This work aims to construct a reduced order model for energy transfer and dissociation in non-equilibrium nitrogen mixtures. The objective is twofold: to present the Coarse-Grain Quasi-Classical Trajectory (CG-QCT) method, a novel framework for constructing a reduced order model for diatom-diatom systems; and to analyze the physics of non-equilibrium relaxation of the nitrogen molecules undergoing dissociation in an ideal chemical reactor. The CG-QCT method couples the construction of the reduced order model under the coarse-grain model framework with the quasi-classical trajectory calculations to directly construct the reduced model without the need for computing the individual rovibrational specific kinetic data. In the coarse-grain model, the energy states are lumped together into groups containing states with similar properties, and the distribution of states within each of these groups is prescribed by a Boltzmann distribution at the local translational temperature. The required grouped kinetic properties are obtained directly by the QCT calculations. Two grouping strategies are considered: energy-based grouping, in which states of similar internal energy are lumped together, and vibrational grouping, in which states with the same vibrational quantum number are grouped together. A zero-dimensional chemical reactor simulation, in which the molecules are instantaneously heated, forcing the system into strong non-equilibrium, is used to study the differences between the two grouping strategies. The comparison of the numerical results against available experimental data demonstrates that the energy-based grouping is more suitable to capture dissociation, while the energy transfer process is better described with a vibrational grouping scheme. The dissociation process is found to be strongly dependent on the behavior of the high energy states, which contribute up to 50% of the dissociating molecules. Furthermore, up to 40% of the energy required to dissociate the molecules comes from the rotational mode, underscoring the importance of accounting for this mode when constructing non-equilibrium kinetic models. In contrast, the relaxation process is governed primarily by low energy states, which exhibit significantly slower transitions in the vibrational binning model due to the prevalence of mode separation in these states.

Modeling the environment surrounding vehicles flying at hypersonic speeds into Earth’s atmosphere is a very challenging task due to the complex interaction between multiple physical processes. In hypersonic flows, the shock heated gases can reach temperatures on the order of tens of thousands of degrees Kelvin. At these extreme temperatures, high energy collisions between the atoms and molecules can lead to excitation of the internal energy modes, dissociation and eventually ionization of the atoms and molecules in the gas.1,2 Thus, at high temperatures, air comprises a large number of chemical components including both neutral and ionized species. This work focuses on the modeling of energy transfer and dissociation of molecular nitrogen in its ground electronic state. Molecular nitrogen is the most abundant species in Earth’s atmosphere and must be accurately modeled for the design of high-speed vehicles. In particular, understanding the chemical processes due to N2–N2 interactions is critical to predicting the initial excitation and dissociation in air flows.

Because the time scale of the flow and the characteristic time of the chemical reactions are comparable, flow calculations require the use of a coupled approach, where chemical kinetics and flow governing equations are solved simultaneously.2,3 Early work on non-equilibrium modeling focused on the empirical multi-temperature models, which were constructed on the assumption that the population of each internal (rotational, vibrational, or electronic) energy mode can be described by means of a Maxwell-Boltzmann distribution at a specific temperature.4–7 The vast majority of multi-temperature models rely on the assumption of equilibrium among the rotational and translational modes at T and equilibrium among the vibrational, electronic, and electron temperature at TV. This assumption stems from the observation that, at low temperatures, rotational excitation occurs very quickly, given the reduced energy spacing among the rotational states. Therefore, the rotational relaxation has reached equilibrium with translation long before the vibrational, electronic, and electron energy modes are significantly excited. Finally, it is worth mentioning that these multi-temperature models require the knowledge of numerous parameters which are calibrated by matching experimental data, collected at ground-based facilities.8–11 

State-to-state (StS) models provide a physics-based alternative to multi-temperature models. In this approach, internal states of atoms and molecules are treated as individual pseudo-species, and their densities are evaluated by considering the detailed kinetic processes.12 This can be done to varying degrees of accuracy, e.g., electronic,13–18 vibrational, and rovibrational state-to-state.

Vibrational StS models generally assume either rotationless molecules or equilibrium of the rotational mode at the local translational temperature and explicitly solve for the individual vibrational state populations.19–30 These models have been used extensively for studying non-equilibrium excitation and dissociation processes. However, the accuracy of these models is limited by the validity of the assumptions made about the rotational mode and the availability of accurate kinetic data for the individual processes. Previous work on computing the kinetic data focused on semi-empirical models such as the ladder-climbing model,31,32 or semi-classical models such as the Schwartz-Slawsky-Herzfeld (SSH) model,33–35 and the Forced Harmonic Oscillator (FHO) model.36–43 Early work on characterizing StS kinetics for three-body systems made use of surprisal analysis based on the work of Procaccia and Levine.44–46 

Finally, rovibrational StS models treat all internal rovibrational states individually as separate species. These models are prohibitively expensive for two or three-dimensional simulations but can be used to study excitation and dissociation in zero-dimensional reactors and 1D Euler flows for validation purposes.47–49 Moreover, a full rovibrational simulation for even simple systems (e.g., atom-diatom interaction) requires a huge amount of kinetic data, which are largely unavailable.

Recent advances in computational quantum chemistry have enabled work detailing the microscopic kinetics for several systems.23–25,27,28,47,48,50–67 This is accomplished in two steps: first determining the interaction potential between two particles and then running scattering calculations to compute transition probabilities. The interaction potential, also known as the potential energy surface (PES), is computed by solving the electronic Schrödinger equation at many geometric arrangements of the atoms.68–71 Next, the transition probability can be computed by simulating many collisions between two particles, using the PES computed from quantum mechanics. In this work, the collisions are simulated using the quasi-classical trajectory (QCT) method, which assumes that the collision occurs classically, solving Hamilton’s equations of motion for the nuclei, but making use of the PES from quantum mechanics to obtain the force between the particles throughout the course of the collision.71–74 The outcome of many collisions is analyzed to obtain the probability that a given reaction occurred. This approach has been used to determine the rovibrational StS kinetics for atom-diatom systems, including the N2(X1Σg+)N(Su4) system. Recent work by Esposito et al.75 coupled QCT with scaling laws to determine the kinetics for the N2(X1Σg+)N2(X1Σg+) system. However, a full QCT characterization of the kinetic data for diatom-diatom systems is impractical because the number of possible transition is on the order of 1015.

As an alternative to the computationally expensive rovibrational StS model, the coarse-grain model (CGM) was developed with the objective of capturing the same physics of StS models at a fraction of the computational cost.76–86 The coarse-grain model, first developed for studying the kinetics of the electronic states of atomic species,14,87 has been recently extended to molecular species and applied to study the non-equilibrium energy transfer and dissociation processes for the N2(X1Σg+)N(Su4) system.80–82,86 In the CGM, energy states are lumped together, and the actual population distribution is approximated with a local reconstruction function derived by maximizing the entropy within the group. This approach is similar to that proposed by Haug et al.,44,45 which lumps energy states, assuming states in the same group to be in equilibrium. Previous work on the CGM has focused on systems with known microscopic StS kinetics. However, for more complex systems, such as interactions between two diatoms, detailing all the microscopic reactions becomes impossible due to the large number of kinetic channels.

The objective of this work is twofold: first to present the Coarse-Grain Quasi-Classical Trajectory (CG-QCT) method, a novel approach to constructing the coarse-grain model for diatom-diatom systems, and second to use the reduced order model constructed to study non-equilibrium energy transfer and dissociation of N2 molecules. The CG-QCT method couples the coarse-grain model with the quasi-classical trajectory calculations to directly obtain the grouped kinetic properties for the CGM, providing a framework for constructing reduced order models for more complex chemical systems. In the proposed method, the quasi-classical trajectory calculations require only information about the energy level grouping and reconstruction within each group to determine the group average reaction rates. The CG-QCT method is used to study the energy transfer and dissociation processes of molecular nitrogen, considering only the interaction between two nitrogen molecules in their ground electronic states, the N2(X1Σg+)N2(X1Σg+) system. Two grouping strategies are considered in this work: energy-based bins and vibrational-based bins. The energy-based grouping strategy lumps energy states with similar internal energy together, regardless of their rotational and vibrational quantum numbers. Vibrational-based binning lumps states together which share the same vibrational quantum number. This is identical to a vibrational StS model assuming the rotational states are in equilibrium at a rotational temperature. The two grouping strategies are compared in a zero-dimensional isothermal and isochoric reactor simulation under severe non-equilibrium conditions.

This work is divided in two parts: The present paper focuses on the model development and application of grouping strategies to the N2–N2 system. In particular, the differences in the results due to the grouping scheme are highlighted. Insights on the strengths and shortcomings of each model are obtained by direct comparison against experimental data. Paper II95 presents a comparison of the results obtained with the CG-QCT method with the ones obtained with the direct molecular simulation (DMS) method.88–95 

This paper is organized as follows: first the physical models, the CGM and CG-QCT methods, will be presented in Sec. II, followed by the results and comparison against the experimental data in Sec. III, a discussion of the physics captured in the results in Sec. IV, and finally the conclusions will be discussed in Sec. V.

In this section, the model reduction framework is described in detail. In Subsection II A, the coarse-grained modeling approach is recalled. This is followed by the Coarse-Grain Quasi-Classical Trajectory (CG-QCT) method in Subsection II B. Finally, the details on the potential energy surface for the N2(X1Σg+)N2(X1Σg+) system are presented in Subsection II C.

State-to-state modeling requires the solution of a large system of equations since different internal energy levels of reacting particles are treated as distinct chemical species. This description although exceedingly accurate has limited applicability to multi-dimensional computational fluid dynamics (CFD) problems due to its prohibitive cost. The present work overcomes this bottleneck through the use of reduced-order models based on the coarse-grain method. The coarse-grain method has been previously applied to different flow problems and chemical systems. The model reduction procedure can be broken down into two key steps:80,82

  1. Local representation and reconstruction: the rovibrational energy states of N2(X1Σg+) are grouped together according to their energy, and the actual distribution within each group is prescribed assuming an equilibrium distribution within the bin at the local translational temperature.

