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Kinetic modeling of high-temperature oxidation and pyrolysis of one-ringed aromatic and alkane compounds

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Kinetic modeling of high-temperature oxidation and pyrolysis of one-ringed aromatic and alkane compounds

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Content KINETIC MODELING OF HIGH-TEMPERATURE OXIDATION AND PYROLYSIS OF ONE-RINGED AROMATIC AND ALKANE COMPOUNDS by Enoch Edward Dames A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfilment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) August 2012 Copyright 2012 Enoch Edward Dames ii This work is dedicated to teachers with the spirit and drive to motivate others to apply the scientific method towards questions about the natural and physical world we live in. iii Table of Contents Dedication .................................................................................................................... ii List of Tables .............................................................................................................. vi List of Figures ........................................................................................................... viii List of Symbols and Abbreviations ........................................................................ xvi Abstract ..................................................................................................................... xix Chapter 1: Introduction ..............................................................................................1 1.1 A recent history of kinetic model development .............................................1 1.1.1 A Jet Surrogate Fuel Model (JetSurF) ...................................................4 1.2 With new molecules and data come new problems and challenges ..............5 1.2.1 Soot formation in the postflame .............................................................5 1.2.2 Fuel structure influence on flame speed and ignition delay ..................6 1.2.3 Preflame dehyrogenation to benzene .....................................................7 1.3 Thesis organization ........................................................................................8 1.4 Chapter 1 References ...................................................................................10 Chapter 2: Computational Methodologies ..............................................................14 2.1 Introduction ..................................................................................................14 2.2 Electronic structure calculations ..................................................................14 2.2.1 The Schrödinger equation ....................................................................14 2.2.1.1 The particle in a box problem .........................................................15 2.2.1.2 The harmonic oscillator problem ....................................................16 2.2.1.3 The rigid rotor problem ..................................................................17 2.2.1.4 The hydrogenic atom ......................................................................17 2.2.2 Hartree-Fock (HF) and post HF approximations .................................19 2.2.2.1 T1 diagnostic ..................................................................................20 2.2.3 Density functional theory (DFT) .........................................................20 2.2.4 Geometries of stationary and critical points ........................................21 2.2.4.1 Molecular point groups and rotational symmetry ...........................24 2.2.4.2 Moments of inertia..........................................................................26 2.3 Thermodynamic Properties ..........................................................................29 2.3.1 Standard thermodynamic state functions .............................................30 2.3.2 Statistical mechanics derivation of thermodynamic functions ............33 2.3.2.1 Molecular partition functions .........................................................33 2.3.2.2 Density and sum of states ...............................................................36 2.4 Reaction rates ...............................................................................................39 2.4.1 Transition state theory (TST) ...............................................................41 iv 2.4.2 The microcanonical rate constant ........................................................42 2.4.3 Pressure dependence ............................................................................43 2.4.3.1 RRKM unimolecular reaction scheme ...........................................45 2.4.4 Kinetic Monte Carlo solution of the Master Equation .........................48 2.4.4.1 Stochastic steps of kinetic Monte Carlo .........................................54 2.4.4.2 Thermal vs. chemically activated reactions ....................................57 2.4.5 Treatment of internal hindered rotational degrees of freedom ............64 2.4.6 Inclusion of channel specific quantum tunneling ................................66 2.4.7 Isodesmic Reactions.............................................................................70 2.4.8 Recent advances in kinetics - roaming radical reactions .....................72 2.4.9 Other aspects of reaction kinetics ........................................................73 2.5 Kinetic Model Development ........................................................................74 2.5.1 Critical rate evaluations .......................................................................75 2.5.2 Rate estimation.....................................................................................83 2.5.3 Elementary rate parameters - methods of determination .....................85 2.5.4 Sensitivity Analysis .............................................................................87 2.6 Computer modeling of experimental combustion phenomena ....................88 2.6.1 Laminar premixed flames ....................................................................88 2.6.2 Burner-stabilized premixed flames ......................................................91 2.7 Chapter 2 References ...................................................................................92 Chapter 3: Case Study on Persistent Aromatic Radicals .....................................100 3.1 Introduction ................................................................................................100 3.2 Methodology for electronic structure calculations ....................................108 3.3 Results and Discussion ..............................................................................112 3.4 Conclusions ................................................................................................125 3.5 Chapter 3 References .................................................................................126 Chapter 4: Kinetics of Isomerization Reactions between Benzylic and Methylphenyl Type Radicals in Single-Ring Aromatics ......................................131 4.1 Introduction ................................................................................................131 4.2 Computational Methodologies ...................................................................140 4.2.1 Quantum Chemical and RRKM Calculations ....................................140 4.2.2 Thermal Rate Calculations .................................................................141 4.3 Results and Discussion ..............................................................................144 4.3.1 Electronic Structure Calculations ......................................................144 4.3.2 Thermal Rate Calculations .................................................................151 4.3.3 Global Reactivity Trends ...................................................................160 4.4 Conclusions ................................................................................................163 4.5 Chapter 4 References .................................................................................164 Chapter 5: Dehydrogenation of Cyclohexane and Methylcyclohexane ..............169 5.1 Introduction and motivation .......................................................................169 5.2 The cyclohexyl system ...............................................................................170 v 5.2.1 Cyclohexene + H ...............................................................................180 5.2.1.1 Methodology .................................................................................184 5.2.2 Competition in cyclohexyl decomposition/isomerization .................192 5.2.3 Unimolecular 5-hexen-1-yl decomposition .......................................200 5.2.3.1 Entropic contributions to 5-exo/6-endo branching .......................206 5.2.3.2 RRKM/ME study for 5-hexen-1-yl ..............................................209 5.3 Benzene formation in 1-hexene and cyclohexene flames ..........................215 5.3.1 Low-pressure premixed burner-stabilized flames ..............................224 5.3.2 Atmospheric nonpremixed coflow flames .........................................229 5.4 Methylcyclohex-2-yl potential energy surface ..........................................232 5.5 Conclusions ................................................................................................239 5.6 Chapter 5 References .................................................................................243 Chapter 6: Summary, Conclusions and Future Work .........................................248 6.1 Concluding remarks ...................................................................................248 Bibliography .............................................................................................................254 Appendix A: Normal Alkane C 5 -C 12 Compounds ................................................278 A1. Reduced and skeletal models ..........................................................................279 Appendix B: JetSurF 1.1 - Cyclohexane and Mono-alkylated Cyclohexane Derivatives ................................................................................................................291 B1. Unimolecular fuel decomposition ..................................................................291 B2. Other initial fuel reactions ..............................................................................292 B3. Comparison with experimental flame speed data ..........................................293 B4. Comparison with shock tube measurements ..................................................300 Appendix C: JetSurF 2.0 .........................................................................................301 C1. Unimolecular fuel decomposition and the carbene mechanism .....................302 Appendix D: Gas-Phase Transport Properties - Methods of Estimation ..........311 Appendix E: Parameters Used in Quantum Chemistry Calculations.................323 B97X-D/6-311G(2d,p) optimized geometries used for the RRKM/ME calculations of Chapter 4 .......................................................................................323 B97X-D/6-311G(2d,p) unscaled frequencies used for the RRKM/ME calculations of Chapter 5 .......................................................................................328 B3LYP/6-311++G(2d,p) parameters for species in Chapter 5 .............................331 Appendix F: Sample RRKM/Master Equation input and source code ..............341 RRKM/Master Equation source code ....................................................................343 vi List of Tables Table 2.1 Selected cyclohexane geometric properties for various basis set sizes, calculated with the B3LYP hybrid density functional. See Figure 2.1 for depictions of listed parameters. Distances in angstroms; angles in degrees. ..............23 Table 2.2 Symmetry point groups and corresponding rotational symmetry numbers [60]. m = 2, 4, 6; n = 2, 3, 4, . . . ................................................................... 26 Table 2.3 Definition of terms used in various levels of approximation for moment of inertia, I (m,n) [65] ...................................................................................................... 29 Table 2.4 Convergence of rate and quality of fits with trial number. Rate corresponds to the RRKM/ME calculated isomerization of 2mp  benzyl at 1400 K and 1 atm. See Figure 4.4 for a depiction of the reaction potential energy surface. ......................................................................................................................... 63 Table 2.5 Summary of rate measurements at various conditions for the reaction of toluene + O( 3 P). ........................................................................................................... 76 Table 2.6 Major reaction classes of n-alkanes considered in JetSurF1.0 [36] ............. 84 Table 3.1 Summary of calculated electronic energies (E 0 ) and zero point energies (ZPE) in Hartrees, sensible enthalpy at 298 K (H ○ (298)-H ○ (0)) in cal/mol, and literature values of enthalpy of formation ( ) of reference species in kcal/mol ..................................................................................................................... 111 Table 3.2 Isodesmic reactions and enthalpies of formation at the standard state. ..... 116 Table 3.3 Standard, (Ph) 3 C-H bond dissociation energy (kcal/mol). ....................... 118 Table 3.4 Standard, central C-C bond dissociation energy (kcal/mol) of HPE (1). . 120 Table 3.5 Single-point, relative B3LYP/6-31G(d) electronic energies E rel (kcal/mol) and eigenvalues of the S 2 operator calculated for triplet diradical acenaphthene derivatives optimized at the M06-2X/6-31+G(d,p) level of theory. ... 123 Table 4.1 Adiabatic flame temperatures of selected aromatic compounds, including the three xylene isomers. Calculated for a stoichiometric fuel-air mixture at T u = 353 K and p = 1 atm. ........................................................................ 136 Table 4.2 Summary of calculated electronic energies (E 0 ) and zero point energies (ZPE) in Hartrees, along with all rotational parameters used in RRKM/Master Equation calculations. ................................................................................................ 144 298 f H  vii Table 4.3 Zero point energy corrected barriers (E o ) for key reactions of interest in this study. ................................................................................................................... 151 Table 4.4 Monte Carlo RRKM/Master Equation results fitted to double three- parameter and conventional Arrhenius expressions for the temperature range 298- 2000 K a ...................................................................................................................... 159 Table 5.1 Summary of absolute rates of H + cyclohexene under ambient temperature and low pressures. .................................................................................. 183 Table 5.2 Summary of calculated zero point energies (ZPE) in Hartrees, along with all rotational parameters used in RRKM/Master Equation calculations. ........... 187 Table 5.3 Unscaled vibrational frequencies used for cyclohexyl RRKM/ME simulations. Calculated at the B3LYP/6-311++G(2d,p) level of theory. See Figure 5.8 for species identification........................................................................... 188 Table 5.4 Temperature variation of total partition functions and entropy of activation (in cal/mol-K) for 5-hexen-1-yl, TS-1-2 (6-endo), and TS-2-6 (5-exo) molecules. .................................................................................................................. 208 Table 5.5 Summary of calculated zero point energies (ZPE) in kcal/mol and zero Kelvin electronic energies (E 0 ) in Hartrees. .............................................................. 235 Table 5.6 Temperature variation for the ratio of vibrational partition functions TS12 (CH 3 -elimination) and TS14 (H-elimination). See Figure 5.29 for species identification, vibrational frequencies calculated at B3LYP/6-311++G(2d,p) level of theory, listed in Appendix. .................................................................................... 238 Table A1 Simplified 1-alkines C k H 2k model for k = 5 and 6 [327] a . ......................... 288 Table C1 Summary of calculated electronic energies (E 0 ), adiabatic singlet triplet gaps (E s-t ), and zero point energies (ZPE) in Hartrees. Hartrees. E s-t of carbenes overestimated, see text. .............................................................................................. 306 Table D1 Estimated Lennard-Jones 12-6 Potential Parameters for Selected Alkane and Alkene Compounds ............................................................................................. 315 viii List of Figures Figure 2.1 Side and top views of the cyclohexane molecule illustrating several characteristic geometric values: axial and equatorial C-H bond lengths, C-C ring bond lengths, CCC bond angles and CCCC dihedral angle. See Table 2.1 for comparison of corresponding experimentally and theoretically (DFT) determined values.......................................................................................................................... 22 Figure 2.2 Density of states calculated for the 2-methylphenyl and benzyl radicals plotted as a function of energy, with a spacing of 10 cm -1 . .......................... 39 Figure 2.3 Rate coefficient of C 7 H 7 + H (+M) = C7H8 (+M) at 2500 K, with Troe parameters fitted from results of [77]. The Lindemann-Hinshelwood derived rate is included to illustrate that higher levels of approximation almost always result in lower rates in the fallow region, as depicted by the blue line. ......... 45 Figure 2.4 Total energy grained activation (solid line) and deactivation (dashed line) probabilities for 2-methylphenyl at T = 1400 K, = 260 cm -1 . An energy grain of 25 cm -1 was used for this calculation. P refers to activation or deactivation as defined in the equations above. More details of the 2- methylphenyl system can be found in Chapter 4. ...................................................... 53 Figure 2.5 Histogram of initial energies selected for 1000 2-methylphenyl molecules at 2000 K in a thermal isomerization reaction. Also shown is the RRKM microcanonical rate at the same temperature for reaction to the benzyl radical. The reaction potential energy surface is depicted in Figure 4.4.................... 56 Figure 2.6 Histogram of initial energies selected for 1000 2-methylphenyl (2mp) molecules at 1400 K in a thermal isomerization reaction. Also shown, but not identified above, are the RRKM microcanonical rates at the same temperature for all possible reactions of the potential energy surface depicted in Figure 4.4. .................................................................................................................. 58 Figure 2.7 Benzyl count history as a function of time from the isomerization of 2-methylphenyl at 1400 K and 1 atm. Results are shown for three different runs, all identical except for total number of stochastic trials. The superimposed orange line on the 10,000 trial data is a least squares fit. Only a fraction of the 10000 trial results are shown as filled circles. Figure 4.4 depicts the reaction potential energy surface for this calculation. ............................................................. 62 Figure 2.8 Form of an arbitrarily defined Eckart potential along the 1D potential energy surface of an exothermic reaction. ................................................................. 68 d E  ix Figure 2.9 Plot of the experimentally determined overall rates of reaction for toluene + O( 3 P). .......................................................................................................... 77 Figure 2.10 Selected data for the rate coefficient of C 6 H 5 CH 3 +H   C 6 H 5 CH 2 +H 2 . , Oehlschlaeger et al. [123]; ■, Ellis et al. [124]; red dashed line, Robaugh and Tsang [125]; black dashed line, Baulch et al. [126]; black dotted line, Rao and Skinner [127]; solid black line, current fit. ............................... 78 Figure 3.1 PAH(3,3) [205] monomer and dimer frontier orbitals calculated at the B97X-D/6-31G(d) level of theory [206]. The estimated binding energy for the dimer is 33 kcal/mol, while the singlet-triplet gap for a monomer is estimated to be 20 kcal/mol via the SF5050 method [207]. ......................................................... 104 Figure 3.2 Structures of selected model compounds (1-14) and the molecule accidentally synthesized by Gomberg [210, 211] 15. .............................................. 106 Figure 3.3 Schematic structures of selected molecules illustrating several key geometric properties. Left: central C-C bond length and C-C-C bond angles. Right: values of dihedral angles between adjacent substituents calculated with M06-2X/6-31+G(d,p). Values in parenthesis are those calculated with B3LYP/6-31G(d). .................................................................................................... 113 Figure 3.4 Central C–C bond lengths computed for selected molecules. ................ 114 Figure 3.5 Standard-state, central C–C BDEs for the class of acenaphthene derivatives studied, HPE and its derivatives. * Open circles represent BDE values estimated directly by subtracting the electronic energy of the diradical from that of the parent molecule, both with zero-point energy and sensible enthalpy corrections. ............................................................................................................... 121 Figure 4.1 Initial decomposition pathways during o-xylene combustion. M typically represents an inert collider species such as N 2 or O 2 . ............................... 136 Figure 4.2 Ignition delay times behind reflected shock tubes for the three xylene isomers [251]. Refelcted shock pressure of 10 atm,  = 1.0, T = 1350 - 1800 K. ... 137 Figure 4.3 Experimental laminar flame speeds of o-, m-, and p-xylene-air mixtures at T u = 353 K and p = 1 atm [252]. The error bars represent 2  uncertainty. ............................................................................................................... 138 Figure 4.4 Potential energy surface for the 2-methylphenyl system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. a Overpredicted by ~10 kcal/mol; see text. b Value taken from [244]. c Intermediate channels not shown. ........................................................................... 148 x Figure 4.5 Potential energy surface for the o-xylene (1,2-dimethyl-6-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. ............................................................................................................. 149 Figure 4.6 Potential energy surface for a m-xylene (1,3-dimethyl-2-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. The full surface is shown for completion, but is symmetric about MX2. ........................................................................................................................ 149 Figure 4.7 Potential energy surface for a m-xylene (1,3-dimethyl-6-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. Kinetics of this system are assumed to be identical to that of the 2- methylphenyl system. ............................................................................................... 150 Figure 4.8 Potential energy surface for the o-xylene (1,2-dimethyl-6-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. ............................................................................................................. 150 Figure 4.9 Arrhenius plots for isomerizations of 2-methylphenyl type radicals to those of benzylic nature. Symbols are rates computed with Monte Carlo RRKM/master equation modeling and lines represent fitted Arrhenius expressions within the recommended temperature range (Table 3). See Figure 4.4-Figure 4.8 for species identifications. Uncertainty bands of  2 kcal/mol in activation energy are included in the high pressure rate constants. ......................... 153 Figure 4.10 Microcanonical high pressure limit rate constants for selected channels in the 2-methylphenyl system. See Figure 2 for species identification. Tunneling contributions not included. ..................................................................... 157 Figure 4.11 : Sites where H-abstraction may result in an isolated free radical. : Sites where H-abstraction may result in an H-shift to a benzylic radical. a Number of sites not adjacent to a methyl group. b Ref.[251]; ignition delay increases with number. c Ref.[248] d Fuel lean flame speed decreases with number [252]. ........................................................................................................... 161 Figure 5.1 Cylcohexyl potential energy surface. Relative energies not included. .. 171 Figure 5.2 Reactions not shown in the cyclohexyl PES of Figure 5.1 ..................... 172 Figure 5.3 Reduced cyclohexyl potential energy surface with 0 K relative energies, ZPE included. Blue: RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p). Red: Knepp and coworkers' modified G2MP2 energies. H atom 298 K has 3/2RT added (0.9 kcal/mol) Note: difference in linear and kinked 5-hexen-1-yl is ~2kcal/mol ................................................................................................................ 175 xi Figure 5.4 Comparison of recommended R 5.1 and R 5.2 reaction rates. Solid line: R 5.1; dashed line: R 5.2; black: Tsang [281]; red: Sirjean et al. [280]; blue: Kneep et al. [279] (thicker line denotes temperature range of interest in study, while thinner lines are extrapolations to higher temperature outside recommended range) ................................................................................................ 176 Figure 5.5 Comparison of branching ratios to cyclohexene + H (solid lines) and 5-hexen-1-yl (dashed lines) as a function of temperature. Black: Tsang [281]; red: Sirjean et al. [280]; blue: Kneep et al. [279] ..................................................... 178 Figure 5.6 Reaction network for the RRKM/ME calculation of cyclohexane + H. Corresponding species nomenclature and names are listed below each species schematic. ................................................................................................................. 185 Figure 5.7 Comparison of RRKM/ME results using the reaction network of Figure 5.6 with 300 K experimentally measured products of the cyclohexene + H reaction in an H 2 bath gas [282]. Filled circles: experimentally measured cyclohexyl and cylopentanemethyl radical product fractions. Dashed lines: simulation results with all vibrational modes treated using the harmonic oscillator assumption. Solid lines: simulation results with CH 2 groups treated as 1D free internal rotors and reduced internal moments of inertia calculated at using the I (3,4) level of approximation. ..................................................................... 189 Figure 5.8 Reaction network for the RRKM/ME calculation of cyclohexyl isomerization/decomposition. Corresponding species nomenclature and names are listed below each species schematic................................................................... 192 Figure 5.9 Arrhenius plot of the overall decomposition of cyclohexyl to products for varying pressure and temperature. ...................................................................... 193 Figure 5.10 Comparison of high-pressure rates through beta-scission (TS12) and H-elimination (TS13) channels. Black: this work; red: Tsang et al. [34]; green, Knepp et al. [279]; blue: Sirjean et al. [280]. ........................................................... 195 Figure 5.11 Arrhenius plots for the reaction channels of cyclohexyl leading to various products in the cyclohexyl system of Figure 5.8. Symbols are rates computed with Monte Carlo RRKM/Master equation modeling and lines drawn to guide the eye. ....................................................................................................... 198 Figure 5.12 Arrhenius plot of summed cyclohexyl isomerization to cyclopentanemethyl and cylopentanemethylene + H. .............................................. 199 xii Figure 5.13 Pressure and temperature branching fractions calculated for reaction channels of reduced cyclohexyl system: (a) 5-hexen-1-yl, (b) cyclohexene + H, (c) cyclopentanemethylene + H, (d) cyclopentanemethyl (nitrogen bath gas). ....... 199 Figure 5.14 Comparison of 6-endo (a) and 5-exo (b) transition state structures. Calculated at the B3LYP/6-311++G(2d,p) level of theory. ..................................... 207 Figure 5.15 Reaction network for the 5-hexen-1-yl radical, used in the RRKM/ME calculation for the thermal isomerization/dissociation of 5-hexen-1- yl. .............................................................................................................................. 209 Figure 5.16 Branching fractions as functions of temperature and pressure (labels in bottom right plot) for the thermal isomerization of 5-hexen-1-yl, as described by the reaction network shown in Figure 5.15 a) cylcohexyl, b) cyclopentanemethyl, c) cyclohexene + H, d) cyclopentanemethylene + H. ............ 210 Figure 5.17 Arrhenius plots for the reaction channels of 5-hexen-1-yl, plotted for T = 500 - 1300 K. Symbols are rates computed with Monte Carlo RRKM/Master equation modeling. ................................................................................................... 212 Figure 5.18 Rate comparisons for the 5-exo (solid lines) and 6-endo (dashed lines) channels in the temperature range of 298 - 1300 K. Blue: Handford- Styring and Walker estimate, T = 600 - 1300 K, 1atm [287]; purple: Tsang high pressure limit, T = 600-1900 K [141]; red: this work, high pressure limit (the red dotted line represents the rate for the 5-exo channel with the activation energy increased by 2 kcal/mol); black: 5-exo, Chatgilialoglu et al., in solution [301] with reported uncertainty, 300 - 355 K; solid triangle: 5-exo, Wu et al. [299]; solid diamond: 5-exo, Walling and Cioffari, in solution [300]. ............................... 213 Figure 5.19 JetSurF model predictions of cyclohexene, 1,3-cyclohexadiene and benzene mole fraction as function of distance from burner for a burner-stabilized premixed flame at P = 30 torr,  = 1.7, 11.13% cyclohexane, 58.87% O 2 in Ar. Flame modeled using an experimentally determined temperature profile, T u = 300 K. ....................................................................................................................... 218 Figure 5.20 Dominant reactions responsible for benzene flux (corresponding rate units on right ordinate) using the JetSurF 2.0 model. Benzene mole fraction superimposed. Burner-stabilized premixed flame conditions: P = 30 torr,  = 1.7, 11.13% cyclohexane, 58.87% O 2 in Ar. Flame modeled using an experimentally determined temperature profile, T u = 300 K. .................................. 219 Figure 5.21 Reactions in JetSurF 2.0 corresponding to those in Figure 5.20 ......... 220 xiii Figure 5.22 298 f H  (kcal/mol) for H-elimination and ring opening of cyclohexadienyl. H atom 298 K has 3/2RT added (0.9 kcal/mol). Black: calculated via RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p); blue: Berho and coworkers [309]; green: CBS-UCCSD(T)//QCISD/6-311G(d,p) of Gao et al. [310]; red: computed from experimental formation enthalpies [222, 310, 311]. ..... 221 Figure 5.23 High-pressure limit rate constants for the unimolecular H- elimination reactions in JetSurF 2.0 leading to respective closed shell stable products (+ H). ......................................................................................................... 222 Figure 5.24 298 f H  for a reduced cyclohexenyl potential energy surface. Black: RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p). H atom 298 K has 3/2RT added (0.9 kcal/mol). Red: computed from experimental formation enthalpies [222, 310, 311]. ................................................................................................................. 223 Figure 5.25 Fitted rate constants at 0.1 atm added or replacing reactions in the JetSurF 2.0 model. Low temperature chemistry also removed from model. * replaces rate recommendation from Tsang [34]. R1 and R2 valid in the range of 500 K - 1300 K. R3-R5 valid in the range of 1000 K - 2000 K. .............................. 225 Figure 5.26 a) Model predictions of cyclohexene (dashed lines) and benzene (solid lines) mole fraction as function of distance from burner for a burner- stabilized premixed flame at P = 30 torr,  = 1.7, 11.13% 1-hexene, 58.87% O 2 in Ar. Flame modeled using an experimentally determined temperature profile, T u = 300 K. Black: JetSurF 2.0 with lumped low T chemistry removed; red: JetSurF 2.0 with lumped low T chemistry removed and new reaction block b) experimentally determined mole fractions of benzene under same flame conditions as in a) [306]. .......................................................................................... 226 Figure 5.27 a) Model predictions of cyclohexene and benzene mole fraction as function of distance from burner for a burner-stabilized premixed flame at P = 30 torr,  = 1.7, 11.13% cC 6 H 12 , 58.87% O 2 in Ar. Flame modeled using a given experimentally determined temperature profile, T u = 300 K. Black: JetSurF 2.0 with lumped low T chemistry removed; red: JetSurF 2.0 with lumped low T chemistry removed and new reaction block b) experimentally determined mole fractions of benzene under same flame conditions as in a) [306]. ........................... 228 Figure 5.28 Dehydrogenation sequence for methylcyclohexane. Grey blocks denote H-abstraction reactions by H, OH, O, CH 3 . Green blocks denoted pressure-dependent H-elimination reactions. Not shown are H 2 eliminations, H- shifting, and -scission ring openings. Also not shown: pathways from cyclohexene and the two methylenecyclohexadiene species that may undergo H- abtraction followed by H-elimination to toluene and benzene. ............................... 233 xiv Figure 5.29 Methylcyclohex-2-yl reaction network with relative energies, calculated at the RCCSD(T)/cc-pVDZ //B3LYP/6-311++G(2d,p) level of theory, ZPE included. Only reactions relevant to the unimolecular thermal isomerization/decomposition of methylcyclohex-2-yl are shown. Pathways with no values were not explored. TS6-9 estimated based on [102]. The bold indices besides each species corresponds to those listed in Table 5.5. ................................ 236 Figure A1 Systematic simplification of JetSurF 1.0 [327]. ..................................... 286 Figure A2 Experimentally [328] (symbols) and numerically (solid line: skeletal model; dash line: simplified model; dotted line: JetSurF v 0.2) determined laminar flame speeds of n-alkane/air mixtures. ....................................................... 289 Figure B1 Experimental and computed ’s of methyl-, ethyl, n-propyl, and n- butyl-cyclohexane/air flames at T u = 353 K and p = 1 atm. Symbols: ( ○) Ji et al. [47] (◇) Dubois et al. [49]. Lines: simulation using JetSurF 1.1. The error bars indicate 2-  standard deviations. ............................................................................. 293 Figure B2 Comparison of experimentally determined ’s of ( ○) cyclohexane/air, ( ◆) methylcyclohexane/air, (■) ethylcyclohexane/air, ( ●) n- propylcyclohexane/air, and (▲) n-butylcyclohexane/air flames relative to those of ( ━) n-hexane/air flames at T u = 353 K [47]. ....................................................... 295 Figure B3 Pseudo-Arrhenius plot of mass burning rate computed of cyclohexane/air and n-propylcyclohexane/air flames at  = 1.0, T u = 353 K, and p = 1 atm using JetSurF 1.1. The lowest temperature results correspond to a fuel-air mixture without argon substitution; the variation of the adiabatic flame temperature is accomplished through nitrogen substitution by argon. .................... 296 Figure B4 Selected intermediate species computed using JetSurF 1.1 for cyclohexane (dashed lines) and n-propylcyclohexane (solid lines) flames at  = 1.0, T u = 353 K, and p = 1 atm. .......................................................................... 297 Figure B5 Ranked logarithmic sensitivity coefficients of with respect to reaction rate coefficients computed using JetSurF 1.1 for cyclohexane/air flames at T u = 353 K and p = 1 atm. .................................................................................... 298 Figure B6 Ranked logarithmic sensitivity coefficients of with respect to reaction rate coefficients computed using JetSurF 1.1 for  = 1.0 n- propylcyclohexane/air flame at T u = 353 K and p = 1 atm. ..................................... 299  S u o  S u o  S u o  S u o xv Figure B7 High pressure cyclohexane shock tube data (1.154% cC 6 H 12 , 20.77% O 2 in N 2 ,  = 0.5, p 5 = 50 atm) [56] along with corresponding predictions from various available kinetic models: black solid line: JetSurF1.1 [37]; dashed red: Silke et al. [335]; dashed blue: El Bakli et al. [52] ; solid red: Sirjean et al. [336]. 300 Figure C1 Potential energy diagrams illustrating selected unimolecular decomposition pathways for (a) cyclohexane and (b) methylcyclohexane. Zero point energy corrections included. Relative energies in kcal/mol. Dashed lines indicate that the barrier required for a H-shift in the formation of corresponding alkenes are not yet determined. ................................................................................ 307 Figure C2 Transition states leading to biradical carbene formation. (a) Cyclohexane. (b) Methylcyclohexane. Bond-breading distance in Angstroms, indicated by arrows. ................................................................................................. 308 Figure C3 High pressure methylcyclohexane shock tube data (1.96% cC 6 H 12 , 20.60% O 2 in N 2 ,  = 1, p 5 = 20 atm) [54] along with corresponding predictions from JetSurF 1.1 (red line) and JetSurF 1.1 with a semi-global carbene initiation step (black dashed line). ........................................................................................... 309 Figure D1 L-J self collision diameter plotted against molecular weight for selected linear alkanes .............................................................................................. 316 Figure D2 L-J well depth plotted against molecular weight for selected linear alkanes ...................................................................................................................... 317 Figure D3 Binary diffusion estimates for dodecane in nitrogen. ............................. 319 xvi List of Symbols and Abbreviations A* - activated complex A † - transition state complex Å - unit of length, Angstrom BDE - bond dissociation energy D ij - binary diffusion coefficient of a species i in bath gas j  - Lennard Jones well depth d E  - downward energy transferred upon collision with a bath gas molecule E - energy E 0 - zero Kelvin barrier height E a - activation energy ESR - Electron Spin Resonance G init - initial energy distribution h - Plank’s constant H - enthalpy HOMO - highest occupied molecular orbital 298 f H  - standard state 298 K formation enthalpy I - moment of inertia I ii - principle moments of inertia, ii = xx, yy, and zz principle axes  - tunneling coefficient K - equilibrium constant or strain rate xvii k - rate constant k a - rate of activation due to collision with another body k(E) - energy dependent microcanonical rate constant uni k  - high pressure limit unimolecular rate constant k B - Boltzmann's constant l - reaction path degeneracy  - reduced mass LJ - Lennard Jones LUMO - lowest unoccupied molecular orbital m - mass [M] - molar concentration of species M M - bath gas molecule ME - Master Equation  is the collision frequency  (2,2) - collision integral  - equivalence ratio P - pressure P c - critical pressure Pij – collision probability for a specific energy level (T and P) PAH - polycyclic aromatic hydrocarbon Q - partition function  ij - molecular collision diameter  - Lennard-Jones collision diameter or rotational symmetry number xviii  u - unburned mixture density (E) the density of states R - ideal gas constant or a random number  - wavefunction S - entropy - laminar flame speed  - angle  - time interval between reactions T - temperature T ad - adiabatic flame temperature T c - critical temperature T u - unburned mixture temperature TST - transition state theory  * - magnitude of imaginary force constant in transition state V - potential barrier height W - sum of states Z –inelastic collision frequency (also known as the L-J collision frequency) ZPE - zero point vibrational energy  S u o xix Abstract Future internal combustion engine design will rely on accurate kinetic models of surrogate fuels that mimic their fossil fuel derived counterparts, having hundreds or thousands of different chemical species. At the same time, technological advances in alternative fuel production will enable tailor made surrogate fuels for use in pre-existing internal combustion engines. For these reasons, a large collaborative effort has been underway to identify and understand key species and their reaction kinetics. This work focuses on two such compound classes - one-ringed aromatics and alkanes, as they make up a significant portion of almost all transportation fuels. Attempts at understanding the combustion characteristics of new compounds frequently results in the identification of missing kinetic pathways and/or key intermediate species. For a kineticist, there is never a lack of new and interesting reactions to explore. This thesis work presents mechanisms seeking to explain observed behavior in the three different premixed flame regions: the postflame region, the flame sheet, and the preflame region. Three topics will be addressed: 1) persistent free radicals in large aromatic compounds, which serve as analogs to systems in nascent and mature soot that may help explain the continued growth of soot in the postflame region where hydrogen radicals are scarce. 2) mutual isomerization through H-atom shifts in benzylic and methylphenyl type radicals xx for toluene, and o-, m-, and p-xylene isomers. The relative structure of xylene isomers and specifically the number of H atoms immediately adjacent to the methyl groups can have a direct impact on their high-temperature oxidation and appear to explain the observed differences in xylene oxidation behind reflected shock waves and in laminar premixed flames. 3) the isomerization and dissociation of cyclohexyl radicals, which marks a necessary step towards a detailed understanding of dehydrogenation from cyclohexane and methylcyclohexane to benzene and toluene. Focus lies on branching fractions of cyclohexyl isomerization/dissociation for various temperatures and pressures, and its relationship to differing observations on the role of early dehydrogenation in benzene formation for various flame configurations. 1 Chapter 1: Introduction 1.1 A recent history of kinetic model development Combustion chemistry of fuels containing hydrocarbons, hydrogen, and oxygen can be modeled as a time dependent complex network of elementary chemical reactions. The diversity and conditional importance of various reaction pathways ultimately leading to both desired and undesired reaction products embodies a level of complexity on par with some of the most elaborate problems in nature. For this reason, in modeling such systems, we frequently only seek applications over a specific range of physical conditions. One of the simplest gas phase combustion mixtures is that of hydrogen and oxygen. For only atmospheric H 2 /O 2 mixtures do we have a complete understanding of the elementary reactions involved in the combustion process. At elevated pressures, our understanding of this chemistry is less certain [1, 2]. The combustion chemistry of hydrocarbons is considerably more complex than hydrogen not only because of the greater number of species and elementary reactions involved, but because it also opens the door to chemical activation and molecular growth processes. The effort necessary for the development of an accurate detailed hydrocarbon chemical model is best exemplified through GRI-Mech 3.0, describing the oxidation of methane and natural gas [3]. This model required detailed chemistry for H 2 , CO, and methane (it also included NO and limited C 2 chemistry). Molecular growth kinetics leading to 2 aromatic molecules was neglected. Nevertheless, the development effort involved over 15 contributors and took a decade. The model, composed of 325 elementary reactions, was tested against over 100 experimental targets. Most of the original rate constants used were either determined experimentally or by combined quantum chemistry - reaction rate theory calculations. The model was optimized within the attendant uncertainty of each rate parameter for roughly 1000 to 2500 K, 10 torr to 10 atm, and equivalence ratio from 0.1 to 5 [3]. Clearly, assembly of GRI-Mech was not a trivial task. Even with seemingly herculean effort, it was still impossible to ensure that the submodel (e.g., H 2 /CO) was predictive for the combustion of syngas [4]. As a decade long effort, GRI-Mech 3.0 was released in the year 2000. During and after that period, small groups and individuals have populated scientific publications with a large number of kinetic models describing the combustion characteristics for almost all representative compounds found in fossil fuels. For example, since 1990, there have been many models proposed for the combustion of benzene and toluene [5-17]. Since the year 2008, there have been over 10 models proposed for the combustion of n-butanol alone [12, 18-31] with little convergence on its underlying chemistry. As a science, this seemingly endless effort presents detailed combustion kinetic modeling as a discipline with no tenable direction or closure in sight. This is not to say the proliferation of kinetic models is without purpose, for they are used to satisfactorily predict global combustion phenomena, albeit almost always under limited 3 conditions. The use of reaction class rate rules can make it a simple matter to assemble a model in a short period. In addition, the flame propagation characteristics of many fuel compounds are most sensitive to the small intermediate species chemistry, for which the elementary rate parameters are more accurately known. Today, a more unified approach to kinetic model development is being proposed and tested. This approach utilizes increased computing capabilities and the ease with which models can be generated 'on-the-fly'. Standard databases (NIST's Kinetic Database, for instance) serve as hubs for well-documented rate parameters. Computer codes allow for automatic generation of models within seconds for many fuels [32, 33]. In general, each program maintains a core set of H 2 /CO and small hydrocarbon chemistry with reaction class rate rules used to generate the remaining reactions for a particular compound. Perhaps considered another school of thought, the Process Informatics Model (PrIMe) does not use reaction classes to generate models, but only uses documented rate parameters with known uncertainties to generate a model based on the available chemistry in the database [34, 35]. PrIMe has the added benefit of using uncertainty quantification for the purpose of model optimization. The strengths of such an approach are obvious - any model generated should be more accurate than one based on rate rules with generally less accuracy. Furthermore, if a model is desired only for application to, say, a gas turbine operating under certain conditions, optimization procedures may be utilized to further refine the model's precision for this application. The drawback, of 4 course, is that the number of possible compounds addressed is limited to the available kinetic and thermodynamic dataset. 