Direct Molecular Simulation of Nitrogen and Oxygen at Hypersonic Conditions

A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY

Maninder S. Grover

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Thomas E. Schwartzentruber, Adviser

A number of people have earned my gratitude for their support, guidance and patience during the course of my graduate career. Firstly, I would like to thank my adviser, Prof. Tom Schwartzentruber for his support, and mentorship, and for affording many opportunities of personal and professional growth throughout my graduate career. Tom has been a great teacher and a source of inspiration for the past four and half years. His enthusiasm and passion for research constantly provided motivation to work hard. I will not forget the overall kindness and support he has shown over the years. Tom will definitely be a role model for me for years to come. I am grateful for the guidance of Dr. Rich Jaffe of NASA’s Ames Research Center, for many valuable discussion and teaching moments that he shared during our collaboration. Working with Rich has been one of the highlights of my time as a graduate student and I am honored and humbled to have had this opportunity. I would like to thank Dr. David Hash for his mentorship, support and encouragement through the years. I would also like to thank Dr. David Schwenke, Dr. Michael Barnhardt, and Dr. Brett Cruden at NASA Ames Research Center for enabling the collaboration between the University of Minnesota and NASA, and for valuable inputs during my time as an intern at Ames Research Center. I would like to thank my collaborators at the University of Illinois at Urbana Cham- paign: Prof. Marco Panesi, Robyn Macdonald, and Simone Venturi. Working with them has certainly been an immense learning experience. I am thankful to my collaborators in the Chemistry Department at the University of Minnesota : Prof. Donald

i Truhlar, Dr. Zoltan Varga, and Dr. Yulia Paukku for providing the potential energy surfaces that most of my work is based on and for always being there to clarify any queries and questions that I had regarding the chemistry aspect of this project. I would like to thank Prof. Graham Candler, Prof. Joseph Nichols, and Prof. Steven Girshick for being a part of my committee and the thesis reviewing process. A special thanks to my colleagues Paolo, Savio, Ross, and Narendra for spirited and valuable discussions that we have had over the years and for all the feed-back they have given about my work. I am grateful to my parents and my sister for always cheering me on and supporting me throughout this endeavor. I am grateful to my friends, here in Minnesota for making me feel at home even when I was half a planet away. Finally, I would like to express my gratitude to the Air Force Oﬃce of Scientiﬁc Research (AFOSR) and the Doctoral Dissertation Fellowship (DDF) for supporting my graduate research.

ii “ Far and away the best prize that life has to oﬀer is the chance to work hard at work worth doing. ” - Theodore Roosevelt

iii Abstract

The objective of this thesis is to characterize the gas-phase thermochemical non-equilibrium that occurs during hypersonic flight for nitrogen and oxygen gases. This thesis uses the direct molecular simulation (DMS) method in conjunction with potential energy surfaces (PESs) to provide an in-depth molecular level analysis of internal energy excitation and dissociation of molecular nitrogen due to N2 +N2 and N2 +N interactions. Characteristic vibrational excitation times and non-equilibrium dissociation rate coefficients are calculated using the ab−initio PESs developed at NASA Ames Research Center. Comparison of these rate coefficients and non-equilibrium vibrational energy distributions is carried out against prior work done with nitrogen using an independently developed ab − initio PES at the University of Minnesota. Good agreement was found between properties predicted by the two PESs. Furthermore, comparative studies were carried out for the nitrogen system between the DMS method and the state-to-state method. The results obtained by the two different methods, are found to be in good agreement. The DMS method is used to calculate benchmark data for vibrational energy excitation and non-equilibrium dissociation due to O2 + O interactions. O2 + O interactions are modeled using nine PESs corresponding to. 11A0, 21A0, 11A00, 13A0, 23A0, 13A00 15A0, 25A0 and 15A00 states, which govern electronically adiabatic (ground-electronic- state) collisions of diatomic oxygen with atomic oxygen. This is the first data set in the aerospace community that incorporates all nine PESs for the O2 + O system and fully describes the dynamics of ground state interactions of diatomic oxygen with atomic oxygen. Characteristic vibrational excitation times are calculated over a temperature range

iv of T = 3000K to T = 15000K. It is observed that the characteristic vibrational excitation time for O2 + O interactions is weakly dependent on temperature and increases slightly with increasing temperature. Vibrational excitation is slowest for interactions in the quintet spin state, with the 15A00 state having the slowest excitation rate, and vibrational excitation is fastest on the 11A0 potential energy surface. Non-equilibrium dissociation rate coeﬃcients are calculated over a temperature range of T = 6000K to T = 15000K during quasi-steady state (QSS) dissociation, and the results agree well with experimental data. For the O2 + O2 system interactions can occur over singlet, quintet and triplet spin states. An in-depth analysis of excitation and dissociation on the quintet and singlet surfaces is provided and bench-mark data for excitation using all three PESs for O2 + O2 interactions is presented for a temperature range of T = 5000K to T = 12000K . Finally, this thesis explores internal energy exchange processes in oxygen and nitrogen. Probability distribution functions for vibrational energy change during collisions are presented (due to N2 + N2 non-reactive collisions, N2 + N2 exchange reactions,

N2 + N non-reactive collisions, N2 + N exchange reactions, O2 + O non-reactive collisions, and O2 + O exchange reactions). It is shown that non-reactive collisions are less eﬃcient in vibrational energy redistribution when compared to exchange reactions. Furthermore, it is observed that the probability distribution functions for vibrational energy change (for both oxygen and nitrogen) are self-similar and may be modeled by simpliﬁed functional forms.

v Contents

Acknowledgementsi

Dedication iii

Abstract iv

List of Tables ix

List of Figuresx

1 Introduction1 1.1 Challenges of Hypersonic Flight...... 1 1.2 Contemporary work...... 4 1.3 Introduction to Direct Molecular Simulation...... 5 1.4 Outline...... 6

2 Potential Energy Surfaces8 2.1 Introduction...... 8 2.2 Spin multiplicity and spatial symmetry...... 12 2.3 Potential energy surfaces for nitrogen...... 15 2.3.1 Ab-initio potential energy surfaces...... 16 2.3.2 Site-to-site potential energy surface...... 20 2.4 Potential energy surfaces for oxygen...... 22

2.4.1 Potential energy surfaces for O3 interactions...... 22

vi 2.4.2 Potential energy surfaces for O4 interactions...... 26

3 Physical Model and Numerical Method 28 3.1 Introduction...... 28 3.2 Direct molecular simulation...... 30 3.2.1 Numerical method...... 30 3.2.2 Trajectory calculations and parallel implementation...... 33 3.2.3 Validation with molecular dynamics...... 40 3.3 Code implementation...... 43 3.3.1 Normal shock waves...... 43 3.3.2 Multidimensional ﬂows...... 45 3.3.3 Chemically reacting isothermal systems...... 46

4 Rovibrational Coupling in Normal Shocks 52 4.1 Introduction...... 52 4.2 Rovibrational coupling in normal shock waves...... 53

5 Direct Molecular Simulation of Nitrogen 57 5.1 Introduction...... 57

5.2 Eﬀect of PES on the N2 + N2 system...... 58 5.2.1 Vibrational energy excitation...... 58 5.2.2 Nonequilibrium dissociation...... 59

5.3 Eﬀect of numerical method on the N2 + N2 system...... 64 5.3.1 Internal energy excitation...... 66 5.3.2 Nonequilibrium dissociation...... 68

5.4 Eﬀect of PES on the N2 + N system...... 71 5.4.1 Vibrational energy excitation...... 71 5.4.2 Non-equilibrium dissociation...... 76

5.5 Eﬀect of numerical method on the N2 + N system...... 80 5.5.1 Internal energy excitation...... 81 5.5.2 Nonequilibrium dissociation...... 83

vii 5.6 Comparison of N4 system to N3 + N4 system...... 85 5.7 Comparison of ab − initio results for the full nitrogen system...... 87 5.8 Conclusions...... 88

6 Direct Molecular Simulation of Oxygen 92

6.1 Direct molecular simulation of O2 + O interactions...... 92 6.1.1 Introduction...... 92

6.1.2 Simulation of O2 + O interactions...... 93

6.1.3 Vibrational energy excitation by O2 + O interactions...... 97

6.1.4 Non-equilibrium dissociation due to O2 + O interactions..... 102

6.2 Direct molecular simulation of O2 + O2 interactions...... 105 6.2.1 Introduction...... 105

6.2.2 Vibrational energy excitation due to O2 + O2 interactions.... 105

6.2.3 Non-equilibrium dissociation due to O2 + O2 interactions.... 108 6.3 Conclusions...... 114

7 Internal Energy Relaxation 116 7.1 Introduction...... 116 7.2 Vibrational energy redistribution in nitrogen...... 117 7.2.1 Vibrational energy redistribution in QSS...... 117 7.2.2 Normalized vibrational energy redistribution probabilities in QSS 122 7.2.3 Multi-quantum level jumps...... 126 7.3 Vibrational energy redistribution in oxygen...... 128 7.3.1 Vibrational energy redistribution in QSS...... 128 7.3.2 Normalized vibrational energy redistribution probabilities in QSS 129 7.4 Conclusions...... 130

8 Summary 133 8.1 Summary of contributions...... 133

Bibliography 139

viii List of Tables

2.1 Properties of diatomic nitrogen...... 17 2.2 Properties of diatomic oxygen...... 22 5.1 Comparison of non-equilibrium dissociation rates for nitrogen...... 80 6.1 Electronic states of molecular oxygen...... 94

6.2 Comparison of QSS dissociation rate coeﬃcients for O2 + O2 interactions 110

ix List of Figures

1.1 Reentry conditions from low Earth orbit, Lunar return, Mars return..2 1.2 Example of a potential energy surface for three oxygen atoms interacting in a collinear arrangement...... 4 2.1 Example of atomic arrangements...... 11 2.2 Comparison of diatomic PESs used in the NASA PESs, and the UMN PES 18 2.3 Comparison of energy contours for a collinear arrangement of three nitrogen atoms produced by the NASA PESand the UMN PES...... 19 2.4 Comparison of energy contours for an in-plane rhombic arrangement of four nitrogen atoms produced by the NASA PES and the UMN PES.. 19 2.5 Analytical diatomic potentials...... 21 2.6 Energy contour for collinear arrangement of three oxygen atoms for the PESs developed by Varga et al...... 23 2.7 Energy contour for collinear arrangement of three oxygen atoms on the PES by Varandas and Pais...... 24 2.8 Potential curves for cuts through the collinear potential surface with one diatomic distance ﬁxed at 1.2 A˚ (Fig. 2.8(a)) and 1.3 A˚ (Fig. 2.8(b)) where the other diatomic distance (the abscissa) is scanned. In each plot shows three curves: XMS-CASPT2 calculations and MR-ACPF calculations for the lowest energy 1A0 surface and also the values calculated from the analytic surface of Varandas and Pais...... 26 2.9 Energy contours for an in-plane rhombic arrangement of four oxygen atoms 27

3.1 Example of N2 + N2 (a) and N2 + N (b) trajectories...... 33

x 3.2 Effect of varying bmax on shock profiles...... 39 3.3 Comparison between shock profiles predicted by DMS and MD...... 41 3.4 Rotational energy distribution functions within the shock wave..... 42 3.5 Non-reacting DMS solutions for shock-wave profiles corresponding to hypersonic flight at approximately 40 km altitude...... 44 3.6 Two dimensional DMS simulation of hypersonic nitrogen flow over a cylinder...... 45 3.7 Example of isothermal relaxation in a box and formation of the QSS for nitrogen...... 48

3.8 Internal energy distributions of N2 molecules in QSS for Tt = 30000K.. 49 3.9 Isothermal simulations starting at various initial conditions...... 49

3.10 DMS isothermal relaxation results for the N2-N2 system using both ab- initio and Ling-Rigby + Morse potentials...... 51 4.1 Non-reacting DMS solutions for shock-wave profiles corresponding to hypersonic flight at approximately 40 km altitude using the Ling-Rigby PES with the harmonic oscillator and Taylor-6 potential...... 54 4.2 Comparison of shock profile obtained from DSMC and DMS (also know as CTC-DSMC)...... 55

5.1 Example vibrational excitation calculation for the N2 + N2 system.... 58 5.2 Comparison of characteristic vibrational relaxation time obtained using the Ames PES (blue) against that obtained using the UMN PES (red) and the Millikan and White correlation (black curve)...... 60

5.3 Composition history comparison for the N2 + N2 system...... 61 5.4 Comparison of non-equilibrium dissociation rate constants obtained for

N2 + N2 collisions...... 62

5.5 Vibrational distribution of N2 molecules in QSS for Ttr = 30000K, 20000K

and 10000K for the N2 + N2 processes...... 64

5.6 Internal energy excitation proﬁles for N2 + N2 using DMS and ME-QCT 66 5.7 Vibrational temperature and rotational temperature excitation proﬁles

for N2 + N2 using DMS and ME-QCT...... 67

xi 5.8 Instantaneous internal energy distribution during the excitation process

for N2 + N2 using DMS and ME-QCT...... 68

5.9 Composition history for the isothermal relaxation calculations for N2+N2 using DMS and ME-QCT...... 69

5.10 Composition histroy for Ttr = 10000 K case with recombination turned oﬀ for ME-QCT...... 70 5.11 Internal energy distribution in QSS...... 70 5.12 Dissociation probability against internal energy...... 71 5.13 Dependance of characteristic excitation time on partial density of molecular nitrogen...... 72 5.14 Example vibrational excitation calculation...... 74 5.15 Comparison of characteristic vibrational time obtained from DMS (blue curve) against Millikan and White formulation (black curve) and contemporary results...... 75 5.16 Evolution of energy modes during isothermal excitation...... 75 5.17 Comparison of non-equilibrium dissociation rate constants obtained for

N + N2 collisions...... 77 5.18 Composition and temperature proﬁles resulting from a dissociation calculation, along with the internal energy distributions of molecules.... 78 5.19 Dissociation probabilities for a given translational temperature over a range of internal energy and vibrational energy...... 79

5.20 Nonequilibrium dissociation rate coeﬃcients obtained for N + N2 and

N2 + N2 using the DMS method with PESs developed at NASA Ames Research Center (blue) and the University of Minnesota (red)...... 80 5.21 nternal energy excitation using DMS and StS methods...... 82 5.22 Instantaneous internal energy distribution during the excitation process using DMS and StS methods...... 82 5.23 Composition history during isothermal relaxation...... 83 5.24 Internal energy distributions obtained during QSS...... 84 5.25 Cumulative probability of dissociation against internal energy...... 84

xii 5.26 Composition history comparison for N3+N4 system and N4 system... 85

5.27 Vibrational distribution of N2 molecules in QSS for Ttr = 30000K, 20000K, 150000K

and 10000K for the N + N2 and N2 + N2 processes...... 86 5.28 Composition and temperature history comparison for N3+N4 system using the NASA PES and the Minnesota PES...... 88

5.29 Vibrational distribution of N2 molecules in QSS for Ttr = 30000K, 20000K

and 10000K for the N + N2 and N2 + N2 processes...... 89 6.1 Temperature and composition history of an isothermal excitation calculation done in DMS for the full O3 system...... 96

6.2 Internal energy distributions of O2 molecules in QSS for Ttr = 12000 K. 97

6.3 Example of excitation calculations with O2 + O interactions...... 98

6.4 Characteristic vibrational excitation time for O2 + O interactions.... 99

6.5 O2 + O QSS dissociation rate coeﬃcients...... 103 6.6 Distributions of the system in QSS and the pre-dissociation distribution

of molecules that dissociate in the O2 + O system...... 104

6.7 Example of excitation calculations with O2 + O2 interactions...... 106

6.8 Characteristic vibrational excitation time for O2 + O2 interactions.... 107

6.9 Comparison of characteristic vibrational excitation time for O2 + O and

O2 + O2 interactions...... 108

6.10 QSS dissociation rate coeﬃcient for O2 + O2 interactions...... 110

6.11 Vibrational energy distribution in QSS given by O2 + O2 interactions.. 112

6.12 Comparison of non-equilibrium dissociation due to O2 + O2 interactions

and O2 + O interactions...... 113 6.13 Comparison of vibrational energy distributions of the system in QSS, and the pre-dissociation distribution of molecules that dissociate due to

O2 + O2, and O2 + O interactions...... 113 7.1 Example of QSS formation in nitrogen...... 117

7.2 Vibrational energy redistribution contours for nitrogen in QSS at Ttr = 30000 K...... 119

xiii 7.3 Conditional vibrational energy transfer probabilities for ﬁxed ﬁnal vibra-

tional energies due to N + N2 and N2 + N2 interactions...... 120 7.4 Conditional vibrational energy transfer probabilities for ﬁxed initial vi-

brational energies due to N + N2 and N2 + N2 interactions...... 121 7.5 Example normalization and curve-collapse of probabilities of vibrational energy redistribution...... 123 7.6 Comparison of normalized vibrational energy transfer probabilities for all

nitrogen interaction events at Ttr = 30000K ...... 124 7.7 Comparison of vibrational energy transfer at diﬀerent temperatures for

N + N2 interactions...... 125 7.8 Comparison of vibrational energy transfer at diﬀerent temperatures for

N2 + N2 interactions...... 125 7.9 Vibrational level jumps as a percentage of interactions that lead to change in vibrational levels...... 127

7.10 Vibrational energy redistribution contours for oxygen in QSS at Ttr = 12000 K...... 129 7.11 Conditional vibrational energy transfer probabilities for ﬁxed initial vi-

brational energies due to O2 + O interactions...... 129 7.12 Probability distribution functions for vibrational energy change due to

O2 + O collisions and O2 + O exchange reactions at Ttr = 12000K and

Ttr = 5000K...... 130

xiv Chapter 1

Introduction

1.1 Challenges of Hypersonic Flight

Flows where the mach number is greater than five are called hypersonic flows. Shock- layer temperatures during hypersonic flight can often exceed thousands of kelvin, and shock stand-off distances are small enough where thermal and chemical equilibrium cannot be reached in the gas. Figure 1.1 shows vehicle trajectory profiles during atmospheric reentry from a low earth orbit corresponding to the space shuttle mission - STS-28, lunar return, and Mars return [1,2]. The abscissa at the bottom of the figure shows the velocity of the vehicle in kilometers per second, the ordinate shows altitude in kilometers and the abscissa on top of the figure shows the stagnation temperature. Furthermore, dashed red curves in the figure highlight the region of the reentry trajectory where important chemical phenomena : oxygen dissociation, nitrogen dissociation and ionization of air occur. This figure exemplifies the extreme conditions that exists during hypersonic flight and highlights the complex physics and chemistry associated with hypersonic flows. Key phenomena to quantify during hypersonic flight in the atmosphere include, vibrational and electronic excitation of the gas, dissociation of molecular nitrogen and oxygen into atomic species, ionization of atmospheric gases, and the consequent radia- tion emitted from the ionized gas. Since, hypersonic fight tests are expensive, there is

1 2

Figure 1.1: Reentry conditions from low Earth orbit, Lunar return, Mars return [1,2]. a need for predictive simulation tools such as computational fluid dynamics (CFD) and direct simulation Monte-Carlo (DSMC). The most widely used thermochemical models used in CFD and DSMC are empirical and rely on limited legacy experimental data [3–6] from the Apollo era (1960s). Experiments are difficult to conduct under the extreme conditions of hypersonic flight; they have significant uncertainties and are unable to exactly recreate flight conditions. For instance experimental data for nitrogen dissociation is only available up to 15000K [7] and the inferred dissociation rates can vary by orders of magnitude between different experiments [7–10]. Additionally, the physics of gases at high temperatures involves strong non-linearities that can make extrapolations from low-temperature experiments inaccurate. Another major short coming of the conventionally used CFD models is in modeling internal energy relaxation and treatment of thermal non-equilibrium. During high-speed flows, behind strong shocks, there is a region where energy in the translational energy mode of the air molecules rapidly increases, whereas the energy in the internal energy 3 modes get excited at a slower rate. Hence, there is thermal non-equilibrium between various energy modes of the gas [11, 12] and a single thermodynamic quantity such as temperature [13] is no longer adequate describe the average energy in all degrees of freedom. Models have been made with three temperatures, describing average energy in the translational mode as translational temperature (Ttr), average energy in the rotational mode of molecules as rotational temperature (Trot) and average energy in the vibrational mode of molecules as vibrational temperature (Tvib). In literature, it is common to find the assumption that the rotational energy mode equilibrates with the translational mode within 1 − 10 mean collision times for the gas and the vibrational energy equilibrates within 103 − 106 mean collision times [14–17]. Hence, the most commonly used CFD models assume trans-rotational equilibrium (Ttr = Trot)[3–6] and only solve for the evolution of vibrational energy using the Landu-Teller equations [18] and vibrational relaxation parameters based in experiments [4,5, 19]. It should be noted the assumption of trans-rotational equilibrium is poorly supported by modern research [20, 21] and ignores coupling between rotational and vibrational modes during the excitation process which has been shown to be important [22, 23]. Furthermore, the Landau-Teller equations are a general result for systems close to equilibrium, in which case relaxation is always exponential, however this model may not be valid more broadly [24]. The motivation of this work is to understand the physics behind internal energy excitation and non-equilibrium dissociation in hypersonic flows at the molecular level and provide data on the quantities of interest that can be implemented into the next generation of CFD and DSMC codes. In particular, excitation and dissociation of molecular nitrogen due to interaction with molecular and atomic nitrogen, and excitation and dissociation of molecular oxygen due to interaction with molecular and atomic oxygen will be studied. 4 1.2 Contemporary work

In recent years there has been a large push to augment experimental data with computational chemistry to characterize the gas under hypersonic conditions. The first step in this method is to calculate a potential energy surface (PES) by solving the electronic Schrödingerequation for various arrangements of atoms of interest to obtain single point energies. The single point energies are interpolated using advanced fitting techniques [25–32] which results in a smooth analytic hypersurface over which the energy and forces acting on the atoms can be calculated. Figure 1.2 shows an example of such a potential energy surface for three oxygen atoms interacting in a collinear arrangement. The figure was created using one of the PESs generated for the O2 + O interactions by Varga et al. [31]. Chapter2 reviews the PESs used in this study.

Figure 1.2: Example of a potential energy surface for three oxygen atoms interacting in a collinear arrangement [31].

A popular way to analyze gas properties with PESs is quasi-classical trajectory (QCT) [28, 33, 34] analysis, which is used to determine state-to-state transition rates for use in master equation analysis [35]. The state-to-state approach, each internal state of 5 the gas is treated as a pseudo-species and during a flow simulation the population of the species is solved using microscopic kinetic data [36]. While this is a powerful approach and is free of empiricism, it can become intractable. For instance, if one accounts for all possible ro-vibrational state transitions, there are about O(107) possible transitions 15 for the N + N2 system and about O(10 ) possible transitions for the N2 + N2 system, making a fully resolved state-to-state model infeasible. Hence, state-to-state models are largely limited to zero- or one-dimensional simulations [20, 36–48]. In an effort to make state-to-state models feasible, binning strategies [49–57] are employed to reduce the computation cost of state-to-state analysis. This is achieved by grouping populations of various states together. In order to determine the kinetic properties for the groups, a sampling based method is used in conjunction with quasiclassical trajectory calculations, bypassing the need to fully characterize the state-to-state kinetics. The manner in which states are grouped together in the model has a large influence on the accuracy of the model and there are often trade offs in the accuracy between characterizing internal energy excitation and dissociation [57]. A detailed discussion on the trade offs and short-comings of binned state-to-state models can be found in chapter5.

1.3 Introduction to Direct Molecular Simulation

The analysis of this thesis uses the direct molecular simulation (DMS) [58] method. The DMS method is similar to the DSMC method [16, 17] and simulates particle collisions in a time-accurate flow field. But instead of using probabilistic collision models, the DMS method performs molecular trajectories using PESs for the desired system. Hence, the only modeling inputs required for the DMS method are the PESs governing the collisions. After a trajectory calculation, the post-trajectory state of the particle becomes its initial state for the next interaction In this manner the DMS method is able to capture non-equilibrium physics and transient energy distributions [58]. This method was first introduced by Koura [59–61] as classical trajectory calculation DSMC (CTC-DSMC) and was extended to rotating and vibrating molecules by Schwartzentruber, Valentini and Norman [22, 62]. 6 Unlike QCT based methods like master equation [35] analysis, DMS implements trajectory integration within a flow field simulation. This negates the construction of a database of all possible state-to-state transitions of the colliding partners. Instead of pre-computing all possible energy transitions, the DMS method automatically simulates only the most dominant energy transitions that actually occur with non-negligible frequency for the conditions of interest. Although computationally demanding, this approach is now tractable for diatom-diatom systems, and in fact, for gas mixtures with a number of species combinations within collisions. Since pre-computation of state- to-state transitions is not required, this makes the DMS method accessible to any gas system for which the PES is available. Furthermore, since the DMS method does not perform binning of energy states, it can used as a benchmark solution on which to check reduced order models for non-equilibrium reacting flow. The DMS method has been used to study rotational energy excitation [62], coupling between rotation and vibrational energy [22, 23]. It has been employed in studying molecular level mechanisms of internal energy excitation and non-equilibrium dissociation in nitrogen [63, 64] and oxygen [65, 66], used to perform comparative analysis of PESs [67, 68] and validate state-to-state binning methodologies [57, 69]. Details of the DMS method will be provided in chapter3.