  2. Macroscopic governing equations: the macroscopic governing equations are obtained by taking moments of the microscopic master equations and applying the reconstruction function based on the assumption of equilibrium within each bin.

1. Local representation and reconstruction

In the present work, the distribution of states within each bin is assumed to be an equilibrium Boltzmann distribution at the local translational temperature,

Fpi(T)=ninp=giQp(T)expEikBT,
(1)

where quantities ni, Ei, and gi are, respectively, the population, energy, and degeneracy of state i, np represents the population of group p, Fpi(T) denotes the distribution of states i within group p at translational temperature T, kB represents the Boltzmann constant, and Qp(T) denotes the partition function of group p, given by

Qp(T)=iIpgiexpEikBT,
(2)

where the set Ip denotes the set of states contained in group p. The general coarse-grain method framework introduces a bin internal temperature, Tp by imposing constraints to maximize the entropy within the group.76,78

One of the most important considerations in the construction of the CGM is how to group the energy states. Previous work has focused on an energy-based grouping method, by which states were grouped according to increasing energy.76,82 In this work, two grouping strategies were considered: uniform width energy-based grouping and vibrational-based grouping.

Uniform width energy-based grouping considers the generic diatomic molecule, M, with internal structure, made up of atoms A and B. The energy states of molecule M are first split into bound and quasi-bound (or pre-dissociated) states. Then, given a number of bound and quasi-bound groups, denoted by NB and NQB, respectively, the energy width of the bins can be determined by

ΔEB=EA+EBNB,ΔEQB=EEAEBNQB,
(3)

where ΔEB and ΔEQB denote the energy width of the bound and quasi-bound bins, respectively, EA and EB represent the formation energy of atoms A and B, and E represents the energy of the rovibrational state with the largest energy. A schematic of the energy-based grouping strategy is shown in Fig. 1(a), considering only three energy bins. In this schematic, the different colors denote different vibrational states, which in the energy-based grouping strategy are lumped together. In this work, the grouping method will be applied to the N2(X1Σg+)N2(X1Σg+) system using 60 bins (40 of bound states, and 20 of quasi-bound or pre-dissociated states). The resulting bin widths are ΔEB = 0.24 eV and ΔEQB = 0.26 eV. However, it is worth recalling that the proposed framework is general and, as such, it can be applied to any diatom-diatom system.

FIG. 1.

Schematic of binning strategies. (a) Microscopic distribution of states and corresponding bins for energy-based binning. (b) Microscopic distribution of states and corresponding bins for vibrational-based binning.

FIG. 1.

Schematic of binning strategies. (a) Microscopic distribution of states and corresponding bins for energy-based binning. (b) Microscopic distribution of states and corresponding bins for vibrational-based binning.

Close modal

Vibrational-based grouping instead considers the vibrational quantum numbers when constructing the groups. In this approach, all the rotational states which share a vibrational quantum number are grouped together. A schematic of this grouping strategy is shown in Fig. 1(b) for three example vibrational states, where the red states belong to v=0, the blue states belong to v=1, and the green states belong to v=2. The resulting bins contain states which span the entire energy spectrum (e.g., the v=0 bin will contain states with all possible rotational quantum numbers or energies).

2. Macroscopic governing equations

The second step of the CGM consists of constructing the macroscopic governing equations to determine group properties np. This is accomplished by summing the microscopic moment equations of order m for states (i, j, k, l) contained in groups (p, q, r, s),

iIpddtniEim  =q,r,sIiIpjIqkIrlIskijklEninjEim+kklijEnknlEim   +q,rIiIpjIqkIrkijkEDninjEim+kkijERnknAnBEim   +qIiIpjIqkijDninjEim+kijRnA2nB2Eim,
(4)

where the k symbols stand for the rate coefficients for excitation, kijklE, de-excitation, kklijE, combined excitation-dissociation, kijkED, combined excitation-recombination, kkijER, dissociation, kijD, and recombination, kijR. Quantities nA and nB denote the number densities of the atomic constituents of molecule M, sets (Ip, Iq, Ir, Is) contain the states within groups (p, q, r, s), and the set I stores all the groups of molecule M. Taking the zeroth-order energy moment of Eq. (4) yields the conservation equations of group mass,

dnpdt=q,r,sIKpqrsEnpnq+KrspqEnrns+q,rIKpqrEDnpnq+KrpqERnrnAnB+qIKpqDnpnq+KpqRnA2nB2.
(5)

Equation (5) constitutes the general conservation equations for the linear CGM. The capitalized symbols K denote the group reaction rate coefficients. The solution of Eq. (5) allows for the determination of the group number densities, which can be then used to reconstruct the population of elementary states via Eq. (1).

The QCT method was used in this work for the scattering calculations. This method makes use Hamilton’s equations of motion to describe the dynamics of the collision but uses the ab initio PES generated by solving the Schrödinger equation at various geometries (i.e., configurations of the nuclei). The QCT code used in this work is a modified version of the Vectorized Variable Timestep Trajectory (VVTC) code written by Schwenke at NASA Ames Research Center.71 The initial states can either be explicitly defined, to determine rate coefficients for specific rovibrational states, or sampled from a thermal distribution at a specific internal temperature. The translational energy of the collisions was sampled from a thermal distribution at prescribed translational temperature values. All other collision parameters (e.g., impact parameter, orientation, vibrational phase, etc.) were sampled from an appropriate distribution using Monte Carlo averaging. The maximum impact parameter was taken to be 7.41 Å, conservatively chosen based on previous work by Valentini et al.92 Further details on the QCT procedure can be found in Karplus et al.72 and Jaffe et al.74 

After simulating a large number of collisions, the reaction cross section can be determined from the probability that a given event occurs. For a general excitation reaction from states (i, j) to states (k, l), the cross section, σijklE, can be written as

σijklE=PijklEπbmax2,PijklE=NijklENtot,
(6)

where PijklE is the probability of the transition (i, j) → (k, l). This quantity is computed from the ratio between the number of trajectories which resulted in the transition, NijklE, and the total number of trajectories simulated, Ntot. The error due to sampling is reduced by the square root of the number of samples (i.e., dσijkl1/Nijkl). Therefore, increasing the number of samples resulting in a given outcome improves the aleatoric uncertainty of a given rate. Quantity bmax denotes the maximum allowed value for the impact parameter. Once the cross section is known, the reaction rate coefficient can be determined by taking a thermal average over the relative velocities of the reaction cross section

kijklE=gσijklEχ1χgfM(T)σijklEdg,
(7)

where quantities g and χ denote, respectively, the relative velocity of the colliding particles and the symmetry factor (which equals 1 when the colliding partners are different and 2 when they are identical). The symbol fM(T) represents a Maxwellian velocity distribution at temperature T.

The CG-QCT method provides the link between the CGM and the QCT simulations and enables the construction of the CGM for diatom-diatom systems. Previous work on the CGM obtained the group-average kinetic parameters by summing the microscopic data and applying the appropriate weighting to obtain the correct distribution within the group. For example, the grouped cross section for an excitation reaction from groups (p, q) to groups (r, s), σpqrsE, can be written as the sum of the microscopic probabilities, PijklE, weighted by the distribution of states within groups p and q, Fpi and Fqj, multiplied by the area of the impact ring, πbmax2,

σpqrsE=iIpjIqkIrlIsPijklEFpi(T)Fqj(T)πbmax2.
(8)

In the CG-QCT method, the summation over the initial states (i, j) is evaluated by sampling these states from groups (p, q) populated according to the distribution functions (Fpi,Fqj). This results in the rate coefficients from pair of groups (p, q) populated according to the prescribed distribution functions (Fpi,Fqj) going to states (k, l). Then the summation over the final states is performed after the QCT calculations are complete. This eliminates the need to directly compute all the elementary cross sections, σijklE, which are impossible for a rovibrational StS model for the N2(X1Σg+)N2(X1Σg+) system. From the grouped cross sections, the grouped reaction rate coefficients can be obtained in the same way as the corresponding microscopic rate coefficients

KpqrsE=gσpqrsEχ.
(9)

For the remaining reactions considered in the diatom-diatom CGM, the cross sections for combined dissociation-excitation and dissociation reactions can be written as

σpqrED=iIpjIqkIrPijkEDFpi(T)Fqj(T)πbmax2,
(10)
σpqD=iIpjIqPijDFpi(T)Fqj(T)πbmax2.
(11)

The reverse reaction rate coefficients can be determined from the forward grouped rates by applying micro-reversibility. Details on computing the reverse reaction rate coefficients can be found in the  Appendix.

In this work, the CG-QCT method has been applied to the N2(X1Σg+)N2(X1Σg+) system to study the dynamics of excitation and dissociation in molecular nitrogen. The N2 molecule in its ground electronic state, N2(X1Σg+), has 9390 rovibrational states, with vibrational quantum numbers v and rotational quantum numbers J. The rovibrational states were determined using quantum mechanics calculations using the WKB approximation71 with the N2(X1Σg+) potential developed by Le Roy et al.96 After sorting the rovibrational states according to increasing energy, these can be assigned a global index i, where the energy and degeneracy of the state can be written Ei and gi. Of these states, most of them have energy below dissociation energy of 9.75 eV, referred to as the bound states. The remaining levels have energy above the dissociation energy but below the J-dependent centrifugal barrier.

