pubs.acs.org/JPCA
Article
Rovibrational-Specific QCT and Master Equation Study on N2(X1Σ+g)
+ O(3P) and NO(X2Π) + N(4S) Systems in High-Energy Collisions
Sung Min Jo, Simone Venturi, Maitreyee P. Sharma, Alessandro Munafo,̀ and Marco Panesi*
Cite This: J. Phys. Chem. A 2022, 126, 3273−3290
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ABSTRACT: This work presents a detailed investigation of the energy-transfer and dissociation mechanisms in N2(X1Σ+g ) + O(3P)
and NO(X2Π) + N(4S) systems using rovibrational-specific quasiclassical trajectory (QCT) and master equation analyses. The
complete set of state-to-state kinetic data, obtained via QCT, allows for an in-depth investigation of the Zel’dovich mechanism
leading to the formation of NO molecules at microscopic and macroscopic scales. The master equation analysis demonstrates that
the low-lying vibrational states of N2 and NO have dominant contributions to the NO formation and the corresponding extinction of
N2 through the exchange process. For the considered temperature range, it is found that nearly 50% of the dissociation processes for
N2 and NO molecules occur in the quasi-steady-state (QSS) regime, while for the Zel’dovich reaction, the distribution of the
reactants does not reach the QSS conditions. Furthermore, using the QSS approximation to model the Zel’dovich mechanism leads
to overestimating NO production by more than a factor of 4 in the high-temperature range. The breakdown of this well-known
approximation has profound consequences for the approaches that heavily rely on the validity of QSS assumption in hypersonic
applications. Finally, the investigation of the rovibrational state population dynamics reveals substantial similarities among different
chemical systems for the energy-transfer and the dissociation processes, providing promising physical foundations for the use of
reduced-order strategies in other chemical systems without significant loss of accuracy.
1. INTRODUCTION
In hypersonic flows, the formation and extinction of nitric
oxide (NO) are important since this chemical species emits
strong ultraviolet radiation inside shock layers. One of the
most well-known reaction pathways contributing to NO
formation in high-temperature air is the following heterogeneous exchange process (i.e., Zel’dovich reaction)
N2 + O → NO + N
In addition to the Zel’dovich mechanism (1), collisional
energy-transfer and dissociation processes of individual
collision pairs in the N2O system (i.e., N2 + O and NO +
N) are also relevant in developing a reliable thermochemical
nonequilibrium model for high-temperature air. Regarding the
N2O system, previous theoretical studies2−11 have been carried
out by means of ab initio potential energy surface (PES)
constructions followed by quasiclassical trajectory (QCT)
calculations for particular chemical reaction channels. Each of
the PESs developed by Bose and Candler2 and Gamallo et al.3
(1)
Due to the fast dissociation of diatomic oxygen (O2), collisions
between molecular nitrogen (N2) and atomic oxygen (O)
control NO formation through reaction 1. The existing
studies1,2 found that the above reaction predominantly occurs
through the lowest triplet surfaces 3A′ and 3A″ in the
hypersonic flow regime, provided that the nonadiabatic
transitions and spin−orbit coupling are neglected. This implies
that the colliding species (e.g., N2 and O) remain in their
electronic ground states after the collision.
© 2022 American Chemical Society
Received: December 6, 2021
Revised: April 26, 2022
Published: May 23, 2022
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Article
O(3P) and NO(X2Π) + N(4S) systems and exclude all other
collisional transition pathways, such as N2(X1Σ+g ) + N(4S),
N2(X1Σ+g ) + N2(X1Σ+g ), and NO(X2Π) + NO(X2Π).
This paper is organized as follows: the related PESs, the
constructed StS kinetic database, the system of master
equations, and the assessment strategy of order reduction
technique for the Zel’dovich mechanism are discussed in
Section 2. In Section 3, the results of the investigation are
presented and it is divided into four subsections: Section 3.1
investigates the internal energy-transfer processes, and Section
3.2 focuses on the analysis of the dissociation−recombination
dynamics. Section 3.3 provides a detailed investigation on the
mechanism of the Zel’dovich reaction. In Section 3.4, the
present macroscopic rate coefficients are compared with the
existing data from the literature. Finally, Section 4 provides the
conclusions from the present work.
was employed to compute thermal rate coefficients for the
Zel’dovich reaction (1). Luo et al.4 computed the same
quantity along with the thermal dissociation rate coefficient in
N2+O collisions by relying on the PESs from Gamallo et al.3
Denis-Alpizar et al.5 proposed global PESs by representing
their ab initio points with a reproducing kernel Hilbert space
(RKHS) scheme. Based on those PESs,5 Koner et al.6
developed a QCT-based state-to-state model for the reverse
process of eq 1, followed by neural network-based coarse-grain
models proposed by Arnold et al.7,8 Koner et al.9 also
improved the accuracy of the PESs of Denis-Alpizar et al.5
using more number of ab initio points having better accuracy.
Lin et al.10,11 constructed global PESs that cover a wide range
of reactive configurations, and they numerically investigated
the reactive trajectories for the Zel’dovich mechanism (1). The
aforementioned studies2,5,9−11 focused on specific portions of
the reaction dynamics that can occur on the N2O surfaces,
instead of the overall energy-transfer and dissociation
mechanisms. In addition, to the authors’ best knowledge, no
detailed state-to-state (StS) master equation analysis has been
carried out for the N2O system. This inhibits a deeper
understanding of the nonequilibrium chemical kinetics in hightemperature air.
By means of rovibrational StS master equation analyses, it is
possible to investigate detailed chemical-kinetic processes in
terms of rovibrational state population dynamics, which
provide a crucial component to the development of reducedorder models. In particular, the study of rovibrational
distributions from the N 2 (X 1 Σ g+ ) + N( 4 S) 12−14 and
O2(X3Σ−g ) + O(3P)15 systems has inspired various coarsegraining strategies15−18 for model reduction in nonequilibrium
chemistry. Although those approaches were developed based
on strong physical foundations and refused ad hoc
assumptions, they have been only tested for N2(X1Σ+g ) +
N(4S),16−18 N2(X1Σ+g ) + N2(X1Σ+g ),19,20 O2(X3Σ−g ) + O(3P),15
CO(1Σ+) + O(3P),21 and CO2(X1Σ+g ) + M22 (M denotes inert
species) chemical systems. This aspect motivates a systematic
comparative investigation of the similarities among different
chemical systems and the general applicability of the coarsegraining strategies, with the ultimate goal of constructing a
comprehensive reduced-order approach for high-temperature
air.
Toward this end, the present study proposes a detailed
investigation of the rovibrational energy-transfer and dissociation processes in the complete N2O molecular system by
means of QCT and master equation analyses. A complete set
of rovibrational-specific rate coefficients, including the
inelastic, dissociation, and homogeneous and heterogeneous
exchange processes, are calculated in a wide range of kinetic
temperatures. Then, they are employed to integrate a set of
master equations for an ideal chemical reactor problem.
