atom during NO₂ pyrolysis.

Nitrous oxide–argon mixtures were prepared from mono-component gas cylinders containing either N₂O or Ar. Nitrogen dioxide–argon mixtures were prepared from a mixture containing 1% of NO₂ in pure argon. High purity gases were employed. The partial pressure method was used to prepare the mixtures. The mixture preparation apparatus was vacuumed to less than 1 × 10⁻⁵ Pa using a turbo-molecular pump. The N₂O and NO₂ contents ranged between 1 and 15 ppmv and between 5 and 40.1 ppmv, respectively. For the NO₂–Ar mixtures, the mixture composition as well as its stability under ambient conditions were checked using a Fourier transform infra-red spectrometer. The mixture composition was found to be in good agreement with the expected composition obtained using the partial pressure method, within the uncertainty of the initial NO₂–Ar blend. The stability of the mixtures containing nitrous oxide was verified in a previous work by the authors.⁴⁷

2.2.2 Shock-tube and optical arrangement. The shock tube used in the present study has been described in previous papers.^25,48,49 Briefly, the apparatus comprised a 78 mm internal diameter stainless-steel pressure-driven shock tube with a 5.5 m-long driven section and a 3.5 m-long driver section. A turbomolecular pump evacuated the test section to less than 2 × 10⁻⁴ Pa, and the typical leak-plus-outgassing rate was on the order of 2 × 10⁻⁶ Pa s⁻¹ or less. Helium was used as the driver gas, and the shock wave was initiated by the bursting of a double diaphragm. The shock wave velocity was measured by piezoelectric pressure gauges mounted flush with the inner wall. Thermodynamic conditions behind the incident and reflected shock waves were computed from the experimental incident shock velocity using the SHOCK routine.⁵⁰ The experiments were performed over the reflected shock temperature range of 1500–2500 K and reflected shock pressure range of 249–994 kPa.

The optical detection technique used for measuring the O atom concentration was an emission line absorption method. A microwave-excited discharge lamp that contained a flowing mixture of 1% O₂ in He maintained at a pressure of 1.4 kPa was used as the light source. This lamp consisted of a 2.45 GHz microwave generator operated at the 100 W power level. A vacuum ultraviolet (UV) monochromator was used to isolate the O triplet wavelength, 130.5 nm, and a solar blind photomultiplier was used to convert vacuum UV photons. The operating conditions were chosen to minimize noise and reflected power without significant loss of signal strength. The absorption measurements were made behind a reflected shock front through two thin MgF₂ windows located 10 mm from the end wall. Details about the O atoms and nitrogen oxide calibrations are presented hereafter.

For the calibration of oxygen atom measurements, the thermal decomposition of highly Ar-diluted nitrous oxide (N₂O) at high temperature was employed. Under these conditions, the decomposition is fast and both recombination and secondary reactions are unimportant so that a constant O atom concentration is achieved. Pyrolysis experiments were performed with mixtures containing 1 to 15 ppmv of N₂O in Ar, in the temperature and pressure ranges of 1882–3976 K and 168–555 kPa, respectively. Fig. 1 presents the evolution of the light absorption as a function of the O-atom concentration. The experimental data could be fitted by the following fourth order polynomial:

where A_O is the absorption by O atoms; and [O] is the O atom concentration in atom per cm³. Note that in the calculation of A_O, the length of the optical path of 7.8 cm is implicitly included. Such a formalism for ARAS calibration curve has been previously employed in ARAS studies.⁵¹ It ensures the extension of the concentration range for which the calibration is valid. Our optical set-up allows for measurements in the range 3 × 10¹² to 2 × 10¹⁴ atom cm⁻³. Subsequent experiments have to be performed in this range of concentration to avoid the saturation of the absorption signal or too low a signal.

Fig. 1 Calibration of absorption by O atoms at 130.5 nm. Experiments were performed using N₂O–Ar mixtures. The experimental conditions were X_N2O = 1–15 ppmv; T = 1882–3976 K and P = 168–555 kPa.