1.1.1 A Jet Surrogate Fuel Model (JetSurF) Future internal combustion engine design will rely on accurate kinetic models of surrogate fuels that mimic their fossil fuel derived counterparts, having hundreds or thousands of different chemical species. At the same time, technological advances in alternative fuel production will enable tailor made surrogate fuels for use in pre-existing internal combustion engines. For these reasons, a large collaborative effort has been underway to identify and understand key species and their reaction kinetics [36-38]. Among jet fuels, roughly 20% are comprised of cycloalkanes, and 15% comprised of aromatic compounds [39]. These two types of compounds are a focus of this thesis work. To this end, a hierarchical approach to model development has been adopted. The foundation model is USC Mech II, which describes the high temperature oxidation and pyrolysis for H 2 /CO /C 1 -C 4 alkanes [40]. JetSurF builds on the chemistry of USCMech II, with the current version including chemistry for H 2 /CO/C 1 -C 12 n-alkanes/cyclohexanes (cyclohexane and methyl- up to butyl-cyclohexane). This work is ongoing as the model is being tested against an ever-increasing number of experimental targets, over hundreds to date [38]. 5 In the process of developing models for new compounds with potentially unknown oxidation/pyrolysis pathways, it is no surprise that challenges arise. It is sometimes necessary to search for and include such pathways in order to allow for the model to predict relevant observations. Thus, the primary focus of this thesis work is not the development of large detailed kinetic models themselves, but to address specific challenges and/or problems that arose as consequences of their development. Nevertheless, aspects of JetSurF model development are still included in the Appendix. 1.2 With new molecules and data come new problems and challenges Attempts at understanding the combustion characteristics of new compounds frequently results in the identification of missing kinetic pathways and/or key intermediate species. For a chemical kineticist, there is never a lack of new and interesting reactions to explore. This thesis work presents mechanisms seeking to explain observed behaviors in three different premixed flame regions: the postflame region, the flame sheet, and the preflame region. One chapter is devoted to a specific issue in each of these flame regions. They are outlined below. 1.2.1 Soot formation in the postflame A generally undesired byproduct of fossil fuel combustion, soot is one of the most prevalent components of anthropogenic aerosols, affecting atmospheric visibility, global climate, and human health. Soot formation in flames is a highly reversible process driven, in part, by competing thermodynamic forces: enthalpy release from chemical 6 bond formation and entropy increase attributed to release of gas phase species accompanying particle formation [41]. The initial growth of polycyclic aromatic hydrocarbons is essentially captured by the hydrogen-abstraction carbon-addition (HACA) mechanism [42, 43]. For example, the presence of acetylene, H atoms, and the high temperature environment inherent to flames facilitate formation of soot precursors such as benzene and naphthalene. However, an entropic resistance to PAH dimerization at these temperatures rules out the self-binding thermodynamically driven growth of PAH stacks and/or clusters [41]. Furthermore, two additional observations conflict with our current understanding of soot formation: (1) nascent soot can be rich in aliphatics and (2) soot mass growth can occur without the presence of H atoms [41, 44-46]. In light of these recent findings, another avenue to soot formation is explored in this work. Chapter 3 explores a certain class of aromatic compounds that are analogues to possible moieties in young and mature soot. These compounds are unique in that steric interactions between aromatic rings result in elongated carbon-carbon bonds with very low strengths. Even at slightly elevated temperatures, these bonds may break, resulting in the formation of resonantly stabilized diradical species. Within the atmosphere, such bonds are subject to photolysis, resulting in diradicals. Under the higher temperatures found in engines and near the spatial domain of flames, equilibrium highly favors the diradical species. In all cases, these diradical species open possibilities for reactions with other molecules and serve as explanations for continued soot growth in the postflame region and the high activity of carbonaceous cloud condensation nuclei. 7 1.2.2 Fuel structure influence on flame speed and ignition delay The chain reactions taking place a premixed flame sheet impacts the flame propagation rate. Chapter 4 explores the role of fuel molecular structure on overall flame phenomena, namely, laminar flame speeds and ignition delay times behind reflected shock waves. By focusing on three isomeric compounds (ortho-, meta-, and para-xylene) with nearly identical thermodynamic properties, the behavioral differences observed under varying conditions can be only attributed to the reaction kinetics of oxidation and/or pyrolysis. In this specific case, dealing with xylene isomers, focus lies on observed differences primarily under high temperature pyrolytic conditions. Intermolecular hydrogen shifting reactions about the aromatic ring and associated methyl groups are shown to play a role in influencing the overall flame speed with which these compounds proceed to final products. 1.2.3 Preflame dehyrogenation to benzene This work was initially motivated by observed discrepancies between JetSurF 1.1 predictions of high pressure ignition delay with their experimentally determined counterparts [37]. JetSurF 1.1 is capable of predicting a wide range of experimental combustion data: laminar flame speeds, plug and jet-stirred flow reactor species profiles, ignition delay times [47-53]. However, the model fails to predict high pressure (>15 atm) shock tube ignition delay times, being too slow by up to several factors in some cases [54-56]. This model-experiment ignition delay discrepancy is considerably greater in the case of mono-alkylated cyclohexane compounds, and is only observed under high 8 pressure. Among the numerous possibilities considered for these mono-alkylated cyclohexane compounds, the lack of dehydrogenation reactions in the model was thought to be partially responsible for the overestimated ignition delay times. Subsequent dehydrogenation reactions early in the flame serve as a source of H radicals which can increase overall reaction progress. Initially, analogous cyclohexane reactions were to be used in estimating the dehydrogenation rates of methylcyclohexane. A review of the currently available dehydrogenation chemistry of cyclohexane revealed that even this chemistry was not satisfactorily understood, particularly because of the strong pressure dependence of the participating reactions. Thus, focus shifted to cyclohexane, and in particular, the first dehydrogenation step - hydrogen elimination from the cyclohexyl radical. Although this reaction is not a kinetic bottleneck to benzene formation, it is a crucial and necessary step to its formation. The underlying theme of Chapter 5 concerns the competition between energetically similar channels - one leading to H- or CH 3 - elimination with the formation of a cyclic alkene, the other an isomerization to either an open ring alkenyl or cyclic radical. Aside from being the most common single ring saturated hydrocarbon found in fuels, and one of the most thoroughly studied molecules in history, a full temperature and pressure mapped understanding of its oxidation and pyrolysis chemistry is still lacking. 9 Some of these issues will be highlighted and a clearer picture of cyclohexane dehydrogenation will be presented. In addition, the role of entropy in the kinetics of methylcyclohexane dehydrogenation will be shown to dominate cyclohexene formation under low pressure combustion conditions. 1.3 Thesis organization Chapter 2 describes the core concepts utilized in electronic structure calculations and gas phase reaction kinetics. These include the topics of quantum chemistry, statistical mechanics, transition state theory, and the application of Monte Carlo algorithms to systems of reacting molecules. Most of the material is only briefly covered. Each working chapter (Chapters 3-5) is prefaced by introductory material emphasizing the objectives of the work and the nature or background from which the particular problem arose. Chapter 6 summarizes the work presented in this thesis and gives suggestions for relevant future work. Chapter 7 contains necessary supplemental information to ensure reproducibility of the work presented herein. 10 Chapter 1 References 1. M. P. Burke; M. Chaos; F. L. Dryer; Y. 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Tsang; N. P. Cernansky; D. L. Miller; A. Violi; R. P. Lindstedt, in: JetSurF 1.1 September 15 ed.; 2009. 36. B. Sirjean; E. Dames; D. A. Sheen; F. N. Egolfopoulos; H. Wang; D. F. Davidson; R. K. Hanson; H. Pitsch; C. T. Bowman; C. K. Law; W. Tsang; N. P. Cernansky; D. L. Miller; A. Violi; R. P. Lindstedt, in: JetSurF 0.1 September 15 ed.; 2009. 37. H. Wang; E. Dames; B. Sirjean; D. A. Sheen; R. Tangko; A. Violi; J. Y. W. Lai; F. N. Egolfopoulos; D. F. Davidson; R. K. Hanson; C. T. Bowman; C. K. Law; W. Tsang; N. P. Cernansky; D. L. Miller; R. P. Lindstedt A high-temperature chemical kinetic model of n-alkane (up to n-dodecane), cyclohexane, and methyl-, ethyl-, n-propyl and n- butyl-cyclohexane oxidation at high temperatures, JetSurF version 2.0. http://melchior.usc.edu/JetSurF/JetSurF2.0/Index.html 38. A. T. Holley; Y. Dong; M. G. Andac; F. N. Egolfopoulos; T. Edwards, Proceedings of the Combustion Institute 31 (2007) 1205-1213 10.1016/j.proci.2006.07.208. 39. H. Wang; X. You; A. V. Joshi; S. G. Davis; ; A. Laskin; F. N. Egolfopoulos; C. K. Law, in: 2007. 13 40. H. Wang, Proceedings of the Combustion Institute 33 (2011) 41-67. 41. M. Frenklach; H. Wang, Symposium (International) on Combustion 23 (1) (1991) 1559-1566. 42. H. Wang; M. Frenklach, Combust. Flame 110 (1-2) (1997) 173-221. 43. B. Oktem; M. P. Tolocka; B. Zhao; H. Wang; M. V. Johnston, Combustion and Flame 142 (4) (2005) 364-373. 44. B. Zhao; K. Uchikawa; H. Wang, Proceedings of the Combustion Institute 31 (2007) 851-860. 45. H. Wang; D. X. Du; C. J. Sung; C. K. Law, Twenty-Sixth Symposium (International) on Combustion, Vols 1 and 2 (1996) 2359-2368. 46. C. Ji; E. Dames; B. Sirjean; H. Wang; F. N. Egolfopoulos, Proceedings of the Combustion Institute 33 (1) (2011) 971-978. 47. S. G. Davis; C. K. Law, Combustion Science and Technology 140 (1-6) (1998) 427-449. 48. T. Dubois; N. Chaumeix; C. E. Paillard, Energy & Fuels 23 (2009) 2453-2466 10.1021/ef801062d. 49. S. S. Vasu; D. F. Davidson; R. K. Hanson, Combustion and Flame 156 (4) (2009) 736-749 10.1016/j.combustflame.2008.09.006. 50. J. P. Orme; H. J. Curran; J. M. Simmie, Journal of Physical Chemistry A 110 (1) (2006) 114-131 10.1021/jp0543678. 51. A. El Bakli; M. Braun-Unkhoff; P. Dagaut; P. Frank; M. Cathonnet, Proceedings of the Combustion Institute 28 (2000) 1631-1638. 52. S. Zeppieri; K. Brezinsky; I. Glassman, Combustion and Flame 108 (3) (1997) 266-286. 53. S. S. Vasu; D. F. Davidson; Z. Hong; R. K. Hanson, Energy & Fuels 23 (1) (2009) 175-185. 54. J. Vanderover; M. A. Oehlschlaeger, International Journal of Chemical Kinetics 41 (2) (2009) 82-91. 55. S. M. Daley; A. M. Berkowitz; M. A. Oehlschlaeger, International Journal of Chemical Kinetics 40 (10) (2008) 624-634. 14 Chapter 2: Computational Methodologies 2.1 Introduction Most of the material presented in this chapter is included for completion of presentation with respect to tools used throughout this work. The subject matter is well developed and represents over a century of contributions from thousands of scientists, many of whom have received Nobel prizes for their work. Some of the examples given in this chapter are taken from work presented in later chapters. Some of them are purely conceptual and illustrative. The sole purpose of this chapter is to introduce and cover the core concepts and tools necessary for a quantitative and accurate description of chemical electronic structure, thermodynamic properties, and gas phase reaction kinetics. Because these topics are in relative stages of maturity, they are generally only briefly covered. 2.2 Electronic structure calculations 2.2.1 The Schrödinger equation The circumstantial failure of classical physics resulted in the development of quantum theory. Specifically, the wave-particle duality served as evidence for quantized behavior of matter. In 1926, Erwin Schrödinger proposed a wave equation for the motion of a particle of mass, m, in one dimension [1]: 2 2 2 d E m dx     15 where x is the spatial coordinate, ћ is Plank's constant, and E is the energy of the particle, which is an eigenvalue of , the wavefunction describing the particle, which can be a function of position and time. Separation of variable allows for the formulation of a time- independent Schrödinger equation. Imposing boundary conditions on the Schrodinger equation forces the solutions to be quantized, as the following sections will illustrate. 2.2.1.1 The particle in a box problem The particle in a box problem is the simplest illustration of the Schrödinger equation. Consider a one-dimensional box of length L containing a particle of mass m. The boundary conditions imposed are that both (x=0) and (x=L) must be zero. It can be shown that the solution is The corresponding allowable energy levels are Separation of variables allows for easy extension of the Schrödinger equation to multiple dimensions, and thus a description of the quantized motion of a particle in free space. As will be seen later, the particle in a '3D box' problem will be used to derive the translational contribution to thermodynamic properties of chemical species. The most striking result in the solution above is the lowest allowed eigenvalue (energy), corresponding to n = 1.   2 sin 0 n nx x x L LL         22 2 1,2,... 8 nh En mL  16 2.2.1.2 The harmonic oscillator problem The harmonic oscillator problem is simply the application Hooke's Law (describing the motion of a mass on a spring with a fixed frame of reference) to the Schrödinger equation. The potential energy function representing the spring is described by a parabola. Thus, the only differences between the quantum harmonic oscillator and the quantum particle in a box is the shape of the potential, or boundaries, containing the mass. The Schrödinger equation for a harmonic oscillator is where k is the force constant of the oscillator. Here, the wavefunction must diminish as x approaches ± . As many quickly realized, this problem makes for a good model describing the vibrational motion of atoms within a molecule. The energy eigenvalues corresponding to the solution of the above equation are The wavefunctions for this problem are described by a product of a Gaussian and a Hermite polynomial. They are not shown since only the energy eigenvalues are used in formulating a description of a molecule's internal vibrational energy. Evident from the equation above is that every polyatomic molecule must have a nonzero vibrational energy, even at absolute zero Kelvin. The vibrational energy at zero Kelvin is known as the 'zero-point energy'. It plays a critical role in accurately describing the internal energy 2 2 2 1 22 d kx E m dx        1 1,2,... 2 v k E v v m    17 of a molecule and leaving it out can lead to substantial errors in predictions of thermodynamic and/or kinetic properties. 2.2.1.3 The rigid rotor problem Implicit in the rigid rotor assumption is that a mass rotating around a fixed axis also maintains a fixed distance. Although this is untrue under some circumstances, the quantum rigid rotor serves as a good model for describing the rotational energy of a molecule. The Schrödinger equation for a rigid rotor is where  represents the position about an axis with a mass, m, fixed at a distance r. The allowable corresponding energies are: I = r 2 is the moment of inertia of the rotating mass, and  the reduced mass (in case there are multiple particles rotating about the same frame of reference). J is the angular momentum quantum number. 2.2.1.4 The hydrogenic atom As one would expect, none of the approximations outlined above can completely describe the motion of an electron around a nucleus. This is the problem of the hydrogenic atom. The Born-Oppenheimer approximation assumes that electrons in a molecule are moving in a field of fixed nuclei. This approximation simplifies the problem in two ways: it 22 2 2 d E Id        2 1 0,1,2,... 2 l m E J J J I    18 allows us to neglect the kinetic energy of the nuclei and assume nuclear-nuclear repulsion to be constant [2]. For this system, it is convenient to write the Schrödinger equation as where H is known as a Hamiltonian, an operator containing descriptions of how the electron, e, and nucleus, N, interact: The first two terms of the Hamiltonian represent the kinetic energy of the electron and nucleus, respectively. The last term represents the Coulomb potential between the electron and a nucleus of atomic number Z separated at a distance r.  0 is the vacuum permittivity, ħ is Plank's constant, and m is the mass of the respective entity. An exact solution exists for this problem and the wavefunctions for the first few energy levels can be found in any relevant text; they are products of radical functions (involving associated Laguerre polynomials) and spherical harmonics. Each wavefunction is a function of three different quantum numbers: n, the principal quantum number, l, the angular momentum quantum number, and m, magnetic quantum number. Acceptable energy levels are given as HE    2 2 2 22 0 2 2 4 eN eN Ze H m m r        4 222 0 1,2,... 8 n e En hn      19 2.2.2 Hartree-Fock (HF) and post HF approximations Although the Schrödinger equation can only be analytically solved for the hydrogen atom, numerous tools exist to obtain chemically accurate relative electronic energies for molecular systems involving H, C, and O atoms. With certain caveats kept in mind, modern day post Hartree-Fock techniques and Density Functional Theory (DFT) are excellent tools for determining relative energies of stationary points on a potential energy surface (PES) of interest. With increasing molecular size comes increased probability for more careful inspection of a reaction path on a PES. An example includes finding the lowest lying electronic state of a species with multiple unpaired electrons. In addition, larger molecules may also undergo reaction from the same reactant to the same product through multiple transition states. Hartree Fock theory is a way to variationally estimate the energy of a system of electrons and nuclei. The form of the wavefunction used is left up to the user. Implementation and advancement of HF theory essentially coincides with computer technology. However, it was soon realized that techniques beyond HF were necessary to make predictions accurate enough to reconcile experimentally determined chemical and physical data. The Born-Oppenheimer approximation helps to simplify the Hamiltonian for a many-electron system by neglecting the motion of nuclei. That is, we assume the nuclei are fixed 20 relative to the electrons, allowing us to remove the nuclear kinetic energy term from the Hamiltonian. Additionally, this allows us to neglect the repulsion between nuclei. Among the post-HF methods utilized in this work, coupled-cluster (CC) theory is used the most [3]. Specifically CC with singles, doubles and perturbative triples excitations are employed with moderately sized basis sets. Known as CCSD(T), this method has become the gold standard for single-reference calculations [4]. 2.2.2.1 T1 diagnostic The T1 diagnostic and its variants give an idea of the multireference character of the wavefunctions describing an electronic system. For coupled cluster calculations, where t is the array of the amplitudes of the singles excitations and N elec are the number of electrons in the occupied and virtual orbitals. T1 values above 0.02 suggest that the coupled cluster results should be used with caution, or possibly some other method utilized instead [4]. Wherever CCSD(T) calculations are mentioned throughout this work, the T1 diagnostics are reported or mentioned. 2.2.3 Density functional theory (DFT) For many combustion-relevant chemical species, especially those of higher molecular weight, experimental structural and force constant data are unavailable. It is thus 1 1 elec Tt N  21 imperative to follow best practices when choosing a model chemistry for a system or potential energy surface of interest. When available, experimental data may be used to validate a particular model chemistry. However, for most hydrocarbon species, DFT has proven to be very reliable and is a current default method for determining both geometries and force constants (vibrational frequencies). All of the geometries presented in this work are obtained using density functional theory, mostly through use of the B3LYP functional [5]. This hybrid scheme has proven invaluable in providing accurate geometries of both stationary and critical points along a potential energy surface. One important caveat, however, is that the right functional must be chosen for the right molecule. For very large molecules like hexaphenylethane (pictured in Figure 3.2), where the combination of steric repulsion and van der Waals interactions play important roles, B3LYP fails to predict the central C-C bond characteristics [6]. Fortunately, the user of modern quantum chemistry software has an essentially endless and expanding database of density functionals from which to choose from, tailored for almost every conceivable purpose. Whenever DFT is used throughout this work, sensitivity analysis and calibration calculations have been used to ensure that the model chemistry is appropriate for the system in question. 2.2.4 Geometries of stationary and critical points HF and post-HF methods greatly benefit from the use of larger basis sets, and techniques like basis set extrapolation of energy. When utilizing DFT however, convergence of 22 geometric and electronic properties with increased basis set size is much steeper. In many cases, the moderately sized 6-31G(d) basis set is chosen when performing calculations on larger molecules because the cost to benefit ratio is optimal at this level [7]. The following example illustrates several points regarding the choice of basis set in calculating the geometry of cyclohexane, with 6-31G(d) being the smallest chosen basis set. Again, this basis set is in fact reasonably large. Comparisons are made with selected experimentally determined bond distances and angles for cyclohexane [8]. Figure 2.1 Side and top views of the cyclohexane molecule illustrating several characteristic geometric values: axial and equatorial C-H bond lengths, C-C ring bond lengths, CCC bond angles and CCCC dihedral angle. See Table 2.1 for comparison of corresponding experimentally and theoretically (DFT) determined values. Table 2.1 compares the cyclohexane geometric properties computed using various basis sets; the properties are identified in Figure 2.1. Bond lengths agree well within one-tenth of an angstrom for all basis sets chosen. All bond angles are in favorable agreement as well. The number of basis functions gives an idea of time required for each geometry optimization (the calculations above were performed in serial, with the final geometry of 23 one job used as input for the subsequent job, rendering a comparison of CPU time usage useless). Although not evident in this example, inclusion of polarization (d, 2d, p, etc.) in basis sets is important for systems with ring strain (cyclohexane has none). It is evident from the above table that the inclusion of diffuse functions (designated by the '+' symbol), has a negligible influence on the overall geometry. This is in fact the case for most pure hydrocarbon compounds of relevance to fossil fuel combustion. Thus, for much of the work in this thesis, exclusion of diffuse functions in basis sets is justified. However, use of diffuse functions becomes necessary when studying systems of ionic nature, van der Waals clusters, and electronic excited states. Table 2.1 Selected cyclohexane geometric properties for various basis set sizes, calculated with the B3LYP hybrid density functional. See Figure 2.1 for depictions of listed parameters. Distances in angstroms; angles in degrees. Basis Set no. basis functions r(c-h) ax r(c-h) eq r(c-c) (ccc) (cccc) 6-31G(d) 114 1.101 1.098 1.537 111.51 54.71 6-31G(d,p) 150 1.100 1.097 1.537 111.47 54.72 6-311G(d,p) 180 1.098 1.095 1.536 111.53 54.53 6-311+G(d,p) 204 1.098 1.095 1.536 111.53 54.53 6-311++G(d,p) 216 1.098 1.095 1.536 111.57 54.54 6-311++G(2d,p) 246 1.097 1.094 1.534 111.56 54.46 6-311+G(2d,p) 234 1.097 1.094 1.534 111.52 54.59 6-311G(2d,p) 210 1.097 1.094 1.534 111.52 54.55 experimental* - 1.098 1.089 1.536 111.08 55.82 *values from [8] 24 Comparison of 6-31G(d) and 6-311G(2d,p) B3LYP results with the experimental data show negligible difference. That is, the differences are so slight no recommendation can be made on basis set choice even though the number of basis functions differs by a factor of two. The computing power available today makes use of even the larger basis set rather quick and routine, especially in the context of DFT. Frequency calculations take longer and, in general, scale linearly in molecular size with CPU time, but such jobs can be left to run overnight. For the varying basis sets above, simple tests show the contribution of force constants to the vibrational partition function result in comparatively negligible differences. In summary, it is not computational inconvenient to use a larger basis set for systems with less than 15 heavy atoms. Additionally, during the course of a study, it may be important to consider electronic excited states for which diffuse functions are beneficial. For instance, isomerizations from six-membered to five-membered ring structures is common under combustion conditions for cyclohexane and mono-alkylated cyclohexane derivatives. Although ring strain is essentially non-existent in cyclohexane, it does exist in 5 and 7 member ring structures, where inclusion of polarization is more important. For the reasons outlined above, much of the work in Chapter 5 employs the B3LYP/6- 311++G(2d,p) level of theory for optimization and frequency calculations. Although not usually an issue, one potential problem with large and diffuse basis sets is that they tend to make geometry optimization difficult [4]. 25 Lastly, it is interesting to note the failure of B3LYP to capture the relative lengths of the axial and equatorial C-H bonds. The equatorial C-H lengths are comparable with those for secondary substitution in propane, while the axial bond lengths are all longer and halfway between those for secondary and tertiary substitution [8]. Although such differences may have more pronounced kinetic (site-specific) effects at lower temperatures, they are negligible under the high-temperature regime that is the focus of much of this work. 2.2.4.1 Molecular point groups and rotational symmetry Modern computational quantum chemistry packages are of great use, but not necessary for determining another very important property directly resulting from the structure of a molecule - symmetry. In fact, knowledge of molecular symmetry is frequently exploited to greatly reduce the overall computation time for high-level quantum chemistry calculations. Rotational symmetry also contributes to an overall reduction to a molecule's entropy by a factor of Rln( ), where  is the rotational symmetry number and R the gas constant. As an example, a C 60 Buckminsterfullerene belongs to the I h point group and has a rotational symmetry of 60. Neglecting the rotational contribution to entropy results in an error of over 8 cal/mol-K in an estimation of its standard state entropy. 26 The concept of rotational symmetry as it pertains to statistical mechanics is also very important, as the quantity 1/  is used ensure that rotational states are not over counted. 1/  is introduced as a multiplicative correction factor to the rotational partition function. Depending on the molecule, this correction factor can have an impact on the rate of a chemical reaction and, usually to a lesser extent, its standard state entropy. Table 2.2 Symmetry point groups and corresponding rotational symmetry numbers [4]. m = 2, 4, 6, . . . ; n = 2, 3, 4, . . . Point Group   C 1 1 C i 1 C s 1 C 2v 2 C v 1 D h 2 S m m/2 C n n C nv n D n 2n D nh 2n D nd 2n T 12 T d 12 O h 24 I h 60 The literature is populated with tables such as the one above, and it never hurts to have one nearby. Although the symmetry number can potentially change a calculated rate constant by a factor of two or more, there are some instances when the concept of symmetry number at high temperature can become somewhat of an art. Consider the cyclohexane molecule in vacuum at 298 K. It lies comfortably in the chair conformation 27 (D 3h point group with a rotational symmetry number of 6) at this temperature, requiring about 10 kcal/mol to isomerize to the less stable twist-boat conformation. The twist boat structure on the other hand belongs to the D 2 point group, and thus  = 4. At 1000 K, the available thermal energy will greatly increase the equilibrium population of the boat conformation. In fact, the chair-boat isomerization may seen as being so quick that the C6 structure can be assumed as dynamically flat, in which case the rotational symmetry would be equal to that of benzene,  = 12. In fact, the National Institutes of Standards and Technology (NIST) group additivity code treats all monocyclic rings as 2D flat geometric structures when determining rotational symmetry. Because consistency is maintained, differences in overall entropy contributions between NIST and Benson group additivity derived standard entropies are usually no greater than a factor of Rln(2). Still, it is in good taste to both be consistent in how symmetry is assigned and to be cognizant of the rationale and treatments used in the literature. 2.2.4.2 Moments of inertia The atoms of a molecule may be modeled as point masses. After assigning a cartesian frame of reference, the three principle moments of inertia can be defined, I ii , where ii denotes the xx, yy, and zz principle axes. The corresponding rotational constants, B, are related to the inverse of the moments of inertia: B = 16.45/I cm -1 , for I in amu. These parameters are used to assign what are denoted as the active and inactive rotational degrees of freedom of a molecule. An active degree of freedom is any of the 3N degrees of freedom that may randomly exchange energy with other internal degrees of freedom. 28 As will be discussed later, this exchange of energy among degrees of freedoms is included in the density of states. These include all internal vibrational, rotational, and torsional modes, and usually one of the external rotational degrees of freedom, but exclude the three translational ones. Inactive degrees of freedom do not exchange internal energy but are still used in the calculation of rate constants, as will be discussed later. In general, the shape of any given molecule can be modeled as a symmetric top (I xx  I yy , and I zz unique), possessing two external inactive degrees of freedom, and one external active degree of freedom. Thus, the 2D inactive external rotational constant is B 2D = sqrt(B 1 B 2 ), where B 1 and B 2 are the external inactive 1D rotation constants. When accounting for internal rotational modes, a number of approximations are available to deal with calculating the relative reduced moment of inertia of the rotating atom/group with respect to the rest of the molecule [9, 10]. Multiple rotors within a molecule are treated separately. For instance, there are always two groups defined: that which is rotating about some bond and the 'rest' of the molecule, which may include other rotors that can be defined in the same manner. In the discussion to follow, the moment of inertia is denoted I (m,n) ; n indicates the level of approximation for a rotor attached to a fixed frame of reference before any reduction of this moment of inertia because of coupling with external or other internal rotations, while m indicates the level of approximation of the coupling reduction [9]. Table 2.3 below describes various levels of approximation in calculating the moment of inertia. 29 Table 2.3 Definition of terms used in various levels of approximation for moment of inertia, I (m,n) [9] Level of approximation Description of moment of inertia, I (m,n) n = 1 I of the rotating group is computed about the axis containing the twisting bond n = 2 I is computed about the axis parallel to the bond but passing through rotating group's center of mass n = 3 I is computed about the axis passing through the centers of mass of both the rotating group and the remainder of the molecule m = 1 I of the rotating group is not reduced m = 2 the reduced moment due to coupling with overall molecular rotation is approximated m = 3 the coupling with molecular rotation is properly computed m = 4 the coupling with molecular rotation is includes an additional term In the simplest approximation, where both m and n = 1, the moment of inertia is defined as that about the rotating (z) axis where L and R represent arbitrarily chosen "left" or "right" groups (simply a conventional bookkeeping notation) . Once calculated, the moments of inertia of the L and R groups may be reduced in the I (2,n) approximation:     1,1 22 j j L or R I m x y          2, 1, 1, 1 1 1 n n n LR I I I  30 The kinetic coupling of one internal rotation with others is incorporated into the m = 4 approximation. These formulae are not included here, but may be found elsewhere [9, 11]. In a study of the reaction between CO and HO 2 , the overall rate was calculated using three levels of approximation (I (2,1) , I (2,3) , I (3,4) ), including treating all internal rotations as harmonic oscillators [11]. This sensitivity analysis showed a fortuitously small difference in computed rate constants for the high pressure limit rate constant across the temperature range of 500 - 2500 K. The authors note that without a careful treatment of the hindered rotors, the uncertainty in the predicted rate constant would have been as large as the scatter in the experimental data they compare their results against. The influence of internal rotor treatment on reaction kinetics is discussed at length in section 2.4.5. 2.3 Thermodynamic Properties Inclusion of accurate thermodynamic properties for the entire temperature range of interest to combustion is critical for a number of reasons. Firstly, the equilibrium constant may be used by the Chemkin [12] suite of utilities to calculate reverse rate constants if they are only provided in one direction, thus necessitating reliable molecular thermodynamic properties as a function of temperature. These data are typically provided in the form of NASA polynomials [13]. Accurate thermochemical parameters are also critical to capturing the evolution of products and byproducts within combustion systems (e.g., CO 2 , HO 2 , soot). 31 2.3.1 Standard thermodynamic state functions The first law of thermodynamics is that of energy conservation. The internal energy, U, of a truly isolated system must remain constant. A system undergoing change in volume because of expansion has an internal energy defined as enthalpy, H It is convenient to rewrite the above equation to reflect the fact that most of the thermodynamic quantities in chemistry are reported on a molar basis. Using the ideal gas assumption, the quantity may be replaced with RT to give and the derivative of both sides of the above equation taken with respect to temperature, T: This equation can be reduced to Lastly, by recognizing that the constant pressure and constant volume specific heats are respectively defined as and we arrive at the relationship between the two specific heats: H U PV  h u Pv  Pv h u RT    RT hu T T T       PV hu R TT                P P h c T       V V u c T       32 The constant pressure specific heat is necessary for calculating the sensible enthalpy  the enthalpy of a substance or molecule at temperatures other than 298 K. At zero Kelvin, the internal energy is necessarily equal to the enthalpy, and both may be defined as where refers to the molar zero point vibrational energy of the molecule (the V is dropped from the acronym). Through the harmonic oscillator approximation, this equals the sum of the vibrational energies for all the internal degrees of freedom. The ZPE for all polyatomic molecules is necessarily greater than zero, and is undefined for monatomic species (e.g., H ). A determination of a systems state at temperatures other than zero Kelvin can be obtained through integrating the specific heat up to the desired temperature. For example, at 298 K, the enthalpy of a molecule may be defined as or A convention of thermodynamics is the designation of a standard state, a reference pressure and composition. The standard state of a substance at a specified temperature is PV c c R      00 elec h u E zpe    zpe     298 0 298 0 P h h c dT     298 0 298 elec P h E zpe c dT     33 its pure form at 1 bar, and is represented by a the symbol, . Consider an arbitrary chemical reaction where  i is the stoiciometric coefficient of species i, which spans all reactants; j of course spans all products. The standard state change in any of the standard thermodynamic state functions, , (e.g., enthalpy, entropy, Gibbs free energy) from reactants to products is then defined as where is the standard thermodynamic state's value of formation f, at temperature T, for species i. The thermodynamic definition of entropy concerns its internally reversible heat change in an isolated system: and the entropy change for any spontaneous process is necessarily greater than or equal to zero: tan ij reac ts products ij    , o rT F  , , , , , tan o o o r T j f T j i f T i products reac ts F F F      ,, o f T i F Q dS T      0 tot S  34 Spontaneity is of course satisfied when change in Gibbs free energy (G = H - TS) is less than zero. Lastly, he standard state entropy is defined as 2.3.2 Statistical mechanics derivation of thermodynamic functions All thermodynamic state properties can be derived from statistical mechanics through the canonical partition function It can be shown that the internal energy of a system is defined as The above equation can then be used to determine a molecule's enthalpy and specific heats. The entropy is defined as The other thermodynamic state properties may of course be derived from the equations above. 2.3.2.1 Molecular partition functions It can be shown that, under the ideal gas assumption, the canonical partition function may be rewritten as 0 T P dT sc T       , ,, i B E N V kT i Q N V T e    2 , ln B NV Q U k T T      , ln ln B NV Q S Q k T T      35 where q is defined as the molecular partition function. The total molecular partition function is the sum over all possible energy states at a given temperature T: Under the [reasonable] assumption that the sum of all energy states is separable, the total molecular partition function can be rewritten in terms of their individual contributions. These energies depend on the assumed quantum mechanical model describing the degree of freedom of interest and are derived from solution of the corresponding Schrödinger equation (see Section 2.2.1). Partition functions for the various degrees of freedom of a molecule, including the molecular electronic partition function, are given below. They can be derived from combining the equation for the molecular partition function with the appropriate expression for the allowable quantized energy states. The electronic partition function can be defined as where g i is the degeneracy of the i th electronic state and i = 1 represents the ground state [14]. Electronically excited states, where i > 1, are typically only encountered under very high temperature and may therefore be neglected under most conditions of relevance to     , ,, ! N q V T Q N V T N      , i B E kT tot i q V T e              ,, tot elec trans rot vib q V T q T q V T q T q T    2 12 B E kT elec q T g g e        36 the work presented here. For the 3N translational degrees of freedom, the corresponding partition function is In practice, the volume is commonly replaced by the factor nRT/P, through the ideal gas law. The vibrational partition function below is written for a single vibrational mode under the harmonic oscillator assumption. For polyatomic molecules, the total vibrational partition function is the product over all internal vibrational modes (3N-5 for linear molecules and 3N-6 for nonlinear molecules). The total molecular vibrational partition function is simply the product of over all vibrational modes. The partition function for a one dimensional rotor can be derived from the allowable energy levels of the rigid rotor and the molecular partition function: Here, B is the rotational constant, as described in 2.2.4.2. In reality, not many molecular entities maintain only one degree of freedom in rotation. For two and three dimensions, the corresponding moments of inertia may be used to calculate the rotational partition function. For instance, the rotational partition function for a nonlinear polyatomic molecule is approximated by   3/ 2 2 2 , B trans Mk T q V T V h       2 2 1 B B hv kT vib hv kT e qT e      ,1 , B rot D kT q V T B     12 ,3 , BBB rot D x y z k T k T k T q V T B B B    37 The last class of commonly utilized partition functions are those describing various types of internal molecular rotors and coupled internal bending/rocking motions. In essence, they are treated using the same approach as all other degrees of freedom: define a Hamiltonian and find the eigenstates, and use the definition of a molecular partition function to yield a tractable solution or approximation. The treatment of internal rotational degrees of freedom will be addressed in 2.4.5. 2.3.2.2 Density and sum of states In order to define a rate constant as a function of a system's vibrational, translational, rotational, and electronic energy, we must know what states are available for energy distribution at a given interval E to dE. For chemically reacting systems, this information must be known or assumed for both reactant and transition state. The quantum partition function is related to the density of states as follows: In this manner, all relevant partition functions may be included in the density of states. This is performed using methods of convolution [15]. The density of states, (E), is defined as   0 exp B E Q E dE kT           0 exp exp i B i B E kT E E dE kT            38 and can be easily interpreted as the degeneracy of all available active configurations at a given energy level E, in a molecule. Recall that these configurations are quantized and determined from what degrees of freedom are designated as active. They typically include vibrational and rotational degrees of freedom. It is frequently necessary to also include internal free or hindered rotational degrees of freedom as well (in place of their harmonic oscillator counterparts). The sum of states is simply the integral of the density of states The sum of states represents the number of available states a molecule may assume up to and including energy level E. The density of states is needed to compute both the microcanonical rate constants and energy transfer probabilities, later to be discussed. The sum and density of states are both computed using the Whitten-Rabinovitch approximation [16]. The figure below gives an example of the density of states for two isomeric species - the resonantly stabilized benzyl radical and the 2-methylphenyl radical.     0 E W E E dE    39 Figure 2.2 Density of states calculated for the 2-methylphenyl and benzyl radicals plotted as a function of energy, with a spacing of 10 cm -1 . Plotting the density of states as shown in Figure 2.2 helps to illustrate several points. It is interesting to observe the quantized nature of available energy states under the region of low energy. However, at around 1000 cm -1 (~3 kcal/mol), the density of states essentially assumes a continuous form with increased energy. The exponential rise is also characteristic. At 10 4 cm -1 (29 kcal/mol), there are already over 100 million available ways to distribute this energy within these molecules. Lastly, a comparison of the density of states between the two species illustrates an important point with kinetic relevance. The benzyl radical is resonantly stabilized, and all of its internal vibrational degrees of 0.1 10 1000 10 5 10 7 10 100 1000 10 4 2mp benzyl  E), states/cm -1 Energy (cm -1 ) E grain = 10 cm -1 40 freedom are relatively high in energy. The CH 2 radical group has a very high rotational barrier and is therefore properly treated by the harmonic oscillator assumption. On the other hand, the 2-methylphenyl radical contains a CH 3 group with a much lower energy barrier to rotation. Lower energy modes have a large influence on the magnitude of molecular partition function. In essence, this is the reason the density of states for 2- methylphenyl is higher for all energy. Furthermore, as a consequence of treating this degree of freedom as a weakly hindered internal rotor (by using a more proper partition function), the difference in this case becomes even greater. In the following work, the counting of ro-vibrational states are performed by using the method described by Gilbert and Smith [15]. The counting of hindered rotational states are performed by using the method described by Knyazev et al. [17]. 