Macroscopic quantities, such as the quasi-steady-state (QSS)
reaction rate coefficient and the internal energy relaxation time,
are evaluated using the rovibrational population distributions
to compare with existing data. The similarity of internal energy
transfer and dissociation among different chemical systems is
investigated by comparing the rovibrational population
dynamics. The present master equation analysis also aims to
understand the mechanism of the Zel’dovich reaction (1) on
the microscopic scale and to assess the validity of the QSS
approximation in modeling that chemical reaction. To achieve
the aforementioned purposes, we pinpoint our investigation on
the reaction channels existing in and between the N2(X1Σ+g ) +
2. PHYSICAL MODELING
2.1. Potential Energy Surfaces for N2O. One of the
earliest kinetic studies for the Zel’dovich reaction was carried
out by Bose and Candler2 using the ab initio data from Walch
et al.1 and Gilibert et al.,23 which were computed from a low
level of theory, complete active space self-consistent field/
contracted configuration interaction (CASSCF/CCI) method.
These ab initio points1,23 were only computed in selected
portions of the surface. Gamallo et al.3 developed the PESs
using second-order perturbation theory on a complete active
space self-consistent-field wave function, CASPT2 method. As
a downside, the CASPT2 method makes the PESs less reliable
for studying dissociation reactions. Furthermore, it was found
in later studies5,9,10 that the 3A′ surface has additional
transition states and local minima that were not accounted
for in the PES by Gamallo et al.3 Accurate consideration of the
stationary points is important since they directly affect the
reaction pathways of the system being studied. The PESs
developed by Lin et al.,10 Denis-Alpizar et al.,5 and Koner et
al.9 use ab initio points computed at the multireference
configuration interaction (MRCI) and MRCI with Davidson’s
correction (MRCI + Q) level of theory, thereby making them
more suitable to study the high-energy collisions of the N2O
system. Lin et al.10 used MRCI energies, which were improved
using the dynamically scaled external correlation method, and
the multibody component of the PES was then fitted using
permutationally invariant polynomials in mixed exponentialGaussian bond order variables. Denis-Alpizar et al.5 and Koner
et al.9 employed the MRCI + Q energies fitted using the RKHS
scheme.
For the purpose of the present study, the PESs developed by
Lin et al.10 are used. The principal reason for using these PESs
is that the range of geometries used in constructing the
analytical surface is adequate to accurately study both energy
transfer and dissociation, except for some regions in the
exothermic exchange channel, NO + N → N2 + O. The PESs
are particularly favorable for representing the highly
endothermic forward exchange reaction (1), for which the
two reaction pathways on the 3A′ surface and the one on the
3
A″ surface are characterized by accurately described stationary
points. The reverse reaction, NO + N → N2 + O, is barrierless
on the 3A″ surface and has a barrier (≈10.5 kcal/mol) on the
3
A′ surface, making it an exothermic reaction. It is worth
mentioning that the PESs by Lin et al.10 are limited in their
accuracy in studying such an exothermic reverse exchange
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reaction, NO + N → N2 + O, since they do not have a
complete representation of the weak long-range interactions in
the NO + N channel. Albeit, as will be shown later in Section
3.4, in the temperature range of interest to this work, these
PESs10 are able to provide a good representation of the energytransfer and dissociation−recombination reactions.
2.2. State-to-State Kinetic Database. In this work, the
following collisional energy-transfer and dissociation−recombination processes are considered by assuming that all
components remain in their electronic ground state:
• Rovibrational excitation and de-excitation through
inelastic and homogeneous exchange
converged and appropriate to study the kinetic processes
relevant to the N2O system.
As discussed in Section 2.1, the 3A′ and 3A″ surfaces are
relevant to the transition between N2(X1Σ+g ) + O(3P) and
NO(X2Π) + N(4S). Transitions through other surfaces, such as
second 3A″, 5A′, and 5A″, result in electronically excited
products, which are out of the present scope. In the
construction of the present StS kinetic database, 1/3
degeneracy is considered for the transition from N2 + O
collisions through each of the two triplet PESs, whereas 3/16
degeneracy is applied for the NO + N interaction to consider
the statistical weight of the electronic state.9
Figure 1 shows the distributions of the energy-transfer rate
coefficients in N2 + O and NO + N systems at 10 000 K. The
kiNO
→j
NO(i) + N Xooo
Y NO(j) + N
NO
kj→i
Article
(2)
2
kmN→
l
N2(m) + O Xoooo
Y N2(l) + O
N
kl →2m
(3)
• Rovibrational dissociation and recombination
kiNO
→c
NO(i) + N Xooo
YN+O+N
NO
kc → i
(4)
2
kmN→
c
N2(m) + O Xoooo
YN+N+O
N
kc →2 m
(5)
• Rovibrational energy transfer through the Zel’dovich
mechanism
kiE→,NO
m
NO(i) + N Xooooo
Y N2(m) + O
E ,N
km → i2
(6)
In eqs 2−6, the k symbol stands for the rate coefficient for the
considered process. The index pairs (i, j) and (m, l) denote the
rovibrational states of NO and N2, respectively. These internal
levels are stored by increasing energy in the sets 0NO and 0N2 .
To each rovibrational state, there corresponds a unique
vibrational (v) and rotational (J) quantum number (e.g., i =
i(v, J)). The subscript c (i.e., continuum) refers to the
dissociated state. Rovibrational state data (e.g., energy levels)
are obtained by solving Schrödinger’s equation based on the
WKB semiclassical approximation.24,25 As a result of these
calculations, N2 has 9093 levels with vmax = 53, whereas NO
has 6739 levels with vmax = 45, including both bound and
quasibound states.
To compute the rovibrational-specific StS rate coefficients,
QCT calculations are carried out using COARSEAIR,15,26 an inhouse QCT code that is a modernized version of the original
VVTC code developed at NASA Ames Research Center by
Schwenke.25 The StS kinetic database constructed here covers
the temperature range of 2500−20000 K. For a particular
initial collision pair, 50 000 trajectories are simulated for the T
up to 5000 K, whereas 30 000 trajectories are employed for the
higher T. For a given trajectory, the maximum iteration time
was set to 4 × 105 au with an initial time step of 5 au. The
impact parameter was determined based on the stratified
sampling with a minimum batch size of 0.25 Å. The level of
statistical convergence according to the variation of the
number of trajectories is presented in Figures S1 and S2 for
selected inelastic transitions at 5000 K. Those figures verify
that the present QCT calculations are statistically well
Figure 1. Distributions of the energy-transfer rate coefficients in N2 +
O (top) and NO + N (bottom) at T= 10 000 K overlayed on the
effective diatomic potentials. The gray dots indicate that the rate
coefficients corresponding to those final rovibrational levels are below
a cutoff value. The black dots represent the initial states.
rate coefficients are overlayed on the effective diatomic
potentials. For N2 + O, the inelastic component is shown,
whereas, for NO + N, the summation of inelastic and
homogeneous exchange rate coefficients is presented. For both
systems, the energy transfer occurs with the highest probability
to the rovibrational states right next to the initial one (i.e., the
black dots). In the low-lying states, most of the transitions
occur within the same vibrational strands, whereas the
transition pathways are spread across different vibrational
levels in the high-lying states. It is interesting to note that the
transition patterns for the energy transfer in Figure 1 are
significantly similar to each other, although they refer to
different chemical systems.
2.3. Master Equations. The StS kinetic database
constructed and discussed in Section 2.2 has been used to
perform a rovibrational master equation analysis of N2(X1Σ+g )
+ O(3P) and NO(X2Π) + N(4S) interactions. The system of
equations used for this purpose describes the evolution of the
various chemical species as a result of inelastic, exchange, and
dissociation processes and reads
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N2
dni
= ωi̇ I −
dt
∑ ωi̇ E,m,NO − ωi̇ D
dnm
= ωṁ I +
dt
coarse-graining the dissociation pathways, and it is employed
here to investigate the accuracy of the existing reduced-order
models15−17,19,30 in describing the Zel’dovich mechanism.