It was shown previously^53–57 in a variety of chemical systems that ARAS measurements could be disturbed by interference absorption by the reactant and/or intermediate species. To account for these perturbations, the absorption cross-sections of NO₂ and NO at 130.5 nm were measured over a range of concentrations and thermodynamic conditions. Fig. 2 and 3 display the evolution of the Napierian logarithm of the normalized absorption inverse as a function of concentration for NO₂ and NO, respectively. Also shown are the results from Nakayama et al.⁵² and Watanabe⁵⁸ respectively for the cross-sections of NO₂ and NO. Cross-sections of 10 ± 2 × 10⁻¹⁸ cm² per molecule for NO₂ and of 3.2 ± 0.5 × 10⁻¹⁸ cm² per molecule for NO were measured. The present measurements and those from the literature, 14 ± 4 × 10⁻¹⁸ cm² per molecule for NO₂⁵² and 3.0 ± 0.3 × 10⁻¹⁸ cm² per molecule for NO,⁵⁸ appear to be in agreement within the experimental uncertainties.

Fig. 2 Calibration of absorption by NO₂ at 130.5 nm. Experiments were performed using NO₂–Ar mixtures. The experimental conditions were X_NO2 = 552 ppmv; T = 289–874 K and P = 6.7–178 kPa. The black dashed lines represent the uncertainty bar of the data from ref. 52.

Fig. 3 Calibration of absorption by NO at 130.5 nm. Experiments were performed with NO–Ar mixtures. The experimental conditions were X_NO = 9960 ppmv; T = 291–901 K and P = 16–142 kPa. The black dashed lines represent the uncertainty bar of the data from ref. 58.

2.3 Combustion modeling

Based on the recent study of Kovacs et al.,⁵⁹ which evaluated the performance of 16 detailed reaction models against a comprehensive database for H₂–O₂–NO_x mixtures, we have selected the three most reliable reaction models: (i) Glarborg et al.,⁶⁰ (ii) Nakamura et al.,⁶¹ and (iii) Zhang et al.⁶² For each model, the impact of replacing the original rate constant of the reaction NO₂ + O = NO + O₂ by the rate constant we calculated has been investigated. The data we employed for assessing this impact include the new data we presently obtained during NO₂ pyrolysis, as well as shock tube data^3,63,64 from the literature obtained for H₂–NO₂(–O₂) mixtures. We also examined the impact of the rate constant update using jet-stirred data⁶⁵ obtained for H₂–NO–O₂ mixtures. For the new data we obtained during NO₂ pyrolysis, the interference of NO and NO₂ at the absorption wavelength of oxygen atom did not enable us to directly extract reliable oxygen atom concentration. Consequently, we compared the experimental total absorption at 130.5 nm with the calculated one which is obtained bywhere A_Tot is the total absorption. As in A_O, the length of the optical path of 7.8 cm is implicitly included in A_Tot.

The shock tube data were modeled using the Cantera package⁶⁶ considering either an adiabatic constant volume or an adiabatic variable-volume reactor model. The variable-volume reactor model was employed when a facility-dependent pressure rise was indicated in the original paper. The jet-stirred reactor data were modeled using the PSR code⁶⁷ from Chemkin II.⁶⁸ Since the jet-stirred experiments are performed at a constant pressure, the PLOG formalism included in the selected reaction models does not pose an issue although Chemkin II cannot handle such a formalism. Reaction models which include the rate constant at the correct pressure were created using the PLOG-replace utility, freely available on the website of the Combustion Chemistry Center of Galway.

3 Results and discussion

In this section, we are successively presenting the updated rate constant for the NO₂ + O = NO + O₂ reaction and its effect on the predictions by three recent reaction models of several fundamental combustion parameters obtained in various mixtures containing NO_x. Indeed, when updating a detailed reaction model, it is important to verify that its performances are at least not deteriorated if not significantly improved. In general, the figures were selected to illustrate the most significant effects. Given the large amount of data we modeled, we cannot show systematically the results obtained for the three reaction models. However, we have included additional modeling results in the ESI.†