1.4 Outline

This dissertation is organized as follows. Chapter2 gives a brief overview of the PESs used in this study for nitrogen and oxygen interactions. Chapter3 explains the direct molecular simulation method, which is the numerical method used to explore thermal and chemical non-equilibrium in this study. The chapter highlights key equations and algorithms used in this method, discusses molecular trajectory integration using PESs, treatment of internal energy in the method, and validates the DMS method against pure molecular dynamics. Chapter4 discusses coupling between rotational and vibrational energy modes during hypersonic ﬂows, by studying rovibrational coupling in normal shocks. Chapter5 discusses internal energy excitation and non-equilibrium dissociation 7 in nitrogen due to N2 + N and N2 + N2 interactions. This chapter performs a comparative study of excitation and dissociation processes by using independently developed PESs for nitrogen interactions at NASA Ames Research Center [25, 26] and the Univer- sity of Minnesota [27] using the DMS method. Furthermore, in chapter5, additionally a comparative analysis between the DMS method and master equation analysis is performed. Chapter6 discusses internal energy excitation and non-equilibrium dissociation in oxygen due to O2 + O and O2 + O2 interactions. Chapter7 discusses inelastic processes that lead to vibrational energy redistribution. Finally, chapter8 provides a brief summary of the dissertation. Chapter 2

Potential Energy Surfaces

2.1 Introduction

In this chapter provides a high-level discussion of potential energy surfaces (PESs) that serve as the only modeling input for the DMS calculations. Potential energy surfaces provide a fit of the potential energy of the atomic arrangements and the gradient of the potential energy is used to perform the molecular trajectories as discussed in section 3.2.2. The potential energy calculation in itself can be based in quantum mechanics or based on experimental data. Early attempts at making PESs are based on a combination of theoretical calculations and empirical data from experiments [70–75]. These PESs are often represented with simple analytical functions which makes the use of these PESs for performing trajectory analysis computationally cheap. However, due to the simplicity of the fitting functions, and use of inadequate data to resolve atomic interactions [67], such PESs are less accurate. Most modern PESs [25–31] are based solely on extensive quantum mechanical calculations. Quantum mechanical interactions between particles are given by the Schrödinger equation:

d 2 i Ψ(x, t) = ~ ∇2 + V (x) Ψ(x, t) = H(x)Ψ(x, t) (2.1) ~dt 2m

Where ~ is the Plank’s constant divided by 2π, Ψ is the wave function, ‘m’ is the mass

8 9 of the particle, V (x) is the potential energy and H(x) is the Hamiltonian operator. The Hamiltonian operator accounts for the total energy of the system (kinetic and potential energy). For a conservative system, both the Hamiltonian (H(x)) and the potential energy (V (x)) are a function of the position of the particle ‘x’ and the wave function can be split into a time dependent component : exp(−iEt/~) and a spatial component: ψ(x). Separating the temporal and the spatial components of equation 2.1 yields the time independent Schr¨odingerequation :

H(x)ψ(x) = Eψ(x) (2.2)

Where ‘E’ is the total energy of the system. For construction of PESs the Schrödinger equation needs to be solved for the interacting nuclei and the electrons around the nuclei. Solving equation 2.2 directly becomes very incredibly complicated for systems involving more than a couple of interacting electrons. Hence, a few assumptions need to be made to solve the time independent Schrödingerequation, the first of these is the Born-Oppenheimer approximation [76]. Under the Born-Oppenheimer approximation the motion of the electrons and the nucleus are assumed to be uncoupled. This assumption is justified by the fact the mass of the nucleus is more than three orders of magnitude larger than the mass of the electrons. This allows us to split equation 2.2 into the electronic Schrödingerequation for fixed nuclear positions, given by :

He(x)ψe(x) = Eeψe(x) , (2.3)

deﬁning,

He(x) = Ke + Vee + VNe + VNN . (2.4)

In equation 2.3 ψe is the wave-function for the electrons and He(x) is the electronic Hamiltonian and includes energy contribution from the kinetic energy of the electrons

(Ke), potential energy due to electron-electron repulsion (Vee), potential energy due to 10 electron-nucleus attraction (VNe), and by convention also includes the term for the potential energy due to inter nuclear repulsion (VNN ). This leaves the nuclear Schr¨odinger equation to have the form :

HN (x)ψN (x) = EψN (x) , (2.5)

and

HN (x) = KN + Ee. (2.6)

Where, ψN is the wave-function of the nucleus and HN (x) is the nuclear Hamiltonian operator. It should be noted that the nuclear Hamiltonian operator gives the total energy of the system (E) and consists of the nuclear kinetic energy term, which describes nuclear motion, and the electronic energy (Ee) obtained from the electronic Schrödinger equation (equation 2.3). Hence, the total energy of the electronic sub-system (Ee) serves as the potential energy governing nuclear motion and by extension defines the PES. The electronic energy is solved for various geometric arrangements of the atoms to generate multiple single point energies, which are then interpolated to form a smooth hypersurface that represents the PES. While equation 2.3 can be solved directly for simple systems, for complex system involving several electrons the equation is becomes difficult to solve directly and a solution is obtained by a series of numerical expansions. From quantum mechanics that the electronic wave-function (ψe) determines the probability density function of the electrons. The first step in this process is to express the electronic wave function of the electrons as a linear combination of atomic orbitals centered at each nucleus of the interacting atoms, the functions describing the electron density about a nucleus are called single particle basis set (this is the first numerical expansion). The Schrödinger equation is solved variationally using these basis sets for the electrons and applying the Hartree-Fock approximation. That is, assuming that each electron experiences the average Coulombic forces due to other electrons. This leads to the generation of molecular orbitals that are a linear combination of the atomic orbitals. Next, the Hartree-Fock 11 assumption is corrected by accounting for electron-electron interactions called electron correlation. Electron correlation effects are calculated by introducing various arrangement of electrons in the molecular orbits (this is the second order expansion of the Schrödingerequation). It should be noted that relativistic effects like spin-orbit coupling are ignored during PES calculations. The calculated electronic energy approaches the exact solution of the Schrödinger equation as the two basis expansions described above become infinite. However, that is not practical. In practice, the single particle basis set consists of a subset set of all possible atomic orbitals and the number of electronic correlations scale as the factorial of the total number of electrons in the system and numerical techniques are required to optimize the number of correlations needed. Hence, this method does not give the exact solution to the Schrödingerequation, and consequently the value of electronic energy Ee. However, the number of basis sets and correlations are increased, and the variational solution to the Schrödingerequation is iterated until the solution becomes sufficiently converged.

(a) Three body arrangement. (b) Four body arrangement.

Figure 2.1: Example of atomic arrangements

In this thesis discusses N2 + N, N2 + N2, O2 + O and O2 + O2 interactions using potential energy surfaces. Before discussing the PESs used in this study it is important to understand the dimensionality of the PESs. As discussed above PESs are based on potential energy calculated for a speciﬁc arrangement of nuclei. For three body systems 12

(N2 + N and O2 + O) an arrangement can be fully described by three parameters (Fig.

2.1(a)). Similarly a four body systems (N2 + N2 and O2 + O2) interactions, the atomic arrangements and hence the PES (Fig. 2.1(b)) can be completely described by six parameters. 1 The dimensionality of a PES is given by 3n − 6, where ‘n’ is the number of interacting atoms.

2.2 Spin multiplicity and spatial symmetry

As evident from the discussion above, the actual arrangement of electrons in a system plays an important role in the determination of the potential energy of an atomic arrangement. Furthermore, a PES can only be made for a particular electronic state. That is, while solving the electronic Schrödingerequation, the overall electronic spin is conserved and spatial symmetry of the electronic structure is held fixed. This implies that if a certain interaction has more than one spin-spatial degenerate state, PESs for each degenerate state need to be constructed to fully describe the system. Spin multiplicity is a measure of the overall electronic spin of a system. Mathemat- ically, spin multiplicity is given by 2S + 1, where S is the total spin angular momentum of the system and is given by the sum of all spin quantum number of un-paired electrons in the system. For example, diatomic nitrogen has no unpaired electrons (S = 0) so it has a spin multiplicity of one and is referred to as a singlet system or a singlet molecule. Similarly, diatomic oxygen in ground electronic state has two unpaired electrons (S = 1) and has a spin multiplicity of three and it is called a triplet system [14, 76]. Usually, the spin multiplicity of a system gives the number of possible orientations of the total spin (S) with respect to the total orbital angular momentum (L). Hence, the spin multiplicity of a system gives the number of near-degenerate levels for that total spin. These near-degenerate levels only differ in their spin-orbit interaction energy, and since the PESs developed for the studies in this thesis ignore the effects of spin- orbit coupling, the spin multiplicity gives the degeneracy of the spin state. However, the equality between spin multiplicity and degeneracy breaks down if the total orbital

1The vertex of the tetrahedral deﬁned by points A, B, C and D is marked ‘O’. 13 angular momentum quantum number (L) is greater than the total spin quantum number (S). In that case the degeneracy of electronic spin states is given by 2L + 1. A famous example of the break down of this equality is atomic nitrogen. Atomic nitrogen has three unpaired electrons (S = 3/2), and has a quartet multiplicity, however, for atomic nitrogen the total orbital angular momentum quantum number is L = 0. Hence, the spin degeneracy for atomic nitrogen is one even though the multiplicity is four.

Characterizing spin-degeneracy for systems studied in this thesis, for N2 + N2 interactions have the total spin angular momentum (S = 0) and the system is a singlet with a spin degeneracy of one, for N2 + N interactions (S = 3/2) and the system is a quartet with a spin degeneracy of one. For the O2 + O2 system in the ground electronic state two triplet diatomic oxygen molecules interacting. This can lead to a total spin quantum number of S = 0, 1, 2. Hence O2 + O2 interactions can occur with an overall singlet, triplet or quintet spin multiplicity, with a degeneracy of 1,3 and 5 respectively. Therefore three unique PESs (one for each overall spin state) are needed to fully describe

O2 + O2 interactions. Similarly, for O2 + O interactions in ground state, when a triplet diatomic oxygen interacts with a triplet atomic oxygen (S = 0, 1, 2) and interactions can occur with singlet, triplet and quintet overall spins with a degeneracy of one, three and ﬁve respectively. Further, the spatial symmetry of electrons also needs to be taken into account while studying atom-molecule interactions. For a non-linear triatomic system there is always a plane of symmetry. Normally a triatomic may have other symmetry elements if the two or more atoms are of the same element and the bond lengths are equal. For a PES, one assumes unequal bond lengths and so the PES is determined with only the symmetry plane. The point group for this is Cs and structures can be symmetric or antisymmetric. A symmetric structure will have same structure above and below the plane and is denoted by the notation ‘a0’ and an antisymmetric structure will have a change of sign above and below the plane of symmetry and is denoted by the notation ‘a00’. So any electron in s-orbitals or p-orbitals lying in the symmetry plane are symmetric and designated a0 and electrons in p-orbitals oriented perpendicular to the symmetry plane are designated as a00 [77]. Overall symmetry of a system is given by the product of 14 individual symmetry elements. The product of the symmetry elements is deﬁned as:

a0 × a0 = a0 , (2.7)

a00 × a00 = a0 (2.8)

and,

a0 × a00 = a00 . (2.9)

By convention, the symmetry of sub-structures like the orientation of electrons in an orbital is given by lower case “a” and the overall symmetry of the system is given by upper case “A”.

Consider N2 +N interactions. For atomic nitrogen the electronic conﬁguration is 1s2 2s2 2p3, hence atomic nitrogen has one electron in each p-orbit. This means that the spatial symmetry with regards to the plane deﬁned by three interacting nitrogen atoms for electrons of atomic nitrogen will always be given by a0 × a0 × a00 = a00. Further, the spatial symmetry of molecular nitrogen is a0, therefore the overall spatial symmetry of 0 00 00 the N3 system will be given as a × a = A . Physically, an overall spatial symmetry of A00 means that regardless of how the atom is oriented there will always be two electrons in the symmetry plane and one will be out of plane. Hence, the N2 + N interactions are a quartet with one spin degeneracy and have only one possible symmetry of A00.

Therefore only one PES is required to understand the dynamics of N2 + N collisions.

Now consider O2 + O interactions. For atomic oxygen the electronic conﬁguration is 1s2 2s2 2p4, hence atomic oxygen has one electron in two p-orbits and a pair of electrons in the third p-orbit. Lets say that the paired electron is in the p-x orbit and the unpaired electrons are in p-y and p-z orbits. Now if the plane of interaction of the three oxygen atoms is the x-y plane, the symmetry of atomic oxygen will be given by (a0 × a0) × a0 × a00 = a00, similarly, for the x-z plane, the symmetry will be a00 as well. However, for the y-z plane the spatial symmetry will be given as a0 × a0 × (a00 × a00) = a0. 15 Hence atomic oxygen has a three fold degenerate spatial symmetry with two a00 states and one a0 state. Now, the symmetry of molecular oxygen in the ground electronic 00 state is a , therefore, the overall spatial symmetry for each O2 + O interaction can be a00 × a0 = A00 and two states with spatial symmetry a00 × a00 = A0, these two states will be referred to as 1A0 and the 2A0 states.

Hence O2 +O interaction can occur with the total spin coupling being singlet, triplet, or quintet with statistical weights of these spin couplings being 1, 3 and 5 respectively. Further, the spatial symmetry of atomic oxygen generates a threefold degenerate ground state for all three spin states. Hence, due to the spin and spatial degeneracies, nine unique PESs are required to fully describe O2 + O interaction dynamics initiated in the ground electronic states of the collision partners. These PESs will have three singlet surfaces : 11A0, 21A0, 11A00, each with a statistical contribution to collision dynamics of 1/9 × 1/3 = 1/27. There will be three triplet surfaces : 13A0, 23A0, 13A00, each with a statistical contribution to collision dynamics of 3/9 × 1/3 = 3/27. Finally, there will be three quintet surfaces : 15A0, 25A0, 15A00, each with a statistical contribution to collision dynamics of 5/9 × 1/3 = 5/27. Such spatial symmetries for molecule-molecule interactions do not exist as geometries for molecule-molecule interactions are deﬁned by four points (atoms) and can be non- planar.

2.3 Potential energy surfaces for nitrogen

In this section, the ab-initio PESs developed for nitrogen at the University of Min- nesota [27] and at NASA Ames Research Center [25, 26] are discussed. These have been used in chapter5 to study the non-equilibrium properties yielded by two independently developed PESs for nitrogen. Next simple site-to-site PES are discussed for the nitrogen system, which are used in chapter3 to demonstrate one-dimensional and multi-dimensional ﬂows, and in chapter4 to study coupling between rotational and vibrational modes during a behind normal shocks. 16 2.3.1 Ab-initio potential energy surfaces

Both NASA Ames Research center and the University of Minnesota have recently developed ab-initio PESs to describe N2 +N and N2 +N2 interactions. NASA Ames Research center developed separate PESs to describe N2 + N interactions [26] and N2 + N2 interactions [25], these will be called the NASA N3 and the NASA N4 PESs respectively.

However, at the University of Minnesota only a single PES for N2 + N2 [27] interactions is constructed, to simulate N2 + N dynamics on this PES one of the four atoms is assumed to be at a large distance and then the interaction between the three bodies is carried out. This PES will be referred to as the UMN PES in this section. The NASA PESs, and the UMN PES were calculated using an open source program MOLPRO [78], the choice of basis sets and variational methods to solve for the electronic energy is discussed below.

Both of the NASA PESs (N3 and N4) were constructed using the augmented correlation- consistent polarized valence triple zeta (aug-cc-pVTZ) [79, 80] basis set. The aug-cc- pVTZ basis set contains five s-orbitals, four p-orbitals, three d-orbitals and two f-orbitals for each atom. The NASA PESs utilized two different methods to compute the electron correlation. The coupled cluster method (CCSD(T)) [81] was used for geometries where there are two distinct N2 molecules and the multi-reference configuration interaction method (MRCI) [82] was used for other geometries. The latter method uses complete active space (CASSCF) [83] molecular orbitals instead of Hartree Fock. This approach includes the most important electron correlation effects in the orbital optimization process, and also speeds up the convergence of the MRCI expansion. For the NASA N4 PES, the CCSD(T) [81] method was used for 3821 geometries and the CASSCF-MRCI method was used for 325 geometries, with energy calculations performed for a total of

4146 geometries. Furthermore, for the NASA N3 PES, the CCSD(T) method was used to calculate 3885 geometries and MCRI method was used for 56 geometries. Addition- ally, both of the NASA N3 and NASA N4 PESs use a diatomic PES based on a diatomic PES proposed by Le Roy et al.[84]. The UMN PES [27] is based on quantum mechanical calculations that are similar 17 Quantity UMN PES NASA PES Dissociation energy at j=0 9.91 eV 9.89 eV Number of vibrational levels for j=0 55 61 Number of rotational levels for v=0 279 279 Number of ro-vibrational levels 9198 9390 Number of bound ro-vibrational levels 7122 7421 Number of quasi-bound ro-vibrational levels 2076 1969

Table 2.1: Properties of diatomic nitrogen from the potential energy surfaces developed at the University of Minnesota [27] and at NASA Ames Research Center [25, 26]

to those described above. The UMN PES uses a minimally augmented correlation- consistent polarized valence triple zeta (maug-cc-pVTZ) basis set [85]. This is the same as the aug-cc-pVTZ basis set used by the NASA PESs but it drops some of the d- and f-orbitals from the augmentation set. A similar CASSCF procedure is used to obtain the molecular orbitals, but second-order perturbation theory was used in place of MRCI. That method is called CASPT2 [86–88]. For generating the PES, 16421 total points were calculated, with 1017 points speciﬁcally being calculated for the N3+N geometries.

As a result the PES can perform N2 + N2 and N2 + N interactions. Table 2.1 compares the properties of the diatomic nitrogen given by the NASA and UMN PESs. The dissociation energy well depth given by the two PESs is slightly different, however, this difference is well within the errors associated with computational methods used to make the PESs. The NASA PESs predict five more vibrational levels at j = 0 for diatomic nitrogen. This can be explained by the shape of the diatomic PES predicted by the NASA and the UMN PESs shown in Fig. 2.2, which shows the diatomic potential energy at j = 0. The diatomic PES used in the given by NASA has a much more gradual ”shoulder” when compared with the UMN PES and exhibits a long range behavior not captured by the UMN PES. This gradual shoulder allows for a more vibrational levels to be accommodated at the higher energy end. This discrepancy at the shoulder of the PES arises from the fact the UMN PES does not include a dispersion term for the diatomic PES. 18

(a) Full diatomic curve. (b) Zoomed view for large r.

Figure 2.2: Comparison of diatomic PESs used in the NASA PESs [25, 26], and the UMN PES [27]

Figure 2.3, shows the energy contours for a collinear arrangement of three nitrogen atoms using the UMN PES [27] and the NASA PES [26]. In reference to Fig. 2.1(a), ◦ the inter-atomic angle is ﬁxed, such that ∠ABC = 180 and inter-atomic distances

RAB and RBC are varied to produce this ﬁgure. Both the PESs give a similar contour, with reactant-product energy valleys (blue) when two of the atoms are close to the equilibrium bond length and the third atom is far away, zero overall energy when all atoms are far away and a repulsive wall when one or both interatomic distances become small. Two small saddle points at are seen RAB = 1.5 A,˚ RBC = 1.2 A,˚ and RAB = 1.2

A,˚ RBC = 1.5 Afor˚ the UMN PES (Fig. 2.3(b)) which are not present in the NASA PES (Fig. 2.3(a)). Figure 2.4 shows the energy contour for an in-plane rhombic arrangement of four atoms using the UMN PES [27] and and the NASA PES [25]. To produce this ﬁgure, in reference to Fig. 2.1(b) inner angles of the rhombus were ﬁxed ∠AOD = ∠AOB = ◦ ∠COD = ∠COB = 90 and distances RAC and RBD are varied. It is observed that when all atoms are far away there is an energy plateau of E = 0 eV, and when one 19

(a) NASA N3 PES. (b) UMN N3 PES.

Figure 2.3: Comparison of energy contours for a collinear arrangement of three nitrogen atoms produced by the NASA PES [26] and the UMN PES [27]

(a) NASA N4 PES. (b) UMN N4 PES.

Figure 2.4: Comparison of energy contours for an in-plane rhombic arrangement of four nitrogen atoms produced by the NASA PES [25] and the UMN PES [27]. or more inter-atomic distances become small a repulsive wall is observed. Product- reactant valleys are produced when RAC is equal to the diatomic nitrogen equilibrium bond length and RBD is large and vice versa for both the PESs. Further, for the

UMN PES there is an energy valley centered around RAC ∼ 2 Aand˚ RBD ∼ 2 A.˚ A 20 corresponding feature does exist in the NASA N4 PES but it appears to be smeared over a larger area than the valley present on the UMN PES. Chapter5 contains discussion on the macroscopic properties calculated for nitrogen using these PESs. These differences maybe attributed the NASA PES having less geometries over which the energy was calculated in comparison to the UMN PES. Clearly, there are differences in the NASA and UMN PESs.As detailed in chapter5 only the difference in the shoulder regions (Fig. 2.2 ) leads to macroscopic difference in dissociation.

2.3.2 Site-to-site potential energy surface

Site-to-site potentials are simpliﬁed potentials that can reduce the cost of trajectory calculation for complex simulations. Instead of having thousands to single point energies ﬁtted by complex functional forms like the ab-initio PESs, these PESs can be expressed using a simple analytical expression. The site-to-site PES used to describe N2 + N2 interactions in chapters3 and4 is from Ling and Rigby [70] and will be referred to as the Ling-Rigby PES. The Ling-Rigby potential is given by the simple expression:

C φ(r) = D exp(−αr − βr2) − f (r) 6 , (2.10) e d r6

−1 −2 where De = 14151.94 kcal/mol, α = 2.2412 A˚ , β = 0.3214 A˚ , C6 = 33.6 (kcal 6 A˚ )/mol, fd(r) is the damping function,

" #! 1 δ 2 f (r) = exp − − 1 , (2.11) d 4 r

where δ = 4.14 A˚ and r is the distance between atoms belonging to diﬀerent N2 molecules. To simulate intra-atomic (diatomic) potential energy while using the Ling-Rigby PES the harmonic oscillator potential and a sixth order Taylor-series approximation of the morse potential (Taylor-6) are used. These these particular diatomic potentials where chosen because they are bound (oscillators) - that is they do not allow for dissociation. This helps in understanding the internal energy excitation process independent 21 of the dissociation process. Further, since the Taylor-6 potential is based on the morse potential this helps in understanding how anharmonicity eﬀects the internal energy excitation process. The schematics for the diatomic potentials can be seen in Fig. 2.5

Figure 2.5: Analytical diatomic potentials.

The Harmonic Oscillator potential is given by:

1 φ(r) = k (r − r )2 . (2.12) 2 f o

The Taylor-6 potential is given by:

7 1 31 φ(r)/D = a2(r−r )2−a3(r−r )3+ a4(r−r )4− a5(r−r )5+ a6(r−r )6 . (2.13) d o o 12 o 4 o 360 o

The Morse potential is given by:

2 φ(r) = Dd (1 − exp {−a[r − ro]}) . (2.14)

In the above equations the coeﬃcient, a, is given by

s k a = f = 2.689A˚−1 , (2.15) 2Dd 22 where Dd = 227.6 kcal/mol is the potential well depth and kf is the modeled spring constant. Finally, r is the distance between atoms belonging to the same N2 molecule, and ro = 1.098 A˚ is the equilibrium bond length.

2.4 Potential energy surfaces for oxygen

This section discusses the nine PESs developed for O2 + O interactions (called the O3

PESs) [31] and the three PESs developed to study O2 + O2 interactions (called the O4

PESs) [29, 30]. Both the O3 and O4 PESs are developed for interactions in the ground electronic state of the collision partners. All twelve PESs use the same diatomic PES. The properties of ground-state diatomic oxygen given by PESs is shown in table 2.2.

Quantity Value Dissociation energy at j=0 5.21 eV Number of vibrational levels for j=0 45 Number of rotational levels for v=0 241 Number of ro-vibrational levels 6115 Number of bound ro-vibrational levels 4581 Number of quasi-bound ro-vibrational levels 1534

Table 2.2: Properties of diatomic oxygen

2.4.1 Potential energy surfaces for O3 interactions

As described in section 2.2 the spin-spatial degeneracy of the O2 + O system require nine PESs to fully describe the dynamics of O2 + O interactions. Varga et al [31] recently developed a suite of nine adiabatic PESs to fully describe electronically adiabatic collisions initiated in the ground electronic states of the two collision partners. These PESs include three surfaces for the singlet spin state: 11A0, 21A0, 11A00 with statistical weight of 1/27 each, three surfaces for the triplet spin state: 13A0, 23A0, 13A00 with a a statistical weight of 3/27 each, and three surfaces for the quintet spin state: 15A0, 25A0, 15A00 with a statistical weight of 5/27 each. Figure 2.6 shows the energy contours for a collinear interaction between three oxygen 23

(a) 11A0. (b) 21A0. (c) 11A00.

(d) 13A0. (e) 23A0. (f) 13A00.

(g) 15A0. (h) 25A0. (i) 15A00.

Figure 2.6: Energy contour for collinear arrangement of three oxygen atoms for the PESs developed by Varga et al. [31] atoms (labeled: A, B and C) using the nine PESs generated by Varga et al. [31], with atom B placed between A and C. There are three main features in these plots. First are the reactant and product potential energy valleys shown in blue. These valleys correspond to the interatomic distance between two atoms being close to the equilibrium bond length of Req = 1.2076 A˚ and the third atom being further away. The second feature is the repulsive regions when one or more interatomic distance gets small. The 24

Figure 2.7: Energy contour for collinear arrangement of three oxygen atoms on the PES by Varandas and Pais [89] third feature is the plateau where all atoms are far away, shown in green. For the singlet surfaces in Fig. 2.6(a) and Fig. 2.6(c), and triplet surfaces in Fig. 2.6(d) and Fig. 2.6(f), there are low-energy paths along which an oxygen atom maybe transferred between the two end atoms. As discussed below, such pathways enable frequent exchange reactions (AB + C → BC + A) which lead to faster internal energy relaxation. However, these low-energy exchange channel do not exists for any of the other five surfaces. Figure 2.7 shows the energy contours for the collinear arrangement for the PES developed by Varandas and Pais [89]. The PES developed by Varandas and Pais corresponds to the 11A0 state and therefore can be compared directly to the PES by Varga et al. [31] shown in Fig. 2.6(a). The PES proposed by Varandas and Pais [89] has twin ∼ ∼ ∼ ∼ saddle points (RAB = 2.4A˚,RBC = 1.2A˚ and RAB = 1.2A˚,RBC = 2.4A)˚ that do not show up in any of the PESs of Varga et al. [31]. Further calculations have been carried out to investigate this discrepancy as the Varandas and Pais PES is extensively used in the aerospace community to characterize O2 + O interactions [46, 48, 90–92]. Varandas and Pais employed the double many-body expansion with input data based on a combination of experimental data and electronic structure calculations [89]. The latter data were taken from the theoretical work of Shih et al. [93], who carried out 25 both single-configuration SCF calculations and multi-configuration, iterative natural orbital (INO), calculations with an unaugmented, unpolarized double zeta basis set and an unaugmented, bond-function-polarized double-zeta basis sets. Their investigation of the bond angle dependence extended to 150 deg and Varandas and Pais extrapolated the existing data points to 180 deg. In contrast, Varga et al. [31] based their fits on extended multi-state second order perturbation [86–88, 94] (XMS-CASPT2) calculations and accurate diatomic potentials [31] . Some cuts are calculated in the region of the Varandas-Pais saddle points by XMS-CASPT2, and as a check even-higher-level calculations with the multi-reference averaged coupled-pair-functional [82, 95] (MR-ACPF) method are carried out. Both the XMS-CASPT2 and MR-ACPF calculations are based on complete active space self-consistent field [83] (CASSCF) calculations where 12 active electrons are distributed in nine active 2p orbitals, and the atoms are described with a minimally augmented correlation-consistent polarized valence triple zeta basis set (maug-cc-pVTZ basis set [79, 80, 85]). All multireference calculations are carried out by the 2012.1 version of the Molpro program [78]. The shapes of the potential along the cuts calculated by XMS-CASPT2 and MR-ACPF agree well. These multi- reference calculations show that the wave function of the lowest energy 1A0 state has a non-negligible multi-configurational character at collinear geometries of three oxygen atoms. The twin saddle points shown in the fit of Varandas and Pais are not supported by either the XMS-CASPT2 calculations or the MR-ACPF calculations. Figure 2.8 shows plots for two cuts through the potential energy surface, one at a bond distance of 1.2 A˚ and one at a bond distance of 1.3 A,˚ which goes right through the Varandas-Pais saddle point region. The XMS-CASPT2 and MR-ACPF curves are for the lowest 1A0 surface, and the energy rises monotonically as the third atom approaches

O2 In contrast, the potential curves computed from the Varandas-Pais surface show a local maximum associated with one of the twin saddle points. It is concluded that the local maxima are spurious artifacts of using inadequate single-configuration reference states. The accurate multi-reference methods used for this analysis were not affordable or available at the time when the study by Shih et al. [93] was published. As shown later in chapter6, such differences between the PES by Varga et al. [31] and the PES 26 by Varandas and Pais [89] lead to significant differences in vibrational excitation rates.

(a) RBC = 1.2A˚ (b) RBC = 1.3A˚

Figure 2.8: Potential curves for cuts through the collinear potential surface with one diatomic distance ﬁxed at 1.2 A˚ (Fig. 2.8(a)) and 1.3 A˚ (Fig. 2.8(b)) where the other diatomic distance (the abscissa) is scanned. In each plot shows three curves: XMS- CASPT2 calculations and MR-ACPF calculations for the lowest energy 1A0 surface and also the values calculated from the analytic surface of Varandas and Pais [89].

2.4.2 Potential energy surfaces for O4 interactions

Paukku et al. [29, 30] recently published potential energy surfaces for the singlet, triplet and quintet. All three surfaces were constructed using the minimally augmented correlation-consistent polarized valence triple zeta basis set (maug-cc-pVTZ basis set [79, 80, 85]) and the energies were calculated using multi-state complete active space second order perturbation theory (MS-CASPT2) [86–88]. The PES was fit to single point energies with 12684 geometries for the singlet spin state, 10100 geometries for the triplet spin state and 10,543 geometries for the quintet spin state. Figure 2.9 shows energy contours for an in-plane rhombic arrangement of four oxygen atoms produced by the PESs for the three spin states of the O4 system. [29, 30]. To produce this figure, in reference to Fig. 2.1(b) inner angles, ∠AOD = ∠AOB = ◦ ∠COD = ∠COB = 90 are fixed and distances RAC and RBD are varied. It is observed 27

(a) O4 singlet. (b) O4 triplet. (c) O4 quintet.

