A master equation study of vibrational relaxation and dissociation of oxygen is conducted using state-specific O_{2}–O transition rates, generated by extensive trajectory simulations. Both O_{2}–O and O_{2}–O_{2} collisions are concurrently simulated in the evolving nonequilibrium gas system under constant heat bath conditions. The forced harmonic oscillator model is incorporated to simulate the state-to-state relaxation of oxygen in O_{2}–O_{2} collisions. The system of master equations is solved to simulate heating and cooling flows. The present study demonstrates the importance of atom-diatom collisions due to the extremely efficient energy randomization in the intermediate O_{3} complex. It is shown that the presence of atomic oxygen has a significant impact on vibrational relaxation time at temperatures observed in hypersonic flow. The population of highly-excited O_{2} vibrational states is affected by the amount of atomic oxygen when modeling the relaxation under constant heat bath conditions. A model of coupled state-to-state vibrational relaxation and dissociation of oxygen is also discussed.

## I. INTRODUCTION

Hypersonic flight experiments, conducted during the past several decades, have revealed a necessity of coupling the experimental and theoretical approaches in order to improve the existing computational models of nonequilibrium air plasma. One of the important aspects of hypersonic aerothemodynamics is the energy exchange among the translational and internal degrees of freedom (DOFs) of species behind a shock wave. Since the characteristic times of excitation of translational and vibrational modes differ by orders of magnitude, the hypersonic flow can be in a state of thermal nonequilibrium. Furthermore, the excitation of internal DOF is coupled to the dissociation process. Thus, it is important to accurately model the energy exchange in order to predict the structure of the shock wave.

Among other phenomena that take place in shock waves, the vibrational excitation and deactivation play an important role in the energy balance. For the past decade, a significant improvement in the fidelity of existing models of air species has been accomplished. Most of the progress has been achieved in modeling of molecular nitrogen,^{1–3} because of its importance in re-entry flows.

Systems that contain oxygen are studied less often, mainly because oxygen quickly dissociates in a strong shock wave. However, a significant amount of molecular oxygen is observed during hypersonic cruise flight in the post-shock region. This flight regime is inherent in hypersonic vehicles which travel at a speed of about 2–3 km/s. The vibrational relaxation in O_{2}–O collisions is known to proceed several orders of magnitude faster than in other types of collisions in air and does not follow the conventional dependence on temperature of the gas flow.^{4} These factors make a state-to-state kinetic approach a very desirable technique, since it is possible to model thermal nonequilibrium in shock and expanding flows without invoking the thermodynamic definition of temperature.

The state-specific transition rates in O_{2}–O collisions were recently computed^{5,6} on the accurate, many-body potential energy surface^{7} using the QCT method. These rates, were employed then in the master equation simulation,^{8} that revealed an anomalously fast vibrational relaxation time in atom-diatom collisions, compared to other molecular systems containing oxygen. These findings were related to the existence of a large attractive component in the O_{3} PES that leads to an extremely efficient energy randomization. The present paper extends the analysis of O_{2}–O vibrationally resolved rates by adding the dissociation channel in the system of master equations.

The state-resolved kinetic models are of significant interest due to their accuracy in describing the nonequilibrium flows over the large range of input parameters. These models are applied to describe kinetics of hypersonic flow in shock waves,^{9,10} nozzle flow,^{11} and in boundary layers.^{12} Under nonequilibrium conditions, the population of the vibrational manifold can be appreciably different from the equilibrium distribution. In shock waves, highly energetic vibrational states are underpopulated and dissociative reactions are dominant. The multi-temperature model is the alternative to the state-to-state models for describing shock flows with the vibrational temperature less than the translational one. However, when the opposite situation takes place, i.e., the flow cools down, the selective recombination pumps energy into high vibrational states, forming the long plateau of the vibrational distribution. The multi-temperature model is not adequate in this regime since the global chemical rates exhibit strong non-Arrhenius behavior. The rates of dissociation in strongly recombining flow are related to either the atomic concentration or to the population of last vibrational state rather than to the vibrational temperature. A comprehensive investigation of N_{2}/N cooling flows was performed by Colonna and coworkers.^{13,14} In the present work, some insight on O_{2}/O recombination flow is obtained by implementing the present O_{2}–O rates.

In order to bring the present simulation closer to real conditions, O_{2}–O_{2} collisions are included as well. The atom-diatom and diatom-diatom collisions are simultaneously modeled in the present work. Because an accurate trajectory simulation of bimolecular collisions in oxygen gas is yet to be conducted, the Forced Harmonic Oscillator (FHO) model^{15} is employed to obtain the O_{2}–O_{2} vibration-translation (VT) and vibration-vibration (VV) transition rates.

To model the state-resolved O_{2}–O_{2} dissociation, the QCT rates of bound-free transitions in O_{2}–O collisions are scaled using the available experimental data.^{16} The idea of scaling is based on the importance of the repulsive brunch of the two-body potential, as follows from the trajectory study of the O_{2}–O system on potential energy surfaces of different fidelities.^{17} An alternative approach, based on the model of preferential dissociation,^{18} is used to obtain dissociation rates in O_{2}–O_{2} collisions. The details of these methods are provided in the main body of the present paper.

This paper is organized as follows. Section II provides background on the O_{2} molecular structure and addresses the main features of the master equations and thermodynamic model. The detailed study of vibrational relaxation and dissociation in oxygen gas is presented in Section III. Conclusions are provided in Section IV.

## II. GOVERNING EQUATIONS

### A. Molecular structure

The O_{2}–O vibration-to-translation (VT) transition rates, employed in the present work, are obtained on the accurate potential energy surface by Varandas and Pais^{7} using the QCT method.^{19} The Varandas PES generates 47 vibrational levels and a maximum of 236 rotational levels for molecular oxygen in the ground electronic state. The total number of rovibrational levels in the ground electronic state of oxygen is 6245, however, taking into account the nuclear spin statistics of a homonuclear molecule, the even numbered rotational levels for the O$ 2 ( X 3 \Sigma g \u2212 ) $ state are forbidden. Since, during a bound-bound rovibrational transition, the symmetry of the initial and final states cannot change, transitions of the O_{2} molecule in the ground electronic state are only allowed between odd–numbered rotational levels.^{20} The energies of rovibrational states are calculated by the Wentzel–Kramers–Brillouin (WKB) approach.