3.1 Rate constant update

We have collected the available rate constant data for the NO₂ + O = NO + O₂ reaction from the kinetic database of NIST. The studies performed before 1970 were not considered. Fig. 4 shows the available rate constant in an Arrhenius plot along with those obtained in the present study. Most of the studies were performed in the low-temperature range and studies at high temperatures are rather scarce. At low temperature, most experimental studies are consistent with each other with a value of around 6 × 10¹² cm³ mol⁻¹ s⁻¹ at 298 K. The CVT rate constants are about two times lower than these experimental data in the low-temperature region. Considering that the CVT calculations are highly sensitive to the accuracy of the potential energy surface at low temperatures but are less affected at high temperatures, we decided to jointly use the experimental low-temperature data and our CVT high-temperature results. Thus, by combining the low-temperature study of Estupiñán et al.²¹ (221–425 K, see Table S2 of the ESI†) and our high-temperature rate constant calculation (1000–3000 K, see Table S1 of the ESI†), we obtained a complex temperature dependence with a decreasing rate constant as temperature increases up to approximately 1000 K and an increasing rate constant as temperature increases above 1000 K which can be described bywhere R = 8.314 J mol⁻¹ K⁻¹. This kind of behavior is typically observed for a reaction with a negative free energy activation barrier. In the high-temperature range, our updated rate constant is significantly different from the one recommended in several reviews^79–85 by extrapolating the low-temperature experimental data to the high-temperature range. Our rate constant is within 45% at 1800 K and within 25% at 2300 K of the value reported by Zuev and Starikovskii.¹⁷ This is in reasonable agreement with the uncertainty of CVT results introduced by the accuracy of potential energy surface calculations. For example, the deviation of ≈3.4 kJ mol⁻¹ for TPSSh energy barrier from the W3X-L benchmark would lead to uncertainties of the calculated rate constant by 26% and 19% at 1800 K and 2300 K, respectively. However, Zuev and Starikovskii reported a temperature-independent rate constant, inconsistent with our calculations.

Fig. 4 Comparison of the available rate constants for the reaction NO₂ + O = NO + O₂. Experimental studies: 2006 T. Dillon;⁶⁹ 2001 E. Estupiñán;²¹ 1999 T. Gierczak;⁷⁰ 1995 S. Paulson;²⁰ 1991 A. Zuev;¹⁷ 1987 R. Geers-Müller;⁷¹ 1986 A. Ongstad;⁷² 1984 A. Ongstad;²² 1979 M. Addison;⁷³ 1975 C. Wu;⁷⁴ 1974 P. Bemand;²⁴ 1973 T. Slanger;¹⁸ 1973 A. Harker;²³ 1973 J. Davis;⁷⁵ 1972 Y. Gershenzon;⁷⁶ 1972 M. A. A. Clyne;⁷⁷ 2003 L. Avallone;⁷⁸ 2004 R. Atkinson;⁷⁹ Reviews.^79–85 The calculated CVT rate constants are our dynamics calculation results. The red solid line plots the final rate constants used in the modeling which corresponds to the combined results of the high-temperature CVT data and low-temperature experimental data from Estupiñán et al.²¹

Only one theoretical study has been previously performed for the reaction NO₂ + O = NO + O₂. Fig. 5 compares the rate constant computed by Zhao et al.⁸⁶ with the one we have calculated. Zhao et al.⁸⁶ computed the reaction rate constant under the framework of transition state theory (TST) based on the electronic structure calculations using the CCSD/6-311++G**//B3LYP/6-311++G** method, which shows that they corrected the single point energies of key reaction species using the single-reference CCSD/6-311++G** method based on the B3LYP/6-311++G** optimized geometries. However, as we mentioned early, this reaction is a strong electron correlation system, whose wavefunction has a moderate multi-reference character, and thus the single-reference CCSD method will overestimate the reaction barrier. They obtained a positive forward barrier of 16.19 kJ mol⁻¹, which is 22.6 kJ mol⁻¹ higher than the value (−6.41 kJ mol⁻¹) we obtained using the TPSSh/MG3S method and 26.02 kJ mol⁻¹ higher than our best estimate obtained using the W3X-L method. Therefore, Zhao et al. significantly underestimated the rate constants compared to the present calculations and experimental data. For example, the rate constant of 4.29 × 10¹¹ cm³ mol⁻¹ s⁻¹ they obtained at 2300 K is over one order of magnitude lower than our value (4.52 × 10¹² cm³ mol⁻¹ s⁻¹) and the experimental value (6.03 × 10¹² cm³ mol⁻¹ s⁻¹) obtained by Zuev et al.,¹⁷ which are the only available experimental data above 2000 K. Their rate constant at 300 K is 5.06 × 10⁷ cm³ mol⁻¹ s⁻¹, over five orders of magnitude lower than 6 × 10¹² cm³ mol⁻¹ s⁻¹ obtained in most experimental studies. More importantly, because of this positive barrier, they predicted a positive temperature dependence of the rate constant at low temperatures, which is in disagreement with the experimental observations.