The potential energy surface for the N2(X1Σg+)N2(X1Σg+) system used in this work was developed by NASA Ames Research Center.59–61 The nuclear positions were divided into regions where both N2(X1Σg+) molecules had bond lengths near equilibrium, and where one or both N2(X1Σg+) molecules had a bond length far from equilibrium. In the first group, where both molecules have bond lengths near equilibrium, electronic structure calculations were performed using the closed-shell coupled-cluster-single-double method to parametrize the wavefunction.68,69 For this region, calculations were performed for 3821 nuclear geometries. In the second region, where one or both bond lengths are far from equilibrium, calculations were performed for 325 nuclear geometries using the multi-reference averaged-coupled-pair-function method.70 Further details on the calculation of the PES can be found in Refs. 59–61.

In this section, an isochoric reactor model is used to study the non-equilibrium dissociation process of nitrogen modeled using the CG-QCT method. In all the simulations, the gas is initially composed of cold nitrogen molecules, populated according to a Boltzmann distribution at TI = 300 K with a density of 1.2 kg/m3 corresponding to a pressure of 1 atm. Under these assumptions, the initial population of the groups is given by

npnN2=Q̃p(T)expEpkBTIpIQ̃p(T)expEpkBTI,
(12)

where nN2 denotes the total number density of nitrogen molecules given by nN2=pInp, Ep is the formation energy of group p, ΔEip is the energy of state i relative to the formation energy of group p such that Ei=Ep+ΔEip, TI is the initial internal temperature, and Q̃p(T) is the partition function of a group relative to the group formation energy: Q̃p(T)=iIpgiexpΔEip/(kBT). At the beginning of the simulation, the translational temperature of the reactor is instantaneously raised and held constant, driving the gas out of equilibrium. Four different translational temperatures are considered: 10 000 K, 13 000 K, 20 000 K, and 25 000 K.

All the simulations are carried out using the energy-based and vibrational-based grouping strategies discussed in Sec. II. The former comprises 60 groups, 40 bound and 20 quasi-bound groups of equal energy width, while the latter considers the 61 vibrational energy levels of N2 as separate groups. In both cases, the collisional processes between two nitrogen molecules considered are excitation, combined excitation-dissociation, and double dissociation. Collisions between atomic and molecular species are ignored.

The section is organized as follows: First, the time evolution of the distribution function predicted by the CG-QCT model will be presented, followed by an analysis of the dissociation and energy transfer processes. Finally, the results obtained with the two different grouping strategies are compared against the available experimental data.

1. Uniform width energy-based grouping

At the beginning of the simulation, only the low-lying energy groups are significantly populated, given the initial values of TI. With time, the random motion of molecules brings about collisions, thus enabling the transfer of kinetic energy into rotational and vibrational energy. Figure 2(a) shows the distribution of energy groups at various times in the 10 000 K reactor simulation. Early in the relaxation process, t = 10−10 s, the distribution is still significantly colder than the final equilibrium distribution, indicating that the gas is still in the midst of the relaxation process. This phase is completed at about t = 10−9 s. Between t = 10−8 s and t = 10−7 s, the distribution is frozen, indicating that the gas has reached the quasi-steady-state (QSS) distribution. After t = 10−7 s, the high energy groups are replenished through recombination, and the gas approaches the final equilibrium distribution after t ≈ 10−5 s. Figure 2(b) shows the time evolution of the population for a subset of the groups. In this figure, the different phases of the thermochemical relaxation can be clearly observed along with the plateau in the population densities which represent the QSS of the gas.

FIG. 2.

Bin distribution for isochoric reactor simulation at Ttr = 10 000 K with 60 energy groups. (a) Bin population from CGM at various times. (b) Time evolution of bin distributions.

FIG. 2.

Bin distribution for isochoric reactor simulation at Ttr = 10 000 K with 60 energy groups. (a) Bin population from CGM at various times. (b) Time evolution of bin distributions.

Close modal

2. Vibrational-based grouping

Figure 3(a) shows the distribution of vibrational groups at various times in the 10 000 K reactor simulation. Because of the difference in grouping strategies, the distribution looks very different from the energy binned results. Early in the relaxation process, t < 10−9 s, the low energy vibrational states (Ev<1.5 eV) appear nearly frozen at a colder temperature than the higher energy states. This bi-modal distribution persists until the QSS is reached at t = 10−7 s. The QSS is clearly observable in Fig. 3(b), which shows a narrow plateau in the population of the high vibrational energy level populations around this time. As observed for the other grouping strategy, the distribution shows significant deviation from equilibrium. It is important to note that the initial energy transfer process predicted by the vibrational-based grouping model is significantly slower than the one predicted by the energy-based grouping method.

FIG. 3.

Bin distribution for isochoric reactor simulation at Ttr = 10 000 K with 61 vibrational specific groups. (a) Bin population from CGM at various times. (b) Time evolution of bin distributions.

FIG. 3.

Bin distribution for isochoric reactor simulation at Ttr = 10 000 K with 61 vibrational specific groups. (a) Bin population from CGM at various times. (b) Time evolution of bin distributions.

Close modal

To better illustrate the differences between the two grouping strategies on the microscopic level, the reconstructed distributions are shown in Fig. 4 at two different times in the 10 000 K reactor. Figure 4(a) shows the distribution during the relaxation process, at t = 1.173 × 10−10 s. At this instant, the total internal energy contained in the molecules is the same; however, due to different assumptions in the grouping strategy, the rovibrational distributions are very different. Since the energy-based grouping assumes equilibrium over a small range of energies (i.e., each energy bin is only 0.25 eV in width), the distribution is mostly continuous across the energy spectrum, with each bin approximating only a reduced number of levels in a narrow energy range. In contrast, the vibrational binning strategy assumes equilibrium at 10 000 K among all rotational states within a given vibrational state. This results in the strand like structure observed in Fig. 4(a).

FIG. 4.

Reconstructed distribution of states from energy and vibrational binning models at 10 000 K. (a) Reconstructed distribution of states at t = 1.173 × 10−10 s. (b) Reconstructed distribution of states at t = 9.869 × 10−8 s.

FIG. 4.

Reconstructed distribution of states from energy and vibrational binning models at 10 000 K. (a) Reconstructed distribution of states at t = 1.173 × 10−10 s. (b) Reconstructed distribution of states at t = 9.869 × 10−8 s.

Close modal

Figure 4(b) shows the reconstructed distribution of states during the dissociation process for both grouping strategies. In both groupings, the distribution of low energy states is very similar and approaches equilibrium. However, nearing the dissociation energy (9.75 eV), the distribution predicted by the energy binning strategy deviates from the Boltzmann distribution. The population predicted by the vibrational grouping model is more complicated. The high-lying vibrational levels contain a reduced number of rotational levels and therefore will capture the depopulation of the high-lying rovibrational states. On the contrary, the low-lying vibrational levels contain a large number of rotational levels characterized by large rotational quantum number. These levels are forced to be in equilibrium with the low-lying rovibrational states by the averaging procedure, thus creating an artificial overpopulation in the tail region of the distribution.

The two models provide a very different representation of rovibrational relaxation. The vibrational grouping appears adequate for the description of the low-lying vibrational level, characterized by a small rotational quantum number, where the mode separation is clearly significant. On the contrary, the energy binning strategy seems more adequate for the description of the relaxation of the high-lying states characterized by low vibrational quantum numbers and high rotational energy content.

The mole fraction of atomic nitrogen predicted by both grouping strategies at various temperatures is shown in Fig. 5. At 25 000 K, the two grouping strategies predict similar dissociation rates, even if the onset of dissociation occurs significantly earlier with the vibrational specific model. As the temperature decreases, the two grouping strategies diverge. At 10 000 K, the energy-based groups predict faster dissociation, with a shorter incubation period. The discrepancy between the two grouping models is due to the treatment of the rotational energy mode. In the vibrational grouping model, the rotational levels are assumed to be populated according to a Boltzmann distribution at the translational temperature. This assumption hinders dissociation at low temperature because it artificially imposes equilibrium between low and high energy states [e.g., (v, J) = (0, 0) and (v, J) = (0, 270)], which is clearly incorrect. On the contrary, the energy-based grouping is able to capture the non-equilibrium between high and low-lying rotational levels since they belong to different groups.

FIG. 5.

N mole fraction as a function of time using energy-based grouping (solid lines) and vibrational-based grouping (symbols) at various temperatures.

FIG. 5.

N mole fraction as a function of time using energy-based grouping (solid lines) and vibrational-based grouping (symbols) at various temperatures.

Close modal

The importance of the high energy states for predicting dissociation is highlighted in Fig. 6. This shows the fraction of molecules which dissociate from a given group when the molecules are in the QSS condition, given by

Pdiss(Ep)=qIrIKpqrEDnppIqIrIKpqrEDnp,
(13)

where p and q are the groups of the dissociating and exciting molecules, respectively, and Pdiss(Ep) is the probability that a molecule in group p dissociates. Figure 6(a) presents the results obtained with the energy-based grouping model. The distribution clearly shows that molecules climb nearly to the dissociation energy before dissociating. This effect is more pronounced at low temperatures, where there is less energy available in the translational mode to facilitate dissociation from the lower energy groups. Thus, the importance of the quasi-bound states increases with decreasing temperature. Table I gives the percentage of dissociation occurring from quasi-bound states as predicted by the model. At low temperatures, nearly 50% of the dissociation occurs from quasi-bound states, highlighting their importance for the prediction of the dissociation process. This conclusion is similar to that reached by Bender et al.97 for the N2(X1Σg+)N2(X1Σg+) system who observed that up to 58% of dissociation events come from trajectories with at least one quasi-bound molecule, as well as Panesi et al. for the N2(X1Σg+)N(Su4) system.48 

FIG. 6.

Fraction of molecules dissociating during QSS region from each group. (a) 60 energy groups. (b) 61 vibrational groups.

FIG. 6.