2.4 Reaction rates Chemical reactions are exceedingly more complex than what is taught at the collegiate level. In fact, most college chemistry courses cover the development of rate theory only up to the end of the 20th century. For most applications, the Arrhenius expression sufficiently describes the temperature dependence of a chemical rate - the time it takes for a molecule of one state to surmount a barrier and transform to another state. This information can be empirically derived by fitting with experimental data. 41 Most of the reactions examined in detail here are unimolecular in nature. The rate constants obtained from the detailed calculations are typically given as a function of temperature, k(T). This is the known as the canonical form of the rate, while the rate as a function of internal energy, k(E), is referred to as the microcanonical rate (since it describes the rate as a function of a specific molecular energy level). The canonical form describes the rate for an ensemble of thermally distributed reactants maintained through collisions. In the following discussion and throughout this work, all references to transition state theory assume the canonical form. 2.4.1 Transition state theory (TST) Evans and Polanyi, and Eyring are credited with the formulation of TST [15, 18, 19]. Central to this theory is the assumption that a molecule following a reaction coordinate passes through a transition state (denoted here as a superscripted dagger, † ) before going on to products. The transition state is a saddle point with positive curvature in all degrees of freedom except the one corresponding to the reaction coordinate, for which it is necessarily negative. This degree of freedom may be properly treated as a translational degree of freedom. For a reaction of first order, the specific rate constant may be written as where l is the reaction path degeneracy,  is the tunneling coefficient, and all other quantities have been previously defined. The TST rate constant is by definition a rate at   † 0 exp tot B B tot Q kT E kT kT hQ       42 the high pressure limit. A major assumption of TST is the equilibrium between reactant and transition state - that the concentration of the forward crossing transition states is the same in the steady state as it would be at total equilibrium where no net reaction was occurring [20]. 2.4.2 The microcanonical rate constant TST assumes the transition states are in equilibrium with the reactant molecules, and is therefore a canonical formulation. The microcanonical rate expression used in RRKM theory is most commonly expressed as where l is the reaction path degeneracy, Q r,in are the rotational inactive partition functions for the transition state ( †) and reactant, W(E † ) the sum of states for the transition state where E † = E – E o for the active degrees of freedom in the activated complex, E o is the energy barrier, (E) the density of states for the active degrees of freedom of the reactant, and h is Plank’s constant. 2.4.3 Pressure dependence Many reactions exhibit strong pressure dependence. Because this pressure dependence typically manifests as a decrease in observed rate relative to that at the high pressure limit, these types of reactions can serve as rate limiting steps within larger chemical       † † , , r in r in WE Q kE Q h E   43 models or mechanisms. Clearly, a theory beyond transition state theory is necessary (although it can be more than sufficient for pressure -independent reactions). Consider a reactant, A, undergoing collision with another body, M, to a rovibrationally excited level A*. This may be represented in a reaction sequence as A + M  A* + M. The excited molecule, A*, may further react and go on to products or be stabilized back to A upon subsequent collision(s). The entire unimolecular process is captured by the well-known Lindemann-Hinshelwood mechanism R 2.1 R 2.2 R 2.3 Immediately clear from the above mechanism is that even 'unimolecular' reactions are really not unimolecular in a strict sense. The presence of a bath gas (referred to as a third body in bimolecular reactions) can profoundly alter the overall rate of reaction from A to products. The law of mass action dictates this rate to be The subscript 'uni' is commonly used for such schemes but henceforth assumed and dropped. Unless otherwise noted, all reactions described in this work are unimolecular. Upon writing the rate equations for the steady state formation of A* and solving for the rate of production of the products, we find * f k A M A M      * b k A M A M      2 * k A products            11 uni d A d products k A dt A dt    44 Equating the above with the overall product formation from A, A  products, the pressure dependent rate of product formation is then In the limit of low pressure, [M]  0, and k = k f [M], while in the limit of high pressure, [M]  , and k = k 2 k f /k b . Thus, the high pressure limit rate constant in this case in independent of pressure, while the low-pressure limit rate is linearly proportional to pressure. The Lindemann-Hinshelwood pressure-dependent rate gives a form as shown in the figure below. In most cases, it overestimates the rate in the 'falloff' region - the region where unimolecular rate transitions from pressure-dependent to pressure-independent.       2 2 b b d products k k A dt k k M     2 2 b b kk k k k M   45 Figure 2.3 Rate coefficient of C 7 H 7 + H (+M) = C7H8 (+M) at 2500 K, with Troe parameters fitted from results of [21]. The Lindemann-Hinshelwood derived rate is included to illustrate that higher levels of approximation almost always result in lower rates in the fallow region, as depicted by the blue line. 2.4.3.1 RRKM unimolecular reaction scheme Introduced in the 1920s, it is not a surprise that we have come a long way in more accurately treating the pressure-dependence of chemically reacting systems. In fact, not long after the introduction of the Lindemann-Hinshelwood mechanism and the introduction of Rice-Ramsperger-Kassel-Markus (RRKM) theory [15, 20], a connection was made between the rate of collision frequency with intermolecular energy transfer. In the simplest approximation, the strong collision model, all of a molecule's internal energy is lost upon a single collision. This is of course incorrect, and as a result, the weak collision model was introduced. Under this assumption, many collisions are required to 1.E-13 1.E-12 1.E-11 1.E-10 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 k (cm 3 molecule -1 s -1 ) Pressure, torr k o k  k Troe k Lindemann 46 activate or deactivate a reacting molecule. To understand these phenomena, reactions R 2.1 - R 2.3 are rewritten as R 2.4 R 2.5 R 2.6 where  is the collision frequency, dk a (E) the energy dependent rate of activation resulting from collision with another body, and k(E) the energy dependent microcanonical rate constant defined in 2.4.2. Note that in the above equations, A † (the critical TST geometry) is introduced. Following the assumption of TST, once this critical geometry is achieved, there is no crossing back to activated reactants. Thus, the rate of formation of A † is by definition equal to the rate of formation of whatever products it is associated with. Another assumption is that any given amount of energy imparted on a molecule is immediately distributed throughout all available active degrees of freedom in a random fashion. This is commonly referred to as the ergodicity assumption, although a more proper title would be 'instantaneous ergodicity'. Ergodicity is indeed not always instantaneous, and in such cases the RRKM derived rate constant will exceed the true rate constant. Such phenomena, not described here, are classified as non-RRKM behavior. For most combustion relevant reactions however, energy randomization is sufficiently faster than molecular isomerization or dissociation. This above mechanism is formally identified as the RRKM reaction scheme. After performing an analogous analysis to that shown for the Lindemann-Hinshelwood     * a dk E A A E       * A E A           * † † kE A E A E     47 mechanism and integrating over all energy levels, the overall rate of production of products is found to be Because the rate is integrated over all available energy states, it is by definition the canonical (thermal) rate constant. The collision frequency can be defined as Under the chemically reacting conditions found in both the atmosphere and combustion systems, the ideal gas law remains valid and allows for calculation of [M] through P/RT. However, the gas-kinetic theory assumption that all collisions are elastic does fail under many such conditions. A more proper definition of collision frequency thus includes particle-particle potential interaction through introduction of the reduced collision integral,  ij (l,s) The reduced collision integral is covered in extensive detail elsewhere [22, 23]. In essence, it captures the non-ideality of real colliding molecules by incorporating aspects of the interaction potential between two species. In returning to the discussion of the strong collision assumption (that each collision completely activates or deactivates a species), it is no surprise it fails in many cases. A natural first step to addressing this was to simply introduce a collision efficiency factor,  (0 <   1), as a multiplicative factor to         0 , 1 E k E P E dE k T P kE        2 8 B ij kT M        , 21 8 ls B ij ij kT Ms     48 the collision frequency. The collision efficiency factor is inherently related to the amount of energy transferred per collision. Many different models and approximations, not covered here, are used to estimate values of  for various reacting molecules and bath gases. Much of this information is fitted to experimental data or derived from experiments themselves. The problem of energy transfer is examined in more detail in the following sections. 2.4.4 Kinetic Monte Carlo solution of the Master Equation The Master Equation (ME) for chemically reacting systems is used to describe the time- evolution of a species on a potential energy surface. In this work, only thermal and chemically activated unimolecular reactions are considered. Solution of the ME is performed using a kinetic Monte Carlo approach based on Gillespie’s exact stochastic algorithm [24, 25]. Necessary to solving it are several things: the same input parameters for RRKM calculations are of course also required in an ME calculation – translational, rotational, vibrational partition functions, internal moments of inertia, active, inactive, and any hindered/coupled rotor information. The molecular weight and LJ parameters of all participating species are also necessary. This information is used to subsequently determine the microcanonical rate constants, the inelastic collision frequencies, densities and sums of states, and collision probabilities. Even though considerable uncertainty can be introduced from the many input parameters and approximations used, solution of the ME is currently one of the most accurate 49 methods of determining a rate constant. Rather than account for pressure dependence through a collision efficiency factor, β, the collision probability is directly calculated for the reactant and every isomer considered, at a specific temperature and pressure. Another beautiful thing about the master equation is that it can be written to account for reacting systems of arbitrary complexity. In contrast, when using TST and RRKM methods, overall rates of decomposition of a molecule, for instance, must be solved for as functions of other microcanonical rate coefficients and collision frequency factors in the reacting systems, commonly through assuming a steady state population of a reactive intermediate. For very complex systems, this process can be quite tedious. Because this work mainly involves reactions with high pressure dependence, they are all treated through use of the Master Equation, written below in discrete form [14]. k ij is the second order rate coefficient for the collisional energy transfer of a molecule from state E j to E i . k m (E i ) is the RRKM microcanonical rate constant for the m th channel of isomerization or dissociation. A(E i ) is a species at a specific energy state E i , and bracketed terms are those in concentration units. The bath gas is denoted by M. This equation describes the time evolution of the competitive process of collision activation and deactivation, isomerization, and dissociation into two or more species. Each of the terms in the above equation can be interpreted as follows:               i ij j ji i m i i j j m d A E k M A E k M A E k E A E dt                     i d A E dt    rate loss of A i due to reaction rate of collisional production of A at energy level j collisional rate loss of A at energy level i - - 50 The failure of the strong collision assumption lead many to intuit a probability-based model of collisional energy transfer. Specifically, a model assuming a probability distribution function for the amount of energy transferred during a collision event. Among the numerous ways to go about solving this equation, we choose to do so by introduction of the energy transition probability where k coll is the collision rate constant. Consider a molecule undergoing collision with a bath gas. P ij is the probability of change in energy from energy state j to state i. Values for the transition probability are subject to the condition that their sum over all energy levels cannot exceed 1 (i.e., ). In discreet form, this is simply . In solving the master equation, is it therefore convenient to rewrite it as such Among the wide variety of collisional energy transfer models available, only the exponential down model is the adopted in this work. It is based on the algorithm described in "Unimolecular Reactions" by Holbrook, Pilling, and Robertson [20]. In general, many of these models produce similar results and the choice is left to the user [15]. ji ji coll k P k  0 1 ij P di    0 1 ij j P                   i ij j ji i m i i jm d A E Z M P A E P A E k E A E dt                 51 The top portion of the equality defines the energy transfer probability for activating collisions, for a transition from energy E i to E j . C i and C j are normalization constants ensuring the integral of the collision probability over all energy levels is equal to unity. The principle of detailed balancing is used to formulate the bottom portion of the equality, defining the energy transfer probability for deactivating collisions. Given enough time, we expect that the following relation to hold: P ij f(j) = P ji f(i), where the function f is the Boltzmann population of species A at energy E i or E j . From this, we may write This relationship ensures the distribution of a non-reacting system over infinite time is indeed the Boltzmann distribution. is the downward energy transferred upon collision of a parent molecule with a bath gas. To date, quite a large number of experimental and computational studies have been performed on the direct determination of energy transfer. Most recently, Jasper and Miller have performed an extensive theoretical study on the collisional energy transfer of CH 4 with eight different bath gases [26]. Their trajectory simulations provide accurate quantitative insight into energy transfer for these bath gases and their results can be easily       1 ji d j i B d E E E j ij E E E k T i i j Ce P Ce                  for j  i for j < i     exp exp j i ji i ij j BB E E P E P E k T k T                 d E  52 extended to make quantitative assumptions about energy transfer for molecules other than just methane. In general, trajectory studies and experimental measurements of energy transfer agree very well. For example, agreement is satisfactorily within 50% for comparison of direct [IR and UV] experimental and theoretical trajectory studies for azulene in Ar at 300 K [27-32]. Some of the first experiments to probe energy transfer suggested that this quantity was independent of temperature for a given molecule-bath gas pair. Later investigations indeed prove this to be untrue. commonly takes the temperature dependent form where is the downward energy transferred at 300 K. increases with temperature, and for most of the work here, a value of 650 cm -1 for T = 2500 K was chosen based on two recent studies [21, 26] with Ar as the bath gas. The value of n is fitted based on the chosen high and low temperature values. Although the principles of energy transfer are intuitive, an explanation of the probability transfer matrix and total normalized probability of deactivation and activation as a function of energy is helpful in grasping a physical understanding of the model. The normalized energy transfer probability matrix is square with the upper diagonal corresponding to activation, and the lower diagonal corresponding to deactivation. The total probability of activation and deactivation may be calculated by appropriate d E    1 ,300 / 300 n dd E E T cm     300 d E  d E  d E  53 summations. For a species at any given energy state, i, the total probability of activation is given by summing the N states included in the specified range while the that for deactivation is given by For example, the figure below illustrates the total probability for an isolated species to undergo activation/deactivation as a function of available energy in the system. Figure 2.4 Total energy grained activation (solid line) and deactivation (dashed line) probabilities for 2-methylphenyl at T = 1400 K, = 260 cm -1 . An energy grain of 25 cm -1 was used for this calculation. P refers to activation or deactivation as defined in the equations above. More details of the 2-methylphenyl system can be found in Chapter 4. The total probability of activation at zero energy is necessarily unity, while necessarily zero at the highest considered energy of 250 kcal/mol. The exponential decrease is , N act i ji ji PP    , N deact i ji ji PP    0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 P E, kcal/mol d E  54 consistent with the definition of energy transfer probability and the notion that many collisions are required to activate or deactivate a molecule at higher energy levels, while the opposite is true for a molecule at an initially low energy level. It is interesting to note that while these formulations are used solely to more accurately describe pressure dependencies of unimolecular reactions, the energy transfer probability is not a function of pressure. The influence of pressure is only captured through the collision frequency, Z. Lastly, the probability matrix can be excessively large and must be computed for each isomer. In addition, because it exponentially vanishes with energy step size, entries far from the diagonal of the matrix are negligibly small. The present work utilizes a previously developed approach using only elements close to diagonal [33]. The accuracy of this method is maintained by ensuring that the normalization constant is met. 2.4.4.1 Stochastic steps of kinetic Monte Carlo The first step in the stochastic algorithm is to determine the initial energy of the thermally or chemically activated complex [20]. The initial energy distribution, G init , for a species undergoing unimolecular isomerization/elimination is In contrast, the initial energy distribution for a chemical activated species/adduct formed through association of two reactants is necessarily defined as   0 exp init B E G E dE kT         55 Further differences between thermal and chemically active reactions will be delineated below in section 2.4.4.2. The stochastic solution of the ME is similar to that implemented in the Multiwell program [34, 35]. Five random numbers are used at various steps in the kinetic Monte Carlo algorithm [24, 25]. With the initial energy distribution defined, a random number R 1 (0  R 1  1) is chosen from the unit interval uniform distribution and the initial energy defined such that Figure 2.4 below depicts a histogram of the initial energies of 1000 molecules at 2000 K for the thermal isomerization of 2-methylphenyl to benzyl. Also shown is the microcanonical rate constant for the isomerization reaction. The barrier height is roughly 43 kcal/mol, indicated by the asymptotic approach of k(E) to zero at this value. With enough initial molecules, this histogram would take the form of a Boltzmann distribution. In fact, it would converge to the G init array computed by the code.     0 exp init B E G k E E dE kT           '' 1 0 E R g E dE   56 Figure 2.5 Histogram of initial energies selected for 1000 2- methylphenyl molecules at 2000 K in a thermal isomerization reaction. Also shown is the RRKM microcanonical rate at the same temperature for reaction to the benzyl radical. The reaction potential energy surface is depicted in Figure 4.4. The collision frequency, along w/ the sum of the rate constants and a second random number, R 2 (0  R 2  1), is used to determine the time interval between reactions: a j is a channel-specific rate of collision or reaction at a given temperature and pressure. For example, in a unimolecular isomerization with only one other reaction channel, k = 2, 0 50 100 150 200 250 0.01 1 100 10 4 10 6 10 8 10 10 k(E), s -1 E, kcal/mol count 0 10 20 30 40 50 T = 2000 K   2 1 ln k j j R a      57 with a 1 and a 2 represented by the rate of isomerization and the collision frequency. The higher the pressure and/or temperature, the smaller the time step between reactions. The collision frequency can dominate the magnitude of the time interval, especially when the initial energy distribution for any given trial is near or below the barriers of other channels. Under such conditions, and as stated above, k(E) approaches zero, and the time interval between reaction becomes more sensitive to collision frequency. This is pictorially illustrated in Figure 2.5, where the energy-grained microcanonical rate constants for the mutual isomerizations between 2-methylphenyl and benzyl, and 2- methylphenyl (2mp) and 3-methylphenyl (3mp), all at 1400K. The inset illustrates the similarity between the 2mp 3mp and 3mp 2mp rates, whose similarities are so great that their microcanonical rate coefficients are nearly indistinguishable. Specifically, the density of states and partition functions of the two isomers are nearly identical, and the transition state is, of course, the same. 58 Figure 2.6 Histogram of initial energies selected for 1000 2-methylphenyl (2mp) molecules at 1400 K in a thermal isomerization reaction. Also shown, but not identified above, are the RRKM microcanonical rates at the same temperature for all possible reactions of the potential energy surface depicted in Figure 4.4. Once the time interval between reaction has been determined, the next step is to determine the event outcome - stabilization, isomerization, or dissociation to bimolecular products. This is performed using the current state n and another random number, R 3 (0  R 3  1), such that the transition takes path through path n 0 20 40 60 80 100 10 -5 10 -3 10 -1 10 1 10 3 10 5 10 7 10 9 10 11 0 50 100 150 200 250 k(E), s -1 E, kcal/mol 10 9 10 10 200 210 220 230 240 250 count T = 1400 K 59 If an isomerization event is chosen, the above process is repeated at the same energy. If the event results in dissociation to bimolecular products, the corresponding bin is marked and the stochastic trial is complete. Any dissociation event is treated as irreversible because of the low probability that two such moieties, once separated, will see each other again. All isomerizations are treated as reversible, with defined corresponding microcanonical rate constants in and out of all channels included in the calculation. If a collisional energy transfer process is chosen, a fourth random number, R 4 (0  R 4  1), dictates whether or not the event is activating or deactivating. If R 4  P act , activation is selected; otherwise, deactivation is selected. The amount of energy transferred is calculated using a final random number, R 5 (0  R 5  1). for activating collisions or for deactivating collisions where n in the above equations denotes the new energy level required to satisfy the corresponding condition. Stabilization is arbitrarily achieved when the total probability of activation equals that of deactivation. This criteria is rationalized on the basis that once a molecule has become stabilized to this level, it has little chance to be energized enough to take part in another reaction event [36]. 1 3 1 nk j k j j k j n a R a a       5 n ji ji RP    5 n ji ji RP    60 Lastly, time averaged reaction rates for every channel are obtained by using counters to keep track of the fate of each stochastic trial (stabilization or dissociation to products). However, distinct methodologies are necessarily used for chemically activated and thermal activation reactions, delineated in the next section. 2.4.4.2 Thermal vs. chemically activated reactions In the case of a thermally activated reaction (or equivalently, a unimolecular reaction/isomerization), the initial energy distribution is simply a function of the Boltzmann energy distribution at a particular temperature. However, in the case of a chemically activated reaction, where by definition, an activated species (adduct) is formed from the association of two reactants, the initial energy distribution is no longer just a function of the Boltzmann energy distribution at a particular temperature. The integral defining the initial energy must now take into consideration the microcanonical rate of the entrance channel. In all cases, this means that the initial energy is greater than if we were to just consider the adduct as a starting reactant in the ‘ground state’. To illustrate this, recall the initial energy distribution for a species undergoing unimolecular isomerization/elimination/dissociation:   0 exp init E G E dE kT         61 As mentioned above, a reactant formed by chemical activation has an initial energy distribution determined by including its corresponding microcanonical formation rate The initial energy therefore does not assume a Boltzmann distribution, until of course enough time passes for steady state to be reached. However, one of the main principles driving chemical activation is that the timescale of reaction dominates that for stabilization. For this reason, the rate constant of any particular channel, k i , can be written as a function of the high-pressure limit rate constant of the entrance channel Where F i denotes the fractional population of molecules in the i th channel and k ,entr is the high pressure limit rate constant of the entrance channel. In this work, the above equation is rewritten with incorporation of the equilibrium constant written from the perspective of the entrance reactants (i.e., A + B  reactant adduct) where k ,out is the high pressure limit rate constant out of the initial adduct back to reactants. The equilibrium constant is calculated in a manner consistent with the methodology used for the corresponding RRKM/ME model. Channel specific rate constants, k i , for thermal activation reactions are calculated in a different manner. The evolving population distribution each isomer or product channel is plotted as a function of time, and a linear least squares fit used to determine the rate     0 exp init E G k E E dE kT               , ,, i i entr k T P F T P k T           , ,, i i out eq k T P F T P k T K T   62 constant. In all cases, enough trials are run to ensure that coefficients of determination, R 2 , for all rates are nearly 1.0. The figure below illustrates the time evolution of benzyl in the isomerization of 2-methylphenyl to benzyl, discussed in more detail in Chapter 4. Figure 2.7 Benzyl count history as a function of time from the isomerization of 2- methylphenyl at 1400 K and 1 atm. Results are shown for three different runs, all identical except for total number of stochastic trials. The superimposed orange line on the 10,000 trial data is a least squares fit. Only a fraction of the 10000 trial results are shown as filled circles. Figure 4.4 depicts the reaction potential energy surface for this calculation. The above figure illustrates several points. For the 10,000 stochastic trial run, just over 10% resulted in isomerization to benzyl. The remaining trials in fact resulted in stabilization of the reactant. Still, enough trials have been run in this example to obtain a reliable estimate for the rate constant at this temperature and pressure. There are clearly many scenarios necessitating an almost prohibitive number of trials to be run. Examples include very high temperatures where the average available energy exceeds that of all the 0 200 400 600 800 1000 1200 0 2 4 6 8 10 12 10000 1000 100 N trials : benzyl count t, msec 63 barriers in the reacting system. In addition, lower temperatures in high barrier isomerizations require many trials since it is improbable that the initially chosen reactant energy will be high enough to surmount reaction barriers. When the upper tail of the Boltzmann distribution lies near a barrier height, the probability that states of these or higher energy will be sample can be prohibitively low. Under such circumstances, the problem may be intractable from this solution method. Fortunately, most reacting systems involve dissociation/isomerization channels and the problem can be reformulated in the context of a chemical activation problem. The principle of microscopic reversibility ensures that the rates calculated are the true rates; true only in the context of the model and all the assumptions, approximations, and input parameters that go into it. The table below illustrates convergence of rate with time, for 1400 K and 1 atm. For this case, the overall convergence of rate with trial number is fast and a calculation with 10,000 trials is sufficient, as illustrated by its unity coefficient of convergence. Table 2.4 Convergence of rate and quality of fits with trial number. Rate corresponds to the RRKM/ME calculated isomerization of 2mp  benzyl at 1400 K and 1 atm. See Figure 4.4 for a depiction of the reaction potential energy surface. N trials k (s -1 ) R 2 100 1.28 0.975 1000 1.13 0.998 10000 1.05 1.000 100000 1.08 1.000 64 2.4.5 Treatment of internal hindered rotational degrees of freedom The crudest approximation is to treat a hindered or free rotation as a harmonic oscillator. Such rotations are usually below 150 cm -1 . Another approximation is to treat the hindered rotor as a free rotor. Although appropriate for groups with little to no barrier to rotation (CH 3 group of toluene, CH 2 group of many radicals), strongly hindered rotations are better treated as harmonic oscillators than free internal rotors. Previous studies on the effect of internal rotor treatment on the reaction of CO + HO 2 have illustrated the importance of properly treating low energy internal modes of freedom [11, 33]. Proper treatment of internal rotors is particularly important for reactions such as the R + O 2  ROO class as well as subsequent H migration and O 2 addition to form hydroperoxyalkylperoxy radicals (HOOQOO) [37]. Such reactions are of course of critical importance in low temperature chemistry, where the kinetic effects of internal rotation are more pronounced. At higher temperatures where more internal energy states are available for population, proper treatment of hindered internal rotations become less important. For example, take the case of a reactant with no internal rotations undergoing reaction with some energy barrier through a transition state with one hindered internal rotor. The overall rate of this reaction is proportional to the density of states of the transition state, among other things. This density of states is in turn dependent on how the internal degrees of freedom are described. A low frequency internal rotation (typically 65 corresponding to a vibrational frequency of < 150 cm -1 ), at the crudest approximation, may be simply described by the harmonic oscillator assumption. Alternatively, the reduced moments of inertia of each rotor may be quantified by varying degrees of approximation and the potential function of the internal rotation defined. Under most circumstances, the difference in contribution to the total partition function describing a molecules' internal degrees of freedom is an overall increase from the harmonic oscillator approximation to that of a hindered or free rotor approximation. This, in turn, changes the density or sum of states by up to an order of magnitude or more, and subsequently, a calculated rate [11, 38]. As one would expect, such effects are particularly strong when only one of the reacting species (reactant or transition state) has internal rotors. However, these differences become less severe with increasing temperature, and therefore usually have a smaller impact on predicted rate constants. For symmetric hindered internal rotations, the potential energy is better described by a cosine potential [9]: In the above equation, V o refers to the height of the potential barrier and n is the symmetry of the rotating group (number of minima) about an angle . A corresponding Schrodinger equation may be defined, the energy eigenvalues obtained, and the subsequent partition functions computed much in a similar fashion as for other degrees of freedom.       1 1 cos 2 o V V n   66 2.4.6 Inclusion of channel specific quantum tunneling Tunneling as a process by which a body with finite mass passes through a barrier with a total energy less than that of the barrier height. This phenomenon is most important at low temperature but its use can significantly affect the overall predicted rates of reactions for many systems, especially those involving the transfer of a proton along a reaction coordinate. Because any physical or quantum chemistry textbook derives the tunneling coefficients for the “particle in a box” problem, only a selected amount of information is covered here. The Schrödinger equation for a particle of mass m in a box is: In the region representing a barrier with height V, the Schrödinger equation is and the solution yields a wavefunction of the form It can be shown that the transmission probability is equal to the ratio of the probability that a particle is traveling towards positive x on the left of a barrier and to the probability that it is traveling to the right on the right of the barrier [39]. The transmission probability is also referred to as the tunneling coefficient and is commonly denoted by the symbol . Various approximations exist to compute the tunneling coefficient for chemical reactions. The simplest is the Wigner approximation [40]: 22 2 8 hd E m dx      22 2 8 hd VE m dx        22 22 ' mE mE i x i x hh Ce C e     67 where  * is the magnitude of the imaginary force constant in the transition state corresponding to the reaction coordinate between reactant and product. A number of other easily applied tunneling approximations have been proposed throughout the years. The tunneling coefficient can be better approximated in a number of ways, one of which is through fitting the one dimensional reaction coordinate to an Eckart potential which takes the form [41]: where y, A, and B are defined below: and L is a characteristic length dependent on the relative barrier heights, the particles mass, and the imaginary frequency of the vibrational mode corresponding to the reaction coordinate. Thus, calculation of the Eckart tunneling coefficient requires the forward and   2 * 1 1 24 B hv T kT        2 () 1 1 Ay By Vx y y        2 exp x y L           12 2 11 22 12 1 11 22 12 * 12 A V V B V V L V V vm                    68 reverse energy barriers ( V 1 and V 2 respectively), as well as the imaginary frequency of the vibrational mode corresponding to the reaction coordinate,  * . The tunneling coefficient through the Eckart barrier is calculated by solving the Schrödinger equation for the Eckart function [42]. Figure 2.8 Form of an arbitrarily defined Eckart potential along the 1D potential energy surface of an exothermic reaction. The above methods of approximating the tunneling coefficients for chemical reactions are all restricted to a one-dimensional potential energy surface. However, a particle may not tunnel through the lowest-lying reaction coordinate. In order to account for such possibilities, multidimensional tunneling corrections have been introduced [43-45]. Such methods are much more computationally costly however, since they require knowledge of -30 -20 -10 0 10 20 30 40 -10 -5 0 5 10 15 Relative Electronic Energy Distance 69 the second derivative of electronic energy with respect to displacement at a sufficient amount of points along the reaction path. Thankfully, use of the Eckart tunneling method is sufficiently accurate for most cases [46]. A Fortran subroutine has been adopted in this work to calculate the Eckart tunneling coefficient as a function of temperature [47]. Tunneling is most commonly applied as a direct multiplier of a rate constant determined through transition state theory: where (T) is the tunneling coefficient, or transmission coefficient. Many times is it desirable or necessary to obtain the rate coefficients as a function of both temperature and pressure. To investigate the validity of the Wigner tunneling approximation used in some of the work here, it was tested by incorporation of a subroutine in the ME/RRKM code with the channel-specific microcanonical rate constants re-written as: This approach is of course less formal than writing the tunneling coefficient as a function of energy in the reaction coordinate, but nonetheless captures the influence of reaction endo/exothermicity.     † exp B o a B k T Q E k T l T kT hQ              † † , , r in a r in WE Q k E T l Q h E    70 The 2-methylphenyl potential energy surface is taken as a test case. The master equation is solved with the above modified channel-specific rate constants. This resulted in only a slight change in the predicted rate constants. The reaction has a barrier of 43 kcal/mol with an exothermicity of -22 kcal/mol. The reverse barrier is thus 65 kcal/mol and the Eckart potential takes the shape of that in Figure 2.7. Use of the newly defined microcanonical rate constant, although improperly implemented, does serve to give an understanding of the role of tunneling in reversible chemical systems at high temperature. In the Master Equation approach, the energy- grained microcanonical rate constant for each channel is computed along with its corresponding Eckart tunneling coefficient. The net effect is an overall decrease in rate constant for each channel compared to that if the traditional microcanonical rate constants for each channel were used and the final derived rate constant multiplied by the tunneling coefficient. It must be noted, however, that this behavior is more prominent in systems where the exothermicity or endothermicity is not very large, or for irreversible reactions. 2.4.7 Isodesmic Reactions Detailed combustion kinetic models contain an increasingly large number of stable and reactive intermediate species. The number of possible species, isomers, and reaction channels exponentially rises with molecule size. The accuracy with which various combustion phenomena are modeled depends on how well the model itself captures chemical heat release and equilibrium product distributions. These characteristics, in 71 turn, solely depend on the thermodynamic properties (discussed in section 2.3) of all species taking part in a chemically reacting process. Benson group additivity has stood the test of time as a tool for thermochemical property estimation [48, 49]. In fact, it accurately predicts the standard formation enthalpies of most hydrocarbon fuels and radicals. However, there are some cases (biradicals, synergistic interaction of many nearby functional groups) where its use can result in errors beyond 2 kcal/mol in formation enthalpy. For this reason, use of isodesmic reactions is an attractive alternative. Additionally, although Benson group additivity remains valuable, correct use of isodesmic reactions is considered a higher level of thermochemical property estimation. The use of isodesmic reactions in computing species thermodynamic properties is one of many that utilizes both ab initio or semi-empirical quantum chemistry calculations and (ideally) accurately measured heats of formation. This is accomplished by first defining a chemical reaction where the types of chemical bonds for the reactants and products are the same. For example, suppose we want to calculate the formation enthalpy of the 2- methylphenyl (2mp) radical. This may be done by first writing the reaction In writing this isodesmic reaction, the bond-by-bond errors in correlation energy are largely cancelled in the computed heat of reaction so that any error should not exceed the largest error associated with experimentally measured heats of formation used [4]. Determination of additionally requires electronic structure and frequency 2mp benzene phenyl toluene       298 2 f H mp  72 calculations for all species in the above equation. By using the reaction enthalpy computed from quantum chemistry and equating it to that determined by known experimental formation enthalpies, we are left with only one unknown parameter, . In the same manner as above, thermodynamic parameters of other C 7 H 7 radicals may be estimated, as performed in Chapter 4. Isodesmic reactions are used throughout this work to estimate thermochemical properties of interest. 2.4.8 Recent advances in kinetics - roaming radical reactions Although the tools here can be traced back over a century, there have been recent advances in the way we quantify chemical reactions, namely the discovery of roaming radical reactions [50, 51]. It is important to mention such reaction mechanisms, as they have been shown to explain otherwise anomalous experimental photoexcitation data [52]. Such reactions do not obey classical (TST) descriptions of reactions. In short, the phenomenon can be explained as such: first, recall traditional TST assumes a chemical reaction to take place through the path of least resistance. However, when the reacting system has an excess of energy, the path of least resistance need not be followed. Such trajectories have been deemed 'quasiclassical', as they deviate from that dictated by TST. These higher energy pathways make other areas of a molecule's potential energy surface available. When this happens, otherwise inaccessible isomerization or dissociation channels also become available.   298 2 f H mp  73 Although roaming radical reactions represent a major advancement in the field of chemical kinetics, their role in ambient to high-pressure combustion kinetics may be limited. For one thing, roaming radical reactions may persist up to, and on the order of, microseconds. Within most internal combustion engines however, collisions between all molecules are taking place very rapidly, and such long roaming radical reactions would not be influential, and indeed, are not necessary to describe the oxidation kinetics of systems like H 2 /CO, the most detailed and accurately known ones to date. Roaming radical reactions are expected to play a more significant role under low pressure - in outer space and within our atmosphere, for example. This said, recent shock tube studies on the high temperature (1400-1800 K) decomposition of propane point towards a minor (10 ± 8%) CH 4 + C 2 H 4 product channel with exclusively available to the roaming reaction pathway for pressures between 5 and 30 torr [53]. The authors found that with increased pressure/gas density, the roaming reaction channel had little influence and its contribution was lower than the uncertainty in the measurements themselves, thus further supporting the notion that roaming reactions are a predominantly low pressure phenomena. 2.4.9 Other aspects of reaction kinetics Lastly, there are several powerful reaction kinetic topics not covered here: variational transition state theory (VTST), variable reaction coordinate - transition state theory (VRC-TST), and direct trajectory simulations. These areas of research remain active with many relevant texts available for consultation. 74 2.5 Kinetic Model Development The previous sections exemplified the tools necessary to elucidate molecular properties with ab initio and semi-empirical quantum chemical methods. Typically, these properties are the ground and excited state total electronic energies and vibrational frequencies. This information can then be used in conjunction with the principles of statistical mechanics to derive the thermodynamic state properties for a molecule of interest. With respect to elementary chemical reactions, quantum chemistry is invaluable as a tool to estimate accurate geometries and energetics of transition state structures, necessary for implementation of most, if not all, reaction rate theories. The same tools described in the previous sections may be used to develop elementary rate parameters for combustion modeling where there are conflicting, scant, or no experimental data. This is necessary to develop complex chemical models with reasonable accuracy. In the combustion community today, there exists a tendency to unknowingly adopt rate parameters with large uncertainties, or those that have been poorly modeled. Such practices result in the propagation of error and uncertainty throughout the scientific community. Another issue involves the fact that, in general, with increasing fuel size comes increasingly complex models. It becomes an arduous task for others to verify that the rate parameters in such large models are sound, and many scientists simply choose not to do so. 75 The following sections describe the main characteristics and principles inherent in model development: critical rate evaluations, the identification and use of reaction classes, the determination of elementary rate parameters, and sensitivity analysis. 