The cross-sectional data of bound-bound transitions, generated by the QCT method,^{6} is modified to take into account the degeneracy of the spin and orbital momentum of reactants. The degeneracy factor of the O(^{3}*P*) and O$ 2 ( 3 X g \u2212 ) $ reactants can be expressed in the form, given by Gross and Billing^{21}:

The degeneracy factor of the bound-bound transition is found as a ratio of the PES degeneracy to the degeneracy of the reactants. Taking into account that the present O_{3} PES is non-degenerate, the total degeneracy for the reaction of bound-bound transitions is given by Eq. (1). At high temperatures, the degeneracy factor asymptotically approaches a factor of 1/27. This means that the reaction takes place on only one of the 27 possible potential energy surfaces.

The dissociation of molecular oxygen in a shock wave with a temperature above a few thousands of K is unlikely to proceed in the adiabatic manner. One way to account for alternative channels of dissociation from excited electronic states is to assume equilibrium between lower vibrational levels of the excited electronic state and high vibrational levels of the ground state.^{22} The contribution of dissociation from the excited electronic level should be added to the dissociation rate from the ground electronic level. It is possible to calculate the degeneracy factor of the dissociation reaction for each rovibrational level of the ground electronic state individually.^{23} In this case, the degeneracy is calculated by assuming only electronic levels with a minimum of energy below the energy of the current rovibrational level. The degeneracy of the bound-free transition for the rovibrational level $ v , j $ is then computed as follows:

where *g ^{X}* = 3 is the degeneracy of the ground electronic state, and

*g*is the degeneracy of the electronically excited state. The assumption of equilibrium between the vibrational levels of the ground and excited states is valid if the following condition holds:

^{E}Assuming that Eq. (3) holds for every rovibrational level of the ground electronic state, one can obtain the degeneracy factor of the bound-free transition, originally proposed in Nikitin:^{22} *g _{BF}* = 16/3.

The vibration-to-translation and VV transition rates in O_{2}–O_{2} collisions are obtained by means of the FHO model.^{15} In order to generate the state-specific VT and VV transition rates the following parameters of the O_{2}–O_{2} system are used: *α* = 4.2 Å^{−1} and *d* = 3.75 Å, where *α* describes the steepness of a purely repulsive intermolecular potential $V r \u223cexp \u2212 \alpha r $ and *d* is the hard sphere diameter of oxygen. Because the probability of multiquantum jumps in the O_{2}–O_{2} system is much lower than that with $ \Delta v =1$, only transitions with $ \Delta v \u22645$ are considered in the present work. The database of O_{2}–O rates, employed in the present work, includes all possible transitions.^{6}

### B. Master equations

#### 1. Vibrational relaxation

The preliminary study of vibrational relaxation of oxygen includes only the bound-bound transitions in a system of master equations. Because at moderate temperatures (below 10 000 K) the trans-rotational equilibrium in hypersonic flows occurs much faster than the trans-vibrational one, the former is assumed throughout the present paper. This step significantly reduces the number of equations to be solved. The resulting equation for the number density of vibrational level *i* can be written as follows:

where $ K i \u2032 \u2192 i O 2 \u2212 O $ and $ K i \u2032 \u2192 i O 2 \u2212 O 2 $ are the VT bound-bound transition rates in O_{2}–O and O_{2}–O_{2} collisions, respectively, $ Q i \u2032 \u2192 i m \u2032 \u2192 m $ is the VV rate of the reaction $ O 2 m + O 2 i \u2192 O 2 m \u2032 + O 2 i \u2032 $, and *N _{v}* is the total number of vibrational states. An implicit method is applied to integrate Eq. (4) with third order accuracy for diagonal and second order accuracy for off-diagonal elements. The initial number density of rovibrational level

*i*,

*j*is evaluated as follows:

where $ Q i , j = 2 j + 1 exp \u2212 e i , 0 / k T vib \xd7exp \u2212 e i , j \u2212 e i , 0 / k T rot $ is the rovibrational factor of level $ i , j $. The initial number density of vibrational level *i* is calculated as a sum of number densities of rotational levels *j*, compatible with *i*. To reduce influence of statistical error of the QCT method, the principle of detailed balance is invoked to generate rates of exothermic transitions,

In order to verify the VT and VV rates, the system of master equations is solved assuming only bound-bound transitions in pure molecular oxygen under constant heat bath conditions with temperatures between 1000 and 30 000 K. The comparison of resulting vibrational relaxation time and the experimental data is provided in Section III. In the present work, the vibrational relaxation process is studied for different compositions of O_{2} and O mixtures. In order to obtain the vibrational relaxation time in pure molecular oxygen, only terms in the second parentheses and the VV relaxation term in Eq. (4) are considered. The terms in the first parentheses are considered if only the O_{2}–O relaxation is of interest. Finally, all terms are taken into account if the relaxation is modeled in the presence of both O_{2} and O projectiles.