Figure 2.9: Energy contours for an in-plane rhombic arrangement of four oxygen atoms produced by the PESs for the three spin states of the O4 system. [29, 30] that when all atoms are far away there is an energy plateau of E = 0 eV, and when one or more inter-atomic distances become small a repulsive wall is observed. A product- reactant valleys is observed when RAC is equal to the diatomic oxygen equilibrium bond length (RAC ∼ 1.2 A)˚ and RBD is large and vice versa for all three PESs. Further, on the

O4 singlet PES there is an energy valley centered around RAC ∼ 2.2 Aand˚ RBD ∼ 2.2

A,˚ a similar structure is present in the O4 triplet PES but the valley is comparatively shallower and and not as wide as it is in the O4 singlet PES. This feature becomes even weaker on the O4 quintet PES. These diﬀerences are attributed to the diﬀerence in electronic arrangements in the surfaces. These PESs are used to study internal energy excitation and non-equilibrium dissociation in chapter6 Chapter 3

Physical Model and Numerical Method

3.1 Introduction

The relevant governing equation for non-equilibrium dilute gases, when the molecular nature of the gas must be explicitly accounted for, is the Boltzmann equation. The Boltzmann equation determines the evolution of the velocity and energy distribution functions of atoms and molecules, locally at any point in the flow. In the near- equilibrium limit where velocity and energy distributions are Maxwell-Boltzmann or Chapman-Enskog, the Boltzmann equation reduces to the continuum Euler or Navier- Stokes equations, respectively. The Boltzmann equation is therefore useful as an accurate model for highly non-equilibrium flow conditions and is also useful to study near-equilibrium flows where molecular physics may have important macroscopic effect, such as the coupling between vibrational energy and dissociation. When all relevant physics are included, the Boltzmann equation becomes highly multidimensional for chemically reacting gas mixtures and partial differential equation (PDE) based numerical solutions become intractable. Instead, the direct simulation Monte Carlo (DSMC) particle-based numerical method [16, 17] has proven to be an accurate and efficient technique to simulate the physics contained in the Boltzmann

28 29 equation. DSMC particles can have a species type, continuous or quantized internal energies (rotation and vibration), and collision probabilities as well as the outcomes of each collision (including chemical reactions) can be determined solely from the properties of atoms and molecules. In the most general implementation, the probabilities of particles transitioning from any initial energy state to any final state, during a collision, can be specified as input to a DSMC simulation (these probabilities are called state-to-state cross-sections). Currently, complete cross-section data cannot be measured experimen- tally. As a result, simplified cross-section models are used within DSMC, which are typically formulated to be consistent with empirical models already used in continuum Navier-Stokes calculations. More recently, computational chemistry is being used to determine cross-sections for DSMC simulations. Theory from quantum mechanics is used to calculate the interactions between atoms and, using large parallel computer clusters, huge numbers of individual collisions are performed. The most common approach is called quasi classical trajectory (QCT) [28, 33, 34] analysis and, in principle, QCT analysis can determine all energy transition cross-sections for use within DSMC. The result would be a predictive simulation capability (i.e. no empiricism) for non-equilibrium reacting flows over complex geometries. However, as discussed in chapter1 section 1.2, the problem is the vast number of possible energy transitions, which make a full state resolved approach infeasible. The large number of required state-to-state cross-sections necessitates further assumptions and simplifications. For example, compared to the ab-initio PESs with large numbers of fitting coefficients, PESs with simpler functional forms can be used in order to obtain more QCT results with less computational resources. Furthermore, many nearby energy states can be binned into a single energy group so that only a limited number of cross-sections for transitions between energy groups are required [49–57]. In some cases, only individual vibrational energy states are considered and probabilities are averaged over all rotational states. Aside from the energy transitions caused by collisions, the collision rate itself (the total cross-section) may be dependent on the internal energy states of molecules, especially at high-temperatures where molecular bonds may 30 be stretched due to rovibrational motion (see Fig. 11 in Ref. [96]); an effect which is commonly neglected. Finally, even for full state-to-state analysis involving all energy state transitions, the decoupling of internal energy into rotational and vibrational modes is not rigorous, especially at high temperatures. Both Master Equation analysis and state-to-state DSMC analysis suffer the same challenges. Of course, it is important to note that such simplifying assumptions may, in fact, be quite accurate. However, without a baseline solution to the full problem it is difficult to quantify the accuracy of such assumptions. The direct molecular simulation method requires none of these simplifications and is computationally tractable for atom-diatom and diatom-diatom systems, even using ab-initio PESs. The approach was first proposed by Koura [59–61], referred to as classical trajectory calculation (CTC)-DSMC. The basic premise is to perform trajectory calculations (on a PES) for each collision within a DSMC calculation. Essentially, instead of pre-computing the probabilities of all possible energy transitions through large numbers of QCT calculations, the DMS method performs collision trajectories “on-the- fly” within a simulation of an evolving gas system. In this manner, the molecular state resulting from one collision becomes the initial state for the next collision. Instead of resolving all possible energy transitions, the DMS method automatically simulates only the most dominant energy transitions that actually occur with non-negligible frequency for the conditions of interest. Although computationally demanding, this approach is now tractable for diatom-diatom systems, and in fact, for gas mixtures with a number of species combinations within collisions.

3.2 Direct molecular simulation

3.2.1 Numerical method

The direct molecular simulation method (DMS) simply embeds trajectory calculations within a standard DSMC simulation. The DSMC method [16, 17] exploits three fun- damental characteristics of dilute gases; (i) molecules move without interaction for timescales given by the local mean-collision-time, (ii) impact parameters and initial 31 orientations of colliding molecules are random, and (iii) there are an enormous number of molecules per cubic mean-free-path and only a statistical representation of these molecules is required. These are accurate and rigorous simplifications for dilute gases. However, the DSMC method then goes further and employs stochastic models to determine the collision rate and the outcome of collisions, thereby introducing empiricism and physical model uncertainty. The DMS method replaces these stochastic collision models with trajectory calculations performed on a PES. As a result, the only model input to a DMS calculation is the PES (or set of PESs) and, as discussed in this chapter, DMS effectively becomes an acceleration technique for Molecular Dynamics (MD) simulation of dilute gases. The DMS method, also referred to as classical trajectory calculation DSMC (CTC- DSMC), was first proposed by Koura [59–61]. The DMS code uses the No-Time-Counter (NTC) collision rate algorithm of Bird [16], which is commonly used in modern DSMC codes [17].

During each DMS time step (∆tDMS), after particles are moved and sorted locally into collision cells, a fraction of the particles (both atoms and molecules) in each cell are selected to undergo trajectory integration. The standard NTC algorithm is used, where a maximum cross-section, (σg)max, is ﬁrst used as a conservative estimate for the number of trajectories to be performed,

1 (σg) W ∆t N = N (N − 1) max p DMS , (3.1) t,max 2 p p V

where Np is the current number of particles in the cell, Wp is the particle weight (the number of real particles represented by each simulation particle), and V is the cell volume. Due to the discrete nature of the simulation this value must be rounded to the nearest integer,

∗ Nt,max = int(Nt,max + 0.5) , (3.2)

and since particles should not undergo more than one collision per timestep, this 32 value should be further limited between

∗ 1 ≤ Nt,max ≤ int(Np/2) . (3.3)

During each timestep and within each cell, the NTC procedure consists of forming ∗ Nt,max particle pairs (randomly from the Np particles within the cell), and accept- ing/rejecting each pair for a trajectory with the following probability (Pt),

! σg Nt,max Pt = ∗ . (3.4) (σg)max Nt,max Here, g is the relative speed between the particle pair and σ is the appropriate cross-section for the pair. Within a DSMC calculation, a cross-section model is used for σ in order to achieve a physically accurate collision rate (and accurate gas transport properties in the limit of near-equilibrium). However, within a DMS calculation, a conservative hard-sphere cross-section is used,

2 σ = πbmax , (3.5)

where it is crucial that the value of bmax is the maximum impact parameter used for each trajectory calculation as described in the next section. The value of (σg)max can be set as a conservatively large constant, or its value is often updated and stored within each cell based on the maximum value experienced previously in the cell, so that the value of Pt remains less than unity. Note that since Pt is applied to each of the ∗ Nt,max pairs, that the total number of trajectories selected for the cell, proportional to ∗ Nt,max × hPti, is not dependent on the value of (σg)max, rather its value is only set to improve computational eﬃciency. Therefore, within each cell and during each timestep, pairs of particles are selected for trajectory calculations using Eqs. 3.1- 3.5. As described by Boyd and Schwartzen- truber [17] the NTC method can be used, un-altered, for gas mixtures and will automatically select appropriate species-pairings for trajectories. For zero-dimensional DMS calculations, only one cell is required, particle movement 33 and position is irrelevant, and therefore Eqs. 3.1- 3.5 are the only additional equations required beyond a standard trajectory code (i.e. a QCT code). Furthermore, for steady- state DMS ﬂows (sections 3.3.1 and 3.3.2), relatively few particles are required per cell

(Np ≥ 20) and quantities can be sampled over many timesteps during steady-state.

Whereas for unsteady ﬂows (section 3.3.3), Np may need to be very large (possibly millions of particles) in order to resolve evolving distribution functions where state populations can span many orders of magnitude.

3.2.2 Trajectory calculations and parallel implementation

Trajectory integration

As depicted in Fig. 3.1, for each particle pair accepted for a trajectory using Eq. 3.4, a trajectory calculation is performed using a PES. In fact, a library of PES subroutines may be available and, depending on the species-pair selected, the appropriate PES subroutine can be used to integrate the trajectory, thereby enabling DMS calculations of gas mixtures.

Figure 3.1: Example of N2 + N2 (a) and N2 + N (b) trajectories.

The trajectory code requires the initial positions and velocities of all atoms for each 34 of the two particles. A relative velocity of g is given to the particles (and atoms that comprise them), the particles are separated at a conservative distance Dcutoﬀ = 15A˚. The trajectories are initialized with a random impact parameter b, such that:

2 2 0 ≤ b ≤ bmax , (3.6)

where bmax must be the same value used in Eq. 3.5, and a random orientation for the interacting molecules. Once the particles are set up for collision, the phase-space coordinates of all the atoms are progressed in time and the trajectory is integrated using the velocity Verlet scheme [97]. The velocity Verlet scheme integrates the equations:

˙ F = m~v = −∇rV (r) (3.7)

and

~v = ~r˙ (3.8)

for every interacting atom. Here, ~r is the atomic position vector, m is the mass of the atom, ~v is the atom’s velocity, and −∇rV (r) is the gradient of the PES which gives the force acting on the atom. The trajectory is integrated until the minimum separation between atoms not bonded in the same molecule is greater than Dcutoﬀ = 15 A.˚

Pre- and post-collision states

Simple rules can be implemented to determine when particles are no longer interacting, and therefore when to stop the trajectory calculation, as well as to determine which atoms are bound within any product molecules. Therefore, any type of trajectory outcome, allowed by the PES, can be simulated and analyzed within a DMS calculation, including exchange, dissociation, swap-dissociation, and double-dissociation reactions [28]. Currently, the DMS method does not model recombination of atomic species into molecules. This will be relevant for discussions involving chemically reactive flows in 35 section 3.3.3, and chapters5,6 and7. Within a DMS calculation, the post-trajectory properties must be associated with (stored in memory for) each DMS particle since, unlike QCT, DMS particles move through the domain and are selected for additional trajectory calculations in subse- quent timesteps. In general, given the post-trajectory atom positions and velocities, new DMS particle velocity vectors (center-of-mass velocities) can be determined and internal energy states (for molecules) can be determined using one of three strategies; (a) determine and store the corresponding quantized state (v,j) using a classical-quantum binning procedure, (b) determine and store the corresponding classical energies and instantaneous bond distance (vib, rot, and dbond), or (c) simply store the positions and velocities of all atoms (i) comprising the particle (xi, vi). Strategy (a) has not yet been tested in DMS. Strategy (b) works well for bound molecules and is used for N2 + N2 interactions in section 3.3.1 and chapter4 . Whereas the angular momentum of a rotating, vibrating molecule is conserved during free-flight, classically defined values of rot and vib oscillate depending on the instantaneous value of dbond (discussed at length in section 5.6.1 of Ref. [98]). Therefore, in addition to storing

rot and vib (standard DSMC variables), storing the additional variable dbond enables the precise molecular state to be regenerated prior to the particle’s next trajectory. It is important to note that strategy (b) does not make any assumptions about decoupling rotational and vibrational energy within the DMS simulation, the variables rot, vib, and dbond are only used to regenerate/record the atom positions within a molecule before/after each trajectory. For simulations involving chemical reactions (section 3.3.3, chapters5,6,7 ), where molecular bonds can be highly stretched, strategy (b) has not proven successful and strategy (c) has been used. Strategy (c) requires the storage of more quantities for each DMS simulation particle, however, has proven to be accurate and robust for a number of PESs and gas conditions. While working with strategy (c), the internal energy of a molecule is calculated using the atomic positions and velocities of the bound atoms. In this work internal energy is split into rotational and vibrational modes using the vibrational prioritization frame work used by Jaﬀe [99]. The division of internal energy 36 mathematically be shown as:

vib(v) = int(v, 0) (3.9)

and

rot(j) = int(v, j) − int(v, 0), (3.10)

where int(v, j) is the internal energy of the molecule, vib(v) is the energy in the vibrational mode v and rot(j) is the energy in the rotational mode j. In practice, the DMS code goes through the following steps to separate internal energy of a molecule into vibrational and rotational energy using the vibrational prioritization framework.

1. First the interatomic distance 0r0 and the interatomic distance vector ~qare calulated

2. Next the diatomic PES with no centrifugal terms (j=0) is used to calculate the potential energy based on the interatomic separation 0r0. Lets call this potential

energy : Vdia

3. Using the atomic velocities of the atoms the linear momentum of the system :

~pdia = m ~v1 − m ~v2 is calculated

4. The angular momentum :~jdia = ~q × ~pdia is calculated next.

2 |~pdia| 5. The total internal energy is calculated as : int(v, j) = 2m + Vdia

6. Next the continuous rotational number ”j” is computed based oﬀ of the angular

momentum ~jdia of the molecule

7. The eﬀective diatomic PES is calculated based on the given angular momentum

~jint

8. Using the eﬀective diatomic PES from the previous step the continuous vibrational quantum number ”v” is calculated 37 9. Now that the continuous vibrational quantum number ”v” and the continuous

rotational quantum number ”j” are known, the internal energy at int(v, 0) is

calculated termed as the vibrational energy of the molecule : vib = int(v, 0)

10. The remaining internal energy is termed as the rotational energy of the molecule

: rot = int(v, j) − vib

This division of internal energy is done as a post processing step after after the trajectory calculation is over. It should be noted that when a molecule is selected for the next trajectory, the trajectory code only uses the atomic velocity and positions of the molecule to perform the trajectory and the method of division of internal energy does not have an eﬀect on the trajectory analysis or the state of molecules in the system. Furthermore, the vibrational and rotational energy modes are coupled and this division of internal energy is arbitrary and is one of many ways one can divide internal energy of a molecule [99]. The average value of the energy in the vibrational mode, converted into temperature units, will be referred to as vibrational temperature (Tvib) and the average energy in the rotational mode, converted into temperature units will be, referred to as the rotational temperature (Trot). These quantities are deﬁned as:

Tvib ≡ hvibi/kB (3.11)

and

Trot ≡ hroti/kB , (3.12)

where kB is the Boltzmann constant. These deﬁnitions for vibrational and rotational temperature will be used in the remainder of the thesis, unless otherwise speciﬁed.

Maximum impact parameter

When describing the DMS method and, in particular, the value of bmax (Eqs. 3.5 and 3.6), it is useful to distinguish a difference between trajectories and “collisions”. 38 Specifically, a collision is a trajectory that results in a finite deflection angle or some

ﬁnite change in particle properties. If a large value is chosen for bmax, then the number of particle pairs accepted for trajectories will be large since Eq. 3.4 is proportional to σ. However, many of these trajectories will be initialized with large impact parameters (Eq. 3.6) and therefore, in many of these trajectories, the two particles will not “collide”

(no change in particle velocities or internal energies). As the value of bmax is lowered, fewer trajectories are simulated, however since these trajectories are initialized with smaller b values, a larger fraction will result in a collision. In fact it is the PES that determines the collision rate, regardless of the number of particles pairs accepted for trajectories. Of course, if bmax is too small then all trajectories will result in a collision and the collision rate is no longer determined by the PES, rather it is now restricted to the hard-sphere collision rate corresponding to the small value of bmax, resulting in an inaccurate collision rate that is too low.

The effect of bmax is clearly shown in Fig. 3.2 for DMS simulations of a normal shock wave in nitrogen (full details of these simulations and the PES are provided later in section 3.3.1). The computed temperature profiles within the shock wave are shown for three different bmax values. As seen in Fig. 3.2(a), the DMS solution does not change when bmax is lowered from 15 A˚ to 4 A.˚ This result is consistent with the QCT analysis of Bender et al. [28], which showed no interaction for N2 + N2 collisions for impact parameters of 6 A˚ and larger. When bmax = 15 A˚ the simulation is computationally inefficient, since a large number of trajectories are simulated in each cell during each timestep that result in no interaction at all. When bmax = 4 A˚ the same solution is obtained with far fewer simulated trajectories. As bmax is lowered below 4 A,˚ noticeable error is introduced. As seen in Fig. 3.2(b), when bmax = 1.75 A,˚ the solution is consistent with a low collision rate (causing increased molecular transport throughout the shock wave) resulting in a thick shock wave that does not even reach thermal equilibrium in the post-shock region.

Clearly, as long as a conservative value of bmax is chosen, it is the PES alone that determines the “collision” rate, regardless of the number of trajectories calculated per timestep. This is perhaps the most crucial aspect of the DMS method. The result is 39

(a) Conservatively large value of bmax. (b) Small value of bmax.

Figure 3.2: Effect of varying bmax on shock profiles. that the only model input to a DMS calculation is the PES (similar to Molecular Dy- namics), which determines both the local collision rate and collision outcomes including all relevant atomic-level physics with no decoupling of rotational and vibrational states. As described in the next section, DMS effectively becomes an acceleration technique for MD simulation of dilute gases.

Parallel implementation strategies

Two parallel implementations of the DMS method are used in this thesis. The first implementation combines trajectory algorithms within the Molecular Gas Dynamic Sim- ulator (MGDS) code (a parallel DSMC code developed at the University of Minnesota [100, 101]). Here, the stochastic DSMC collision algorithms are simply replaced by the trajectory algorithms outlined above, and extra variables are created to store the new particle information required by techniques (b) or (c) discussed above. All other aspects of the MGDS code, including the parallelization strategies, are unaltered and 2D/3D DMS simulations over complex geometry can be performed. The second implementation is designed specifically for time-accurate, zero-dimensional (0D), DMS calculations using high fidelity ab-initio PESs. In this case, particle position 40 and movement is not relevant, and a single partition stores the details of all simulation particles and selects particle pairs for trajectories using Eqs. 3.1- 3.5. The particle pairs selected at each timestep are then communicated to, and distributed among, a large number of partitions/processors for efficient parallel computation of the trajectories. The post collision states are then communicated back to the main partition in preparation for the next timestep iteration. In fact, this DMS implementation and the REAQCT code [28], used for large batches of QCT calculations, use the same source code for parallelization. The important point is that, given an existing QCT code, it is a trivial extension to perform 0D DMS calculations, essentially requiring only the allocation of an array of particle structures and the addition of Eqs. 3.1- 3.5.

3.2.3 Validation with molecular dynamics

A series of studies were performed by Valentini et al. where shock waves in argon [102, 103], mixtures of argon, helium, and xenon [104], and shock waves and expansions in diatomic nitrogen [105] were studied using pure molecular dynamics (MD) in the dilute gas regime. Such MD calculations are deterministic where the position and momenta of all atoms in the flow volume are advanced by solving the equations of motion with femtosecond scale timesteps. Since the same PES can serve as the sole model input for both DMS and MD calculations, comparison between DMS and pure MD can directly verify the techniques/assumptions (see (i)-(iii) discussed in section 3.2) employed by the DMS method. Norman et al. demonstrated that DMS (CTC-DSMC) exactly reproduces pure MD simulations, at the level of the velocity and rotation energy distribution functions, for normal shock waves in argon and diatomic nitrogen [62]. For example, a Mach 7 nitrogen 3 shock wave with pre-shock conditions of ρ1 = 0.1 kg/m and T1 = 28.3 K was simulated using DMS and pure MD. The site-to-site Lennard-Jones (LJ) potential was used to determine forces between atoms of different nitrogen molecules, and for atoms of the same molecule the bond length was held fixed using the RATTLE algorithm [106]. There- fore, only translational-rotation energy transfer was simulated in both DMS and MD. The temperature and density profiles within the shock wave are shown in Fig. 3.3(a) 41

(a) Rotational energy excitation [62]. (b) Rovibrational energy excitation.

Figure 3.3: Comparison between shock proﬁles predicted by DMS and MD.

where precise agreement is found for both parallel (Tx) and normal (Tyz) translational temperature components, as well as the rotational temperature and density profiles. Figure 3.3(b) presents DMS and MD calculations of diatomic nitrogen where the Ling- Rigby potential is used to determine forces between atoms of different molecules and a harmonic oscillator (HO) potential is used to determine forces between atoms of the same molecule. Since the HO potential ensures bound molecules, both DMS and MD model translational-rotational-vibrational energy transfer, but no chemical reactions. 3 The shock wave conditions are M1 = 7, T1 = 2500K and ρ1 = 0.5kg/m . As evident from Fig. 3.3(b), the temperature profiles match up precisely, within statistical uncertainty, which is very low for these simulations. Note that the relatively high density and high freestream temperature result in fewer molecules per cubic mean- free-path and a shorter vibrational relaxation time, and therefore shorter domain length. This is necessary in order for the pure MD simulations to remain computationally tractable. In section 3.3.1, DMS alone is used to simulate shock wave conditions directly relevant to hypersonic vehicles. In Fig. 3.4, the rotational energy distribution functions corresponding to various positions within the shock wave from Fig. 3.3(a), computed using pure MD (black 42

(a) MD results (black lines) compared to DSMC results (red lines).

(b) MD results (black lines) compared to CTC-DSMC (DMS) results (blue lines).

Figure 3.4: Rotational energy distribution functions within the shock wave in Fig. 3.3(a). Further details regarding these simulations can be found in Refs. [62] and [107]. lines), are compared against DSMC results (red lines in Fig. 3.4(a)) and against DMS results (blue lines in Fig. 3.4(b)). The distributions exhibit non-Boltzmann behavior within the shock and reach a Boltzmann distribution as the gas nears the post-shock region. The most important aspect of Fig. 3.4 is that the DSMC results used a new translational-rotational collision model, called the Non-equilibrium Direction Dependent (NDD) model, that was developed specifically to reproduce the MD simulation results [107]. Noticeable discrepancy between DSMC and MD is observed, even using a DSMC model tailored to the MD result (Fig. 3.4(a)), while the agreement between DMS and MD is exact (Fig. 3.4(b)). This fact supports the conclusion that DMS acts as an acceleration technique for the MD simulation of dilute gases. Finally, DMS has also been shown to capture long-lived “orbiting” collisions. When 43 the collision energy is sufficiently low, relative translation energy can be temporarily transferred to rotational energy, leaving insufficient translational energy for the pair to escape the weak attractive force between them. This results in long interaction times before the molecular orientations and translational energy are such that the pair sepa- rates. Such long-lived collisions are thought to be important for three-body collisions and therefore recombination reactions. In fact, preliminary algorithms to extend DMS to three-body collisions have been proposed [108] and it seems possible that DMS may be able to reproduce pure MD simulations that include three-body interactions and recombination reactions. The DMS method has many similarities with pure Molecular Dynamics and may have application outside of the high-temperature reacting flows discussed in this thesis. For example, Bruno et al. [109] have recently used DMS (referred to as CT-DSMC) to study oxygen transport properties.

3.3 Code implementation

3.3.1 Normal shock waves

This subsection shows example DMS simulation results for normal shock waves with free-stream conditions relevant to hypersonic ﬂight vehicles. Speciﬁcally, conditions were extracted from the 1976 Standard Atmosphere model at 40 km altitude (ρ1 = 0.0037 3 kg/m and T1 = 251.05 K). Normal shock waves were generated for free stream velocities of u = 3599.6 m/s, 5154.4 m/s, 6459.6 m/s, and 8206.9 m/s, which produce post- shock temperatures of approximately T2 = 5000 K, 10,000 K, 15,000 K, and 25,000 K, respectively. These conditions are relevant to hypersonic aircraft and reentering spacecraft, as well as missile applications [1,2, 110, 111]. For the calculations shown above, the interatomic behavior was modeled using a simple harmonic oscillator PES and the intermolecular interaction was modeled using an analytical PES by Ling and Rigby [70] 1 . A harmonic oscillator PES enforces the fact that no dissociation reaction was allowed during these simulations. Certainly, dissociation reactions would occur under the conditions studied, however, it is useful to

1The Ling-Rigby PES has been discussed in section 2.3.2 44

(a) Post-shock temperature of 5000 K . (b) Post-shock temperature of 10,000 K.

Figure 3.5: Non-reacting DMS solutions for shock-wave profiles corresponding to hypersonic flight at approximately 40 km altitude. prohibit chemical reactions in order to study rotational and vibrational excitation across a shock. More detailed analysis of vibrational and rotational energy excitation across a shock will be provided in chapter4. Finally, to manage pre- and post-trajectory molecular states and atom positions/velocities, the DMS simulations presented in this section employ strategy (b) discussed in section 3.2.2 and the parallel MGDS implementation discussed in section 3.2.2 45 3.3.2 Multidimensional flows

Since DMS has been implemented within the MGDS parallel DSMC code, two and even three-dimensional DMS simulations are possible [58]. An example DMS solution, using the Ling-Rigby and harmonic oscillator potentials, for Mach 20 nitrogen flow over a 6 cm diameter cylinder (with freestream values of ρ = 0.8 × 10−4 kg/m3 and T = 600 K), is shown in Fig. 3.6(a). Analogous to a DSMC simulation, a DMS solution is first obtained on a uniform grid, after which adaptive mesh refinement (AMR) is performed and a new solution is obtained. The AMR process was repeated twice, resulting in the grid shown in Fig. 3.6(a). Each cell is refined below the local mean free path scale, variable timesteps are prescribed in cells (set below the local mean collision time scale), and each cell contains at least 10 molecules.

(a) 2D temperature ﬁeld and simulation (b) Stagnation line proﬁles. grid.

Figure 3.6: Two dimensional DMS simulation of hypersonic nitrogen ﬂow over a cylinder.

During the movement step, molecules’ center of mass velocities are specularly reflected if they hit the symmetry boundary and atom velocities are updated appropriately. Particles that hit a surface element are diffusely reflected with full thermal accommoda- tion. Specifically, atom positions and velocities relative to the center of mass are re-set 46 to match rotational and vibrational energies sampled from Boltzmann distributions corresponding to the wall temperature (set to 1000 K). The simulations use strategy (c) to store atom positions and velocities that define the molecular state. Temperature profiles along the stagnation line are plotted in Fig. 3.6(b). The expected trend of coupled translational-rotational-vibrational excitation within a diffuse shock wave, is observed, followed by the approach to thermal equilibrium in the high density region next to the cylinder surface. While a pure DSMC simulation of this flow requires approximately 1 CPU for a few hours, the DMS calculation requires approximately 312 core processors for 5 hours. Clearly, 2D/3D DMS calculations are computationally expensive. However, this example simulation demonstrates that it is now possible to simulate macroscopic dilute gas flows over complex geometries where a PES acts as the sole model input.

3.3.3 Chemically reacting isothermal systems

This section discussed DMS of chemically reacting systems by using the nitrogen system as an example. For the discussion of chemically reacting nitrogen the ab-initio PES developed by Paukku et al. [27] in the Department of Chemistry at the University of Minnesota is used. After initial construction, the PES was modiﬁed slightly for use in a QCT study of nitrogen dissociation by Bender et al. [28]. The PES code is freely available online [112] and this exact source code is called as a subroutine within the DMS code to evaluate forces between atoms during DMS trajectories. The PES is accurate for both N2 + N2 collisions and N2 + N collisions [27], where for N2 + N collisions, the position of one N atom is held ﬁxed at a far distance, while forces on the remaining three N atoms are evaluated [64]. The ab-initio PES is far more computationally expensive to evaluate compared to simple analytical PESs such as the Ling-Rigby and harmonic oscillator PESs described in the previous subsections. As a result, the computational cost of performing molecular trajectory goes up. This restricts the analysis of chemically reacting systems with ab-initio PESs to 0D isothermal simulations. While suing the ab- initio PESs the second parallelization strategy for the DMS implementation discussed in section 3.2.2 is used and is found to scale very well on large numbers of CPUs. 47 A full nitrogen DMS simulation will account for the following interactions as a direct consequence of trajectory calculation:

1.N 2(v1,j1) + N2(v2,j2) N2(v1’,j1’) + N2(v2’,j2’) (non-reactive collision)

2. NN(v1,j1) + N’N’(v2,j2) NN’(v1’,j1’) + NN’(v2’,j2’) (exchange reaction)

3.N 2(v1,j1) + N2(v2,j2) N + N + N2(v2’,j2’) (simple dissociation)

4.N 2(v1,j1) + N2(v2,j2) N + N + N + N (double dissociation)

5. NN(v1,j1) + N’N’(v2,j2) N + N’ + NN’(v2’,j2’) (exchange + dissociation)

6.N 2(v1,j1) + N N2(v1’,j1’) + N (non-reactive collision)

7. NN(v1,j1) + N’ NN’(v1’,j1’) + N (exchange reaction)

8.N 2(v1,j1) + N N + N + N (simple dissociation)

An example of a reactive isothermal relaxation simulation is shown in Fig 3.7. The blue curve shows the translational temperature (Ttr), the black curve shows the rota- 2 tional temperature (average rotational energy in temperature units) (Trot) , the red curve shows the vibrational temperature (average vibrational energy in temperature 3 units) (Tvib) , and the green curve shows the composition histroy of molecular nitrogen in the system as a fraction of the initial amount of nitrogen in the system. In this example the gas is initialized with Ttr = Trot = Tvib = 30000K and the translational temperature is kept constant at Ttr = 30000K by resampling the center of mass velocities of the simulated particles from a Maxwell-Boltzmann distribution at Ttr = 30000K. The density was set to ρ = 1.28kg/m3 and at t = 0, the box only has molecular nitrogen. As the system evolves, molecules dissociate. Initially, the dissociation is only due to N2 + N2 collisions, but as the population of atomic nitrogen increases, N2 + N collisions start playing an important role in the dissociation process. Since recombination is not modeled, the system progresses until all molecules have dissociated and the system comprises of only atomic nitrogen.

