A rovibrationally state-specific collision model for the $O2(\Sigma g\u22123)+O(P3)$ system is presented for direct simulation Monte Carlo, including rotation–vibration–translation energy transfer, exchange, dissociation, and recombination processes. The two-step binary collision approach is employed to model recombination reactions. Two available cross section databases by Andrienko/Boyd and Esposito/Capitelli are employed for the rovibrationally resolved model (rv-STS) and vibrationally resolved model (v-STS), respectively. The difference between rv-STS and v-STS comes from two contributions: the multisurface factor of dissociation (*f*_{MS}) and the rotational averaging process. The dissociation cross section with the constant *f*_{MS} is typically larger than with the variable *f*_{MS}, especially for the low vibrational energy states. On the other hand, the cross sections resulting from the rotationally averaged database are found to underpredict the dissociation rate coefficient at low temperatures. In the rovibrational heating case, the rv-STS predicts faster relaxation than the v-STS, which also shows a lower quasi-steady-state temperature than v-STS. In the rovibrational cooling case, the rv-STS shows a faster relaxation than v-STS, which also presents a thermal non-equilibrium between rovibrational and translational mode during the cooling process.

## I. INTRODUCTION

Using state-specific models to investigate the flow field surrounding hypersonic vehicles has gained significant attention due to the improved accuracy over multi-temperature models, as well as the availability of potential energy surface data and corresponding collision cross section or rate coefficients calculated by quasi-classical trajectory (QCT) methods. Due to the complex thermochemical processes such as internal energy transfer, dissociation, recombination, ionization, and radiation, regions of the flow including shock layers, boundary layers, and the wake surrounding the vehicle are in thermal and chemical non-equilibrium. A number of systems have been investigated in previous studies due to their importance in Earth atmospheric reentry, such as the $N2(\Sigma g+1)+N2(\Sigma g+1)$ system,^{1–5} the $N2(\Sigma g+1)+N(Su4)$ system,^{6–10} and the $O2(X3\Sigma g\u2212)+O(P3)$ system.^{11–13} Nonequilibrium effects have often been found to manifest through a complex coupling of relaxation processes, including dissociation coupled with internal energy relaxation.^{14–17}

Comparing with the N_{2} + N system, the O_{2} + O system exhibits several unique characteristics. First of all, the energy of the first lowest electronic excitation states of O_{2} is significantly below that of its dissociation energy. Therefore, a multisurface factor is usually applied to dissociation cross sections/rate coefficients to capture the dissociation from the electronically excited states of O_{2}.^{18,19} Second, the characteristic relaxation time of the rotational and vibrational mode of the O_{2} + O system is of similar order through a wide temperature range,^{11} which indicates the importance of capturing the rotational non-equilibrium for the O3 system. It should be pointed out that in the most recent study by Varga *et al.*, nine potential energy surfaces of the ground state O_{2} + O system are presented,^{20} and the state-specific rate coefficients of vibrational energy transfer and dissociation have been computed by Andrienko based on Varga’s PESs.^{21} The scope of this work focuses on the single potential surface by Varandas and Pais,^{22} which corresponds to the 1^{1}*A*′ surface,^{20} for the consistent comparison to our previous work.^{23} This lays the groundwork for future comparison to the O3 system based on the more recent PESs by Varga *et al.*^{20}

Direct simulation Monte Carlo (DSMC) is a widely used method to study rarefied, hypersonic flows and has been extensively developed for reacting flows in the past few decades. Recently, much of this work has focused on the extension of DSMC for state-specific databases^{17,23–29} and comparison with the widely used legacy phenomenological models. Such studies have revealed, for example, that the total-collision-energy (TCE) model underestimates the vibrational preference to the dissociation.^{23,25} Similarly, the Larsen–Borgnakke model results in an internal energy redistribution that favors a transition to low energy states, regardless of the initial state, unlike state-specific models that show preference for mono-quantum transitions.^{30,31}