For the grouped internal states, the following collisional
heterogeneous exchange process is analyzed
(7)
m
NO
∑ ωi̇ E,m,NO − ωṁ D
(8)
i
NO
NO(p) + N ↔ N2(q) + O,
dnN
=
dt
∑ ωi̇ D + 2 ∑ ωṁ D − ∑ ∑ ωi̇ E,m,NO
i
i
NO
N2
i
m
where p and q denote the grouped states of NO and N2,
respectively. For the p-th and q-th groups/bins, the equation
governing the time rate of change of the grouped population is
obtained by summing the master equations over the states
within each group (i.e., zeroth-order moment)
(9)
m
NO N2
i
(10)
m
where nX denotes the number density of the X species/level
and t identifies time. The mass production rates for excitation,
ω̇ I, Zel’dovich reaction, ω̇ E, and dissociation, ω̇ D, are evaluated
based on the zeroth-order reaction rate theory.27,28 Their
expressions are provided in Appendix A.
The master equations (eqs 7−10) are numerically integrated
using PLATO (PLAsmas in Thermodynamic nonequilibrium),29
a library for nonequilibrium plasmas developed within The
Center for Hypersonics and Entry System Studies (CHESS) at
the University of Illinois at Urbana-Champaign. In solving the
master equations using the kinetic database in Section 2.2, the
exothermic rate coefficients are used as a reference for the
inelastic and homogeneous exchange processes, while the
endothermic ones are reconstructed based on the microreversibility. For the Zel’dovich mechanism, however, the
endothermic rate coefficient (i.e., for N2 + O → NO + N) is
employed, while the reverse rate coefficients are reconstructed.
This is due to the fact that the reverse rate coefficients have a
higher level of uncertainty (see Figure 15b for more details).
The global rate coefficients can be computed over the time
domain from the microscopic state-specific rate coefficients
and the rovibrational population distributions. The forward
and backward global heterogeneous exchange rate coefficients
corresponding to eq 6 are defined as
1
E ,NO
=
k̃
nNO
N2 NO
∑∑
m
dnp
dt
,NO
E ,N2
= −kpE→
q npnN + kq → p nqnO
,NO
kpE→
q =
∑∑
i
∑ ∑
m ∈ 0 qN2
kqE→,Np2 =
p ,NO
kiE→,NO
m -i
i ∈ 0NO
p
∑ ∑
m ∈ 0 qN2
(15)
,N2 q ,N2
kmE→
i- m
i ∈ 0NO
p
(16)
2
where - ip,NO and - q,N
m are, respectively, the prescribed internal
distribution functions within groups p and q. Here, for
simplicity, they are taken as thermal equilibrium (i.e.,
Maxwell−Boltzmann) distributions at the local translational
temperature T
( ) =n,
n
g exp(− )
e
gi exp − k Ti
- ip ,NO =
i
B
∑i ∈ 0NO
p
ei
kBT
i
i ∈ 0NO
p
p
( ) =n ,
n
g exp(− )
(17)
e
nikiE→,NO
m ,
m ∈ 0N2,
i ∈ 0NO
-
q ,N2
m
gm exp − k mT
m
B
=
∑m ∈ 0 N2
i
NO N2
(14)
The grouped Zel’dovich reaction rates kp → qE, NO and kq → pE,N2
are defined as
q
(11)
k̃
q ∈ 0N2
(13)
∑ ωi̇ D + ∑ ∑ ωi̇ E,m,NO
1
=
n N2
p ∈ 0NO,
NO N2
dnO
=
dt
E ,N2
Article
m
em
kBT
m ∈ 0 qN2
q
(18)
,N2
nmkmE→
i,
m ∈ 0N2,
i ∈ 0NO
where em and gm denote the internal energy and degeneracy
of m-th state, and kB stands for the Boltzmann constant. The
form of gm can be found in eq 35 in Appendix A. The
multiplication of both sides of eq 14 by - ip,NO leads to
m
(12)
At equilibrium, eqs 11 and 12 become the thermal
heterogeneous exchange rate coefficients as the internal states
follow a Maxwell−Boltzmann distribution. The global
dissociation rate coefficients of NO and N2 are defined
following the previous work by Panesi et al.12
2.4. Assessment of Reduced-Order Modeling for the
Zel’dovich Mechanism. A previous study by Venturi et al.15
proposed an encoding−decoding strategy for assessing the
accuracy of reduced-order models in representing a particular
channel of the rovibrational-specific kinetics. The approach
was based on comparing the results of fully StS simulations
with those obtained from master equation studies in which the
state-specific rate coefficients for the channel of interest were
first grouped and then reconstructed, while all of the
coefficients for the remaining processes were kept at the StS
level. The strategy was there adopted to analyze the effects of
dni
,NO
E ,N2
p ,NO
= −kpE→
,
q ninN + kq → p nqnO - i
dt
i ∈ 0NO
p
(19)
In eq 19, np is replaced by ni by invoking the local equilibrium
relation (17). Similarly, the group number density nq can be
transformed to the rovibrational-specific one, nm, via multi2
plication by - q,N
m , which gives
- qm,N2
dni
,NO q ,N2
E ,N2 p ,NO
= −kpE→
nmnO ,
q - m ninN + kq → p - i
dt
i ∈ 0NO
p ,
m ∈ 0 qN2
(20)
From eq 20, the grouped-reconstructed Zel’dovich reaction
E,N2
rate coefficients k̅E,NO
i→m and k̅m→i can be defined as follows
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ki̅ → m
E ,NO
,NO q ,N2
=kpE→
q -m ,
E ,N
=kqE→,Np2 - ip ,NO,
km̅ → i2
,
i ∈ 0NO
p
m ∈ 0 qN2
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(21)
This means that the grouped-reconstructed rate coefficient
can be obtained from the respective grouped rate coefficient by
weighting it based on the final state’s contribution to its group
partition function.
2
By spanning eq 20 over m for fixed i, the - q,N
in the leftm
hand side goes to that based on the local equilibrium relation
(18). Finally, the time rate of change of rovibrational-specific
population, dni/dt, can be denoted using the groupedreconstructed Zel’dovich reaction rate coefficients as follows
dni
E ,NO
E ,N
= − ki̅ → m ninN + km̅ → i2nmnO ,
dt
i ∈ 0NO
p ,
m ∈ 0 qN2
(22)
Figure 2. Temporal evolution of rotational and vibrational temperatures of N2 and NO at different heat-bath temperatures (dissociation
and Zel’dovich mechanisms excluded).
Equation 22 allows us to investigate the performance and
effectiveness of reduced-order methods in reproducing the
heterogeneous exchange dynamics, as the remaining excitation
and dissociation channels are treated rovibrational-specifically.
The corresponding results are presented and discussed in
Section 3.3.
increases, the vibrational and rotational relaxation profiles for
the two molecules get closer to each other.