2As deﬁned in equation 3.12 3As deﬁned in equation 3.11 48

Figure 3.7: Example of isothermal relaxation in a box and formation of the QSS for nitrogen.

Due to dissociation, the population of higher vibrational and rotational levels are depleted and there is a loss of energy in the rotation and vibrational modes, which man- ifests as an initial decrease of the vibrational and rotational temperatures. Eventually, the inelastic collision processes catch up with the loss of energy due to dissociation. This leads to formation of a Quasi-Steady State (QSS) [113]. At the QSS condition, the rate of dissociation from high rotational and vibrational levels is equal to the rate of repopulation of those levels by inelastic processes. Due to this balance, the distribution of energy in the vibrational and rotational energy modes becomes time invariant. These time invariant, non-equilibrium distributions can be seen in Figs. 3.8(a) and 3.8(b). These depleted, time invariant distributions manifest themselves macroscopically as lowered but constant vibrational and rotational temperatures (Fig. 3.7). Additionally, the distribution of total internal energy in QSS can be seen in Fig.3.8(c). It should be noted that the QSS is solely a function of the translational temperature and the density of the gas, and is independent of the initial state of the gas. This is shown in Fig. 3.9, where the same QSS state is obtained in three diﬀerent isothermal simulations, each having diﬀerent initial conditions. An important advantage of the DMS method, compared to Master Equation and 49

(a) Vibrational Energy. (b) Rotational Energy. (c) Total internal Energy.

Figure 3.8: Internal energy distributions of N2 molecules in QSS for Tt = 30000K.

Figure 3.9: Isothermal simulations starting at various initial conditions. state-resolved approaches, is that one can quickly compare macroscopic and microscopic results produced by diﬀerent PESs without the need to construct multiple databases of transition rates (or cross-sections). In Fig. 3.10, the DMS method is used to compare rovibrational relaxation and dissociation predicted by both the ab-intio PES, and an analytical PES made by describing inter-molecular interactions by the potential proposed by Ling and Rigby [70] and intra-molecular interactions being described by the Morse potential. Since the Morse + 50

Ling-Rigby PES combination is not suitable for N2 + N collisions, only N2 + N2 collisions are considered, as described in Ref. [63, 67]. Specifically, if N atoms are produced by a dissociating trajectory, they are simply removed from the simulation. Therefore, the rovibrational relaxation and dissociation results in Fig. 3.10 are interpreted as due to N2 + N2 collisions only. In order to study such differences, N2 system is initialized in thermal equilibrium (at 15,000, 20,000, and 30,000 K) and allow it to evolve into QSS. It is evident that macroscopically, the two PESs produces slightly different results for the internal temperatures during QSS, and for the rate of dissociation in QSS. The vibrational energy distributions in QSS predicted by both ab-initio and Morse + Ling- Rigby PESs, seen in Fig. 3.10, appear similar, however, it is important to note that they are plotted on a log scale over many orders of magnitude. Upon close inspection, the differences in edfs are consistent with the differences in average energy profiles (Trot and

Tvib). For example, in Fig. 3.10(b), the vibrational energy population is higher for the ab-initio PES, which is consistent with the higher value of Tvib in Fig. 3.10(a) compared to the corresponding Morse + Ling-Rigby results. In general, directly comparing results produced by diﬀerent PESs may be useful in understanding what features of a PES are responsible for a macroscopic eﬀect. Along these lines, DMS has recently been used to compare dissociation predictions for two ab-intio PESs developed at the University of Minnesota and NASA Ames Research Center [67, 68] (Chapter5). 51

(a) Macroscopic variables for 15,000 K. (b) Molecular distributions for 15,000 K.

(e) Macroscopic variables for 30,000 K. (f) Molecular distributions for 30,000 K.

Figure 3.10: DMS isothermal relaxation results for the N2-N2 system using both ab- initio and Ling-Rigby + Morse potentials. Chapter 4

Rovibrational Coupling in Normal Shocks

4.1 Introduction

As described previously in section 3.2.2, no decoupling of rotational and vibrational energy occurs during a DMS calculation. Rather, the method operates only on atom positions and velocities. Rotational/vibrational energies or quantized states are only obtained as a post-processing step. Thus, the method accurately simulates rovibrational coupling according to the PES with no further assumptions. In a paper by Valentini et al. [22], rovibrational coupling was studied using DMS for zero-dimensional relaxation calculations in nitrogen. It was found that, under high temperature conditions, the Jeans and Landau-Teller models could not reproduce the DMS results, which clearly showed additional rotation-vibration coupling. For that study Valentini et al. [22] used the Ling-Rigby PES [70] to simulate molecule-molecule interactions and a harmonic oscillator to simulate the diatomic potential. Addition- ally, Valentini et al. [22] also performed zero-dimensional relaxation calculations where they replaced the harmonic oscillator potential with the Taylor-6 potential (deﬁned in section 2.3.2) and concluded that the addition of anharmonicity to the diatomic potential enables larger bond stretching for a speciﬁc vibrational energy, and therefore

52 53 greater coupling with the rotational energy. The zero-dimensional results, exhibiting rovibrational coupling, motivated the studies involving shock waves presented in this chapter.

4.2 Rovibrational coupling in normal shock waves

This chapter present DMS simulation results for normal shock waves in nitrogen gas with free-stream conditions relevant to hypersonic flight vehicles. Specifically, conditions were extracted from the 1976 Standard Atmosphere model at 40 km altitude (ρ1 = 0.0037 3 kg/m and T1 = 251.05 K). Normal shock waves were generated for free stream velocities of u = 3599.6 m/s, 5154.4 m/s, 6459.6 m/s, and 8206.9 m/s, which produce post- shock temperatures of approximately T2 = 5000 K, 10,000 K, 15,000 K, and 25,000 K, respectively. These conditions are relevant to hypersonic aircraft and reentering spacecraft [1,2] , as well as missile applications [110, 111]. Similar to the analysis done by Valentini et al. [22] the Ling-Rigby PES [70] for molecule-molecule interaction, along with the harmonic oscillator and Taylor-6 potentials do simulate the diatomic potential are used. Hence, the interacting molecules will remain bound during the simulation and no chemical reactions would occur. Certainly, dissociation reactions would occur under the conditions studied, however, it is useful to prohibit chemical reactions in order to study rovibrational coupling. Finally, to manage pre- and post-trajectory molecular states and atom positions/velocities, the DMS simulations presented in this section employ strategy (b) discussed in section 3.2.2 and the parallel MGDS implementation discussed in section 3.2.2. The temperature profiles resulting from each normal shock DMS calculation are shown in Fig. 4.1. The red curve shows the variance in molecular center of mass velocity in the direction of the shock and is multiplied by the factor mN2 /3kB, and is labeled as ‘Tx’. The black curve shows the translational temperature, the orange line shows the rotational temperature and the blue line shows the vibrational temperature. The curves shown with dots are results obtained using the harmonic oscillator as the diatomic potential and the solid lines are the results obtained using the Taylor-6 potential as the 54

(a) Post-shock temperature of 5000 K . (b) Post-shock temperature of 10,000 K.

Figure 4.1: Non-reacting DMS solutions for shock-wave profiles corresponding to hypersonic flight at approximately 40 km altitude using the Ling-Rigby PES with the harmonic oscillator and Taylor-6 potential. diatomic potential. For a post-shock temperature of 5000 K (Fig. 4.1(a)), rotational energy excitation is orders of magnitude faster than vibrational excitation. Rotational energy quickly equilibrates with translational energy and vibrational energy proceeds to excite slowly towards the trans-rotational energy. As shown in Fig. 4.1(a), DMS simulations using both the harmonic oscillator and Taylor-6 potentials produce nearly identical results and 55 no rovibrational coupling effects are observed. For post-shock temperatures of 10,000 K and 15,000 K (Figs. 4.1(b) and 4.1(c)), rovibrational coupling effects are now evident. The identifying feature is that the rotational temperature does not immediately reach the translational temperature, rather the rotational temperature peaks and begins to decrease before equilibrating with translational temperature, this effect is seen to be more pronounced for the simulations with the Taylor-6 potential. The Jeans equation, which only models translation-rotation energy transfer cannot capture this trend and, instead, would result in the rotational energy always increasing towards the translational energy. This trend is most clear for the DMS calculation with a post-shock temperature of 25,000 K (Fig. 4.1(d)) where rotational and vibrational temperatures appear to equilibrate with each other prior-to equilibrating with the translational temperature for the simulation using the Taylor-6 potential.

Figure 4.2: Comparison of shock proﬁle obtained from DSMC and DMS (also know as CTC-DSMC)

Figure 4.2 compares a shock simulation done using standard DSMC to that using

DMS (also called CTC-DMSC), where the post shock temperature is T2 = 25000K. The absence of rovibrational coupling in the DSMC simulation yields a noticeably different shock profile, where the translational (black) and rotational (orange) temperature equilibrate with each-other first and the vibrational (blue) temperature catches up later. 56 For the DMS simulation using the Taylor-6 diatomic potential, it is evident that the shock layer, the rotational temperature does not peak as much as it does for the DSMC simulation. Further, as stated above, for this shock case there is equilibration between rotational and vibrational modes before equilibration with the translational mode. If a more realistic Morse potential, shown as the black curve in Fig. 2.5, is used, the degree of rovibrational coupling is expected to increase and be noticeable for lower post-shock temperatures, although the overall temperature profiles are not expected to change significantly. Practically, it would be interesting to quantify how changes in rotational and vibrational energies due to coupling actually influence the rate of dissociation, in comparison to models that decouple rotational and vibrational energy transfer. This is difficult since for high-temperatures where coupling is significant, dissociation is likely occurring during rovibrational excitation. In order to investigate all processes simultaneously, zero-dimensional DMS simulations of dissociating nitrogen and oxygen, using ab-initio PESs, are presented in chapters5 and6 Chapter 5

Direct Molecular Simulation of Nitrogen

5.1 Introduction

This chapter explores internal energy excitation and non-equilibrium dissociation of nitrogen under hypersonic conditions, due to N2 + N and N2 + N2 interactions using ab-initio PES. In this chapter, ﬁrst, N2 + N2 interactions are studied using the DMS method with PESs developed at NASA Ames Research Center [25, 26] (called the Ames PES in the chapeter) and the PES developed at the University of Minnesota for the

N4 system (called the UMN PES in the chapter) . Keeping the numerical method for calculating excitation and dissociation the same while changing the PES that governs the molecular interactions helps understand the eﬀect a PES can have on macroscopic properties (section 5.2). Next the eﬀect of numerical methods is studied by performing a comparative analysis of non-equilibrium energy transfer and dissociation of nitrogen molecules due to N2 + N2 interactions by using DMS and the state-to-state method, where both methods are based on molecular interactions using the NASA N4 PES [25]

(section 5.3 ). A similar analysis is done for N2 + N interactions in sections 5.4 and 5.5.

57 58

5.2 Eﬀect of PES on the N2 + N2 system

5.2.1 Vibrational energy excitation

This compares characteristic vibrational excitation time calculated using the NASA PES to those calculated by Valentini et al. [63] using the DMS method. To calculate vibrational energy excitation isothermal excitation calculations as discussed in section

3.3.3 are carried out. However, during the excitation calculation, only N2 + N2 interactions were allowed and any N2 + N interactions were ignored as to isolate the eﬀect of

N2 + N2 interactions during excitation. These excitation calculations started with an initial rotational and vibrational temperature Trot = Tvib = 2000 K. Excitation calculations were carried out with translational temperature maintained at Ttr = 10000K and

Ttr = 25000K.

(a) Temperature and composition history. (b) Evolution of the vibrational distribution function for the case in Fig: 5.14(a).

Figure 5.1: Example vibrational excitation calculation for the N2 + N2 system.

As an example, the temperature and composition history for the case with Ttr = 25000K case in Fig. 5.1(a). The internal energy excitation process and dissociation of molecular nitrogen occur concurrently. The rotational and vibrational temperatures 59 excite and level oﬀ below the translational temperature - as the gas reaches Quasi- Steady State (QSS)1. Furthermore, in Fig. 5.1(b) shows the evolution of the vibrational energy distribution function. It is shown that the vibrational energy distribution starts at the Boltzmann vibrational energy distribution at Tvib = 2000K and excites through a series of non-Boltzmann distribution towards the QSS distribution. This non-Boltzmann excitation process has recently been described by Singh and Schwartzentruber [114]. The composition histories obtained from the above calculations were ﬁtted to a Landau-Teller like equation [18]:

Tvib(t) = Tvib|final − ∆Tvib[exp(−t/τ)] (5.1)

where,

∆Tvib = Tvib|final − Tvib|t=0 . (5.2)

This is a general result for systems close to equilibrium, in which case relaxation is always exponential, however this model may not be valid more generally [24]. Figure 5.2 compares the characteristic vibrational excitation time calculated using the Ames PES with those calculated by Valentini et al. [63] and those given by the Millikan and White correlation [115] with the correction proposed by Park [3]. It can be seen that the vibrational excitation time calculated in this study match well with those obtained by using the UMN PES [63] and the Millikan and White data[115].

5.2.2 Nonequilibrium dissociation

Dissociation rate coefficients during QSS with the Ames PES are calculated and compared them to the QSS dissociation rate coefficients calculated calculated using the UMN PES [63]. To calculate the dissociation rate coefficients, isothermal relaxation calculations like the one shown in Fig. 3.7 were carried out at Ttr = 10000 K, 15000K, 20000K and 30000K. All simulations started with a box of pure diatomic nitrogen in

1formation of QSS has been discussed in section 3.3.3 60

Figure 5.2: Comparison of characteristic vibrational relaxation time obtained using the Ames PES (blue) against that obtained using the UMN PES (red) [63] and the Millikan and White correlation (black curve) equilibrium with the translational temperature. As the simulation progressed, dissociation occurred and the amount of diatomic nitrogen was decreases. The density of the 3 gas was kept at ρ = 1.283 kg/m and the particle weight (Wp) for these simulations was ﬁxed at one. A comparison of the composition histories of the gas during the DMS simulation can be seen in Fig. 5.3. It can be seen that the composition histories agree remarkably well for cases with Ttr = 30000K and 20000K. A small, but small noticeable are seen for the 15000K and 10000K cases, where the predicted dissociation rates are higher using NASA PES than using the Minnesota PES. To quantify the rate of dissociation, dissociation rate coeﬃcients were calculated using the rate law:

d[N ] 2 = −kN2+N2 [N ]2 − kN+N2 [N][N ] . (5.3) dt d 2 d 2

However, since N2 + N interactions were ignored in these cases, equation 5.3 is 61

Figure 5.3: Composition history comparison for the N2 + N2 system. reduced to:

d[N ] 2 = −[N] kN+N2 [N ] . (5.4) dt 0 d 2 The composition histories during the QSS were fit to Eq. 5.4 to back out the non-equilibrium dissociation rate coefficients. Figure 5.4 compares these dissociation rate coefficients obtained using the Ames PES (blue) and the UMN PES (red). It is observed that the dissociation rate coefficients obtained by using the Ames and UMN 62

PES agree well over the temperature range investigated, with the Ttr = 10000 K case showing the most difference between the dissociation rate coefficients of about ∼ 51%. Additionally, comparison of the dissociation rate coefficients obtained from this study with experimental data from Appleton et al. [7], and experimental data by Byron [10]. The experimental fit for the dissociation rate coefficient by Appleton et al. [7] is shown in the solid black line and the green symbols show the range of temperatures over which the experiment was conducted, Appleton et al. reported a ±37% uncertainty in the reported values. It is seen that the dissociation rate coefficients reported by Appleton et al. [7] are fairly close to the DMS results. Similarly, the dashed line shows the experimental fit for the dissociation rate coefficients reported by Byron [10] and the orange symbols denote the range over which the experiments were conducted, no uncertainty in the data was reported by Byron [10].

Figure 5.4: Comparison of non-equilibrium dissociation rate constants obtained for N2 + N2 collisions using the DMS method using the NASA PES (blue) and the UMN PES (red). Also shown are, rate coeﬃcients from experiments[7, 10]. The black lines represent the extrapolation obtained from the experimental data and the circular symbols show the range of experimental data.

It should be noted, that the nonequilibrium dissociation rate coefficients obtained by the DMS method using two independently developed ab-initio PESs agree very well and differ at most by ∼ 51%, while the fits for the dissociation rate coefficients reported 63 by experiments vary by orders of magnitude. Next, the vibrational energy distribution functions of molecules in QSS are analyzed. These distributions are shown for three translational temperatures in Fig. 5.5 and compared with the corresponding thermal equilibrium distribution (i.e., the Boltz- mann distributions). It should be noted that the two PESs [25, 27] were developed independently and have several differences as discussed and shown in section 2.3.1. For all three temperatures, the vibrational energy distribution functions of diatomic nitrogen for the N2 +N2 system in QSS are non-Boltzmann due to a depletion of higher vibrational levels. Therefore, most of the dissociation occurs from molecules with high internal energy. This is more pronounced for the Ttr = 30000K and Ttr = 20000K case (Fig. 5.5(a) and 5.5(b)). For the bulk of the molecules in QSS, the two PESs predict almost the same vibrational energy populations. For the Ttr = 10000 K case the system distributions do not precisely coincide but are very similar. The Ames PES result has a greater population than the UMN PES for vibrational energy vib >∼ 4 eV. This larger population of higher vibrational energies is consistent with the higher dissociation rate predicted by the NASA PES (a factor of 1.51 times higher than the Minnesota PES, as noted earlier). This higher population for vibrationally excited molecules in results obtained by the NASA PES is due to the NASA diatomic PES having a gradual shoulder which enabling more vibrational levels to exist in the high energy region (as discussed in section 2.3.1). The vibrational distributions of the pre-collision states of dissociating molecules are shown as solid curves in Figure 5.5. As seen in prior publications [63, 64] , the pre-collision distributions for the dissociating species become flatter as temperature increases. For Ttr = 10000 K (5.5(c)) most of the dissociation occurs from the population of molecules with high vibrational energy. Since the underlying populations of such high vibrational energy are very small at that temperature, this leads to a strong vibrational favoring in the dissociation process as documented in detail by Valentini et al. [64]. However, at higher temperatures (Fig. 5.5(a) and 5.5(b)), dissociation occurs from all levels due to the higher translational energies available in the collisions. All distribution functions in the QSS predicted by both the Ames and UMN PESs agree remarkably 64 well. Upon close inspection, it can be seen that the NASA PES has less dissociation from lower v levels and overshoots the distribution obtained by the Minnesota PES in the middle before dropping down at the distribution obtained by the Minnesota PES at the tail end of the distribution.

(a) Ttr = 30000K. (b) Ttr = 20000K. (c) Ttr = 10000K.

Figure 5.5: Vibrational distribution of N2 molecules in QSS for Ttr = 30000K, 20000K and 10000K for the N2 + N2 processes

Such remarkable agreement between the results obtained from two independently developed PES and experimental data lends conﬁdence in using computational chemistry for developing the next generation of CFD models for hypersonic applications.

5.3 Eﬀect of numerical method on the N2 + N2 system

This section presents a comparative analysis of non-equilibrium energy transfer and dissociation of diatomic nitrogen using DMS [58] and using the maximum-entropy quasiclassical trajectory (ME-QCT) method [33, 35]. Like the section above, both methods were used to simulate isothermal excitation calculations at Ttr = 10000 K and

25000 K, where the initial internal temperature was Tvib = Trot = 2000 K and density 3 ρ = 1.283kg/m . Both methods make use of the Ames N4 PES to model N2 + N2 interactions. This study was done in collaboration with Prof. Marco Panesi and Robyn Macdonald at the University of Illinois - Urbana Champagne, who performed the ME- QCT calculations. Details of this work can be found in a joint paper [57]. The basis of the state-to-state approach is to consider each individual ro-vibrational 65 state of diatomic nitrogen a as pseudo-species and solve directly for their population using the microscopic kinetic data. However, diatomic nitrogen as characterized by 2 the NASA-Ames N4 PES has about 9390 rovibrational levels. This implies that a fully resolved state-to-state approach will have to consider 9390 pseudo-species and consider O(1015) possible state-to-state interactions3. This makes a fully resolved state- to-state approach infeasible. To make this problem tractable, ro-vibrational energies were binned, hence reducing the overall number of pseudo-species. In this study two types of grouping strategies were used:

1. Energy based grouping: This strategy groups states close in ro-vibrational energy. The assumption of this grouping strategy is that, states with similar energy will equilibrate quickly with each other. However, this strategy neglects any information about the quantum conﬁguration of the states. In this approach the population of each ro-vibrational state is reconstructed based on the average internal energy or internal temperature of the group. In this study analysis using energy based grouping used 60 energy bins.

2. Vibrational state based grouping: This strategy groups ro-vibrational states with the same vibrational quantum number together. The population of the rotational states within each vibrational state can be reconstructed from the rotational

temperature (Trot(v)) or average rotational energy of the vibrational group. In this work, due to the lack of energy transfer coeﬃcients required to solve for the rotational temperature, the rotational temperature within the groups is ﬁxed at

the translational temperature (Trot(v) = Ttr). This is analogous to the vibrational state-to-state model assuming equilibrium between rotational and vibrational modes. In this study analysis using vibrational based grouping used 61 energy bins.

2As shown in table 2.1 in chapter2 39390 possible initial states of the two interacting molecules and 9390 ﬁnal states for the two interacting molecules, gives (93904)/2 possible interacting combinations. 66 5.3.1 Internal energy excitation

Internal energy excitation proﬁles using both DMS and ME-QCT can be seen in Fig.

5.6. For Ttr = 10000K. The internal energy relaxation predicted by the energy based ME-QCT model is significantly faster than both the DMS method and vibrational specific ME-QCT model. The vibrational specific ME-QCT method starts with higher internal energy because the rotational mode is already excited; however, the energy of the molecules does not increase until 109 seconds, indicating that the molecules are not gaining energy from the translational mode until that point. Similarly, at 25000K, the internal energy relaxation rate predicted by the energy based ME-QCT model is faster than the DMS method. In this case the vibrational ME-QCT model again starts with higher energy than the DMS method, but the internal energy in this case leads the DMS method. In both cases the internal energy at the final time, corresponding to the QSS energy, is very close, differing by less than 5%. This difference is due to differences in the rotational and vibrational energy in the QSS region.

Figure 5.6: Internal energy excitation proﬁles [57]

In addition to the total internal energy, the rotational and vibrational temperatures predicted from the DMS and vibrational based ME-QCT model are compared. For the vibrational ME-QCT model, the rotational temperature is assumed to be fixed at the translational temperature. The temperatures for both cases are shown in Fig. 5.7. The 67 vibrational specific ME-QCT model matches very well with the vibrational temperature predicted by the DMS method. However, this is helped by the assumption of equilibrium between the rotational and translational temperatures. If this assumption were relaxed, the vibrational relaxation time predicted by the vibrational specific model would most likely become much slower because the rotational mode would need to become excited first. Moreover, the assumption of equilibrium between rotation and translation is seen to breakdown particularly at higher temperatures, where the rotational temperature predicted by the DMS method only reaches Trot = 21200 K.

Figure 5.7: Vibrational temperature and rotational temperature excitation proﬁles [57].

Figure 5.8 shows the instantaneous internal energy distribution during internal energy excitation, here the distributions predicted by the energy and vibrational based ME-QCT models and the DMS methods are shown. The energy based ME-QCT method predicts significantly faster relaxation than the DMS method at both temperatures. This is due to the pseudo-ladder climbing model for internal energy relaxation observed in the energy based ME-QCT model. The vibrational specific ME-QCT model predicts very different excitation processes at Ttr = 10000 K and Ttr = 25000 K. At Ttr = 10000, the relaxation process predicted by the vibrational specific ME-QCT model matches well with the DMS data for the bound energy states. The quasi-bound4 states remain

4Quasi-bound states refer to states where the total internal energy is greater than the diatomic 68 over-populated by the vibrational speciﬁc ME- QCT model. However, this is not of great concern, because at this time the eﬀect of dissociation is minimal. In contrast, at

Ttr = 25000 K the vibrational specific ME-QCT model predicts significantly more energy in the molecules at this time. This is an artifact of the fixed rotational temperature within the vibrational groups. As a result of this assumption, which was observed to break down at this temperature, the population of the energy states is poorly predicted by the vibrational specific ME-QCT model.

Figure 5.8: Instantaneous internal energy distribution during the excitation process [57].

5.3.2 Nonequilibrium dissociation

Figure 5.9 shows the mole fraction of atomic nitrogen as a function of time for both temperatures studied. The DMS prediction matches well with the ME-QCT energy based grouping model at both temperatures. However, the vibrational specific ME- QCT model predicts significantly slower dissociation because of the way energy states are grouped. Since all rotational states within a vibrational group are in equilibrium at a common rotational temperature (Trot = Ttr ), the quasi-bound states are lumped with bound states. This is inconsistent with the stark differences in the kinetics that characterize the dissociation process from these internal states. For instance, a molecule energy barrier for j=0 69 with state (v,J) = (0,0) is very unlikely to dissociation, while a molecule with (v,J) = (0,270) is extremely likely to dissociate. The vibrational based grouping, lumps these two states together. This effect is amplified for the Ttr10000K case. As at low temperatures, quasi-bound molecules are more likely to dissociate.

Figure 5.9: Composition history for the isothermal relaxation calculations [57].

Figure 5.10 shows the energy based ME-QCT model without recombination for the

Ttr10000K. As the DMS method can not currently simulate recombination reactions, running ME-QCT simulations without recombination provides a better apples-to-apples comparison with the DMS data. Figure 5.10 shows results from the ME-QCT simulation when recombination is turned oﬀ for the Ttr = 10000K case, here the ME-QCT simulations agree even better with the DMS simulation. Comparing ﬁgure 5.10 and

5.9 it can easily be concluded that for the isothermal simulation at Ttr10000K, after about t = 10−6s of simulated time, recombination eﬀects start to play a role. This study was not done for the Ttr = 25000K case as the dissociation process there is very overwhelming and recombination would not have a meaningful eﬀect. Internal energy distribution during QSS are compared in Fig. 5.11. It is important to note that the DMS and ME-QCT vibrational binned results were mapped to the ME-QCT energy bins for comparison purposes. At both temperatures, the energy based ME-QCT distributions match the DMS quite well across the whole energy range. 70

Figure 5.10: Composition histroy for Ttr = 10000 K case with recombination turned oﬀ for ME-QCT [57].

The distribution bends down at the dissociation energy in both the DMS and ME- QCT results. However, the ME-QCT vibrational bins signiﬁcantly over-predict the population of the quasi-bound states. This is due to the very strong assumption that the rotational temperature of the bins is the same as the translational temperature. Furthermore, even the low energy groups in the ME-QCT vibrational speciﬁc model are overpopulated compared to the DMS method.

Figure 5.11: Internal energy distribution in QSS [57]. 71

Figure 5.12: Dissociation probability against internal energy [57].