Gas-phase recombination has been found to play an important role in the net heat flux to hypersonic vehicle surfaces,^{32} which has been demonstrated to account for up to 50% of the surface heat flux to the vehicle wall. In addition, providing a compatible recombination model for state-specific DSMC is crucial in order for the system to reach the proper equilibrium state. This work expands the two-step binary collision (TSBC) framework in our previous work^{23} to include a rovibrationally resolved database, which provides the required cross sections for dissociation and inelastic processes. This information is then used to derive the orbiting-pair (OP) formation cross section and stabilization cross section for the closure of the recombination model, based on detailed balance and microscopic reversibility principles. By employing the TSBC framework, the overall balance and state-specific detailed balance are able to be satisfied in the DSMC simulation.

The objective of this work is to develop a rovibrationally resolved implementation in DSMC, which is then compared to our previous rotationally averaged model.^{23} Furthermore, the comparison between Esposito/Capitelli and Andrienko/Boyd databases is conducted, with emphasis on the effect of rotationally averaging on cross sections and the multisurface factors.^{18,19} The rest of this paper is organized as follows: The rovibrationally resolved cross section database is introduced in Sec. II. In Sec. III, the two-step binary collision approach for modeling recombination with a rovibrationally resolved database is developed. The implementation of the rovibrationally state-specific collision model in DSMC is then proposed in Sec. IV. In Sec. V, the comparison of the Esposito/Capitelli database and the Andrienko/Boyd database is presented. The equilibrium dissociation/recombination rate coefficients are first investigated. The effects of employing the variable multisurface factor for dissociation and employing rotationally averaging on cross sections are then discussed. The discussion of the heating/cooling relaxation test cases in the 0-D heat bath is then presented, followed by conclusions.

## II. ROVIBRATIONALLY RESOLVED STATE-SPECIFIC DATABASE

The rovibrationally resolved database of collision cross sections for the $O2(\Sigma g\u22123)+O(P3)$ system by Andrienko/Boyd^{11} is employed in this work, which was obtained by quasi-classical trajectory (QCT) calculations on the single potential energy surface, which was constructed using the double many-body expansion (DMBE) method for the ground electronic state of ozone by Varandas and Pais.^{22} The rovibrational energy levels were computed using the WKB approximation for the O_{2} potential,^{22} which comprises 3110 rovibrational energy levels of O_{2}($v$, *J*), where $v$ and *J* indicate the vibrational and rotational quantum number, respectively. The rest of this paper shall, however, use a single index, *i*, to denote the rovibrational energy level of the system, where

The cross section data of O_{2} + O is available within the range of relative translational energy from 0.01 eV to 20 eV with 32 grids and variable intervals. The database contains elastic, rotational–vibrational translational (RVT) transition, and dissociation cross sections for each rovibrational state. It should be pointed out that the elastic collision cross sections are computed by setting up the cutoff impact parameter to a value where no more inelastic collision (RVT transition + dissociation) is observed^{33} in the QCT calculation.

The degeneracy factor, *g*_{BB} = 1/27, is applied to the cross sections of RVT energy transition cross sections to account for the statistical weight 1/27 of the 1 ^{1}*A*′ surface when the detailed information from the other eight surfaces is not available.^{11,19} It represents a case that neglects the contribution from other eight surfaces and has been demonstrated as a reasonable approximation at low temperature.^{34} In the QCT calculation, whole processes are assumed adiabatic on the given PES. It is noted, however, that the dissociation of O_{2} molecules is significantly higher than the first excited molecular state, ^{1}Δ, which is about 1 eV above the molecular ground state. Therefore, the multisurface factor is usually introduced. Nikitin^{18} proposed a constant multisurface factor *f*_{MS} = 16/3 for every rovibrational state to account for the dissociation from the electronically excited states of O_{2} whose energy is less than dissociation energy, as listed in Table I. This factor assumes the ratio of the dissociation cross section from an electronically excited state to the ground state is equal to the ratio of the degeneracy of the excited state to the degeneracy of the ground state.