2.5.1 Critical rate evaluations An important aspect of model development inherent within the combustion community is the continual growth of experimental and theoretical data for the same reaction or system. For instance, Table 2.5 illustrates the wealth of information for just one reaction - the reaction of toluene + O( 3 P). Selected corresponding rates are shown in Figure 2.8. One benefit of such data proliferation is that scientific consensus may converge with time. This is not always the case however, especially with complex reactions containing a multitude of product channels. In the reaction of toluene + O( 3 P) for instance, much uncertainty remains regarding the product channels of this reaction, although there is considerably precise agreement in the overall rate. The dominant product channels are   3 6 5 3 C H CH O P     6 5 3 6 5 3 6 5 3 C H O CH OC H CH H HOC H CH   76 76 Table 2.5 Summary of rate measurements at various conditions for the reaction of toluene + O( 3 P). Products Experiment Conditions Sloane [54] 108, 106 and 80 amu: benzaldehyde + H 2, phenol, CO (major), CPD, or 3-penten-1- yne (minor) reddish-yellow tar Crossed Molecular Beams Collision energy at 298 K: 0.619 kcal/mol Baseman et al. [55] 99% CH3 + Phenoxy, 1% H + Cresoxy Crossed Molecular Beams Collision energy at 298 K: 9.7 kcal/mol Jones and Cvetanovic [56] o-cresol, p-cresol, m-cresol (>5%) CO, H2O, reddish-yellow matl of low volatility Hg photosensitized decomposition of N 2O T: 298  2, Rates at: 393,495 K ; P: 1 atm Gaffney et al. [57] Phenol, cresols, CO; minor – CH4 C2H6, reddish-yellow tar Cons V reactor T: 373  2 K P: 80-400 Torr Nicovich et al. [58] Unspecified, CO < 5% of O 3 Flash photolysis(193nm)-resonance fluorescence T: 298-950 K P: Parker and Davis [59] Those with ketene functionalities (2,4, and 6-MHO), cresols, benzyl alcohol Photolysis of toluene/ozone mixtures in argon matrix ( 280nm) T: 12 K Furuyama and Ebara [60] Overall rate Microwave discharge-fast-flow method, O generated from N+NO=N2+O T: 373-648 K Sol et al. [61] Cresols, phenol, benzaldehyde Microwave discharge of N 2O T: 298 K Hoffmann et al. [62] C7H8O* -> prod, C7H7 + OH (<10% at 1300 K) Glass shock tube, O atom detected through VUV absorption at 130.5 nm T: 1100-1350 K Atkinson and Pitts [63] Overall rate Flash photolysis of toluene/NO 2 in Hg T: 299-440  1 K Colussi et al. [64] N/A Flash photolysis of toluene/NO 2 in Hg T: 298-462  K Mani and Sauer [65] N/A Pulsed-radiolysis in CO 2 and N 2O T: * P: 97 atm McLain et al. [66] CO, CO2 Shock Wave in O2/Ar, UV and IR emissions of CO, CO2 and the product [O] [CO] T: 1700-2800 P: 1.1-1.7 atm Tappe [67] Need paper Low pressure discharge flow with molecular beam and mass spectral analysis T: 305-873 K 77 Figure 2.9 Plot of the experimentally determined overall rates of reaction for toluene + O( 3 P). With much information available for much parent fuel – radical reactions, the critical evaluations of individual rate parameters becomes invaluable. Consider the following reaction of toluene with atomic hydrogen: -14 -13 -12 -11 -10 0 0.5 1 1.5 2 2.5 3 3.5 4 Hoffmann et al 1990 Tappe et al 1989 Nicovich et al 1982 Atkinson and Pitts 1979 Colussi et al 1975 Furuyama and Ebara 1975 Grovenstein and Mosher 1970 Mani and Sauer 1968 Jones and Cvetanovic 1961 log (k / cm 3 molecule -1 s -1 ) 1000/T (K) O( 3 P) + C 6 H 5 CH 3 --> Products 78 R3.1 This reaction consumes an H radical and creates the resonantly stabilized benzyl radical, effectively slowing ignition/oxidation. It represents one of the initial and most important reactions in the combustion of toluene. Figure 3.1 illustrates selected rates of R3.1, and is referred to in the following discussion. Figure 2.10 Selected data for the rate coefficient of C 6 H 5 CH 3 +H   C 6 H 5 CH 2 +H 2 . , Oehlschlaeger et al. [68]; ■, Ellis et al. [69]; red dashed line, Robaugh and Tsang [70]; black dashed line, Baulch et al. [71]; black dotted line, Rao and Skinner [72]; solid black line, current fit. 6 5 3 6 5 2 2 C H CH H C H CH H      10 11 10 12 10 13 10 14 0.60 0.80 1.00 1.20 1.40 k / cm 3 mole -1 s -1 1000/T 79 This reaction was most recently examined in shock tubes by Oehlschlaeger et al. [68, 69], who suggested rates for two different temperature ranges: 700-1800 K with a reported rate uncertainty of  25%, and an experimentally determined rate for the temperature range of 1256-1667 K. The former rate was fitted to a three-parameter Arrhenius expression because a two-parameter expression including only their data had a pre- exponential factor exceeding the gas kinetic theory dictated collision limit. Additionally, the low temperature datum of Ellis et al. [69] was available, enabling a fit over a larger temperature range. Oehlschlaeger points out that their new data are similar to a previous study at similar temperatures by Hippler et al. [73] (not shown in Figure 3.1), but significantly reduced in scatter and therefore allowing for a more accurate rate determination at higher temperatures. Given the level of experimental precision, it is of interest to note that the calculated activation energy of 14.9 kcal/mol can be viewed as an upper limit, since reducing the pre-exponential factor to one within the collision limit necessarily requires a reduction in activation energy. Although most high temperature rates agree well, there remains considerable uncertainty at lower temperatures. Ellis et al. [69] conducted experiments at 773 K and 500 Torr in an aged pyrex reactor and used a root mean squares method to determine rate ratios for reactions of interest. Both Ellis et al. and Oehlschlaeger et al. argue that the Baulch recommendation may be unsound, representing a middle of the road approach to account for discrepancies at low and high temperatures. Specifically, this middle of the road approach is heavily based on the high temperature data of Hippler et al. (similar to the data of Oehlshlaeger et al.) and 80 the lower temperature data of Robaugh and Tsang [70], which was fitted to a curve. As seen by the slope of this curve in Figure 3.1, the shock tube data of Robaugh and Tsang are indicative of an activation energy too low (~8 kcal/mol) for this reaction, making the reliability of the suggested rate in this region questionable. Additionally, the low temperature datum of Ellis et al. is also uncertain for a number of reasons. With regards to R3.1, the 2003 study of Ellis et al. is essentially a replica of the same study performed by the same group in 1986 [74]. The empirical method of rate determination in these studies is relative to the rate of the overall reaction of H + C 2 H 6 : where The rates on the right hand side of the above equation are those pertaining to the production of the species in parenthesis. The first channel was measured with respect to the rate of the H + O 2  OH + O reaction, known to a sufficient level of accuracy and precision [71, 75]. Additionally, the rate of the first channel as determined by Ellis et al. agrees well with the theoretically computed value reported by Tokmakov and Lin [76]. Thus, it is expected to be reliable. At 773 K, no other channels are expected to be of significance.     6 5 3 26 39 overall overall k H C H CH k H C H          6 5 3 6 6 3 6 6 2 2 overall prod prod k H C H CH k C H CH k C H CH H      81 Immediately evident from above proportion relationship is that the overall rate of H + C 6 H 5 CH 3 is greatly dependent on the rate used for the relative H + C 2 H 6 reaction, at 773 K. This rate itself still remains considerably uncertain, as scatter in corresponding experimental data illustrate [71]. Furthermore, different H + C 2 H 6 rates were used between the 1986 [77] and 2003 [75] publications, as the authors more recently adopted a 1994 recommendation by Baulch et al, 16% larger than the value used in the 1986 publication. Because the measure proportions is found to be so large (~40), small differences in the H + C 2 H 6 reaction rate have significant affects on the H + C 6 H 5 CH 3 overall rate. This is one drawback of determining rates using indirect methods. In the 1986 study, Baldwin and coworkers indirectly measured an overall rate of H + C 6 H 5 CH 3 to be (5.8 ± 0.76) x 10 11 cm 3 mole -1 s -1 , compared to (5.05 ± 0.8) x 10 11 cm 3 mole -1 s -1 in the more recent 2003 study. Use of this higher rate increases the methyl H-abstraction reaction substantially, from 2.4x10 11 to 3.2x10 11 cm3mole -1 s -1 . This subsequently calculated rate in fact agrees better with the suggestions of Baulch et al, Hippler et al, and Rao and Skinner, at 773 K. However, the H + C 2 H 6 rate used in the 1986 publication represents an upper limit in the scatter of available experimental data [71]. Unfortunately, this issue was not raised in the most recent 2003 study. Furthermore, propagating Baulch’s most recent H + C 2 H 6 rate uncertainty values into the methodology of Ellis et al would result in an erroneous R3.1 rate in the range of (-0.71 – 9.48) x 10 11 cm 3 mole -1 s -1 , greatly exceeding the uncertainty reported by the authors. This uncertainty is illustrated in Figure 2.9. 82 Finally, the H + toluene rate of 2.2 x10 11 cm 3 mole -1 s -1 suggested in Table 5 of Ellis et al. does not agree with that directly calculated elsewhere in the paper (2.2 x10 11 vs 2.4 x10 11 cm 3 mole -1 s -1 ), an almost 10% difference. In addition, a visual inspection of Fig 8 in the work of Oehlshlaeger et al. suggests they chose to use the lower incorrect value. Thus, the data of Oehlshlaeger et al. and Ellis et al. were refit with the constraint that it pass through the low temperature datum of Ellis et al, even though the reliability of the rate determining method is questionable. The above discussion clearly demonstrates the need for further research regarding this channel at temperatures below 1000 K. Although this particular reaction represents an extreme case, it illustrates both the need for critical rate evaluations, and the tedium that sometimes accompanies it. 2.5.2 Rate estimation Among the typically many reactions that may occur in a reaction with one or more molecules, some fall into a certain reaction classes – a set of chemical reactions exhibiting similar rates, potential energy surfaces, reaction coordinates, or any combination these. Well know reaction classes common to combustion are -scission and H-abstraction by another radical (e.g., H , OH , O , HO 2 , CH 3 ). Because kinetic rate parameters of the same reaction class are very similar, identification of such classes can be of great benefit to a kineticist. For example, in modeling fuels for which little or 83 no experimental or theoretical elementary rate parameters are available, it is common practice to use the rate of an analogous reaction from the same reaction class to estimate the parameter of interest. With respect to linear and cyclic alkanes, the kinetics of initial fuel cracking are decoupled from the oxidation/pyrolysis of the cracked intermediates they form [78, 79]. Thus, the kinetic bottleneck to overall fuel combustion lies in chemistry of cracked intermediates. Consider the combustion of a fuel such as dodecane, a normal alkane with a 12-carbon chain backbone. An H-abstraction from either end of this molecule will produce the 1-dodecyl radical. Under combustion conditions, the subsequent -scission of this radical is considerably faster than the combustion kinetics of the subsequently formed intermediates [78]. This rule, and others, are applied to linear alkanes of varying lengths (C k H 2k+2 , 5 ≤ k ≤ 12) in a recently published mechanism, JetSurF 1.0 [80]. Table 3.1 illustrates the major reaction classes for the parent n-alkane fuels within JetSurF 1.0. 84 Table 2.6 Major reaction classes of n-alkanes considered in JetSurF1.0 [80] Reaction type Source and Method of Rate Estimation C-C bond fission in n-alkane Back rate constant from 2C2H5 → n-C4H10 (k ∞ ) H-abstraction by H, O, OH, O2, and CH 3 Cohen’s method using the rate constants of C 3 H 8 + X → n-C 3 H 7 or i-C 3 H 7 + HX (X = H, O, O 2 & CH 3 ). C x H 2x+2 + OH → C x H 2x+1 + H 2 O: Sivaramakrishnan & Michael [81] Mutual isomerization of alkyl radicals (1,4, 1,5 and 1,6 H-shift) Tsang, Manion and co-workers. n-pentyl [82], n-hexyl [83], n-heptyl [84], n-octyl [85]. Rate parameters for the C k H 2k+1 (9≤ k ≤ 12) radicals are equal to those of n-octyl [84]. All possible C-C bond beta-scission in alkyl radicals See above The C-C bond fission reactions kinetic parameters for C 5 -C 12 alkenes as well as mutual isomerization and C-C bond b-scission of alkenyl radicals are based on the work of Tsang and coworkers [86, 87] on 1-hexenyl and 1-pentenyl radicals. The H-abstractions kinetic parameters are based on those of analogous reactions of C 3 and C 4 species in USC-Mech II. Use of the reactions classes in Table 3.1 allowed for a more expedited development of the JetSurf 1.0 mechanism. The identification and accurate treatment of such ‘reaction classes’ has the potential to save considerable amounts of time when developing chemical mechanisms. 85 The high-temperature pyrolysis and oxidation of cyclohexane and several of its monoalkylated derivatives was also recently addressed by our group, and published online as a mechanism - JetSurF 1.1 [88]. The identification of several reactions classes, shown in Table 3.1, made the development of this mechanism considerable easier. A novel reaction path involving the treatment of methylphenyl radicals for aromatic systems is proposed in Chapter 4. 2.5.3 Elementary rate parameters - methods of determination Another crucial aspect of mechanism development is the calculation of elementary rate parameters, especially those with conflicting rates reported in literature or those not yet investigated. Typically, the following methods are used in the determination of elementary rate parameters:   experimental methods (i.e., shock tubes, flow/static reactors)   analogous reactions (use of reaction classes)  ab initio/semi-empirical methods with transition state theory and/or RRKM/master equation modeling Some of the most reliable and accurate experimentally determined rate constants come from shock tube experiments with sophisticated laser diagnostics [89-96]. The nearly adiabatic and isochoric process is among the most ideal encountered in the experimental combustion community. In addition, the frequently used low fuel concentrations minimize unwanted side reactions. Elementary rate parameters are most frequently back- 86 calculated by fitting to a complex chemical mechanism, with the only unknown being the rate of interest. Recent advances in shock tube techniques allow for intermediate species time histories behind reflected shocks; some of these include H 2 O, CO, CO 2 , OH, and more recently, C 2 H 4 [97-100]. The uncertainty of these measurements typically increases with the temperature of the reflected shock, as the uncertainty is roughly 1% [96]. At 1500 K, an uncertainty of ± 15 K may change a model prediction of intermediate species concentration by up to 30% within the first 50 s after the reflected shock. The implications are obvious for kineticists using such data to validate chemical mechanisms at early times behind the reflected shock. At the same time, these species time histories may be used to constrain some rate constants within a model using methods of uncertainty quantification and minimization [101-103]. Already discussed earlier in this chapter are the necessary tools for ab initio determinations of electronic energy, vibrational frequencies, and geometry optimizations of reactants, products, and transition states involved in elementary reactions. This information is used in conjunction with statistical mechanics to derive thermodynamic state functions of a molecule of interest. Subsequent elementary rate parameters may be determined using additional methods. Well established techniques such as Transition State Theory, Rice–Ramsperger–Kassel– Marcus (RRKM), and Master Equation modeling are frequently used to determine the overall thermally-averaged rates and branching ratios for reactions with complex 87 potential energy surfaces and multiple adducts/products [11, 33, 104]. However, especially with respect to master equation modeling, the proper treatment of a reaction over a large temperature and pressure range requires considerable effort and meticulous care. 2.5.4 Sensitivity Analysis Sensitivity analysis is an invaluable tool for model development because it allows for the identification of the overall rate-limiting reactions with respect to flame phenomena such as laminar flame speed or ignition delay. In doing so, it also enables identification of elementary reactions to be considered by experimentalists [105]. It additionally aids in the reduction of large detailed model into skeletal forms [106, 107]. Lastly, the influence of reactions with considerable uncertainty (greater than one order of magnitude) on model performance may be addressed through sensitivity analysis. A brief formulation for sensitivity analysis is given below. A dependent variable, such as laminar flame speed, may be described as where x is an independent variable vector consisting of reaction scalars such as temperature, pressure, and species concentrations. k is a set of parameters (e.g. preexponential factors, diffusion coefficients) [105]. The absolute and relative sensitivities, S and S rel ij s,k , of s with respect to the parameter set k are then respectively defined as:   ;  s s x k 88 S rel ij s,k is a single value of the relative sensitivity of the dependent variable i with respect to parameter j. Typically, a normalized sensitivity is reported. For example, the normalized sensitivity of a species i to an initial concentration of species n, , is: Sensitivity analysis is most commonly performed on ignition delay or laminar flame speed with respect to rate parameters and/or diffusion coefficients. 2.6 Computer modeling of experimental combustion phenomena 2.6.1 Laminar premixed flames A large part of the modeling work presented here is necessarily supplemented with by carefully obtained experimental data. Full descriptions of the physical and mathematical model of premix flames can be found elsewhere. Only some of the fundamental concepts and experimental methods are covered here. , , , , j rel i ij ij k s S sk     s k s k s S k o n x           ln ln o i i n i o i n n xt xt x x t x x     S 89 The laminar flame speed is a result of the conceptual formulation of an ideal flame with the following characteristics [108]: 1. 1-dimensional 2. planar 3. adiabatic 4. at steady state 5. laminar Because the five conditions listed above are impossible to reproduce and probe in the laboratory, the laminar flame speed, , is not directly measured, but must be extrapolated from other observables. A common way to do this is by measuring the flame extinction strain rate [109]. The extrapolated flame speed can also be quite sensitive to the extrapolation scheme [110]. For instance, in a study of C5-C12 normal alkane compounds, compared to linearly extrapolated values, the laminar flame speeds obtained using non-linear extrapolations were found to be 1 to 4 cm/s lower depending on the equivalence ratio [111]. Full details of the experimental determination for the cyclohexane and mono- alkylated cyclohexanes flames may be found elsewhere [112]. Under ambient conditions, most of the fuels of relevance to this body of work are in the liquid phase. As such, it is necessary to vaporize them prior to introduction in to flow stream. This is commonly accomplished with a syringe pump. The fuel/air mixture is then guided to a  S u o  S u o 90 burner, specifically a symmetric twin-flame, opposed-jet configuration. All measurements are performed in the vicinity of the centerline. Axial flow velocities are measured along the stagnation streamline using digital particle image velocimetry, the flow tracers being 0.3- m silicon oil droplets. To determine , the minimum point of the velocity profile upstream of the flame is measured and defined as a reference flame speed, S u,ref [109]. The absolute value of the maximum velocity gradient in the hydrodynamic zone is defined as the imposed strain rate, K. By monitoring the variation of S u,ref with K, is determined by extrapolation of S u,ref to K = 0. For all of the flame studies mentioned in this work, the extrapolation procedure was carried out non-linearly, using a recently developed computationally- assisted technique [110]. For equivalent fuels, experimentally derived flame speeds are valuable in verifying the integrity of a kinetic model. They are especially useful when provided in tandem with data from isomeric compounds. Because many isomeric species exhibit the same peak burning temperatures and effectively have equivalent thermodynamic state properties, differences in laminar flame speeds are indicative of unique or dominant kinetic pathways. Well established computer codes were used to model shock tubes, flow reactors, burner stabilized flames, and premixed laminar flames, the specifics of which will not be described  S u o  S u o 91 here, but may be found in corresponding manuals to the codes briefly described. The Sandia CHEMKIN [113] was used for all modeling purposes. The code is coupled with the Sandia PREMIX code [114] and Transport [115] subroutine libraries for simulation of laminar flame speed. Key binary diffusion coefficients involving atomic and molecular hydrogen are based on [116] and [117]. Diffusion coefficients for larger normal alkane species are estimated based on a method described in the Appendix. 2.6.2 Burner-stabilized premixed flames Low pressure burner stabilized flames with multiplexed chemical kinetics photoionization mass spectrometry aided by synchrotron radiation is capable of providing a wealth of data concerning fuel, product, and intermediate species [118]. 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Wang, Journal of Physical Chemistry a 114 (9) (2010) 3355-3370. 100 Chapter 3: Case Study on Persistent Aromatic Radicals A generally undesired byproduct of fossil fuel combustion, soot is one of the most prevalent anthropogenic aerosols, affecting atmospheric visibility, global climate, and human health. Soot formation in flames is a highly reversible process driven, in part, by two competing thermodynamic forces: enthalpy release from chemical bond formation and entropy increase resulting from release of gas phase species accompanying particle formation. The initial growth of polycyclic aromatic hydrocarbons is essentially captured by the hydrogen-abstraction carbon-addition (HACA) mechanism. For example, the presence of acetylene, H atoms, and the high temperature environment inherent to flames facilitate formation of soot precursors such as benzene and naphthalene. However, an entropic resistance to PAH dimerization at these temperatures rules out the self-binding thermodynamically driven growth of PAH stacks and/or clusters. Furthermore, two additional observations conflict with our current understanding of soot formation: (1) nascent soot can be rich in aliphatics and (2) soot mass growth can occur without the presence of H atoms. In light of these recent findings, other avenues to soot formation must be explored. 3.1 Introduction Aside from the release of greenhouse gases, soot is another major product of combustion. Soot, generally described as aggregates of roughly spherical, primary particles coexisting with graphite flakes, can serve as nuclei for atmospheric aerosols, otherwise known as 101 cloud condensation nuclei [1], [2]. It is therefore considered a pollutant. Although the most prominent atmospheric aerosol nucleation phenomena involve sulfuric acid, ammonia, or ionic particles, carbon-based aerosols are also known to act as cloud condensation nuclei. For example, soot exhaust from aircraft engines is thought to be responsible, at least partially, for contrail formation in the wake of an aircraft [3-5]. Combustion soot has also been observed to show various wetting properties [4-7]. More specifically, depending on the conditions of soot formation, the material may be either hydrophobic or hydrophilic, with the ability to uptake substantial amounts of water on both surface sites and internally within pores [5, 8]. What has not been adequately addressed though is the mechanism of water uptake by a soot surface traditionally thought to be hydrophobic. In particular, the potential role of free-radical species on the surface of a carbon particle and/or within its pores has not yet been considered. Although the oxidation of soot surfaces by ozone, OH and other oxidants can lead to the formation of polar sites capable of water uptake [2, 9-25] persistent free radicals can also form tight hydrogen bonds with water molecules, and thus act as molecular sites for water uptake. Theoretically, the binding energy of hydrogen-like bonds between a methyl radical and water is 1.5 kcal/mol [26, 27]. This binding energy increases to over 2, 3, and 4 kcal/mol for ethyl, isopropyl and tert-butyl radicals, respectively [27, 28]. Likewise, some oxygenated hydrocarbon free radicals exhibit similar tendencies to form hydrogen bonds with water. For example, the phenoxyl-water binding energy was reported to be ~4 kcal/mol [29]. 102 Persistent free radicals may also impact the mechanism of soot nucleation and surface growth in hydrocarbon combustion. Currently, the modeling of soot formation in flames generally follows the hydrogen-abstraction-carbon-addition (HACA) mechanism [30, 31]. In this mechanism, soot surface growth requires the existence of gas-phase free radicals, notably the H atom. The addition of gas-phase, hydrocarbon species, e.g., acetylene, to an aromatic carbon site requires H-atom abstraction of an aryl-H to produce a radical site, which subsequently reacts with acetylene or other hydrocarbon species. Experimental evidence suggests that soot can continue to grow in mass even in the absence of gas-phase free radicals (e.g., H ). In ethylene-air counterflow diffusion flames, the soot volume fraction observed towards the stagnation surface of the opposed jets can be predicted only if soot retains a free radical characteristic while they are transported towards the stagnation surface [32]. In more recent studies of burner- stabilized ethylene-oxygen-argon flames, the soot volume fraction was found to continually increase in the post flame region, where the temperature drops below 1500 K and few gas-phase free radicals can survive [33, 34]. Ample evidence suggests persistent free radicals exist in carbonaceous materials with little to no oxygenates. Electron Spin Resonance (ESR) spectra of anthracite, a coal containing little to no oxygenated compounds show a measurable concentration of free radicals [35]. Furthermore, soot formed from the pyrolysis of ethylene, acetylene, and various jet fuel surrogates also show measurable concentrations of radicals likely 103 associated with carbon atoms [36]. The reported Landé g-factors for such soot samples suggest that the radicals are of   nature, most of which are associated with aromatic compounds. For free radicals to persist on the surface of or within the structure of soot, they must maintain a certain level of thermodynamic stability; yet, to bind with water or a hydrocarbon species, they must be active enough to form hydrogen or covalent bonds. There are certainly many possibilities for stable radical types and classes. Examples are given in a review by Hicks [37]. Localized unpaired electrons in their ground states have also been detected in the ESR spectra of 4,4’-polymethylenebistriphenylmethyl, (1,4- phenylene) bisdiarylmethyl, and (4,4’-biphenylene) bisdiarylmethyl in suspensions of benzene [38]. Additionally, singlet diradical ground states have been shown to exist in polyacenes with eight or more rings [39]. These radicals are known to be of  nature. Even graphene sheets of varying size are now well known exhibit peculiar characteristics many now hope to exploit. For example, Figure 3.1 illustrates one of many examples of newly realized electronic properties of a class of polycyclic aromatic hydrocarbons (graphene flakes), defined by the number of aromatic ring units, n, spanning both dimensions of the planar PAH(n,n) sheet [40]. 104 Figure 3.1 PAH(3,3) [40] monomer and dimer frontier orbitals calculated at the B97X-D/6-31G(d) level of theory [41]. The estimated binding energy for the dimer is 33 kcal/mol, while the singlet-triplet gap for a monomer is estimated to be 20 kcal/mol via the SF5050 method [42]. The figure above depicts the frontier orbitals; these are the second to highest occupied molecular orbital (HOMO-1), the highest occupied molecular orbital (HOMO), and the lowest unoccupied molecular orbital (LUMO). Visualizing the electron distribution in these orbitals gives an indication of how these graphene flakes are likely to interact with other gas phase species. It is clear from the HOMO that the highest energy valence electron is mainly distributed along the outer portion of the PAH. As a result, the self interaction energy of two PAH(3,3) monomers is quite strong, at 33 kcal/mol. The strength of this binding is considerable greater than those of their corresponding PAH molecules [43] that lie on the stabilomer grid, first proposed by Stein and Fahr [44]. However, it is still unlikely to be strong enough to keep the two flakes together at the temperatures found just beyond the post-flame region of most flames. This is why electronically exicted states of these and similar molecules present itself as an attractive alternative. At only 20 kcal/mol, the singlet-triplet gap of PAH(3,3) can easily be 105 overcome through photoexcitation or even intersystem crossing. Although exploring such possibilities is planned for the future, the focus of the current work lies on weak covalent bonds and soot analogues with highly resonant radical characteristics. Using density functional theory, molecular analogues to possible aromatic diradical systems existing in young and mature soot are explored. Specifically, a class of persistent free radicals is identified that can be dynamically generated from aromatics with extremely weak bonds, namely, the hybrid derivatives of acenaphthene and hexaphenylethane (HPE, 1), as depicted in Figure 3.2. 106 1 (HPE) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 3.2 Structures of selected model compounds (1-14) and the molecule accidentally synthesized by Gomberg [45, 46] 15. 107 These compounds were chosen from the consideration that HPE-like molecules, including pyracenes, are known to have weak central C-C bonds with lengths around or greater than 1.7 Å because of steric repulsion [47]. The acenaphthene derivatives with increasing numbers of substituted phenyl groups are conceptual tools for probing the question of how carbonaceous aromatic radicals may persist in soot. For example, tetraphenylacenaphthene, 14, represents a hybrid structure between acenaphthene and HPE. Its molecular structure is similar to HPE, but with two of the phenyl groups replaced by acenaphthene. HPE has long-standing history as a molecule with an extremely weak central carbon- carbon bond. It is generally recognized that the first literature occurrence of this molecule coincides with a failed attempt to synthesize it by Gomberg [45, 46]. On the other hand, acenaphthene, the base molecule for the target species studied here, which may be formed through the hydrogenation of acenaphthylene, has direct relevance to the evolution of soot precursors and soot formation itself [30, 48-50]. The likelihood that structures 7, 9, 11, and 13 would form in the gas phase of a flame is extremely small. However, similar moieties containing very weak C-C bonds are likely to exist in many types of soot, especially near graphitic edges at the interfaces of condensed polycyclic aromatic hydrocarbons (PAHs) and the neck regions of aggregates of primary particles. Sites on and within soot, possibly represented as boundaries between PAH conglomerates and/or graphitic edges, may facilitate the existence of these 108 types of C-C bonds. Such bonds may freely open and close with photoexcitation, attack of a polar molecule, or even as a result of molecular strain and thermal fluctuations. Thus, the purpose of this study is not to illustrate formation pathways of the model compounds in soot structures, but to demonstrate how high levels of steric strain can substantially reduce specific C-C bond strengths. 3.2 Methodology for electronic structure calculations Electronic structure calculations were performed on an in-house cluster using the computational chemistry package QChem 3.1 [51]. Geometries were optimized using the B3LYP [52, 53] and M06-2X [54] density functionals. The 6-31G(d) basis set used for B3LYP calculations. For M06-2X, the 6-31+G(d,p) basis set was used to account for long-range interactions for which diffuse and polarization functions are expected to be important [54]. The M06-2X functional was used in this study because it was specifically designed to take into account long-range noncovalent interactions typical of large PAHs, and includes roughly twice the amount of Hartree-Fock exchange (54%) compared to the B3LYP functional. The M06-2X functional has been validated and tested against a wide range of molecules [54]. For purposes of comparison, calculations were performed at two other levels of theory to determine the bond dissociation energy (BDE) of the C-H bond in triphenylmethane. These are: B3LYP/6-31+G(d) and B3LYP/6-311+G(d,p). 109 Local minima were confirmed by the lack of imaginary frequencies, while transition state structures were confirmed by the presence of one, and only one, imaginary frequency. In most cases, the default Direct Inversion of the Iterative Subspace (DIIS) [55] convergence algorithm was used; in cases where convergence failed, the algorithm was switched to either Geometric Direct Minimization (GDM [56] or DIIS_GDM, in which the first SCF cycle employs DIIS and subsequent cycles utilize GDM. All frequency calculations were performed at the B3LYP/6-31G(d) level of theory, and multiplied by a factor of 0.96 to account for vibrational anharmonicity [57]. Because of the large computational demand of force analyses at the M06-2X/6-31+G(d,p) level of theory, all enthalpy values were calculated utilizing frequencies taken from B3LYP/6-31G(d) calculations. In any case, the zero-point energy differences between B3LYP and M06- 2X amounted to less than 1% for several closed and open shell systems tested here. Calculations involving acenaphthene derivatives in their diradical forms were performed in their triplet states (the eigenvalue of the S 2 operator was confirmed to be within a few percent of the theoretical value, 2); reasoning for this is discussed later. Symmetry was not enforced in any calculations. Therefore, the symmetry of most optimized structures belonged to the C 1 point group. All of the optimized geometries and corresponding vibrational frequencies used in this work may be found in the supporting information section of online publication [58]. 110 Isodesmic reactions were used to estimate the enthalpies of formation for derivatives of HPE (1) and the triphenylmethyl radical. In most cases, reaction schemes were homodesmic – not only was bond type conserved, but so was aromaticity. Most experimental standard enthalpies of formation of the reference species are well documented [59-62], as shown in Table 3.1. Standard-state BDE values were determined directly from zero-point corrected electronic energies and sensible enthalpies at 298 K and indirectly from the enthalpy of formation of a parent molecule and its dissociated products or diradicals. 111 Table 3.1 Summary of calculated electronic energies (E 0 ) and zero point energies (ZPE) in Hartrees, sensible enthalpy at 298 K (H ○ (298)-H ○ (0)) in cal/mol, and literature values of enthalpy of formation ( ) of reference species in kcal/mol  Species B3LYP/6-31G(d) M06-2X/6-31+G(d,p) ref E 0 ZPE a E 0 target species b 1 (C 38 H 30 , HPE) –1466.02914 0.54054 –1465.55335 33.38 2 (C 19 H 16 , TPM) –733.65829 0.27987 –733.39873 17.28 3 (C 19 H 15 , TPM  –733.03119 0.26781 –732.76509 16.60 4 (C 42 H 32 ) –1619.67351 0.58353 –1619.13520 35.19 5 (C 46 H 34 ) –1773.31634 0.62980 –1772.73444 39.16 6 (C 23 H 17 , DPNM ) –886.67866 0.31214 –886.35608 19.59 7 (C 18 H 14 ) –694.36209 0.25373 –694.11356 14.77 8 (C 18 H 14 :) –694.28015 0.24861 –694.01692 15.78 9 (C 24 H 18 ) –925.41206 0.33190 –925.08636 20.25 10 (C 24 H 18 :) –925.34145 0.32777 –924.98093 20.62 11 (C 30 H 22 ) –1156.44158 0.40947 –1156.05306 25.15 12 (C 30 H 22 :) –1156.38703 0.40517 –1155.95174 25.16 13 (C 36 H 26 ) –1387.46578 0.48669 –1387.01039 30.11 14 (C 36 H 26 :) –1387.43668 0.48380 –1386.96031 30.47 reference species H  –0.50027 –0.49667 0.89 52.1 [59] CH 4 –40.51826 0.04352 –40.48874 3.49 –17.8 0.1 [60] C 2 H 6 –79.83016 0.07228 –79.78293 4.65 –20.0 0.1 [60] tert-C 4 H 9 • (tert-butyl) –157.79778 0.11278 –157.71592 7.48 11.0 0.7 [61] C 6 H 6 (benzene) –232.24777 0.09670 –232.15393 5.87 20.0 0.2 [62] C 7 H 8 (toluene) –271.56563 0.12322 –271.45391 7.68 12.1 0.1 [62] C 7 H 7  (benzyl) –270.91516 0.11028 –270.79480 7.05 49.5 1.0 [61] C 8 H 18 (hexamethylethane) –315.70006 0.23680 –315.56185 13.20 –54.0 0.5 c C 10 H 8 (naphthalene) –385.89144 0.14194 –385.89144 8.60 35.9 0.4 [62] C 12 H 10 (acenaphthene) –463.31311 0.17567 –463.13309 10.37 37.3 0.7 [62] a Multiplied by 0.96 to account for anharmonicity. b See Figure 3.2 for structures. c Average of the values reported in refs [63] and [64]. 298 f H  298 0 HH  298 f H  112 3.3 Results and Discussion Figure 3.3 depicts HPE and two similar molecules, in which one or more phenyl groups are replaced by the naphthyl groups. These geometries were optimized at the M06-2X/6- 31+G(d,p) level of theory. They represent the lowest energy conformations and illustrate the uniqueness of this class of molecules. Not only are central bond lengths longer than a typical single C-C bond by ~0.2Å, their most stable conformations are opposite of what is conventionally expected. The Newmann projections of Figure 3.3 depict the lowest energy states of these conformations - a nearly eclipsed position. The stability of the eclipsed conformers is a manifestation of the attractive dispersion forces of the aromatic groups across the C-C axis. These dispersion forces counteract the steric repulsion among the same aromatic groups, which cause the central C-C bond to elongate. 113 Figure 3.3 Schematic structures of selected molecules illustrating several key geometric properties. Left: central C-C bond length and C-C-C bond angles. Right: values of dihedral angles between adjacent substituents calculated with M06-2X/6-31+G(d,p). Values in parenthesis are those calculated with B3LYP/6-31G(d). As expected, longated C-C bonds are also prevalent in acenaphthene derivatives. Beginning with the central C–C bond in acenaphthene (1.57 Å), Figure 3.4 illustrates a clear trend between the number of phenyl group additions and bond length. In general, the central C–C bond length increases with additions of phenyl groups. For 6.2 (9.0) 112.5 (111.9) 1.70 (1.74) 1.71 (1.74) 10.3 (8.8) 6.5 (11.5) 112.2 (112.0) 111.2 (112.0) 1 4 5 1.74 (1.75) 114 tetraphenylacenaphthene 13, the central C-C bond increases to 1.7 Å, as determined by the M06-2X/6-31+G(d,p) calculation. As expected, both M06-2X/6-31+G(d,p) and B3LYP/6-31G(d) predict the same C–C length for acenaphthene, a molecule with comparatively minimal long range interactions. With phenyl substitution however, M06- 2X/6-31+G(d,p) calculations exhibit consistently shorter C–C bond lengths than their B3LYP/6-31G(d) counterparts, resulting from an enhanced stability through dispersion forces. Figure 3.4 Central C–C bond lengths computed for selected molecules. Although direct electronic energy calculations allow estimates of the central C-C bond energies readily, there is a cause for concern with regard to the accuracy of these estimates, simply because the errors in the electronic energies calculated for the closed- shell parent molecule and its open-shell diradical generally do not cancel out. For this reason, isodesmic reactions were employed to obtain estimates for BDE through enthalpies of formation estimated for the parent molecule and its diradical resulting from 1.55 1.60 1.65 1.70 1.75 0 2 4 6 8 B3LYP/6-31G(d) Length M06-2X/6-31+G(d,p) Length Central Bond Length (A) 115 central C-C bond fission. However, the use of isodesmic reactions with large molecules can also lead to some errors. Because these systems generally have large sizes, compounding errors may result from the need to use a large number of small reference species whose enthalpies of formation are uncertain to an extent. Conversely, large species may be used as reference species, but their enthalpies of formation are usually associated with even larger uncertainties. In addition, it is not trivial to write reactions that preserve the amount and type of electron correlation present in large target species. Hence, the goal here is not to provide accurate estimates for the enthalpy of formation, but rather to explore the bond energies using two separate approaches. 116 Table 3.2 Isodesmic reactions and enthalpies of formation at the standard state. (kcal/mol) Species Isodesmic Reaction a B3LYP/6-31G(d) M06-2X/6-31+G(d,p) 1 + 6CH 4  6C 6 H 6 + C 8 H 18 199 169 2 + 2CH 4  3C 7 H 8 73 62 3 + 3CH 4  3C 6 H 6 + tert-C 4 H 9  96 90 4 + 6CH 4  5C 6 H 6 + C 8 H 18 + C 10 H 8 212 185 5 + 6CH 4  4C 6 H 6 + C 8 H 18 + 2C 10 H 8 231 195 6 + C 6 H 6  3 + C 10 H 8 107 102 7 + CH 4  C 12 H 10 + C 7 H 8 65 57 8 + 2C 7 H 8 + C 2 H 6  7 + 2C 7 H 7 + 2CH 4 121 112 9 + 2CH 4  C 12 H 10 + 2C 7 H 8 94 82 10 + 2C 7 H 8 + C 2 H 6  9 + 2C 7 H 7 + 2CH 4 143 135 11 + 3CH 4  C 12 H 10 + 3C 7 H 8 135 111 12 + 2C 7 H 8 + C 2 H 6  11 + 2C 7 H 7 + 2CH 4 174 161 13 + 4CH 4  C 12 H 10 + 4C 7 H 8 178 145 14 + 2C 7 H 8 + C 2 H 6  13 + 2C 7 H 7 + 2CH 4 203 165 a See Table 3.1 for species nomenclatures and Figure 3.2 for species structures. 298 f H  117 Results from the isodesmic reactions at the B3LYP/6-31G(d) and M06-2X/6-31+G(d,p) levels of theory are presented in Table 3.2. One isodesmic reaction was written for each target species. Concerning the acenaphthene derivatives M, isodesmic reactions use CH 4 , acenaphthene (C 12 H 10 ), and toluene (C 7 H 8 ) as the reference species and are written as follows, . For their diradical counterparts (R ), we utilized the enthalpy of formation estimated for the parent molecule M and isodesmic reactions given by, , where C 7 H 7 • is the benzyl radical. The coefficients n and m are given in Table 3.2. In all cases, the values of enthalpy of formation determined from M06-2X/6-31+G(d,p) are smaller than those from B3LYP/6-31G(d), by as much as 30 kcal/mol. In general, the M06-2X/6-31+G(d,p) values are expected to be more reliable because they more accurately describe the dispersion forces in these systems. For molecular structures without significant dispersion interactions (e.g., triphenylmethane 2), both levels of theory are reasonably accurate in predicting BDE, i.e., within 5 kcal/mol from the corresponding experimental values. This accuracy is adequate for the purpose of the current study. Table 3.3 illustrates BDE values estimated for the quaternary C-H bond dissociation in 2, forming the hydrogen and triphenylmethyl 3 radicals. The available experimental BDE values of (Ph) 3 C-H are 75±2 and 80.8±3.0 kcal/mol [65, 66]. The latter value was determined in a dimethyl sulfoxide solution from 4 12 10 7 8 M CH C H C H nm    2 6 7 7 4 R 2C H 2C H 2CH M       118 equilibrium acidity and electrochemical data, while the former value was obtained through a pyrolytic and photochemical process, whereby the author measured the activation energy associated with the unimolecular dissociation of a parent molecule in a toluene bath gas. Also available is the BDE value determined with the ONIOM method (75.9 kcal/mol) [67]. The BDE values estimated from different approaches, including the use of isodesmic reactions and the M06-2X/6-31+G(d,p) all fall in the range of current experimental and theoretical uncertainties with small effects arising from the basis set sizes. Table 3.3 Standard, (Ph) 3 C-H bond dissociation energy (kcal/mol). B3LYP/6-31G(d) 71.5 B3LYP/6-31+G(d) 72.6 B3LYP/6-31G(d) Isodesmic 74.9 a B3LYP/6-311+G(d,p) 73.2 M06-2X/6-31+G(d,p) 77.9 M06-2X/6-31+G(d,p) Isodesmic 79.8 a ONIOM [67] 75.9 expt [65] 75±2 expt [66] ,b 80.8±3.0 a See Table 3.2. b In Me 2 SO solution. From the M06-2X/6-31+G(d,p) isodesmic calculations, we estimate the central C-C BDE of HPE (1) to be 11.3  1.4 kcal/mol (Table 3.4). The uncertainty value was determined from the uncertainties in the enthalpies of formation of the reference species; they do not 119 indicate the accuracy of the computational method. The BDE was also calculated directly from the electronic energies with zero-point and sensible enthalpy corrections. The value determined in this manner is 11.6 kcal/mol, in close agreement with the results obtained from isodesmic reactions. The only available gaseous phase experimental value for the BDE of HPE comes from Szwarc [65]. They reported a BDE value of 11  2 kcal/mol, which is in close agreement with the M06-2X/6-31+G(d,p) results. Admittedly, the value reported by Szwarc [65] is close to that of the intradimer C–C BDE in 15, the molecule Gomberg originally and accidentally synthesized. That C-C BDE is incidentally also around 11 kcal/mol [68]. A theoretical study performed by Vreven and Morokuma [67], for which they calibrate a three-layer hybrid quantum mechanical scheme known as the ONIOM method, predicted the BDE of the central C-C bond of HPE (1) to be 16.6 kcal/mol, somewhat larger than the M06-2X/6-31+G(d,p) results presented here. The discrepancy may be attributed, at least in part, to the use of the B3LYP functional for geometry optimizations in the ONIOM calculation. In comparison to the results obtained for triphenylmethane, the B3LYP/6-31G(d) calculations fail miserably when predicting the central C-C BDE in HPE, as shown in Table 3.4. The BDE values are underestimated by as much as 30 kcal/mol. Use of isodesmic reaction improves the prediction only slightly. Overall, this reflects the inability of B3LYP to accurately account for long-range dispersion interactions. In essence, the difference in the BDE values predicted by B3LYP/6-31G(d) and M06-2X/6-31+G(d,p) reflects the overall stabilization energy gained from the long- 120 range interactions of the phenyl groups. In what follows, the M06-2X/6-31+G(d,p) results are exclusively used in the discussion of central C-C BDE. Table 3.4 Standard, central C-C bond dissociation energy (kcal/mol) of HPE (1). B3LYP/6-31G(d) –23.8 B3LYP/6-31G(d) isodesmic a –8.1±1.4 M06-2X/6-31+G(d,p) 11.6 M06-2X/6-31+G(d,p) isodesmic a 11.3±1.4 ONIOM [67] 16.6 expt [65] 11±2 a See Table 3.2. The uncertainty values are determined from the uncertainties in the enthalpies of formation of the reference species. They do not indicate the accuracy of the computational method. Figure 3.5 shows the variation of standard, central C-C BDE values for the acenaphthene and hexaphenylethane series of compounds. For comparison, the plot also shows the BDE values estimated from taking the difference of the electronic energies of the diradical and its parent molecule, both corrected for zero-point energy and sensible enthalpy. It is seen that the BDE values of acenaphthene derivatives generally decreases with an increase in phenyl substitution. For example, the BDE value is 55-60 kcal/mol for phenylacenaphthene 7, and decreases to about 20 kcal/mol in tetraphenylacenaphthene 13. 