For in-depth interpretation of the similar structure of the
internal energy transfer among the different chemical systems,
observed in Figure 2, the rovibrational distributions for the
three chemical systems, N2 + N, N2 + O, and NO + N, are
compared against each other in Figure 3. The comparison is
made at TV = 2240 K, in which 20% of vibrational relaxation is
completed. It is worth mentioning that the rovibrational rate
coefficients for N2+N are taken from the previous work by
3. RESULTS
In this section, master equation simulations, conducted for a
wide range of initial and heat-bath conditions, are analyzed and
discussed in detail. Macroscopic quantities such as global rate
coefficients, relaxation times, and chemistry-energy coupling
parameters are calculated from the zeroth- and first-order
moments of the rovibrational population distributions. Also,
particular attention is devoted to investigating similarities in
the population dynamics among different chemical systems
and the effect of the Zel’dovich reaction on internal energy
transfer at both state-specific and macroscopic levels.
In the analysis to be presented, the chemical compositions
are initialized to have 50% of diatomic target species and 50%
of atomic colliding particles unless they are explicitly
mentioned. The initial gas pressure and internal temperature
are set to 1000 Pa and 300 K, respectively. The above initial
conditions are utilized in all simulations unless otherwise
stated. As per the heat-bath temperatures, the values of 2500,
5000, 7500, 10 000, 15 000, and 20 000 K are considered.
3.1. Energy-Transfer Processes in Isolated Systems. In
this section, energy transfer by the inelastic and homogeneous
exchange in N2 + O and NO + N systems is studied in detail
by neglecting dissociation and Zel’dovich mechanisms. As a
result, there is no interaction between the two chemical
systems.
Figure 2 shows the temporal evolution of the extracted
rotational (TR) and vibrational (TV) temperatures of N2 and
NO at the three distinct heat-bath temperatures. The overall
structure of evolution of the internal (i.e., rotational and
vibrational) temperatures is similar for both molecules: at the
lowest temperature, vibrational relaxation is much slower
compared to rotational relaxation, whereas at higher temperatures, the two processes occur at comparable time scales. At
lower temperatures, the vibrational relaxation rates are
significantly different while the rotational relaxation rates are
close for the two molecules. However, as the temperature
Figure 3. Rovibrational distributions in N2 + N (top), N2+O
(middle), and NO + N (bottom) systems at TV = 2240 K, in which
20% of vibrational relaxation completed at T = 10 000 K (dissociation
and Zel’dovich mechanisms excluded). The distributions are colored
by the vibrational quantum number.
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Panesi et al.12 Overall, the rovibrational distributions have
similar structures even though they stem from different
chemical systems. This fact is justified by the similarities
between the energy-transfer rate coefficients, shown in Figure
1. The internal energy transfer of the diatomic species is mostly
governed by the behavior of the low-lying energy states, which
have a strand-like structure during the early stage of energy
transfer.12,17 The inset figures highlight the common existence
of vibration-specific strands in the low-lying energy states and
the similarity among the three chemical systems. This implies
the possibility of applying the existing reduced-order method
for energy transfer by Sahai et al.,17 originally tested on the N2
+ N interactions, to other chemical systems without a
significant loss of accuracy.
One of the prevalent ways of describing energy transfer in
hypersonic environments is via the Landau−Teller model,31
which requires specifying a relaxation time τ for the kinetic
process being considered. The value of τ can affect the
behavior of thermal energy relaxation in the nonequilibrium
region of hypersonic shock layers. Figure 4 compares the
rotational (pτRT) and vibrational (pτVT) relaxation times of the
N2 + O and NO + N systems as a function of the kinetic
temperature. Here, p denotes the partial pressure of the
colliding atomic species. In the present study, the relaxation
time is evaluated based on the e-folding method. To the best of
the authors’ knowledge, there is no available experimental data
for rotational and vibrational relaxation times for both
chemical systems. Thus, the comparison is performed against
existing theoretical models.32−35
For both chemical systems, the correlation-based model by
Millikan and White,32 later modified by Park for hightemperature effects,33 shows a strong departure from the
values computed here. For N2 + O, the temperature
dependence of pτVT from the present result is different from
those of the previous studies,32−34 especially below 5000 K.
For NO + N, the temperature dependence of the present result
is similar to those of existing studies,32−34 although the
absolute value of the present pτVT is larger by a factor of 5. As
per rotational relaxation, the present results are in disagreement with the Parker model35 by around a factor of 3 for the
NO + N system. In the Parker model, the rotational relaxation
time was derived by classical mechanics using the rigid-rotor
model that cannot account for the effect of a permanent dipole
in diatom−atom systems.35,36 Consistent with the temperature
evolution of Figure 2, the rotational and vibrational relaxation
times of both chemical systems converge to a common
asymptote as the heat-bath temperature increases. The
asymptote of the rotational and vibrational relaxation times
at the high-temperature range was also observed in the
previous studies of the N2 + N system.12,37 In those references,
the asymptote was justified by the presence of the
homogeneous exchange reaction. However, it is also observed
here for the N2 + O system, in which a homogeneous exchange
process does not occur. The relaxation times obtained from the
present master equation analyses are curve-fitted and labeled as
present (fit) in the figure. The form and parameters of the
curve-fit are summarized in Appendix B.
3.2. Dissociation/Recombination Processes in Isolated Systems. In this section, dissociation and recombination processes in N2 + O and NO + N systems are investigated
by disregarding the Zel’dovich mechanism. Hence, the two
systems evolve independently as in the previous section.
Article
Figure 4. Comparison of predicted rotational and vibrational
relaxation times with existing theoretical models:32−35 (a) N2 + O
and (b) NO + N (dissociation and Zel’dovich mechanisms excluded).
Figure 5 shows the temporal evolution of the average
rotational (ER) and vibrational (EV) energies of N2 and NO at
three different kinetic temperatures, T. In the present study,
the average rotational and vibrational energies of N2 are
correspondingly defined as
N
E R,N2 =
∑m2 er(m)nm
N
∑m2 nm
(23)
N
E V,N2 =
∑m2 e v(m)nm
N
∑m2 nm
(24)
where er(m) and ev(m) are the rotational and vibrational
energies of the given rovibrational state m, respectively. The
plateau of the energy curves indicates the QSS periods of each
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shown in Figure 2, the energy evolution trends of N2 and NO
are close to each other, while NO represents the faster onset of
the QSS period than N2 due to the lower dissociation limit and
the faster internal energy transfers within NO. Furthermore,
since N2 has a larger bonding energy than NO, the QSS period
of N2 is longer than that of NO.
The comparison among the internal state population
distributions from the different chemical systems, shown in
Figure 3, is re-examined here for the dissociation and
recombination processes. In Figure 6, a comparison of the
internal energy state population distributions is presented for
the N2 + N, N2 + O, and NO + N systems during the QSS
period. The main figures (i.e., those on the left side) are
colored by the vibrational quantum number v, while the
subfigures (i.e., those on the right side) are colored by the
state-specific energy deficit from the centrifugal barrier. In the
present study, the energy deficit of N 2 and NO is
correspondingly defined as15
Figure 5. Average rotational and vibrational energy distributions of
N2 and NO at the three different kinetic temperatures (Zel’dovich
mechanism excluded). The arrows indicate the molecular QSS
periods in cases where the plateau regime is not explicitly shown. The
solid arrow indicates the N2 QSS, while the dashed one denotes the
NO QSS.