Fraction of molecules dissociating during QSS region from each group. (a) 60 energy groups. (b) 61 vibrational groups.

Close modal
TABLE I.

Percentage of dissociation from quasi-bound states in energy-based grouping model.

T (K)Dissociation from QB (%)Δevibdiss/Δetotdiss (%)Δerotdiss/Δetotdiss (%)
10 000 46.9 60.1 39.9 
13 000 45.0 59.0 41.0 
20 000 40.8 57.2 42.8 
25 000 38.6 56.4 43.6 
T (K)Dissociation from QB (%)Δevibdiss/Δetotdiss (%)Δerotdiss/Δetotdiss (%)
10 000 46.9 60.1 39.9 
13 000 45.0 59.0 41.0 
20 000 40.8 57.2 42.8 
25 000 38.6 56.4 43.6 

The vibrational grouping method, shown in Fig. 6(b), predicts very different behavior. At the lowest temperature, 10 000 K, the molecules generally climb to higher vibrational states (≈7–8 eV) before dissociating, as predicted by the ladder climbing model. However, at the highest temperature, 25 000 K, the molecules are much more likely to dissociate from low energy vibrational states (≈1–4 eV). This behavior can be explained as follows: at high temperatures, the low vibrational states are rotationally excited (given the assumption of rotational equilibrium), and the high lying rotational states are therefore significantly populated, thus significantly contributing to dissociation. In other words, the dissociation energy of a molecule decreases as J increases. However at low temperatures, the Boltzmann weighting favors the lower lying rotational energy levels, characterized by significantly lower dissociation probability, thus hindering dissociation from the high-lying rotational states of a given group.

To gain more insight on the relative contribution of rotational and vibrational energy to dissociation, Fig. 7 shows the fraction of energy lost by the two modes during dissociation at 10 000 K, normalized by the total internal energy lost, given by

Δerot,pdissΔetotdiss=iIpniEirotPdiss(Ep)pIiIpniEirot+EivibPdiss(Ep),Δevib,pdissΔetotdiss=1Δerot,pdissΔetotdiss,
(14)

where ΔErot,pdiss and ΔEvib,pdiss denote, respectively, the energy lost in dissociation from the rotational and vibrational modes of group p, ΔEtotdiss denotes the total energy lost during dissociation, and Eirot and Eivib denote, respectively, the rotational and vibrational energies of state i. It is interesting to observe that the energy-based bins predict a similar contribution between rotation and vibration, with the latter contributing slightly more. The percentage of energy lost from each mode in the energy-based grouping model is also summarized in Table I, highlighting that at high temperature, the energy loss from each mode becomes comparable. In contrast, the vibrational grouping provides a completely different picture: the contribution of rotation does not exceed 10% of the total internal energy lost. In other words, as a result of the assumptions made, the importance of rotation on the kinetics is downplayed at low temperatures in the vibrational-based averaging. The work of Bender et al.97 obtained similar results to what is observed in the vibrational grouping method, which finds that the rotational energy contribution to dissociation is significantly smaller than the vibrational contribution. However, this may be an artifact of the assumptions made in the model (e.g., rotation and vibrational modes are decoupled).

FIG. 7.

Fraction of rotational and vibrational energy lost by the molecules during QSS region from each group at 10 000 K. (a) 60 energy groups. (b) 61 vibrational groups.

FIG. 7.

Fraction of rotational and vibrational energy lost by the molecules during QSS region from each group at 10 000 K. (a) 60 energy groups. (b) 61 vibrational groups.

Close modal

The group distribution during QSS for the energy and vibrational-based bins is shown in Fig. 8 for various temperatures. In both cases, significant departures from the QSS temperature Boltzmann distributions are observed. The low energy groups appear to be close to equilibrium, whereas the distribution of the high energy tail appears significantly depopulated. The behavior of the distribution is consistent with the dissociation probability functions discussed above. The high energy molecules, more likely to dissociate, are responsible for the departures of the distribution from equilibrium. Between the two models, the vibrational energy model seems to predict lower non-equilibrium of the high energy tail.

FIG. 8.

Quasi-steady-state group distribution for various temperatures; symbols indicate the QSS distribution and lines indicate the equilibrium distribution at the QSS temperature. (a) 60 energy bins. (b) 61 vibrational groups.

FIG. 8.

Quasi-steady-state group distribution for various temperatures; symbols indicate the QSS distribution and lines indicate the equilibrium distribution at the QSS temperature. (a) 60 energy bins. (b) 61 vibrational groups.

Close modal

Figure 9 shows the local dissociation rate profiles during the relaxation. These have been calculated from the production rate of atomic nitrogen as follows:

KD=121nN2nN2dnNdt.
(15)
FIG. 9.

N2–N2 local dissociation rate computed from energy-based groups (solid lines) and vibrational-based groups (symbols).

FIG. 9.

N2–N2 local dissociation rate computed from energy-based groups (solid lines) and vibrational-based groups (symbols).

Close modal

The rate profiles are monotonically increasing in the early stage of the relaxation for both models, until the onset of the QSS distribution, responsible of the formation of a plateau in the profiles. A comparison of the nitrogen concentration profiles and the dissociation rates shows that, across a range of conditions analyzed, the entire dissociation process proceeds under QSS conditions for the N2–N2 processes. This was not the case for the N2–N system, where significant part of the dissociation occurred in non-QSS conditions.48 The comparison of the rate profiles obtained with the two different grouping strategies shows that the QSS dissociation rate predicted at low temperatures by the vibrational grouping is approximately half of those predicted by the energy-based grouping. Moreover, the onset of the QSS region is significantly delayed for the vibrational grouping model. This indicates that not only is the dissociation process different between the two groupings but also the energy transfer process has significant differences. As the temperature increases, the vibrational grouping QSS dissociation rate overshoots the rate predicted by the energy groups. This is due to the over-population of high energy rovibrational states in the vibrational grouping model discussed earlier.

The internal energy of the molecules, shown in Fig. 10, can be computed as

Eint=iIniEiiIni.
(16)

The energy-based grouping starts with very low internal energy and excites quickly to the QSS, by 10−8 s at all temperatures. At the low temperatures, the total internal energy of the molecules is the same as the thermal internal energy, indicating that the QSS distribution is not much different from the thermal Boltzmann distribution. This is not surprising since most of the energy is contained in the low energy states, whose population is close to equilibrium at low temperatures, and the high energy states do not significantly contribute to the internal energy content. At higher temperatures, the QSS has significantly lower internal energy from the final equilibrium value, indicating the presence of stronger non-equilibrium effects.

FIG. 10.

Internal energy of N2 molecules as a function of time from energy (solid lines) and vibrational (symbols) grouping.

FIG. 10.

Internal energy of N2 molecules as a function of time from energy (solid lines) and vibrational (symbols) grouping.

Close modal

The vibrational specific bins exhibit very different behavior throughout the relaxation process. Despite the initial excess of internal energy, due to the assumption of T = TR, the vibrational grouping model predicts significantly slower relaxation across the range of temperatures, if compared with the energy-based model.

Further insights on the energy transfer processes can be obtained by computing the second order moment of the transition rates for both models. To this aim, the master equation is expressed as diffusion or Fokker-Planck type equation,2,98 for which the dynamics of relaxation is controlled by the diffusion coefficients, expressed in the function of the second order moment of the transition rates. The expression of the second order moment for the group p is

M(p)=12rIpr2K(p,r),
(17)

where K(p, r) is the effective rate of energy relaxation from group p to group r,

K(p,r)=sIqIfqKpqrse,
(18)

where fq=nqnN2 denotes the fraction of particles in group q.

Figure 11(a) shows the second order moment for the energy-based bins formulation at 10 000K. Initially, the moment exhibits a monotonically increasing behavior with the bin energy. At later times, however, the efficiency of the first few groups increases creating a shallow bottleneck between 1 and 2 eV. At high energies, the dependence of the moment on the group energy is nearly exponential and appears to be unchanged during the relaxation.

FIG. 11.

Moment at various times at 10 000 K. (a) 60 energy bins. (b) 61 vibrational bins.

FIG. 11.

Moment at various times at 10 000 K. (a) 60 energy bins. (b) 61 vibrational bins.

Close modal

Figure 11(b) shows the second order moment for the vibrational bins at 10 000 K. In general, the moments appear several orders of magnitude smaller across the entire energy spectrum. In particular, below 4 eV the coefficients are extremely small. This explains the formation of the bimodal distribution shown in Fig. 3(a) and in general slow vibrational relaxation observed in Fig. 10. It is important to stress that, contrary to what observed by other researchers, the vibrational relaxation does not exhibit a bottleneck.99 This is due to the effect of rotation, that provides additional channels to vibrational processes, hence enhancing the relaxation.

Figure 12 shows the excitation rate contour plots, as a function of the energies of the products, for given pairs of initial groups. The analysis was repeated for the energy and vibrational grouping models. To facilitate the comparison between the models, we selected reactants with similar energy groups. In both cases, the energy of the second reactant was increased to analyze the behavior of the rates for the lower and the upper part of the distribution.

FIG. 12.

Excitation contours from a pair of fixed groups with energy and vibrational bins at 10 000 K. (a) 60 energy bins N2(6) + N2(10) ⇌ N2(r) + N2(s). (b) 61 vibrational bins N2(4) + N2(11) ⇌ N2(v) + N2(w). (c) 60 energy bins N2(6) + N2(26) ⇌ N2(r) + N2(s). (d) 61 vibrational bins N2(4) + N2(30) ⇌ N2(v) + N2(w).

FIG. 12.