In order to understand the relative importance of energy states for dissociation, the distribution of dissociating molecules in QSS was compared and shown in Fig. 5.12. While the trend is well matched in both approaches, molecules climb to high energy states before dissociating, the actual peak and width are diﬀerent in both approaches. This indicates that the lumping of quasi-bound states with bound states as done in the vibrational based ME-QCT model can contribute to large errors in the dissociation behavior, due to the importance of these states for predicting dissociation.

5.4 Eﬀect of PES on the N2 + N system

5.4.1 Vibrational energy excitation

This section discusses the effect of atom-molecule collisions on the vibrational excitation of the gas. Isothermal vibrational excitation are carried out for translational temperatures ranging from Ttr = 5000K to Ttr = 30000K. For all cases the initial conditions are 3 generated such that at t = 0, Tvib = Trot = 3000K, density was set to ρ = 1.28kg/m , 6 the particle weight is set to unity (Wp = 1), and 8 × 10 DMS particles are simulated. These calculations where carried out in an isothermal heat bath, however, to isolate the effect of N + N2 collisions, the partial density of molecular nitrogen (ρN2 /ρtotal) 72 is reduced until the characteristic excitation time becomes independent of the partial density of molecular nitrogen. To determine the effect of ρN2 /ρtotal on characteristic excitation time, a series of excitation calculations were run with ρN2 /ρtotal varying from

1 to 0.001. For these calculations, at t = 0, Tvib = Trot = 3000K and Ttr = 30000K. Fig. 5.13 shows the characteristic vibrational excitation time as function of the partial density of nitrogen. As can be seen from the ﬁgure, the characteristic vibrational excitation time is independent of the partial density of nitrogen for ρN2 /ρtotal ≤ 0.05. To be conservative, the work presented in this section uses ρN2 /ρtotal = 0.01. Hence, in a simulation of 8×106 DMS particles ∼ 40, 200 are molecular nitrogen with the remainder being atomic nitrogen.

Figure 5.13: Dependance of characteristic excitation time on partial density of molecular nitrogen.

Figure 5.14(a) shows an example of an excitation simulation. In the example, the blue curve represents the translational temperature, the black curve is the rotational temperature and the red curve is the vibrational temperature. The green curve shows the fraction of molecular nitrogen present, with respect to the initial amount of molecular nitrogen in the system. A Landau-Teller type expression is fit to the red curve to give the characteristic vibrational excitation time (τvib). For cases with Ttr ≤ 10000K vibrational temperature equilibrates with translational temperature before dissociation becomes 73 prominent. However, for cases with Ttr > 10000K, there is significant dissociation before the internal energy modes attain a steady state. For these cases the relaxation of internal energy is strongly coupled with dissociation. Figure 5.14(b) shows the vibrational excitation mechanism at the molecular level for the case presented in Fig. 5.14(a). The solid purple curve shows the vibrational distribution function at t = 0, the solid blue curve shows the distribution function at QSS and the solid black curve shows the equilibrium vibrational distribution at 30000K. The symbols show instantaneous vibrational distribution functions at various times. One can see, as the vibrational excitation begins, initially, there is an over population of the high energy tail as seen at t=τ/8, τ/4, τ/2. As the system progresses, the lower energy levels get populated until eventually the system reaches the QSS distribution. Millikan and White developed correlations for vibrational excitation [115] for non- reactive collisions. Later corrections were made to the Millikan and White formulation by Park et al. [3] to make the correlation appropriate for high temperature conditions by limiting the excitation rates. In Fig. 5.15 comparison of the characteristic relaxation time obtained from the DMS method (blue curve with circular symbols) with the corrected Millikan-White correlation (black curve) is made. It can be seen that the characteristic vibrational time obtained from the DMS calculations is an order of magnitude lower than the predictions made using the Millikan and White correlation. This is because the Millikan and White study only considers non-reactive interactions between collision partners and ignores the effect of exchange reactions on the vibrational relaxation process. It has been shown that exchange reactions are very efficient at redistributing internal energy [64, 116–118]. The results obtained are remarkably close to earlier DMS work done by Valentini et al. [64] using the Minnesota PES[27]. Addi- tionally, there is good agreement with other contemporary results obtained by Panesi et al.[20] and Kim and Boyd[47]. Figure 5.16 shows the evolution of internal energy modes for the isothermal excitation calculations at translational temperature, Ttr = 5000K (Fig. 5.16(a)) and

Ttr = 30000K (Fig. 5.16(b)). The blue curve shows the average translational energy, the green curve show the average internal energy, the black curves shows the average 74

(a) Temperature and composition history. (b) Evolution of the vibrational distribution function for the case in Fig: 5.14(a).

Figure 5.14: Example vibrational excitation calculation. rotational energy and red curves show the average vibrational energy. As discussed in chapter3 no a-priori assumption is made about de-coupling of rotational and vibrational energy. It can be seen that for the Ttr = 5000K case the rotational energy excites about two orders of magnitude faster than the vibrational energy, whereas for the Ttr = 30000K case, the rotational and vibrational energy modes excite almost simultaneously. The ﬁgures show that at lower temperatures the rotational and vibrational modes are fairly de-coupled and at higher temperatures the coupling between the rotational and vibrational modes becomes more prominent. This shows up in the excitation of internal energy, where it is evident that for the low temperature case (Fig. 5.16(a)) internal energy excitation is bi-modal; the internal energy initially excites rapidly as the rotational energy excites and then slows down as vibrational energy is not yet exciting. On the other hand, for the high temperature case the internal energy excitation occurs monotonically and does not exhibit the bi-modal characteristic. It is evident that the rate at which internal energy is excited is primarily dictated by vibrational energy as it is the slowest component of the internal energy to excite. Hence, the characteristic vibrational excitation times provide a good measure of overall internal energy excitation. 75

Figure 5.15: Comparison of characteristic vibrational time obtained from DMS (blue curve) against Millikan and White formulation (black curve) and contemporary results[20, 47, 64]

(a) Ttr = 5000K. (b) Ttr = 30000K.

Figure 5.16: Evolution of energy modes during isothermal excitation 76 5.4.2 Non-equilibrium dissociation

This section discusses dissociation due to N + N2 collisions. As described in the previous section (section 5.4.1), to isolate the eﬀects of atom-molecule collisions in DMS, the partial density of molecular nitrogen (ρN2 /ρtotal) was reduced so that N + N2 col- 6 lisions dominate. As in section 5.4.1, ρN2 /ρtotal = 0.01 was used with 8 × 10 DMS particles and the particle weight was set to unity (Wp = 1). The molecules in the system were initialized to Trot = Tvib = Ttr and simulations were carried out for

Ttr = 10000K, 15000K, 20000K, 25000K and 30000K. An example of such a calculation can be see in Fig. 5.18(a). Once the system is in QSS, dissociation rate constants are calculated by ﬁtting composition histories to the following equation:

d[N ] 2 = −kN2+N2 [N ]2 − kN+N2 [N][N ] . (5.5) dt d 2 d 2

Since, [N2] << [N], the ﬁrst term in Eq. 5.5 can be ignored and it is assumed that [N] does not change in time because, at t = 0, [N] >> [N2]. Therefore, [N](t) ∼

[N](t = 0) = [N]0. This reduces Eq. 5.5 to a pseudo-ﬁrst order rate law given by:

d[N ] 2 = −[N] kN+N2 [N ] , (5.6) dt 0 d 2 which can be integrated to give:

N+N2 [N2](t) = [N2]0exp(−kd [N]0t) . (5.7)

QSS composition histories obtained from DMS were ﬁtted to Eq. 5.7 to back out

N+N2 dissociation rate constants for N + N2 collisions (kd ). Fig. 5.17 shows the non- equilibrium dissociation rate constants obtained in this manner. Figure 5.17 compares current results with dissociation rate constants obtained from an earlier DMS study done by Valentini et al. [64] using the Minnesota PES. There is remarkable agreement between the dissociation rate constants that are obtained from the DMS method using two independently developed ab − initio PESs (at NASA and UMN). Additionally, comparison id made with the experimental dissociation rate coefficients obtained by 77 Appleton et al. [7], Hanson and Baganoff [8] and those by Kewley and Hornung [119]. The curve fit from these experiments can be seen in Fig. 5.17 as the solid black curve, the dashed-dot black curve and the dashed curve respectively, the symbols on the curves represent the temperature range over which the experiments were conducted.

Figure 5.17: Comparison of non-equilibrium dissociation rate constants obtained for N + N2 collisions using the DMS method using the NASA PES (blue) and the UMN PES (red). Also plotted are rate coeﬃcients from experiments[7,8, 119]. The black lines represent the extrapolation obtained from the experimental data and the circular symbols show the range over which experimental data was taken

The DMS method provides insight into the dissociation process at the molecular level as well, because the method provides the distribution of internal energy of the molecules in the simulation. Figure 5.18(b) gives an example of such a distribution taken for a case with Ttr = 30000K (shown in Fig. 5.18(a) ). The red curve shows the distribution of internal energy (f(int)) of the system in QSS and the blue curve shows the distribution of pre-collisional internal energy for the molecules that have dissociated (fd(int)) while the system was in QSS. The blue curve peaks at int ∼ 9.9eV , which corresponds to the well-depth for the diatomic potential at j = 0. Similar distribution curves determined made for Ttr = 10000K, 15000K, 20000K and analysis of the data is used to construct dissociation probabilities as a function of internal energy and translational temperature, as seen in Fig. 5.19(a). 78

(a) Composition and temperature proﬁles with (b) Internal energy distributions for molecules in Ttr = 30000K QSS (f(int)) and for molecules that dissociated (fd(int)) at Ttr = 30000K

Figure 5.18: Composition and temperature proﬁles resulting from a dissociation calculation, along with the internal energy distributions of molecules.

Figure 5.19(a) shows the QSS dissociation probability at a given internal energy for translational temperatures ranging from 10000K to 30000K. It is observed that for

int < 9.9eV the probability of dissociation increases with increasing internal energy.

Further, the rate of the slope of the dissociation probability goes as 1/Ttr. Figure 5.19(a) also shows that the dissociation probability curves become closely banded and relatively ﬂat for int > 9.9eV . This shows that the probability of dissociation become less dependent on translational temperature for molecules with int > 9.9eV . For the nitrogen PESs used in this study [25, 26], the well depth of the diatomic potential for j = 0 is ∼ 9.89eV . Hence the molecules with int > 9.9eV are quasi- bound. These molecules are trapped in metastable energy levels due to the centrifugal energy barrier. From Fig. 5.19(a) one can conclude that the probability of dissociation of quasi-bound molecules is large and weakly dependent on the average translational energy (Ttr) and therefore weakly depandant on the translational energy of the collision partner of the quasibound molecule. Fig. 5.19(b) shows the probability of dissociation over a range of vibrational energies for a given translational energy and averaged over rotational energy. A similar trend is seen to what is observed in Fig. 5.19(a) for internal 79

(a) Dissociation probability in QSS versus internal (b) Dissociation probability in QSS versus vibra- energy for a given translational temperature. tional energy for a given translational temperature.

Figure 5.19: Dissociation probabilities for a given translational temperature over a range of internal energy and vibrational energy. energy, where the dissociation probability rises with increase in vibrational energy and the slope of the logarithm of the probability is proportional to 1/Ttr. Further, a sharp rise in dissociation probability after vib > 9.66 eV is seen. Figure 5.20 compares the nonequilbrium dissociation rate coeﬃcients obtained for the N + N2 collisions to that obtained for N2 + N2 collisions in the previous section.

One can see that the dissociation rate coeﬃcients for the N + N2 collisions are higher than those for N2 + N2 collisions. The reason behind this diﬀerence can be explained by Fig. 5.27 Figure 5.27 shows the QSS vibrational energy distributions obtained for the system at Ttr = 30000K, 20000K, 15000K, 10000K obtained with N + N2 (N3) collisions and compares them with QSS vibrational energy distributions obtained with N2 + N2 (N4) collisions. It is shown that the distributions for the N3 process deviate less from the equilibrium distribution than those obtained for the N4 process. This higher population at the high energy tail is due to exchange reactions that are likely to result in greater translation-to-internal energy transfer[64]. The higher population at the high energy 80

Figure 5.20: Nonequilibrium dissociation rate coeﬃcients obtained for N + N2 and N2 + N2 using the DMS method with PESs developed at NASA Ames Research Center (blue) and the University of Minnesota (red).

N3 N3 N4 N4 Temperature (K) kd−Ames/kd−UMN kd−Ames/kd−UMN 10000 1.50 1.51 15000 - 1.09 20000 1.26 1.19 30000 1.10 1.04

Table 5.1: Comparison of non-equilibrium dissociation rates for nitrogen

tail results in more molecules having enough energy to dissociate and consequently a higher dissociation rate coeﬃcient than the N4 process. This phenomenon was observed in the earlier work done by Valentini et al. [64] using the Minnesota PES. Table 5.1 shows a comparison for the non-equilibrium dissociation rate coeﬃcients obtained from the NASA PES and Minnesota PES.

5.5 Eﬀect of numerical method on the N2 + N system

Similar to section 5.3, the thermo-chemical properties for N2 + N interactions obtained using the Ames PES [26] are now compared with the state-to-state method. However, 15 unlike N2+N2 interactions that had O(10 ) possible state to state interactions, the N2+ 81 N system only has ∼ O(107) possible state interactions5. With modern computational resources it is tractable to perform a full state resolved analysis of the N2 + N system. Hence, for the analysis shown in this section no energy grouping was performed. Comparative analysis between DMS and the fully resolved state-to-state (StS) method was performed for zero-dimensional isothermal excitation calculations, where, Ttr =

10000 K and 20000 K, the internal temperature Trot = Tvib = 3000 K, and the density was set to ρ = 1.283kg/m3. Just like the previous section (5.4), the DMS simulations were carried out in a heat bath of atomic nitrogen with a trace amount of molecular nitrogen (ρN2 /ρtotal = 0.01) to isolate N2 + N interactions. For the StS simulations only reactions between atomic and diatomic nitrogen were considered. Since, the DMS method does not perform re-combination reactions, the recombination reactions for the StS analysis where turned oﬀ. This study was done in collaboration with Prof. Marco Panesi and Robyn Macdonald at the University of Illinois - Urbana Champagne, who performed the StS calculations. Details of this work can be found in a joint paper [69].

5.5.1 Internal energy excitation

Figure 5.21 shown internal energy excitation proﬁles obtained from the DMS and StS simulations for the two cases. The DMS method consistently predicts higher internal energy of the molecules at the end of the simulation time. Figure 5.22 shows an instantaneous distribution of energy states during the relaxation process at both translational temperatures. Due to the small amount of simulated molecules the energy distribution obtained from DMS lacks deﬁnition in the high energy tail. This is because DMS is a sampling based method. There are only 40200 diatomic nitrogen molecules at t = 0 owing to the low mass fraction maintained to isolate N2 +N interactions, and as the simulation progresses, more and more diatomic nitrogen is dissociated. The horizontal strands seen at the higher energies are due to the few number of particles present in the simulation. At the high energy states, there may only be one or two molecules in that energy range, resulting in a horizontal strand structure. For

5 2 9390 initial states for N2 and 9390 ﬁnal states for N2 giving (9390) total rovibrational state-to-state transition 82

Figure 5.21: Internal energy excitation using DMS and StS methods. [69]. high temperature, the strand structure occurs for higher energy states because of the increase in internal energy. Despite the sampling resolution of the DMS results, the agreement between the StS and DMS methods is generally good during the excitation process

Figure 5.22: Instantaneous internal energy distribution during the excitation process using DMS and StS methods [69]. 83 5.5.2 Nonequilibrium dissociation

The composition history from the two dissociating cases is shown in Fig. 5.23. Due to the very low number of nitrogen molecules initially, the amount of dissociation which is small. Nonetheless, the composition history predicted by both methods is in excellent agreement. It is important to note that because recombination reactions are not captured by the DMS method and in this case ignored by the StS method, this system will never retrieve an equilibrium composition.

Figure 5.23: Composition history during isothermal relaxation [69].

Figure 5.24 shows the QSS distribution obtained for the two cases using DMS and StS. In both cases the distribution predicted by the DMS method has been averaged over several time-steps within the QSS region to improve statistics. Since, the QSS distributions are time invariant, time-averaging does not effect the nature of the distribution. In general, the trend observed in the distribution in QSS is similar between the two methods. It is observed that the StS distribution has scatter. However, this is because the StS method works on quantized energy levels and the DMS method is continuous. Due to the varying degeneracy of quantized states, the actual population of various states which are energetically near each other can be significantly different. As a result, the StS data exhibits large fluctuations from energy group to group. In contrast, the DMS data corresponds to a continuous energy distribution for internal energy. In 84 general the same features are observed at both Ttr = 10000 K and Ttr = 20000 K.

Figure 5.24: Internal energy distributions obtained during QSS.

Figure 5.25: Cumulative probability of dissociation against internal energy.

Finally, the cumulative distribution of dissociating molecule energy is shown in Fig. 5.25. The results from the StS method and the DMS method are in excellent agreement for both temperatures, with molecules primarily dissociating from energies close to the dissociation limit. At higher temperature it can be seen that molecules from lower energies are able to dissociate more readily. This is due to the increased translational energy 85 available to facilitate dissociation. Therefore, it appears that at high temperature, the importance of low energy states for dissociation increases.

5.6 Comparison of N4 system to N3 + N4 system

The N4 system for the NASA PES has been haracterized in section 5.2, and the N3 system for the NASA PES has been chracterized in section 5.4. This section will discuss isothermal relaxation for a box ﬁlled with molecular nitrogen in which both N3 and N4 processes are allowed to occur concurrently and compare this to work shown only for the N4 system. An example of an isothermal relaxation with the N3 + N4 system has been shown in Fig. 3.7.

Figure 5.26: Composition history comparison for N3+N4 system and N4 system.

Figure 5.26 shows the comparative composition history of the isothermal relaxations with the N4 and N3+N4 systems. The comparative isothermal relaxations are carried 86

(a) Ttr = 30000K. (b) Ttr = 20000K.

Figure 5.27: Vibrational distribution of N2 molecules in QSS for Ttr = 30000K, 20000K, 150000K and 10000K for the N + N2 and N2 + N2 processes

out for Ttr = 30000K, 20000K and 15000K. For results shown in Fig. 5.26, the isothermal box was initialized such that Ttr = Tvib = Trot. One can see that the initial dissociation rate in each case corresponds to the dissociation rate for the pure N4 system. However, once the concentration of molecular nitrogen drops to roughly 90% of the original concentration ([N2]/[N2]0 = 0.9) the system deviates from the pure N4 behavior and dissociation becomes faster, a result of the dissociation process being faster 87 when N atoms are present.

Further an example of a 0-D isothermal excitation calculation where both N2 + N and N2 + N2 collisions were allowed in Fig. 3.9. In this case, the system starts oﬀ with a box full of pure molecular nitrogen; with initial Tvib = Trot = 3000K (red and black dash-dot lines respectively) and the translation temperature was kept constant at

Ttr = 30000K. I can be observed that, in contrast to the N + N2 vibrational excitation profile seen in Fig. 5.14(a), the vibrational temperature excitation profile in Fig. 3.9 has an inflection point. The inflection point can be attributed to the fact that vibrational excitation due to N2 + N2 is about an order of magnitude slower than vibrational excitation due to N2 + N interactions. In the case shown in Fig. 3.9 the system starts with pure molecular nitrogen and the excitation rate is due to N2 + N2 interactions. As the system evolves in time, atomic nitrogen is introduced due to dissociation. The build up of atomic nitrogen rapidly accelerates the excitation process and eventually the excitation rate is dictated by N2 + N interactions. This change in excitation rate of over an order of magnitude introduces the inflection point.

5.7 Comparison of ab − initio results for the full nitrogen system

In precious sections (section 5.4.1 and section 5.4.2) it has been shown that the characteristic excitation times and the non-equilibrium dissociation rates obtained with the NASA PES and Minnesota PES for the N3 process are agree remarkably well. Further, section 5.2 has shown agreement between dissociation rates for the N4 processes between the two PESs. This section will compare the combined N3+N4 process for isothermal relaxation of nitrogen for the two PESs. For the comparative study, three cases were analyzed and can be seen in Fig. 5.28. For the first two cases (shown in Fig.5.28(a) and 5.28(b)) isothermal box relaxation was initialized with the system containing nitrogen molecules at Trot = Tvib = 3000K with no atomic nitrogen present. In these cases the translational temperature was set to Ttr = 30000K and Ttr = 20000K respectively. As the system evolves, one can see 88 dissociation of molecular nitrogen and excitation of internal energy modes to QSS levels. The excitation curves and the QSS temperatures for the internal energy modes produced by both the PESs are identical and the composition history of molecular nitrogen is comparable. The third case presented is for Ttr = 10000 K. Here the isothermal box relaxation was initialized with nitrogen molecules having Trot = Tvib = Ttr and with no atomic nitrogen present. This choice of initial condition was done for computational efficiency, however, as shown in Fig. 3.9 initial values of vibrational and rotational temperature have no bearing on the final QSS condition. For this case too, one sees that the two PESs predict the same QSS temperatures but the NASA PES result is more dissociative than the Minnesota PES. A similar trend was observed for the N4 system at Ttr = 10000K section 5.2.

(a) Ttr = 30000K. (b) Ttr = 20000K. (c) Ttr = 10000K.

Figure 5.28: Composition and temperature history comparison for N3+N4 system using the NASA PES [25, 26] and the Minnesota PES [27, 28]. For Ttr = 30000K, 20000K, 150000K and 10000K

Figure 5.29 shows the QSS vibrational energy distribution functions for the cases shown in Fig. 5.28. The QSS vibrational energy distributions given by the two PESs are nearly identical. This shows that the results produced by the two PESs agree at the molecular level as well.

5.8 Conclusions

Direct molecular simulations using PESs developed at NASA Ames Research Center were performed to capture internal energy excitation and non-equilibrium dissociation 89

(a) Ttr = 30000K. (b) Ttr = 20000K. (c) Ttr = 10000K.

Figure 5.29: Vibrational distribution of N2 molecules in QSS for Ttr = 30000K, 20000K and 10000K for the N + N2 and N2 + N2 processes

due to N2+N and N2+N2 interactions under isothermal conditions. The results of these simulation were compared with isothermal DMS simulations carried out using PESs developed, independently, at the University of Minnesota for nitrogen interactions. Good agreement was found for macroscopic quantities like characteristic vibrational energy excitation time and non-equilibrium dissociation rate coefficients. Furthermore, the PESs agreed remarkably in defining the molecular-level energy distribution of the system. Given that the two independently developed PESs give results in close agreement, this confirms the physical mechanisms involved in nitrogen ro-vibrational excitation and dissociation reported in recent articles, and lends confidence in using computational chemistry for hypersonic flow modeling. Characteristic vibrational excitation times were calculated at T = 10000 K and

T = 25000 K for N2 + N2 interactions and for temperatures ranging from T = 5000K to Ttr = 30000K for N2 + N interactions using the Ames PES. It was observed that the characteristic vibrational excitation times due to N2 + N interactions were an order of magnitude lower than those predicted by the Millikan and White theory and agreed well with characteristic excitation times calculated using PESs developed at the University of Minnesota and other contemporary results. Further, it was shown that during vibrational excitation the high energy tail of the vibrational energy distribution gets populated ﬁrst and as the excitation continues the gas evolves through a series of non-Boltzmann distributions to the QSS distribution. 90

This mechanism of excitation was observed for both N2 + N2 interactions and N2 + N interactions and has a two-fold consequence for hypersonic ﬂight. Firstly, the vibrational energy excitation is a non-Boltzmann process. The Landau-Teller equations that are used in most CFD codes are based on near-Boltzmann assumptions. Secondly, since the vibrational energy excitation mechanism over-populates the high energy tail of the vibrational energy distribution, it produces more vibrationally excited molecules than anticipated by a Boltzmann distribution. Hence, during the excitation process there may be more dissociation than previously assumed with Landau-Teller theory. DMS calculations were used to calculate non-equilibrium (QSS) dissociation rate coeﬃcients over temperatures ranging from Ttr = 10000K to Ttr = 30000K due to both,

N2 + N2 and N2 + N interactions. For the N2 + N interactions, it was observed that the logarithm of the dissociation probability goes as 1/Ttr for molecules with int < 9.9eV .

For molecules with int > 9.9eV (quasi-bound molecules) the probabilities of dissociation become weakly dependent on translational temperature. One can extend the above conclusion to say that the dissociation of quasi-bound molecules is weakly dependent on the translational energy of the collision partner. Further, it was observed that the logarithm of the dissociation probability goes as 1/Ttr against vibrational energy as well. These well deﬁned trends could be used to develop new dissociation models for DSMC and CFD simulations Further, comparative studies between DMS and ME-QCT with vibrational and energy groups for N2 + N2 interactions, and between DMS and StS method for N2 + N interactions while using the Ames PES were carried out. Comparison with StS method for N2 +N interactions with the DMS method was found to be good. This validated the results produced by the DMS method and demonstrated the applicability of the DMS method to studying non-equilibrium excitation and dissociation processes in cases like

N2 + N2 interactions where a full scale StS solution will be prohibitively expensive. For the N2 +N2 interactions it was found that for ME-QCT the grouping technique plays an important role. The comparison revealed that the energy based grouping strategy can accurately predict both the microscopic and macroscopic properties throughout dissociation process across a range of temperatures. However, this approach predicts faster 91 energy transfer than the DMS method exhibits. In contrast, the vibrational specific grouping was able to match macroscopic properties during the energy transfer process, but failed to capture both the composition profile and the microscopic distribution of the high lying levels. Since DMS calculations do not involve any assumptions about energy grouping, decoupling of internal mores, and uses the PES as its only modeling input, it can be used as a baseline solution for any future grouping methodology for master equation analysis or any other reduced order models for hypersonic flows. Chapter 6

Direct Molecular Simulation of Oxygen

6.1 Direct molecular simulation of O2 + O interactions

6.1.1 Introduction

The focus of this section is to characterize O2 + O interactions. As discussed in chapter 2, molecular and atomic oxygen in their ground electronic states each have two unpaired electrons spin coupled as a triplet. Hence an O2 + O interaction can occur with the total spin coupling being singlet, triplet, or quintet with statistical weights of these spin couplings being 1, 3 and 5 respectively. Further, the spatial symmetry of atomic oxygen generates a threefold degenerate ground state for all three spin states. Hence, due to the spin and spatial degeneracies, nine unique PESs are required to fully describe O2 + O interaction dynamics initiated in the ground states of the collision partners. Most work done on characterizing the O2 + O system for hypersonic application [46, 48, 90–92] has used the singlet global potential energy surface for the O3 system provided by Varandas and Pais [89], which was shown to have spurious features in section 2.4.1. Furthermore, while the singlet state (11A0) is important for studying the formation of ozone, it describes only 1/27 of all possible interactions and does not provide a complete

92 93 picture of the dynamics for the O2 + O system. Recently, Varga et al.[31] developed the nine adiabatic ab − initio PESs describing the electronically adiabatic collisions of ground electronic-state triplet O2 with ground electronic state triplet O. This section characterizes vibrational energy excitation and dissociation of diatomic oxygen due to

O2 + O interactions using all nine PESs. Results will be compared with experimental data and prior work.

6.1.2 Simulation of O2 + O interactions

The DMS method as discussed in chapter3 was used to simulate zero-dimensional isothermal calculations (described is section 3.3.3) discussed in this chapter. However, due to the presence of nine surfaces describing collisions in the ground state of the collision partners, and presence of low lying electronically excited states for oxygen a few modiﬁcations and assumptions to the standard DMS approach were made to address

O2 + O collisions.