121 Figure 3.5 Standard-state, central C–C BDEs for the class of acenaphthene derivatives studied, HPE and its derivatives. * Open circles represent BDE values estimated directly by subtracting the electronic energy of the diradical from that of the parent molecule, both with zero-point energy and sensible enthalpy corrections. Structures 4 and 5 were studied to explore the effect of naphthyl substitution in HPE. The M06-2X/6-31+G(d,p) derived BDEs of both are very low, at 7.7 and 10.5 kcal/mol respectively. Not surprisingly, the central C–C bond lengths of 4 and 5 (1.71 and 1.745 Å, respectively) are larger than that of HPE. As can be deduced from the BDE values, no simple correlation can be made between the number of phenyl groups added to the parent HPE molecule and its corresponding BDE value. The addition of larger aromatic groups to HPE does not necessarily lead to lower BDEs. The central C–C bond strength is a result of competition between steric repulsion and dispersive attraction. In comparing tetraphenylacenaphthene 13 with HPE, although the central bond lengths are approximately equal, the BDE in 13 is substantially higher than that in HPE, by 0 10 20 30 40 50 60 70 0 2 4 6 8 M06-2X/6-31+G(d,p) electronic energy* M06-2X/6-31+G(d,p) Isodesmic expt 67 ONIOM 69 Central BDE (kcal/mol) 7 9 11 13 1 4 5 122 roughly 10 kcal/mol. This difference is related to the naphthalene group in 13 that enhances central C-C bonding strength through resonance. Compared to the typical C–C bond strength of 90 kcal/mol [69] found in many hydrocarbons, a bond strength of 20 kcal/mol still illustrates the significance of steric repulsions in 13. For acenaphthene derivatives with fewer phenyl groups than in 13, results here show that the central BDEs are about 50-60 kcal/mol. As mentioned earlier, the reported BDEs for acenaphthene derivatives are based on calculations for their triplet diradicals. Recent CASSCF and CBSQB3 calculations for some smaller hydrocarbon diradicals show that triplet energy levels lie above their singlet counterparts [70]. In the current work, attempts to perform geometry optimizations for diradical acenaphthene derivatives in the open-shell singlet states failed in all but one case, because the geometries converged toward those of closed-shell, parent molecules. Therefore, single-point B3LYP/6-31G(d) calculations for diradical acenaphthene derivatives were carried out to compare the energies of closed-shell singlets, open-shell singlets and triplets. These calculations used the geometries optimized at the M06-2X/6- 31+G(d,p) level of theory. To obtain the correct atomic spin densities for open-shell singlet systems, spin symmetry was broken in the initial guess of the wavefunction by mixing 50% of the LUMO with the HOMO. In each case, the eigenvalues of the S 2 operator were confirmed to be within a few percent of their respective, expected values, as shown in Table 3.5. A value close to unity suggests a substantial amount of triplet 123 state contamination and a small singlet–triplet gap. In all cases, the triplet state lies <3 kcal/mol below the open-shell singlet, and well below the closed shell singlet (>10 kcal/mol). The tests performed above suggest that with proper geometry optimization the open-shell singlets would probably lie lower in energy than the triplets, and hence the BDEs reported here for acenaphthene derivatives should be considered as upper limit values. Table 3.5 Single-point, relative B3LYP/6-31G(d) electronic energies E rel (kcal/mol) and eigenvalues of the S 2 operator calculated for triplet diradical acenaphthene derivatives optimized at the M06-2X/6-31+G(d,p) level of theory. species a state b E rel <S 2 > 14 CSS 10.7 0.00 OSS 0.0 1.01 T 0.0 2.05 12 CSS 16.6 0.00 OSS 1.4 1.05 T 0.0 2.07 10 CSS 13.9 0.00 OSS 1.6 1.01 T 0.0 2.08 8 CSS 22.1 0.00 OSS 2.7 1.05 T 0.0 2.08 a See Table 3.2 for species nomenclatures. b CCS: closed-shell singlet; OSS: open-shell singlet; T: triplet. Equilibrium constants for central C-C fission were calculated as a function of temperature using the values of the enthalpy of formation from isodesmic reactions, in addition to the molecular parameters calculated at B3LYP/6-31G(d) level of theory. At 1000 K, 13 → 124 14 has an equilibrium constant on the order of 10 -4 . Within the realm of radical concentrations, this is rather high. Near room temperature, however, the equilibrium constant of 13 → 14 reduces to ~10 -15 . These results are not interpreted as discouraging because in any soot-like system, it is likely that molecular analogues to the systems here will experience enhanced steric interactions from presence of larger aromatic structures than phenyl, resulting in even longer bond lengths, and possibly in smaller bond strengths than that found in HPE. Lastly, the current theoretical results underscore the importance of competition between steric repulsion and dispersive attraction in influencing the strength of aliphatic C-C bonds that join several small aromatic units together (e.g., phenyl and naphthyl). This competition dictates the actual strength of these bonds. Extrapolation of this findings to soot with constituent molecules bound to the edges of adjacent graphene sheets or at the interfaces of condensed PAHs suggests that extremely weak C–C bonds could indeed exist in otherwise strongly bound systems. Stronger dispersive forces in soot structure, coupled with the facts that aromatic ring sizes are generally larger than phenyl [33] and that additional steric strain may result from imperfectly annealed graphetic structures, all point towards the possible existence of persistent free radicals formed dynamically through repetitive cleavage and formation of unique aliphatic C-C bonds holding aromatic structures together. This dynamical process is, of course, molecular vibrational and thermal fluctuational in nature. 125 3.4 Conclusions Analogues representing a class of possible molecular scenarios in young and mature soot are presented as an explanation for the existence of persistent free radicals found in products of combustion. 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In the high temperature pyrolysis and oxidation of toluene, three possible methylphenyl radical isomers can be formed (2-, 3-, and 4-methylphenyl). The 2-methylphenyl radical may undergo a fascicle isomerization to benzyl through a pathway not directly accessible to 3- and 4- methylphenyl. Electronic structure calculations show that 2-methylphenyl isomerization to benzyl is preferred energetically by at least 20 kcal/mol over other possibilities. Monte Carlo RRKM/Master Equation simulations illustrates that, at temperatures ≥ 1400 K, and for almost all pressures, the 2-methylphenyl radical has a lifetime  1 s, whereas 3- and 4-methylphenyl radicals have a substantially longer lifetime, allowing them to react with other species during toluene oxidation. This is also found to be the case for all dimethylphenyl radicals in the three xylene isomers. As a result, the structure of the xylene isomers and specifically the number of H atoms immediately adjacent to the methyl groups can have a direct impact on high-temperature oxidation of the xylene isomers and appear to explain the observed differences in xylene oxidation behind reflected shock waves and in laminar premixed flames. 132 4.1 Introduction Of the main components found in petroleum derived transportation fuels, aromatics exhibit the most complex overall combustion chemistry. Consequently, there remains considerable uncertainty regarding the dominant combustion pathways of such compounds. For this reason, reaction kinetic models for the combustion of even simple aromatic fuels, e.g., benzene and toluene, have undergone continued evolution over the last two decades, with new models still emerging to this day [1-13]. A less extensive evolution has taken place regarding the high-temperature oxidation kinetics of the xylene isomers [13-20]. In the high-temperature oxidation of toluene, the initial reactions predominantly involve the production of the resonantly stabilized benzyl radical from unimolecular C-H fission or H-abstraction. Unimolecular aryl C-H fission leading to three possible methylphenyl radical isomers is usually not considered owing to their large bond energies (~110 kcal/mol) [21]. However, H-abstraction from the ring can result in the formation of these isomeric radicals, although such reactions are usually not considered in kinetic models. Recent quantum chemistry calculations suggest that the production of methylphenyl radicals is not negligible. The branching ratio of the H abstraction by OH with toluene forming H 2 O + methylphenyl and H 2 O + benzyl and is roughly 1:3 at 1000K, increasing to 1:2 at 2000K [22]. Another common source of methylphenyl type radicals is in the combustion of 2,4,6-tri-nitrotoluene (TNT) [23]. Under explosive conditions, methylphenyl radicals may form in high concentrations, as one of the dominant initiation reactions is the unimolecular decomposition of TNT to dinitromethylphenyl + NO 2 , with an activation energy of only 61 kcal/mol. 133 In the context of poly-substituted aromatics, methylphenyl radicals may result from one additional source – aryl C-CH 3 fission. The decomposition of xylenes, for instance, to a methylphenyl radical + CH 3 is expected to occur with a similar energy barrier to that of toluene decomposition to phenyl + CH 3 . The lower-right pathway of Figure 4.1 illustrates such a reaction. Indeed, this channel in toluene accounts for over 20% of decomposition products at temperatures above 1200 K, increasing to 40% at and above 1500 K [24, 25]. Figure 4.1 Initial decomposition pathways during o-xylene combustion. M typically represents an inert collider species such as N 2 or O 2 . There are subtle, but important kinetic differences between the phenyl-type and benzyl- type radicals during high-temperature oxidation of toluene and xylene isomers. The phenyl radical can undergo quite rapid reaction with molecular oxygen, producing phenoxy (C 6 H 5 O) and benzoquinone (C 6 H 4 O 2 ), C 6 H 5 + O 2 → C 6 H 5 O + O (R1a) → C 6 H 4 O 2 + H (R1b)  + H + CH 3  + H  + H  +M, H, O, OH +M +M, H, O, OH +M, H, O, OH  + H + CH 3    + H + CH 3   + H  + H  + H  + H  +M, H, O, OH +M +M, H, O, OH +M, H, O, OH 134 both of which are expected to assist in radical pool growth through radical chain branching (R1a) or the production of the H atom (R1b). The benzyl radical, on the other hand, does not react with molecular oxygen rapidly. It is known that the rate coefficient of the reaction C 6 H 5 CH 2 + O 2 → products (R2) is a factor of 13 and 160 smaller than k1 at 700 and 1500 K. Rather, resonance stabilization allows the benzyl radical to be rather long-lived, leading to radical termination through its recombination with the H atom, C 6 H 5 CH 2 + H (+M) → C 6 H 5 CH 3 (+M) (R3) A methylphenyl radical immediately adjacent to a methyl group has several fates. For example, the 2-methylphenyl radical may undergo CH 3 or H elimination to form o- benzyne or methylbenzyne, but the energy barriers for these reactions are rather large, roughly 80 kcal/mol [26-29], for H elimination, and around 70 kcal/mol for CH 3 elimination, based on a comparison of toluene bond strengths. On the other hand, H- shifting about the ring occurs with an energy barrier of approximately 60 kcal/mol [27]. Finally, H-shifting to form resonantly stabilized benzyl is expected to be favored on the basis of reaction exothermicity. Under oxidative conditions, a methylbenzyl radical may undergo reaction with molecular oxygen. Such reactions have been recently shown to enable chain branching pathways in one-ring aromatics with two or more adjacent methyl groups [30, 31]. Oxygen addition to methylbenzyl forming the peroxyl radical is largely independent of temperature and 135 with a rate coefficient ~ 10 12 cm 3 mol -1 s -1 . da Silva and Bozzelli [30] have recently shown how successive O 2 addition to o-methylbenzyl can lead to chain branching, eventually forming o-quinone methide + 2OH + H. Such a reaction pathway appears to be available exclusively to o-xylene in which the two methyl groups are adjacent to each other and has been used to explain the overall greater reactivity of o-xylene over m- and p-xylene. The above mechanism also appears to explain the empirical correlations between the structures of various substituted aromatics and their Research Octane Number (RON) [30, 32]. In general, RON decreases with the number of adjacent methyl pairs in these aromatic fuels. For instance, in the xylene class of compounds, o-xylene is the only species with an adjacent methyl pair (see Table 1). Its RON is lower than its m- and p- counterparts, which are roughly equivalent. Global flame and ignition phenomena demonstrate that the observed differences between the three xylene isomers must be kinetic in nature. This is supported through a comparison of their adiabatic flame temperatures, shown below in Table 4.1. 136 Table 4.1 Adiabatic flame temperatures of selected aromatic compounds, including the three xylene isomers. Calculated for a stoiciometric fuel-air mixture at T u = 353 K and p = 1 atm. Chemical name Formula Structure T ad (K) benzene C 6 H 6 2373 toluene C 7 H 8 2359 o-xylene C 8 H 10 2349 m-xylene C 8 H 10 2349 p-xylene C 8 H 10 2349 137 o-xylene also has noticeably lower ignition delay times than m- and p-xylene over a wide range of conditions (996 ≤ T (K) ≤ 1408 K, p = 10 and 40 atm,  = 0.5 and 1.0 in air) [33]. For example, the ignition delay time measured for a stoichiometric o-xylene-air mixture at 1200 K temperature and 10 atm pressure is a factor of 1.6 and 1.9 shorter than m- and p-xylene-air mixtures, respectively, under comparable conditions. Figure 4.2 below compares the ignition delay times for the three isomers at unity equivalence ratio for the temperature range of 1350 - 1800 K and a pressure of 10 atm [33]. Figure 4.2 Ignition delay times behind reflected shock tubes for the three xylene isomers [33]. Refelcted shock pressure of 10 atm,  = 1.0, T = 1350 - 1800 K. 10 2 10 3 0.67 0.75 0.82 0.90 ortho meta para 1000 K / T 1.962% xylene, 20.60% O 2 , and 77.44% N 2 Ignition Delay Time ( s) 138 Likewise, the laminar flame speed of fuel-lean o-xylene-air mixtures was found also to be higher than that of m-xylene by about 1 cm/s, and that of m-xylene is higher than that of p-xylene also by 1 cm/s [34]. This trend is visible in Figure 4.3, which compares the premixed laminar burning velocities of the three isomers for a wide range of equivalence ratio. Figure 4.3 Experimental laminar flame speeds of o-, m-, and p- xylene-air mixtures at T u = 353 K and p = 1 atm [34]. The error bars represent 2  uncertainty. The differences in the measured ’s cannot be explained by the variations in T ad . As shown in Table 4.1, the difference in T ad for  = 1 flames of the various aromatic compounds is rather small. Excluding benzene and toluene, T ad values are well within 7 K of each other. For the benzene/air and toluene/air flames, T ad is somewhat higher 0.6 0.8 1.0 1.2 1.4 1.6 10 20 30 40 50 o-xylene/air m-xylene/air p-xylene/air Laminar Flame Speed, S o u , cm/s Equivalence Ratio,   S u o 139 than that of other flames by as much as 30 K and 17 K respectively. Hence, the propagation rates of aromatic flames must be influenced quite significantly by the finite rate reaction kinetics. Furthermore, the global, high-temperature oxidation experiments appear to suggest that the peroxyl route is not the only channel that may explain the differential oxidative reactivity among the xylene isomers, as such a pathway does not support the difference observed between m- and p-xylene. In fact, the predominantly low-temperature peroxyl route is expected to be unimportant to the overall oxidation rates primarily because O 2 addition to o-methylbenzyl is only 22-23 kcal/mol exothermic [35, 36], and is therefore highly reversible above 1000 K. Such isomerizations have been observed experimentally in the past, but with little consensus in the rates [37-39]. H-shift leading to benzyl is energetically feasible only for the 2-methylphenyl radical, i.e., The only available kinetic rate estimate is given by Gail et al. [15-17] in a detailed kinetic mechanism for o-xylene oxidation:  M  M   1 13 1040/ 3 s 6.80 10 T ke   140 Under the jet stirred reactor condition they studied (T = 1300 K,  = 1.0,  = 1 s), the above rate essentially leads to instantaneous isomerization of all 2-methylphenyl radicals produced to form benzyl [17]. There is currently no clear description of how fast methylphenyl radicals isomerize to those of benzylic nature. Thus, the goal of this work is to provide accurate rate estimates for the isomerization of methylphenyl radicals to those of benzylic nature. This theoretical study examines the H-shift reactions in methylphenyl and dimethylphenyl through ab initio electronic structure and RRKM/Master Equation analysis. This allows for a description of various 2-methylphenyl – benzyl isomerization kinetic parameters which have not yet been fully quantified up to now. The rate estimates should be of some benefit to those in the combustion modeling community focusing on substituted aromatic compounds. 4.2 Computational Methodologies 4.2.1 Quantum Chemical and RRKM Calculations Electronic structure calculations were performed on both an in-house cluster and University of Southern California's Center for High-Performance Computing and Communications (HPCC), using the computational chemistry package QChem 3.2 [40]. All geometries and vibrational frequencies were determined using a recently developed long-range corrected hybrid density functional with damped atom-atom dispersion 141 corrections [41]. Termed B97X-D, this functional has been tested using a wide array of training sets with average statistical errors significantly smaller than those obtained using the B3LYP-D functional. The 6-311G(2d,p) basis set was also used for all species in this study. Although there are not yet any studies suggesting proper scaling factors for the level of theory used here, vibrational frequencies were multiplied by 0.968, suggested for B3LYP/6-311G(2d,p) [42]. A computational grid with 75 radial points and 434 angular points per radial point was used in the calculations for all species. The effect of tunneling was accounted for through the Wigner treatment [43]. 4.2.2 Thermal Rate Calculations Solution of the master equation for the mutual isomerization reactions in this study were obtained through a kinetic Monte Carlo algorithm. It is described in brief here, as details may be found in Chapter 2. The stochastic approach is similar to that implemented in Multiwell [44, 45]. The time evolution of the systems studied here are described using the discrete form of the Master Equation: where A denotes a species, E i an energy level, and M a collider species (N 2 in this case). The brackets, [], denote concentrations of the species within. k m is a channel-specific microcanonical rate constant at a specific energy level, calculated using Equation 2 below. For mutual isomerizations, all channels are treated as reversible. Z is the collision frequency calculated using a Lennard-Jones potential interaction, with               i ij j ji i m i i jm d A E Z M P A E P A E k E A E dt                 142 parameters taken from the early work of Hippler and coworkers [46]. All Lennard-Jones parameters excluding the bath gas were assumed to be equal to that of toluene. P ij is the probability of a species to be transferred from energy level j to energy level i. These values are calculated using the exponential down model, with  E down  = 350(T/298) 0.3 cm -1 for Argon (assumed equivalent to N 2 in this study) [47]. A maximum energy of 87,500 cm -1 was assigned, with energy grains of 25 cm -1 . The microcanonical rate constant corresponding to each channel was determined using the following conventional expression described in more detail in Chapter 2. The density and sum of states of all active internal vibrational modes were computed with the Whitten–Rabinovitch approximation [48]. Active free rotors were treated with the method of Astholz et al [49]. One-dimensional hindered rotors were treated with Knyazev’s formulae [50, 51]. The methyl group of toluene was treated as a one- dimensional active free internal rotor, as it exhibits nearly barrierless rotation [52-55]. The same is true for methyl rotations in m- and p-xylene, while methyl rotations in o- xylene are hindered with barriers of 1.22 kcal/mol [55, 56]. The methyl group of o- methylbenzyl has a hindered rotational barrier of 2.16 kcal/mol [56]. The rotation of CH 2 about the phenyl ring in benzyl has a significantly higher barrier to rotation. Relaxed       † † , , r in r in WE Q kE Q h E   143 scans at the B97X-D/6-31G(d,p) level of theory put its value at around 13 kcal/mol. Thus, this rotor is adequately described as a harmonic oscillator. Experimentally fitted reduced internal rotational constants of methyl were taken from Lin and Miller [56]. For all internal free and hindered rotors, the corresponding low frequency vibrational modes were identified and removed. Selected parameters are shown in Table 4.2; the reader is referred to the appendix for molecular geometries and unmodified vibrational frequencies. An in house code was used to solve for the time evolution of a thermally excited reactant at a given temperature and pressure. Convergence in rate with number of stochastic trials was ensured. The rate constants are then obtained through linear least squares fitting of the desired species count as a function of time. Coefficients of determination, R 2 , for all rates were nearly 1.0. 144 Table 4.2 Summary of calculated electronic energies (E 0 ) and zero point energies (ZPE) in Hartrees, along with all rotational parameters used in RRKM/Master Equation calculations. external rotors internal rotors d species a E 0 ZPE inactive b (cm -1 ) active c (cm -1 ) no. of rotors V 0 (kcal/mol) benzyl -270.88549 0.11511 0.075 0.187 0 - 2MP -270.85032 0.11556 0.071 0.200 1 0 3MP -270.85034 0.11536 0.072 0.200 1 0 TS-1-2 -270.77680 0.11000 0.075 0.188 0 - TS-1-3 -270.74509 0.10919 0.071 0.205 1 0 OX6 -310.16796 0.14370 0.057 0.112 2 1.22 OX1 -310.20274 0.14314 0.058 0.109 1 2.16 OX5 -310.16727 0.14360 0.058 0.108 2 1.22 TS-6-1 -310.09491 0.13747 0.059 0.099 1 0 TS-6-5 -310.06169 0.13708 0.057 0.111 2 1.22 MX2 -310.16739 0.14319 0.049 0.124 2 0 MX1 -310.20255 0.14258 0.051 0.123 1 0 TS-2-1 -310.09484 0.13753 0.053 0.111 1 0 PX2 -310.16732 0.14296 0.043 0.194 2 0 PX1 -310.20271 0.14249 0.045 0.181 1 0 TS-2-1 -310.09320 0.13738 0.045 0.180 1 0 TS-2-3 -310.06374 0.13685 0.043 0.197 2 0 species σ (Å) ε/k B (K) N 2 3.74 82 all other 5.68 495.3 a See Figure 4.4-Figure 4.8 for species identification. b Two-dimensional (all symmetry number σ =1); c One-dimensional (symmetry number σ =2 for benzyl, MX2, PX1, TS-2-3 (PX); σ =1 otherwise). d One-dimensional (all symmetry numbers σ =3, rotational constants = 5.2 cm -1 ). Those with no barrier are treated as free internal rotors. 4.3 Results and Discussion 4.3.1 Electronic Structure Calculations The large and diverse number of semi-empirical density functionals available today enables computationally fast and reasonably accurate energies for systems tailored to different applications. However, some caution must be taken to ensure that the right functional is being used for the right system(s). Although the B97X-D functional was 145 developed and tested against a diverse training set with thermochemistry and kinetics in mind for large molecular systems, several independent tests were performed to validate the use of it here in obtaining accurate barrier heights. Although no theoretical studies have yet been performed on the isomerization kinetics of methylphenyl radicals, the isomerization of phenylethen-2-yl to 2-styrenyl through a five- member ring transition state has been investigated using a large number of quantum chemical methods, including PM3, MP2, B3LYP, CASPT2, and G2MP2 [57]. The authors choose G2MP2 as the method of choice. The similarity of this reaction with the mutual isomerization of methyl phenyl radicals, in conjunction with the large body of data available for this reaction, make it a good candidate for validating use of the B97X-D functional for the systems in this study. Indeed, not only were predicted geometries very similar to those of determined by G2MP2, but the 0 Kelvin activation energy was found to agree within 2.1 kcal/mol ( 26.3 compared to 28.4 kcal/mol for G2MP2, including zero-point energy). Additionally, the barrier height of 2-methylphenyl isomerization to benzyl through a four-membered ring transition state was compared at several levels of theory with varying basis set sizes. All values agree to within ~1 kcal/mol. The reader is referred to the Appendix illustrating a comparison of these values. In general, the B97X-D method predicts an isomerization barrier lower by 1 kcal/mol, relative to that determined at the CCSD(T)/aug-cc-pVDZ level of theory. Energy extrapolation to the complete basis set 146 limit of the couple cluster calculations would reduce this difference further, but as will be discussed later, this is not the largest source of uncertainty in barrier heights. Furthermore, such an approach was taken by Derudi and co-workers who recently investigated benzyl decomposition kinetics at the CCSD(T)/CBS//B3LYP/6-31+G(d,p) level of theory [58]. Compared to the B97X-D calculations performed here, barrier heights for 2-methylphenyl isomerization to benzyl agree to within 0.3 kcal/mol. The choice of the 6-311G(2d,p) basis set accompanying the B97X-D was also found to be sufficient, with the inclusion of diffuse functions having negligible effect on calculated barrier heights. Unfortunately, use of B97X-D, and density functional theory in general, is not without concern. Zero point energy corrected H and CH 3 elimination barriers from 2- methylphenyl to the formation of a benzyne are predicted to be 86.0 and 79.1 kcal/mol respectively, depicted in Figure 4.4. The barrier of H elimination from phenyl forming o- benzyne has been measured at 77.5  3.1 kcal/mol [29]. A high level ab initio treatment of this reaction predicts a barrier of 76 kcal/mol, in good agreement with experimental measurements [28]. Calculated at the B97X-D/6-311G(2d,p) level of theory however, this barrier is predicted to be 87 kcal/mol. A difference of 11 kcal/mol is unreasonably large. The artificially high barrier for these types of elimination/dissociation reactions is a known problem with DFT [59]. Relative to other possibilities though, these barriers are so high that the corresponding channels need not be included in the kinetic study. Thus, while the isomerization energetics are sufficiently accurate using the B97X-D/6- 147 311G(2d,p) level of theory, the barriers to H or CH 3 elimination are not reliable. For completion however, they are shown in Figure 4.4 along with a truncated pathway for benzyl decomposition. The reader is referred to the literature for more elaborate descriptions of this system [58, 60, 61]. Benzyl decomposition has also been investigated in shock tubes and found to result in the formation of a C 7 H 6 fragment and H, with an overall activation energy of 80.7 kcal/mol. A comprehensive analysis of the benzyl potential energy surface, involving 15 accessible energy wells connected by 25 transition states, suggests the C 7 H 6 fragment to be fulvenallene [25]. Recent RRKM/Master equation simulations by the same group suggest benzyl decomposes via two pathways to benzyne + CH3 and fullvallene + H with nearly equal branching ratios between 1500 and 2000 K [58]. 148 Figure 4.4 Potential energy surface for the 2-methylphenyl system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. a Overpredicted by ~10 kcal/mol; see text. b Value taken from [25]. c Intermediate channels not shown. As shown in Figure 4.4, the isomerization barrier from 2-methylphenyl to the resonantly stabilized benzyl is clearly favored, with an activation energy of only 43 kcal/mol. In fact, for all the systems studied, barriers to such isomerizations ranged from 42-43 kcal/mol, with corresponding reverse barriers around 64-67 kcal/mol. The forward barriers of interest are listed in Table 4.3 and also shown in all corresponding potential energy surface figures. Barriers of H-shifting about the ring are also consistently similar, ranging from 61-63 kcal/mol. Figure 4.4-Figure 4.8 depict potential energy surfaces for the corresponding reactions in the xylene class of compounds.   42.8 -22.3 62.2 Relative Energy (kcal/mol) 0 0 + CH 3 + H 86.0 a 79.1 a + H c 86.8 b 3mp 2mp TS-1-3 TS-1-2 benzyl   42.8 -22.3 62.2 Relative Energy (kcal/mol) 0 0 + CH 3 + H 86.0 a 79.1 a + H c 86.8 b     42.8 -22.3 62.2 Relative Energy (kcal/mol) 0 0 + CH 3 + H 86.0 a 79.1 a + H c 86.8 b 3mp 2mp TS-1-3 TS-1-2 benzyl 149 Figure 4.5 Potential energy surface for the o-xylene (1,2-dimethyl-6-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. Figure 4.6 Potential energy surface for a m-xylene (1,3-dimethyl-2-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. The full surface is shown for completion, but is symmetric about MX2.     42.1 -22.2 62.7 Relative Energy (kcal/mol) 0 0 OX5 OX6 TS-6-5 TS-6-1 OX1     42.1 -22.4 42.1 Relative Energy (kcal/mol) -22.4 0 MX1 MX2 TS-2-1 TS-2-1 MX1      42.1 -22.4 42.1 Relative Energy (kcal/mol) -22.4 0 MX1 MX2 TS-2-1 TS-2-1 MX1   150 Figure 4.7 Potential energy surface for a m-xylene (1,3-dimethyl-6-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy. Kinetics of this system are assumed to be identical to that of the 2-methylphenyl system. Figure 4.8 Potential energy surface for the o-xylene (1,2-dimethyl-6-phenyl) system. Calculated at the B97X-D/6-311G(2d,p) level of theory. Includes zero point energy.    42.7 -22.7 62.2 Relative Energy (kcal/mol) 0 0 MX5 MX6 TS-6-5 TS-6-1 MX1       42.7 -22.7 62.2 Relative Energy (kcal/mol) 0 0 MX5 MX6 TS-6-5 TS-6-1 MX1    43.1 -22.5 61.3 Relative Energy (kcal/mol) 0 0 PX3 PX2 TS-2-3 TS-2-1 PX1       43.1 -22.5 61.3 Relative Energy (kcal/mol) 0 0 PX3 PX2 TS-2-3 TS-2-1 PX1 151 Table 4.3 Zero point energy corrected barriers (E o ) for key reactions of interest in this study. 4.3.2 Thermal Rate Calculations The Master Equations for each system consider two reversible channels leaving the reactant methylphenyl species: mutual isomerization to a benzylic species and H-shift to the next ring site. Other possibilities, like those shown in Figure 4.4, are not competitive with the aforementioned channels. As will be discussed shortly, even the H-shift channel has only a slight affect on the overall predicted rate constants; the other channels, 10-20 kcal/mol above this, have no influence on the overall kinetics of the system.                     43.1 42.1 42.7 42.1 42.8 E o (kcal/mol) Product Reactant 43.1 42.1 42.7 42.1 42.8 E o (kcal/mol) Product Reactant 152 As shown in Figure 4.9, for all temperatures and pressures, isomerization rates from the reactant to a benzylic species across all systems are all roughly within a factor of two of each other. This is of course expected. Firstly, the central aromatic ring of every species dominates the rotational contributions to their corresponding partition functions. The ratio of the inactive external rotors between the transition state and reactant across all systems is nearly the same. Similarities between the 2-methylphenyl and MX6 (1,3- dimethyl-6-phenyl) systems shown in Figure 4.4 and Figure 4.7 respectively are so great that calculations were not needed for this system. Thus, the suggested rate constants for the MX6 system are set to those calculated for the 2-methylphenyl system. Because the four-membered ring transition states for all isomerizations are nearly identical, so are their tunneling contributions, which increase isomerization rates by around 40 % at 1000K, decreasing to 20% at 1500 K. 153 Figure 4.9 Arrhenius plots for isomerizations of 2-methylphenyl type radicals to those of benzylic nature. Symbols are rates computed with Monte Carlo RRKM/master equation modeling and lines represent fitted Arrhenius expressions within the recommended temperature range (Table 3). See Figure 4.4-Figure 4.8 for species identifications. Uncertainty bands of  2 kcal/mol in activation energy are included in the high pressure rate constants. To investigate the validity of the Wigner tunneling approximation in this work (as a direct multiplier of the overall isomerization rates) the channel-specific microcanonical rate constants are re-defined as: 10 3 10 4 10 5 10 6 10 7 10 8 10 9 0.5 0.6 0.7 0.8 0.9 1 2mp benzyl MX6 MX1 k0.1 k1 k10 k50 kinf k, 1/s P (atm) 10 3 10 4 10 5 10 6 10 7 10 8 10 9 0.5 0.6 0.7 0.8 0.9 1 OX6 OX1 10 3 10 4 10 5 10 6 10 7 10 8 10 9 0.5 0.6 0.7 0.8 0.9 1 PX2 PX1 1000 K / T k, 1/s 10 3 10 4 10 5 10 6 10 7 10 8 10 9 0.5 0.6 0.7 0.8 0.9 1 MX2 MX1 1000 K / T         † † , , r in a r in WE Q k E T l Q h E    154 This approach is of course less formal than writing the tunneling coefficient as a function of energy in the reaction coordinate [62], but nonetheless captures the influence of reaction endo/exothermicity on the tunneling coefficients and overall rate constants in the ME formalism. With the 2-methylphenyl system is taken as a test case, the master equation is solved with the above modified channel-specific rate constants. For these calculations, the Eckart tunneling correction was applied [63, 64]. For mutual isomerization reactions, this approach has been shown to be comparable to higher level multidimensional corrections [65]. The net effect of including tunneling in the Master Equation framework is an overall decrease in isomerization rate compared to that if the conventional RRKM channel-specific rate constants were used and the final derived rate constant multiplied by the tunneling coefficient. For instance, the 2MP  benzyl isomerization rate increased by less than 20% at 1400 K. Although, the differences are expected to be larger at lower temperatures, there are numerous sources of uncertainty and large discrepancies between various tunneling approximations at temperatures below 500 K [65]. More importantly, at such temperatures, the systems studied here are essentially unreactive. And at temperatures above 1000 K, all tunneling approximations converge to within the uncertainty of the predicted rates. Thus, for the reactions in this study, use of the simple Wigner tunneling coefficient as a direct multiplier of the Master Equation derived rate constants is justified. However, for systems with more pronounced reaction endo/exothermicity or with irreversible channels, use of the above approach may not be sufficient. 155 Figure 4.9 also illustrates the high pressure dependence of the mutual isomerization reactions in this study. At 2000 K, calculated rates span over two orders of magnitude between 0.1 atm and the high pressure limit. For all systems at 0.1 atm, isomerization rates slightly decrease above 1700 K. At high temperatures, the average initial energy of the reactant is well above all barriers in the system, at which point pseudo-equilibrium between all isomers is established. In addition, the smaller frequency of collisions at 0.1 atm reduces the stabilization count in the lowest energy lying wells. Another reason there is a slight decrease in rate with increasing temperature at low pressures is the subtle influence of the H-shifting channels, as some of the stochastic trials end with stabilization occurring in these wells. However, the fractional population of these species is so small that the overall rates are still negligible for all temperatures and pressures. For instance, at 2000K and 0.1 atm, 0.3% of 2-methylphenyl undergo H-shifting to 3-methylphenyl, this representing an upper bound for all T and P. In the context of a methylphenyl radical adjacent to a methyl group, H-shifting to other phenyllic ring sites effectively does not occur. However, for ‘isolated’ methylphenyl radicals at high temperatures, H-shifting may play a more pronounced role. An isolated methylphenyl radical is defined here as one that is not adjacent to a methyl (or any alkyl) group. These sites are designated by filled circles in Figure 4.11. As mentioned earlier, these radical species readily form through H- abstraction reactions. Figure 4.10 displays the high pressure limit rate constants for selected channels in the 2-methylphenyl system. At and above 1800 K, the lifetime to H- 156 shifting about the ring is less than a microsecond. Thus, isolated methylphenyl radicals formed may quickly find their way to any available alkyl group. Under oxidative conditions though, this H-shifting about the ring can be expected to compete with O 2 addition only at high temperatures and pressures. For pressures and temperatures at and above 1atm and 1400 K, the lifetimes of all methylphenyl radicals adjacent to methyl groups is less than a microsecond, decreasing by over two orders of magnitude at 2000 K. Under such conditions, it is reasonable to assume that all 2-methylphenyl type radicals, once formed, immediately isomerize to their corresponding benzyl isomers. On the other hand, all isolated methylphenyl type radicals should be explicitly considered, since the overall H-shifting to benzyl is not nearly as fast. The equilibrium constant of 2-methylphenyl  benzyl is also plotted in Figure 4.10 to illustrate the overall stability of benzyl with change in temperature. It’s clear that for all temperatures, equilibrium lies strongly on the side of benzyl. Lastly, the 2-methylphenyl  benzyl rate used in the o-xylene mechanism of Gail and coworkers is also plotted in Figure 4.10. As mentioned earlier, the rate has a pre-exponential factor of similar magnitude to those computed here (shown in the high-pressure limit values of Table 4.4), but an activation energy of 2.1 kcal/mol is too low by over an order of magnitude. This rate is therefore not valid for any temperature of interest to combustion. 157 Figure 4.10 Microcanonical high pressure limit rate constants for selected channels in the 2-methylphenyl system. See Figure 2 for species identification. Tunneling contributions not included. The lack of corresponding experimental data for these reactions makes it difficult to quantify the uncertainty in the estimated rate parameters. Clearly, reaction barriers are one of the main contributors to uncertainty in chemical kinetics. In this study, isomerization barriers of 2MP  benzyl differed by only ~1kcal/mol between several other quantum chemical methods tested (see table S1). However, use of the B97X-D/6- 311G(2d,p) level of theory to reproduce the G2MP2 chosen barrier for isomerization of hydrogen in the phenylethen-2-yl radical resulted in a discrepancy of 2.1 kcal/mol [57]. Thus, an uncertainty of  2 kcal/mol is assigned to the activation energies in the 10 0 10 2 10 4 10 6 10 8 10 10 10 12 10 14 0.5 0.6 0.7 0.8 0.9 1 2mp -> benzyl benzyl -> 2mp Keq, 2mp <=> benzyl 2mp -> 3mp 2mp -> benzyl (Gail et al., 2008) 1 100 10 4 10 6 10 8 10 10 10 12 10 14 k , 1/s 1000K / T K eq , unitless 8 ? 158 recommended rate expressions. This translates to an overall rate uncertainty factor of 2-3 for the temperature and pressure conditions of this study. Double three-parameter modified Arrhenius expressions are recommended in Table 4.4. Two modified Arrhenius expressions are necessary to capture the pressure dependence at higher temperatures and the many orders of magnitude in rate spanned over the temperature range of 298 - 2000 K. Corresponding high pressure limit rate constants are also recommended in conventional Arrhenius form. The high pressure rate constant for 2-methylphenyl to 3-methylphenyl H-shift is also suggested in Table 4.4, and may be used as general rate of H-shifting about aromatic rings. It is important to note that, even though rates are recommended down to 298 K, the reactions studied here are inherently high temperature phenomena. The rate constants for all isomerizations at room temperature are effectively zero. 159 Table 4.4 Monte Carlo RRKM/Master Equation results fitted to double three-parameter and conventional Arrhenius expressions for the temperature range 298-2000 K a k(T) = A 1 T n1 exp(–C 1 /T) + A 2 T n2 exp( –C 2 /T) Reaction channel P (atm) A 1 n 1 C 1 A 2 n 2 C 2 2MP → benzyl and 0.1 2.01×10 116 -30.186 51114 6.41×10 47 -11.150 26942 MX6 → MX1 1 5.34×10 102 -25.808 50000 9.42×10 45 -10.471 26942 10 1.57×10 91 -22.039 50000 2.36×10 43 -9.519 26943 50 3.59×10 82 -19.274 50000 1.37×10 41 -8.711 26943 k  3.34×10 13 0 45925 OX6 → OX1 0.1 1.13×10 124 -32.586 51114 7.9×10 52 -12.723 26942 1 6.21×10 112 -28.812 51114 2.19×10 51 -12.140 26942 10 8.39×10 100 -24.912 51114 3.59×10 48 -11.117 26942 50 5.88×10 91 -21.964 51115 1.11×10 46 -10.209 26942 k  4.35×10 13 0 44081 MX2 → MX1 0.1 1.14×10 120 -31.335 51114 1.94×10 51 -12.340 26942 1 2.65×10 110 -28.081 51114 6.14×10 49 -11.779 26942 10 3.10×10 100 -24.783 51114 3.16×10 47 -10.933 26943 50 7.78×10 92 -22.315 51115 1.52×10 45 -10.085 26943 k  6.87×10 13 0 45387 PX2 → PX1 0.1 4.09×10 116 -30.298 51114 5.24×10 47 -11.177 26942 1 9.70×10 105 -26.747 51114 1.76×10 45 -10.258 26943 10 6.93×10 93 -22.785 51114 2.21×10 42 -9.194 26943 50 5.43×10 84 -19.875 51115 1.12×10 40 -8.367 26972 k  3.42×10 13 0 46324 2MP → 3MP (ring H-shifting) k  1.42×10 14 0 32732 a Units are s -1 , cal/mol, and K. Rates computed with N 2 as the bath gas. b See Figure 4.4 through Figure 4.8 for species identification. These rates may also be used to confidently estimate rates in analogous systems of larger size, with two or more aromatic rings. Such information may be useful in describing the kinetics of large PAH like those formed during soot formation. Although only methyl substituents are considered here, the isomerization kinetics for larger hydrocarbon chains (i.e., ethyl, propyl) may be even more rapid because of a decrease in ring strain of the 160 transition states involved and subsequently lower activation energy. This is indeed the case for ethyl benzene [57]. 4.3.3 Global Reactivity Trends As noted in the introduction and illustrated in Figure 4.11, general trends of reactivity have been established for mono- and poly-substituted aromatics. Earlier observations have shown that RON decreases with the number of adjacent methyl pairs in these aromatic fuels [32]. In particular, da Silva and Bozzelli have further suggested the potential role of 2-methylphenyl + O 2 branching reactions as an explanation for why o- xylene has a lower RON than its counterparts [30]. However, the same trend is also observed under high temperature shock tube conditions [33], where isomerization of 2- methylphenyl to benzyl is expected to compete with O 2 addition. Consequently, other possibilities for the o- > m- > p-xylene RON and ignition delay trends must be entertained. 161 161 Type of substituted one-ring aromatic No. isolated sites a 3 2 1 0 0 Reactivity in Shock Tubes b III I II III N/A Fraction of analogous C 6 H 5 reaction with other species 1/2 1/3 1/6 0 0 Research Octane Number c 124 120 146 146 170 Flame speed d I II III IV V Figure 4.11 : Sites where H-abstraction may result in an isolated free radical. : Sites where H-abstraction may result in an H-shift to a benzylic radical. a Number of sites not adjacent to a methyl group. b Ref.[33]; ignition delay increases with number. c Ref.[30] d Fuel lean flame speed decreases with number [34]. 162 As shown in Figure 4.11, o-xylene has 2 isolated sites, while m-xylene and p-xylene have 1 and 0, respectively. Although the bond strength of an aryl C-H is higher than both the methyl C-H and aryl C-CH 3 bonds, they more readily undergo scission at high temperatures [18]. Additionally, they undergo H-abstraction reactions with other radicals (H, OH, HO2, etc.) further forming isolated methylphenyl radicals. These isolated radicals can then directly react with molecular oxygen leading to secondary chain branching. Such a possibility is in stark contrast to da Silva and Bozzelli’s explanation for xylene oxidation trends at lower temperatures, although the two phenomena are not mutually exclusive. A reliable detailed kinetic model including all the above kinetic parameters and those already available would be necessary to quantitatively probe reasons behind the relative reactivity of xylenes. Not only is such a mechanism is beyond the scope of this work, but the fundamental kinetics of benzene/toluene pyrolysis and oxidation must first be understood to a more accurate level than our current level of understanding. However, we expect that, based on the discussion above and number of available isolated sites, the reactivity of xylenes may be ordered as ortho > meta > para, as indeed observed in the laminar flame speeds [66], rapid compression machines [32], jet-stirred reactors [15-17], flow reactors [14], and shock-tube ignition delays [33] measured for these compounds under comparable conditions. 163 4.4 Conclusions The potential energy surfaces for several analogous systems of toluene and xylene nature were investigated through the use of density functional theory. Barriers for the exothermic isomerization of a methylphenyl radical to one of benzylic nature were determined to be 42-43 kcal/mol, with an assigned uncertainty of  2 kcal/mol. Barriers of H-shifting from isolated sites about a six-membered aromatic ring were calculated to be 61-63 kcal/mol. 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Eckart, Physical Review 35 (11) (1930) 1303-1309 64. R. L. Brown, Journal of Research of the National Bureau of Standards 86 (4) (1981) 2 65. B. Sirjean; E. E. Dames; H. Wang; W. Tsang, The Journal of Physical Chemistry A (2011), in press. 66. C. Ji; E. Dames; H. Wang; F. N. Egolfopoulos, Combustion and Flame, submitted. 169 Chapter 5: Dehydrogenation of Cyclohexane and Methylcyclohexane 5.1 Introduction and motivation This work was initially motivated by observed discrepancies between detailed kinetic model predictions of high pressure ignition delay with their experimentally determined counterparts. The kinetic model in question is JetSurF 1.1 [1]. JetSurF 1.1 is capable of predicting a wide range of experimental combustion data: laminar flame speeds, plug and jet-stirred flow reactor species profiles, ignition delay times [2-8]. However, the model fails to predict high pressure (>15 atm) shock tube ignition delay times, being too slow by up to several factors in some cases [9-11]. This model-experiment ignition delay discrepancy is considerably greater in the case of mono-alkylated cyclohexane compounds, and is only observed under high pressure. Among the possibilities considered for these mono-alkylated cyclohexane compounds, the lack of dehydrogenation reactions in the model was thought to be partially responsible for the overestimated ignition delay times. Subsequent dehydrogenation reactions serve as a source of H radicals which can increase overall reaction progress. Initially, analogous cyclohexane reactions were to be used in estimating the dehydrogenation rates of methylcyclohexane. After inspection of available dehydrogenation chemistry of cyclohexane, we realized that even this chemistry not satisfactorily understood, particularly because of the high pressure dependence of the participating reactions. Thus, focus shifted to cyclohexane, and in particular, the first dehydrogenation step - hydrogen 170 elimination from the cyclohexyl radical. Although this reaction is not a kinetic bottleneck to benzene formation, it is a crucial step to its formation. Another motivation for this work is a desire to understand the thus far unknown kinetic and/or thermodynamic reasons for differing sooting tendencies between cyclohexane and its mono-alkylated counterparts. Recall that soot formed in flames roughly scales linearly with the benzene concentration. As such, benzene formation is conventionally defined as a sooting precursor and used to judge how much any particular fuel or burning condition will soot. The sooting tendency of mono-alkylated cyclohexanes has been long known to be significantly greater than for cyclohexane. Additionally, recent low pressure burner stabilized methylcyclohexane flame experiments have allowed for a closer look at benzene formation. A number of possibilities will be discussed and the methyl group will be seen to enhance sooting propensity of methylcyclohexane relative to cyclohexane. 