Fig. 5 Comparison of the theoretical rate constants for the reaction NO₂ + O = NO + O₂. 2018 Zhao.⁸⁶

Fig. 6 compares the rate constants included in the reaction models of Glarborg, Nakamura, and Zhang, with the one obtained in the present study. The temperature-dependence of the rate constants included in the reaction models is consistent with the experimental data in the low-temperature range. However, in the high-temperature range, the temperature-dependence is opposite to the one we obtained in our calculations. This leads to a discrepancy of as high as 3.6 times between the rate constant we obtained and the rate constant included in Glarborg's model at a temperature of 3000 K. A large discrepancy is also observed between our rate constant and the one included in Zhang's model since at temperatures in the range 800–1300 K, our rate constant is approximately one third that of Zhang's model. The rate constant included in the model of Glarborg et al. corresponds to the rate constant measured by Bemand et al.⁸⁷ over the temperature range 298–1055 K. The rate constant included in Nakamura's model comes from the rate constant review of Tsang and Herron.⁸⁴ The rate constant recommended by Tsang and Herron is based on experimental data obtained at low temperatures and it is stated that the temperature dependence of this rate constant must be very low given the large branching ratio between the exchange channel, i.e. NO₂ + O = NO + O₂, and the addition channel, i.e. NO₂ + O = NO₃. As seen from the Atkinson et al. review⁷⁹ and the kinetics database of NIST, the rate constant of the addition channel has been studied only at temperatures below 400 K. An important fall-off behavior was emphasized at atmospheric pressure. At 300 K, the high-pressure limit of the rate constant of the addition channel is approximately twice that of the exchange channel. The rate constant included in the model of Zhang et al. is the rate constant recommend by Tsang and Herron multiplied by 1.5. From this comparison and the examination of the sources of the rate constants included in the models, we conclude that these rate constants are not chemically consistent for the modeling of combustion chemical kinetics.

Fig. 6 Comparison of the rate constants for the reaction NO₂ + O = NO + O₂ included in the three selected reaction models with the one derived from combining low-temperature experimental data from the literature²¹ and new high-temperature quantum calculations.

3.2 Nitrogen dioxide pyrolysis

Fig. 7 compares the total absorption profiles during NO₂ pyrolysis predicted by the three reaction models we have selected. The experimental profiles are shown as symbols. It is noted that (i) the results shown in Fig. 7 were obtained without any modification of the reaction models; and (ii) the experimental signals were post-treated using a Savitzky–Golay filter to remove some noise.⁵⁵ At low temperatures, significant discrepancies are observed between the predictions of the models. For example, at 1492 K, the model of Glarborg predicts a total absorption that is twice larger than that of Zhang's model. At high temperatures, the predictions of the models converge although the shape of the profiles can be somehow different during the first part of the experiment.

Fig. 7 Experimental and calculated total absorption profiles during NO₂–Ar mixture pyrolysis.