Treatment of Internal Energy

As discussed earlier, the DMS method only operates on the positions and velocities of atoms present in the system. The positions and velocities of atoms bound in a molecule are used to determine the internal energy of the molecule during post processing. In this work the internal energy of the molecule is divided into vibrational and rotational modes using the vibrational prioritization frame work used by Jaﬀe [99]. The details of this division can be found in section 3.2.2. In addition to ro-vibrational internal energy modes, while considering the diatomic oxygen molecule, one has to account for electronically excited states of diatomic. Table 6.1 lists the electronically excited states of diatomic oxygen as reported by Saxon et al. [120]. One can see that accounting for electronically excited states becomes especially important while considering dissociating molecules as there are ﬁve electronically excited states before the diatomic dissociation limit and additionally, two additional electronically excited levels above the dissociation limit. 94 State Energy (eV) Degeneracy 3 − X Σg -5.21 (-4.96) 3 1 a ∆g -3.84 2 1 + b Σg -3.15 1 1 − c Σu -1.02 1 3 C ∆u -0.81 6 3 + A Σu -0.72 3 3 − B Σu 1.14 3 3 2 Πg 1.98 6

Table 6.1: Electronic states of molecular oxygen and their energies and degeneracies[120]. The ground state energy value in parenthesis is the value reported in Ref.[120], this value has been replaced by the diatomic well depth of O2 given in Ref.[31]

Nikitin [121] proposed that the excitation rate to the lower electronic levels is on the order of the vibrational relaxation time and the higher electronic levels get populated at higher temperatures. He assumed that there will be equilibrium among vibrational levels of the electronically excited states. With these assumptions, dissociation can be assumed to occur concurrently from the ground state and electronically excited states, and a very rough estimate of the overall dissociation rate can be obtained by multiplying the rate obtained from the ground-state calculation by the sum of degeneracies of all the electronic states (including ground state) and dividing it by the degeneracy of the ground state. This multiplication factor comes out to be η = 16/3 for classically bound molecules and to ηQB = 25/3 for quasibound molecules. Since quasi-bound molecules are statistically insigniﬁcant it is common in literature to use η = 16/3. This method has been used in contemporary work with oxygen [46, 48, 90–92] to account for the eﬀect of electronically excited states. This is a very approximate way to account for electronically excited states. A more accurate method to account for electronically excited states would be to calculate electronically non-adiabatic trajectories[122, 123] and to explicitly include electronically excited O2 states in the simulation. However, this requires PESs for all excited states and state couplings to be available which is not the case at the time of this study. 95 Isothermal Relaxations

Results discussed in this chapter are obtained using zero-dimensional isothermal excitation simulations. Constant translational temperature is maintained in the simulation by sampling the center-of-mass velocities of particles in the DMS calculation from a Maxwell-Boltzmann distribution at the desired temperature after every time step. To isolate the eﬀect of O2 + O interactions, the system is initialized such that

ρO2 /ρtotal = 0.01 (similar to section 5.4 and 5.5 for N2+N). All simulations were started with eight million total particles of which 40200 are diatomic oxygen. The simulation 3 density is set to ρ = 1.28kg/m and the particle weight is set to unity (Wp = 1)

Since, there are nine PESs on which O2 + O interactions occur, first isothermal simulations with individual PESs corresponding to a particular spin-spatial degeneracy were done. The individual PES simulations were carried out at T = 5000 K, 8000 K and 12000 K for the first adiabatic A0 surfaces, at T = 5000 K and 12000 K for the first adiabatic A00 surfaces and only preliminary simulations were carried out for the second adiabatic A0 surfaces. Next, simulations where all spatially degenerate PESs for a single spin state were accounted for in the same simulation were performed. For these simulations, once the particles are paired to undergo trajectory integration they are randomly assigned one of the three spatially degenerate PESs for that spin state, with each spatially degenerate PES having a 1/3rd probability of selection. This kind of simulation was were carried out at T = 5000 K, 8000 K and 12000 K for all spin states. Finally, simulations that incorporated all nine PESs were performed. For these simulations after the particles are selected to undergo trajectory integration the trajectory has a 1/27 probability to be governed by the 11A0, 11A00, 21A0 PESs a 3/27 probability to be governed by the 13A0, 13A00, 23A0 PESs and a 5/27 probability to be governed by the 15A0, 15A00, 25A0 PESs. Results from these simulations will be referred to as the ”full O3 system” in the rest of the thesis. Full O3 system calculations were carried out at T = 3000 K, 5000 K, 6000 K, 8000 K, 10000 K, 12000 K and 15000 K. An example of an isothermal excitation calculation for the full O3 system can be seen in Fig. 6.1. The blue curve shows the translational temperature that is kept 96 constant at Ttr = 12000 K. The black curve represents the rotational temperature and the red curve represents the vibrational temperature. The rotational and vibrational temperatures are initialized at 1400 K. The green curve shows the fraction of diatomic oxygen present in the system as a fraction of the initial diatomic oxygen in the system. Recombination is not currently allowed in DMS, hence the simulation continues until all molecular oxygen is dissociated.

Figure 6.1: Temperature and composition history of an isothermal excitation calculation done in DMS for the full O3 system.

One can see that the vibrational and rotational temperatures of the gas excite from

Tvib|t=0 = Trot|t=0 = 1400 K and level oﬀ at Tvib = Trot ≈ 11400 K. This non-equilibrium state of the gas is called the quasi steady state (QSS). In QSS the rotational and vibrational energy lost due to dissociation is balanced by internal energy excitation due to inelastic collisions and exchange reactions. These balancing processes lead to the energy distribution functions in the QSS becoming time invariant [113]. These time invariant QSS distributions can be seen in Fig. 6.2, which shows that the energy distributions in QSS are depleted when compared with the equilibrium energy distributions. The ag- gregate eﬀect of depletion of the vibrational and rotational energy distributions in QSS is seen in Fig. 6.1, where one can see that the vibrational and rotational temperatures in QSS are less than the equilibrium temperature of T = 12000 K. It has been shown in 97 section 3.3.3 that for a given translational temperature the QSS is independent of the initial internal energy of the gas in section 3.3.3 and reference [68].

(a) Vibrational Energy Distribu- (b) Rotational Energy Distribu- (c) Internal Energy Distribution tion tion

Figure 6.2: Internal energy distributions of O2 molecules in QSS for Ttr = 12000 K.

6.1.3 Vibrational energy excitation by O2 + O interactions

This section discusses vibrational excitation of molecular oxygen by O2 +O interactions. Figure 6.3(a) shows an example of excitation calculations where the initial vibrational temperature is set to Tvib|t=0 = 1400 K and the translational temperature is set to

Ttr = 12000 K. The red curve shows the vibrational temperature profile for a simulation that used all nine oxygen PESs (full O3 system) and completely characterizes the excitation behavior of ground-electronic-state oxygen due to O2 + O interactions, under the assumption that all energy transfer collisions of O2 are due to collisions with O. This figure also shows the vibrational temperature profiles for excitation calculations carried 1 0 5 00 out using 1 A , 1 A PESs and for all the spin states for the O3 system. As described in Sec. 6.1.2 in the simulations corresponding to a particular spin state all spatially degenerate PESs for that spin state are used. It was found that among the individual PESs the excitation process is fastest on the 11A0 surface and is the slowest on the 15A00 surface. Also it is observed that interactions in the quintet spin state excite oxygen significantly more slowly than interactions in the singlet and the triplet spin states. Figure 6.3(b) shows the evolution of the vibrational distribution function during the 98

(a) Vibrational temperature excitation proﬁles (b) Evolution of vibrational distribution function for the full O3 system (red curve) shown in Fig6.3(a)

Figure 6.3: Example of excitation calculations with O2 + O interactions. excitation process for the full O3 system. In particular, the vibrational temperature profile given by the red curve in Fig.6.3(a). As observed in studies for nitrogen [64, 68] and for theO2 + O2 interactions in oxygen [66], during the excitation process the high- energy tail of the vibrational energy distribution gets populated first, then the system gradually evolves through a series of non-Boltzmann distributions to reach the QSS distribution. Characteristic vibrational excitation times are calculated by fitting the vibrational temperature excitation profile to the Landau-Teller expression [18]:

Tvib(t) = Tvib|final − ∆Tvib[exp(−t/τ)] (6.1)

where,

∆Tvib = Tvib|final − Tvib|t=0 . (6.2)

This is a general result for systems close to equilibrium, in which case relaxation is always exponential, however this model may not be valid more generally [24]. While 99 calculating the characteristic vibrational excitation time, Tvib|final is set to the vibrational temperature under equilibrium conditions. For example, for the case shown in

Fig.6.3(a), Tvib|final = 12000 K. Furthermore, Eq. 6.1 is fit to the initial part of the vibrational temperature excitation profile to eliminate any coupling between vibrational excitation and energy lost due to dissociation. This fit for the full O3 system of the example case is shown as the pink dashed line in Fig.6.3(a).

(a) Comparison of vibrational excitation time (b) Comparison with literature[3, 124, 125] among O3 PESs by Varga et al. [31] and characteristic excitation time by Andrienko and Boyd.

Figure 6.4: Characteristic vibrational excitation time for O2 + O interactions.

Characteristic vibrational excitation times are calculated for the full O3 system over temperatures ranging from T = 3000 K to T = 15000 K. Characteristic excitation times for the singlet, triplet, quintet spin states and individual PESs are calculated for temperatures ranging from T = 5000 K to T = 12000 K. Figure 6.4(a) shows the characteristic vibrational excitation time calculated for the full O3 system, for singlet, triplet and quintet spin states, and for the 11A0 and 15A00 PESs. In Fig. 6.4(a), one can see that for the full O3 system (red curve with circular symbols), the characteristic excitation time increases with increasing temperature. However, this increase is minuscule, with the difference between the characteristic excitation time 100 at T = 3000 K and T = 15000 K being ∼ 25%. The 11A0 PES allows for the fastest excitation and the 15A00 PES has the slowest excitation among all nine PESs. It is evident that at T = 5000 K excitation on the 11A0 is 53 times faster than that on 15A00. However, this difference narrows down as the temperature increases with the difference between excitation on 11A0 and 15A00 PESs at T = 12000 K being a factor of 17. It is observed that, over the whole range of temperatures, the oxygen molecule is excited much more slowly when the system is in the quintet spin state than when it is in the singlet and triplet spin states. Additionally, while the characteristic excitation time for the singlet and triplet states increases with temperature, the characteristic excitation time for the quintet decreases with increase in temperature. These differences can largely be attributed to the fact that the quintet PESs (15A0, 15A00 and 25A0 PESs) allows for significantly fewer exchange reactions than the singlet and triplet PESs (11A0, 11A00, 21A0, 13A0, 13A00 and 23A0 PESs) [126] and it is well known that exchange reactions are more efficient at redistributing vibrational energy [116–118]. Examples of these exchange barriers were shown earlier in Fig. 2.6 in chapter2 Finally, Fig. 6.4(a) compares the characteristic excitation time calculated in this study to contemporary simulations by Andrienko and Boyd [48]. The difference with the predictions of Andrienko and Boyd [48] is a factor of four at the low temperature range and about a factor of seven at the high-temperature range with respect to the full O3 system (red curve). While the data presented in this study stays fairly flat through the temperature ranges, the characteristic excitation time proposed by Andrienko and Boyd increases slightly with temperature. It should be noted that Andrienko and Boyd [48] used the PES developed by Varandas and Pais [89] for th 11A0 state whereas the data presented by the red curve in Fig. 6.4(a) shows the excitation for the full O3 system, incorporating all nine PESs and covers all possible O2 + O interactions for the ground electronic state. Hence, a more direct comparison of the characteristic excitation time obtained by Andrienko and Boyd [48] is with the characteristic excitation time obtained from the 11A0 surface given by the dashed blue curve with triangular symbols.The results obtained by Andrienko and Boyd [48] differ by more than an order of magnitude from the findings of this study. Compared to the results of this study, the much larger 101 characteristic relaxation time (i.e. slower relaxation) predicted by Andrienko and Boyd [48] is consistent with the spurious features in the PES by Varandas and Pais discussed in section 2.4.1, which might prevent frequent exchange reactions that are efficient at vibrational excitation. Since these features are not present in the 10A0 PES developed by Varga et al. [31], used in this study, exchange reactions on the surface may be more frequent, leading to a smaller characteristic relaxation time (i.e. faster relaxation). Figure 6.4(b) compares the characteristic excitation time for full O3 system with the characteristic excitation times for O2 + O interactions present in the literature [3, 124, 125]. The solid black curve in the figure shows the experimental fit for the characteristic vibrational excitation time by Ibraguimova et al.[124]; the brown symbols on the black curve show the temperature range (4000 K to 10800 K) over which the experiment was conducted. The experimental data, like the data in this study, also predict a relatively flat profile for the characteristic excitation time. The characteristic excitation times calculated in this work are about two times lower than the experimental data.

Kiefer and Lutz[125] measured the relaxation of O2 in a mixture containing O and evaluated the vibrational energy relaxation of O2 due to O2 + O collisions by using the linear mixture rule, whose validity has been studied with variable results[127, 128]. The dashed black curve in Fig.6.4(b) shows their experimental fit, and the purple symbols show the temperature range (1600 K to 3300 K) over which these experiments were conducted. It is observed that while the experimental values of the characteristic excitation time matches with the findings of this study at low temperatures the trend proposed by Kiefer and Lutz[125] does not match. The orange curve in the figure shows the characteristic excitation times proposed by Park [3]. Park [3] used data from Kiefer and Lutz [125] to propose the fit for the characteristic vibrational excitation time. Hence, like the experimental data by Kiefer and Lutz [125], one can see that the values proposed by Park [3] are close to the data calculated in this study at the low end of the temperature range; the fit proposed by Park [3] matches the data in this study at the high end of the temperature range too. However, neither the trend of the Park data nor the intermediate values match with the 102 findings of this study.

6.1.4 Non-equilibrium dissociation due to O2 + O interactions

This section discusses dissociation in the QSS due toO2 + O collisions. The rate law for dissociation of diatomic oxygen in a simulation with O2 and O is given as:

d[O ] 2 = −kO2+O2 [O ]2 − kO2+O[O][O ] . (6.3) dt d 2 d 2 Since, the isothermal simulations conducted for this study are carried out in a bath of atomic oxygen with a trace amount of diatomic oxygen, it can be assumed that

O2 + O2 interactions are negligible. Further, since [O]>>[O2], it is assumed that the concentration of atomic oxygen does not change signiﬁcantly with time and can be treated to be constant: [O](t) ≈ [O](t = 0) =[O]0. This reduces equation 6.3 to a pseudo ﬁrst order equation:

d[O ] 2 = −[O] kO2+O[O ](t) . (6.4) dt 0 d 2 Which can be integrated to the form:

O2+O [O2](t) = [O2]|t=0 exp(−kd [O]0t) (6.5)

After the isothermal system reaches a QSS the composition history of the system is fitted to Eq. 6.5 to obtain the dissociation rate coefficients. Rate coefficients obtained this way are multiplied by the multi-surface correction factor (η = 16/3). This study is done for temperatures ranging from T = 6000 K to T = 15000 K for the full O3 system. Non-equilibrium dissociation rate coefficients obtained are plotted in Fig. 6.5 (red curve), along with QSS rates published by Andrienko and Boyd [48] and non-equilibrium experimental rate coefficients by Shatalov [129]. It can be seen that there is agreement between the DMS rate coefficients obtained in this study and experimental data by Shatalov [129]. Andrienko and Boyd [48] present two sets of non-equilibrium rate coefficients, first 103

Figure 6.5: O2 + O QSS dissociation rate coefficients. assuming only vibrational non-equilibrium represented by the green line labeled “QSS- VT” and another set assuming rotational and vibrational non-equilibrium represented by the blue lines, labeled“QSS-RVT”. It should be noted that QSS dissociation rates are sensitive to the vibrational energy distribution in the gas [64, 68]. As shown in the previous section vibrational energy excitation due to O2 + O interactions using the Varandas and Pais PES [89] is slower than vibrational excitation due to O2 + O interactions using the PESs of Varga et al. [31] due to spurious features on the Varandas and Pais PES. Consequently, O2 + O interactions occurring on the Varandas and Pais PES are expected to created fewer vibrationally excited molecules and as a result the PES gives a smaller QSS dissociation rate coefficient than calculated in this study. An apples-to-apples comparison for the QSS dissociation rate coefficients calculated here (red curve) would be with the “QSS-RVT” (blue) curve, because in the DMS method does not assume translation-rotation equilibrium. Hence the QSS achieved in DMS is for both, rotational and vibrational energies. Furthermore, as discussed in section 5.5, assuming rotational thermal equilibrium conditions artificially increases the population of the quasi-bound molecules. Hence, the “QSS-VT” rates shown here are incorrect and the DMS data should only be compared to the “QSS-RVT” solution, which is less than 104 the DMS result presented here due to the use of an incorrect PES.

(a) 6000 K . (b) 8000 K.

Figure 6.6: Distributions of the system in QSS and the pre-dissociation distribution of molecules that dissociate in the O2 + O system

Figure 6.6 shows the vibrational energy distribution of the gas in QSS (solid red curve). Even at the lowest temperature (Ttr = 6000K) used to study the dissociation process, the high-energy tail of the vibrational energy distribution is depleted due to dissociation. Hence, even as low as Ttr = 6000K, dissociation is a non-Boltzmann 105 process. Further, Fig. 6.6 also show the pre-dissociation distribution of the molecules that dissociated in QSS (dashed red curves).

6.2 Direct molecular simulation of O2 + O2 interactions

6.2.1 Introduction

As noted in chapter2, O2 + O2 interactions in the ground electronic state can have an overall singlet, triplet, or quintet spin state. To this eﬀect, three PESs to fully describe

O2 + O2 interactions were calculated at the University of Minnesota by Paukku et al.

[29, 30]. Similar to the analysis done for the O2 + O system, isothermal calculations with each of the PESs was carried out to understand the behavior of individual spin states. Finally, isothermal excitation calculations were carried out with all three surfaces incorporated in the same calculation, these calculations will be termed as the “full O4 calculations”. During the full O4 calculations, when two molecules were selected for trajectory integration, the probability that the trajectory will be calculated on the singlet surface is 1/9, the probability that the trajectory will be calculated on the triplet surface is 3/9, and the probability that the trajectory will be calculated on the quintet surface is 5/9. Finally, similar to the calculations done in section 5.2, atom-molecules interactions were ignored in order to focus on O2 + O2 interactions only.

It should be noted that the triplet O4 PES is under peer-review [30] and the results shown with the triplet surface and the full O4 system incorporating the triplet PES should be considered preliminary. However, the results presented in this section are not expected to change.

6.2.2 Vibrational energy excitation due to O2 + O2 interactions

Vibrational energy excitation was studied for O2 + O2 interactions using isothermal excitation calculations. For these calculations, the initial rotational and vibrational temperature was Tvib = Trot = 1400 K. For the analysis done with the singlet and the quintet surface, excitation was studied by maintaining the translational temperature 106 at Ttr = 5000K, 8000K, 10000, 12000K and 15000K. For the triplet PES, excitation was studied by maintaining the translational temperature at Ttr = 5000K, 8000K and 12000K. For the full O4 system excitation was studied by maintaining the translational temperature at Ttr = 5000K, 6000K, 8000K and 12000K.

(a) Composition and temperature history for an (b) Evolution of vibrational distribution function isothermal excitation with the full O4 system for the full O4 system (red curve) shown in Fig 6.3(a)

Figure 6.7: Example of excitation calculations with O2 + O2 interactions.

Figure 6.7(a) shows an example excitation calculation done with the full O4 system, where Ttr = 12000K. The profiles for the internal temperatures (average rotational and vibrational energy represented in temperature units) excites and levels off th QSS condition. In Fig. 6.7(b) the evolution of the vibrational energy distribution during the excitation process can be seen. It is observed that excitation due to O2 + O2 1 interactions follows the same mechanism as what is observed for N2 + N2, N2 + N , and 2 O2 + O interactions seen earlier, where during the vibrational excitation process the high energy tail of the vibrational energy distribution gets excited first and then the system reaches the QSS distribution through a series of intermediate non-Boltzmann distributions.

1 N2 + N2 and N2 + N excitation mechanisms shown in chapter5 2 O2 + O excitation mechanisms shown in section 6.1.3 107 Figure 6.8(a) shows the characteristic vibrational excitation times obtained using the singlet (purple), quintet (red), and the triplet (blue) surfaces along with the characteristic vibrational excitation time for the full O4 system. It can be seen that for

Ttr ≤ 10000K the vibrational excitation is faster on the quintet PES as compared to the singlet and triplet PESs. However, above Ttr = 10000K the characteristic excitation time given by the three PESs are similar. Since the quintet PES is responsible for

(5/9) of all O2 + O2 interactions the overall characteristic vibrational excitation time given by the full O4 system is closer to the quintet characteristic vibrational excitation time. When compared to excitation due to O2 + O interaction seen in section 6.1.3 the dependance of characteristic excitation time on the PES is not as strong for the O2 +O2 system.

(a) Comparison of vibrational excitation time (b) Comparison with literature[3, 124, 130] among O4 PESs by Paukku et al. [29, 30]

Figure 6.8: Characteristic vibrational excitation time for O2 + O2 interactions.

Figure 6.8(b) shows characteristic excitation rates presented by Park [3] in orange and experimental characteristic vibrational times obtained by Losev [130] and Ibraguimova [124]. The circular symbols show the experimental data, while the black lines show the proposed ﬁts based on the experiments. It is evident that the experimental data agrees well with the computational results found in this study. 108

Figure 6.9: Comparison of characteristic vibrational excitation time for O2 + O and O2 + O2 interactions.

Figure 6.9 compares characteristic vibrational excitation time obtained for O2 + O2 and O2 + O interactions. It is observed that the vibrational energy excitation is faster for O2 + O interactions by about a factor of ﬁve at he high temperature end and by an order of magnitude at the low temperature end of this study. This trend is consistent with the trend observed for nitrogen. Again, this is because atom-molecule interactions allow for more exchange reactions to take place than molecule-molecule interactions and exchange reactions in-turn are more eﬃcient in transferring energy than simple collisions [116–118]

6.2.3 Non-equilibrium dissociation due to O2 + O2 interactions

This section will discuss dissociation of oxygen due to O2 + O2 interactions in QSS. Isothermal calculations are carried out using both the singlet and quintet PESs for

Ttr = 8000K, 10000K, 12000K and 15000K. Since the PES for the triplet spin state has only recently been available and is still under peer-review no dissociation analysis has been done for the triplet spin state, however this chapter does discuss two dissociating cases for the full O4 system at Ttr = 8000K and Ttr = 12000K. Dissociation rate 109 coeﬃcients were calculated by ﬁtting the composition history of the system in QSS to the rate law:

d[O ] 2 = −kO4[O ]2 − kO3[O][O ]. (6.6) dt d 2 d 2 Since atomic oxygen produced by dissociation reactions are removed from the simulation, Eq. 6.6 reduces to:

d[O ] 2 = −kO4[O ]2. (6.7) dt d 2 Figure 6.10 shows the non-equilibrium dissociation rate coefficients obtained using the DMS method along with the dissociation rate coefficients obtained by Andrienko and Boyd [131] and experimental data by Ibraguimova et al. [124] and Shatalov et al. [129]. The non-equilibrium dissociation rate coefficient obtained from the singlet PES (blue) are slightly higher than those obtained using the quintet PES (red), while the dissociation rate coefficients obtained by the full O4 system lay close to the singlet and quintet results. All DMS rates presented in this figure have been corrected by multiplying the dissociation rate coefficient by the multi-surface correction factor (η = 16/3). Further, it can be seen that the QSS rates published by Andrienko and Boyd [131] agree well with the computational results presented in this study. However, it should be noted that Andrienko and Boyd obtained their results using a triplet O4 PES developed by Varandas and Pais [132] and did not multiply their results with a multi-surface correction factor. The experimental dissociation rate coefficients by Ibraguimova et al. [124] and Shat- alov et al. [129] compare nicely with the QSS dissociation rates obtained in this study. This can be explained by Fig. 4 in Ref. [124] The figure shows the maximum vibrational temperature (Tv) (average vibrational energy) observed behind the shock front 0 0 versus T0, the temperature behind the shock. The curve labeled 1 shows the case of equilibrium between vibrational and translational modes. One can see that for stronger shocks with higher temperatures the experimental data shows that the vibrational and 110

Figure 6.10: QSS dissociation rate coeﬃcient for O2 + O2 interactions

Table 6.2: Comparison of QSS dissociation rate coeﬃcients for O2 + O2 interactions

translational temperatures did not equilibrate. Hence it may be possible that the high temperature rate coefficients inferred from the data may correspond to non-equilibrium (QSS) rate coefficients, as simulated in this section. A similar argument can be made for the experimental data obtained by Shatalov [129]. Table 6.2 compares the non-equilibrium dissociation rate coefficients obtained from the quintet and singlet PESs. While the two PESs give similar dissociation rate coefficients, the singlet PES is more dissociative than the quintet PES. The reason behind this phenomenon can be explained by the molecular level details obtained from the DMS simulation. Figure 6.11 shows the ground electronic state vibrational distribution functions for the O2 +O2 system (dashed lines) and the vibrational energy distribution of the 111 ground electronic state molecules that dissociated (solid lines). Figure 6.11 shows that for a gas in QSS the vibrational energy distribution has a depleted population at the high energy tail as compared to the equilibrium vibrational energy distribution at the temperature of the isothermal simulation. It is shown that this departure form the equilibrium distribution is true even at our lowest simulation temperature of Ttr = 8000K

(Fig. 6.11(d)). Showing that dissociation is a non-equilibrium process at Ttr = 8000K as well. It can be observed from the distributions of dissociating molecules, that for the molecules interacting on the singlet PES, the dissociating species vibrational populations remain nearly constant over a range of vibrational levels. On the other hand the distribution of dissociating molecules obtained from the simulation using the quintet PES is more sensitive to the vibrational energy of the molecules, with fewer molecules dissociating from the lower vibrational energy end of the distribution. For the high temperature cases presented in Fig. 6.11(a) and 6.11(b) it can be seen that the distribution of dissociating molecules for the singlet PES flairs up at low vibrational levels. Since, there are more molecules in the low vibrational levels this increase in dissociation from the lower vibrational levels is representative of a higher population at low vibrational energy. However, there is no such increase in dissociation from molecules with lower vibrational energy for the quintet PES. This shows that molecules with less vibrational energy are easily dissociated while interacting on the singlet PES. This results in the singlet PES being more dissociative than the quintet PES and consequently having a higher dissociation rate coefficient. Figure 6.12 shows the comparison of non-equilibrium dissociation rate coefficients obtained by the full O4 system3 with those obtained for the full O3 system4. It is observed that dissociation of oxygen in QSS is approximately two times higher for the full O3 system. Again, the reason behind this difference is apparent from the QSS vibrational energy distribution for the two systems in Fig. 6.13. One can see that the vibrational energy distribution for the full O4 system is more depleted than that of the

3 interactions including all spin degeneracies of O2 + O2 interactions 4 interactions including all spin-spatial degeneracies of O2 + O interactions 112

(a) Ttr = 15000K. (b) Ttr = 12000K.

Figure 6.11: Ground electronic state vibrational energy distributions of O2 molecules in QSS given by O2 + O2 interactions.

full O3 system. This is because O2 +O interactions are more likely to result in exchange reactions than O2 + O2 interactions [126]. Since exchange reactions are more likely to redistribute internal energy [116–118], the O3 system is able to re-populate the high energy tail of the vibrational energy distribution to a larger extent. Hence, in the O3 system there is a greater population of vibrationally excited molecules which leads to the higher QSS dissociation rate constant. 113

Figure 6.12: Comparison of non-equilibrium dissociation due to O2 + O2 interactions and O2 + O interactions

(a) 8000 K . (b) 12000 K.

Figure 6.13: Comparison of vibrational energy distributions of the system in QSS, and the pre-dissociation distribution of molecules that dissociate due to O2 +O2, and O2 +O interactions. 114 6.3 Conclusions

Internal energy relaxation and dissociation is studied for O2 + O interactions using nine PESs that govern electronically adiabatic collisions of ground-electronic-state O2 with ground-electronic-state O. Characteristic vibrational excitation times for O2 + O collisions are calculated and found to vary by over an order of magnitude among the nine PESs. Specifically, it is observed that the vibrational excitation process is the slowest for interactions occurring on the first adiabatic quintet A00 (15A00) PES and the fastest for the interactions occurring on the first adiabatic singlet A0 (11A0) PES. When running simulations where all spatial degeneracies are accounted for, interactions with quintet spin lead to the slowest vibrational excitation and, interactions with singlet spin the fastest. These relaxation rate results are qualitatively consistent with scans of the nine PESs, some of which showed low-energy exchange reaction pathways that lead to efficient vibrational energy exchange. For the 11A0 state, which governs 1/27 of the collisions, significant differences between the results using the PES from Varga et al.[31] compared to contemporary results using the PES of Varandas and Pais[89] were observed. The PES developed by Varandas and Pais is found to have features not supported by new electronic structure calculations, and these differences explain the difference in vibrational relaxation rates computed. A number of recent studies model O2 + O interactions under hypersonic conditions [46, 48, 90–92] have been performed using the PES developed by Varandas and Pais [89].