Energy (eV) . | Degeneracy . | State . |
---|---|---|

−5.1 | 3 | $X3\Sigma g\u2212$ |

−3.84 | 2 | a^{1}Δ_{g} |

−3.15 | 1 | $b1\Sigma g+$ |

−1.02 | 1 | $c1\Sigma a\u2212$ |

−0.81 | 6 | C^{3}Δ_{u} |

−0.72 | 3 | $A3\Sigma u+$ |

Energy (eV) . | Degeneracy . | State . |
---|---|---|

−5.1 | 3 | $X3\Sigma g\u2212$ |

−3.84 | 2 | a^{1}Δ_{g} |

−3.15 | 1 | $b1\Sigma g+$ |

−1.02 | 1 | $c1\Sigma a\u2212$ |

−0.81 | 6 | C^{3}Δ_{u} |

−0.72 | 3 | $A3\Sigma u+$ |

Esposito and Capitelli proposed an alternative way to evaluate the contribution of O_{2} on the electronically excited states to dissociation, which is expressed as the “variable multisurface factor.”^{19} Unlike Nikitin who assumes that dissociation from all electronic energy states are possible for every rovibrational energy state, Esposito and Capitelli assumed that for a rovibrational states ($v$, *j*) with energy *E*($v$, *j*) of O_{2}, only the electronic energy states with energy less than *E*($v$, *j*) can be considered as the possible pathway for the dissociation of O_{2}, while the contribution from an electronic state *i* is again proportional to its degeneracy *g*_{i} based on Nikitin’s model.^{18} Therefore, the variable multisurface factor *f*_{MS} at the state ($v$, *j*) can be evaluated by

where *g*_{0} = 3 denotes the degeneracy of the electronic ground state of O_{2} and *g*_{i} is the degeneracy of the electronic energy state i. The variable multisurface factor is employed in the Esposito/Capitelli database. In this work, the constant *f*_{MS} introduced above is used unless otherwise stated.

### A. State-specific detailed balance

The cross section database computed by QCT calculation usually contains both excitation and de-excitation RVT energy transition cross sections. However, using both excitation and de-excitation cross sections in DSMC may lead the simulation toward the incorrect steady state.^{31} To avoid this issue, the principle of microscopic reversibility is applied to each pair of excitation and de-excitation cross sections. Consider a transition from the lower rovibrational energy state *i* to the higher energy state *i*′ (that is, *E*_{i′} > *E*_{i}), where the rovibrational energy of each state is denoted as *E*_{i} and *E*_{i′}, respectively. In this work, the de-excitation cross section *σ*_{i′i} is applied directly from the database, and the corresponding excitation cross section (*σ*_{ii′}) is computed based on the principle of microscopic reversibility with energy conservation,

where *g*_{i} denotes the rotational degeneracy of the rovibrational energy state *i* and *E*_{tr} is the relative translational energy.

### B. Total collision cross sections

For O_{2}(*i*) + O collision, the total collision cross section, $\sigma TO2(i)+O$, is defined as the sum of elastic cross sections (*σii*), inelastic cross sections ($\sigma ^ii\u2032$), and dissociation cross sections ($\sigma ^d,i$),

where *i*′ is the post-collision rovibrational state of O_{2}. All *σ*_{x} (without “hat”) on the right-hand side of the above equations represent the raw data of cross sections from QCT calculation. The total collision cross section in Eq. (5) is then used in the DSMC simulation. It should be noted that, rigorously speaking, the elastic collision cross section *σ*_{ii} could only be computed with scattering theory based on quantum mechanics.^{35} However, the scope of this work only focuses on the internal energy transfer and dissociation/recombination, and all simulations are conducted under the equilibrium or heat-bath conditions where the results are not influenced by elastic cross sections.