#### 2. Dissociation and recombination

The master equation in the presence of dissociation and recombination processes has the following appearance:

where $ D i O 2 \u2212 O 2 $, $ D i O 2 \u2212 O $, $ R i O 2 \u2212 O 2 $, and $ R i O 2 \u2212 O $ are the state-specific dissociation and recombination rates in O_{2}–O_{2} and O_{2}–O collisions. The number density of atomic oxygen is given by the following equation:

### C. Vibrational-translational model

The solution of Eq. (4) describes the time-dependent population of each vibrational level during the relaxation to thermal equilibrium. Because the large number of vibrational states complicates further analysis, the energy-equivalent vibrational temperature is used to describe the population of the entire vibrational ladder. The average vibrational energy is evaluated in the following manner:

The vibrational temperature is evaluated from $ e \u0304 vib $ by solving the following implicit equations:^{3}

where

Due to the centrifugal forces arising from the rotation of a molecule, the vibrational and rotational levels of energy are coupled. In order to calculate the vibrational temperature of the flow, the energies of the internal degrees of freedom should be decoupled. In the present work, the “vibration-prioritize” framework^{3} is adopted. In Eq. (11), the vibrational energy *e*_{i,vib} is defined as the rovibrational energy of the corresponding state at the ground rotational level, and vibrational temperature is defined as a function of *e _{vib}* only.

The solution of Eq. (10) is obtained by the bisection method. It is important to note that the vibrational temperature, defined by Eqs. (9)-(11), is evaluated only in the post-processing step, after the solution of the master equations has been obtained. It is possible to define the vibrational relaxation time based on the temporal evolution of internal energy under heat bath conditions. There are several ways to define a characteristic relaxation time.^{24} One approach is to calculate the relaxation time as if the equilibration process follows the Landau-Teller equation:

The vibrational energy, given by Eq. (12), evolves exponentially in time. The vibrational relaxation time, *τ _{vib}*, is then defined by the corresponding vibrational energy

*e*

_{v,efold}:

where *e*_{vib,efold} is the average vibrational energy at the time *τ _{vib}*, $ e vib T 0 $, and $ e vib T $ are the average vibrational energies evaluated at initial and final temperatures, respectively. This approach is called the e-folding method.

^{24}

### D. State-specific dissociation rates

An important aspect of coupling the vibrational relaxation and dissociation processes is an accurate state-specific O_{2} dissociation rates in molecule-molecule and molecule-atom collisions. For the latter, accurate data are available for all vibrational states, obtained via the QCT method.^{6} However, the complete database of O_{2}–O_{2} transition rates is not yet available. The concept of preferential dissociation, based on the Treanor-Marrone model,^{18} can be used to overcome this difficulty. The state-specific depletion rates are readily obtained by multiplying the global rate of dissociation by nonequilibrium factor *Z*,

where parameter *Z* is calculated as follows:

The adjustable parameter *U* is referred to as the “characteristic” dissociation temperature and should be chosen according to the best description of available data. A low value of *U* corresponds to the situation where quantum effects play an important role in dissociation, while infinite *U* means that dissociation is equally probable from any vibrational state. Some empirical values of *U*, such as *D*/6*k*, *D*/3*k*, and 3*T*, have been suggested previously. In the present work, a combination of them is chosen by comparing the dissociation rates generated in this way with the available O_{2}–O QCT data. Namely, in the present work, the parameter *U* is calculated as follows:

Equation (16) reflects the quantum behavior of the dissociation process for low lying vibrational states at low temperatures and an increase of depletion probability with temperature and for excited states. The comparison of *D _{i}* in O

_{2}–O collisions, obtained from the Treanor-Marrone model with the O

_{2}–O QCT data is shown in Fig. 1. One can see that the constant value

*U*=

*D*/6

*k*gives a strong underestimation of rates at high temperatures, where quantum effects are less pronounced. At the same time, the value of

*U*= 3

*T*gives a strong underestimation of rates at low temperatures, which can be explained by the fact, that the “characteristic” dissociation temperature under these conditions is larger than 3

*T*. The hybrid value of these two values of

*U*, given by Eq. (16), provides the best description of the O

_{2}–O QCT data and is used in the present work to obtain the O

_{2}–O

_{2}state-specific dissociation rates.

Another way to estimate the value of *D _{i}* in O

_{2}–O

_{2}collisions is based on the fact that the dissociation rates in O

_{2}–O collisions are weakly dependent on the attractive part of the potential energy surface and only affected by its repulsive brunch. In Andrienko and Boyd,

^{17}two potential energy surfaces were adopted: the first PES was obtained by the summation of the pairwise interactions in the O

_{2}–O complex, described by the Hulburt–Hirshfelder (HH) potential.

^{25}The second PES accounted for the three-body interaction,

^{7}in addition to the two-body term, included in the HH PES. The comparison of these two approaches revealed a very small difference between dissociation rates, which indicates a minor influence of the three-body interaction term. Indeed, the dissociation process occurs at high collisions energies, when colliding particles interact mostly via the repulsive branch of the two-body potential.

Thus, it appears reasonable to scale the accurate state-resolved O_{2}–O dissociation rates in order to obtain the bound-free transition rates for the O_{2}–O_{2} system. In the present work, the scaling factor is calculated as the ratio of the global dissociation rate for atom-diatom and diatom-diatom collisions. The state-specific O_{2}–O_{2} dissociation rates are calculated as follows:

Several O_{2} global dissociation rates were previously reported.^{16,26–28} These data differ by the value of pre-exponential and temperature dependence factors in the generalized Arrhenius form, shown in Table I.