Excitation contours from a pair of fixed groups with energy and vibrational bins at 10 000 K. (a) 60 energy bins N2(6) + N2(10) ⇌ N2(r) + N2(s). (b) 61 vibrational bins N2(4) + N2(11) ⇌ N2(v) + N2(w). (c) 60 energy bins N2(6) + N2(26) ⇌ N2(r) + N2(s). (d) 61 vibrational bins N2(4) + N2(30) ⇌ N2(v) + N2(w).

Close modal

The contours for the energy-based grouping, shown in Figs. 12(a) and 12(c), exhibit a maximum corresponding to the “elastic” processes in which neither of the groups change. The magnitude of the rates decays exponentially for increasing energies of the products. The behavior is nearly isotropic, indicating that both reactants have equal probability of being excited. It is interesting to note that all the possible energy transfer reactions follow this behavior, and it is possible to fit the rates with a unique exponential function with three parameters (A, σ, γ), based on the work of Barker et al.,100 

KpqrsE=Aexp(EpEr)2+(EqEs)2σγ.
(19)

In Figs. 12(b) and 12(d), the reaction rates for the vibrational energy groups exhibit a different behavior. At low energies, the vibrational-translational (VT) energy transfer processes are very inefficient compared with the high energy states. This is consistent with the results shown in Fig. 11(b). In contrast, due to the anharmonicity of the vibrational states, multi-quantum jumps are very probable for the high lying vibrational energy levels [Fig. 12(d)]. This justifies the establishment of a multimodal distribution in the early part of the relaxation. Moreover, at low energies, vibrational-vibrational (VV) energy transfer appears very efficient, thus favoring the thermalization of the distribution.

Sections III A– III C focused on the analysis of the dynamics of dissociation and energy transfer obtained with the two different grouping strategies. In order to assess the validity of each model, the results obtained are now compared against the available experimental data. Two different observables are used: phenomenological dissociation rate coefficient and an energy transfer relaxation time.

The thermal and QSS dissociation rate coefficients predicted by both the grouping models are compared against experimental data in Fig. 13. The numerical values of the rates in function of temperature can be found in Table II.

FIG. 13.

N2–N2 dissociation rate coefficients compared with previous experimental work from Appleton et al.9 and Kewley and Hornung.102 (a) Thermal equilibrium, compared with previous work of Jaffe et al.101 and Bender et al.97 (b) Quasi-steady-state.

FIG. 13.

N2–N2 dissociation rate coefficients compared with previous experimental work from Appleton et al.9 and Kewley and Hornung.102 (a) Thermal equilibrium, compared with previous work of Jaffe et al.101 and Bender et al.97 (b) Quasi-steady-state.

Close modal
TABLE II.

Dissociation rate coefficients in cm3/s.

Thermal equilibriumQSS
Energy binsVibrational binsEnergy binsVibrational bins
10 000 K 8.6844 × 10−14 8.6693 × 10−14 3.5509 × 10−14 2.0462 × 10−14 
13 000 K 1.0397 × 10−12 1.0376 × 10−12 3.4085 × 10−13 2.2250 × 10−13 
20 000 K 1.7563 × 10−11 1.7552 × 10−11 3.9152 × 10−12 3.7318 × 10−12 
25 000 K 4.7894 × 10−11 4.7800 × 10−11 9.3343 × 10−12 1.1089 × 10−11 
Thermal equilibriumQSS
Energy binsVibrational binsEnergy binsVibrational bins
10 000 K 8.6844 × 10−14 8.6693 × 10−14 3.5509 × 10−14 2.0462 × 10−14 
13 000 K 1.0397 × 10−12 1.0376 × 10−12 3.4085 × 10−13 2.2250 × 10−13 
20 000 K 1.7563 × 10−11 1.7552 × 10−11 3.9152 × 10−12 3.7318 × 10−12 
25 000 K 4.7894 × 10−11 4.7800 × 10−11 9.3343 × 10−12 1.1089 × 10−11 

The thermal dissociation rate, shown in Fig. 13(a), is computed by assuming an equilibrium distribution between all the groups

KD=pqqIrIKpqrEDQp(T)QN2(T)Qq(T)QN2(T)+2pqqIKpqDQp(T)QN2(T)Qq(T)QN2(T).
(20)

Although in excellent agreement with the work of Jaffe et al.101 and Bender et al.,97 both predictions grossly overestimate the experimental rates from Appleton et al.,9 and Kewley and Hornung.102 This is not surprising, since the population of the high-lying energy levels for both models was found to strongly deviate from equilibrium.

The QSS rates shown in Fig. 13(b) for both models are more consistent with the experimental data. In particular, the energy-based groups show excellent agreement with the Appleton9 data across the range of experimental conditions. The vibrational specific grouping predicts a lower rate coefficient at low temperatures (e.g., 10 000 K), and the slope of the rate does not agree with the experimental fit. At higher temperatures, both grouping strategies predict similar rate coefficients. The Kewley and Hornung data102 predict a significantly different temperature dependence, inconsistent with what was predicted by the CG-QCT method, demonstrated by the different slope in the data in Fig. 13. It is the authors’ opinion, based on preliminary analysis, that the discrepancy in the Kewley and Hornung data compared with the CG-QCT method is due to the non-equilibrium model adopted by the authors to interpret the experimental data (not shown). Additional experimental data, not included in the figure, have been used in the comparison. Hanson and Baganoff103 predict the same temperature dependence as Kewley and Hornung, but their rates are significantly higher than all other experimental results. Park104 reinterpreted the experimental data of Appleton, by including non-equilibrium effects in the post-processing. As a result, the Park dissociation rate coefficient appears larger than the original value given by Appleton; however, the temperature dependence is similar.

The vibrational relaxation time was computed using the e-folding method (the time required for the mode to reach 63.2% of its steady state energy) from the vibrational bins. The vibrational relaxation time is shown in Fig. 14, compared with the Millikan-White correlation,105 and high-temperature correlations developed by Park6 and Boyd.106 At low temperatures, the vibrational relaxation matches well with the Millikan-White correlation. The high-temperature corrections to Millikan-White match well at high temperatures with the vibrational binning data.

FIG. 14.

τVT computed with various models and experimental data from Appleton and Steinberg.107 

FIG. 14.

τVT computed with various models and experimental data from Appleton and Steinberg.107 

Close modal

The results have clearly shown how the choice of grouping strategy has a profound impact on the characteristics of the thermochemical relaxation. The two grouping strategies adopted in this work are based on two fundamentally different assumptions: the vibrational grouping is constructed on the assumption of rigid separation between the rotational and vibrational energy modes, while the energy-based grouping, by lumping the levels independently of their vibrational and rotational characteristics, assumes exactly the opposite. Each of the governing assumptions has its merits: the separation of modes was clearly demonstrated experimentally and theoretically for the low lying energy levels. This justifies the use of a vibrational-based grouping strategy. On the contrary, the high energy states exhibit much stronger rovibrational coupling, which implies that energy-based binning provides a more accurate description of their behavior.

The energy-based grouping model predicts that molecules must climb to high energy rovibrational levels before dissociating and that a significant amount of energy lost in dissociation comes from rotationally excited molecules. By dissecting the energy ladder into narrow rovibrational energy groups, the contribution of vibration and rotation to dissociation can be separated without introducing significant artificial bias. The energy grouping model predicts nearly equal contribution of rotation and vibration to dissociation. This is not the case for the vibrational grouping method, which strongly underpredicts the role of rotation. Moreover, the narrow width of the energy bins (0.25 eV) limits the negative influence of the assumption of thermal equilibrium between translation and the group internal temperature.

In contrast, the vibrational binning strategy assumes equilibrium among rotational states across a wide range of energy, especially for the low vibrational energy groups (e.g., v=0 spans nearly 15 eV). As a result of the averaging process, the importance of dissociation from the high energy states is overwhelmed by the improbability of the low energy states dissociating. The weakness of the vibrational grouping strategy originates from the fact that low and high energy states are governed by different kinetics. To be successful, rovibrational states with similar rates should be grouped together (or averaged together).86 

The energy transfer process predicted by the energy level grouping was very fast across the range of temperatures. Because states with very different quantum configurations are lumped together in this approach, the mode separation known to exist for low energy states during the relaxation is not captured. This problem is not present in the vibrational-based grouping because the difference in the dynamics of slow and fast processes (i.e., vibrational and rotational relaxation, respectively) is correctly captured. Since internal energy relaxation and dissociation do not overlap in the thermochemical relaxation process, errors in the modeling of the relaxation are unlikely to affect the dynamics of dissociation.

The objective of this work is twofold: first to present a novel approach to constructing reduced order models (CG-QCT) for diatom-diatom interactions and second to study the non-equilibrium relaxation of nitrogen molecules due to N2–N2 kinetic processes. The CGM is constructed by lumping rovibrational states together into groups and by determining the group averaged kinetic properties by means of the quasi-classical trajectory calculations. Moments of the master equations are used to compute the population of the groups during the relaxation. Two very different grouping strategies, energy and vibrational-based grouping, were used to investigate the non-equilibrium relaxation of molecular nitrogen in zero-dimensional isothermal and isochoric reactors. It was found, based on comparison with experimental data, that both approaches captured the dissociation process,9 while the vibrational bins better captured the energy transfer process.6,105,106

The comparison of the energy transfer relaxation rates, predicted by the vibrational grouping method, against available experimental data,6,105,106 validates the classical picture of relaxation based on the rigid separation among the rotational and vibrational energy modes. It was found that the relaxation data provided by the Millikan and White correlation are in good agreement with the theoretical predictions provided by the vibrational grouping method for temperatures below 10 000 K. On the contrary, any attempt to ignore the mode separation by grouping rovibrational levels characterized by different vibrational quantum number has lead to an overestimation of the relaxation rate.