### C. Rotational temperature and vibrational temperature

In this work, the energy-equivalent method^{8} is used to compute the rotational and vibrational temperature for a given rovibrational energy distribution function. Consider a rovibrational energy distribution at state *i*, denoted as *f*_{i}. The averaged energy $\u0113x$ is computed as

where *x* can be “r” or “v,” which represents “rotational” or “vibrational,” respectively. *E*_{x,i} is the corresponding energy of rovibrational state *i*. The rotational and vibrational temperatures can be computed by solving the following equations:

where

and the Newton’s method is employed for solution of the equations.

### D. Rotational and vibrational relaxation times

In this work, the e-folding method is adopted to calculate relaxation times^{11,36} in 0-D heat-bath simulation, which assumes relaxation that follows the Landau–Teller equation,

where $ex*(Thb)$ is the averaged energy at the heat-bath temperature *T*_{hb}, *e*_{x}(*t*) corresponds to the averaged energy at time *t*, and the subscript *x* represents either the rotational or vibrational mode. The characteristic relaxation time, *τ*_{x}, is defined as the required time of the system to reach the average energy, *e*_{x,efold},

which is derived from integrating Eq. (15), and *T*_{x,0} is the initial rotational or vibrational temperature.

## III. ROVIBRATIONALLY RESOLVED GAS-PHASE RECOMBINATION MODEL

To model the gas-phase recombination in DSMC, the two-step binary collision (TSBC) approach is adopted, which is initially developed by Koura.^{37} The TSBC model was later extended for the vibrationally resolved QCT data in our previous work.^{23} While the derivation of TSBC is similar to the derivation for a vibrationally resolved (rotationally averaged) database,^{23} the TSBC definition of the orbiting pair (OP) is different under a rovibrationally resolved database, and the derivation is presented below. Consider the state-specific dissociation/recombination of oxygen in TSBC, as shown in the following equation:

where $IBQ$ denotes the set of indices of rovibrational energy states of molecular oxygen. The forward process of dissociation indicated by far left-hand and right-hand sides in Eq. (17) is modeled as usual in DSMC, in which the cross section is used to determine a dissociation probability, resulting in the post-collision formation of three distinct atoms (discussed further in Sec. IV). To retrieve the recombination cross sections from the known dissociation cross sections, both microscopic reversibility and detailed balance must be satisfied in order to reach the proper equilibrium steady-state condition. To accomplish this, the recombination process is considered to occur in two steps [backward process shown in Eq. (17)]: (i) two colliding atoms form an orbiting pair (OP) and (ii) a third atomic oxygen stabilizes OP to form an oxygen molecule at the rovibrational level *i*.

To derive the state-specific recombination probability for DSMC, several assumptions need to be made: (i) The OP is assumed as a unique state, which is internal energy free, and Eq. (17) is assumed as the only possible dissociation/recombination channel in the model; (ii) the OP formation and splitting [forward and backward reactions between stages (2) and (3) in Fig. 1] are assumed to be rapid and in equilibrium;^{37} and (iii) the velocity distribution function of oxygen atoms are Maxwellian. In addition, to make the recombination model compatible with dissociation, the principle of detailed balance and microscopic reversibility is included, as shown in Fig. 1. With those constraints, we want to derive the OP formation cross section ($\sigma OPf$) and the OP stabilization cross section (*σ*_{fi}), which can then be used to evaluate the recombination probability in DSMC.

By definition, the overall recombination rate coefficient (*k*_{r}) can be expressed as

Based on assumption (ii), the second step of the backward reaction [from stage (2) to (1) in Fig. 1] is the rate-determining step of recombination. Therefore, Eq. (18) can be further modified to

where *k*_{stab} is the overall stabilization rate coefficient, which is defined as the sum of de-excitation rate coefficient from OP to each rovibrational energy level i (*k*_{stab} ≡ *∑*_{i}*k*_{fi}). The number density of OP in Eq. (19) is evaluated by considering the equilibrium between OP formation and splitting based on assumption (ii), where

$kOPf$ and $kOPb$ denotes the rate coefficient from stage (3) to (2) and its backward reaction in Fig. 1, respectively. (The direction of the recombination process is considered as “forward” as naming.) Combining Eqs. (19) and (20), the overall recombination rate coefficient can be expressed as

where *τ*_{OP} is introduced as $1/kOPb$.