Source . | A (cm^{3}/s)
. | B . | C (K) . |
---|---|---|---|

Bortner, 1969 | 1.37 × 10^{−5} | −0.83 | 59 400 |

Johnston, 1968 | 4.980 × 10^{−6} | −1.00 | 59 500 |

Park, 1989 | 3.320 × 10^{−3} | −1.50 | 59 500 |

Ibraguimova, 2003 | 1.627 × 10^{1} | −2.50 | 59 380 |

Source . | A (cm^{3}/s)
. | B . | C (K) . |
---|---|---|---|

Bortner, 1969 | 1.37 × 10^{−5} | −0.83 | 59 400 |

Johnston, 1968 | 4.980 × 10^{−6} | −1.00 | 59 500 |

Park, 1989 | 3.320 × 10^{−3} | −1.50 | 59 500 |

Ibraguimova, 2003 | 1.627 × 10^{1} | −2.50 | 59 380 |

The O_{2}–O_{2} state-specific dissociation rates, calculated by means of the Treanor-Marrone model for selected vibrational states from the global rates by Johnston, Park, and Ibraguimova, are shown in Fig. 2. While the low-lying states have the conventional behavior for dissociation rate, the result of the Treanor-Marrone model for highly excited states has a non-physical behavior, suggesting the decrease of dissociation rate at high temperatures. This decrease is larger for the larger absolute value of parameter *B* in Table I. It is interesting to note, that in the work by Capitelli *et al.*^{29} similar behavior was observed for N_{2}–N dissociation rates for *ν* ≥ 60, when compared to the results of the QCT method. Apparently, the choice of temperature dependence factor, *B*, plays an important role here. Those works, who report the larger absolute value of *B*, demonstrate the more significant decrease of *D _{i}* with temperature; this drop can be as large as two orders of magnitude in the temperature range between 1000 and 10 000 K. In light of this finding, the present work adopts the O

_{2}–O

_{2}global dissociation rate by Bortner,

^{16}which has the smallest absolute value of

*B*and was originally obtained with an emphasis on hypersonic temperatures. The comparison of scaled rates and those obtained by the Treanor-Marrone model for the O

_{2}–O

_{2}system is shown in Fig. 3. The Treanor-Marrone rates, calculated using the Eq. (16), again demonstrate the best agreement with the rates obtained by means of the scaling procedure, using Eq. (17).

### E. Quasi-steady state treatment

Because of the large difference in the rates of vibrational relaxation and dissociation, that is typically observed in hypersonic flows, the population of the vibrational ladder remains relatively constant during the time when dissociation processes start to play a noticeable role. This situation is usually referred to as the Quasi-Steady State (QSS) and appears to be an important step in the analysis of the state-to-state kinetics.

The population of the vibrational ladder can be approximately described by some nonequilibrium temperature during the QSS phase, that is typically lower than the equilibrium one in the case of heating flow. Because of this fact, the QSS dissociation rate is lower than that at equilibrium conditions. In fact, the investigation of the QSS state is very important for O_{2}–O collisions, since the vibrational relaxation becomes less efficient with temperature,^{4,6} unlike in other molecular systems of interest. Since the population of states is nearly constant during the QSS phase, the system of master equations, given by Eq. (7), can be simplified by setting the left hand side to zero. The system of master equations then has the following appearance:

where all terms, related to O_{2}–O_{2} collisions, are omitted in order to study the O_{2}–O dissociation process. The first line in Eq. (18), given by *i* = 0, is substituted by the mass conservation equation in the following form:

After some manipulations, the rate of change of *n*_{O2} can be written as follows:

where *ρ*_{i,h} is the homogeneous solution of Eq. (18). Then the QSS dissociation rate can be defined as

## III. RESULTS

### A. Vibrational relaxation in O_{2}–O and O_{2}–O_{2} collisions

The O_{2}–O vibrational relaxation time, previously obtained in Andrienko and Boyd^{6} from the solution of master equations, is compared to the vibrational relaxation time derived from the rate of monoquantum deactivation, reported by Esposito *et al.*^{5} and Ibraguimova, Sergievskaya, and Shatalov^{30} in Fig. 4. The solid line describes the vibrational relaxation time by the VT model. The relaxation time, given by the short dashed line, is taken from experimental work.^{30} The long dashed line represents the relaxation time derived from the rate of monoquantum deactivation by Esposito and Capitelli.^{31} The symboled line represents Park’s curve fit to the Millikan-White (MW) equation^{32} using the data by Kiefer and Lutz.^{33} Theoretical calculations by Quack and Troez^{4} are shown by squares.

The vibrational relaxation time, obtained from the VT thermodynamic model, is in good agreement with the existing experimental data in the range from 1000 K to 3600 K. The largest difference with Breen *et al.* data is observed in the high temperature region and is not higher than 15%. Taking into account the overall uncertainty of shock tube facilities, the agreement with the experimental data is very satisfactory. To the authors’ knowledge, other simulations that adopt a master equation, do not exist. The direct extrapolation of experimental data, performed by Park,^{27} suggests that the vibrational relaxation of oxygen by the parent atom becomes more efficient with temperature.^{27} The present data support the directly opposite temperature dependence: the vibrational relaxation time becomes smaller at low temperatures. This observation is in agreement with the results by Quack and Troe.^{4} The relaxation times, reported in Ibraguimova, Sergievskaya, and Shatalov,^{30} provide a good description of vibrational temperature behind shock waves in the range of temperatures from 4000 K to 10 800 K. The agreement between results in Ibraguimova, Sergievskaya, and Shatalov^{30} and those generated by the present VT model is very satisfactory and both results are substantially different from the MW relation. As pointed out in Andrienko and Boyd,^{6} the increase of O_{2}–O vibrational relaxation time with temperature is due to the barrierless O_{3} PES. The energy randomization in the O_{3} complex is especially efficient at low collision energies, while at high temperatures particles interact mostly via the repulsive part of potential.

The large attractive components in the O_{3} potential energy surface have significant influence on the rates of mono- and multiquantum transitions as well. The rates of *v* = 1 → *v*′ = 0, *v* = 10 → *v*′ = 9, and *v* = 10 → *v*′ = 5 transitions for O_{2}–O and O_{2}–O_{2} are shown in Fig. 5. For reference, the global dissociation rate coefficients are also shown by dashed symboled lines. While the rates of O_{2}–O_{2} bound-bound transitions demonstrate a strong temperature dependence, the O_{2}–O rates are nearly constant in the considered temperature range. Moreover, the O_{2}–O rates of multiquantum transitions are only slightly lower than that of a monoquantum jump. Also, the rate of dissociation in collisions with oxygen radical is substantially higher than in O_{2}–O_{2} and comparable to that of bound-bound transitions, which can be important in high temperature flow with an abundance of oxygen atoms. This subject is investigated in greater detail below.