Fig. 8 illustrates the effect of temperature on the total absorption during NO₂ pyrolysis. The effect of the update of the rate constant of the reaction NO₂ + O = NO + O₂ on the predictions of the model of Nakamura is also shown. As temperature increases, faster and faster increase of the total absorption is observed. At the highest temperature, the total absorption rapidly reaches a plateau. Above 2000 K, the total absorption profiles are similar to the one shown in Fig. 8 for T = 2019 K. Overall, the original model of Nakamura under-estimates the rate of increase as well as the long term value of the absorption. On the contrary, the updated model of Nakamura predicts these two features better. At the highest temperature, the effect of the reaction rate update becomes rather weak and the updated reaction model over-estimates the absorption value within the plateauing period. This is due to the lower importance of the reaction NO₂ + O = NO + O₂ in the high-temperature range. A similar behavior was observed for the model of Zhang whereas for the Glarborg model, the reaction rate constant update has an opposite effect, i.e. it induces a slightly lower rate of increase at early times and lower plateauing value at long times.

Fig. 8 Effect of temperature on the total absorption during NO₂–Ar mixture pyrolysis. Solid lines: Nakamura model. Dashed lines: updated Nakamura model. Conditions: X_NO2 = 20 ppm.

Fig. 9 illustrates the effect of pressure on the total absorption during NO₂ pyrolysis. The effect of the update of the rate constant of the reaction NO₂ + O = NO + O₂ on the predictions of the model of Zhang is also shown. Similar to the temperature increase, the pressure increase induces a higher rate of total absorption increase at early times and a higher long term plateau. The lower consumption rate of oxygen atom induced by the rate constant update tends to worsen the prediction of the experimental total absorption using the model of Zhang. A similar behavior was observed for the model of Nakamura whereas slightly better predictions were induced by the update in the model of Glarborg.

Fig. 9 Effect of pressure on the total absorption during NO₂–Ar mixture pyrolysis. Solid lines: Zhang model. Dashed lines: updated Zhang model. Conditions: X_NO2 = 5 ppm.

During NO₂ pyrolysis, the two most sensitive reactions are R₁: NO₂(+M) = NO + O(+M) and R₂: NO₂ + O = NO + O₂. Such a behavior is typically observed in ARAS experiments. This is because the reactant decomposition reaction is largely controlling the rate of production of the transient species which are measured.⁸⁸Fig. 10 shows the effect on the predictions of the updated Glarborg's model of multiplying and dividing the rate constants of R₁ and R₂ by two. It is seen that both reactions have a strong influence on the predicted total absorption profiles. Reaction (1) controls both the early time rate of increase and the long term value of total absorption. Reaction (2) has more impact on the final plateau value of the total absorption. The impact of the reaction rate constants change is stronger at low temperature, see the ESI.† The fact that both reactions (1) and (2) have strong impact on the absorption profile prevented us from extracting experimentally the rate constant of R₂. Nevertheless, our high-accuracy quantum calculation enabled us to propose a consistent rate constant and, overall, improved the predictive capability of the reaction models we have selected under NO₂ pyrolysis conditions. It is noted that the rate constants used in the three reaction models for the thermal decomposition of NO₂ are not the same. In both Glarborg's and Zhang's models, the reaction is declared in the backward direction, i.e. NO + O + M = NO₂ + M, and has a Troe formalism. In the Glarborg's model, the rate parameters for N₂ as the third body come from the review of Tsang and Herron⁸⁴ but a specific expression for the low-pressure limit is employed for Ar as the third body.⁸⁹ In Zhang's model, the expression recommended by Tsang and Herron is also employed with a collision efficiency for Ar equal to 0.6. The rate constant included in the model of Nakamura is for the forward direction. It is given as a second-order rate constant, and seems to be consistent with the values of Troe¹⁴ and of Hiraoka and Hardwich.¹⁵ Comparing the low-pressure rate constants in the temperature range 1500–2500 K, the values in Glarborg's and in Zhang's models agree within 9%. As for Nakamura's model, the rate constant for NO₂ decomposition is twice lower than the one in Glarborg's model at 1500 K. This discrepancy reduces to 8% at 2500 K.

Fig. 10 Effect of k₁ and k₂ on the total absorption during NO₂–Ar mixture pyrolysis. The updated Glarborg model. k₁ corresponds to the rate constant of NO₂(+M) = NO + O(+M). k₂ corresponds to the rate constant of NO₂ + O = NO + O₂. Conditions: T = 1511 K; P = 330 kPa; X_NO2 = 40.1 ppm.