Using DMS, it is possible to simulate the full O2 + O system using all nine PESs within an excitation and dissociation calculation. The characteristic vibrational excitation time for diatomic oxygen calculated from simulations that accounted for all nine spin-spatial degeneracies of ground state O2 + O interactions is weakly dependent on temperature, with the characteristic excitation time increasing slightly as temperature increases. Characteristic vibrational excitation times for O2 + O2 interactions are obtained. The characteristic excitation times obtained in this study using DMS compared well with experimental data. It was observed that vibrational energy excitation is faster 115 due to O2 + O interactions than O2 + O2 interactions. This was attributed to the fact that O2 +O interactions are more prone to performing exchange reactions [126]. During vibrational excitation, it is observed that the high-energy tail of the vibrational energy distribution gets populated ﬁrst, and then the system’s state distribution gradually progresses towards the QSS. This is consistent with the excitation mechanism observed for nitrogen in chapter5.

Non-equilibrium dissociation rate coefficients for the O2 + O system were obtained. The non-equilibrium dissociation rate coefficients obtained from simulations that incorporated all nine spin-spatial degeneracies of the O2 + O system agree well with experimental data. Furthermore, non-equilibrium dissociation rate coefficients for the O2 +O2 system are obtained for the singlet, quintet and the full O4 system. There is agreement between the dissociation rate coefficients obtained in this study and experimental data.

It was observed that the QSS dissociation rate coeﬃcients for O2 + O interactions were about two times higher than those obtained for the O2 + O2 system. Chapter 7

Internal Energy Relaxation

7.1 Introduction

As described in chapters5 and6, dissociation of nitrogen and oxygen molecules is strongly coupled to their internal energies. In chapters5 and6, Landau-Teller theory [18] was used to analyze vibrational energy excitation rates. This chapter presents a more general analysis of internal energy relaxations processes, also referred to as inelastic collision processes. Speciﬁcally, the vibrational energy relaxation is studied during QSS dissociation, where four types of interactions are deﬁned:

1. Molecule-molecule collisions: These interactions refer to the non-reactive elastic and inelastic collisions between two molecules.

2. Molecule-molecule exchange reactions: These interactions refer to the reactive collisions between two molecules where the atoms bound to the molecules are swapped during a trajectory integration. Here, the vibrational energy change is measured as the diﬀerence in the vibrational energy of the reactant and product molecules.

3. Molecule-atom collisions: These interactions refer to the non-reactive elastic and inelastic collisions between a molecule and an atom.

116 117 4. Molecule-atom exchange reactions: These interactions refer to the reactive collisions between a molecules and an atom where the reactant atom displaces one of the atoms bound in the reactant molecule during a trajectory integration. Here, the vibrational energy change is measured as the diﬀerence in the vibrational energy of the reactant and product molecules.

(a) Isothermal relaxation at T = 30000K . (b) Vibrational energy distribution in QSS for the example case is Fig. 7.1(a).

Figure 7.1: Example of QSS formation in nitrogen.

Recall that during QSS dissociation, the above stated internal energy relaxation processes are balanced by loss of internal energy due to dissociation. An example DMS result showing the QSS state and depleted vibrational energy distribution function for nitrogen are shown in Fig. 7.1.

7.2 Vibrational energy redistribution in nitrogen

7.2.1 Vibrational energy redistribution in QSS

This section uses the example case shown in Fig. 7.1, to illustrate the internal energy relaxation due to N2 +N and N2 +N2 interactions in QSS. The translation temperature for this case was maintained at Ttr = 30000K, both N2+N and N2+N2 interactions were 118 allowed during the isothermal relaxation simulation, and the ab-initio PES developed at the University of Minnesota was used for trajectory integration in DMS. Once the gas in the simulation reached QSS, vibrational energy of each interacting molecule was monitored by recording initial and final vibrational energy of molecules undergoing trajectory integration. This data was used to construct Fig. 7.2, which shows the vibrational energy redistribution in QSS. The abscissa shows the range of initial vibrational energy of a molecule, the ordinate shows the final vibrational energy of the molecule and the contour shows the number of transitions from an initial vibrational energy to a final vibrational energy normalized by total number of events (collisions or exchange reactions). This quantity is the conditional vibrational energy redistribution probability, this will be referred to simply as “probabilities” for convenience. It should be noted that these probabilities are calculated by considering the change in vibrational energy of every molecule, regardless of the rotational energy of the molecule. Hence, the probabilities presented in this chapter are averaged over the rotational energy mode

The contours for N2 + N collisions (Fig.7.2(a)) and N2 + N2 collisions (Fig.7.2(c)) peak along the line where vib−i = vib−f . Implying that the ﬁnal vibrational energy of the molecule is close to the initial vibrational energy. Figure 7.2(b) shows vibrational energy redistribution due to N2 +N exchange reactions and is also symmetric about the

vib−i = vib−f line but with a larger spread, showing that N2 + N exchange reactions are eﬃcient at redistributing vibrational energy allow for larger energy jumps. Figure

7.2(d) shows the vibrational energy redistribution due to N2 + N2 exchange reactions. Unlike the other vibrational energy redistribution mechanisms, vibrational energy redistribution due to N2 + N2 exchange reactions shows weak dependance of the ﬁnal vibrational energy on the initial vibrational energy of the molecule. Hence, N2 + N2 exchange reactions are the most eﬃcient at redistributing vibrational energy in a system.

However, since N2 + N2 exchange reactions are extremely rare [28] these interactions do not play a large role in vibrational energy redistribution in the system. Figure 7.3 shows curves extracted along a fixed final vibrational energy from the contours shown in Fig. 7.2 (cuts along the abscissa). It is evident that vibrational energy redistribution profiles form self similar curves that scale with the vibrational 119

(a) N + N2 collisions. (b) N + N2 exchange reaction.

Figure 7.2: Contour plots of vibrational energy redistribution due to N2+N and N2+N2 interactions at Ttr = 30000K energy distribution of the gas.

Figure 7.3(b) shows that N2+N exchange reactions result in probabilities following a banded structure a week maxima at vib−i = vib−f . For N2+N2 exchange reactions there is a banded structure of the conditional vibrational energy redistribution probability as well, with the probability of transitioning to a ﬁnal energy being directly correlated with the value of the QSS distribution at that energy . On the other hand, Fig 7.3(a) shows the conditional probabilities for change in vibrational energy due to N2 + N 120

(a) N2 + N collisions. (b) N2 + N exchange reaction.

Figure 7.3: Conditional vibrational energy transfer probabilities for fixed final vibrational energies due to N + N2 and N2 + N2 interactions. collisions. The probability distributions look like self-similar functions that spike sharply at vib−i = vib−f , implying that for most collisions the vibrational energy of the molecule remains the same as its pre-collision state. A similar trend is observed for the vibrational energy redistributions due to N2 + N2 collisions in Fig. 7.3(c). Figure 7.4 shows the probability curves extracted for a fixed initial energy from the contours shown in Fig. 7.2 (cuts along the ordinate). It can be seen that, like the contour cuts seen in Fig. 7.3, the conditional vibrational energy redistribution 121

(a) N2 + N collisions. (b) N2 + N exchange reaction.

Figure 7.4: Conditional vibrational energy transfer probabilities for ﬁxed initial vibrational energies due to N + N2 and N2 + N2 interactions. probabilities for a ﬁxed initial vibrational energy scale with the QSS distribution of the gas. Furthermore, a similar trend for vibrational energy redistribution due to N2 + N2 collisions, N2 + N collisions and N2 + N exchange reactions is observed. However, for

N2 + N2 exchange reactions, the vibrational energy redistribution curves in Fig. 7.4 are observed to have slightly more dependence on the QSS distribution in comparison to the conditional vibrational energy redistribution curves for a fixed final vibrational energy shown in Fig.7.3(d). 122 It should be noted that since the conditional vibrational energy redistribution probabilities scale with the vibrational energy distribution of the gas, the effect of vibrational energy redistribution at the low energy levels will dominate the overall behavior of the gas because that population of the low energy states is so high.

7.2.2 Normalized vibrational energy redistribution probabilities in QSS

The conditional probabilities of vibrational energy redistribution shown in the previous section show clear dependance on the vibrational energy distribution of the gas. Figure 7.5(a) shows the conditional probabilities of vibrational energy redistribution due to

N2 + N collisions shown in Fig. 7.4(a) normalized by the QSS vibrational energy distribution. By normalizing the conditional probability curves by the QSS vibrational energy distribution an identical curve representing the vibrational energy change in a molecule is obtained. This curve is displaced by the value of the initial vibrational energy level the curve was extracted from. This implies, that regardless of the initial vibrational energy of a molecule, the function describing the change in vibrational energy of the molecule is the same. Figure 7.5(b) shows that after being normalized by the vibrational energy distribution of the gas, the curves describing the probability of vibrational energy change of a molecule collapse into the same curve. This singular curve describes the probability distribution function (pdf) of change in vibrational energy due to N2 + N collisions. To eliminate the sampling scatter the pdfs obtained at all stations are averaged. Similar pdfs were obtained for N2 + N2 collisions N2 + N exchange reactions and N2 + N2 exchange reactions. The results are shown in Fig. 7.6. Figure 7.6 shows the probability distribution function of vibrational energy change due to N2 + N and N2 + N2 interactions at Ttr = 30000K, obtained by normalizing the conditional probability curves seen in Fig. 7.4. The green line shows the probability of change in vibrational energy per N2 +N2 exchange reaction. It can be seen that N2 +N2 exchange reactions are more likely to increase the vibrational energy of the molecule.

This is consistent with the study by Bender et al. [28] where N2 +N2 exchange reactions formed vibrationally excited molecules. The blue line shows the pdf for the change in vibrational energy per N2 + N exchange reaction. The N2 + N exchange reactions have 123

(a) Normalization of conditional probabilities of (b) Curve-colllapse of normalized conditional vibrational energy redistribution due to N2 + N probabilities of vibrational energy redistribution interactions by the QSS vibrational energy distri- due to N2 + N interactions. bution

Figure 7.5: Example normalization and curve-collapse of probabilities of vibrational energy redistribution.

a slight peak at ∆vib = 0 but with a bias of adding vibrational energy to the newly formed molecule (∆vib > 0). The purple and red curves show the probability change in vibrational energy per N2 +N2 and N2 +N collision. There is a sharp peak at ∆vib = 0 and a sharp decrease in probability as the variation in energy increases. Further, the spread of the probability distribution function associated with N2 +N2 collisions is wider in the immediate vicinity of ∆vib = 0. Implying the N2 + N2 collisions cause more change in vibrational energy than N2 + N collisions. Clearly, exchange reactions cause large change in vibrational energy, and hence aid the vibrational relaxation process. This is consistent with prior research [28, 64, 68, 116, 117]. Further analysis of these mechanisms of vibrational energy redistribution is done using isothermal relaxation calculations at Ttr = 10000K, 20000K, and 30000K. Figure

7.7(a) compares the pdf of vibrational energy change due to N2 + N exchange reactions at Ttr = 30000K and Ttr = 20000K. It is observed that the spread of the probability distribution function narrows down as temperature decreases. Figure 7.7(b) compares the probability distribution function of vibrational change due to N2 + N collisions 124

Figure 7.6: Comparison of normalized vibrational energy transfer probabilities for all nitrogen interaction events at Ttr = 30000K

at Ttr = 30000K, Ttr = 20000K, and Ttr = 10000K. Again, it is observed that the spread of the probability distribution function decreases and the peak (at ∆vib = 0) becomes more prominent a temperature decreases. Additionally, the probability distribution function corresponding to ∆vib > 0 drops oﬀ more gradually than the probability distribution function corresponding to ∆vib < 0. This is expected since the translational temperature is held ﬁxed at a high value in these isothermal calculations.

Vibrational energy redistribution due to N2 + N exchange reactions at Ttr = 10000K is not shown due to poor statistical convergence of the data.

Figure 7.8(a) shows the pdf for vibrational energy change per N2 + N2 exchange reaction at Ttr = 30000K and Ttr = 20000K. The probability distribution function increases monotonically, showing that N2 + N2 exchange reactions are more likely to produce vibrationally excited molecules. As the temperature decreases the slope of the probability distribution function increases. Figure 7.8(b) shows the probability distribution function of vibrational energy change due to N2 + N2 collisions at Ttr =

30000K, Ttr = 20000K, and Ttr = 10000K. As observed for N2+N collisions, the spread of the pdf narrows and the peak at ∆vib = 0 becomes more prominent with decrease 125

(a) N + N2 exchange reaction. (b) N + N2 collisions.

Figure 7.7: Comparison of vibrational energy transfer at diﬀerent temperatures for N + N2 interactions

(a) N2 + N2 exchange reaction. (b) N2 + N2 collisions.

Figure 7.8: Comparison of vibrational energy transfer at diﬀerent temperatures for N2 + N2 interactions.

in temperature. Also, the probability distribution function corresponding to ∆vib > 0 drops oﬀ more gradually than the probability distribution function corresponding to

∆vib < 0, which is attributed to the fact that the translational temperature is held ﬁxed 126 at a high value during these isothermal calculations. Vibrational energy redistribution due to N2 +N2 exchange reactions at Ttr = 10000K is not shown due to poor statistical convergence of the data.

Both N2 + N and N2 + N2 exchange reactions have a bias noted towards producing vibrationally excited molecules. This is because exchange reactions are associated with an energy barrier and only interactions with sufficient collision energy are able to perform to result in exchange. This energy is then redistributed during the interaction and can be distributes among internal energy modes. However, it should be noted that the probability distribution functions calculated in this study are normalized by the number of times the event occurred. Hence, any overall impact in vibrational energy redistribution exchange reactions should be scaled by the exchange reaction rate constant. Bender et al. [28] have shown that the N2 + N2 exchange reaction rate to be small. Hence the overall effect of N2 + N2 exchange reactions on vibrational energy redistribution will be negligible. However, Valentini et al. [64] have shown the N2 + N exchange reaction rate is not negligible and N2 + N exchange reactions have a significant effect on overall vibrational relaxation of nitrogen[64, 68]

7.2.3 Multi-quantum level jumps

This subsection discusses jumps in the quantum vibrational number caused by N2 + N and N2 +N2 interactions. Figure 7.9 shows the change in vibrational level due to N2 +N and N2 + N2 interactions as a function of the percentage of interactions that lead to change in vibrational levels. On the abscissa, to the right of the change in vibrational level of ten, represents the cumulative eﬀect of change in vibrational levels more than ten. Figure 7.9(a) shows the vibrational level change due to exchange reactions. The vibrational level change due to N2+N exchange reactions decreases with temperature. A change in ∆v = 1 accounts for ∼ 13% of all exchange reactions and the cumulative eﬀect of ∆v > 10 accounting for ∼ 24% of all exchange reactions Ttr = 30000K. In contrast at Ttr = 10000K, ∆v = 1 accounts for ∼ 27% and ∆v > 10 only accounts for ∼ 6% of exchange reactions. The trend for vibrational level change due to N2 + N2 exchange 127

(a) Exchange reaction. (b) Collisions

Figure 7.9: Vibrational level jumps as a percentage of interactions that lead to change in vibrational levels.

reactions is weekly dependent on temperature with ∼ 40% of all N2 + N2 exchange reactions causing change in vibrational levels of ∆v > 10. Hence, while considering vibrational level change due to exchange reactions, it is important to consider multi- quantum jumps in vibrational levels. Additionally, weak dependance of vibrational level change with temperature indicates that vibrational level change due to exchange reactions is largely a function of the energy barrier associated with the exchange reaction and only weakly depends on the temperature at which the reaction takes place. Figure 7.9(b) shows vibrational level change due to non-reactive inelastic collisions for N2 + N and N2 + N2. It can be seen that most inelastic collisions lead to a ∆v = 1, and there is a sharp decline in percentage of inelastic collisions causing a change in vibrational level ∆v > 1. Furthermore, there is a strong temperature dependent trend with the percentage of multi-quantum jumps in vibrational levels increasing with temperature. At Ttr = 30000K, ∼ 73% of N2 + N inelastic collisions that lead to a change in vibrational level caused a ∆v = 1, while only ∼ 3% collisions cause a ∆v > 10.

Of N2 + N2 inelastic collisions ∼ 69% cause the vibrational level to change by one and only ∼ 2% of collisions cause a change in vibrational level of greater than ten levels.

At Ttr = 10000K, ∼ 90% of N2 + N inelastic collisions, changed the vibrational level 128 by one and only ∼ 0.3% changed vibrational levels by more than ten levels. While for

N2 + N2 inelastic collisions, ∼ 97% changed the vibrational level by one and ∼ 0.02% changed vibrational level by more than ten levels.

7.3 Vibrational energy redistribution in oxygen

This section presents preliminary results for vibrational energy redistribution due to 1 O2 + O interactions. The full O3 system as described in chapter6 was used to study the vibrational energy redistribution process under QSS conditions.

7.3.1 Vibrational energy redistribution in QSS

Figure 7.10 shows the conditional probability of vibrational energy redistribution due to O2 + O interactions during QSS conditions where Ttr = 12000K. This is the same QSS condition shown in Fig. 6.1. Similarity is observed between vibrational energy redistribution in oxygen due to O2 + O interactions and the results shown for N2 + N interactions in Fig. 7.2. It is observed that like N2 + N collisions, vibrational energy redistribution due to O2 + O collisions is fairly localized (vib−i = vib−f ). Similarly, for O2 + O exchange reactions some symmetry is observed about vib−i = vib−f , but generally, O2 + O exchange reactions allow for larger changes in vibrational energy. Figure 7.11 shows the curves extracted along a ﬁxed initial vibrational energy from the contours shown in Fig. 7.10. As seen in Fig. 7.4, the conditional vibrational energy redistribution curves taken at various initial vibrational energy states are self similar and scale with the vibrational energy distribution of the gas in QSS. Furthermore, the overall shape of the conditional vibrational energy redistribution functions due to

O2 + O collisions and O2 + O exchange reactions is similar to the those seen for N2 + N collisions and N2 + N exchange reactions shown in Fig. 7.4. Similar to the analysis for nitrogen, the conditional vibrational energy redistribution probabilities were used to construct probability distribution functions for change in vibrational energy due to

O2 + O collisions and O2 + O exchange reactions.

1 Simulations using all nine PESs for O2 + O interactions 129

(a) O2 + O collisions. (b) O2 + O exchange reaction.

Figure 7.10: Contour plots of vibrational energy redistribution due to O2+O interactions at Ttr = 12000K

(a) O2 + O collisions. (b) O2 + O exchange reaction.

Figure 7.11: Conditional vibrational energy transfer probabilities for ﬁxed initial vibrational energies due to O2 + O interactions.

7.3.2 Normalized vibrational energy redistribution probabilities in QSS

Figure 7.12 shows the comparison between probability distribution functions for change in vibrational energy due to O2 +O collisions (solid lines) and O2 +O exchange reactions

(dashed lines) at Ttr = 12000K (blue curves) and Ttr = 5000K (red curves). It can 130

Figure 7.12: Probability distribution functions for vibrational energy change due to O2 + O collisions and O2 + O exchange reactions at Ttr = 12000K and Ttr = 5000K.

be seen that the pdf for O2 + O collisions peaks at ∆vib = 0 and becomes more prominent with decrease in temperature. Also, the probability distribution function corresponding to ∆vib > 0 drops oﬀ more gradually than the probability distribution function corresponding to ∆vib < 0, which is attributed to the fact that the translational temperature is held ﬁxed at a high value during these isothermal calculations. For

O2 + O exchange reactions the pdf describing change in vibrational energy is broader than that for collisions. Implying that exchange reactions are once again more eﬃcient in redistributing vibrational energy than non-reactive collisions. These pdfs are very similar to those for shown for nitrogen in the previous section and the same functional form could be used to parameterize the pdfs characterizing the change in vibrational energy due to inelastic processes for the two gases.

7.4 Conclusions

This chapter presented vibrational energy redistribution probability distribution functions for N2 + N collisions, N2 + N exchange reactions, N2 + N2 collisions, N2 + N2 131 exchange reactions, O2 +O collisions and O2 +O exchange reactions obtained from from DMS calculations simulating the gas in QSS. It is observed that non-reactive collisions have narrow probability distribution functions for vibrational energy redistribution, centered at ∆vib = 0eV . The probability distribution functions for collisions are biased to produce molecules with higher vibrational energy as the probability distribution function at ∆vib > 0 drops off more gradually than the probability distribution function at ∆vib < 0. It is shown that the probability distribution functions for N2 + N and O2 + O exchange reactions have a slight peak at ∆vib = 0eV but allow for large changes in the vibrational energy of the molecule, with a bias towards making more vibrationally excited molecules. Further- more, the probability distribution function for vibrational energy change due to N2 +N2 exchange reactions increases monotonically, showing that N2 + N2 exchange reactions are more likely to produce vibrationally excited molecules. Finally, it was observed that the width of all the probability distribution functions decreases as temperature decreases. There are three degrees of asymmetry in vibrational energy redistribution trends during QSS. First, all vibrational energy change pdfs (atom-molecule collision, molecule- molecule collision, atom-molecule exchange reaction, molecule-molecule exchange reaction) are asymmetric and are biased towards producing vibrationally excited molecules. Second, the vibrational energy redistribution pdfs are truncated at lower energies (since a molecule cannot have negative vibrational energy), meaning that molecules with lower vibrational energy are more likely to increase vibrational energy. Third, all the vibrational energy change pdfs scale with the distribution of the gas. This amplifies the first two asymmetries for molecules in the lower vibrational energy end of the population and drives the production of vibrationally excited molecules. This chapter also characterized vibrational quantum level jumps due to atom-molecule and molecule-molecule interactions in nitrogen. It was shown that exchange reactions allow for large changes in the vibrational quantum levels. Hence, accounting for multi- quantum vibrational level jumps is important while treating exchange reactions. On the other hand, for inelastic collisions, single vibrational level jumps dominate. 132

The pdfs characterizing vibrational energy change due to N2 + N interactions and

O2 +O interactions are found to be very similar and could potentially be modeled using the same functional form. Chapter 8

Summary

8.1 Summary of contributions

In this thesis, a recently developed numerical method, called Direct Molecular Simula- tion (DMS), was used to study internal energy excitation and nonequilibrium dissociation in high temperature nitrogen and oxygen gases. The DMS method is shown to be an acceleration technique for Molecular Dynamics (MD) simulations of dilute gases, and therefore, the evolution of a gas system can be simulated using a potential energy surface (PES), or set of PESs, as the only model input. Therefore DMS acts as a multiscale method that links computational chemistry directly to macroscopic aerothermodynamics. Results from the first-principles calculations, presented in this thesis, can be used to develop nonequilibrium thermochemistry models for use in CFD and DSMC solvers applied to hypersonic flows. A number of contributions to this research field were described in this thesis, and are summarized in this section.

1. DMS Code Development

(a) The DMS code was extended to handle an arbitrary set of PESs as input. This enables the simulation of gas mixtures and also enables the simulation of all possible spin-state interactions for a given species pair. As an example,

133 134 DMS calculations were carried out using 12 PESs within a simulation of a full oxygen system. Such a ﬁrst-principles simulation (including diatom- diatom interactions with all possible spin-states) is intractable by master- equation analysis due to the vast number of possible energy state transitions [58, 65, 68].

2. Potential Energy Surface Analysis

(a) The O2+O PES of Varandas and Pais, widely used in the aerothermodynamic community, was found to have spurious (un-physical) energy maxima that inhibit exchange reactions. As a result, vibrational relaxation times predicted using the Varandas and Pais PES were found to be more than one order of

magnitude lower than predicted by new accurate PESs for the O2 +O system [65].

(b) Two ab-intio PESs for nitrogen (N2 +N2 and N2 +N interactions) developed independently at the University of Minnesota and NASA Ames Research Center were found to have very similar topology, however, had diﬀerent long- range behaviors for the diatomic portion of the PESs [67, 68]

3. Rovibrational Coupling in Normal Shock Waves

(a) DMS simulations of one-dimensional shock waves revealed rovibrational coupling eﬀects, not captured by standard Landau-Teller and Jeans equations. Coupling eﬀects on dissociation are estimated to be minimal, however, it was shown that the degree of coupling increases with the anharmonicity of the diatomic potential [23].

4. DMS Results for Nitrogen

(a) Previous DMS calculations for nitrogen using the PESs from Minnesota were repeated using the PES from NASA Ames Research Center. The internal energy excitation and non-equilibrium dissociation results, predicted by both sets of PESs, where in excellent quantitative agreement down to the level 135 of the rotational and vibrational energy distribution functions. Such a comparison is not feasible using master-equation analysis and was enabled by the DMS method. The close agreement found using two independently constructed PESs lends conﬁdence to the use of computational chemistry to understand hypersonic ﬂow physics [67, 68].

(b) DMS results for nitrogen were compared to state-resolved, master-equation

predictions for the N2 + N system, in order to verify both numerical approaches. DMS results were also compared to binned master-equation pre-

dictions for the N2 + N2 system in order to determine the most accurate binning technique. It was found that while vibrational binning was most accurate during initial excitation, it was inaccurate during QSS dissociation, where rovibrational binning was most accurate [57, 69].

5. DMS Results for Oxygen

(a) Benchmark DMS data was produced for the O2 + O system using the 9 PESs developed by Varga et al. [31] This was the ﬁrst time that all spin- spatially degenerate states were accounted for (previous research only con-

sidered 1/27th of real O2 + O interactions). Vibrational relaxation time constants and non-equilibrium dissociation rate constants were determined and found to agree within the uncertainty of experimental shock tube data [65].

(b) DMS calculations were performed for each spin-spatial degenerate state in- dividually and order-of-magnitude diﬀerences in vibrational relaxation times were found. This is due to certain spin-states being more (or less) favor- able for exchange reactions, and the results indicate that accounting for all spin-spatially degenerate states is indeed required to study oxygen gas [65].

(c) Benchmark DMS data was produced for the O2 + O2 system using the three PESs developed by Paukku et al. [29, 30] This was the ﬁrst time that all spin generate states were accounted for. Results showed minimal variation 136 between the three PESs. Vibrational relaxation time constants and non- equilibrium dissociation rate constants were determined and found to agree within the uncertainty of experimental shock tube data [66].

(d) Overall, the DMS results for oxygen (O2 + O and O2 + O2 interctions) were

qualitatively very similar to the results for nitrogen (N2+N, N2+N2). Specif- ically, internal energy relaxation is found to be much faster for atom-diatom collisions, compared to diatom-diatom collisions, due to frequent exchange reactions. Dissociation rates are found to be higher when signiﬁcant atomic species are present. Most importantly, this higher dissociation rate is found to be an indirect eﬀect due to the increased vibrational excitation rate, which repopulates the high-energy tail of the vibrational energy distribution, leading to more dissociation. Therefore, non-equilibrium dissociation rates are found to be sensitive to non-Boltzmann internal energy distribution functions and internal energy exchange is the dominant physical mechanism that determines such non-Boltzmann distributions. The similarities between nitrogen and oxygen systems, indicates that a common model may work well for both gases.

6. Internal Energy Relaxation

(a) DMS calculations were post-processed to analyze the probability distribution functions (pdfs) for the change in vibrational energy during various types of collisions. It was found that the pdfs for inelastic collisions were self- similar when scaled with the underlying vibrational energy distribution in

the gas. The pdfs were narrow with a peak at ∆vib = 0, implying that only small changes in vibrational energy per collision result from typical inelastic collisions. For exchange reaction collisions, the pdfs were also self similar. However, the pdfs were broad, implying that large changes in vibrational energy per collision are probable for exchange collisions. 137 (b) Results for oxygen were very similar to the results for nitrogen. The self- similar pdfs imply that a simple functional form might capture all state- resolved transition probabilities in both nitrogen and oxygen, however, the functional form should be diﬀerentiated between regular inelastic collisions and exchange collisions [118].

The above contributions to the understanding of nitrogen and oxygen under hypersonic ﬂow conditions may be useful in developing non-equilibrium aerothermodynamics models for use in CFD and DSMC solvers for hypersonic ﬂows. Following is the list of journal articles resulting from this work

1. Macdonald, R.L., Grover, M.S., Schwartzentruber, T.E. and Panesi, M., 2018. Construction of a coarse-grain quasi-classical trajectory method. II. Comparison against the direct molecular simulation method. The Journal of Chemical Physics, 148(5), p.054310.

2. Schwartzentruber, T.E., Grover, M.S. and Valentini, P., 2017. Direct molecular simulation of nonequilibrium dilute gases. Journal of Thermophysics and Heat Transfer, pp.1-12.

3. Jaﬀe R.L., Grover M.S., Venturi S., Schwenke D.W., Valenti P., Schwartzen- truber T.E., Panesi, Comparison of quantum mechanical and empirical potential energy surfaces and computed rate coeﬃcients for N2 dissociation. Journal of Thermophysics and Heat Transfer, submitted 2017, [under review].