N2
emD = V max
(J ) − em
(25)
NO
eiD = Vmax
(J ) − ei
(26)
2
VNmax
(J)
where
and VNO
max(J) denote the maximum values of the
effective diatomic potential at the given rotational state J of N2
and NO, respectively. As observed from the v contours, the
vibrational states, which are in the high-lying energy region
around the dissociation limit, are not in equilibrium with each
other. This indicates that the collisional dissociation processes
are not governed by a vibration-specific dynamics. On the
other hand, the subfigures clearly show that the equilibrium
distributions of the high-lying energy states are highly
correlated with the levels’ energy deficit. This is in line with
the previous study by Venturi et al.15 regarding the O2 + O
species. As discussed in the following paragraphs, most of the
dissociation occurs during the QSS periods. The following
evolution from the plateau to the equilibrium state is governed
by the recombination processes. From the figure, it is noted
that the time scale of NO recombination is faster than that of
N2 over T = 10 000 K. Similar to the energy-transfer processes,
Figure 6. Rovibrational distributions in N2 + N (top), N2 + O (middle), and NO + N (bottom) systems in the midst of the QSS region at T =
10 000 K (Zel’dovich mechanism excluded). The distributions are colored by the vibrational quantum number v and the energy deficit eD.
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was defined by summing up and normalizing the ω̇ D at a given
time step with the total amount of chemical production rate
due to dissociation and recombination. It needs to be noted
that the initial species mole fractions for the heat-bath
simulations are different from each other to investigate the
two different isolated chemical systems. For the dissociation of
N2 by N2 + O collisions, about 70% of the dissociation takes
place during the QSS period of N2, as shown in Figure 7a. On
the other hand, about 50% of NO dissociation occurs within
the QSS period. For NO + N, three different reaction channels
exist for the dissociation due to the possibility of the
homogeneous and heterogeneous exchanges prior to the
dissociation. This aspect degrades the validity of the QSS
assumption for NO dissociation, as shown in Figure 7b. The
contribution of the exchanged pairs to dissociation leads to
about 10% difference in the amount of dissociation taking
place in the QSS period. It needs to be noted that the levels of
influence from the heterogeneous and the homogeneous
exchanged pairs are close to each other. The influence of the
exchange pair on the dissociation of NO shown in Figure 7b is
consistently observed at T = 5000 and 20 000 K as well (see
Figure S3).
In Figure 8, the internal energy loss ratio due to the
dissociation, CD, is presented as a function of T. The definition
system, and it is another evidence of the existence of similarity
among different chemical systems, especially for the collisional
dissociation process.
In multidimensional hypersonic flow-field simulations, one
of the most popular strategies to model the collisional chemical
production rates ω̇ is to employ the macroscopic QSS rate
coefficient by assuming that the participating species are in the
QSS region.30 The rovibrational-specific master equation
analysis performed in the present study allows us to evaluate
the actual amount of chemical reactions taking place during the
QSS periods of the participating species and to verify the QSS
assumption for the chemical reaction modeling. In this section,
it is performed for the collisional dissociation processes in the
isolated chemical systems, while the complete chemical system
including the Zel’dovich mechanism is investigated in Section
3.3.
Figure 7 shows the time-cumulative chemical production
rate distributions for the dissociation and recombination
processes at T = 10 000 K, together with the global
dissociation rate coefficients. The time-cumulative quantity
Figure 8. Rotational and vibrational energy loss ratios due to the
dissociation as a function of kinetic temperatures (Zel’dovich
mechanism excluded).
of CD is taken from the previous study by Panesi et al.,12 and
the quantity is defined at the mid-point of the species QSS
periods. The vibrational energy loss ratios are very similar
between N2 and NO, provided that the NO + N system has
both the inelastic and homogeneous exchange processes. On
the other hand, the NO dissociation has a larger portion of
rotational energy loss than N2, especially in the high T range. A
part of this trend is attributed to the existence of the
homogeneous exchange reaction in the NO + N system. This
fact accelerates the thermalization among the vibrational states,
resulting in more populated high-lying J levels (see Figure S4).
Without the homogeneous exchange, the rotational energy loss
ratio of NO becomes closer to that of N2, whereas the
Figure 7. Time-cumulative chemical production and global
dissociation rate distributions at T = 10 000 K (Zel’dovich mechanism
excluded). The solid lines indicate the chemical production rates,
while the dashed lines represent the global dissociation rate
coefficients. The shaded regions indicate the QSS periods of diatomic
species. (a) N2 + O and (b) NO + N.
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whereas the extinction of NO is mostly controlled by the
dissociation of NO rather than the exchange kinetics into N2.
The mechanism of NO formation is worth to be explored in
detail since NO is a strongly radiating species in a hypersonic
flow regime, and it has an important role in high-temperature
combustion and related air pollution study. Figure 10 provides
the rovibrationally resolved heterogeneous exchange production rates relevant to N2 extinction and NO production. The
reported quantity was computed from the result of the master
equation study at T = 10 000 K and t = 4 × 10−7 s of Figure 9
in the sense that T = 10 000 K is a representative condition of
interest to hypersonic applications. The time t = 4 × 10−7 s was
chosen to analyze the NO formation mechanism. The statespecific N2 extinction and NO formation rates are correspondingly defined by summing up and normalizing eq 31 as follows
vibrational energy loss shifts to lower values with a certain
amount of offset due to the less population of high-lying v
states. As T increases, the rotational and vibrational energy loss
ratios move in the opposite direction. This is because the
contribution from the internal states characterized by lower v
and higher J becomes dominant as T increases. It is interesting
to note that, for both N2 and NO, the vibrational energy loss
ratios CDV are rather different from the previously estimated
value of 0.3 using the SSH theory38 in the present considered
temperature range.
3.3. Zel’dovich Mechanism. In this section, the results of
master equation analyses are presented by considering the
complete set of chemical-kinetic processes (eqs 2−6). This
allows us to investigate the role of the collisional heterogeneous exchange process (eq 6) through the forward and
backward Zel’dovich mechanisms in the internal energy
transfers of the N2 + O and NO + N systems and to
characterize the NO formation and extinction. For this
purpose, the set of master equations from eqs 7 to 10 are
integrated, and none of the individual components are set to
zero in this case. In total, approximately an order of 108 kinetic
processes is considered for solving the set of master equations
in the considered temperature range.
Figure 9 shows the temporal evolution of species mole
fractions χ at the three different kinetic temperatures T. As T
NO
N
Ω̇m2 =
100 × ∑i
N
NO
∑m2 ∑i
ωi̇ E, m,NO
ωi̇ E, m,NO
(27)
N
NO
Ω̇i = −
100 × ∑m2 ωi̇ E, m,NO
N
NO
∑m2 ∑i
ωi̇ E, m,NO
(28)
Figure 10a indicates that the low-lying states of N2 around v
= 0 and J = 65 have major contributions to the conversion into
NO during the NO formation phase. Similarly, the NO
production occurs with the predominant influence from the
low-lying states near v = 0 and J = 45, as shown in Figure 10b.
After the conversion of N2 into the low-lying states of NO, the
inelastic and homogeneous exchange by the NO + N collision
governs the ladder-climbing excitation process to the highlying energy levels.