The dissociation process is controlled by the tail of the rovibrational distribution. For the high-lying states, the strong rovibrational coupling is well captured by the energy-based grouping. So molecules have to climb the rovibrational energy ladder before dissociating, thus acquiring a significant vibrational and rotational energy. In all the cases analyzed, rotational energy accounts for more than 40% of the total energy required to dissociate the N2 molecules. While vibrational bias is introduced in the conventional non-equilibrium models, no models include bias to account for effect of rotational excitation of the molecules.97 Finally, it is important to note that during the dissociation process, the distribution of the N2 molecules is in QSS. This is very different from what is observed in the N2–N case, where significant part of the dissociation takes place in non-QSS conditions.

To complete the analysis of these two grouping strategies, the second part of this paper presents a detailed comparison of the CG-QCT method with the Direct Molecular Simulation (DMS)92–94 method.95 

The authors would like to thank AFOSR and in particular Dr. Ivett Leyva for supporting this research. Ms. R. Macdonald was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program. Dr. M. Panesi was supported by the Air Force Office of Scientific Research Young Investigators Program No. FA9550-15-1-0132 with Program Office Dr. Ivett Leyva. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the AFOSR or the U.S. government.

The reverse reaction rates are obtained by applying microreversibility. Microreversibility holds for the StS reactions

kijkEDninj=kkijERnknAnB,
(A1)

where the ⋆ denotes the equilibrium composition at the local translational temperature of each species. The internal temperature of the group is assumed to be in equilibrium with the translational mode, giving an equilibrium distribution at the local translational temperature. Rearranging this expression and using the definition of the grouped rate, KrpqER,

KrpqER=iIpjIqkIrkkijERFrk(T)=iIpjIqkIrkijkEDninjnAnBnr.
(A2)

Multiplying and dividing the entire expression by the equilibrium population of the groups p and q allows the expression to be re-written in terms of the forward group rate, KpqrED,

KrpqER=iIpjIqkIrkijkEDninpnjnqnpnqnAnBnr=npnqnAnBnriIpjIqkIrkijkEDFpi(T)Fqi(T),
(A3)
KrpqER=npnqnAnBnrKpqrED.
(A4)

The equilibrium composition is obtained using the partition functions

KrpqER=Qptot(T)Qqtot(T)QAtot(T)QBtot(T)Qrtot(T)KpqrED.
(A5)

For the molecules in groups p, q, and r, the partition function can be written as the product of the internal and translational partition functions

Qptot(T)=2πmpkBThP23/2iIpgiexpEikBT,
(A6)

where mp denotes the mass of the molecule containing group p, and hP denotes Planck’s constant. For the atoms A and B, assuming that they both are in the ground electronic state, which have degeneracy given by gA and gB, respectively, the partition function is written as

QAtot(T)=gAexpEAkBT2πmAkBThP23/2,
(A7)

where mA denotes the mass of atom A. The partition function of atom B is of the same form. This framework can also be applied to determine the recombination rates of the KpqR reaction

KpqR=iIpjIqkijR=iIpjIqkijDninjnA2nB2,
(A8)
KpqR=npnqnA2nB2KpqD.
(A9)

Similar to the combined excitation-dissociation reaction, the partition functions are used to compute the equilibrium composition of groups p and q and atoms A and B.