In equilibrium, the overall recombination rate coefficient *k*_{r}(*T*) can be determined in terms of the equilibrium dissociation rate coefficient and the equilibrium constant. The equilibrium dissociation rate coefficient at temperature *T* can be evaluated by the sum of state-specific dissociation rate coefficients weighted by the Boltzmann rovibrational energy distribution, *f*_{i},

where *Q*_{i} is the rovibrational partition function. The overall equilibrium dissociation rate coefficient can be determined as follows:

where *k*_{d,i} is the dissociation rate coefficient at the energy level *i*.

Recall that the OP formation and splitting are assumed to be rapid. Therefore, the excitation process of the molecule is the rate-determining step of the dissociation process, and the dissociation rate coefficient from the rovibrational energy level *i* is approximately the excitation rate coefficient *k*_{if} [from state (1) to (2) in Fig. 1], that is, *k*_{d,i} ≈ *k*_{if}. The ratio of equilibrium dissociation and recombination rate coefficients is expressed based on the law of mass action,^{38}

where *K*_{eq} is the equilibrium constant. Combining Eqs. (21), (23), and (24) and employing the detailed balance between *k*_{if}(*T*) and *k*_{fi}(*T*),

where *g*_{f} = 1 since OP is assumed as a unique state. Finally, we can get

The reaction rate coefficient and the cross section can be related via kinetic theory,

where $\sigma OPf$ is the OP formation cross section. The OP formation cross section retrieves the equilibrium OP formation rate when integrated over Maxwellian distributions for each reactant atoms at the temperature *T*:

where *ε*_{tr} is the relative translational energy between two oxygen atoms and *μ* is their reduced mass. By introducing the average value of the quantity *E*_{tr}*σ*_{OP}, the integration in Eq. (28) can be expressed as

where $C(T)=12(8kT\pi \mu )1/2(kT)\u22121$. Equation (30) indicates that in the TSBC model, the overall detailed balanced could be satisfied if

The overall stabilization cross section is defined analogously to the overall stabilization rate coefficient

where *E*_{tr} denotes the relative translational energy between OP and the third oxygen atom. These cross sections can be expressed in terms of the dissociation cross sections via microscopic reversibility and energy conservation. Microscopic reversibility relates the two processes by^{39}

where *E*_{tr}′ is the final relative translational energy between recombined O_{2} and O.

In this section, the TSBC model is derived starting from assuming that both rotational and vibrational energy can contribute to dissociation, and then, the OP state is defined as a state that is rovibration free, which are the main differences from the TSBC for vibrationally resolved databases in our previous work.^{23}

## IV. IMPLEMENTATION OF ROVIBRATIONALLY STATE-SPECIFIC MODEL IN DSMC

Since the scope of this work focuses on the internal energy transfer and dissociation by O_{2} + O collisions and O + O + O recombination, the elastic collision between O + O and the collision between O_{2} + O_{2} are not considered in this work.

### A. O_{2} + O collisions

The no-time-counter (NTC) scheme is employed in this work. Expressions of number of collision candidates in each time step and the collision probability are from the Ref. 23. Considering that a simulation particle O_{2} with state *i* collides with another simulation particle O, the total collision cross section is computed based on Eq. (5). The workflow of the O_{2} + O collision procedure is shown in Fig. 2. Once the pair is selected to collide, the post-collision state of O_{2} is first determined by drawing a random number to compare with the cumulative probability array. The probability toward a state is computed by *σ*_{x}/*σ*_{T}, where x denotes as each pathway. If the RVT transition to the post-state *i*′ is chosen, the rovibrational energy of the O_{2} particle is updated to *E*_{rv}(*i*′). If the dissociation pathway is chosen, the simulation particle O_{2} is deleted from the particle list and generates two particles with species O. It is important to note that the energy re-distributing process of dissociation in this DSMC scheme is not a closed problem. The energy re-distribution of the dissociation process is described as follows: consider the equation of energy conservation before and after the dissociation,

A uniformly distributed random number, *γ*, is adopted to distribute *E*′_{tr} among three O particles. The fraction of post-collision relative translational energy *γE*′_{tr} is assigned to OP + O, and the rest of energy (1 − *γ*)*E*′_{tr} is assigned to the final dissociation products. Finally, the post-collision velocities of particles are determined by the conservation of energy and assuming isotropic scattering.