3.3 Hydrogen oxidation by nitrogen dioxide

We selected experimental shock tube data for H₂–NO₂–Ar mixtures from the literature^63,64 to determine the impact of the update of the rate constant of the reaction NO₂ + O = NO + O₂ under oxidative conditions. Fig. 11 shows the delay-time predicted by the original and updated models of Zhang. The rate constant update induces a decreased competition for oxygen atom consumption between NO₂ + O = NO + O₂ and the chain branching reaction O + H₂ = OH + H. This leads to an increase of the overall reactivity but does not modify significantly the global activation energy of the oxidation process. It is noted that the rate constants used for the O + H₂ = OH + H reaction in the three models are significantly different at temperature below 1500 K but are within 25% at temperatures above 1500 K. Glarborg's model includes the rate constant recommended in the review of Baulch et al.⁹⁰ whereas both Nakamura's and Zhang's models include the rate constant measured by Sutherland et al.⁹¹ These rate constants are in overall agreement with other experimental⁴⁸ and optimized⁹² values in the high-temperature range, i.e. T > 1500 K. However, at lower temperatures, significant discrepancies can be observed. Table 2 summarizes the impact of the rate constant update on the predicted delay-times and normalized OH* peak heights. The predictions of the model of Glarborg are the least modified whereas those of the models of Nakamura and Zhang are the most impacted. Overall, the rate constant update tends to improve the predictions of all the reaction models.

Fig. 11 Effect of the rate constant update on the time to half decrease of OH* signal in H₂–NO₂–Ar mixtures. Conditions: X_H2 = 0.222%; X_NO2 = 0.392%; P = 105–124 kPa. Experimental data taken from ref. 63.

Table 2 Effect of the rate constant update on the mean error (%) in the models prediction with respect to the experimental data of Mulvihill et al.^63,64

	Delay OH*	Peak height OH*	Delay H2O
Glarborg	25.8	42.7	24.8
Updated Glarborg	24.4	41.7	23.1
Nakamura	72.5	43.3	64.6
Updated Nakamura	60.2	40.2	58.9
Zhang	28.9	37.6	26.8
Updated Zhang	18.5	32.2	19.5

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Fig. 12 shows the influence of the rate constant update on the H₂O and OH* temporal profiles predicted by the models of Zhang and of Nakamura, respectively. This figure gives an overview of the models prediction before and after the update of the rate constant for the reaction NO₂ + O = NO + O₂. In both cases, the overall shape of the profiles is not modified but the reactivity is slightly increased. Again, this change of reactivity is attributed to the decreased competition between NO₂ + O = NO + O₂ and O + H₂ = OH + H.

Fig. 12 Effect of the rate constant update on the H₂O and OH* profiles in H₂–NO₂–Ar mixtures. Conditions for the top plot: X_H2 = 0.222%; X_NO2 = 0.375%; T = 1441 K; P = 112 kPa. Conditions for the bottom plot: X_H2 = 0.222%; X_NO2 = 0.392%; T = 1612 K; P = 123 kPa. Experimental data taken from ref. 63 and 64.

In addition, we have examined the impact of the rate constant update on the sensitivity coefficients on H₂O and OH* concentration. The sensitivity coefficient for the i^th reaction was defined as C_i = ∂ ln[j]/∂ ln k_i, where [j] is the concentration of the j^th chemical species, and k_i is the rate constant of the i^th reaction. For this analysis, we have employed the model of Zhang whose predictions are the most affected by the update. These results are shown in Fig. 13. It is noted that only the reactions which demonstrate the highest and lowest sensitivity coefficients are shown in Fig. 13. For the sensitivity coefficients of H₂O concentration, the values are only slightly affected by the update and no change of sign was observed. The reaction NO₂ + O = NO + O₂ is not within the 10 most sensitive reactions anymore after the rate constant update is applied. Before the update is applied, the coefficient for this reaction is weakly negative. For OH* concentration, the sensitivity coefficients are not significantly modified by the rate constant update. The reaction NO₂ + O = NO + O₂ demonstrates an important negative coefficient both before and after the reaction rate constant update.