1.
J. D.
Anderson
,
Hypersonic and High Temperature Gasdynamics
(
McGraw-Hill
,
New York, NY
,
1989
).
2.
C.
Park
,
Nonequilibrium Hypersonic Aerothermodynamics
(
Wiley
,
New York, NY
,
1990
).
3.
W. J.
Vincenti
and
C. H.
Kruger
,
Introduction to Physical Gasdynamics
(
Wiley
,
New York, NY
,
1965
).
4.
P. A.
Gnoffo
,
R. N.
Gupta
, and
J. L.
Shinn
, “
Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium
,” in
NASA Technical Paper 2867
,
1989
.
5.
C.
Park
, “
Assessment of two-temperature kinetic model for ionizing air
,”
J. Thermophys. Heat Transfer
3
,
233
244
(
1989
).
6.
C.
Park
, “
Review of chemical-kinetic problems of future NASA missions. I. Earth entries
,”
J. Thermophys. Heat Transfer
7
,
385
398
(
1993
).
7.
C.
Park
,
J. T.
Howe
,
R. L.
Jaffe
, and
G. V.
Candler
, “
Review of chemical-kinetic problems of future NASA missions. II. Mars entries
,”
J. Thermophys. Heat Transfer
8
,
9
23
(
1994
).
8.
J. P.
Appleton
, “
Shock-tube study of the vibrational relaxation of nitrogen using vacuum-ultraviolet light absorption
,”
J. Chem. Phys.
47
,
3231
3240
(
1968
).
9.
J. P.
Appleton
,
M.
Steinberg
, and
D. J.
Liquornik
, “
Shock-tube study of nitrogen dissociation using vacuum-ultraviolet light absorption
,”
J. Chem. Phys.
48
,
599
608
(
1968
).
10.
K.
Miki
,
M.
Panesi
,
E. E.
Prudencio
, and
S.
Prudhomme
, “
Estimation of the nitrogen ionization reaction rate using electric arc shock tube data and Bayesian model analysis
,”
Phys. Plasmas
19
,
023507
(
2012
).
11.
K.
Miki
,
M.
Panesi
,
E.
Prudencio
, and
S.
Prudhomme
, “
Probabilistic models and uncertainty quantification for the ionization reaction rate of atomic nitrogen
,”
J. Comput. Phys.
231
,
3871
3886
(
2012
).
12.
M.
Capitelli
,
C. M.
Ferreira
,
B. F.
Gordiets
, and
A. I.
Osipov
,
Plasma Kinetics in Atmospheric Gases
(
Springer
,
2000
).
13.
M.
Panesi
,
T. E.
Magin
,
A.
Bourdon
,
A.
Bultel
, and
O.
Chazot
, “
Fire II flight experiment analysis by means of a collisional-radiative model
,”
J. Thermophys. Heat Transfer
23
,
236
248
(
2009
).
14.
M.
Panesi
,
T. E.
Magin
,
A.
Bourdon
,
A.
Bultel
, and
O.
Chazot
, “
Electronic excitation of atoms and molecules for the FIRE II flight experiment
,”
J. Thermophys. Heat Transfer
25
,
361
374
(
2011
).
15.
M. G.
Kapper
and
J.-L.
Cambier
, “
Ionizing shocks in argon. Part I: Collisional-radiative model and steady-state structure
,”
J. Appl. Phys.
109
,
113308
(
2011
).
16.
R. L.
Macdonald
,
A.
Munafò
,
C. O.
Johnston
, and
M.
Panesi
, “
Nonequilibrium radiation and dissociation of co molecules in shock-heated flows
,”
Phys. Rev. Fluids
1
,
043401
(
2016
).
17.
A.
Bultel
,
B. G.
Chéron
,
A.
Bourdon
,
O.
Motapon
, and
I. F.
Schneider
, “
Collisional-radiative model in air for earth re-entry problems
,”
Phys. Plasmas
13
,
043502
(
2006
).
18.
A.
Bultel
and
J.
Annaloro
, “
Elaboration of collisional-radiative models for flows related to planetary entries into the Earth and Mars atmospheres
,”
Plasma Sources Sci. Technol.
22
,
025008
(
2013
).
19.
G.
Colonna
,
I.
Armenise
,
D.
Bruno
, and
M.
Capitelli
, “
Reduction of state-to-state kinetic to macroscopic models in hypersonic flows
,”
J. Thermophys. Heat Transfer
20
,
477
486
(
2006
).
20.
E.
Josyula
,
W. F.
Bailey
, and
S. M.
Ruffin
, “
Reactive and nonreactive vibrational energy exchanges in nonequilibrium hypersonic flows
,”
Phys. Fluids
15
,
3223
3235
(
2003
).
21.
A.
Aliat
,
P.
Vedula
, and
E.
Josyula
, “
State-to-state modeling of radiation coupled to vibration-translation relaxation and dissociation in nonequilibrium gas flows
,”
Phys. Rev. E
83
,
067302
(
2011
).
22.
A.
Aliat
,
P.
Vedula
, and
E.
Josyula
, “
State-specific dissociation modeling with multiquantum vibration-translation transitions
,”
Phys. Rev. E
83
,
037301
(
2011
).
23.
D. A.
Andrienko
and
I. D.
Boyd
, “
High fidelity modeling of thermal relaxation and dissociation of oxygen
,”
Phys. Fluids
27
,
116101
(
2015
).
24.
D.
Andrienko
and
I. D.
Boyd
, “
Investigation of oxygen vibrational relaxation by quasi-classical trajectory method
,”
Chem. Phys.
459
,
1
13
(
2015
).
25.
D. A.
Andrienko
and
I. D.
Boyd
, “
Master equation study of vibrational and rotational relaxations in oxygen
,”
J. Thermophys. Heat Transfer
30
,
533
552
(
2015
).
26.
D. A.
Andrienko
and
I. D.
Boyd
, “
Kinetic models of oxygen thermochemistry based on quasi-classical trajectory analysis
,”
J. Thermophys. Heat Transfer
0
,
1
13
(
2016
).
27.
D. A.
Andrienko
and
I. D.
Boyd
, “
Thermal relaxation of molecular oxygen in collisions with nitrogen atoms
,”
J. Chem. Phys.
145
,
014309
(
2016
).
28.
D. A.
Andrienko
and
I. D.
Boyd
, “
State-specific dissociation in O2–O2 collisions by quasiclassical trajectory method
,”
Chem. Phys.
491
,
74
81
(
2017
).
29.
K.
Neitzel
,
D.
Andrienko
, and
I. D.
Boyd
, “
Aerothermochemical nonequilibrium modeling for oxygen flows
,”
J. Thermophys. Heat Transfer
31
,
634
645
(
2017
).
30.
D. A.
Gonzales
and
P. L.
Varghese
, “
Vibrational relaxation models for dilute shock heated gases
,”
Chem. Phys.
195
,
83
91
(
1995
).
31.
S. W.
Benson
and
T.
Fueno
, “
Mechanism of atom recombination by consecutive vibrational deactivations
,”
J. Chem. Phys.
36
,
1597
1607
(
1962
).
32.
J.
Keck
and
G.
Carrier
, “
Diffusion theory of nonequilibrium dissociation and recombination
,”
J. Chem. Phys.
43
,
2284
2298
(
1965
).
33.
R. N.
Schwartz
,
Z. I.
Slawsky
, and
K. F.
Herzfeld
, “
Calculation of vibrational relaxation times in gases
,”
J. Chem. Phys.
20
,
1591
1599
(
1952
).
34.
D.
Rapp
and
T.
Kassal
, “
Theory of vibrational energy transfer between simple molecules in nonreactive collisions
,”
Chem. Rev.
69
,
61
102
(
1969
).
35.
D. B.
Landrum
and
G. V.
Candler
, “
Development of a new model for vibration-dissociation coupling in nitrogen
,” AIAA Paper No. 92-2853,
1992
.
36.
E. H.
Kerner
, “
Note on the forced and damped oscillator in quantum mechanics
,”
Can. J. Phys.
36
,
371
377
(
1958
).
37.
C. E.
Treanor
, “
Vibrational energy transfer in high energy collisions
,”
J. Chem. Phys.
43
,
532
538
(
1965
).
38.
A.
Zelechow
,
D.
Rapp
, and
T. E.
Sharp
, “
Vibrational-vibrational-translational energy transfer between two diatomic molecules
,”
J. Chem. Phys.
49
,
286
299
(
1968
).
39.
J. D.
Kelley
, “
Vibrational energy transfer processes in collisions between diatomic molecules
,”
J. Chem. Phys.
56
,
6108
6117
(
1972
).
40.
I. V.
Adamovich
,
S. O.
Macheret
,
J. W.
Rich
, and
C. E.
Treanor
, “
Vibrational energy transfer rates using a forced harmonic oscillator model
,”
J. Thermophys. Heat Transfer
12
,
57
65
(
1998
).
41.
E. V.
Kustova
and
E. A.
Nagnibeda
, “
Kinetic model for multi-temperature flows of reacting carbon dioxide mixture
,”
Chem. Phys.
398
,
111
117
(
2012
).
42.
I. V.
Adamovich
,
J. W.
Rich
, and
S. O.
Macheret
, “
Existence of the bottleneck in vibrational relaxation of diatomic molecules
,”
J. Thermophys. Heat Transfer
11
,
261
265
(
1997
).
43.
S. O.
Macheret
and
I. V.
Adamovich
, “
Semiclassical modeling of state-specific dissociation rates in diatomic gases
,”
J. Chem. Phys.
113
,
7351
7361
(
2000
).
44.
K.
Haug
,
D. G.
Truhlar
, and
N. C.
Blais
, “
Monte Carlo trajectory and master equation simulation of the nonequilibrium dissociation rate coefficient for Ar + H2 → Ar + 2H at 4500 K
,”
J. Chem. Phys.
86
,
2697
2716
(
1987
).
45.
K.
Haug
,
D. G.
Truhlar
, and
N. C.
Blais
, “
Erratum: Monte Carlo trajectory and master equation simulation of the nonequilibrium dissociation rate coefficient for Ar + H2 → Ar + 2H at 4500 K [J. Chem. Phys. 86, 2697 (1987)]
,”
J. Chem. Phys.
96
,
5556
5557
(
1992
).
46.
I.
Procaccia
and
R. D.
Levine
, “
Vibrational energy transfer in molecular collisions: An information theoretic analysis and synthesis
,”
J. Chem. Phys.
63
,
4261
4279
(
1975
).
47.
M.
Panesi
,
T. E.
Magin
,
A.
Munafò
,
A.
Bourdon
,
R.
Jaffe
, and
D. W.
Schwenke
, “
Rovibrational internal energy excitation and dissociation of nitrogen in hypersonic flows
,” in
Proceedings of the Summer Program
(
Center for Turbulent Research, Stanford University
,
Stanford, CA
,
2010
).
48.
M.
Panesi
,
R. L.
Jaffe
,
D. W.
Schwenke
, and
T. E.
Magin
, “
Rovibrational internal energy transfer and dissociation of N(Su4)+N2(Σg+1) system in hypersonic flows
,”
J. Chem. Phys.
138
,
044312
(
2013
).
49.
J. G.
Kim
and
I. D.
Boyd
, “
State-resolved master equation analysis of thermochemical nonequilibrium of nitrogen
,”
Chem. Phys.
415
,
237
246
(
2013
).
50.
F.
Esposito
and
M.
Capitelli
, “
Quasi-classical molecular dynamic calculations of vibrationally and rotationally state selected dissociation cross sections: N + N2(v, J) → 3N
,”
Chem. Phys. Lett.
302
,
49
54
(
1999
).
51.
F.
Esposito
,
C.
Gorse
, and
M.
Capitelli
, “
Quasi-classical dynamics calculations and state-selected rate coefficients for H + H2(v, j) → 3H processes: Application to the global dissociation rate under thermal equilibrium conditions
,”
Chem. Phys. Lett.
303
,
636
640
(
1999
).
52.
F.
Esposito
and
M.
Capitelli
, “
Quasi-classical dynamics and vibrational kinetics of N + N2(v) system
,”
Chem. Phys.
257
,
193
202
(
2000
).
53.
F.
Esposito
,
I.
Armenise
, and
M.
Capitelli
, “
N–N2 state-to-state vibrational relaxation and dissociation rate coefficients based on quasi-classical calculations
,”
Chem. Phys.
331
,
1
8
(
2006
).
54.
F.
Esposito
and
M.
Capitelli
, “
QCT calculations for the process N2(v) + N → N2(v′) + N in the whole vibrational range
,”
Chem. Phys. Lett.
418
,
581
585
(
2006
).
55.
F.
Esposito
and
M.
Capitelli
, “
The relaxation of vibrationally excited O2 molecules by atomic oxygen
,”
Chem. Phys. Lett.
443
,
222
226
(
2007
).
56.
F.
Esposito
,
I.
Armenise
,
G.
Capitta
, and
M.
Capitelli
, “
O–O2 state-to-state vibrational relaxation and dissociation rates based on quasiclassical calculations
,”
Chem. Phys.
351
,
91
98
(
2008
).
57.
D.