The accuracy of O_{2}–O_{2} VT and VV FHO rates is an important subject in the present work. Unfortunately, rates obtained by trajectory simulation on an accurate PES are not available for temperatures relevant to hypersonic flows. However, the O_{4} system was extensively studied at conditions corresponding to atmospheric chemistry.^{34–36} The present FHO rates are compared with the available experimental and theoretical data in Fig. 6. Reactions of VT energy transfer ($ O 2 v + O 2 \u2192 O 2 v \u2212 1 + O 2 $) and VV energy transfer ($ O 2 v + O 2 0 \u2192 O 2 v \u2212 1 + O 2 1 $) are considered. Solid lines describe the present FHO data, long dashed lines correspond to rates by Coletti and Billing,^{35} and dashed-dotted line presents results by Campoz-Martínez *et al.*^{36} Short dashed line gives the total (VT+VV) rate of energy removal based on the experimental measurements by Rogaski *et al.*^{37} The difference in theoretical calculations by Coletti and Billing and Campoz-Martínez *et al.* is first of all due to the O_{4} PES adopted for the calculations. The former work used a modified potential surface based on molecular beam experiments,^{34} where particles behave like rigid rotors. The work by Campoz-Martínez *et al.* used the PES by Varandas, obtained via the DMBE method. Unlike the former PES, the O_{4} DBME PES introduces the open reaction channel for ozone formation. This particular fact leads to the appreciable difference in the VV energy rate constants for high *v*.^{35} The present VV+VT FHO rate has good agreement with the experimental data at *v* > 20. For low *v*, the agreement of the VT and VV FHO rates with those by Coletti and Billing is satisfactory.

The e-folding O_{2}–O_{2} vibrational relaxation time, obtained from Eq. (13), is shown in Fig. 7 by the solid black line. The summary of experimental measurements, reflected in Park,^{27} is shown by square symbols. The present FHO rates satisfactorily describe the vibrational relaxation time in O_{2}–O_{2} collisions in the considered temperature range as well as the deviation of relaxation time from the linear dependence at high temperatures due to the breakdown of the Landau-Teller theory. In the high temperature limit, the VT and VV processes become coupled and the VVT energy transfer should be considered instead. However, due to the lack of theoretical studies, this step has been left for future investigation.

The influence of a small concentration of atomic oxygen, admixed with the pure O_{2}, on vibrational relaxation time is shown in Fig. 7 with the short-dashed and long-dashed lines that correspond to 1% and 5% molar fractions of atomic particles. The long-dashed line with diamond symbols corresponds to the vibrational relaxation time in O_{2}–O collisions. The presence of even a small concentration of atoms greatly reduces the relaxation time at moderate hypersonic temperates. This is the direct consequence of the anomalously fast vibrational energy randomization in ozone.^{4} As temperature increases, the O_{2}–O relaxation becomes less efficient and the average relaxation time in the O_{2}–O mixture converges to the conventional MW dependence.

The variation of vibrational temperature with time in pure oxygen and in the 5%O–95%O_{2} mixture under the heat bath temperatures of 1000, 2000, 3000, and 10 000 K is shown in Fig. 8. The initial vibrational temperature is set to 100 K. No dissociation is considered. A significant difference in the vibrational temperature is seen during the relaxation to thermal equilibrium at temperatures less than 5000 K. However, at high temperatures, the admixing of atomic oxygen does not introduce any visible difference into the vibrational energy of the ensemble. This can be explained by the relatively inefficient vibrational deactivation of oxygen by the parent atom at high temperatures, as follows from Fig. 4.

Because the vibrational temperature is typically defined by the first few low-lying vibrational states, Fig. 8 does not reflect the influence of O_{2}–O collisions on the population of excited vibrational states. For this purpose, the instantaneous vibrational distribution is studied at the moment of active vibrational relaxation and during the QSS phase at different temperatures of the heat bath. Three cases are considered: dissociation and vibrational relaxation occur by means of (1) only O_{2}–O_{2} collisions, (2) O_{2}–O_{2} and O_{2}–O collisions in the initially pure molecular oxygen, and (3) in the case of a 5% molar fraction of atomic oxygen as an initial condition. The last case is important for computational simulation of test cases, discussed in Holden, Wadhams, and MacLean.^{38}

The populations of the vibrational ladder are shown in Figs. 9 and 10 for T = 3000 K and 11 for 10 000 K for the case of heating flow. Figures 9 and 11 reflect the populations at t = 10^{−9} s, which corresponds to the early stage of relaxation prior to the QSS phase. Figure 10 describes the population during the QSS phase.

During the process of vibrational relaxation at T = 3000 K the populations of excited vibrational states significantly exceeds the equivalent equilibrium population, given by the current vibrational temperature of the gas. Accounting for the atom-molecule collisions noticeably changes the number density of highly excited states because of the large probability of multiquantum jumps, compared to O_{2}–O_{2} collisions. This situation takes place even when no oxygen atoms were introduced in the initial composition. The admixture of atomic oxygen significantly increases the population of excited states, depopulating the low-lying vibrational states. The equilibrium population of the vibrational ladder also changes in the presence of atomic oxygen due to the smaller vibrational relaxation time, as follows from Fig. 7.

The relaxation at T = 10 000 K is similar to that at T = 3000 K. Because the vibrational relaxation time in O_{2}–O increases with temperature, the equilibrium populations in the presence of O atoms do not change significantly, as can be seen from Fig. 11. Nevertheless, the atom-diatom collisions define the populations of excited states in many ways, because of the large amount of dissociated oxygen atoms at these temperatures.

The influence of oxygen atoms on the population of the vibrational states in cooling flows is shown in Figs. 12 and 13 for two cases: T$ vib 0 =10\u2009000$ and 20 000 K, respectively. In both cases, the translational temperature is set to 3000 K. Atom-molecule collisions lead to underpopulation of the excited states, as opposed to the heating flow, and this effect is stronger at higher vibrational temperatures. Accounting for O_{2}–O collisions reduces the population of excited states by an order of magnitude; the presence of oxygen radicals, admixed to the flow initially, increases this effect.