5.2 The cyclohexyl system Like other typical first fuel radical species, cyclohexyl is formed either from a direct scission of an H-atom or H-abstraction from various radicals (e.g., H, O, OH, HO 2 , and CH 3 ). Under low temperature and/or high-pressure conditions, cyclohexyl may react with any available O 2 , thus initiating low-temperature chain branching pathways [12, 13]. Such a possibility of course does not exist under pyrolytic conditions, and is less important under high temperature combustion. Rather, cyclohexyl may react with itself, another hydrocarbon compound, other gas-phase radicals, or it may undergo 171 unimolecular isomerization/decomposition. The last step is a prerequisite for eventual dehydration to benzene. There are two major direct channels available for the unimolecular isomerization/decomposition of cyclohexyl: isomerization to 1-hexen-5-yl and a hydrogen elimination to cyclohexene + H. The figure below illustrates those and other channels on the cyclohexyl potential energy surface. Figure 5.1 Cylcohexyl potential energy surface. Relative energies not included. Not included in Figure 5.1 are the following isomerization and subsequent decomposition reactions, shown below in Figure 5.2: CH3    H   H  H  H  172 Figure 5.2 Reactions not shown in the cyclohexyl PES of Figure 5.1 As the focus here lies in the thermal decomposition/isomerization of cyclohexyl, not all of the channels outlined above need be considered because barriers for the major channels of interest are considerably smaller than those for other minor channels. In addition, for the conditions of this work (thermal isomerization from cyclohexyl), sensitivity calculations were performed showing that the higher energy channels contribute roughly 5% of the total overall cyclohexyl consumption at 2500 K and 1 atm. Neglecting them in these calculations is therefore justified. However, such channels may necessary for modeling shock tube conditions where the initial internal molecule (vibrational) temperature can be much great than that of the bath gas. In an effort to further understand cyclohexane oxidation and pyrolysis under combustion relevant conditions, a number of recent studies have been performed regarding the cyclohexyl system. Knepp and coworkers recently performed a TST study on the decomposition of cyclohexyl using modified a G2MP2 composite model chemistry on a spin restricted surface [14]. The high pressure rate constants for selected channels were calculated using the rigid-rotor harmonic oscillator assumptions for all vibrational      173 degrees of freedom. Among O 2 + cyclohexyl pathways investigated, the following reactions were also studied: R 5.1 and R 5.2 with corresponding rate coefficients R 5.1: R 5.2: These rates were used in a larger kinetic model for cyclohexyl + O 2 , valid in the temperature range of 533-780 K. It is important to note that the geometry of the open 5- hexen-1-yl radical used in the study by Knepp and coworkers maintains a ring like structure and is roughly two kcal/mol above the lowest lying normal conformation. The choice of this geometry is motivated by the fact that the isomerization of cyclohexyl to cyclopentylmethyl does not require a fully elongated intermediate since both critical geometries out of the low 5-hexen-1-yl well also maintain a ring structure. This geometry is only about 2 kcal/mol higher than the fully elongated structure. Although such difference is of little kinetic consequence at high temperature (the energy difference of barrier heights out of the 5-hexen-1-yl well are more important), this should be kept in mind when comparing relative energy results across studies. 6 11 6 10 cC H cC H H  6 11 5-hexen-1-yl cC H      13 1 17862 6.91 10 exp ks T        13 1 15995 6.63 10 exp ks T    174 Sirjean and coworkers also studied the same reactions (including other size cycloalkyl radicals undergoing ring opening reactions) using TST with energies calculated at the CBSQB3 level of theory [15]. High-pressure rate recommendations for R 5.1 and R 5.2 are given below R 5.1: R 5.2: valid in the temperature range of 500-2000 K. In addition, isodesmic reactions were utilized to recommend standard 298 K formation enthalpies for the stationary structures. Free alkyl groups were treated with the HinderedRotor option in Gaussian 03, while constrained torsions of the cyclic structures were treated as harmonic oscillators. The authors also accounted for tunneling using the Wigner approximation. In comparison to available experimentally determined standard 298 K formation enthalpies, the agreement is satisfactory. From an experimental point of view, the cyclohexyl system can be more conveniently studied from the chemically activated perspective of cyclohexene + H. For this reaction, and that of cyclohexyl thermal isomerization/decomposition, many of the channels in Figure 5.1 have negligible influence on the overall reaction progress, allowing for a simpler depiction of the cyclohexyl PES, as depicted in Figure 5.3.     11 0.834 1 18289 8.91 10 exp k T s T        12 0.624 1 15506 2.75 10 exp k T s T    175 Figure 5.3 Reduced cyclohexyl potential energy surface with 0 K relative energies, ZPE included. Blue: RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p). Red: Knepp and coworkers' modified G2MP2 energies. H atom 298 K has 3/2RT added (0.9 kcal/mol) Note: difference in linear and kinked 5-hexen-1-yl is ~2kcal/mol Most recently, Tsang performed shock tube experiments on the thermal decomposition of cyclohexyl using t-butylcyclohexane as a radical source [16]. He also studied the cracking of 5-hexen-1-yl from the decomposition of 1,8-nonadiene. Although no comment can be made regarding the ME/RRKM methodology of this results presented in the work-in-progress poster, Tsang and coworkers performed a similar study on cyclopentyl cracking [17] and found model agreement with experimental data was best when the 0 K electronic energy barriers to ring opening and H-elimination were altered from their model chemistry derived values. In addition, slight adjustments of the low frequency vibrations in the transition states were necessary to create a model best fitting the experimental data. A similar conclusion can be assumed regarding the analogous  H H   0 21.7 21.6 31.6 28.8 36.6 34.8 31.4 32.4 32.0 30.9 6.0 5.6 43.2 41.1 35.2 35.7 Relative Energy, kcal/mol 176 cyclohexyl channels, as the recommended activation energies are up to three kcal/mol below those recommended by others. The tuning to this degree is near or above the limit of the chemical accuracy in the computational model chemistries used by other groups. Nevertheless, the recommended high-pressure limit rates are R 5.1: R 5.2: Figure 5.4 below compares various recommended rates of R 5.1 and R 5.2. Figure 5.4 Comparison of recommended R 5.1 and R 5.2 reaction rates. Solid line: R 5.1; dashed line: R 5.2; black: Tsang [17]; red: Sirjean et al. [15]; blue: Kneep et al. [14] (thicker line denotes temperature range of interest in study, while thinner lines are extrapolations to higher temperature outside recommended range)     11 0.69 1 17085 3.34 10 exp k T s T        12 0.07 1 14083 6.03 10 exp k T s T    100 1000 10 4 10 5 10 6 10 7 10 8 10 9 10 10 0.4 0.8 1.2 1.6 2 Rate, s -1 1000 / T, 1/K 177 From the above figure, it is clear the rates of H-elimination agree quite well between the studies of Tsang and Knepp and coworkers, with the discrepancy reaching up to only a factor of three at 300 K. However, the CBSQB3 derived apparent activation energy of the H-elimination channel is slightly higher than the others. For the ring-opening channel, discrepancies across the included studies vary by up to an order of magnitude at higher temperature. The activation energies between the studies of Knepp and Sirjean agree well, although the differences in the pre-exponential factors are large, roughly an order of magnitude over the temperature range of 300-2000 K. In comparing the rate recommendation from Tsang against those from Sirjean and coworkers, the overall difference for both channels is an effective reduction in Tsang's activation energies. These differences have a large impact on the predicted branching ratios for the two channels, depicted below in Figure 5.5. 178 Figure 5.5 Comparison of branching ratios to cyclohexene + H (solid lines) and 5-hexen- 1-yl (dashed lines) as a function of temperature. Black: Tsang [17]; red: Sirjean et al. [15]; blue: Kneep et al. [14] At temperatures above 1500 K, the predicted branching ratios of Tsang differ by 20-30 percent from those of Sirjean and Knepp. Such a difference would have an impact in the context of a detailed kinetic model and predicted concentrations of cyclohexene, cyclohexadiene, and benzene throughout various flame configurations. Furthermore, inclusion of the ring-opening channel in a detailed kinetic model may make modeling of the flame phenomena a more numerically stiff problem. Although the rate of cyclohexyl to 5-hexen-1-yl can certainly be defined and quantified, the rates of 5-hexen-1-yl out of its low-lying energetic well are much faster than the rates into them. Tsang's combined experimental and ME/RRKM study included almost all channels presented in Figure 5.1 0 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Branching Ratio 1000 / T, 1/K 179 and Figure 5.2. Thus, the respective rates from 5-hexen-1-yl to cyclohexyl and cyclopentylmethyl are and In essence, the lifetime of 5-hexen-1-yl can be very short, thus explaining why Stein and Rabinovitch did not detect it in their experiments of cyclohexene + H. If this is true, there may be no gain or purpose in defining the rate of its formation from cyclohexyl isomerization. One would be better off treating this species as a pseudo transition state. However, with enough downward energy transfer and/or collisions, such a species may have time to stabilize. Such a possibility is dependent on the radical well depth in addition to the magnitude and rate of collisional energy transfer (i.e., temperature and pressure). For this reason, and because large discrepancies in literature exist for the rates of cyclohexyl isomerization/decomposition, this system is re-evaluated here. Additionally, as stated above, the H-elimination reaction from cyclohexyl to cyclohexene is the first dehydrogenation step to benzene under some flame conditions. The following sections cover the reaction of cyclohexene + H as it pertains to the early work of Stein and Rabinovitch, the thermal isomerization of cyclohexyl, and the thermal isomerization of the 5-hexen-1-yl radical, which happens to also be a dominant initial intermediate during the decomposition/oxidation of 1-hexene.     4 1.33 1 2375 5.50 10 exp k T s T        8 0.36 1 5387 9.55 10 exp k T s T    180 5.2.1 Cyclohexene + H In many cases, the reactions of cyclic alkyl radicals can be studied from the corresponding H + olefin species, resulting in a chemically activated reaction [18-20]. The early work of Stein and Rabinovich showed that in the low pressure (with H 2 as a bath gas) room temperature chemically activated reaction of H + cyclohexene, dominant products are the cyclohexyl and cyclopentanemethyl radicals. Their data were taken in the pressure range of 0.01 to 100 torr.. Although 5-hexen-1-yl is formed, its lifetime is fleeting under these conditions. In fact, Stein and Rabinovich did not observe any 5- hexen-1-yl. However, using fitted RRKM parameters, they were able to deduce the relative enthalpies of reaction shown in Figure 5.3, calculating that the initial cyclohexyl adduct lies 35.5 kcal/mol below the entrance channel, and subsequent isomerization to cyclopentanemethyl is 6.1 kcal/mol in height. Finally, they deduced that the low lying 5- hexen-1-yl radical lies 20.2 kcal/mol above cyclohexyl. Similarly, Gierczak and coworkers authors studied the chemically activated reaction of cyclohexene and 1,5-hexadiene undergoing reaction with excited hydrogen atoms [20]. The experiments were conducted at ambient temperature and pressures ranging from 8.6 to 77 torr, with H-atoms generated by the photolysis of dihydrogen sulfide. Products were then analyzed using gas chromatography. Only trace amounts of ethylene were found in the reaction of H with cyclohexene, while the reaction of H with 1,5-hexadiene resulted in substantial amounts of smaller fragments (ethylene, propylene, and 1-butene). This suggests that ring opening followed by -scission in the H + cyclohexene system is 181 not significant under these conditions. However, large amounts of cyclohexane were observed in the reaction of H with 1,5-hexadiene, suggesting ring closure to cyclohexyl. An accompanying RRKM study was carried out, with pre-exponential factors, barrier heights, transition state structures, and energy transfer all treated as variables to fit both their data and that of Stein and Rabinovitch. In fact, the data of Stein and Rabinovitch was decisive in determining the parameters for the rate of cyclohexyl-cyclopentylmethyl isomerization, given below, with activation energy in cal/mol: Interestingly, they assigned a barrier of cyclohexyl ring opening to normal 5-hexen-1-yl to be 35.5 kcal/mol, proposing the aforementioned isomerization proceeds through a different transition state. This distinct six-membered ring to five-membered ring intramolecular rearrangement was first suggested by Gordon [21, 22], who postulated the isomerization might occur without the initial ring opening step. However, this suggestion is a result of his assertion that 5-hexen-1-yl may undergo a 1,5 H transfer to an allylic radical with a barrier of only 5-6 kcal/mol. This barrier is in fact underestimated by about 25 kcal/mol [23], negating this hypothesis. Instead, 1-hexene-5-yl radical simply undergoes a large amplitude bending motion facilitating the transition to cyclopentylmethyl. This is a very quick process, with barriers of only about 10 kcal/mol. Several measurements on the absolute rate of cyclohexene + H have also been made. Furukawa and coworkers study the reaction of H + cyclohexene and cyclohexadiene under ambient temperature and pressures of 3.5, 16, and 22 torr in a discharge flow   13.6 1 30800 10 expks RT    182 reactor in an Argon bath gas [24]. The reaction progress was monitored by following H atom concentration via an ESR spectrometer. The average rates of reaction of H with cyclohexene and cyclohexadiene were respectively measured to be: H + cyclohexene: k = (6 ± 1) 10 11 cm 3 mol -1 sec -1 and H + cyclohexadiene: k = (9 ± 2) 10 11 cm 3 mol -1 sec -1 The authors note that because no difference in rate constant was observed across all pressures (a six fold increase altogether), the chemically active species is effectively stabilized and that the measured rates are those at the high-pressure limit. Additionally, Knutti and Buhler measured the rates of H addition to various cycloalkenes in the millibar region with rate constants determined from a larger kinetic model [25]. The rates measured are: H + cyclohexene: k = (4.9 ± 0.9) 10 11 cm 3 mol -1 sec -1 H + 1,3-cyclohexadiene: k = (1.41 ± 0.26) 10 12 cm 3 mol -1 sec -1 H + benzene: k = (1.8 ± 0.2) 10 10 cm 3 mol -1 sec -1 Melville and Robb also measured the rates of H atoms with various alkenes through colorimetry [26]. The rate of H + cyclohexene was measured to be: H + cyclohexene: k = (5.8) 10 11 cm 3 mol -1 sec -1 Hoyermann and coworkers also measured the rate of H + various cyclic hydrcarbons at temperatures between 296 and 493 K. 183 H + cyclohexene: k = (3.2 ± 0.4)10 13 exp((-2.5 ± 0.6)/RT) with units in kcal, mol, s, with k 298 = (4.7 ± 0.9) 10 11 cm 3 mol -1 sec -1 H + 1,3-cyclohexadiene: k = (3.2 ± 0.4)10 13 exp((-1.5 ± 0.3)//RT) with units in kcal, mol, s, with k 298 = (2.6) 10 12 cm 3 mol -1 sec -1 H + benzene: k = (6.9 ± 1.5)10 12 exp((-3.7 ± 0.7)/RT) with units in kJ, mol, s, with k 298 = (2.5) 10 12 cm 3 mol -1 sec -1 Lastly, Kerr and Parsonage recommended the rate of H + cyclohexene to be 4.0x10 11 cm 3 mol -1 sec -1 at 298 K and under low pressure [27]. The aforementioned rates of H + cyclohexene are summarized in Table 5.1, and provide a range of 4(10 11 ) - 8(10 11 ) cm 3 mol -1 sec -1 for the overall association of H + cyclohexene near ambient temperature. Table 5.1 Summary of absolute rates of H + cyclohexene under ambient temperature and low pressures. k 300 (10 -11 ) cm 3 mol -1 sec -1 Pressure torr Method Bath Gas Reference 6 ± 1 3.5-22 ESR Argon Furukawa et al. 4.9 ± 0.9 0.3 mass spectrometry Helium Knutti and Buhler 5.8 Evacuated Colorimetry H 2 Melville and Robb 7.9 evacuated Colorimetry H 2 Allen et al. 4.7 ± 0.9 2 - 15 ESR Helium Hoyermann et al. 4.0 evacuated recommendation n/a Kerr Because the reaction of H + cyclohexene is assigned as the entrance channel for RRKM/ME study of the chemically activated cyclohexyl system, its accurate 184 determination is important. The range of rates defined by the experimental studies of Table 5.1 are used in the current cyclohexyl RRKM/ME analysis to confirm the predicted high pressure rate constant for the reaction of H + cyclohexene. Since the experimentally determined range of rates span only a factor of two, they provide a good constraint on the corresponding predicted value, which is calculated as such: 5.1 In the equation above, everything but the equilibrium constant defines the high pressure rate constant of cyclohexyl dissociating into cyclohexene + H. The equilibrium constant is defined as the ratio of the forward and reverse rates from the perspective of cyclohexene + H acting as reactants. The product of the two aforementioned quantities is thus the high pressure limit rate constant of cyclohexene + H. 5.2.1.1 Methodology Zero Kelvin electronic energies of all stationary points and transition states in the reduced cyclohexyl system were taken from the work of Knepp and coworkers [14]. Knepp and coworkers used a modified G2MP2 composite method. The modification involved using a larger basis set (B3LYP/6-311++G(d,p)) for the geometries and zero point energies [14]. The relative energies are depicted in Figure 5.3 along with RCCSD(T)/cc- pVDZ//B3LYP/6-311++G(2d,p) zero Kelvin ZPE corrected energies. The reasons for this inclusion will become apparent in a later discussion. Nevertheless, unrestricted calculations were found to have considerable spin contamination, and as a result, spin     † 3 1 1 0 , exp tot B entr eq tot Q kT E k K T cm mol s RT hQ        185 restricted calculations were performed on all species. In addition, spin restricted relative energies resulted in more accurate values. Because not all of the necessary structural information was not included in the study (i.e., the supplemental information), geometries and vibrational frequencies of all molecules were recalculated at the B3LYP/6- 311++G(2d,p) level of theory. Unscaled vibrational frequencies are presented in Table 5.3, while geometries are presented in the Appendix. Vibrational frequencies were multiplied by a factor of 0.97 to account for vibrational anharmonicity. RRKM/ME calculations were carried out for the chemically activated reaction of cyclohexene + H and include the isomers and products shown in Figure 5.3. The reaction network is shown in the figure below. Figure 5.6 Reaction network for the RRKM/ME calculation of cyclohexane + H. Corresponding species nomenclature and names are listed below each species schematic. Details of the calculation methodology are described in Chapter 2 and only necessary points are reiterated here. H 2 is designated as the bath gas in calculations comparing branching ratios against those experimentally determined by Stein and Rabinovitch. The CH 2 rotors were treated as one dimensional free rotors, with the reduced moments of inertia calculated at the I (3,4) level of approximation. The minimum number of stochastic  H  H  TS12 TS13 TS26 TS68 PXC6H11 5-hexen-1-yl C5H9CH2 cyclopentanemethyl cC6H11 cyclohexyl cC6H10+H cyclohexene + H C5H8CH2+H 186 trials across all T and P calculations necessary for converged results was no less than 10,000. Tunneling was accounted for via the Eckart approximation. The equilibrium constant of the entrance channel was calculated as a function of temperature using the modified G2MP2 energies of Knepp et al. along with the structural and vibrational information determined at the B3LYP/6-311++G(2d,p) level of theory. To confirm the accuracy of this method, several comparisons were made. Using the above parameters and the statistical mechanics tools described in Chapter 2, the equilibrium constant for the reaction of H + cyclohexene  cyclohexyl is computed to be with the activation energy in units of kcal/mol. At 300K, and using equation 5.1, the high pressure limit rate constant for the entrance channel is therefore computed to be k 300 = 3.19 (10 11 ) cm 3 mol -1 sec -1 which is just outside the range of experimentally reported values summarized in Table 5.1. This value also compares well with the rate rule recommendation of the high pressure limit H addition to an internal double bond: k 300 = 1.54 (10 11 ) cm 3 mol -1 sec -1 [28]. The work of Stein and Rabinovitch was used to verify various model assumptions - internal rotor treatment and the energy transfer model, specifically, the downward energy transferred per deactivating collision, , at 300 K. Stein and Rabinovitch derived relative yields of cyclohexyl and cyclopentanemethyl, shown in Figure 5.7 below along         3 0.612 3 1 33.21 7.95 10 exp eq K T T cm mol RT   d E  187 with RRKM/ME results. H 2 was used as bath gas in both the experiments and simulations. Table 5.2 Summary of calculated zero point energies (ZPE) in Hartrees, along with all rotational parameters used in RRKM/Master Equation calculations. external rotors internal rotors e species a ZPE b inactive c (cm -1 ) active d (cm -1 ) no. of rotors V 0 (kcal/mol) I (3,4) (cm -1 ) cC 6 H 11 97.3 0.083 0.148 0 - pxC 6 H 11 94.2 0.465 0.045 1 0 10.06 TS12 94.8 0.077 0.135 0 - TS13 92.5 0.084 0.147 0 - C 5 H 9 CH 2 96.1 0.211 0.088 1 0 9.92 TS26 94.6 0.176 0.086 0 - TS68 92.0 0.212 0.087 0 - cC 6 H 10 91.4 0.085 0.114 0 - Species σ (Å) ε/k B (K) N 2 /AR f 3.41 120 H 2 2.83 60 all other 5.29 465 a See Figure 5.8 for species identification; electronic energies taken from [14]. b Unscaled, calculated at the B3LYP/6-311++G(2d,p) level of theory. c Two-dimensional (all symmetry number σ =1); d One-dimensional ( σ =1). e One-dimensional (all symmetry numbers σ =2). Rotor hindrance estimated from analogous species [29]. Barrierless rotors treated as free internal rotors. AR assumed equivalent to N 2 ; LJ parameters and their sources: H 2 : [19]; N 2 /AR: [30]; all other: [31] 188 188 Table 5.3 Unscaled vibrational frequencies used for cyclohexyl RRKM/ME simulations. Calculated at the B3LYP/6-311++G(2d,p) level of theory. See Figure 5.8 for species identification. species vibrational frequency (cm -1 ) cC 6 H 10 171.08 272.56 402.07 452.98 504.1 654.37 728.85 818.96 828.21 880.09 907.28 935.3 996.43 1020.9 1044.8 1070.94 1100.76 1160.78 1163.76 1249.7 1270.1 1293.01 1356.93 1369.22 1369.31 1383.99 1427.6 1475.54 1482.66 1492.06 1499.89 1706.47 2988.37 2988.46 3007.04 3012.93 3029.33 3029.78 3055.63 3059.2 3127.77 3150.34 cC 6 H 11 184.63 216.55 324.77 398.05 442.21 460.03 611.51 785.96 803.57 856.76 870.95 876.31 928.33 1016.94 1032.47 1054.81 1097.39 1114.06 1128.68 1151.67 1255.97 1283.08 1295.69 1342.19 1349.18 1354.75 1377.27 1391.61 1395.88 1469.58 1475.31 1489.58 1492.94 1502.71 2895.92 2900.68 2993.7 3011.97 3012.06 3045.12 3047.75 3049.1 3050.59 3054.38 3164.23 C 5 H 9 CH 2 22.3 82.76 186.39 298.63 427.38 521.64 539.86 629.24 762.79 820 840.42 883.46 920.42 939.88 982.56 1008.45 1038.81 1047.81 1109.61 1203.23 1211.35 1233.57 1257.49 1297.49 1315.59 1327.85 1329.62 1337.74 1375.57 1468.53 1490.16 1498.81 1502.02 1523.4 3003.2 3020.87 3031.27 3035.08 3048.95 3063.44 3078.16 3081.41 3094.24 3129.26 3232.82 pxC 6 H 11 78.48 90.63 104.66 133.09 182.88 345.76 358.29 446.84 548.11 659.37 738.63 794.45 920.57 930.57 945.61 983.65 1021.62 1029.36 1052.87 1072.99 1189.32 1239.02 1253.67 1309.1 1319.45 1329.79 1339.23 1365.18 1458.42 1468.09 1485.25 1487.28 1503.92 1700.58 2997.86 3000.06 3020.55 3031.22 3038.45 3068.17 3116.41 3126.92 3135.05 3208.24 3233.9 TS13 445.93i 156.97 236.96 319.26 366.64 419.42 461.04 540.71 674.14 747.54 804.3 823.84 880.59 922.03 927.57 942.49 960.63 1008.33 1032.71 1074.96 1159.47 1206.15 1237.73 1269.73 1293.09 1340.43 1362.79 1375.12 1436.29 1462.19 1472.04 1484.84 1491.69 1579.92 2939.92 3001.65 3008.92 3014.53 3049.49 3059.19 3105.78 3123.79 3138.35 3193.8 3206.67 TS12 546.77i 134.63 236.18 293.17 420.13 441.91 464.8 502.22 684.62 744.79 819.82 829.29 878.13 904.84 933.24 1004.86 1022.38 1049.51 1067.73 1097.31 1164.57 1167.39 1244.68 1271.09 1295.85 1358.86 1370.81 1372.27 1385.7 1426.77 1467.79 1481.38 1498.9 1506.78 1637.51 2981.82 3009.2 3015.24 3017.29 3027.72 3040.02 3059.87 3063.27 3139.05 3159.67 TS26 469.53i 40.67 243.04 269.98 369.61 435.66 502.58 570.21 598.08 783.39 810.13 821.78 845.95 893.31 918.76 933.3 951.7 1000.77 1048.59 1062.63 1163.55 1184.22 1234.19 1265.18 1284.61 1343.65 1346.8 1359.27 1426.99 1469.73 1485.4 1493.95 1506.74 1543.67 2983.66 3016.34 3024.33 3037.89 3058.53 3069.68 3116.24 3125.6 3139.57 3204.59 3214.5 TS68 723.18i 77.84 253.7 353.3 472.42 484.6 495.2 570.95 590.29 642.68 761.44 846.78 858.02 890.65 902.34 905.76 937.42 991.97 1022.21 1052.47 1153.66 1173.04 1226.41 1244.11 1273.06 1308.76 1317.95 1343.18 1347.19 1445.6 1475.56 1481.14 1497.67 1512.9 1625.78 3013.1 3026.68 3035.56 3044.48 3076.9 3082.42 3089.64 3094.09 3130.09 3213.08 189 Figure 5.7 Comparison of RRKM/ME results using the reaction network of Figure 5.6 with 300 K experimentally measured products of the cyclohexene + H reaction in an H 2 bath gas [18]. Filled circles: experimentally measured cyclohexyl and cylopentanemethyl radical product fractions. Dashed lines: simulation results with all vibrational modes treated using the harmonic oscillator assumption. Solid lines: simulation results with CH 2 groups treated as 1D free internal rotors and reduced internal moments of inertia calculated at using the I (3,4) level of approximation. The parameter for cyclohexyl in the H 2 bath gas at 300 K was set to 250 cm -1 . Although tuning of to yield better agreement with the data of Stein and Rabinovitch is possible, such an approach was not justified based on the relative insensitivity of the results to variation in at 300K and the uncertainties in both the computational and experimental methods. More importantly, and as illustrated in Figure   0.0001 0.001 0.01 0.1 1 0.01 0.1 1 10 100 Yield P, torr d E  d E  d E  190 5.7, agreement with the current methodology is surprisingly good and well within both uncertainty bounds. Thus, a of 250cm -1 at 300K is sufficient. commonly takes the temperature dependent form Various studies have shown how increases with temperature, and a value of 650 cm -1 for T = 2500 K was chosen based on two recent studies [32, 33]. Based on two values above, the coefficient n is calculated to be 0.45. Thus, the temperature dependent used for the cyclohexyl system, with H 2 as the bath gas, is Choice of this above relationship is motivated by the fact that the higher temperature simulations of this work use N 2 /Ar as the bath gas. At 2500 K, the above equation gives = 650 cm -1 , the same as that given by the analogous parameters for N 2 recommended by Jasper and Miller at the same temperature [32]. Use of the above temperature dependent expression is thus adequate for the simulations in this work. Lastly, the difference in predicted branching fractions of cylcopentanemethyl and cyclohexyl when treating just one internal degree of freedom as an internal rotor are significant at low pressure and negligible at higher pressures. Although more appropriate to treat low energy coupled internal torsions (involving two carbon atom groups), the d E  d E    1 ,300 / 300 n dd E E T cm     d E  d E    0.45 1 250 / 300 d E T cm   d E  191 good agreement with a proper treatment of just the CH 2 rotor is sufficient for this system. Thus, all RRKM/ME simulations for the cyclohexyl system followed this approach. In the context of the RRKM/ME model parameters used here, the work of Stein and Rabinovitch enabled verification of several things. The relative electronic energies of cyclohexyl and cyclohexene + H in conjunction with the current rotor treatment were used to give an estimate of the high pressure limit rate of the entrance channel that is in good agreement with available rates found in literature. Thusly, the modified G2MP2 electronic energies of Knepp et al are of satisfactory accuracy, at least concerning this part of the PES. The level rotor treatment is also justified and is more than adequate for higher temperature applications, where their effects become dampened relative to the harmonic oscillator treatment. Secondly, the choice of energy transfer parameters at 298 K results in very good agreement with the measured branching fractions of cyclohexyl and cyclopentanemethyl. The choice of at 298 K is therefore sufficient. However, to verify accuracy of the RRKM/ME model at higher temperatures and pressures, and to verify accuracy of the remaining parts of the cyclohexyl PES, other comparisons are necessary, and will be addressed in the following sections. 5.2.2 Competition in cyclohexyl decomposition/isomerization All cyclohexyl unimolecular thermal isomerization simulations are conducted with N 2 /Ar as the bath gas. The potential energy surface shown in Figure 5.3 was used to study the pressure and temperature dependent competition between H-elimination to cyclohexene + d E  192 H and isomerization to other major channels, including the possibility for another H- elimination to cyclopentanemethylene + H. The corresponding reaction network is shown below. Figure 5.8 Reaction network for the RRKM/ME calculation of cyclohexyl isomerization/decomposition. Corresponding species nomenclature and names are listed below each species schematic. As will be shown, the H-elimination to cyclopentanemethylene + H channel contributes little to cyclohexyl consumption under all T and P considered here. And as mentioned earlier, calculations were performed to test the sensitivity of predicted results on the inclusion of two high-energy channels leaving the 5-hexen-1-yl radical: -scission to C 2 H 4 +C 4 H 7 and H-shift to an allylic C 6 H 11 radical. The results indicate these higher energy channels contribute roughly 5% of the total overall cyclohexyl consumption at 2500 K and 1 atm. Their inclusion is therefore unnecessary and does not influence the overall results. For investigating the competition between various isomerization/decomposition channels available to cyclohexyl, omission of higher energy channels like those shown in Figure 5.1 and Figure 5.2 is justified. Pressure dependent thermal rate constants were calculated using the RRKM/ME method as described in Chapter 2. All input parameters for the model simulation are given in Table 5.2 and  H  H  TS12 TS13 TS26 TS68 PXC6H11 5-hexen-1-yl C5H9CH2 cyclopentanemethyl cC6H11 cyclohexyl cC6H10+H cyclohexene + H C5H8CH2+H 193 Table 5.3. Zero Kelvin electronic energies were taken from the work of Knepp and coworkers [14]. Figure 5.9 Arrhenius plot of the overall decomposition of cyclohexyl to products for varying pressure and temperature. The above figure illustrates the total rates of cyclohexyl consumption for various T and P. The high pressure limit rates through TS-1-2 (ring-opening) and TS-1-3 (H-elimination) are also included. The total rate is defined as the sum of the rates for all channels, each of which is depicted below in Figure 5.11. 10 3 10 5 10 7 10 9 10 11 0.6 0.8 1 1.2 1.4 1.6 cC 6 H 11 products 0.01 0.1 1 10 100 1000 TS12 TS13 k, 1/s P (atm) 1000 / T, 1/K 194 Available high-pressure rates through beta-scission (TS-1-2) and H-elimination (TS-1-3) channels are plotted in Figure 5.10 along with results from this work. Recalling that relative electronic energies for the current work are taken from the calculations of Knepp et al., the slight differences between high pressure rate constants can be attributed to a number of possibilities: vibrational frequencies used in the calculations and/or how the total rotational partition function was defined. Calculations in this work also take into account doubly degenerate reaction paths leading out of cyclohexyl. All things considered, the difference in high pressure rate constants is not large. Overall, agreement in the high pressure rate for H-elimination cyclohexene + H is quite good across all studies. The high pressure -scission rates calculated in this work agree well with those predicted by Sirjean and coworkers. As mentioned earlier, the activation energy used for the corresponding reaction by Tsang is likely too low, and may have been tuned to better reproduce experimental results. Considering the existence of pressure falloff even at 1000 atm (as shown in Figure 5.9), it is now understandable why such fitting may have been performed. Figure 5.11 below illustrates the pressure dependence of various channels for the isomerization/decomposition of cyclohexyl for pressures from 0.01 - 1000 atm. 195 Figure 5.10 Comparison of high-pressure rates through beta-scission (TS12) and H-elimination (TS13) channels. Black: this work; red: Tsang et al. [16]; green, Knepp et al. [14]; blue: Sirjean et al. [15]. 100 1000 10 4 10 5 10 6 10 7 10 8 10 9 10 10 0.4 0.8 1.2 1.6 2 Tsang et al. 2008 Knepp et al 2002 Sirjean et al. 2008 this work Rate, s -1 1000 / T, 1/K 196 196 Figure 5.11 Arrhenius plots for the reaction channels of cyclohexyl leading to various products in the cyclohexyl system of Figure 5.8. Symbols are rates computed with Monte Carlo RRKM/Master equation modeling and lines drawn to guide the eye. 10 2 10 4 10 6 10 8 10 10 cC 6 H 11 cC 6 H 10 + H 0.01 0.1 1.0 10 100 1000 kinf k, 1/s P (atm) 10 4 10 6 10 8 10 10 cC 6 H 11 PXC6H11 10 3 10 5 10 7 10 9 cC 6 H 11 CH2cC5H9 0.4 0.6 0.8 1 1.2 1.4 k, 1/s 1000 K / T, 1/K 10 3 10 5 10 7 10 9 0.4 0.6 0.8 1 1.2 1.4 cC 6 H 11 CH2cC5H8 + H 1000 K / T, 1/K 197 A comparison of the pressure dependence in the H-elimination and -scission rates illustrates the nature of the products being formed. H-elimination results in bimolecular product formation and is therefore set as an irreversible reaction with no subsequent pathways. On the other hand, -scission leads to the 5-hexen-1-yl radical which resides in a low lying well. Stabilization of this species requires sufficient energy transfer, resulting in a higher pressure dependence and substantially decreased rate of formation at lower pressures. The lower pressure environment allows for further isomerization to cyclopentanemethyl and cyclopentanemethylene + H. This, in turn, is reflected in the generally monatomic increase in the rates of formation of these aforementioned species. Rates leading to the formation of cyclopentanemethylene + H reflect the higher 40 kcal/mol barrier. However, the peculiar pressure dependence of the cyclopentanemethyl channel is because of competition with the corresponding H-elimination channel. This is best illustrated by viewing the sum of the rates of the two channels, as shown in Figure 5.12. 198 Figure 5.12 Arrhenius plot of summed cyclohexyl isomerization to cyclopentanemethyl and cylopentanemethylene + H. Through inspection of the above figure, it is evident that the lower temperature results asymptotically approach the high pressure limit rate constant corresponding to the direct isomerization of cyclohexyl to cyclopentanemethyl. At the highest temperatures, the results approach the high pressure limit rate constant for the direct isomerization from cyclohexyl to cylopentanemethylene + H. Figure 5.13 plots the branching fractions for the isomers and products in the cyclohexyl system for various T and P. The log 10 of the pressure is displayed, and the branching fractions for all species are plotted on the z-axis. Color indicates the level of branching fraction, with lighter hues corresponding to higher fractions. 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 0.4 0.6 0.8 1 1.2 1.4 0.01 0.1 1.0 10 100 1000 k, 1/s P (atm) cC 6 H 11 CH 2 cC 5 H 8 + H + CH 2 cC 5 H 9 1000 / T, 1/K 199 Figure 5.13 Pressure and temperature branching fractions calculated for reaction channels of reduced cyclohexyl system: (a) 5-hexen-1-yl, (b) cyclohexene + H, (c) cyclopentanemethylene + H, (d) cyclopentanemethyl (nitrogen bath gas). Under temperatures of 1000 K it becomes difficult to obtain enough statistics to achieve accurate data. For instance, at 1000 atm and 1000 K, a simulation of 100,000 trials results in less than 100 molecules leaving the cyclohexyl well and going on to other isomers/products; this is because an extremely small portion of the Boltzmann energy distribution exceeds barrier heights to reaction. -2 -1 0 1 2 3 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 log(P), torr T, K a b c d 200 In summary, RRKM/ME calculations for the cyclohexyl system revealed a rich distribution of products highly dependent on both temperature and pressure, as shown in Figure 5.13. For experimental studies in which chemically activated cyclohexene + H reactions were examined (300 K and < 0.1 atm), the lack of 5-hexen-1-yl radical being detected is supported by the present RRKM/ME simulations: at temperatures lower than 1000 K, the production of 5-hexen-1-yl from cyclohexyl was calculated to be negligible. At higher temperatures, the stabilization of the chemically activated cyclohexyl becomes less efficient, allowing for a greater extent of ring opening, leading a greater production of the 5-hexen-1-yl radical. Under the conditions studied, the 5-hexen-1-yl radical can comprise up to and over 60% of the total product distribution. The significance of this result lies in the fact that 5-hexen-1-yl cannot be considered fleeting under all cases. Similar behavior is expected for analogous ring systems. The large amount of cyclopentanemethyl formed at lower temperatures is predominantly because of entropic effects and also consistent with experimental observations by Gierczak and coworkers [20]. This behavior is more easily understood through the perspective of thermal 5-hexen-1-yl isomerization. In this way, the competition between 5-exo and 6-endo ring closers (to cyclopentanemethyl and cyclohexyl, respectively) can be more closely examined. 201 5.2.3 Unimolecular 5-hexen-1-yl decomposition Prior to consideration of other pathways, the initial discussion focuses on the two ring closure possibilities of 5-hexen-1-yl, shown below. The terminology for the above reactions follows the guidelines initially proposed by Baldwin [34]. In the context of the above possibilities, 6-endo ring closure indicates the new radical site lies within the new 6-membered ring, while 5-exo ring closure indicates formation of a 5-membered ring whose radical site lies outside the ring. An initial look at this system might leave one assuming a roughly 50-50 split between cyclopentanemethyl and cyclohexyl. The 6-membered ring has less ring strain, the radical site is more stable on a secondary carbon compared to when on a primary carbon, and the overall reaction enthalpy to cyclohexyl is more exothermic (see Figure 5.3). However, the energy barrier to cyclohexyl is slightly higher, by 2.1 kcal/mol. One might hypothesize these effects to cancel out and the branching out of the 5-hexen-1-yl to be roughly equal. This is indeed not the case - overall branching is highly pressure and temperature dependent. The competitive cyclization of 5-hexen-1-yl has been extensively studied in the past under ambient and elevated temperatures (T usually  200 C) and in solution [23, 35-    6-endo 5-exo 202 42]. Synthetic chemists have long made use of the ring closure competition to create functionalized ring compounds. In particular, Beckwith spent much of his career characterizing and exploiting 5-hexen-1-yl radical containing systems, leading to what is now a universal synthetic methodology for a wide variety of compounds [43]. As a result, for lower temperature conditions, the kinetics of this system are well described. Under ambient temperature and all pressures, thermal isomerization of 5-hexen-1-yl leads almost exclusively to cyclopentanemethyl. From a molecular orbital standpoint, there exists favorable orbital overlap in the 5-exo ring closure, and unfavorable orbital overlap in the 6-endo closure. This is manifested as a slightly lower activation energy for the 5- exo closure. Although this reasoning is sound, it is only part of the picture. It will soon be clear that another contributor to cyclopentanemethyl formation is its relatively higher entropy. Steric hindrance to the 5-exo cyclization has long been known to significantly decrease the yield of cyclopentanemethyl. For instance, incorporation of a methyl group to the carbon 5 position of 5-hexen-1-yl, making 5-methyl-5-hexen-1-yl, yields a 2:3 ratio of the 5-exo/6-endo products at 65 C [41]. This is not true for methyl substituents at other positions however [44]. The ability to control selectivity in this manner has obvious benefits. This realization of controlled selectivity will be of value later when considering competing pathways in methylcyclohexyl isomerization/decomposition. 203 The low temperature 5-exo isomerization has been extensively measured in solution. Walling and Cioffari determined the room temperature rate to be 1.1(10 5 ) s -1 [41]. Chatgilialoglu et al performed a number of studies on this reaction, with the most recent recommendation being 2.2(10 5 ) s -1 at 298 K, determined by laser flash photolysis [42]. For the range of 298 - 355 K, the 5-exo cyclization can be expressed as with energy in kcal/mol (i.e., E a = 6.85 kcal/mol). A limited number of corresponding computational kinetic studies have been performed. Wu and coworkers perform a TST study on the 5-exo rate at 298 K, recommending a value of 3.6(10 5 ) s -1 , in good agreement with the above experimental values [40]. Matheu and coworkers used an automated approach to study the temperate and pressure dependence of the cyclohexyl system, using rate rules for the high pressure limit rate constant and an approximate QRRK formalism for microcanonical rates. The modified strong collision approximation is used to capture pressure dependence. Satisfactory agreement was obtained with the low pressure data of Stein and Rabinovitch discussed earlier, and the reported overall uncertainties in predicted pressure dependent rates an order of magnitude. The authors use rate rules and B3LYP/cc-pVTZ energies to calculate the high pressure limit rate of the 6-endo channel:       1 5 log 10.37 0.32 6.85 0.42 / 2.3 exo k RT s         8 0.86 1 6 5900 10 exp endo k T s RT     204 The high pressure rate for the 5-exo channel was adopted from Newcomb, who in turn used the experimentally determined rates by Chatgilialoglu et al., discussed above [42, 45]. In addition to long standing and continued interest from the organic synthesis community, this system has seen some experimental interest from the gas-phase kinetics and combustion community. Recall that Gierczak and coworkers studied the chemically activated reaction of 1,5-hexadiene with excited hydrogen atoms [20]. The experiments were conducted at ambient temperature and pressures ranging from 8.6 to 77 torr, with H- atoms generated by the photolysis of dihydrogen sulfide. Products were then analyzed using gas chromatography. Major products include methylcyclopentane, cyclohexane and 1-hexene. Use of highly excited H atoms reacting with 1,5-hexadiene also resulted in substantial amounts of smaller fragments (ethylene, propylene, and 1-butene) formed from various scission pathways. A small amount of cyclohexene was observed, and attributed to a multistep pathway from the initial H -abstraction from the central carbon in 1,5-hexadiene to form 1,5-hexadien-3-yl which subsequently undergoes a 1,6-cyclization to cyclohexene. Methylcyclopentane and cyclohexane are attributed to the H addition to 5-hexen-1-yl and subsequent isomerizations. Although the authors use their experimental data and RRKM calculations to recommend a rate of isomerization for various channels, no recommendation was made for the two isomerization channels out of the 5-hexen-1-yl well. Nevertheless, their data serve to illustrate the wide range in product distribution obtained when using highly excited H-atoms reacting with 1,5-hexadiene. 205 Handford-Styring and Walker discussed the cyclization of 5-hexen-1-yl in work primarily focused on cyclopentane and cyclopentyl [23]. The authors estimate rates for the cyclization between 600 and 1300 K at 1 atm (in O 2 ), concluding that the dominant product should be cyclohexyl radicals. They further suggest successive oxidation to cyclohexene, cyclohexadiene, and then benzene. While the latter may be true, the authors did not consider the 5-exo isomerization in their calculations. In fact, at 600 K and 1 atm (in N 2 ), roughly 95% of the product is represented by cylcopentanetmethyl, decreasing to about 10% at 1300 K. The dominant product at lower temperatures is cylcopentanetmethyl, not cyclohexyl. At higher temperatures, product distribution becomes competitive among all channels excluding the cyclopentanemethylene + H elimination. Nevertheless, a rate for 6-endo was estimated for the range of 600-1300 K and at 1 atm. Lastly, and as previously mentioned, Tsang and coworkers performed a combined shock tube and computational study on the isomerization and decomposition of cyclohexyl, also recommending rates for the 5-exo and 6- endo cyclizations [46]. The respective high pressure limit rate constants are     10 1 6 4210 10 exp endo ks T     206 and valid in the temperature range of 600 - 1900 K. 5.2.3.1 Entropic contributions to 5-exo/6-endo branching The role of entropy in dictating branching for this system is quite clear when investigating the partition functions of the transition states leading out of the 5-hexen-1-yl well. Looking at the PES of Figure 5.3, recall that 5-exo channel requires 2.1 kcal/mol less energy than the 6-endo channel. Again, these energies were calculated using a modified G2MP2 scheme [14]. The accuracy of the relative energetics for these channels can in fact be verified by the experimental 98:2 branching ratio of cyclopentanemethyl:cyclohexyl at lower temperatures [41]. It will also soon be clear that use of the CCSD(T)/cc-pVDZ energies for this system would have incorrectly predicted branching at lower temperatures because of the lower relative energy differences in transition states leading out of the 5-hexen-1-yl well. In addition, we will see how a comparison of the partition functions of the two transition states clearly illustrates the higher entropy of the 5-exo channel.     