Bose
and
G. V.
Candler
, “
Thermal rate constants of the N2 + O → NO + N reaction using ab initio 3A′′ and 3A′ potential energy surfaces
,”
J. Chem. Phys.
104
,
2825
(
1996
).
58.
D.
Bose
and
G. V.
Candler
, “
Thermal rate constants of the O2 + N → NO + O reaction using ab initio 2A′ and 4A′′ potential energy surfaces
,”
J. Chem. Phys.
107
,
6136
6145
(
1997
).
59.
R. L.
Jaffe
,
D. W.
Schwenke
,
G.
Chaban
, and
W.
Huo
, “
Vibrational and rotational excitation and relaxation of nitrogen from accurate theoretical calculations
,” AIAA Paper 2008-1208,
2008
.
60.
G.
Chaban
,
R. L.
Jaffe
,
D. W.
Schwenke
, and
W.
Huo
, “
Dissociation cross-sections and rate coefficients for nitrogen from accurate theoretical calculations
,” AIAA Paper 2008-1209,
2008
.
61.
R. L.
Jaffe
,
D. W.
Schwenke
, and
G.
Chaban
, “
Vibration-rotation excitation and dissociation in N2–N2 collisions from accurate theoretical calculations
,” AIAA Paper 2010-4517,
2010
.
62.
D. A.
Andrienko
and
I. D.
Boyd
, “
Rovibrational energy transfer and dissociation in O2–O collisions
,”
J. Chem. Phys.
144
,
104301
(
2016
).
63.
I. S.
Ulusoy
,
D. A.
Andrienko
,
I. D.
Boyd
, and
R.
Hernandez
, “
Quantum and quasi-classical collisional dynamics of O2–Ar at high temperatures
,”
J. Chem. Phys.
144
,
234311
(
2016
).
64.
H.
Luo
,
M.
Kulakhmetov
, and
A.
Alexeenko
, “
Ab initio state-specific N2 + O dissociation and exchange modeling for molecular simulations
,”
J. Chem. Phys.
146
,
074303
(
2017
).
65.
J.
Li
,
P. J. S. B.
Caridade
, and
A. J. C.
Varandas
, “
Quasiclassical trajectory study of the atmospheric reaction N(2D) + NO(X2Π) → O(1D) + N2(X1Σg+)
,”
J. Phys. Chem. A
118
,
1277
1286
(
2014
).
66.
A.
Lombardi
,
N.
Faginas-Lago
,
L.
Pacifici
, and
A.
Costantini
, “
Modeling of energy transfer from vibrationally excited CO2 molecules: Cross sections and probabilities for kinetic modeling of atmospheres, flows, and plasmas
,”
J. Phys. Chem. A
117
,
11430
11440
(
2013
).
67.
M.
Kulakhmetov
,
M.
Gallis
, and
A.
Alexeenko
, “
Ab initio-informed maximum entropy modeling of rovibrational relaxation and state-specific dissociation with application to the O2 + O system
,”
J. Chem. Phys.
144
,
174302
(
2016
).
68.
G. D.
Purvis
and
R. J.
Bartlett
, “
A full coupled-cluster singles and doubles model: The inclusion of disconnected triples
,”
J. Chem. Phys.
76
,
1910
1918
(
1982
).
69.
K.
Raghavachari
,
G. W.
Trucks
,
J. A.
Pople
, and
M.
Head-Gordon
, “
A fifth–order perturbation comparison of electron correlation theories
,”
Chem. Phys. Lett.
157
,
479
483
(
1989
).
70.
R. J.
Gdanitz
and
R.
Ahlrichs
, “
The averaged coupled-pair functional (ACPF): A size-extensive modification of MR CI(SD)
,”
Chem. Phys. Lett.
143
,
413
420
(
1988
).
71.
D. W.
Schwenke
, “
Calculations of rate constants for three-body recombination of H2 in the presence of H2
,”
J. Chem. Phys.
89
,
2076
2091
(
1988
).
72.
M.
Karplus
,
R. N.
Porter
, and
R. D.
Sharma
, “
Exchange reactions with activation energy. I. Simple barrier potential for (H, H2)
,”
J. Chem. Phys.
43
,
3259
3287
(
1965
).
73.
D. G.
Truhlar
and
J. T.
Muckerman
, “
Reactive scattering cross sections. III: Quasiclassical and semiclassical methods
,” in
Atom-Molecule Collision Theory: A Guide for the Experimentalist
, edited by
R. B.
Bernstein
(
Springer US
,
1979
), pp.
505
566
.
74.
R. L.
Jaffe
,
D. W.
Schwenke
, and
M.
Panesi
, “
First principles calculation of heavy particle rate coefficients
,” in
Hypersonic Nonequilibrium Flows: Fundamentals and Recent Advances
, edited by
E.
Josyula
(
American Institute of Aeronautics and Astronautics
,
2015
), Chap. 3, pp.
103
158
.
75.
F.
Esposito
,
E.
Garcia
, and
A.
Laganà
, “
Comparisons and scaling rules between N + N2 and N2 + N2 collision induced dissociation cross sections from atomistic studies
,”
Plasma Sources Sci. Technol.
26
,
045005
(
2017
).
76.
Y.
Liu
,
M.
Vinokur
,
M.
Panesi
, and
T. E.
Magin
, “
A multi-group maximum entropy model for thermo-chemical nonequilibrium
,” AIAA Paper 2010-4332,
2010
.
77.
Y.
Liu
,
M.
Panesi
,
A.
Sahai
, and
M.
Vinokur
, “
General multi-group macroscopic modeling for thermo-chemical non-equilibrium gas mixtures
,” AIAA Paper 2014-3205,
2014
.
78.
Y.
Liu
,
M.
Panesi
,
A.
Sahai
, and
M.
Vinokur
, “
General multi-group macroscopic modeling for thermo-chemical non-equilibrium gas mixtures
,”
J. Chem. Phys.
142
,
134109
(
2015
).
79.
A.
Munafò
,
M.
Panesi
,
R. L.
Jaffe
,
G.
Colonna
,
A.
Bourdon
, and
T. E.
Magin
, “
QCT-based vibrational collisional models applied to nonequilibrium nozzle flows
,”
Eur. Phys. J. D
66
,
188
(
2012
).
80.
A.
Munafò
,
M.
Panesi
, and
T. E.
Magin
, “
Boltzmann rovibrational collisional coarse-grained model for internal energy excitation and dissociation in hypersonic flows
,”
Phys. Rev. E
89
,
023001
(
2014
).
81.
A.
Munafò
and
T. E.
Magin
, “
Modeling of stagnation-line nonequilibrium flows by means of quantum based collisional models
,”
Phys. Fluids
26
,
097102
(
2014
).
82.
A.
Munafò
,
Y.
Liu
, and
M.
Panesi
, “
Physical models for dissociation and energy transfer in shock-heated nitrogen flows
,”
Phys. Fluids
27
,
127101
(
2015
).
83.
A.
Munafò
,
N. N.
Mansour
, and
M.
Panesi
, “
A reduced-order NLTE kinetic model for radiating plasmas of outer envelopes of stellar atmospheres
,”
Astrophys. J.
838
,
126
(
2017
).
84.
H. P.
Le
,
A. P.
Karagozian
, and
J.-L.
Cambier
, “
Complexity reduction of collisional-radiative kinetics for atomic plasma
,”
Phys. Plasmas
20
,
123304
(
2013
).
85.
A.
Guy
,
A.
Bourdon
, and
M.-Y.
Perrin
, “
Consistent multi-internal-temperatures models for nonequilibrium nozzle flows
,”
Chem. Phys.
420
,
15
24
(
2013
).
86.
A.
Sahai
,
B.
Lopez
,
C. O.
Johnston
, and
M.
Panesi
, “
Adaptive coarse graining method for energy transfer and dissociation kinetics of polyatomic species
,”
J. Chem. Phys.
147
,
054107
(
2017
).
87.
M.
Panesi
and
A.
Lani
, “
Collisional radiative coarse-grain model for ionization in air
,”
Phys. Fluids
25
,
057101
(
2013
).
88.
K.
Koura
, “
Monte Carlo direct simulation of rotational relaxation of diatomic molecules using classical trajectory calculations: Nitrogen shock wave
,”
Phys. Fluids
9
,
3543
3549
(
1997
).
89.
K.
Koura
, “
Monte Carlo direct simulation of rotational relaxation of nitrogen through high total temperature shock waves using classical trajectory calculations
,”
Phys. Fluids
10
,
2689
2691
(
1998
).
90.
H.
Matsumoto
and
K.
Koura
, “
Comparison of velocity distribution functions in an argon shock wave between experiments and Monte Carlo calculations for Lennard-Jones potential
,”
Phys. Fluids A
3
,
3038
3045
(
1991
).
91.
P.
Norman
,
P.
Valentini
, and
T.
Schwartzentruber
, “
GPU-accelerated classical trajectory calculation direct simulation Monte Carlo applied to shock waves
,”
J. Comput. Phys.
247
,
153
167
(
2013
).
92.
P.
Valentini
,
T. E.
Schwartzentruber
,
J. D.
Bender
,
I.
Nompelis
, and
G. V.
Candler
, “
Direct molecular simulation of nitrogen dissociation based on an ab initio potential energy surface
,”
Phys. Fluids
27
,
086102
(
2015
).
93.
P.
Valentini
,
T. E.
Schwartzentruber
,
J. D.
Bender
, and
G. V.
Candler
, “
Dynamics of nitrogen dissociation from direct molecular simulation
,”
Phys. Rev. Fluids
1
,
043402
(
2016
).
94.
T. E.
Schwartzentruber
,
M. S.
Grover
, and
P.
Valentini
, “
Direct molecular simulation of nonequilibrium dilute gases
,”
J. Thermophys. Heat Transfer
0
,
1
12
(
2017
).
95.
R. L.
Macdonald
,
M. S.
Grover
,
T. E.
Schwartzentruber
, and
M.
Panesi
, “
Construction of a coarse-grain quasi-classical trajectory method. Part II: Comparison against the direct molecular simulation method
,”
J. Chem. Phys.
148
,
054310
(
2018
).
96.
R. J.
Le Roy
,
Y.
Huang
, and
C.
Jary
, “
An accurate analytic potential function for ground-state N2 from a direct-potential-fit analysis of spectroscopic data
,”
J. Chem. Phys.
125
,
164310
(
2006
).
97.
J. D.
Bender
,
P.
Valentini
,
I.
Nompelis
,
Y.
Paukku
,
Z.
Varga
,
D. G.
Truhlar
,
T.
Schwartzentruber
, and
G. V.
Candler
, “
An improved potential energy surface and multi-temperature quasiclassical trajectory calculations of N2 + N2 dissociation reactions
,”
J. Chem. Phys.
143
,
054304
(
2015
).
98.
M.
Capitelli
, in
Nonequilibrium Vibrational Kinetics
, Topics in Current Physics, edited by
M.
Capitelli
(
Springer
,
Berlin
,
1986
), Vol. 39.
99.
S. P.
Sharma
,
W. M.
Huo
, and
C.
Park
, “
Rate parameters for coupled vibration-dissociation in a generalized SSH approximation
,”
J. Thermophys. Heat Transfer
6
,
9
21
(
1992
).
100.
J. R.
Barker
,
L. M.
Yoder
, and
K. D.
King
, “
Vibrational energy transfer modeling of nonequilibrium polyatomic reaction systems
,”
J. Phys. Chem. A
105
,
796
809
(
2001
).
101.
R. L.
Jaffe
,
D. W.
Schwenke
,
M.
Grover
,
P.
Valentini
,
T. E.
Schwartzentruber
,
S.
Venturi
, and
M.
Panesi
, “
Comparison of quantum mechanical and empirical potential energy surfaces and computed rate coefficients for N2 dissociation
,” AIAA Paper 2016-0503,
2016
.
102.
D.
Kewley
and
H.
Hornung
, “
Free-piston shock-tube study of nitrogen dissociation
,”
Chem. Phys. Lett.
25
,
531
536
(
1974
).
103.
R. K.
Hanson
and
D.
Baganoff
, “
Shock-tube study of nitrogen dissociation rates using pressure measurements
,”
AIAA J.
10
,
211
215
(
1972
).
104.
C.
Park
, “
Two-temperature interpretation of dissociation rate data for N2 and O2
,” AIAA Paper No. 1988-458,
1988
.
105.
R. C.
Millikan
and
D. R.
White
, “
Systematics of vibrational relaxation
,”
J. Chem. Phys.
39
,
3209
3214
(
1963
).
106.
I. D.
Boyd
and
E.
Josyula
, “
State resolved vibrational relaxation modeling for strongly nonequilibrium flows
,”
Phys. Fluids
23
,
057101
(
2011
).
107.
J. P.
Appleton
and
M.
Steinberg
, “
Vacuum-ultraviolet absorption of shock-heated vibrationally excited nitrogen
,”
J. Chem. Phys.
46
,
1521
1529
(
1967
).