Figure 11 shows the time-cumulative chemical production
rate distributions by the Zel’dovich mechanism at T = 5000,
10 000, and 20 000 K. It is important to note that the red lines
indicate the limiting boundaries of the variation of chemical
reactions. At a given temperature and pressure, the master
equation results for different initial sets of mole fractions fall
within the cyan region in Figure 11, provided that the mole
fraction ratio of the target molecules and colliding atoms is
one. As shown in the figures, almost all of the Zel’dovich
processes occur outside the molecular QSS periods of the
considered temperature range. This implies that the QSS
assumption is an inadequate approximation for the Zel’dovich
reaction N2 + O ↔ NO + N. In addition, the feasibility of the
QSS assumption further deteriorates as more NO exist at the
initial condition.
Since the majority of the Zel’dovich reaction occurs outside
of the molecular QSS regime, this process requires further
investigations to better understand which processes contribute
to QSS. To this aim, we analyze the contribution of
dissociation to the Zel’dovich mechanism. This is based on
the fact that the previous studies12,37 have revealed that the
dissociation predominantly promotes the formation of QSS.
Figure 12 compares the global rate coefficient for N2 + O →
NO + N and the corresponding rovibrational distributions
with and without dissociation. The definition of k̃E,N2 can be
found in eq 12. As shown in Figure 12a, the global rate
coefficient without the dissociation does not have the plateau
region (i.e., from t = 6 × 10−6 s to t = 4 × 10−4 s for the black
solid line). This means that the system does not reach the QSS
period if the dissociation process is not considered, as
supported by the rovibrational distribution shown in Figure
Figure 9. Species mole fraction distributions at T = 7500 K (dasheddotted lines), 10 000 K (solid lines), and 20 000 K (dashed lines).
The species initial mole fractions are set to χN2 = χO = 0.5.
increases, the level of NO formation increases due to the larger
contribution from the Zel’dovich mechanism. In the early
stages, nearly until the peaks of χNO, the mole fraction profiles
of the diatomic species are aligned together with their atomic
collision partners because the collisional heterogeneous
exchange actively occurs prior to the onset of the molecular
QSS (see Figure 11 for more details). Once most of the
heterogeneous exchange reaction occurs, the mole fractions of
each collision pair are split into two branches. This fact
indicates that the dissociation starts to control the overall
chemical reactions. It is interesting to note that the formation
of NO is governed by the heterogeneous exchange from N2,
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Figure 10. Percentage distributions of the heterogeneous exchange production rates at t = 4 × 10−7 s of T = 10 000 K in Figure 9. The empty gray
dots indicate that the production rate is below or above a cutoff value. (a) N2 extinction rate with a cutoff value above −0.01% and (b) NO
production rate with a cutoff value below 0.02%.
value. Given the fact that the majority of multidimensional
hypersonic CFD codes employ QSS rate coefficients to model
the Zel’dovich reaction, the results in Figures 11−13 imply that
further investigation on the reliability of such an approximation
is required, as it was found to be not valid for the exchange
process.
The accuracy of the QSS-approximated modeling for the
Zel’dovich reaction can be investigated by comparing it with
the fully StS results. For this purpose, the groupedreconstructed method in eq 22 is employed to integrate the
set of master equations with several reduced-order models,
including the QSS approximation for the heterogeneous
exchange process. Figure 14 shows the NO mole fraction
distributions for the fully StS (i.e., rovibrational state-specific)
and the grouped-reconstructed methods for the Zel’dovich
mechanism. As the grouped-reconstructed methods, the
vibration-specific (VS),16 the energy-based group (energy),19
the adaptive coarse-graining (adaptive),17 and a hybrid model
of the adaptive17 and the energy deficit-based15 group methods
are employed together with the QSS model. It should be
mentioned that all of the reduced-order models employed here
12b. At the same time instant, the case without dissociation
already reaches the thermal equilibrium, whereas the rovibrational states are aligned as the QSS distribution in the case
with dissociation. The result in Figure 12 is clear evidence of
the breakdown of the QSS assumption for modeling the
chemical reaction by the Zel’dovich mechanism (1) and that
the QSS period is primarily controlled by the dissociation
process. Results for a similar investigation for T = 5000 and
20 000 K can be found in Figures S5 and S6.
As observed in Figures 11 and 12, the QSS approximation
breaks down for the Zel’dovich mechanism. This requires
further investigation since the accurate prediction of NO
formation and extinction through the exchange process plays
an important role in nonequilibrium hypersonic flow and
radiation modeling.39,40 Figure 13 shows the evolution of the
global Zel’dovich reaction rate coefficient k̃E,N2 and NO mole
fraction at the three different T. In the considered temperature
range, the peak of χNO occurs before the onset of the molecular
QSS periods. Moreover, at the time instants at which the NO
formation starts to be relevant, the magnitude of k̃E,N2 is
between 3 and 7 times smaller than the corresponding QSS
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Figure 11. Time-cumulative chemical production rate distributions for the heterogeneous exchange process NO + N ↔ N2 + O. The magenta- and
gray-shaded regions indicate the QSS periods of N2 and NO, respectively. The cyan-shaded areas represent the region where the chemical reaction
occurs: (a) T = 5000 K, (b) T = 10 000 K, and (c) T = 20 000 K. The red lines indicate the two different limiting cases of initial mole fractions. The
figures share the plot legend.
Figure 12. Comparisons of (a) the global rate coefficient for N2 + O → NO + N and (b) the corresponding rovibrational distributions of N2 at t =
4 × 10−5 s at T = 10 000 K with and without dissociation kinetics. The initial mole fraction is set to χN2 = χO = 0.5.
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20 000 K. In the comparison, the QSS rate coefficient is not
presented due to the following two reasons: (1) it is almost
identical with the thermal rate coefficient, as shown in Figure
13, and (2) the QSS assumption is not valid for the
heterogeneous exchange, as discussed in Figures 11 and 12.
It is worth mentioning that the present thermal rate
coefficients shown in Figure 15 were computed from the
QCT rate coefficients corresponding to each chemical system.
In detail, the QCT result for N2 + O → NO + N was employed
to calculate the thermal rate coefficient shown in Figure 15a,
whereas the QCT results for NO + N → N2 + O were used to
obtain the results in Figure 15b. This enables the comparison
of QCT-based rate coefficients, obtained from the different
PESs,2−5,9 with the experimental data for both forward and
backward directions of the Zel’dovich reaction N2 + O ↔ NO
+ N.
As shown in Figure 15a, the present thermal heterogeneous
exchange rate coefficient is in reasonable agreement with the
existing experimental41−43 and theoretical3,4,9 data. At temperatures over 5000 K, the present result agrees with the thermal
rate coefficients by Bose and Candler,2 Luo et al.,4 and Koner
et al.,9 whereas a departure is observed for the semiempirically
determined value by Park.30 Below 5000 K, the present result
is close to the data by Gamallo et al.,3 and it is within the range
of experimental error.41−43
As shown in Figure 15b, the heterogeneous exchange rates
for NO + N → N2 + O range within a factor of 4. The
discrepancy between the present and the existing QCT-based
results3,5,9 implies the uncertainty of the global triplet PESs for
predicting the heterogeneous exchange kinetics through the
highly exothermic reactive channel, NO + N → N2 + O. From
the comparison between the present rate and that using the
PESs by Denis-Alpizar et al.,5,6 it is worth mentioning that the
macroscopic thermal rates from the different PESs can be in
good agreement, even though the relevant microscopic
transition probabilities represent a certain discrepancy (see
Figure S7). This possibly implies an uncertainty of the
corresponding region on the PESs. The present result is
enveloped by the existing QCT-based results;3,5,9 however, it
shows a certain discrepancy against the measurements by
Davidson and Hanson, 44 obtained from a resonance
absorption spectrophotometry for N in a shock tube.