### B. O + O + O collision for recombination

## V. RESULTS

### A. Comparison of databases between Esposito/Capitelli and Andrienko/Boyd

The difference of databases from Esposito/Capitelli^{19} and Andrienko/Boyd^{11} is investigated in this section. The database by Esposito/Capitelli contains the rotationally averaged cross sections, which is employed in our previous work to study the vibrationally resolved state-specific implementation for DSMC.^{23} To make a direct comparison, the rovibrationally resolved cross sections by Andrienko/Boyd is first averaged with respect to the rotational mode. Equations (36) and (37) are used to rotationally average the inelastic and dissociation cross sections, respectively,

where *Q*_{r} denotes the rotational partition function. The degeneracy factor *g*_{BB} = 1/27 is employed in both databases. For the multisurface factor *f*_{MS}, the database by Esposito/Capitelli uses the variable *f*_{MS} they proposed.^{19} In this section, applying both the constant *f*_{MS}^{18} and variable *f*_{MS} on Andrienko/Boyd database is discussed.

Figure 4 shows the VT transition cross sections from $v$ = 1 → 0 and $v$ = 20 → 10 at *T*_{rot} = 3000 K and 20 000 K. It is discovered that for VT transition cross section, both databases show good agreement. It also noted that the cross section data from Andrienko/Boyd has less noise than that from Esposito/Capitelli.

Figure 5 compares dissociation cross sections between two databases from v = 0, v = 20, and v = 40 at *T*_{rot} = 3000 K and 20 000 K. The results of applying constant and variable multisurface factors on the dissociation cross sections by Andrienko/Boyd’s database are both plotted. When the variable *f*_{MS} is employed on Andrienko/Boyd’s data, it again shows a reasonable agreement with Esposito/Capitelli’s data.

The effect of different *f*_{MS} is also shown in Fig. 5. The dissociation cross sections with constant *f*_{MS} = 16/3 show much higher values than using variable *f*_{MS}, especially for the low energy states and low temperatures. It is because the variable *f*_{MS} approaches to 16/3 only when the contribution from high-lying rovibrational states becomes important.

### B. Investigation of dissociation/recombination

The overall dissociation and recombination rate coefficients in the range of equilibrium temperature 4000–20 000 K are shown in Fig. 6. The rate coefficients from experiment by Ibraguimova *et al.*^{40} and our previous work^{23} are also included in Fig. 6 for comparison. It should be emphasized that only the expression of the dissociation rate coefficient is directly provided by Ibraguimova *et al.* in Ref. 40, and the corresponding recombination rate coefficients are retrieved by the dissociation rate coefficient and the equilibrium constant via the detailed balance principle, which is denoted as “Ibraguimova + d.b.” in the following plots. It should also be noted that in Ref. 40, Ibraguimova *et al.* provided two expressions for the dissociation rate coefficients in the low and high temperature range,^{40} while in this work, only the low-temperature expression is indicated for comparison. This is because the high-temperature results by Ibraguimova *et al.* may be influenced by non-equilibrium effects, which shows a significant discrepancy to the current studies, as observed by Kulakhmetov *et al.*^{13} and our previous work.^{23}

In Figs. 6(a) and 6(b), the DSMC simulation results with rovibrationally resolved database (current work) and vibrationally resolved database are denoted as rv-STS and v-STS (Ref. 23), respectively. The simulation is conducted by generating 1 × 10^{5} of O and O_{2} simulation particles in a box with a fixed equilibrium temperature and lasts for a sufficient time (when about 1 × 10^{4} dissociation/recombination events are recorded). The number of dissociation and recombination collisions and spending time are recorded and converted to the corresponding rate coefficients.