4 1.33 1 5 2375 5.50 10 exp exo k T s T         8 0.36 1 6 5387 9.55 10 exp endo k T s T     207 The entropic differences between the two transition states can be intuitively understood by visual inspection of their geometries, shown below in Figure 5.14. CH 2 rotations are tied up in the 6-endo transition state, unlike in the 5-exo transition state which has one free CH 2 rotor. Figure 5.14 Comparison of 6-endo (a) and 5-exo (b) transition state structures. Calculated at the B3LYP/6-311++G(2d,p) level of theory. The entropic contributions can in fact be easily quantified. Consider a 5-hexen-1-yl system where the barrier heights of the 5-exo and 6-endo channels are identical. Under such circumstances, the only factors affecting the rate out of this channel are the relative differences in the partition functions, the tunneling coefficients, and reaction path degeneracies. For this system, the last two conditions are similar or identical and therefore negligible, leaving the molecular partition function as the sole reason for any kinetic selectivity. Using the same input parameters as in the cyclohexyl system (given in Table 5.2), the 5-exo:6-endo branching fractions were recomputed with equal exit barrier heights. For T = 300-700 K, branching is roughly 3:1 for 5-exo:6-endo, in stark contrast to the 98:2-95:5 ratios for the same temperature range using the correct PES. b a 208 This simple calculation allows us to conclude entropy indeed plays a role in selectivity, but to a lesser degree than the electronic energy differences. A further understanding of the entropic effect can be gained by a comparison of differences between reactant and each channels' barrier. The material presented in Chapter 2 can be used to derive an entropy of activation for any given reaction: R is the universal gas constant and S † is the entropic difference between reactant and transition state ( † ), in units of cal/mol-K. Q refers to the total molecular partition function. In the calculation of activation entropy, it can be shown that the ratio in the above equation is dominated by internal degrees of freedom. Table 5.4 Temperature variation of total partition functions and entropy of activation (in cal/mol-K) for 5-hexen-1-yl, TS-1-2 (6-endo), and TS-2-6 (5-exo) molecules. T, K Q 5-hexen-1-yl Q † 6-endo Q † 5-exo S † 6-endo S † 5-exo 300 5.0(10 7 ) 1.3(10 6 ) 3.9(10 6 ) -7.3 -5.1 400 5.1(10 8 ) 8.1(10 6 ) 2.6(10 7 ) -8.2 -5.9 500 4.7(10 9 ) 5.4(10 7 ) 1.8(10 8 ) -8.9 -6.5 600 4.2(10 10 ) 3.7(10 8 ) 1.3(10 9 ) -9.4 -6.9 700 3.5(10 11 ) 2.6(10 9 ) 9.2(10 9 ) -9.7 -7.2 1200 6.2(10 15 ) 2.8(10 13 ) 1.0(10 14 ) -10.7 -8.2 Table 5.4 illustrates differences in the partition functions and corresponding entropy of activations for the 5-exo:6-endo channels. The partition function of the 5-exo transition † † ln Q SR Q     209 state (TS-2-6) is over a factor of three greater than that for the 6-endo transition state (TS- 1-2) for all T. In fact, even at 1200 K, the total partition function TS-2-6 is still 3.5 times greater than that for TS-1-2. A comparison of activation entropies shows that the 5-exo channel results in the smallest amount of entropy loss. Thus, while some of the branching calculated using the proper electronic energies can be attributed to relative barrier heights, the above discussion quantitatively illustrates entropy's significant role. At higher temperatures, the increased available free energy in the system reduces both entropic and energetic preferences, and branching no longer favors cyclopentanemethyl. Synthetic chemists could indeed exploit this knowledge if formation of 6-membered alkane rings is desired. They could additionally alter the pressure of their reactor to achieve the same goal. 5.2.3.2 RRKM/ME study for 5-hexen-1-yl Figure 5.15 Reaction network for the 5-hexen-1-yl radical, used in the RRKM/ME calculation for the thermal isomerization/dissociation of 5-hexen-1-yl. A RRKM/ME study was carried out on the thermal isomerization of the 5-hexen-1-yl radical, using the reaction network shown in Figure 5.15. The calculation differs from  H  H  TS12 TS13 TS26 TS68 PXC6H11 5-hexen-1-yl C5H9CH2 cyclopentanemethyl cC6H11 cyclohexyl cC6H10+H cyclohexene + H C5H8CH2+H 210 the thermal isomerization cyclohexyl only in assignment of the reactant. Pressure and temperature branching fractions for the 5-hexen-1-yl system are shown below in Figure 5.16. Figure 5.16 Branching fractions as functions of temperature and pressure (labels in bottom right plot) for the thermal isomerization of 5-hexen-1-yl, as described by the reaction network shown in Figure 5.15 a) cylcohexyl, b) cyclopentanemethyl, c) cyclohexene + H, d) cyclopentanemethylene + H. -2 -1 0 1 2 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 d c b a , TK   log , P torr 211 The above figure illustrates how increasing temperature and decreasing pressure result in greater selectivity of the cyclohexyl radical, albeit not much. Even higher temperatures result in preferential H-elimination to cyclohexene + H. The negligible branching fraction for all temperatures and pressures of the cyclopentanemethylene + H channel is because of its higher energy barrier of 41.1 kcal/mol. Figure 5.17 illustrates the relative temperature independence of the cyclohexyl and cyclopentanemethyl channels. On the other hand, the pressure dependence is quite high. As expected, the H-elimination channels are only competitive with the other two channels at higher temperatures, above 1000 K. The importance of pressure dependence and overall branching for this system will be highlighted shortly, and will be shown to influence early formation of benzene under certain conditions in 1-hexene flames 212 212 Figure 5.17 Arrhenius plots for the reaction channels of 5-hexen-1-yl, plotted for T = 500 - 1300 K. Symbols are rates computed with Monte Carlo RRKM/Master equation modeling. 10 4 10 6 10 8 10 10 k, 1/s 5-hexen-1-yl CH 2 cC 5 H 9 5-hexen-1-yl cC 6 H 11 10 4 10 6 10 8 10 10 0.8 1 1.2 1.4 1.6 1.8 2 k, 1/s 5-hexen-1-yl cC 6 H 10 + H 1000 / T, 1/K 0.01 0.1 1 10 100 0.8 1 1.2 1.4 1.6 1.8 2 5-hexen-1-yl CH 2 cC 5 H 8 + H 1000 / T, 1/K P (atm) 213 Figure 5.18 Rate comparisons for the 5-exo (solid lines) and 6-endo (dashed lines) channels in the temperature range of 298 - 1300 K. Blue: Handford- Styring and Walker estimate, T = 600 - 1300 K, 1atm [23]; purple: Tsang high pressure limit, T = 600-1900 K [46]; red: this work, high pressure limit (the red dotted line represents the rate for the 5-exo channel with the activation energy increased by 2 kcal/mol); black: 5-exo, Chatgilialoglu et al., in solution [42] with reported uncertainty, 300 - 355 K; solid triangle: 5-exo, Wu et al. [40]; solid diamond: 5-exo, Walling and Cioffari, in solution [41]. Figure 5.18 illustrates the predicted 5-exo and 6-endo rates calculated in this work along with those mentioned earlier. The room temperature experimental data (taken in solution) are in fair agreement with each other. The uncertainty in the rates determined by Chatgilialoglu et al. roughly spans an order of magnitude. None of the computationally determined predictions fall within this uncertainty band. The higher rates predicted in this study can be attributed to the electronic energy of the 5-hexen-1-yl radical. Recall that electronic energies were adopted from the modified G2MP2 10 2 10 4 10 6 10 8 10 10 0.8 1.2 1.6 2 2.4 2.8 3.2 k, 1/s 5-hexen-1-yl cC 6 H 11 and CH 2 cC 5 H 9 1000 / T, 1/K 214 calculations of Knepp and coworkers [14]. The electronic energy for 5-hexen-1-yl corresponds to a cyclic geometry, as opposed to the lower energy normal (elongated) alkyl radical. This cyclic species is roughly 2 kcal/mol higher in energy. Thus, a 2 kcal/mol reduction in barrier height of the 5-exo channel brings the corresponding rate down and within better agreement of the low temperature experimental studies. This is also illustrated in Figure 5.18. More importantly, this disagreement confirms the 5- hexen-1-yl radical in the experimental studies was of the lower energy isomeric form. The high-pressure limit rate for the 6-endo channel is of course also overestimated at low temperatures. However, under the temperatures of interest to combustion, these differences have a smaller effect on the overall rate constant. For example, for a reaction whose activation energy is 10 kcal/mol, a 2 kcal/mol increase only decreases the rate constant by less than a factor of two at 1500 K. At 298 K, the difference in barrier height results in an absolute rate difference by almost an order of magnitude. The above study helped to quantify the entropic contribution to 5-exo/6-endo competition. By setting the barrier heights equal to one another, the room temperature predicted branching fraction changed from 98:2-98:5 to 3:2, illustrating the importance of entropy and also confirming the dominant role of relative electronic energies in controlling branching fraction. The good agreement with available room temperature experimental studies also verifies that the sufficient level of accuracy achieved through use of the modified G2MP2 relative energies. In addition, it illustrates that the corresponding CCSD(T)/cc-pVDZ relatives energies (a model chemistry used later for 215 methylcyclohexane) does not sufficiently capture this subtle difference. Lastly, although the electronic energy of the 5-hexen-1-yl used in this work reflects a less stable cyclic structure and has significant kinetic consequence at room temperature, the differences at combustion-relevant temperatures are expected to be far less and therefore of no consequence to the results of the cyclohexyl thermal isomerization/decomposition RRKM/ME study. In conclusion, RRKM/ME calculations for the 5-hexen-1-yl system show that although this radical species lies in a shallow energy well (~10 kcal/mol), it can become collisionally stabilized under high pressure and at elevated temperatures. Under such conditions, reactions with other gas phase radicals and unimolecular isomerization/decomposition are expected to dominate its consumption. In addition, the RRKM/ME results illustrate that under the conditions of low pressure and high temperature, the 5-hexen-1-yl radical preferentially undergoes a formally direct H- elimination reaction through cyclohexyl, to cyclohexene + H. At lower temperatures, however, the 5-hexen-1-yl radical almost exclusively undergoes 5-exo ring closure to cyclopentanemethyl. The effect of entropy in this process was quantified, and reaction path energetics was identified as the main contributor to the dominance of the 5-exo closure at 298 K. At temperatures relevant to combustion, the increased available free energy in the system reduces both entropic and energetic preferences, and branching no longer favors cyclopentanemethyl. 216 5.3 Benzene formation in 1-hexene and cyclohexene flames Other than direct and indirect kinetic studies on the 5-hexen-1-yl radical, the combined use full detailed kinetic models and tools like sensitivity analysis can aid in understanding the detailed chemistry pathways in flames. For instance, the role of 5- hexen-1-yl the combustion kinetics of 1-hexene is quite clear. This radical is one of the first formed in the pyrolysis/oxidation of 1-hexene. Under high temperature combustion conditions, this radical is expected to predominantly undergo -scission to smaller intermediate species. Under low temperature conditions, well known O 2 addition and subsequent isomerization/decomposition are expected to dominate. The work presented in previous sections illustrated how 5-hexene-1-yl may undergo a formally direct H- elimination to cyclohexene + H; this can have a significant effect on the amount of cyclohexene, benzene, and, possibly, soot predicted to form in 1-hexene flames. Currently however, no kinetic model includes such a pathway. It is also important to recognize that H-elimination to cyclic alkene is possible not just for 5-hexene-1-yl, but other similar radicals as well. One way to test the influence of this chemistry on global flame phenomena is of course to model them. Low pressure burner stabilized flames with multiplexed chemical kinetics photoionization mass spectrometry aided by synchrotron radiation are capable of providing a wealth of data concerning fuel, product, and intermediate species [47]. These flames are modeled here using Chemkin-II [48] with experimental temperature profiles used as input. Sensitivity tests show that, compared to use of the mixture averaged 217 transport formulation, the multicomponent transport formulation has negligible effect on the predicted mole fractions for species of interest here. Thus, the mixture averaged transport formulation was used for all simulations shortly discussed. Attention is first turned towards the dominant pathways to benzene formation for these low-pressure flame conditions [49-52]. The JetSurF 2.0 model is used in the following examples, for illustrative purposes. The influence of the cyclohexyl isomerization/decomposition on benzene formation will be discussed after a general overview of dehydrogenation kinetics. The role of dehydrogenation from cyclohexane to benzene is well quantified, and supported by peak mole fractions of two stable cyclic alkenes leading up to benzene: cyclohexene and cyclohexadiene. This is illustrated below in Figure 5.19. As expected, cyclohexene is formed before cyclohexadiene, which in turn is formed before benzene. 1,4-cyclohexadiene is not included in this analysis. Although it does play a role in the eventual formation of benzene under conditions conducive to dehydrogenation, its exclusion from this discussion in no way affects the overall conclusions to be shortly drawn. 218 Figure 5.19 JetSurF model predictions of cyclohexene, 1,3-cyclohexadiene and benzene mole fraction as function of distance from burner for a burner-stabilized premixed flame at P = 30 torr,  = 1.7, 11.13% cyclohexane, 58.87% O 2 in Ar. Flame modeled using an experimentally determined temperature profile, T u = 300 K. The reactions predominantly responsible for benzene formation can be revealed through its flux at various positions in the flame. Using the JetSurF 2.0 model, Figure 5.20 illustrates the reactions dominating benzene formation for a low pressure fuel rich premixed flame. The corresponding reactions are shown in Figure 5.21. For this particular flame, H-elimination from the resonantly stabilized cC 6 H 7 radical is responsible for benzene formation at small distances from the burner, while later in the flame direct H 2 elimination from 1,3-cyclohexadiene makes up a significant portion of the benzene production. Propargyl-propargyl recombination (R b ) contributes a negligible amount to benzene formation under these flame conditions. 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 0.1 0.2 0.3 0.4 0.5 0.6 Mole Fraction Distance from burner, cm 219 Figure 5.20 Dominant reactions responsible for benzene flux (corresponding rate units on right ordinate) using the JetSurF 2.0 model. Benzene mole fraction superimposed. Burner-stabilized premixed flame conditions: P = 30 torr,  = 1.7, 11.13% cyclohexane, 58.87% O 2 in Ar. Flame modeled using an experimentally determined temperature profile, T u = 300 K. Among the routes to benzene from cyclohexane, the model includes two relevant H 2 elimination reactions leading to 1,3-cyclohexadiene + H 2 and benzene + H 2 , both of which have similar pre-exponential factors and activation energies, at around 60 kcal/mol. The H 2 elimination from 1,3-cyclohexadiene is an especially large contributor to benzene formation. This is easily understood by recognizing that the peak 1,3- 0 1 10 -4 2 10 -4 3 10 -4 4 10 -4 5 10 -4 6 10 -4 7 10 -4 -6 10 -7 -4 10 -7 -2 10 -7 0 2 10 -7 4 10 -7 6 10 -7 8 10 -7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 benzene mole fraction Reaction Rate, mol cc -1 s -1 Distance from burner, cm R a R b R c R d R e R f 220 cyclohexadiene mole fraction is located where temperature approaches and exceeds 1300 K, while the peak reaction rate occurs where the temperature is between 1500 - 1600 K. Figure 5.21 Reactions in JetSurF 2.0 corresponding to those in Figure 5.20 Figure 5.23 depicts the high-pressure limit rate constants for three H-elimination reactions leading to benzene. Among the three relevant H-elimination reactions leading to cyclic alkenes, the H-elimination reaction of cyclohexenyl (cC 6 H 9 ) to 1,3- cyclohexadiene + H represents a kinetic bottleneck. The resonance stability of the parent cyclic allyl radical results in a comparatively higher barrier height to H-elimination, at ~50 kcal/mol. However, in the case of H-elimination from cyclohexadienyl, which is even more resonantly stabilized, one of the products is benzene, which unlike cyclohexadiene, is greatly stabilized by resonance. The corresponding standard state 298 R a R b R c R d R e R f 2C3H3  H OH H2O C2H3 H2 H  H2 H  H2 221 K reaction enthalpy was determined via UV spectroscopy to be 21.1  2.9 kcal/mol [53]. The barrier height to this reaction is considerably smaller than that for H-elimination from cyclohexenyl, at ~25kcal/mol. Furthermore, the ring opening barrier of the cyclohexadienyl radical is prohibitively high, more than twice the barrier to H- elimination. Figure 5.22 (kcal/mol) for H-elimination and ring opening of cyclohexadienyl. H atom 298 K has 3/2RT added (0.9 kcal/mol). Black: calculated via RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p); blue: Berho and coworkers [53]; green: CBS-UCCSD(T)//QCISD/6-311G(d,p) of Gao et al. [54]; red: computed from experimental formation enthalpies [54-56]. The aforementioned energetics of cyclohexadienyl are shown in Figure 5.22. The activation energy for the addition of H + benzene is overestimated by about 2 kcal/mol since the experimentally reported activation energies fall in the range of 3-5 kcal/mol Relative Energy, kcal/mol  H  60.1 0 52.2 26.9 20.2 21.1 2.9 22.0 22.2 298 f H  222 [57-64]. Although the reliability of the present CCSD(T) calculations is poor, largely because of the chosen basis set size, the relative energies do serve to gain an understanding of the competition (or lack thereof) between H-elimination and ring opening for the cyclohexadienyl radical. Figure 5.23 High-pressure limit rate constants for the unimolecular H-elimination reactions in JetSurF 2.0 leading to respective closed shell stable products (+ H). Ring opening of the cyclohexenyl radical is likely to occur with similar kinetics (albeit slower overall) and pressure dependence as the cyclohexyl system. The PES is quite similar in fact, the main difference being larger barrier heights for all channels, excluding that for 1,3-cyclohexadiene + H. Figure 5.24 below shows the cyclohexenyl PES calculated at the RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p) level of theory. 10 -3 10 -1 10 1 10 3 10 5 10 7 10 9 10 11 0.4 0.6 0.8 1 1.2 1.4 Rate, s -1 1000 / T, 1/K 223 Figure 5.24 for a reduced cyclohexenyl potential energy surface. Black: RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p). H atom 298 K has 3/2RT added (0.9 kcal/mol). Red: computed from experimental formation enthalpies [54-56]. A transition state corresponding to the H-elimination reaction of cyclohexenyl could not be located. Nevertheless, the barrier is expected to be small compared to the energies of the products. Hoyermann and coworkers reported the activation energy for this reaction to be 1.5  0.3 kcal/mol [25, 61]. The measured room temperature rates of H + 1,3- cyclohexadiene (see section 5.2.1 for reported values) also agree well with those determined by two other groups, suggesting the recommend value for activation energy is reliable [24, 25]. As demonstrated earlier with the cyclohexyl system, the level of theory used in calculating the electronic energies of the cyclohexenyl system is not sufficient for an accurate determination of competitive branching and kinetics via RRKM/ME. It is likely a larger basis set would give satisfactory results, but these calculations are beyond the scope of this work.  H H   0 44.9 48 33.1 30 44.3 19.8 45.9 45.8 52.5 Relative Energy, kcal/mol 298 f H  224 A look at the overall competing dehydrogenation pathways to benzene from cyclohexane give a clear indication of relative role of all likely reactions. In summary, the following observations can be made in the context of cyclohexane dehydrogenation to benzene:  the formation of cyclohexene via cyclohexyl is a key step towards dehydrogenation to benzene;  the competitive isomerization/decomposition of cyclohexenyl is likely to occur with similar kinetics (albeit slower overall) and pressure dependence as in the cyclohexyl system;  H-elimination of cyclohexenyl (cC 6 H 9 ) to 1,3-cyclohexadiene + H represents a kinetic bottleneck to benzene formation via dehydrogenation;  direct H 2 elimination from 1,3-cyclohexene dominates benzene formation under flame conditions conducive to dehydrogenation;  the H-elimination from cyclohexadienyl to benzene is fascicle. 5.3.1 Low-pressure premixed burner-stabilized flames Using the results of the RRKM/ME studies for both cyclohexyl and 5-hexen-1-yl, several rates were fitted at 0.1 atm to most closely the model low pressure burner stabilized flame configurations necessary chemical intermediate speciation through the use of probing techniques. Most of the experiments are performed at pressures between 10-50 torr. 225 Figure 5.25 Fitted rate constants at 0.1 atm added or replacing reactions in the JetSurF 2.0 model. Low temperature chemistry also removed from model. * replaces rate recommendation from Tsang [16]. R1 and R2 valid in the range of 500 K - 1300 K. R3- R5 valid in the range of 1000 K - 2000 K. Figure 5.25 gives rates for several reactions studied in this work, for pressures of 0.1 atm. They were obtained by choosing the best fitting form (a constant, Arrhenius, or modified three-parameter Arrhenius) for the rate in question. For example, the rate of R1 at 0.1 atm is nearly constant for all temperatures considered, as seen in Figure 5.17. For the purpose of this work, a constant value of 10 6 s -1 is sufficient. This reaction block is used to investigate the role of these reactions on the predicted cyclohexene and benzene mole fractions in the 1-hexene and cyclohexane flames of Hansen and coworkers [50]. H       H   R1 * R2 R3 R4 * R5 reactant product rate (s -1 ) 10 6 1.05(10 -26 )T 10.69 exp(193/1.987/T) 1.8(10 12 )exp(-33,725/1.987/T) 2.9(10 18 ) T -2.65 exp(-20,326/1.987/T) 1.7(10 123 )T -30.5 exp(-132,150/1.987/T) 226 Figure 5.26 a) Model predictions of cyclohexene (dashed lines) and benzene (solid lines) mole fraction as function of distance from burner for a burner-stabilized premixed flame at P = 30 torr,  = 1.7, 11.13% 1-hexene, 58.87% O 2 in Ar. Flame modeled using an experimentally determined temperature profile, T u = 300 K. Black: JetSurF 2.0 with lumped low T chemistry removed; red: JetSurF 2.0 with lumped low T chemistry removed and new reaction block b) experimentally determined mole fractions of benzene under same flame conditions as in a) [50]. 0.0 5.0 10 -5 1.0 10 -4 1.5 10 -4 2.0 10 -4 2.5 10 -4 3.0 10 -4 0 0.2 0.4 0.6 0.8 1 Mole Fraction 0.0 5.0 10 -5 1.0 10 -4 1.5 10 -4 2.0 10 -4 2.5 10 -4 3.0 10 -4 0 0.2 0.4 0.6 0.8 1 Mole Fraction Distance from burner, cm a b 227 The experimental benzene species profile for the 1-hexene flame shows a relatively high initial non-zero mole fraction which is also captured by the model predictions, but to a lesser extent. The influence of 5-hexen-1-yl cyclization and subsequent decomposition to cyclohexene + H is evidenced by the large jump in cyclohexene mole fraction between the original and modified JetSurF 2.0. The increase in the initial and peak mole fraction of predicted cyclohexene concentration is over a factor of two. Looking back at the branching ratios for 5-hexen-1-yl decomposition shown in Figure 5.16, we see how cyclohexene + H dominates the product distribution at the low pressure and high temperature regime. For the system studied here, under high temperature (greater than 1000 K), two H-elimination channels dominate consumption of 5-hexen-1-yl, with roughly 80% of the consumption taking place through cyclohexene + H. This behavior supports the early formation of benzene in these flames. Inclusion of this new chemistry has a similar effect on the predicted cyclohexene and benzene mole fractions for a cyclohexane flame under the same burner configurations. These comparisons are shown in Figure 5.27. 228 Figure 5.27 a) Model predictions of cyclohexene and benzene mole fraction as function of distance from burner for a burner-stabilized premixed flame at P = 30 torr,  = 1.7, 11.13% cC 6 H 12 , 58.87% O 2 in Ar. Flame modeled using a given experimentally determined temperature profile, T u = 300 K. Black: JetSurF 2.0 with lumped low T chemistry removed; red: JetSurF 2.0 with lumped low T chemistry removed and new reaction block b) experimentally determined mole fractions of benzene under same flame conditions as in a) [50]. 0 0.001 0.002 0.003 0.004 0.005 0 0.1 0.2 0.3 0.4 0.5 0.6 Mole Fraction Distance from burner, cm 0 0.001 0.002 0.003 0.004 0.005 0 0.1 0.2 0.3 0.4 0.5 0.6 Mole Fraction Distance from burner, cm a b 229 As expected, a flux analysis for cyclohexene early in this flame indeed shows the majority of it formed from the H-elimination of cyclohexyl, in addition to the formally direct H-elimination from the 5-hexen-1-yl radical. Furthermore, the relative influence of cyclohexane concentration on corresponding peak concentrations of benzene can also be inferred from the above plots. A comparison of relative changes in cyclohexene and benzene mole fractions between JetSurF 2.0 and the revised model confirm the influence and importance of early dehydrogenation in these flames, through the initial H- elimination of cyclohexyl. Further H-abstraction results in formation of cyclohexenyl, which subsequently undergoes H-elimination to 1,3-cyclohexadiene. Once formed, 1,3- cyclohexadiene predominantly undergoes direct H 2 elimination forming benzene. 5.3.2 Atmospheric nonpremixed coflow flames The roles of dehydrogenation in the formation of benzene under different flame configurations than those discussed above have been explored for some time. McEnally and coworkers have shown dehydrogenation to be unimportant for some nonpremixed flames, where -scission dominates initial fuel consumption pathways [65, 66]. Specifically, McEnally and Pfefferly studied 1-hexene doped (4000 ppm) methane flames in an atmospheric coflow nonpremixed configuration, using photoionization/time-of- flight mass spectrometry (PTMS) to measure centerline species concentrations for benzene and various C3-C5 sized intermediate species [67]. Based on product measurements and 1-hexene decomposition rates, the authors concluded benzene 230 formation in these flames is predominantly because of unimolecular 1-hexene decomposition and subsequent recombination reactions involving C3 + C3 and C4 + C2 species. McEnally and Pfefferly additionally studied sooting tendencies of cyclohexane, cyclohexene, 1,3-cyclohexadiene, 1,4-cyclohexadiene, and benzene doped (2000 ppm) methane flames in an atmospheric coflow nonpremixed configuration, using PTMS to measure centerline species concentrations for benzene and various C3-C5 intermediate species [65]. PTMS measurements for the cyclohexane and cyclohexene flames show peak benzene concentrations that are a factor of eight smaller than that for the 1,3- cyclohexadiene doped flames. In addition, no initial benzene was detected for the cyclohexane and cyclohexene flames, while substantial amounts were detected for the two cyclohexadiene flames. Finally, based on product analysis, the authors concluded benzene formation in the cyclohexane flame was also attributed to fuel decomposition and subsequent recombination reactions involving C3 + C3 and C4 + C2 species. The RRKM/ME analyses for cyclohexyl previously discussed seem to corroborate the general observations made by McEnally and Pfefferly, that benzene formation in the atmospheric nonpremixed cyclohexane flame does not originate from dehydrogenation reactions. Recall the high pressure dependence of the isomerization/decomposition rates for both the cyclohexyl and 5-hexen-1-yl systems. 231 The observed products in the cyclohexane flame of McEnally and Pfefferly support pathways to the formation of cyclohexyl [65]. Looking back at Figure 5.13, at and above 1000 K, we see that cyclohexyl is predominantly consumed by 5-hexen-1-yl and cyclopentanemethyl. These species may subsequently undergo -scission and H-shifting reactions. Less than 20% of the product distribution is composed of cyclohexene + H at 1000 K. Thus, the RRKM/ME analysis illustrates why dehydrogenation to benzene is unlikely under the cyclohexane flame conditions of McEnally and Pfefferly. In the case of the atmospheric nonpremixed 1-hexene flame, no such correlations can be drawn from the results of the 5-hexen-1-yl RRKM/ME study. This is due to the fact that most of the 1-hexene fuel under these flame conditions was observed to undergo direct unimolecular C-C scission to smaller species, and that H-abstraction reactions play a very limited role [67]. It is interesting to note, however, that cyclohexene can also undergo direct C-C fission, but to 1-hexene. This process may allow for substantial amount of 1- hexene to exist closer to the diffusion flame front than in an analogous 1-hexene flame, thus increasing the likelihood for H-abstraction reactions with other gas-phase radicals, resulting in a formally direct H-elimination to cyclohexene+H. In contrast, a more oxidizing environment readily enables the proliferation of H abstraction reactions by OH, O, and HO 2 , leading to cyclohexyl. In essence, the work here confirms the above observation. However, two points must be reinforced. The reason dehydrogenation plays little role in nonpremixed flames in simply because of the 232 relative decrease in H-abstraction reactions taking part compared to in a premixed flame. Secondly, the RRKM/ME analysis illustrates why dehydrogenation to benzene is unlikely under the cyclohexane flame conditions of McEnally and Pfefferly 5.4 Methylcyclohex-2-yl potential energy surface Methylcyclohexane has a higher sooting propensity than cyclohexane under comparable conditions. It has both a higher threshold sooting index (TSI) in both premixed and diffusion flames, and a higher micropyrolysis index (MPI) [68, 69]. The reasons for this are not yet clear, and are hindered by the complex combustion chemistry of these compounds. The detailed kinetic modeling of methylcyclohexane is significantly more challenging than cyclohexane because of the increased number of kinetic avenues available, which result from the presence of a methyl group. For this reason, lumping of similar species is frequently performed [6, 70, 71]. Focus here lines on methylcyclohexane's dehydrogenation chemistry. The dehydrogenation of methylcyclohexane to benzene and toluene can take place through many different pathways, as illustrated below. 233 Figure 5.28 Dehydrogenation sequence for methylcyclohexane. Grey blocks denote H- abstraction reactions by H, OH, O, CH 3 . Green blocks denoted pressure-dependent H- elimination reactions. Not shown are H 2 eliminations, H-shifting, and -scission ring openings. Also not shown: pathways from cyclohexene and the two methylenecyclohexadiene species that may undergo H-abtraction followed by H- elimination to toluene and benzene. Clearly evident from the above figure is that there are many intermediates with nearly identical reaction paths and/or thermochemical properties. Use of kinetic Monte Carlo in calculating all these rates would be nonsensical. Alternative options include isomeric lumping to reduce the total number of reactions, and/or leaving out negligible channels. 234 Another possibility is to estimate the rates based on similar reactions. For instance, many of the pressure dependent H-elimination reactions could be estimated from the cyclohexyl system. Rate rules taking into consideration ring strains and any resonance stabilization could be equally useful. However, the focus here is not to assemble such a model, but rather to attempt to use knowledge of cyclohexyl isomerization/decomposition to infer reasons for methylcyclohexane's greater sooting propensity over cyclohexane. Recent experiments, similar to the low pressure burner stabilized flames of cyclohexane and 1-hexene, have been performed for methylcyclohexane as well [52]. Early dehydrogenation of methylcyclohexane is observed for the three different flame configurations. In general, the peak cyclohexene concentration for three flames is five times greater than for 1,3-hexadiene.. Consider the H-elimination of methylcyclohex-2-yl, leading to cyclohexyl (indicated by the arrow below the middle species of the second row in Figure 5.29). This species is chosen as a test case because it allows for CH 3 -elimination reaction to cyclohexene, in addition to H-eliminations resulting in two different methylcyclohexenes. The PES of the methylcyclohex-2-yl radical was explored at the RCCSD(T)/cc-pVDZ//B3LYP/6- 311++G(2d,p) level of theory (see Table 5.5 for electronic energies and vibrational energies). Although this level of theory is not suitable for accurate rate calculations, it does serve to provide a reasonable picture of the relative electronic energies in this system. 235 Table 5.5 Summary of calculated zero point energies (ZPE) in kcal/mol and zero Kelvin electronic energies (E 0 ) in Hartrees. index species a ZPE b E 0 1 MCHeq 113.94 -273.72464 2 CH3+cC6H10 18.62 -273.68090 3 3CH3cC6H10+H 109.01 -273.16492 4 2CH3cC6H10+H 108.74 -273.16882 5 PX7-2C7H13 * 111.48 6 PXCH251C6H11 111.62 -273.68260 7 SXC2H4cC5H8 114.00 -273.71446 8 CH3PXCH212cC5H8 113.44 -273.71402 9 PXCH231C6H11 111.93 -273.68264 10 PX1-2C7H13 * 112.52 CH3 18.62 -39.71586 H n/a -0.49928 TS12 112.31 -273.66608 TS13 109.75 -273.66120 TS14 109.85 -273.66266 TS15 112.35 -273.67096 TS16 112.31 -273.66966 a See Figure 5.29 for species identification; electronic energies calculated at the RCCSD(T)/cc-pVDZ//B3LYP/6-311++G(2d,p) level of theory. b Unscaled, calculated at the B3LYP/6-311++G(2d,p) level of theory. * Electronic energy not calculated. Although the relative energy of methyl + cyclohexane lies almost 10 kcal/mol lower than its H-elimination counterparts, all three elimination reactions from methylcyclohex-2-yl have similar barriers. It can therefore be concluded that CH 3 bond strength cannot be responsible for or contribute towards to an increased production of cyclohexene relative to that produced in corresponding cyclohexane flames. In addition, both ring opening barriers are lower in energy than the elimination barriers, similar to corresponding 236 barriers in the cyclohexyl system. Subsequent 5-exo ring closure barriers are also similar in energy. Thus, there is nothing unique about the electronic energetics of the methylcyclohexyl radicals relative to those in the cyclohexyl system that can explain an increased production of cyclic alkenes relative to cyclohexane. Figure 5.29 Methylcyclohex-2-yl reaction network with relative energies, calculated at the RCCSD(T)/cc-pVDZ //B3LYP/6-311++G(2d,p) level of theory, ZPE included. Only reactions relevant to the unimolecular thermal isomerization/decomposition of methylcyclohex-2-yl are shown. Pathways with no values were not explored. TS6-9 estimated based on [29]. The bold indices besides each species corresponds to those listed in Table 5.5. 1 2 3 4 5 6 7 8 9 10 ~37      H CH3  H  23.6 30.5 33.2 24.1 6.2 6.4 24.4 35.2 32.1 34.9 29.2 32.9 35.8 237 Another possible reason could involve the ring opening channels. If, for instance, ring opening was unfavorable compared to H- or CH 3 -elimination, a higher concentration of cyclohexene would be expected relative to that produced under the same temperature and pressure for cyclohexane. The kinetics and preferential branching of the 5-hexen-1-yl radical serve as an excellent analogy to this system. Recall that 5-hexen-1-yl almost completely isomerizes to cyclopentanemethyl under lower temperatures. A substituent placed at the carbon 5 position greatly affects the branching ratio, changing it from 98:2 to 2:3 under similar conditions. This can be attributed to steric hindrance to ring 5-exo ring closure. However, no such steric effect is observed for substituents placed at other locations along the carbon backbone. Extending this discussion to methylcyclohexane, of all possible radicals formed from the -scission ring opening reactions of various methylcyclohexyl radicals, there is only one pathway to a sterically hindered 5-exo ring closure: a -scission ring opening of the methylcyclohex-1-yl radical followed by the 5- exo closure. Even with this available pathway, its overall effects on cycloalkene formation are expected to be negligible, and the steric effects, in general, are expected to be less influential at combustion-relevant temperatures. Thus, any steric influence of the methyl group indirectly contributing to increased cycloalkene formation can be ruled out. Regarding in increased cycloalkene production in for methylcyclohexane compared to cyclohexane, we have seen that the system energetics are not responsible, nor are any sterically induced effects. The last remaining possibility lies in a thermodynamic driving force to preferential cyclic alkene production, namely, entropy. The loose nature of the 238 transition state leading to CH 3 elimination is responsible for this higher entropy of activation, and a well-known phenomena in reaction kinetics. This hypothesis can be confirmed simply by comparing the ratio of partition functions for the transition states of the H- and CH 3 - elimination channels in methylcyclohex-2-yl. Such a comparison essentially illustrates the difference in pre-exponential factors for these corresponding rates. Table 5.6 Temperature variation for the ratio of vibrational partition functions TS12 (CH 3 -elimination) and TS14 (H- elimination). See Figure 5.29 for species identification, vibrational frequencies calculated at B3LYP/6-311++G(2d,p) level of theory, listed in Appendix. T, K Q † TS12 / Q † TS14 500 24.4 700 12.3 1000 7.1 1300 5.0 1500 4.4 1700 3.8 The above calculation assumes the methyl rotor to be described by a harmonic oscillator. A more proper treatment (free internal rotor) would indeed result in an even greater rate, by roughly another factor of two. This conclusion is expected to be valid for direct C-H or C-CH 3 fission from methylcyclohexane, and direct H 2 or CH 4 eliminations from 2,4-methylcyclohexadiene. Thus, the increased entropy of activation for CH 3 eliminations results in substantially 239 increased rates, even though the barrier height to reaction is similar to those for H- or H 2 - elimination. In conclusion, this seems to explain methylcyclohexane's increased production of benzene relative to cyclohexane. 5.5 Conclusions In conclusion, RRKM/ME calculations for the 5-hexen-1-yl system show that although this radical species lies in a shallow energy well (~10 kcal/mol), it can become collisionally stabilized under high pressure and at elevated temperatures. Under such conditions, reactions with other gas phase radicals and unimolecular isomerization/decomposition are expected to dominate its consumption. In addition, the RRKM/ME results illustrate that under the conditions of low pressure and high temperature, the 5-hexen-1-yl radical preferentially undergoes a formally direct H- elimination reaction through cyclohexyl, to cyclohexene + H. At lower temperatures, however, the 5-hexen-1-yl radical almost exclusively undergoes 5-exo ring closure to cyclopentanemethyl. The effect of entropy in this process was quantified, and reaction path energetics was identified as the main contributor to the dominance of the 5-exo closure at 298 K. At temperatures relevant to combustion, the increased available free energy in the system reduces both entropic and energetic preferences, and branching no longer favors cyclopentanemethyl. 240 RRKM/ME calculations for the cyclohexyl system revealed a rich distribution of products highly dependent on both temperature and pressure. For experimental studies in which chemically activated cyclohexene + H reactions were examined (300 K and < 0.1 atm), the lack of 5-hexen-1-yl radical being detected is supported by the present RRKM/ME simulations: at temperatures lower than 1000 K, the production of 5-hexen- 1-yl from cyclohexyl was calculated to be negligible. At higher temperatures, the stabilization of the chemically activated cyclohexyl becomes less efficient, allowing for a greater extent of ring opening and higher production of the 5-hexen-1-yl radical. Under the conditions studied, the 5-hexen-1-yl radical can comprise up to and over 60% of the total product distribution. The significance of this result lies in the fact that 5-hexen-1-yl cannot be considered fleeting under all cases. Similar behavior is expected for analogous ring systems. Furthermore, the RRKM/ME analysis for the unimolecular decomposition of cyclohexyl shows that H-elimination to cyclohexene + H dominants under low pressure and high temperature conditions. A reaction set was introduced into the JetSurF model with the purpose of exploring how sequential H elimination impacts benzene production in burner-stabilized premixed 1-hexene and cyclohexane flames. Comparisons of the original and modified models in conjunction with rate analyses allow the following observations to be made in the context of cyclohexane dehydrogenation to benzene: 241  the formation of cyclohexene via cyclohexyl is a key step towards dehydrogenation to benzene;  the competitive isomerization/decomposition of cyclohexenyl is likely to occur with similar kinetics (albeit slower overall) and pressure dependence as in the cyclohexyl system;  H-elimination of cyclohexenyl (cC 6 H 9 ) to 1,3-cyclohexadiene + H represents a kinetic bottleneck to benzene formation via dehydrogenation;  direct H 2 elimination from 1,3-cyclohexene dominates benzene formation under flame conditions conducive to dehydrogenation;  the H-elimination from cyclohexadienyl to benzene is fascicle. A comparison of relative changes in cyclohexene and benzene mole fractions between JetSurF 2.0 and the revised model confirm the influence and importance of early dehydrogenation of cyclohexane in low-pressure premixed flames, through the initial H- elimination of cyclohexyl. Further H-abstraction results in the formation of cyclohexenyl, which subsequently undergoes H-elimination to 1,3-cyclohexadiene. Once formed, 1,3-cyclohexadiene undergoes direct H 2 elimination, forming benzene. Reaction flux analyses suggest that this last step to benzene formation dominates over the H- elimination from cyclohexadienyl. 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Simmie, Proceedings of the Combustion Institute 31 (2007) 267-275. 248 Chapter 6: Summary, Conclusions and Future Work 6.1 Concluding remarks Analogues representing a class of possible molecular scenarios in young and mature soot are presented as an explanation for the existence of persistent free radicals in soot structures. It is postulated that these persistent free radicals may exist within the turbostratic framework of soot particles and are diradical in nature. Computational results show that increasing the number of phenyl groups in acenaphthene results in increased central bond lengths and a decrease in the bond dissociation energy. The origin of this behavior lies in an increase in steric repulsion that accompanies phenyl group additions. Ab initio calculations involving these structures illustrate they may facilitate the existence of indefinitely persistent free radicals formed dynamically through repetitive bond cleavage and formation. The potential energy surfaces for several analogous systems of toluene and xylene nature were investigated through the use of density functional theory. Energy barriers for the exothermic isomerization of a methylphenyl radical to one of benzylic were determined to be 42-43 kcal/mol, with an assigned uncertainty of 2 kcal/mol. Barriers to H-shifting from isolated sites about a six-membered aromatic ring were calculated to be 61-63 kcal/mol. RRKM/Master Equation calculations were carried out through a kinetic Monte Carlo approach for the mutual isomerization of methylphenyl radicals in toluene and the xylene class of compounds. For nearly all temperatures and pressures, methylphenyl radicals adjacent to methyl groups exclusively isomerize to a more stable radical of 249 benzylic nature. Corresponding rate expressions are proposed over the temperature and pressure ranges of interest to combustion modeling. Across systems, rate coefficients are all similar, exceeding 10 6 s -1 at 1400 K, indicating that the isomerization is facile under combustion temperatures. Lastly, the number of isolated sites across the xylene family is used to explain the high temperature global oxidation reactivity trends observed experimentally, with ortho- > meta- > para-xylene. RRKM/ME calculations for the 5-hexen-1-yl system show that although this radical species lies in a shallow energy well (~10 kcal/mol), it can become collisionally stabilized under high pressure and at elevated temperatures, where the downward energy transfer is sufficiently great. Under such conditions, reactions with other gas phase radicals and unimolecular isomerization/decomposition are expected to dominate its consumption. In addition, the RRKM/ME results illustrate that under the conditions of low pressure and high temperature, the 5-hexen-1-yl radical preferentially undergoes a formally direct H-elimination reaction through cyclohexyl, to cyclohexene + H. At lower temperatures, however, the 5-hexen-1-yl radical almost exclusively undergoes 5-exo ring closure to cyclopentanemethyl. The effect of entropy in this process was quantified, and reaction path energetics was identified as the main contributor to the dominance of the 5- exo closure at 298 K. At temperatures relevant to combustion, the increased available free energy in the system reduces both entropic and energetic preferences, and branching no longer favors cyclopentanemethyl. 