Following is the list of refereed conference publications resulting from this work:

1. Grover M.S., Schwartzentruber, T.E., Varga, Z. and Truhlar, D.G., 2018. Dy- namics of vibrational energy excitation and dissociation in oxygen from direct molecular simulation. In 2018 AIAA Aerospace Sciences Meeting (p. 0238).

2. Grover M.S. and Schwartzentruber, T.E., 2018. Redistribution of Vibrational Energy: Mechanisms and Transition Probabilities. In 2018 AIAA Aerospace Sci- ences Meeting (p. 1231). 138 3. Grover M.S.. and Schwartzentruber, T.E., 2017. Internal energy relaxation and dissociation in molecular oxygen using direct molecular simulation. In 47th AIAA Thermophysics Conference (p. 3488).

4. Grover M.S., Schwartzentruber, T.E. and Jaﬀe, R.L., 2017. Dissociation and internal excitation of molecular nitrogen due to N2-N collisions using direct molecular simulation. In 55th AIAA Aerospace Sciences Meeting (p. 0660).

5. Grover M.S., Valentini, P. and Schwartzentruber, T.E., 2015. Coupled rotational- vibrational excitation in shock waves using trajectory-based direct simulation Monte Carlo. In 53rd AIAA Aerospace Sciences Meeting (p. 1656).

6. Jaﬀe, R.L., Schwenke, D.W., Grover M.S., Valentini, P., Schwartzentruber, T.E., Venturi, S. and Panesi, M., 2016. Comparison of quantum mechanical and empirical potential energy surfaces and computed rate coeﬃcients for N2 dissociation. In 54th AIAA Aerospace Sciences Meeting (p. 0503).

7. Macdonald, R.L., Grover M.S., Schwartzentruber, T.E. and Panesi, M., 2017. Coarse grain modeling and direct molecular simulation of nitrogen dissociation. In 47th AIAA Thermophysics Conference (p. 3165).

8. Macdonald, R.L., Grover M.S., Schwartzentruber, T.E. and Panesi, M., 2018. State-to-State and Direct Molecular Simulation Study of energy transfer and dissociation in nitrogen mixtures. In 2018 AIAA Aerospace Sciences Meeting (p. 0239).

9. Chaudhry, R.S., Grover M.S., Bender, J.D., Schwartzentruber, T.E. and Can- dler, G.V., 2018. Quasiclassical Trajectory Analysis of Oxygen Dissociation via O2, O, and N2. In 2018 AIAA Aerospace Sciences Meeting (p. 0237). Bibliography

[1] J. H. Grinstead, M. C. Wilder, D. C. Reda, C. J. Cornelison, B. A. Cruden, and D. W. Bogdanoﬀ. Shock tube and ballistic range facilities at nasa ames research center. 2010.

[2] J. S. Robinson, K. E. Wurster, and J. C. Mills. Entry trajectory and aeroheating environment deﬁnition for capsule-shaped vehicles. Journal of Spacecraft and Rockets, 46(1):74, 2009.

[3] C. Park. Assessment of a two-temperature kinetic model for dissociating and weakly ionizing nitrogen. Journal of Thermophysics and Heat Transfer, 2(1): 8–16, 1988.

[4] C. Park. Review of chemical-kinetic problems of future nasa missions, i: Earth entries. Journal of Thermophysics and Heat transfer, 7(3):385–398, 1993.

[5] C. Park, J. T. Howe, R. L. Jaﬀe, and G. V. Candler. Review of chemical-kinetic problems of future nasa missions, ii: Mars entries. Journal of Thermophysics and Heat transfer, 8(1):9–23, 1994.

[6] C. Park. Assessment of two-temperature kinetic model for ionizing air. Journal of Thermophysics and Heat Transfer, 3(3):233–244, 1989.

[7] J. P. Appleton, M. Steinberg, and D. J. Liquornik. Shock-tube study of nitrogen dissociation using vacuum-ultraviolet light absorption. The Journal of Chemical Physics, 48(2):599–608, 1968.

139 140 [8] R. K. Hanson and D. Baganoﬀ. Shock-tube study of nitrogen dissociation rates using pressure measurements. AIAA Journal, 10(2):211–215, 1972.

[9] D. J. Kewley and H. G. Hornung. Free-piston shock-tube study of nitrogen dissociation. Chemical Physics Letters, 25(4):531–536, 1974.

[10] S. Byron. Shock-tube measurement of the rate of dissociation of nitrogen. The Journal of Chemical Physics, 44(4):1378–1388, 1966.

[11] J. D. Anderson Jr. Hypersonic and High-Temperature Gas Dynamics, AIAA Edu- cation Series, 2nd Ed. American Institute of Aeronautics and Astronautics, Inc., Reston, VA,, 2006.

[12] W. G. Vincenti and C. H. Kruger Jr. Introduction to Physical Gas Dynamics. Krieger Publishing Company, Malabar, FL, 2002.

[13] D. A. McQuarrie. Statistical Mechanics. University Science Books, Mill Valley, CA, 2000.

[14] R. D. Levine. Molecular Reaction Dynamics. Cambridge University Press, Cam- bridge, England, 2005.

[15] C. F. Curtiss J. O. Hirshfelder and R. B. Bird. Molecular Theory of Gases and Liquids, Structure of Matter Series. John Wiley and Sons Inc., Hoboken, NJ, 1964.

[16] G. A. Bird. Molecular Gas Dynamics and Simulation of Gas Flows. Cambridge University Press, Cambridge, England, 1994.

[17] I. D. Boyd and T. E. Schwartzentruber. Nonequilibrium Gas Dynamics and Molec- ular Simulation. Cambridge University Press, New York, NY, 2017.

[18] L. Landau and E. Teller. Zur theorie der schalldispersion. Physik Z. Sowjetunion, 10(34), 1936. 141 [19] R. C. Millikan and D. R. White. Systematics of vibrational relaxation. The Journal of chemical physics, 39(12):3209–3213, 1963.

[20] M. Panesi, R. L. Jaﬀe, D. W. Schwenke, and T. E. Magin. Rovibrational internal energy transfer and dissociation of n 2 (1 σ g+)- n (4 s u) system in hypersonic ﬂows. The Journal of chemical physics, 138(4):044312, 2013.

[21] M. Panesi, A. Munafò,T. E. Magin, and R. L. Jaffe. Nonequilibrium shock-heated nitrogen flows using a rovibrational state-to-state method. Physical Review E, 90 (1):013009, 2014.

[22] P. Valentini, P. Norman, C. Zhang, and T. E. Schwartzentruber. Rovibra- tional coupling in molecular nitrogen at high temperature: An atomic-level study. Physics of Fluids, 26(5):056103, 2014.

[23] M. S. Grover, P. Valentini, and T. E. Schwartzentruber. Coupled rotational- vibrational excitation in shock waves using trajectory-based direct simulation monte carlo. AIAA SciTech, Kissimmee, FL, 2015.

[24] D. G. Truhlar and N. C. Blais. Test of the linear sum rule for vibrational energy transfer by trajectory calculations. The Journal of Chemical Physics, 77(5):2430– 2431, 1982.

[25] R. Jaﬀe, D. Schwenke, and G. Chaban. Vibrational and rotational excitation and dissociation in n2-n2 collisions from accurate theoretical calculations. In 10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, volume 4517, 2010.

[26] G. Chaban, R. Jaﬀe, D. W. Schwenke, and W. Huo. Dissociation cross sections and rate coeﬃcients for nitrogen from accurate theoretical calculations. AIAA paper, 1209:2008, 2008.

[27] Y. Paukku, K. R. Yang, Z. Varga, and D. G. Truhlar. Global ab initio ground- state potential energy surface of n4. The Journal of chemical physics, 139(4): 044309, 2013. 142 [28] J. D. Bender, P. Valentini, I. Nompelis, Y. Paukku, Z. Varga, D. G. Truhlar, T. E. Schwartzentruber, and G. V. Candler. An improved potential energy surface and multi-temperature quasiclassical trajectory calculations of n2+ n2 dissociation reactions. The Journal of chemical physics, 143(5):054304, 2015.

[29] Y. Paukku, K. R. Yang, Z. Varga, G. Song, J. D. Bender, and D. G. Truhlar. Potential energy surfaces of quintet and singlet o4. The Journal of chemical physics, 147(3):034301, 2017.

[30] Y. Paukku, Z. Varga, and D. G. Truhlar. Potential energy surface of triplet o4. The Journal of chemical physics, submitted, 2017.

[31] Z. Varga, Y. Paukku, and D. G. Truhlar. Potential energy surfaces for o+ o2 collisions. The Journal of chemical physics, 147(15):154312, 2017.

[32] J. D. Bender. Multiscale Computational Analysis of Nitrogen and Oxygen Gas- Phase Thermochemistry in Hypersonic Flows. PhD thesis, University of Min- nesota, Minnesota, MN, 2016.

[33] D. G. Truhlar and J. T. Muckerman. Atom-Molecule Collision Theory: A Guide for the Experimentalist. Plenum Press, New York, NY, 1979.

[34] Gert Due Billing and ER Fisher. Vv and vt rate coeﬃcients in n2 by a quantum- classical model. Chemical Physics, 43(3):395–401, 1979.

[35] A. Fern´andez-Ramos,J. A. Miller, S. J. Klippenstein, and D. G. Truhlar. Modeling the kinetics of bimolecular reactions. Chemical reviews, 106(11):4518–4584, 2006.

[36] Mario Capitelli, Carlos M Ferreira, Boris F Gordiets, and Alexey I Osipov. Plasma kinetics in atmospheric gases, volume 31. Springer Science & Business Media, 2013.

[37] G. Colonna and M. Capitelli. Self-consistent model of chemical, vibrational, electron kinetics in nozzle expansion. Journal of thermophysics and heat transfer, 15 (3):308–316, 2001. 143 [38] M. Capitelli, I. Armenise, D. Bruno, M. Cacciatore, R. Celiberto, G. Colonna, O. De Pascale, P. Diomede, F. Esposito, C. Gorse, et al. Non-equilibrium plasma kinetics: a state-to-state approach. Plasma Sources Science and Technology, 16 (1):S30, 2007.

[39] E. Josyula and W. F. Bailey. Vibration-dissociation coupling using master equations in nonequilibrium hypersonic blunt-body ﬂow. Journal of thermophysics and heat transfer, 15(2):157–167, 2001.

[40] A. Bultel, B. van Ootegem, A. Bourdon, and P. Vervisch. Inﬂuence of ar 2+ in an argon collisional-radiative model. Physical review E, 65(4):046406, 2002.

[41] G. Colonna, L. Daniela Pietanza, and M. Capitelli. Recombination-assisted nitrogen dissociation rates under nonequilibrium conditions. Journal of Thermophysics and Heat Transfer, 22(3):399–406, 2008.

[42] M. Panesi, T.E. Magin, A. Bourdon, A/ Bultel, and O. Chazot. Electronic excitation of atoms and molecules for the ﬁre ii ﬂight experiment. Journal of Ther- mophysics and Heat Transfer, 25(3):361–373, 2011.

[43] A. Munaf`o,A. Lani, A. Bultel, and M. Panesi. Modeling of non-equilibrium phenomena in expanding ﬂows by means of a collisional-radiative model. Physics of Plasmas, 20(7):073501, 2013.

[44] J. G. Kim and I. D. Boyd. Monte carlo simulation of nitrogen dissociation based on state-resolved cross sections. Physics of Fluids, 26(1):012006, 2014.

[45] K. L. Heritier, R. L. Jaﬀe, V. Laporta, and M. Panesi. Energy transfer models in nitrogen plasmas: Analysis of n 2 (x σ g+ 1)–n (4 s u)–e- interaction. The Journal of chemical physics, 141(18):184302, 2014.

[46] T. J. Wilson and K. A. Stephani. State-to-state vibrational energy modeling in dsmc using quasiclassical trajectory calculations for o+ o2. In 46th AIAA Thermophysics Conference, page 3839, 2016. 144 [47] J. G. Kim and I. D. Boyd. State-resolved master equation analysis of thermo- chemical nonequilibrium of nitrogen. Chemical Physics, 415:237–246, 2013.

[48] D. A. Andrienko and I. D. Boyd. Rovibrational energy transfer and dissociation in o2–o collisions. The Journal of chemical physics, 144(10):104301, 2016.

[49] G. Colonna, I. Armenise, D. Bruno, and Ma. Capitelli. Reduction of state-to-state kinetics to macroscopic models in hypersonic ﬂows. Journal of thermophysics and heat transfer, 20(3):477–486, 2006.

[50] Y. Liu, M. Vinokur, M. Panesi, and T. Magin. A multi-group maximum entropy model for thermo-chemical nonequilibrium. AIAA paper, 4332:2010, 2010.

[51] Y. Liu, M. Panesi, A. Sahai, and M. Vinokur. General multi-group macroscopic modeling for thermo-chemical non-equilibrium gas mixtures. The Journal of chemical physics, 142(13):134109, 2015.

[52] M. Panesi and A. Lani. Collisional radiative coarse-grain model for ionization in air. Physics of Fluids, 25(5):057101, 2013.

[53] A. Munaf`oand T. E. Magin. Modeling of stagnation-line nonequilibrium ﬂows by means of quantum based collisional models. Physics of Fluids, 26(9):097102, 2014.

[54] E. V. Kustova and E. A. Nagnibeda. Kinetic model for multi-temperature ﬂows of reacting carbon dioxide mixture. Chemical Physics, 398:111–117, 2012.

[55] A. Guy, A. Bourdon, and M. Perrin. Consistent multi-internal-temperatures models for nonequilibrium nozzle ﬂows. Chemical Physics, 420:15–24, 2013.

[56] R. L. Macdonald, R. L. Jaﬀe, D. W. Schwenke, A. Munaf`o,and M. Panesi. Ab initio based rovibrational grouping model for n2 (1σ+ g)-n2 (1σ+ g) energy transfer and dissociation. In 47th AIAA Thermophysics Conference, page 3164, 2017. 145 [57] R. L. Macdonald, M. S. Grover, T. E. Schwartzentruber, and M. Panesi. Coarse grain modeling and direct molecular simulation of nitrogen dissociation. The Journal of chemical physics, Accepted, 2018.

[58] T. E. Schwartzentruber, M. S. Grover, and P. Valentini. Direct molecular simulation of nonequilibrium dilute gases. Journal of Thermophysics and Heat Transfer, pages 1–12, 2017.

[59] 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. Physics of Fluids A: Fluid Dynamics, 3(12):3038–3045, 1991.

[60] K. Koura. Monte carlo direct simulation of rotational relaxation of diatomic molecules using classical trajectory calculations: Nitrogen shock wave. Physics of Fluids, 9(11):3543–3549, 1997.

[61] K. Koura. Monte carlo direct simulation of rotational relaxation of nitrogen through high total temperature shock waves using classical trajectory calculations. Physics of Fluids, 10(10):2689–2691, 1998.

[62] P. Norman, P. Valentini, and T. Schwartzentruber. Gpu-accelerated classical trajectory calculation direct simulation monte carlo applied to shock waves. Journal of Computational Physics, 247:153–167, 2013.

[63] 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. Physics of Fluids, 27(8):086102, 2015.

[64] P. Valentini, T. E. Schwartzentruber, J. D. Bender, and G. V. Candler. Dynamics of nitrogen dissociation from direct molecular simulation. Physical Review Fluids, 1(4):043402, 2016.

[65] M. S. Grover, T. E. Schwartzentruber, Z. Varga, and D. G. Truhlar. Dynamics of vibrational energy excitation and dissociation in oxygen from direct molecular simulation. In 2018 AIAA Aerospace Sciences Meeting, page 0238, 2018. 146 [66] M. S. Grover and T. E. Schwartzentruber. Internal energy relaxation and dissociation in molecular oxygen using direct molecular simulation. In 47th AIAA Thermophysics Conference, page 3488, 2017.

[67] 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. In 54th AIAA Aerospace Sciences Meeting, page 0503, 2016.

[68] M. S. Grover, T. E. Schwartzentruber, and R. L. Jaﬀe. Dissociation and internal excitation of molecular nitrogen due to n2-n collisions using direct molecular simulation. In 55th AIAA Aerospace Sciences Meeting, page 0660, 2017.

[69] R. L. Macdonald, M. S. Grover, T. E. Schwartzentruber, and M. Panesi. State-to- state and direct molecular simulation study of energy transfer and dissociation in nitrogen mixtures. In 2018 AIAA Aerospace Sciences Meeting, page 0239, 2018.

[70] M. S. H. Ling and M. Rigby. Towards an intermolecular potential for nitrogen. Molecular Physics, 51(4):855–882, 1984.

[71] H. Eyring and M. Polanyi. Uber einfache gasreaktionen. Z Phys Chem Abt B, 12: 279–311, 1931.

[72] P. London. Sommerfeld festschrift (1928). Z. Electrochemie, 35:552, 1929.

[73] S. Sato. On a new method of drawing the potential energy surface. The Journal of chemical physics, 23(3):592–593, 1955.

[74] S. Sato. Potential energy surface of the system of three atoms. The Journal of Chemical Physics, 23(12):2465–2466, 1955.

[75] M. Karplus, R. N. Porter, and R. D. Sharma. Exchange reactions with activation energy. i. simple barrier potential for (h, h2). The Journal of Chemical Physics, 43(9):3259–3287, 1965.

[76] I. N. Levine. Quantum Chemistry. Prentice Hall, Upper Saddle River, NJ, 2000. 147 [77] F. A. Cotton. Chemical applications of group theory. John Wiley & Sons, 2008.

[78] H. J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Sch¨utz,P. Celani, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, et al. Molpro, version 2012.1, a package of ab initio programs, cardiﬀ, uk, 2012.

[79] T. H. Dunning Jr. Gaussian basis sets for use in correlated molecular calculations. i. the atoms boron through neon and hydrogen. The Journal of chemical physics, 90(2):1007–1023, 1989.

[80] R. A. Kendall, T. H. Dunning Jr, and R. J. Harrison. Electron aﬃnities of the ﬁrst-row atoms revisited. systematic basis sets and wave functions. The Journal of chemical physics, 96(9):6796–6806, 1992.

[81] G. D. Purvis III and R. J. Bartlett. A full coupled-cluster singles and doubles model: The inclusion of disconnected triples. The Journal of Chemical Physics, 76(4):1910–1918, 1982.

[82] R. J. Gdanitz and R. Ahlrichs. The averaged coupled-pair functional (acpf): A size-extensive modiﬁcation of mr ci (sd). Chemical physics letters, 143(5):413–420, 1988.

[83] B. O. Roos, P. R. Taylor, P. E. M. Si, et al. A complete active space scf method (casscf) using a density matrix formulated super-ci approach. Chemical Physics, 48(2):157–173, 1980.

[84] R. J. Le Roy, Y. Huang, and C. Jary. An accurate analytic potential function for ground-state n 2 from a direct-potential-ﬁt analysis of spectroscopic data. The Journal of chemical physics, 125(16):164310, 2006.

[85] E. Papajak, H. R. Leverentz, J. Zheng, and D. G. Truhlar. Efficient diffuse basis sets: cc-pv x z+ and maug-cc-pv x z. Journal of chemical theory and computation, 5(5):1197–1202, 2009. 148 [86] K. Andersson, P. Malmqvist, and B. O. Roos. Second-order perturbation theory with a complete active space self-consistent field reference function. The Journal of chemical physics, 96(2):1218–1226, 1992.

[87] H. J. Werner. Third-order multireference perturbation theory the caspt3 method. Molecular Physics, 89(2):645–661, 1996.

[88] J. Finley, P. Malmqvist, B. O. Roos, and L. Serrano-Andr´es. The multi-state caspt2 method. Chemical physics letters, 288(2-4):299–306, 1998.

[89] A. J. C. Varandas and A. A. C. C. Pais. A realistic double many-body expansion (dmbe) potential energy surface for ground-state o3 from a multiproperty ﬁt to ab initio calculations, and to experimental spectroscopic, inelastic scattering, and kinetic isotope thermal rate data. Molecular Physics, 65(4):843–860, 1988.

[90] M. Kulakhmetov, M. Gallis, and A. Alexeenko. Eﬀect of o 2+ o ab initio and morse additive pairwise potentials on dissociation and relaxation rates for nonequilibrium ﬂow calculations. Physics of Fluids, 27(8):087104, 2015.

[91] F. Esposito and M. Capitelli. Quasiclassical trajectory calculations of vibrationally speciﬁc dissociation cross-sections and rate constants for the reaction o+ o2 (v) 3o. Chemical physics letters, 364(1-2):180–187, 2002.

[92]

[93] S. Shih, R. J. Buenker, and S. D. Peyerimhoﬀ. Theoretical investigation of the cyclic conformer of ozone. Chemical Physics Letters, 28(4):463–470, 1974.

[94] T. Shiozaki, W. Gy˝orﬀy, P. Celani, and H. J. Werner. Communication: Extended multi-state complete active space second-order perturbation theory: Energy and nuclear gradients, 2011.

[95] H. J. Werner and P. J. Knowles. A comparison of variational and non-variational internally contracted multiconfiguration-reference configuration interaction calculations. Theoretica chimica acta, 78(3):175–187, 1990. 149 [96] T. E. Schwartzentruber and I. D. Boyd. Progress and future prospects for particle- based simulation of hypersonic flow. Progress in Aerospace Sciences, 72:66–79, 2015.

[97] D. Frenkel and B. Smit. Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, San Diego, CA, 2002.

[98] P. E. Norman. Atomistic Simulation of Nonequilibrium Phenomena in Hypersonic Flows. PhD thesis, University of Minnesota, Minnesota, MN, 2013.

[99] R. L. Jaﬀe. The calculation of high-temperature equilibrium and nonequilibrium speciﬁc heat data for n2, o2 and no. In 22nd Thermophysics Conference, page 1633, 1987.

[100] D. Gao, C. Zhang, and T. E. Schwartzentruber. Particle simulations of planetary probe ﬂows employing automated mesh reﬁnement. Journal of Spacecraft and Rockets, 48(3):397–405, 2011.

[101] I. Nompelis and T. E. Schwartzentruber. A parallel implementation strategy for multi-level cartesian grid based dsmc codes. In 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, page 1204, 2013.

[102] P. Valentini and T. E. Schwartzentruber. A combined event-driven/time-driven molecular dynamics algorithm for the simulation of shock waves in rareﬁed gases. Journal of Computational Physics, 228(23):8766–8778, 2009.

[103] P. Valentini and T. E. Schwartzentruber. Large-scale molecular dynamics simulations of normal shock waves in dilute argon. Physics of Fluids, 21(6):066101, 2009.

[104] P. Valentini, P. A. Tump, C. Zhang, and T. E. Schwartzentruber. Molecular dynamics simulations of shock waves in mixtures of noble gases. Journal of Ther- mophysics and Heat Transfer, 27(2):226–234, 2013. 150 [105] P. Valentini, C. Zhang, and T. E. Schwartzentruber. Molecular dynamics simulation of rotational relaxation in nitrogen: Implications for rotational collision number models. Physics of Fluids, 24(10):106101, 2012.

[106] H. C. Andersen. Rattle: A velocity version of the shake algorithm for molecular dynamics calculations. Journal of Computational Physics, 52(1):24–34, 1983.

[107] C. Zhang, P. Valentini, and T. E. Schwartzentruber. Nonequilibrium-direction- dependent rotational energy model for use in continuum and stochastic molecular simulation. AIAA journal, 52(3):604–617, 2014.

[108] P. Norman and T. E. Schwartzentruber. Classical trajectory calculation direct simulation monte carlo: Gpu acceleration and three body collisions. In 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, page 1200, 2013.

[109] D. Bruno, A. Frezzotti, and G. P. Ghiroldi. Oxygen transport properties estima- tion by classical trajectory–direct simulation monte carlo. Physics of Fluids, 27 (5):057101, 2015.

[110] D. N. Morris and P. Benson. Data for icbm re-entry trajectories. Technical report, RAND CORP, Santa Monica, CA, 1963.

[111] D. N. Morris. Charts for determining the characteristics of ballistic trajectories in a vacuum. Technical report, RAND CORP, Santa Monica, CA, 1964.

[112] Volobuev Y. L. Lynch G. C. Jasper A. W. Truhlar D. G. Allison T. C. Wagner A. F. Garrett B. C. Espinosa-Garcia J. Duchovic, R. J. and POTLIB Corchado, J. C. URL http://comp.chem.umn.edu/potlib.

[113] C. Lim and D. G. Truhlar. Nonequilibrium eﬀects in chemical kinetics. straight- line paths for homonuclear diatomic dissociation-recombination process. The Journal of Physical Chemistry, 87(15):2683–2699, 1983. 151 [114] N. Singh and T. E. Schwartzentruber. Nonequilibrium internal energy distributions during dissociation. Proceedings of the National Academy of Sciences, 115 (1):47–52, 2018.

[115] R. C. Millikan and D. R. White. Systematics of vibrational relaxation. The Journal of chemical physics, 39(12):3209–3213, 1963.

[116] S. H. Bauer and S. C. Tsang. Mechanisms for vibrational relaxation at high temperatures. The Physics of Fluids, 6(2):182–189, 1963.

[117] D. L. Thompson. On a classical trajectory study of energy transfer in some atom- diatomic molecule systems. The Journal of Chemical Physics, 56(7):3570–3580, 1972.

[118] Maninder S Grover and Thomas E Schwartzentruber. Redistribution of vibrational energy: Mechanisms and transition probabilities. In 2018 AIAA Aerospace Sciences Meeting, page 1231, 2018.

[119] D. J. Kewley and H. G. Hornung. Free-piston shock-tube study of nitrogen dissociation. Chemical Physics Letters, 25(4):531–536, 1974.

[120] R. P. Saxon and B. Liu. A binitio conﬁguration interaction study of the valence states of o2. The Journal of Chemical Physics, 67(12):5432–5441, 1977.

[121] E. E. Nikitin. Theory of elementary atomic and molecular processes in gases. Oxford, Clarendon Press, 1974.

[122] A. W. Jasper and D. G. Truhlar. Non-bornoppenheimer molecular dynamics for conical intersections, avoided crossings, and weak interactions. In Conical Inter- sections: Theory, Computation and Experiment, pages 375–414. World Scientiﬁc, 2011.

[123] D. S. Sholl and J. C. Tully. A generalized surface hopping method. The Journal of chemical physics, 109(18):7702–7710, 1998. 152 [124] L. B. Ibraguimova, A. L. Sergievskaya, V. Y. Levashov, O. P. Shatalov, Y. V. Tunik, and I. E. Zabelinskii. Investigation of oxygen dissociation and vibrational relaxation at temperatures 4000–10 800 k. The Journal of chemical physics, 139 (3):034317, 2013.

[125] J. H. Kiefer and R. W. Lutz. The eﬀect of oxygen atoms on the vibrational relaxation of oxygen. In Symposium (International) on Combustion, volume 11, pages 67–76. Elsevier, 1967.

[126] R. S. Chaudhry, M. S. Grover, J. D. Bender, T. E. Schwartzentruber, and G. V. Candler. Quasiclassical trajectory analysis of oxygen dissociation via o2, o, and n2. In 2018 AIAA Aerospace Sciences Meeting, page 0237, 2018.

[127] C. Lim and D. G. Truhlar. Study of mixture eﬀects in the nonequilibrium kinetics of homonuclear diatomic dissociation and recombination. The Journal of Physical Chemistry, 88(4):778–792, 1984.

[128] S. Raynor and K. Burke. Ab-initio study of h2 dissociation and vibrational relaxation in mixtures of he and ar: Implications for the linear mixture rule. The Journal of chemical physics, 87(10):5788–5793, 1987.

[129] O. P. Shatalov. Molecular dissociation of oxygen in the absence of vibrational equilibrium. Combustion, Explosion and Shock Waves, 9(5):610–613, 1973.

[130] S. A. Losev and N. A. Generalov. A study of the excitation of vibrations and dissociation of oxygen molecules at high temperatures. In Soviet Physics Doklady, volume 6, page 1081, 1962.

[131] D. A. Andrienko and I. D. Boyd. State-speciﬁc dissociation in o2–o2 collisions by quasiclassical trajectory method. Chemical Physics, 491:74–81, 2017.

[132] A. Varandas and A. Pais. Double many-body expansion potential energy surface a p a for o4 (3 ), dynamics of the o (3 )+ o3 (11) reaction, and second virial coeﬃ- cients of molecular oxygen. In Theoretical and Computational Models for Organic Chemistry, pages 55–78. Springer, 1991.