In Fig. 6(a), it is found that the dissociation rate coefficient of rv-STS is slightly larger than v-STS when the equilibrium temperature is less than 10 000 K. Similarly, the recombination rate coefficient of rv-STS is found higher than v-STS at low temperatures, as shown in Fig. 6(b). It is also discovered that the recombination rate coefficients of rv-STS show better agreement with the values retrieved from the experiment by Ibraguimova *et al.*^{40} Recall that the main differences between rv-STS and v-STS are the rotationally averaging process and different multisurface factor *f*_{MS}, and Fig. 7 investigates the influence of each of these. The recombination rate coefficients shown in Fig. 7(b) are computed by integrating the dissociation cross sections with corresponding Maxwellian distribution at the given temperature to get the dissociation rate coefficients first, as shown in Fig. 7(a); then, the recombination rate coefficients are computed based on detailed balance. Consider the solid red line shown in Fig. 7(b), which denotes the use of Andrineko/Boyd database with constant *f*_{MS} = 16/3, as introduced in Sec. II. When we change the constant *f*_{MS} to variable *f*_{MS} (red dashed line), it is found that the recombination rate coefficient decreases uniformly at all temperatures. When it further undergoes the rotationally averaging (equivalent to the Esposito/Capitelli database, blue dashed line), the recombination rate coefficient decreases further, especially at the low temperature range. Therefore, the reason of observing the underestimation of recombination rate coefficients at low temperature in Fig. 6(a) is mainly due to the use application of rotationally averaging in the QCT database.

### C. Rotational and vibrational relaxation times

The rv-STS implementation in DSMC is first verified by evaluating the rotational and vibrational relaxation times under the heat-bath conditions, where the dissociation/recombination are turned off in these simulations. The initial rovibrational temperatures are 100 K with varying heat-bath temperature from 1000 K to 20 000 K. The initial number density of oxygen atoms is 9.99 × 10^{23}/m^{3}, and 1.0 × 10^{21}/m^{3} for molecular oxygen. 1 × 10^{6} simulation particles are employed in these simulations, where the initial rovibrational energy state of particles is drawn from the Boltzmann distribution at temperature 100 K. The time step of 1.0 × 10^{−11} s is applied, and the simulation lasts 1 × 10^{5} time steps. The relaxation time for DSMC is evaluated as follows: after a period of simulation time, when the averaged rotational (or vibrational) energy equals to the quantity computed by Eq. (16), that period is defined as the rotational (or vibrational) relaxation time for DSMC heat-bath simulation.

The results are compared with the master equation solution by Andrienko and Boyd,^{11} as shown in Fig. 8. It shows that the current database with microscopic reversibility treatment shows a reasonable agreement with Andrienko and Boyd results^{11} for both rotational and vibrational relaxation times in the temperature range of interest.

### D. Relaxation process with dissociation/recombination in 0-D heat bath

The rv-STS is further adopted to study the rotation–vibration–translation coupling effect in 0-D heat-bath simulations. Both heating and cooling of the rovibrational mode are investigated, and the initial condition in each simulation is listed in Table II. About 10 000 simulation particles are applied initially whose internal energy is drawn from a Boltzmann distribution at initial rovibrational temperature. The time step 5 × 10^{−13} s is used for both heating and cooling cases. The results of using v-STS^{23} are compared in both relaxation cases.