### B. Multiquantum transitions

Existing kinetic theories, such as the FHO model, that are currently used to evaluate the probabilities of mono- and multiquantum transitions, predict the latter to be much smaller compared to that with $ \Delta v =1$ at low and moderate temperatures, relevant to hypersonic flow. This result rests on the assumption of a strong repulsion between colliding particles that govern the trans-vibrational energy exchange. The probability of multiquantum transitions increases with temperature, nevertheless, for most practical applications the inclusion of transitions with $ \Delta v \u22645$ is usually enough to perform accurate state-to-state shock flow simulations.^{39} However, because of the large attractive component in the O_{3} potential energy surface, the probability of multi- and single quantum jumps in O_{2}–O collisions has the same order of magnitude, as can be seen in Fig. 5. This section investigates the influence of transitions with $ \Delta v >1$ on the process of thermal relaxation.

The variation of the population of the vibrational ladder with time is shown in Figs. 14 and 15 for the cases of heating and cooling flows. The translational temperature is set to a constant value of 3000 K. The initial vibrational temperature is set to 100 K and 10 000 K, respectively. The simulations are performed using the complete set of state-to-state rates as well as including transitions with $ \Delta v \u22645,\u20092$, and 1. The initial concentrations of atoms and molecules are set to 0.9 × 10^{18} and 0.1 × 10^{18} cm^{−3}. In both cases of heating and cooling flows, the multiquantum jumps play a significant role in the vibrational relaxation process. For the former, the influence of multiquantum jumps is more pronounced for the excited vibrational states due to the larger number of open relaxation channels. Specifically, for the lower vibrational states the multiquantum transitions result in the overpopulation and faster relaxation to equilibrium (*v* = 5 and 10). For highly excited states, the multiquantum jumps have the opposite effect of faster depopulation (*v* = 20 and 30). The influence of multiquantum transitions on the relaxation in the expanding flow is large for the entire vibrational ladder. In this case, failure to account for multiquantum jumps will lead to the underestimation of the population of ground and low-lying states and to the overestimation of the population of highly excited states.

The evolution of vibrational temperature to thermal equilibrium for the considered heat bath conditions is shown in Figs. 16 and 17 for translational temperatures of 3000, and 10 000 K, respectively. The upper and lower parts of the plot correspond to the cooling and heating flows. Due to the fast vibrational relaxation and relatively slow dissociation at 3000 K, the QSS state is reached only at the equilibrium temperature. In the case of T = 3000 K the influence of multiquantum vibrational jumps is less pronounced in the heating flow due to the low population of excited states. However, for the expanding flow calculations with transitions $ \Delta v \u22645$ lead to a difference compared to those using the complete database. For both types of flows, the difference in the relaxation time and the duration of the QSS state between the full state-to-state approach and the one with $ \Delta v =1$ is approximately one order of magnitude.

The variation of vibrational temperature with time in the case of T = 10 000 K has a different pattern. Because in this case dissociation is faster, the nonequilibrium QSS state can be clearly observed in both types of flows. The QSS vibrational temperature and the duration of the QSS phase are similar for heating and shock flows. Since the initial concentration of atomic oxygen, set to 0.9 × 10^{18} cm^{−3}, is less than that in equilibrium, the QSS state is accompanied by strong dissociation, which leads to a drop of vibrational temperature below the translational one for the cooling flow, as can be seen in Fig. 17. Accounting for multiquantum jumps leads to a higher vibrational temperature and a shorter duration of the QSS phase.

### C. Dissociation

Experimental measurements of dissociation rates in nitrogen^{40} and oxygen^{41} indicate that the depletion of gas proceeds more efficiently in the parent atom-diatom rather than in the diatom-diatom collisions. As pointed out in Valentini *et al.*,^{42} one of the reasons for this is a “scrambling” effect of pre-collisional vibrational states that takes place during the exchange reaction in N_{2}–N collisions. Because the insertion mechanism in the N_{2}–N system is more efficient than that in N_{2}–N_{2}, the former implicitly affects the populations of higher vibrational states that ultimately lead to dissociation.

It was recently shown^{6} that the exchange reaction has a large impact on the energy randomization in the O_{3} complex at collisions energies below 1 eV. Furthermore, the exchange mechanism is responsible for the large multiquantum jumps during the vibrationally inelastic collisions. Taking these facts into account, one can expect a large influence of oxygen radicals on not only the vibrational relaxation but also on the duration of the QSS phase. Presently adopted O_{2}–O and O_{2}–O_{2} dissociation rates support the conclusions, made in Valentini *et al.*:^{42} the ratio of D^{O2−O} and D^{O2−O2} is equal to 3.90 at T = 2000 K, increasing up to 9.43 at T = 10 000 K. Thus, it is important to accurately model the amount of oxygen atoms in the flow and the population of the vibrational ladder prior to the QSS regime.

The instantaneous dissociation rate can be defined for O_{2}–O and O_{2}–O_{2} collisions, as well as for these two concurrent processes, in the evolving gas mixture. The global dissociation rate coefficient, *D _{inst}* for the latter can be calculated as follows:

where Δ*t* is the time step, adopted for the solution of the master equations. Dissociation rate coefficients for atom-diatom ($ D inst O 2 \u2212 O $) and diatom-diatom ($ D inst O 2 \u2212 O 2 $) collisions, calculated in a manner, similar to Eq. (22), are shown in Figs. 18 and 19, respectively. For reference, $ D inst mix $ is shown in Fig. 19 by the dashed line.