The fidelity of the present rovibrational-specific QCT rate
coefficients and the employed triplet PESs10 is further
validated by comparing the macroscopic equilibrium constant
for the N2 + O ↔ NO + N reaction with the existing data9,47
and the exact values obtained based on the partition functions.
Figure 16 shows the equilibrium constants along the
considered kinetic temperature range. For the sake of
consistency with the present QCT result, the effect of excited
electronic states is neglected when computing the exact
equilibrium constant (i.e., green dashed line). The present
QCT-based equilibrium constant, obtained from the ratio of
the thermal QCT rate coefficients in Figure 15, is in good
agreement with the exact value and the literature data.9,47,48
Figure 17 compares the present dissociation rate coefficients
with the existing data.4,9,33,37,43,47−58 For both chemical
systems, the QSS rate coefficients have slightly lower values
than the equilibrium ones, which is consistent with the results
obtained for the N2 + N12,37 and O2 + O59 systems. For the N2
+ O system, the measured N2 + N dissociation rate
coefficients49−52 are employed in the comparison due to the
lack of experimental data for the N2 + O system. The present
Figure 13. Time evolution of the global rate coefficient for N2 + O →
NO + N (solid lines) and the corresponding NO mole fractions
(dashed lines) at T = 5000, 10 000, and 20 000 K. The empty dots
point out the middle of the N2 QSS region. The initial mole fraction
was set to χN2 = χO = 0.5.
are based on the idea of clustering the levels in groups. Levels
in the same group are assumed in local equilibrium between
each other. What differentiates the reduced-order models is
(1) the number of groups. The QSS model has one single
group, while the remaining ones have a number of clusters
equal to the amount of distinct vibrational states in the
molecule. (2) The strategy adopted for clustering the levels.
The macroscopic QSS rate coefficient obtained from the
master equation analysis (e.g., the empty dots in Figure 13) is
utilized to construct the QSS-assumption-based groupedreconstructed rate coefficients in eq 21. It should be noted
that the other kinetic processes, except for the Zel’dovich
reaction, remain in the resolution of fully StS for all of the
grouped-reconstructed computations. In the considered
temperature range, the QSS method overpredicts the level of
χNO, especially in the NO formation phase and the peak value.
In hypersonic nonequilibrium flow-radiation computations, the
predictive accuracy on the population of electronic groundstate NO affects the calculation of the radiative emission
profile by NO bands, which strongly contribute to the radiative
heat flux. As a consequence, the overestimation of the NO
mole fraction by the QSS method could cause overprediction
of the NO emission profile. The remaining groupedreconstructed calculations reasonably reproduce the fully StS
profiles, except for the VS model at T = 20 000 K. Among the
different grouping strategies, the energy-based group and the
hybrid approach present better accuracy than the others. The
results shown in Figure 14 imply that the existing reducedorder models15−17,19 can be employed for a better description
of the collisional heterogeneous exchange process of N2 + O ↔
NO + N, instead of the QSS approximation.30
3.4. Comparison of Reaction Rate Coefficients. In this
section, the macroscopic reaction rate coefficients, obtained
from the present QCT and master equation analyses, are
compared with existing data. Figure 15 compares the present
heterogeneous exchange rate coefficients with data in the
literature2−5,9,30,41−46 for temperatures ranging from 2500 to
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Figure 14. Distributions of NO mole fraction (left). The ratio of NO mole fractions by the grouped-reconstructed methods to the fully StS
approach (right). The left and right figures share the plot legend. (a) T = 5000 K, (b) T = 10 000 K, and (c) T = 20 000 K. As initial mole fractions,
χN2 = χO = 0.5 is considered.
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Figure 16. Comparison of the macroscopic equilibrium constant KE
for N2 + O ↔ NO + N.
equilibrium rate coefficient is in good agreement with the
previous QCT results for N2 + O by Luo et al.4 and Koner et
al.,9 although the employed PESs are different. Some
discrepancies appear when comparing with the data by
Park,33 especially below 10 000 K. This is most probably due
to the fact that the data by Park were estimated from the
measurements of N2 + N.49−51
To perform comparisons for the NO + N system,
measurements for NO + Ar43,53−56 are considered due to
the lack of experimental data for dissociation in NO + N
interactions. The overall spread of the measured NO + Ar
dissociation rate coefficients43,53−56 is around 4 orders of
magnitude. Interestingly, the QSS rate coefficient computed
here is in very good agreement with the measurements by
Wray et al.,54 Thielen et al.,43 and the review data by Tsang
and Herron.58 On the other hand, the suggested value by
Park33,47 for computing hypersonic shock layers is 2 orders of
magnitude larger than the present calculations. In the recent
study by Kim and Jo,34 it was found that the NO dissociation
rate coefficient critically affects the prediction of nonequilibrium air radiation for earth re-entry conditions. Their
study employed the NO + Ar dissociation rate coefficient
recommended by Tsang and Herron58 as dissociation rates for
NO + N and NO + O, instead of the Park value.33,47 This fact
improved the accuracy of the nonequilibrium radiation
prediction, and the measured NO radiation bands60 were
better reproduced. These findings are in line with the present
QCT-based dissociation rate coefficient calculations for the
NO+N system shown in Figure 17b.
Figure 15. Comparisons of the heterogeneous exchange rate
coefficients with the existing experimental41−44 and theoretical2−5,9,30,45,46 data: (a) N2 + O → NO + N and (b) NO + N →
N2 + O.
calculations predict lower values than the measured ones for
both QSS and equilibrium. Interestingly, the present QSS
dissociation rate coefficients in N2 + O interactions are close to
those for N2 + N.37,57 This seems to confirm the possible
similarity in physical behavior among different chemical
systems that have been discussed in detail in Sections 3.1
and 3.2. In Figure S8, the PES cuts for N2 O and N3 are
provided around one of the local minima locations of N2 O
3
A′. The similar locations and depths of the local minima
between the two different chemical systems can support the
close dissociation rates shown in Figure 17a since the
dissociation of diatomic species in the considered systems is
characterized by the diatomic potential and interactions with a
drifting atom that can be depicted by the geometry of local
minima. In the considered temperature range, the present
4. CONCLUSIONS
This work has performed the rovibrational-specific QCT and
master equation analysis of the kinetics of N2(X1Σ+g ) + O(3P)
and NO(X2Π) + N(4S) systems at conditions of interest to
hypersonic applications. The complete sets of the StS kinetic
database on the N2O system allow the investigation of detailed
chemical-kinetic processes, including the Zel’dovich mechanism, at both microscopic and macroscopic scales. The
phenomenological rate coefficients for the Zel’dovich and
dissociation reactions derived in this analysis agree with
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models for air chemistry. It is also found that the formation of
NO and the corresponding extinction of N2 through the
Zel’dovich mechanism are dominated by the low-lying
vibrational states. Furthermore, the distribution of the N2
molecules does not reach any QSS conditions without
dissociation. These findings demonstrate the invalidity of the
QSS approximation widely adopted when modeling the
chemical reactions in hypersonic flow simulations. A coarsegrained description of the Zel’dovich reaction is proposed to
overcome the limitation of the QSS approach. This allows us
to accurately describe the details of the thermochemical
relaxation and the concentration profiles driven by the
exchange process.