Case . | T_{tr} (K)
. | T_{r,0} (K)
. | T_{v,0} (K)
. | n_{O,0}
. | $nO2,0$ . |
---|---|---|---|---|---|

Heating | 7000 | 2000 | 2000 | 1.48 × 10^{24} | 6.79 × 10^{24} |

Cooling | 2000 | 7000 | 7000 | 1.00 × 10^{25} | 2.51 × 10^{24} |

Case . | T_{tr} (K)
. | T_{r,0} (K)
. | T_{v,0} (K)
. | n_{O,0}
. | $nO2,0$ . |
---|---|---|---|---|---|

Heating | 7000 | 2000 | 2000 | 1.48 × 10^{24} | 6.79 × 10^{24} |

Cooling | 2000 | 7000 | 7000 | 1.00 × 10^{25} | 2.51 × 10^{24} |

#### 1. Rovibrational heating case (dissociation regime)

In this test case, the evolution of the system can be divided into four distinct stages: (i) rovibrational heating, (ii) quasi-steady-state (QSS), (iii) final relaxation, and (iv) equilibrium, where the detailed description can be found in our previous work.^{23} In this work, we focus on the comparison between rv-STS and v-STS models. The evolution of energy-equivalent vibrational temperatures^{8,11} are shown in Fig. 9(a). In Fig. 9(a), QSS is found in both cases. However, the QSS vibrational temperature of rv-STS is lower than the v-STS by about 500 K, and the duration is shorter than v-STS. It is also discovered that the relaxation of rv-STS is faster than the v-STS: it indicates that the rovibrational model predicts stronger dissociation during QSS, which also leads to the system reaching equilibrium state faster (∼4.5 × 10^{−7} s) than v-STS (∼6.0 × 10^{−7} s), as shown in Fig. 9(b).

#### 2. Rovibrational cooling case (recombination regime)

In the cooling case, the initial gas composition is set to promote recombination. Figure 10(b) shows the early stage during the full cooling process since the most interesting phenomenon is occurred at this stage. It is found that the recombination predicted by rv-STS is much stronger than v-STS when *t* < 5 × 10^{−}7 s, which leads to a larger difference between vibrational and translational temperature, as shown in Fig. 10(a), and increases O_{2} faster, as shown in Fig. 10(b).

## VI. CONCLUSION

In this work, a rovibrationally resolved state-specific model (rv-STS) in DSMC is presented based on the cross section database by Andrienko/Boyd. The rovibrational energy transfer, dissociation, and recombination in the O_{2} + O system are studied in particular. To model the recombination in DSMC, the two-step binary collision framework is applied, which ensures that the overall detailed balance, as well as the state-specific detailed balance between dissociation and recombination with the rovibrationally resolved cross sections, is satisfied. The concept of the no-time-counter scheme is applied for the binary collisions and is also extended for the state-specific recombination model.

The comparison of the Andrienko/Boyd database used in current work and the Esposito/Capitelli database used in our previous work is first proposed to investigate the effect of applying rotationally averaging and different multisurface factors on the collision cross sections. While the RVT transition cross sections of two databases match each other, the dissociation cross section of Andrienko/Boyd is much larger than Esposito/Capitelli due to the use of constant *f*_{MS}. In the equilibrium test case, both dissociation and recombination rate coefficients are underestimated by v-STS at the low temperature range, which is mainly due to the rotationally averaging process. However, it should be aware that while the rv-STS result is closer to the result by Ibraguimova *et al.*, the O_{2} + O dissociation coefficient proposed in their paper is not measured directly from the experiment. Instead, it is a secondary result and involves models and assumptions.

The rv-STS and v-STS were next compared in 0-D rovibrational heating and cooling cases. In the heating case, the rv-STS predicts a shorter time to reach the QSS temperature and final equilibrium than the prediction by v-STS. The rv-STS model also predicts a smaller QSS temperature than the v-STS model. When it comes to the cooling case, rv-STS also predicts stronger recombination than v-STS, which leads to the observation of thermal non-equilibrium.

## ACKNOWLEDGMENTS

This work was supported by an Early Career Faculty grant from the NASA’s Space Technology Research Grants Program and the Air Force Office of Scientific Research under Award No. FA9550-17-1-0127. The authors gratefully acknowledge Dr. D. A. Andrienko for providing the QCT database and Dr. F. Esposito for many useful discussions.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.