All three dissociation rate coefficients change throughout the relaxation because the state-specific transition rates depend on the vibrational quantum number and the population of vibrational states also changes with time. A meaningful value of the dissociation rate can be defined only when the relaxation process reaches the QSS phase due to the relatively constant population of the vibrational ladder. At low temperatures, the QSS phase requires a much longer time to be established than at high temperatures. The duration of the QSS phase in O_{2}–O collisions is much shorter, compared to that when only O_{2}–O_{2} collisions are considered, which is explained by the difference in the global dissociation rate coefficient. When both types of collisions are considered, the QSS phase becomes shorter and the instantaneous dissociation rate coefficient increases due to the abundance of oxygen radicals at high temperatures, as follows from Fig. 19.

The evolution of vibrational temperature and species concentration with time is shown in Figs. 20 and 21 when only O_{2}–O and O_{2}–O_{2} collisions are considered, respectively. In both cases the initial T_{vib} is set to 100 K, and translational temperature is set to a constant in the range of 2000–30 000 K. In the case of only atom-diatom collisions, the vibrational relaxation occurs mostly prior to the QSS phase at temperatures below 5000 K, and the dissociation takes place from the vibrational ladder, populated at a nearly equilibrium temperature (see Fig. 10). As temperature increases further, the dissociation becomes faster, and the vibrational temperature during the QSS phase starts to deviate from the equilibrium value. Now, because the O_{2}–O vibrational relaxation time increases with temperature, unlike in O_{2}–O_{2} collisions, and the dissociation rates follow the conventional Arrhenius form, the further increase of translational temperature does not lead to an increase of vibrational temperature during the QSS phase. The plateau, that is typically observed at low temperatures, is shorter and smeared. In other words, the increase of vibrational relaxation time with temperature in O_{2}–O collisions has a limiting effect on vibrational temperature during the QSS phase. One should use this observation with caution, because electronic excitation, that appears to be an additional channel of O_{2}–O collisions at high temperatures, is not included in the derivation of the present vibrational relaxation time.

Solutions of the master equations when only diatom-diatom collisions are considered are shown in Fig. 21. In contrast to the case with O_{2}–O relaxation, the vibrational temperature during the QSS phase increases monotonically in the considered interval of translational temperatures. The dissociation at temperatures not exceeding 5000 K occurs only after the vibrational relaxation is complete. This is true for both O_{2} and O heat bath conditions. At higher temperatures, the depletion of molecules may occur prior to thermalization of the vibrational mode. At T = 10 000 K nearly 20% of O_{2} is depleted in the case of a molecular heat bath. Under the same conditions of an atomic heat bath, the increase of n_{O} is comparable with that of the equilibrium composition. This is due to the faster dissociation in O_{2}–O collisions compared to the pure O_{2} case. The separation of the QSS and relaxation phases is justified for the O_{2}–O system only at temperatures lower than 10 000 K.

The QSS dissociation rate coefficient in O_{2}–O collisions, calculated using Eq. (21), is shown in Fig. 22 by the short dashed line. The dissociation rate, calculated using the QCT data assuming thermal equilibrium, is show by the long dashed line. For reference, the data by Shatalov, Park, and Bortner are shown by circles, diamonds, and crosses, respectively. The QSS rate coefficient is smaller than the equilibrium dissociation rate in the range of temperatures between 1000 and 10 000 K. This is explained by the underpopulation of highly excited vibrational states during the QSS regime, as can be seen in Fig. 10. The QSS rate is on the order of two times lower than the equilibrium rate at T = 1000 K, increasing to a factor of six at T = 10 000 K. The experimental and theoretical results,^{16,27,28} used for modeling of composition of nonequilibrium flows, generally demonstrate a better agreement with the QSS rate coefficient, than with the equilibrium one. The two-temperature rate coefficient derived by Park, has the closest agreement with the QSS dissociation rate, slightly underestimating the present results at T = 10 000 K.

The recombination process can be important in the nozzle flow with *T _{vib}* >

*T*when the atomic concentration substantially exceeds the equilibrium composition. Because the population of the vibrational ladder exhibits a strongly non-Boltzmann dependence, the global rates of chemical reactions have a non-Arrhenius behavior. The multi-temperature model,

^{27}popular due to its simplicity, is inadequate for describing the chemical composition in rapidly expanding flows for this reason. The system of master equations, coupled to the accurate rates of energy exchange in O

_{2}–O

_{2}and O

_{2}–O collisions, is an accurate tool for studying nonequilibrium kinetics in a strongly recombinative flow.

In the present paper, the nozzle flow kinetics is simulated at two initial vibrational temperatures of 3000 and 5000 K and at a fixed translational temperature of 1000 K. The initial molar fraction of atomic oxygen is set to 0.9. The evolution of vibrational population with time is shown in Fig. 23. In the case of lower T$ vib 0 $, the initial phase of thermal relaxation can be characterized by the formation of a long plateau above the initial distribution. This mechanism is attributed to a selective recombination to higher vibrational states. Because the excited states are strongly overpopulated, it is preferable to describe the rates of chemical transformation in terms of the population of the last vibrational state.^{14} The lower vibrational states initially maintain a nearly constant population, while the extensive energy exchange takes place. For this reason, the rate of dissociation, described by the vibrational temperature, would give misleading results in these simulations. The depopulation of lower states occurs only at the late stage of relaxation when the vibrational temperature of the system converges to the equilibrium value. It is worth to note that at stronger nonequilibrium conditions, T$ vib 0 =5000$ K, the up pumping of excited states does not occur, since those states are already densely populated. The formation of a plateau can be achieved by increasing the initial concentration of oxygen atoms.

### D. Vibrational-dissociation coupling

In order to describe the coupled processes of vibrational relaxation and dissociation, it is convenient to define the energy rate constant, that describes an average loss of vibrational energy in a single dissociation event. This step is important for development of reduced order thermochemistry models. The coupling coefficient, C_{DV,i}, can be defined for each vibrational level as well as for the entire vibrational ladder. For the former, the following equation is used for evaluation of the energy rate constant in collisions O$ 2 v = i $–O_{2}:

A similar expression is used to define the energy rate constant in atom-molecule collisions, $ C DV , i O 2 \u2212 O $. The total coupling coefficient is calculated as follows:

In the present work, the energy rate coefficients, given by Eqs. (23) and (24), are normalized by the classical dissociation energy *D _{e}*.