■
APPENDIX
ÅÄÅ
ÑÉÑ
Å 1
ÑÑ
n − ni ÑÑÑÑnN
I j
ÑÑ
ÅÅÅÇ K i , j
ÑÖ
A. Mass Production Rates
NO
I
ω̇i =
∑
j
ω̇mI
N2
=
∑
l
ω̇iE, m,NO
=
ÅÅ
kiNO
ÅÅ
→j Å
ÄÅ
ÉÑ
ÅÅ 1
ÑÑ
Ñ
nl − nmÑÑÑnO
I
ÑÑ
ÅÅÇ K m , l
ÑÖ
Å
2 Å
kmN→
lÅ
ÅÅ
ÅÄÅ
Å
ÅÅ
kiE→,NO
m Å
ÅÅninN
ÄÅ
Å
ÅÅÇ
ÑÉÑ
1
ÑÑ
− E nmnOÑÑÑ
ÑÑ
Ki,m
ÑÖ
ÉÑ
ÑÑ
1
− D nNnOÑÑÑÑnN
ÅÅÅ
ÑÑÑ
Ki
Ç
Ö
ω̇i =
ÅÅ
Åni
kiNO
→ cÅ
Å
ω̇mD
Å
2 Å
Ånm
kmN→
cÅ
Å
D
=
(29)
ÅÄÅ
ÅÅÅ
Ç
(30)
(31)
(32)
ÑÉ
1 2 ÑÑÑÑ
− D nN ÑÑnO
K m ÑÑÑÖ
(33)
,where K stands for the equilibrium constant of a particular
collisional process to evaluate reverse rate coefficients by
invoking the microreversibility. For the collisional excitation
process between m-th and l-th states of N2, the equilibrium
constant KIm,l is defined as
Figure 17. Comparisons of the dissociation rate coefficients with the
existing experimental43,49−56 and theoretical4,33,37,47,57,58 data: (a) N2
+ O → 2N + O and (b) NO + N → 2N + O.
( )
exp(− )
e
K mI , l
available data from the literature. In contrast, marked
differences are observed when comparing the results of the
ab initio calculations with the widely adopted reaction rate
coefficients (i.e., the Park model). The relaxation rate
parameters describing rovibrational energy transfer demonstrate the inadequacy of the correlation formulas often used in
practical applications. Comparing the dynamics of relaxation
among the N2 + O, NO + N, and N2 + N systems, investigated
in detail by examining the time evolution of their rovibrational
state population dynamics, reveals important similarities in the
evolution of their kinetics. The three systems show a similar
strand-like structure of the low-lying rovibrational states and
similar evolution of the internal energy (i.e., energy-transferchemistry coupling) during dissociation. This is an important
finding as it facilitates the construction of reduced-order
=
gl exp − k lT
B
gm
em
kBT
(34)
The degeneracy of the level, gm, can be denoted as
N2
gm = geN2gnuc
(2J(m) + 1)
(35)
with ge and gnuc being the electronic ground and the nuclear
spin degeneracy, respectively. J(m) is the rotational quantum
number of the m-th state. For N2(X1Σ+g ), gnuc is correspondingly set to 6 and 3 for the even and odd J states, respectively.
For NO(X2Π), gnuc is equal to 3. For the collisional
heterogeneous exchange process between i-th and m-th states
of NO and N2, the equilibrium constant KEi,m is expressed as
3287
https://doi.org/10.1021/acs.jpca.1c10346
J. Phys. Chem. A 2022, 126, 3273−3290
The Journal of Physical Chemistry A
K iE, m
=
e
eO
B
B
■
( )
( )
g exp(− )Q g Q
gm exp − k mT Q tN2geOexp − k T Q tO
ei
kBT
NO N
t
e
K mD
=
(geNQ tN)2
gmQ tN2
AUTHOR INFORMATION
Marco Panesi − Center for Hypersonics and Entry Systems
Studies (CHESS), University of Illinois at UrbanaChampaign, Urbana, Illinois 61801, United States;
orcid.org/0000-0002-8650-081X; Email: mpanesi@
illinois.edu
ij E ND − em yz
zz
expjjjj− 2
z
j
kBT zz
k
{
Authors
Sung Min Jo − Center for Hypersonics and Entry Systems
Studies (CHESS), University of Illinois at UrbanaChampaign, Urbana, Illinois 61801, United States;
orcid.org/0000-0001-6687-0867
Simone Venturi − Center for Hypersonics and Entry Systems
Studies (CHESS), University of Illinois at UrbanaChampaign, Urbana, Illinois 61801, United States
Maitreyee P. Sharma − Center for Hypersonics and Entry
Systems Studies (CHESS), University of Illinois at UrbanaChampaign, Urbana, Illinois 61801, United States
Alessandro Munafò − Center for Hypersonics and Entry
Systems Studies (CHESS), University of Illinois at UrbanaChampaign, Urbana, Illinois 61801, United States
(37)
B. Curve-Fit of Relaxation Times
The forms of curve-fitted vibrational and rotational relaxation
times can be defined as
τV T = τMW + τc
(38)
pτRT = exp(A r T−1/3 + Br )
Complete contact information is available at:
https://pubs.acs.org/10.1021/acs.jpca.1c10346
(39)
Notes
where τMW and τc denote the Millikan−White and hightemperature correction terms, respectively. They can be
expressed as
101325
exp(A v (T −1/3 − Bv ) − 18.42)
τMW =
p
(40)
The authors declare no competing financial interest.
The rovibrational-specific database of ab initio rate coefficients
for the N2O system that supports the findings of this work is
openly available at https://www.chess.aerospace.illinois.edu/
databases.61
■
ij
8kBT yzz
zz
τc = jjjjnsσv
z
π
m
r
(41)
k
{
In the equations, ns and mr are the number density of the target
species and reduced mass of collision pairs, respectively. The
forms of curve-fit in eqs 38−41 are selected since they have
been widely adopted by several hypersonic CFD codes. Av, Bv,
σv, Ar, and Br are the curve-fit parameters for the pτVT and pτRT.
They are computed from the present master equation analysis
and are summarized in Table B1. The curve-fit is valid for
temperatures ranging from 2500 to 20 000 K.
−1
ACKNOWLEDGMENTS
This work was supported by ONR Grant No. N00014-21-12475 with Dr. Eric Marineau as the Program Manager. The
authors would like to thank Dr. R. L. Jaffe (NASA AMES
Research Center) for the useful discussions about QCT
calculations.
■
type of
collision
Av
Bv
σv [m2]
Ar
Br
N2 + O
NO + N
250.0
77.5
0.0432
0.0205
5.5 × 10−25T0.95
6 × 10−21T0.05
−51.1
−46.9
−15.3
−15.1
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N
t
D
D
ji E NO − E N2 zyz
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(36)
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ASSOCIATED CONTENT
sı Supporting Information
*
The Supporting Information is available free of charge at
https://pubs.acs.org/doi/10.1021/acs.jpca.1c10346.
Additional master equation results were obtained for
other initial heat-bath conditions (PDF)
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J. Phys. Chem. A 2022, 126, 3273−3290
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