Fig. 13 Effect of the rate constant update on the normalized sensitivity coefficients on H₂O (top) and OH* (bottom) in H₂–NO₂–Ar mixtures. Blue: Zhang model. Red: updated Zhang model. Based on experimental conditions taken from ref. 63 and 64.

It is noted that we also examined the effect of the reaction rate constant update on the ignition delay-time obtained by Mathieu et al.³ for H₂–O₂–NO₂–Ar mixtures and on the species profiles obtained by Mueller et al.⁹³ in a flow reactor for H₂–NO₂–O₂–N₂ mixtures. No significant changes were observed in the predictions of the three reaction models we have selected.

3.4 Hydrogen oxidation by nitrogen oxide and dioxygen

We have also verify the impact of the update of the rate constant of the reaction NO₂ + O = NO + O₂ on the modeling of species profile obtained in a jet-stirred reactor by Dagaut and Dayma⁶⁵ for H₂–NO–O₂–N₂ mixtures. Fig. 14 compares the predicted H₂ and H₂O profiles of the original and updated models of Zhang. Overall, the simulated profiles were not much modified by the rate constant update although an increased reactivity was observed under some conditions. It is noted that the modifications induced for the models of Glarborg and Nakamura were even smaller than those for the model of Zhang. At 800 K, which is the temperature at which half of the H₂ is consumed, the net rate of consumption of H₂ and the net production rate of H₂O are increased by 11%. It is due to the more than twice lower rate constant for the reaction NO₂ + O = NO + O₂ in the updated Zhang's model. This induces a reduced competition between NO₂ + O = NO + O₂ (rate of production is decreased by 53%) and H₂ + O = H + OH which is the second contributor to H₂ consumption. In addition, the OH production rate is increased and the hydroxyl radical preferentially reacts with H₂, H₂ + OH = H + H₂O, which is the main reaction for H₂ consumption and H₂O production.

Fig. 14 Effect of model update on the H₂ and H₂O profiles in a H₂–NO–O₂–N₂ mixture in a jet-stirred reactor. Conditions: X_H2 = 0.01; X_O2 = 0.05; X_NO2 = 245 ppm; N₂: balance; residence time is 0.24 s; P = 101 kPa. Solid lines: Zhang's model. Dashed lines: updated Zhang model. Experimental data taken from ref. 65.

4 Conclusions

In the present study, low-temperature experimental data from the literature and new high-temperature quantum calculations were combined to obtain a chemically consistent rate constant in the temperature range 221–3000 K for the reaction NO₂ + O = NO + O₂. The rate constant demonstrates a negative temperature-dependence in the low-temperature range and a positive dependence in the high-temperature range. This is in contrast to the rate constants used in many detailed reaction models in which only a negative temperature-dependence is included. Updating the rate constant of the reaction NO₂ + O = NO + O₂ in three recent reaction models enables the improvement of their predictive capabilities of the total absorption at 130.5 nm during NO₂ pyrolysis. In addition, the predictions for other experimental data obtained for H₂–NO₂–Ar mixtures were also improved for the three models. In particular, in the low-temperature range of the shock tube data, the predicted ignition delay-times obtained for the updated models were decreased due to the reduced competition between NO₂ + O = NO + O₂ and H₂ + O = OH + H. In the future, the impact of this rate constant update on the prediction of various kinetics targets during the oxidation of hydrocarbon by NO₂ should be investigated. We recommend our updated rate constant to be included in existing and future detailed reaction models in order to improve their chemical consistency.

Conflicts of interest

The authors have no conflicts to declare.

Acknowledgements

This work was partly supported by the National Natural Science Foundation of China (91841301 and 21973053). The numerical part of the present study was performed at Tsinghua University whereas the experimental part was performed at ICARE-CNRS Orléans. The authors are grateful to Pr Guillaume Dayma for his help with the FTIR experiments. The authors also thank Dr Clayton Mulvihill for providing his shock tube data.

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Footnote

† Electronic supplementary information (ESI) available: Tables containing the calculated and experimental rates used to obtain the rate constant in the Arrhenius form. Additional modeling results. See DOI: 10.1039/d0cp05131d

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