The state-resolved coupling coefficient at heat bath temperatures of 3000 and 10 000 K is shown in Fig. 24. The energy rate coefficients are calculated during the QSS phase where most energy transfer occurs, and the population of the vibrational ladder is relatively constant. At T = 3000 K the maximum coupling coefficient is shifted toward the highly excited vibrational states. *C*_{DV,i} reaches a maximum at *v* = 21 in O_{2}–O collisions and at *v* = 26 for the O_{2}–O_{2} system. Lower states are coupled more strongly in the O_{2}–O interaction, while high lying states have significantly higher *C*_{DV,i} in O_{2}–O_{2} collisions. In both cases the coupling coefficient approaches zero at low and high *v* because of the low dissociation probability and a low equilibrium population, respectively. At T = 10 000 K the maximum of *C*_{DV,i} shifts toward the low-lying states: *v* = 5 for O_{2}–O and *v* = 14 for O_{2}–O_{2}. This is explained by a larger number of energetic collisions that lead to dissociation from lower states, compared to the low temperature case.

The total energy rate coefficients are shown in Fig. 25 at kinetic temperatures between 1000 and 10 000 K. At low temperatures, both types of collisions, O_{2}–O and O_{2}–O_{2}, have *C _{DV}* close to unity, since the dissociation proceeds mostly from excited vibrational states. As the temperature increases, lower levels become involved. Because the dissociation in O

_{2}–O collisions is much more efficient than that in O

_{2}–O

_{2}, the $ C DV O 2 \u2212 O $ energy rate coefficient is substantially lower than $ C DV O 2 \u2212 O 2 $.

In the presence of both types of collision, the coupling coefficient will take an intermediate value. The variation of vibrational temperature and *C _{DV}* in the O

_{2}–O mixture in heating flow at T = 10 000 K is shown in Fig. 26. Initially, the gas contains only oxygen molecules at a concentration of 10

^{18}cm

^{−3}. By the time t = 10

^{−8}s the mixture contains a considerable amount of oxygen atoms, that greatly reduces the duration of the dissociation processes. Because oxygen atoms lead to more efficient dissociation, a decrease of vibrational temperature during the QSS phase occurs. The energy rate constant reaches the maximum during the QSS phase and then converges to a constant value, defined by the amount of oxygen atoms in the mixture.

Figure 27 demonstrates the concurrent influence of vibrational relaxation and dissociation in a two-species gas. Because in the high temperature limit the rates of monoquantum transitions in O_{2}–O_{2} collisions are comparable or less than the rate of O_{2}–O dissociation (Fig. 5), the latter significantly affects the population of the vibrational ladder. Thus, the term QSS, applied to the considered conditions, should be used with caution.

## IV. CONCLUSION

Vibrational relaxation and dissociation of molecular oxygen are studied by means of master equations in the presence of atom-molecule and bimolecular collisions. Accurate bound-bound and bound-free transition rates, generated by the QCT method, are adopted to model the O_{2}–O interaction. Meanwhile, the FHO model is used to generate the VT and VV bimolecular relaxation rates. A strong dependence of vibrational relaxation time, obtained by the e-folding method, on the amount of atomic oxygen in the gas mixture is observed. The average relaxation time in the presence of a small (1%-5% molar fraction) amount of oxygen atoms is several orders of magnitude lower than in pure oxygen at temperatures below 3000 K. As temperature increases, this difference becomes smaller.

The efficient energy randomization in O_{2}–O collisions causes a strong over- and underpopulation of highly-excited vibrational states in heating and cooling flows, respectively. Moreover, the state-resolved populations are strongly defined by the multiquantum transitions. In the case of cooling flow, the multiquantum jumps noticeably affect even the population of the ground state. Failure to account for all possible transitions that take place during O_{2}–O collisions may lead to a strong overestimation of relaxation time in the gas mixture.

Comparison of O_{2}–O and O_{2}–O_{2} state-specific dissociation rates revealed the former as a more effective channel of dissociation, possibly because of the efficient exchange reaction in the triatomic system. For this reason, the average loss of vibrational energy during a single dissociation event, evaluated for the O_{2}–O system, is substantially lower than that in O_{2}–O_{2} collisions. This fact should be taken into account when developing reduced order thermochemical models of partially dissociated flows. Since the O_{2}–O dissociation rate coefficient is somewhat comparable to the O_{2}–O_{2} VT transition rates at high temperatures (about 10 000 K), a constant population of the vibrational ladder during the active dissociation is not always observed.

## Acknowledgments

The authors gratefully acknowledge funding for this work through Air Force Office of Scientific Research Grant No. FA9550-12-1-0483. DA would like to thank Dr. Jae Gang Kim for the O_{2}–Ar QCT program code and numerous fruitful discussions.

## REFERENCES

_{2}collisional dissociation and rotation-vibration energy transfer

_{2}(v) system

_{2}state-to-state vibrational relaxation and dissociation rates based on quasiclassical calculations

_{3}from a multiproperty fit to ab initio calculations, and to experimental spectroscopic, inelastic scattering, and kinetic isotope thermal rate data

_{2}+ O → O + O

_{2}and O

_{2}+ O + Ar → O

_{3}+ Ar on a modified ground-state potential energy surface for ozone

_{2}= 3O

_{2}behind a strong shock wave

_{2}–O

_{2}gas mixture with electronically excited species O

_{2}(Δ), O(D), OH(

^{2}Σ) involved

_{2}(i) + N = 3N: A comparison of trajectory calculations and the treanor–marrone model

_{2}molecules by atomic oxygen

_{2}–O

_{2}dimer

_{2}: Reaction or enhanced vibrational relaxation?

_{2}and implications for its atmospheric fate