250 RRKM/ME calculations for the cyclohexyl system revealed a rich distribution of products highly dependent on both temperature and pressure. For experimental studies in which chemically activated cyclohexene + H reactions were examined (300 K and < 0.1 atm), the lack of 5-hexen-1-yl radical being detected is supported by the present RRKM/ME simulations: at temperatures lower than 1000 K, the production of 5-hexen- 1-yl from cyclohexyl was calculated to be negligible. At higher temperatures, the stabilization of the chemically activated cyclohexyl becomes less efficient, allowing for a greater extent of ring opening, leading a greater production of the 5-hexen-1-yl radical. Under the conditions studied, the 5-hexen-1-yl radical can comprise up to and over 60% of the total product distribution. The significance of this result lies in the fact that 5- hexen-1-yl cannot be considered fleeting under all cases. Similar behavior is expected for analogous ring systems. Furthermore, the RRKM/ME analysis for the unimolecular decomposition of cyclohexyl show that H-elimination to cyclohexene + H dominants under low pressure and high temperature conditions. A reaction set was introduced into the JetSurF model with the purpose of exploring how sequential H elimination impacts benzene production in burner-stabilized premixed 1-hexene and cyclohexane flames. Comparisons of the original and modified models in conjunction with rate analyses allow the following observations to be made in the context of cyclohexane dehydrogenation to benzene: 251  the formation of cyclohexene via cyclohexyl is a key step towards dehydrogenation to benzene;  the competitive isomerization/decomposition of cyclohexenyl is likely to occur with similar kinetics (albeit slower overall) and pressure dependence as in the cyclohexyl system;  H-elimination of cyclohexenyl (cC 6 H 9 ) to 1,3-cyclohexadiene + H represents a kinetic bottleneck to benzene formation via dehydrogenation;  direct H 2 elimination from 1,3-cyclohexene dominates benzene formation under flame conditions conducive to dehydrogenation;  the H-elimination from cyclohexadienyl to benzene is fascicle. A comparison of relative changes in cyclohexene and benzene mole fractions between JetSurF 2.0 and the revised model confirm the influence and importance of early dehydrogenation of cyclohexane in low-pressure premixed flames, through the initial H- elimination of cyclohexyl. Further H-abstraction results in the formation of cyclohexenyl, which subsequently undergoes H-elimination to 1,3-cyclohexadiene. Once formed, 1,3-cyclohexadiene undergoes direct H 2 elimination, forming benzene. Reaction flux analyses suggest that this last step to benzene formation dominates over the H- elimination from cyclohexadienyl. The above conclusion is expected to be valid for direct C-H or C-CH 3 fission from methylcyclohexane, and direct H 2 or CH 4 eliminations from 2,4-methylcyclohexadiene. 252 In addition, the reason methylcyclohexane readily produces more benzene than cyclohexane was elucidated: the increased entropy of activation for CH 3 eliminations results in substantially increased rates, even though the barrier height to reaction is similar to those for H- or H 2 -elimination. Relative PES traits and the possibility for steric hindrance induced from the methyl group were found to have no role on increased production of cylclohexene relative to cyclohexane. Recall one of the early motives for the dehydrogenation work - a possible explanation for discrepancies between high-pressure shock tube data (Daley et al, 2008, Vanderover and Oehlschlaeger 2009) and the JetSurF 1.1 model. However, attention was necessarily shifted towards competitive cyclohexyl isomerization/decomposition as its chemistry is necessary to dehydrogenation chemistry yet was poorly understood. Although a detailed dehydrogenation model was not assembled for methylcyclohexane, preliminary tests suggest H radicals formed from dehydrogenation reduces ignition delay by about 10-15% under high pressure shock tube conditions. Although inclusion of semi-global low temperature chemistry can reconcile agreement to some extent, predicted ignition delays are still too long, suggesting that there remain undiscovered oxidation/decomposition pathways, or that the experimental data in question are in err. Regarding the detailed modeling of mono-alkylated cyclohexanes, a full dehydrogenation scheme could be accurately developed based on only a few carefully selected reaction kinetic studies, using RRKM/ME for instance. In particular, a RRKM/ME study for the 253 methylcyclohex-2-yl system will enable sufficiently accurate pressure and temperature dependent rate constants for the all of the isomerization/decomposition pathways available to different methylcyclohexyl radicals. In the same manner, another RRKM/ME study will enable sufficiently accurate pressure and temperature dependent rate constants for all methylcyclohexenyl radicals. The reaction kinetics calculations that make up the last two working chapters illustrate the importance of a balance approach to detailed model development. Although the future of model development will undoubtedly rely on some level of automation, novel compounds and fuels will frequently reveal gaps in our knowledge. The strong pressure dependence of many reactions will remain a challenge in this aspect. 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Laskin (2005) Hydrophilic properties of aged soot. Geophysical Research Letters, 32. 285 Appendix A: Normal Alkane C 5 -C 12 Compounds Although not the focus of the current work, a full detailed kinetic description of cyclohexane combustion chemistry is not complete unless the chemistry of its ring- opening products like 1-hexene and 5-hexen-1yl are also fully understood. For this reason, it was necessary to first develop a model describing the pyrolysis and oxidation of normal alkane compounds beyond those included in the 'base model', USC Mech II (111 species, 784 reactions) which that describes the oxidation of H 2 and CO and the high- temperature chemistry of C 1 -C 4 hydrocarbons [1]. The alkane compounds considered are normal pentane up to, and including, normal dodecane (undecane was not included). The initial efforts resulted in JetSurF v0.2 and 1.0 [2]. They each consist of 194 species and 1459 reactions. The models describe the pyrolysis and oxidation kinetics of normal alkanes up to n-dodecane at high temperatures. It is important to note that these models are “un-tuned” and in fact periodically updated. The development effort centers on achieving consistent kinetic parameter assignment and predictions for a wide range of hydrocarbon compounds. The base model is appended with a set of reactions (83 species and 675 reactions) to describe high-temperature pyrolysis and oxidation of normal alkanes (C k H 2k+2 , 5 ≤ k ≤ 12). More details on the kinetic parameters of the reaction classes used in the n-alkane model can be found on the JetSurF website. The JetSurF 1.0 release maintains all the kinetic, thermodynamic, and transport property data of v0.2, but features a preliminary determination of the model uncertainty through the MUM-PCE method [3] and its quantitative impact on predicted combustion properties. 286 A1. Reduced and skeletal models The goal of this work was to make available to the CFD community a tractable model for applications in reacting flow simulations. Furthermore, the validity of this approach was made possible by the realization that the cracking kinetics of the initial high temperature decomposition for these fuels is independent of the kinetics of the smaller intermediates [4]. Additionally, the focus of this model is in accurately capturing the decomposition of the fuel, and therefore no sooting chemistry is included. The C5-C12 chemistry was reduced to a model containing only 9 species, but still for application to high temperature conditions only. Figure A1 depicts the steps considered when constructing the simplified and reduced models. Figure A1 Systematic simplification of JetSurF 1.0 [5]. n-C n H 2n+2 with 12  n  5 Chain length? Radical type? 1,5 H-shift if  C 6 -scission H-abstraction: OH, H, CH 3 , O, O2, HO 2 Unimolecular initiation C 0 – C 4 C 0 – C 4 C 5 – C 12 C 5 – C 12 Secondary Primary Alkenes Alkyl,  C 5 USC Mech II 1-alkene simplified 287 From parent fuel n-alkane (C n H 2n+2 ), initial C-C bond fissions lead to different sets of two primary alkyl radicals C i H 2i+1 + C j H 2j+1 with i+j = n. The chain length of the formed alkyl radicals are then tested: 1. If i and/or j ≤ 4, the alkyl radical oxidation reactions are described by the detailed kinetic model, USC-Mech II 2. If i and/or j ≥ 4 then the chain length is tested to determine if a 1,5 H-shift isomerization through a 6-member ring transition state is allowed (if i and/or j ≥ 6). This step leads to primary and secondary radicals that further decompose by  - scission leading to primary alkyl radicals and 1-alkenes (1-C k H 2k ). 3. Primary radicals formed in step 2 are tested again in steps 1 and 2 until they decompose to fragments that can be described by USC-Mech II. Depending on the size of 1-alkenes formed, 1-C k H 2k , decomposition reactions of these species are described by USC-Mech II (if k ≤ 4) or by a simplified 1-hexene / 1-pentene sub- mechanism if 5 ≤ j ≤ 6. 4. Subsequent small radicals formed during the process react through H-abstraction by OH, H, CH 3 , O, O 2 , HO 2 , leading to primary and secondary radicals., whose further decomposition reactions are described in steps 1 to 3. This simplification procedure leads to a model composed of semi-global reactions whose rate parameters are taken from JetSurF v.0.2. Table A1 illustrates the some of the reactions used in the simplified model. 288 Table A1 Simplified 1-alkenes C k H 2k model for k = 5 and 6 [5] a . No. Reaction A N E a 1 C 6 H 12  aC 3 H 5 + nC 3 H 7 1.07E+23 -2.03 74960 2 C 6 H 12 + H  C 2 H 4 + pC 4 H 9 8.00E+21 -2.39 11180 3 C 6 H 12 + H  C 3 H 6 + nC 3 H 7 1.60E+22 -2.39 11180 4 C 6 H 12 + H  C 4 H 6 + C 2 H 5 + H 2 5.40E+04 2.50 -1900 5 C 6 H 12 + O  C 2 H 4 + nC 3 H 7 + HCO 3.30E+08 1.45 -402 6 C 5 H 10  aC 3 H 5 + C 2 H 5 7.24E+22 -1.94 75470 7 C 5 H 10 + H  C 2 H 4 + nC 3 H 7 8.00E+21 -2.39 11180 8 C 5 H 10 + H  C 3 H 6 + C 2 H 5 1.60E+22 -2.39 11180 9 C 5 H 10 + H  C 4 H 6 + CH 3 + H 2 5.40E+04 2.50 -1900 10 C 5 H 10 + O  pC 4 H 9 + HCO 3.30E+08 1.45 -402 289 289 10 20 30 40 50 60 70 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Equivalence Ratio,  n-Dodecane-air T u = 403 K, P = 1 atm Laminar Flame Speed (cm/s) 10 20 30 40 50 60 70 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Equivalence Ratio,  n-Decane-air T u = 403 K, P = 1 atm Laminar Flame Speed (cm/s) 10 20 30 40 50 60 70 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Equivalence Ratio,  n-Nonane-air T u = 403 K, P = 1 atm Laminar Flame Speed (cm/s) 10 20 30 40 50 60 70 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Equivalence Ratio,  n-Octane-air T u = 353 K, P = 1 atm Laminar Flame Speed (cm/s) Figure A2 Experimentally [6] (symbols) and numerically (solid line: skeletal model; dash line: simplified model; dotted line: JetSurF v 0.2) determined laminar flame speeds of n-alkane/air mixtures. 290 Figure A2 shows comparisons of the full and reduced JetSurF used in computations of premixed laminar burning velocities [5, 6]. Although these are the only comparisons shown here, the simplified n-alkanes models were extensively validated against a large set of experimental data including: fuel pyrolysis in a jet-stirred reactor, laminar flame speeds, and ignition delay times behind reflected shock waves and for a wide range of conditions for each n-alkane from pentane to n-dodecane [5]. Global combustion parameters, including conversion of the fuel, ignition delays and flame speeds as well as detailed species profiles of pyrolysis products and transient free radicals can be well described using the simplified n-alkane model along with a detailed base model describing the oxidation of H 2 and CO, and the high-temperature chemistry of C 1 -C 4 hydrocarbons. In addition, a skeletal model is presented that describes the combustion of n-alkanes from C 1 to C 12 , and allows for the description of global parameters in good agreement with experiments, except for transient free radical profiles. This study confirms our former conclusion from the study of n-dodecane [4]. For practical fuels, such as JP-7 and JP-8, the cracking should lead to the production of a small number of relevant components, containing mostly hydrogen, methane and C 2 -C 4 compounds. Thus, the alternative approach to understand the quantitative combustion behaviors of jet fuel seem to be feasible: the initial pyrolytic process of the fuel is described by a simplified model and the reaction kinetics of the cracked products are treated fundamentally. 291 Appendix B: JetSurF 1.1 - Cyclohexane and Mono-Alkylated Cyclohexane Derivatives JetSurF 1.1 consists of 352 species and 2083 reactions. JetSurF 1.1 adds the pyrolysis and oxidation kinetics of n-butylcyclohexane at high temperatures to JetSurF 1.0. The development effort centers on n-butyl-cyclohexane, but the model includes also the high- temperature chemistry of n-propylcyclohexane, ethylcyclohexane, methylcyclohexane, and cyclohexane. JetSurF 1.1 satisfactorily predicts a wide range of combustion phenomena [7]. The selection of model - data comparisons presented in this work illustrate key mechanistic kinetic details regarding new kinetic insights or new information on fuel structure effects on corresponding combustion kinetics. B1. Unimolecular fuel decomposition In the JetSurF1.1 model, rate coefficients of unimolecular isomerization of cycloalkanes to linear alkenes were adopted from the analogous cyclohexane reaction: 6 12 6 12 1 cC H C H  Cyclohexane isomerization to 1-hexene was first studied by Tsang in a single-pulse shock tube at temperatures over 1000 K and pressures up to 7 atm [8]. Based on these measurements, cyclohexane was proposed to isomerize to 1-hexene with an activation energy of 88 kcal/mol. Ab initio calculations by Sirjean et al. support this hypothesis through a biradical intermediate corresponding to C-C fission of the cyclohexane ring 292 followed by a H-shift through a six membered ring [9]. A more recent combined theoretical-shock tube study further supports the direct C-C fission biradical mechanism [10], with overall decomposition rates to 1-hexene in agreement with those originally determined by Tsang. Therefore, Tsang's recommendation was used for all unimolecular ring opening reactions in JetSurF 1.1. For all mono-alkylated cycloalkanes, ring opening to all possible normal alkenes were considered. For C-C scission pathways adjacent to the alkyl group, the activation energy was reduced by 1 kcal/mol to account for the increased stability of an unpaired electron located at a secondary carbon relative to a primary carbon. B2. Other initial fuel reactions Hydrogen abstraction by H, O, HO2, and CH3 were determined by analogous propane reactions in USC Mech II. Such rates are thought to be reasonable since cyclohexane exhibits little to no ring strain. Additionally, the current rates agree well with recent available ab initio predictions [11]. Rate parameters for hydrogen ring-abstraction by OH are adopted from a recent combined ab initio and shock tube study on the reaction of cyclohexane and methylcylcohexane with OH [12]. Following the creation of cycloalkyl radical species, the dominant pathways are various ring-opening and -scission reactions. All of these reactions were taken or estimated from recent shock tube experiments by Tsang focusing on the isomerization and 293 decomposition of the cyclohexyl radical [13]. Most of these reactions include pressure dependence. B3. Comparison with experimental flame speed data A brief description of experimental flame methods are given in section 2.6 and are not repeated here. A comparison of the laminar flame speeds between all newly included fuel components of JetSurF 1.1 is illustrated in Figure B1. Figure B1 Experimental and computed ’s of methyl-, ethyl, n-propyl, and n-butyl- cyclohexane/air flames at T u = 353 K and p = 1 atm. Symbols: ( ○) Ji et al. [14] ( ◇) Dubois et al. [15]. Lines: simulation using JetSurF 1.1. The error bars indicate 2-  standard deviations.  S u o 294 A comparison of ’s of cyclohexane and mono-alkylated cyclohexanes with those of n- hexane flames reveals interesting thermal and kinetic behaviors. Figure B2 depicts the experimental ’s of cyclohexane and mono-alkylated cyclohexanes relative to n- hexane/air flames [16]. Cyclohexane/air flames exhibit somewhat larger ’s than n- hexane/air, by 1 to 3 cm/s, in agreement with Davis and Law [17]. This difference is caused by the difference in the adiabatic flame temperature, T ad , between cyclohexane and n-hexane. For example, for  = 1.0, T u = 353 K, and p = 1 atm, T ad = 2312 K for cyclohexane/air and 2303 K for n-hexane/air flames. Computational tests show that if the T ad of the n-hexane/air flame is made equal to that of cyclohexane/air (e.g., by substituting nitrogen by argon), would be 53.7 cm/s, close to 54.1 cm/s as computed for cyclohexane/air. Hence, the observed difference is of thermal nature. Figure B2 Comparison of experimentally determined ’s of ( ○) cyclohexane/air, ( ◆) methylcyclohexane/air, (■) ethylcyclohexane/air, ( ●) n-propylcyclohexane/air, and (▲) n-butylcyclohexane/air flames relative to those of ( ━) n-hexane/air flames at T u = 353 K [14].  S u o  S u o  S u o  S u o  S u o 0.85 0.9 0.95 1 1.05 1.1 0.6 0.8 1.0 1.2 1.4 1.6 S u o /S u o (n-hexane/air) Equivalence Ratio,   S u o 295 With alkyl group substitution, decreases by 4 to 5 cm/s, but the alkyl chain size has no measurable effect. Reaction path analysis reveals that during the oxidation of mono- alkylated cyclohexanes, from methyl- to n-butyl-cyclohexane, the distribution of intermediates is nearly the same. Hence, the overall kinetic behavior is similar. Not yet explained is the difference in between cyclohexane and mono-alkylated cyclohexane flames. The cause must be kinetic because the difference in T ad is no larger than 2 K among all fuels. Plotting the mass burning rate, o uu S  , as shown in Figure 7.5, the pseudo-activation energy E a may be derived as     2 ln 1 o a u u ad E R S T      [18]; R is the ideal gas constant,  u the unburned mixture density, and the variation of T ad is accomplished through nitrogen substitution by argon. For both cyclohexane and n- propylcyclohexane flames, E a = 68 kcal/mol over a broad range of T ad . Therefore, the differences between cyclohexane and its alkylated counterparts are independent of flame temperature and more closely rooted in chemical kinetics.  S u o  S u o  S u o 296 Figure B3 Pseudo-Arrhenius plot of mass burning rate computed of cyclohexane/air and n-propylcyclohexane/air flames at  = 1.0, T u = 353 K, and p = 1 atm using JetSurF 1.1. The lowest temperature results correspond to a fuel-air mixture without argon substitution; the variation of the adiabatic flame temperature is accomplished through nitrogen substitution by argon. Similarly to cyclohexane flames, was found be insensitive to cracking chemistry specific to the alkylated fuels. Figure B6 depicts the sensitivity coefficients of with respect to kinetics for a  = 1.0 n-propylcyclohexane/air flame. Among these reactions, none involves n-propylcyclohexane and its immediate cracking intermediates. Similarly to cyclohexane, methylcyclohexane is consumed by H-abstraction to form various methylcyclohexyl radicals. In the context of JetSurF 1.1, most of these radicals undergo rapid -scission reactions, leading to the opening of the ring. The products include straight and branched-chain intermediates.  S u o  S u o 297 0 0.5 1 1.5 2 2.5 3 3.5 0.05 0.06 0.07 0.08 0.09 Mole Fraction (x10 3 ) Distance (cm) aC 3 H 5 (x5) 1,3-C 4 H 6 C 3 H 6 Figure B4 Selected intermediate species computed using JetSurF 1.1 for cyclohexane (dashed lines) and n-propylcyclohexane (solid lines) flames at  = 1.0, T u = 353 K, and p = 1 atm. Cracking of the branched-chain intermediates produces large amounts of propene (C 3 H 6 ) and allyl (aC 3 H 5 ). As shown in Figure B4, n-propylcyclohexane produces roughly twice more propene but only one third as much 1,3-butadiene compared to cyclohexane flames. Not shown in Figure B4 are methyl, methane, and ethylene, as their peak concentrations are nearly identical between cyclohexane and n-propylcyclohexane flames. The sensitivity analysis of Figure B5 reveals that 1,3-C 4 H 6 + H → C 2 H 4 + C 2 H 3 exhibits a notable influence on of cyclohexane flames, since further reaction of vinyl with O 2 is effectively a secondary chain-branching step.  S u o 298 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 H+O 2 =O+OH CO+OH=CO 2 +H HCO+M=CO+H+M H+O 2 (+M)=HO 2 (+M) C 4 H 6 +H=C 2 H 4 +C 2 H 3 HCO+O 2 =CO+HO 2 C 2 H 3 +H=C 2 H 2 +H 2 C 2 H 3 +O 2 =HCO+CH 2 O C 2 H 3 (+M)=C 2 H 2 +H(+M) CH 3 +HO 2 =CH 3 O+OH HCO+H=CO+H 2 HO 2 +H= 2 OH C 2 H 2 +O=HCCO+H H 2 +O 2 =HO 2 +H C 2 H 4 +OH=C 2 H 3 +H 2 O O+H+M=OH+M H+OH+M=H 2 O+M HCO+H 2 O=CO+H+H 2 O C 2 H 3 +O 2 =CH 2 CHO+O 2CH 3 =H+C 2 H 5 Logarithmic Sensitivity Coefficient Figure B5 Ranked logarithmic sensitivity coefficients of with respect to reaction rate coefficients computed using JetSurF 1.1 for cyclohexane/air flames at T u = 353 K and p = 1 atm. In contrast, the abundance of propene and allyl in n-propylcyclohexane causes secondary chain termination. Allyl is resonantly stabilized and not easily oxidized by O 2 , rather recombining with the H atom (aC 3 H 5 + H + M → C 3 H 6 + M) to form propene, which serves as a secondary but notable chain terminating step, as shown in Figure B6. Additionally, propene reacts with H to regenerate allyl (C 3 H 6 + H → aC 3 H 5 + H 2 ), representing a net sink for H. Hence, although the initial fuel cracking is not rate limiting and impacts negligibly, the distribution of the cracking products matters, owing to the different kinetic behaviors of C 3 and C 4 intermediates in the overall chain reaction process. This conclusion is consistent with an earlier study which determined the  S u o  S u o 299 distribution of the cracked products in n-alkane flames is most critical in predicting propagation rates of laminar flames for normal alkanes [4]. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 H+O 2 =O+OH CO+OH=CO 2 +H HCO+M=CO+H+M H+O 2 (+M)=HO 2 (+M) HCO+O 2 =CO+HO 2 CH 3 +HO 2 =CH 3 O+OH C 2 H 3 (+M)=C 2 H 2 +H(+M) C 2 H 3 +H=C 2 H 2 +H 2 HCO+H=CO+H 2 C 2 H 3 +O 2 =HCO+CH 2 O aC 3 H 5 +H(+M)=C 3 H 6 (+M) HO 2 +H=2OH CH 3 =H+C 2 H 5 H 2 +O 2 =HO 2 +H C 3 H 6 +H=aC 3 H 5 +H 2 HCO+H 2 O=CO+H+H 2 O C 2 H 4 +OH=C 2 H 3 +H 2 O O+H+M=OH+M CH 2 *+O 2 =H+OH+CO C 2 H 2 +O=HCCO+H Logarithmic Sensitivity Coefficient Figure B6 Ranked logarithmic sensitivity coefficients of with respect to reaction rate coefficients computed using JetSurF 1.1 for  = 1.0 n-propylcyclohexane/air flame at T u = 353 K and p = 1 atm. B4. Comparison with shock tube measurements JetSurF 1.1 is capable of predicting a wide range of experimental combustion data: laminar flame speeds, plug and jet-stirred flow reactor species profiles, ignition delay times [14, 15, 17, 19-22]. However, the model fails to predict high pressure (>15 atm) shock tube ignition delay times, being too slow by up to several factors in some cases  S u o 300 [23-25]. This model-experiment ignition delay discrepancy is considerably greater in the case of mono-alkylated cyclohexane compounds, and is only observed under high pressure. Consider the Figure B7 below, seemingly illustrating the inadequacy of several detailed kinetic models in predicting high-pressure cyclohexane ignition delay, particularly at lower temperatures. Figure B7 High pressure cyclohexane shock tube data (1.154% cC 6 H 12 , 20.77% O 2 in N 2 ,  = 0.5, p 5 = 50 atm) [25] along with corresponding predictions from various available kinetic models: black solid line: JetSurF1.1 [7]; dashed red: Silke et al. [26]; dashed blue: El Bakli et al. [21] ; solid red: Sirjean et al. [27]. All selected models over estimate ignition day by over a factor of three at temperatures lower than 1000 K. Attempts to bring the JetSurF model agreement closer to the measured ignition delay times eventually lead to the development of the most recent version of JetSurF, version 2.0, discussed in the next section. 10 1 10 2 10 3 10 4 0.80 0.85 0.90 0.95 1.00 1.05 1.10 Ignition Delay ( s) 1000 K / T 301 Appendix C: JetSurF 2.0 Although JetSurF 1.1 performs well against a wide range of experimental targets (laminar flame speeds, plug and jet-stirred flow reactor species profiles, ignition delay times), there remain inconsistencies with ignition delay time comparisons at high pressures [24]. Among other things, JetSurF 2.0 addresses this issue. Although JetSurF 2.0 is the latest model release, it does not introduce any new fuel compounds. Rather, this version includes the following updates from JetSurF 1.1: - The base model was changed to the unoptimized USC Mech II [1] - The rate of HO 2 + OH = H 2 O + O 2 was updated to that recommended in the Baulch compilation [28], based on Michael's 2006 and Hanson's 2010 studies - The rate of C 2 H 4 (+M) = H 2 CC+H 2 (+M) was changed to that by Pilla and coworkers [29] - The global low-T chemistry of n-dodecane is tuned to give better NTC region results - Hydrogen abstraction reactions by H, O, CH3 modified are from Violi and Lai (personal communication, 2010) - The rate coefficients of C 3 H 8 +HO 2 was re-evaluated from Baldwin's 2002 value with the T exponents set to 2.65 for nC 3 H 7 and 2.77 for iC 3 H 7 - Addition of low temperature chemistry for cyclohexane based on the work of Fernandes and coworkers [30]. This chemistry was propagated throughout the alkylated cyclohexanes as well 302 - Initiation by carbene reactions were added, with cyclo  alkene reactions based on a singlet carbene mechanism (this work [31]) Selected changes are discussed in more detail below. C1. Unimolecular fuel decomposition and the carbene mechanism The unimolecular fuel decomposition reactions in JetSurF for all cycloalkane compounds are derived from the early shock tube experiments of Tsang [8]. No such measurements have yet been performed for mono-alkylated cyclohexanes, and corresponding rate coefficients must be determined through analogous reactions or computed from theory. However, their barriers to ring opening can be inferred from recent shock tube studies of cis-1,2-dimethylcyclohexane, which has a barrier to decomposition of 84 kcal/mol [32]. Three different straight-chain octane isomers were identified, indicating that initial decomposition takes place solely at the C-C bond between the two methyl groups. Because the overall activation energy for unimolecular decomposition of cis-1,2- dimethylcyclohexane is 84 kcal/mol, the H-shifts of all diradicals to stable alkenes must reflect this. In addition, Brown and King conducted very low pressure pyrolysis (VLPP) experiments for cyclohexane and methylcyclohexane [33]. The authors fitted Arrhenius expressions for the rate of ring opening through the C-C bonds adjacent to the methyl group, with uni k  = 10 16.4  0.3 exp(-82.5  2.4/RT) sec -1 , where activation energy is in units of kcal/mol. Additionally, their results for cyclohexane are in good agreement 303 In order for cyclohexane and alkylated cyclohexanes to isomerize to a normal or branched alkene, two steps are necessary. The first is of course some kind of C-C scission resulting in a biradical species. The subsequent process is a rate limiting H-shift, resulting in the alkene. Calculations at the CBSQB3 level predict this second step has only a slightly higher barrier than that for the first ring opening scission step [9]. Under combustion conditions, a quasi equilibrium between a closed shell cyclic species and its ring opened biradical counterpart can be rationalized. Collisional stabilization of such diradicals may also allow for chain initiation reactions that have been proposed in the past for ethylene, propyne, allene, and acetylene [34-36]. For example, the diradicals formed in the decomposition of methylcyclohexane may survive long enough to react with molecular oxygen and decompose into free radicals through a chemically activated process. 2 3 6 11 O M CH cC H diradical free radicals    For methylcyclohexane, the possible diradicals formed are where path a and b denote bond scission to two different radical types: formation of a carbene and formation of a radical where the two unpaired electrons lie on separate carbons. Under conditions of high pressure and temperature, stabilization through 304 collisional energy transfer is expected to be significant enough to allow for some degree of stabilization. As stated earlier, such overall mechanisms have been shown to significantly decrease predicted high-temperature ignition delay times in various combustion kinetic models. Such pathways are thought to be available to cycloalkanes as well, especially those with one or more alkyl groups, and that these chain reaction mechanisms may help to reconcile shock tube – model predictions observed at high temperatures and pressures (like that shown in Figure B7). The carbene mechanism, in particular, was seen as an attractive potential pathway. In a study of the isomerization of ethylene, to H 3 CCH: for instance, Pople found estimated the 0 K enthalpy difference to be 75 kcal/mol and also suggested a higher barrier transition state to the carbene does not exist. The clear differences observed in experimental ignition delay times between cyclohexane and its mono-alkylated derivatives raise the possibility of a faster initial decomposition/initiation mechanism than is currently understood for alkylated cyclohexanes. Specifically, the presence of an alkyl group allows for other more energetically favorable pathways. One possibility not yet considered involves the production of singlet carbene species. This mechanism has been shown to significantly decrease predicted high-temperature ignition delay times in combustion kinetic models of ethylene, propyne, allene, and acetylene [34-36]. This mechanism is expected to be more prevalent in unsaturated hydrocarbons [34]. As such, its potential role in the initiation of 305 radical chain processes in the oxidation of cyclohexane and its mono-alkylated derivatives was explored. Quantum chemistry calculations were carried out in an effort to elucidate the energetics of the carbene mechanism [31]. CCSD(T)/6-31G(2d,p) single-point calculations were performed on all B3LYP/6-311G(2d,p) optimized structures. T1 diagnostic values of all exclusively carbon and hydrogen containing species were below the standard value of 0.02. The energetics of diradical species were computed for their well-behaved triplet electronic states and then corrected by adding singlet-triplet energy gaps following the high-spin protocols described in [37]. Following this scheme, CCSD(T)/6-31G(2d,p) single point energies of open-shell singlet intermediates were calculated at their triplet states. Vertical singlet-triplet gaps were then calculated using spin-flip time dependent hybrid density functional theory with 50-50 functional (SF-DFT/5050) [37], at B3LYP/6- 311G(2d,p) singlet optimized structures. Spin contamination was minimal, with expectation values of the S 2 operator for open shell species all well within 5% of their theoretical values. Final open-shell singlet electronic energies were obtained by subtracting the vertical singlet-triplet gaps from CCSD(T)/6-31G(2d,p) total electronic energies. To assess the affect of basis set size on predicted energy barriers, a large number of CCSD(T) single point calculations were performed using the triple zeta basis set, 6-311G(2d,p). Use of this larger basis set consistently results in lower energy barriers between reactants and intermediates by ~1 kcal/mol. Although not performed 306 here, it would reasonable to lower the CCSD(T)/6-31G(2d,p) barriers by this corresponding amount. Table C1. Summary of calculated electronic energies (E 0 ), adiabatic singlet triplet gaps (E s-t ), and zero point energies (ZPE) in Hartrees. Hartrees. E s-t of carbenes overestimated, see text. species E 0 E s-t ZPE cyclohexane -235.24115 - 0.16971 cyclohexane TS -235.06819 0.03281 0.16081 c6h12: -235.05594 0.05165 0.16024 c6h12  -235.09379 0.00064 0.15807 1hexene -235.20640 - 0.16478 methylcyclohexane -274.45233 - 0.19751 methylcyclohexane TS -274.27123 0.05468 0.18895 c7h14: (primary) -274.26488 0.05197 0.18872 c7h14: (secondary) -274.27642 0.05338 0.18897 c7h14  -274.30550 0.00062 0.18703 1heptene -274.41376 - 0.19342 2heptene -274.41570 - 0.19311 dimethylcyclohexane -313.66188 - 0.22463 dmchcarbene : -313.48448 0.05360 0.21730 All relative energies presented here were determined from the values shown in Table C1. Selected decomposition pathways for cyclohexane and methylcyclohexane are shown in Figure C1. 307 Figure C1 Potential energy diagrams illustrating selected unimolecular decomposition pathways for (a) cyclohexane and (b) methylcyclohexane. Zero point energy corrections included. Relative energies in kcal/mol. Dashed lines indicate that the barrier required for a H- shift in the formation of corresponding alkenes are not yet determined. The energy difference between cyclohexane and its biradical isomer resulting from C-C fission is computed to be 85.0 kcal/mol. This is in good agreement with previously computed values of 83.5 [9] and 86.4 kcal/mol [10]. The corresponding energy difference in methylcyclohexane for direct C-C fission at the alkyl group is roughly the same, 85.4 kcal/mol. Transition states to the above diradicals were not explore, as their geometries and energies are expected to be similar to previously reported values. However, transition states leading to the diradical carbenes in cyclohexane and methylcyclohexane decomposition were computed and shown in Figure C2. 0 78.0 85.0 82.5 (a) 18.8 0 71.7 85.4 79.7 74.1 (b) 20.3 21.7 0 71.7 85.4 79.7 74.1 (b) 20.3 21.7 0 71.7 85.4 79.7 74.1 (b) 20.3 21.7 0 71.7 85.4 79.7 74.1 (b) 20.3 21.7 308 Figure C2 Transition states leading to biradical carbene formation. (a) Cyclohexane. (b) Methylcyclohexane. Bond-breading distance in Angstroms, indicated by arrows. As mentioned above, choice of the SF5050 methodology for these systems turned out to give incorrect results. A re-evaluation of the methodology revealed its particular failure for this class of carbene diradicals. This re-evaluation essentially put the carbene intermediates at relative energies above the corresponding separated biradicals. Specifically, the computed singlet-triplet gaps were being overestimated, and use of an spin flip methodology better suited to these carbene radicals resulted in substantially smaller singlet-triplet gaps (Krlov, personal communication). The overall result was of course in increase in the relative energies of those carbenes. Use of this improper method fortunately has little influence on the energetics of the separated diradicals since their separation distance significantly reduces their singlet-triplet gaps (see Table C1). However, even if the above results were found to be valid, their influence on high pressure ignition delay predictions was found to be negligible. A carbene mechanism 3.302 3.492 (a) (b) 309 including possible O 2 addition followed by chain branching also had no influence, as shown in Figure C3. Figure C3 High pressure methylcyclohexane shock tube data (1.96% cC 6 H 12 , 20.60% O 2 in N 2 ,  = 1, p 5 = 20 atm) [23] along with corresponding predictions from JetSurF 1.1 (red line) and JetSurF 1.1 with a semi-global carbene initiation step (black dashed line). The above comparison illustrates several important points. First, the carbene ring opening mechanism is not important for methylcyclohexane. Analogous tests show the same can be said for cyclohexane. Even a fictitiously low activation energy has no influence on the predicted high-pressure ignition delay. It can be concluded that the carbene ring opening mechanism is likely not competitive with direct C-C -scission to a diradical, and subsequent isomerization to an alkene. Secondly, the carbene + O 2 initiation mechanism does not have an effect on predicted ignition delay times, 10 2 10 3 10 4 10 5 0.85 0.90 0.95 1.00 1.05 1.10 1.15 Ignition Delay ( s) 1000 K / T 310 suggesting such a mechansim, even if it were to exist, cannot reconcile the model with the above data. Lastly, ring opening isomerization of cylcohexane to 1-hexene rate suggestions by Tsang have been corroborated by a number of studies [10, 33]. recent shock tube experiments by Peukert and coworkers [10], confirming the overall energetics of the ring opening pathway. 311 Appendix D: Gas-Phase Transport Properties - Methods of Estimation To date, the only set of Lennard-Jones (LJ) 12-6 potential parameters available for normal alkanes is that of Mansoori et al. [38], who employed the method of corresponding states and liquid properties, including molar volume, viscosity, and thermal conductivity, to estimate the potential parameters. Considering the non-spherical nature of the molecules and that the molecular configurations and interactions are not identical between the liquid and vapor states, the validity of these LJ parameters for calculating gaseous diffusion coefficients remains questionable. For this reason, the correlations of corresponding states of Tee et al. [39], as used in an earlier study on the transport properties of aromatics [40], were employed here to estimate these parameters. A brief overview of the governing theoretical framework is first given. The binary diffusion coefficient of a species i in bath gas j, D ij , can be reasonably approximated using formulations provided in many texts.     1,1 * 2 21 1 5sin 4 s r kT D m NLD     312 The methodology of Lennard-Jones (LJ) parameters estimation is similar to that proposed by Wang and Frenklach [40]. In this method, the LJ self-collision diameter  and well depth  are related to the critical pressure P c (atm) and temperature T c (K), 1/3             c c P a b T       B c a b k T where  is the acentric factor, and a’s and b’s are empirically derived coefficients. A consideration of 14 substances ranging from noble gases to benzene and n-heptane yielded [39]. 2.3511   a , 0.0874   b , 0.7915   a and 0.1693   b . The L-J parameters are determined from empirical relations adapted from Tee et al., who used viscosity, , and second virial coefficient, B, data of 14 substances ranging from inert gases to benzene and normal heptane. The empirical constants in the above equations derived from estimations of the simultaneous  and B least squares reduction. The estimated correlations, although with greater weighted sums of squares deviations when compared to calculated correlations, were preferentially chosen because of the nature of the data used in deriving the empirical relations. For example, 54% of the data 313 were represented by  and B values of the following substances: Ar, Kr, Xe, CH 4 , N 2 , and CO 2 . Thus, the rationale behind choosing estimated correlations lies in the observation that their fits more accurately represent the linear alkane and alkene chemical series. Recognizing that these coefficients may not be optimized for hydrocarbon compounds, we tested alternative values of the coefficients for closer predictions of n-alkanes up to n- heptane, as presented by Tee et al. [28]. These values are , 0.3955   b , 0.8063   a and 0.6802   b . The use of the first and second sets of coefficients is referred hereafter as methods 1 and 2, respectively. The acentric factor  in Eq. (1) and Eq. (2) was evaluated using the Lee-Kesler vapor- pressure relations [41]:   16 16 ln 5.927 6.096 1.289ln 0.169 15.252 15.688 13.472ln 0.436 c P                   (3) where  = T b /T c, and T b is the boiling point at the ambient pressure. With the exceptions of 1-nonene and 1-undecene, values of critical temperature T c and critical pressure P c were obtained from the CRC Handbook [42]. Boiling point temperatures were obtained from the CRC Handbook as well. 1-nonene and 1-undecene critical constants were 2.3511   a 314 obtained from Steele and Chirico [43]. The estimated values of  and  /k B are displayed in Table D1. 315 Table D1 Estimated Lennard-Jones 12-6 Potential Parameters for Selected Alkane and Alkene Compounds Formula Compound T b (K) a T c (K) a P c (atm) a    k B (K)   (Å)  n-C 5 H 12 n-pentane 309.2 469.7 33.3 0.249 458 5.45 n-C 6 H 14 n-hexane 341.9 507.6 29.9 0.298 512 5.74 n-C 7 H 16 n-heptane 371.6 540.2 27.0 0.350 564 6.00 n-C 8 H 18 n-octane 398.8 568.7 24.6 0.400 613 6.25 n-C 9 H 20 n-nonane 424.0 594.6 22.6 0.447 660 6.47 n-C 10 H 22 n-decane 447.3 617.7 20.8 0.492 705 6.68 n-C 11 H 24 n-undecane 469.1 639.0 19.5 0.541 750 6.83 n-C 12 H 26 n-dodecane 489.5 658.0 18.0 0.58 790 7.05 n-C 13 H 28 n-tridecane 508.6 675.0 16.6 0.621 829 7.24 n-C 14 H 30 n-tetradecane 526.7 693.0 15.5 0.643 862 7.44 1-C 5 H 10 1-pentene 303.1 464.8 35.1 0.233 449 5.34 1-C 6 H 12 1-hexene 336.6 504.0 31.7 0.287 505 5.63 1-C 7 H 14 1-heptene 366.8 537.3 28.8 0.341 558 5.88 1-C 8 H 16 1-octene 394.4 567.0 26.4 0.392 608 6.10 1-C 9 H 18 1-nonene 420.1 593.7 24.2 0.438 655 6.33 1-C 10 H 20 1-decene 443.7 617.0 21.9 0.478 698 6.58 1-C 11 H 22 1-undecene 469.1 638.0 20.4 0.575 764 6.69 1-C 12 H 24 1-dodecene 487.0 658.0 18.0 0.547 775 7.09 a All property values were taken from [42] except for the critical temperature and pressure values for 1-nonene and 1-dodecene, which were taken from [43]. 316 The LJ parameters recommended in Table D1 are used in calculating binary diffusion coefficients via the SANDIA code [44]. Molecular species with fewer than five carbon atoms are neglected, as they already exist with the transport database. Diffusivities of polycyclic aromatic hydrocarbons, for which no T c and P c values exist, were determined as outline by Wang and Frenklach [40]. 3 4 5 6 7 8 9 0 50 100 150 200 Method 1 Method 2 Mansoori et al. (1980)  (Å) MW (g/mol) Figure D1 L-J self collision diameter plotted against molecular weight for selected linear alkanes 317 0 250 500 750 1000 0 50 100 150 200 Method 1 Method 2 Mansoori et al. (1980) MW (g/mol)   B (K) Figure D2 L-J well depth plotted against molecular weight for selected linear alkanes For purpose of comparison, Figure D1 and Figure D2 depict L-J parameters plotted against molecular weight for three different methodologies, based on liquid thermal conductivity and liquid molar volume data correlations. Again, method 1 refers to the use of a calculated correlation by Tee et al., with critical constants obtained from the CRC Handbook. Method 2 employs an estimated correlation, Eq. (1) and Eq. (2), in addition to the use of critical constants found in the CRC Handbook. Mansoori et al. [38] also employ a method of corresponding states, using combinations of liquid molar volume data, liquid viscosity data, and liquid thermal conductivity data to estimate L-J parameters of many linear alkanes. 318 Evident from Figure D1 that method 2 predicts larger well depths for the entire range of molecular weight. In practice however, the binary diffusion coefficient is insensitive to this value since its contribution only appears in the calculation of the collision integral. Figure D3 depicts the estimates based on three methodologies for the C 12 H 26  N 2 Binary Diffusion Coefficient (BDC). Method 2, recommended here, symbolizes a middle road approach for one of large uncertainty. Although the accuracy of these methods cannot be verified because there is a lack of experimental diffusion data for many larger hydrocarbons, Figure D3 illustrates the degree of uncertainty when predicting binary diffusion coefficients for these molecules. At the upper temperature limit in Figure D3, the percentage differences of Method 1 and Mansoori et al. compared to Method 2 are 5.0% and 9.7% respectively. Furthermore, the uncertainty in BDC estimations could be even larger considering that dodecane may only be poorly described using the spherical potential assumption. 319 0 0.2 0.4 0.6 0.8 200 400 600 800 1000 1200 Method 1 Method 2 Mansoori et al. (1980) D n-C12H26-N2 (cm 2 /s) T (K) Figure D3 Binary diffusion estimates for dodecane in nitrogen. 320 Article References 1. H. Wang; X. You; A. V. Joshi; S. G. Davis; ; A. Laskin; F. N. Egolfopoulos; C. K. Law, in: 2007. 2. B. Sirjean; E. Dames; D. A. Sheen; X.-Q. You; C. Sung; A. T. Holley; F. N. Egolfopoulos; H. Wang; S. S. 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Ribaucour, Journal of Physical Chemistry A 111 (19) (2007) 3761-3775. 27. B. Sirjean; F. Buda; H. Hakka; P. A. Glaude; R. Fournet; V. Warth; F. Battin- Leclerc; M. Ruiz-Lopez, Proceedings of the Combustion Institute 31 (2007) 277-284. 28. D. L. Baulch; C. T. Bowman; C. J. Cobos; R. A. Cox; T. Just; J. A. Kerr; M. J. Pilling; D. Stocker; J. Troe; W. Tsang; R. W. Walker; J. Warnatz, Journal of Physical and Chemical Reference Data 34 (3) (2005) 757-1397. 322 29. G. L. Pilla; D. F. Davidson; R. K. Hanson, Proceedings of the Combustion Institute 33 (2011) 333-340. 30. R. X. Fernandes; J. Zador; L. E. Jusinski; J. A. Miller; C. A. Taatjes, Physical Chemistry Chemical Physics 11 (9) (2009) 1320-1327. 31. E. Dames; A. Krylov; H. Wang in: Proceedings of the US National Technical Meeting of the Combustion Institute Atlanta, GA, 2011; Atlanta, GA, 2011. 32. C. M. Rosado-Reyes; W. Tsang, in: 21st International Symposium on Gas Kinetics, 2010. 33. T. C. Brown; K. D. King, International Journal of Chemical Kinetics 21 (4) (1989) 251-266. 34. H. Wang, International Journal of Chemical Kinetics 33 (11) (2001) 698-706. 35. A. Laskin; H. Wang, Chemical Physics Letters 303 (1-2) (1999) 43-49. 36. J. H. Kiefer; W. A. Vondrasek, International Journal of Chemical Kinetics 22 (7) (1990) 747-786. 37. Y. H. Shao; M. Head-Gordon; A. I. Krylov, Journal of Chemical Physics 118 (11) (2003) 4807-4818. 38. G. A. Mansoori; V. K. Patel; M. Edalat, Int. J. Thermophys. 1 (3) (1980) 285-298. 39. L. S. Tee; S. Gotoh; W. E. Stewart, Industrial & Engineering Chemistry Fundamentals 5 (3) (1966) 356. 40. H. Wang; M. Frenklach; Nf, Combustion and Flame 96 (1-2) (1994) 163-170. 41. B. I. Lee; M. G. Kesler, Aiche Journal 21 (3) (1975) 510-527. 42. D. R. Lide, Handbook of Chemistry and Physics, 71th ed., CRC Press, Boca Raton, FL, 1990-1991. 43. W. V. Steele; R. D. Chirico; Ky, Journal of Physical and Chemical Reference Data 22 (2) (1993) 377-430 44. R. J. Kee; J. Warnatz; J. A. Miller, in: Sandia National Laboratories Report, 1983 323 Appendix E: Parameters Used in Quantum Chemistry Calculations Zero Kelvin energy barriers of 2-methylphenyl -> benzyl. wB97X-D/6-311G(2d,p) 46.1 wB97X-D/6-311+G(2d,p) 46.0 M06/6-311G(2d,p) 47.3 M06/6-311++G(2d,p) 47.3 CCSD(T)/cc-pVDZ 47.2 CCSD(T)/aug-cc-pVDZ 47.0 QCISD(T)/cc-pVDZ 47.2 B97X-D/6-311G(2d,p) optimized geometries used for the RRKM/ME calculations of Chapter 4 Benzyl C 0.00000000 0.00000000 0.00000000 C -1.40006800 -0.00000300 0.00000400 C -2.13746500 1.21132900 0.00000500 C -3.51462600 1.20477000 0.00000100 C -4.21513800 0.00000100 0.00000100 C -3.51463000 -1.20476800 0.00000500 C -2.13746700 -1.21133100 0.00000600 H -1.59735600 -2.15174400 0.00001000 H -4.05686500 -2.14318600 0.00001100 H -5.29825500 0.00000200 -0.00000400 H -4.05686100 2.14318900 -0.00000300 2MP C 0.00000000 0.00000000 0.00000000 C -1.50355000 -0.06033700 -0.00190400 C -2.29500500 1.06475000 -0.00181100 C -3.66090900 1.10398100 -0.00086100 C -4.32593600 -0.12336300 0.00004600 C -3.59209500 -1.29950400 -0.00050400 C -2.20280800 -1.26947900 -0.00161300 H -1.64344200 -2.20011000 -0.00283400 H -4.10469100 -2.25396200 -0.00046200 H -5.40953800 -0.15148500 0.00080800 H -4.21041100 2.03828100 -0.00096300 324 H -1.59734900 2.15173900 0.00000800 H 0.55718800 0.92804600 0.00003800 H 0.55719400 -0.92804200 -0.00004000 H 0.36927300 0.56341000 -0.85919100 H 0.43058300 -1.00153500 -0.03733300 H 0.36872200 0.49628200 0.90021400 3MP C 0.00000000 0.00000000 0.00000000 C -1.50212000 0.10011100 0.00407900 C -2.12310600 1.35641600 0.00357700 C -3.48878700 1.38923000 0.00130100 C -4.31849600 0.29865600 0.00024400 C -3.69266600 -0.94535700 0.00149100 C -2.30615200 -1.03531800 0.00353200 H -1.83810200 -2.01361500 0.00523000 H -4.29223300 -1.84897200 0.00160400 H -5.39819900 0.38654600 -0.00097800 H -1.52635900 2.26296500 0.00534800 H 0.43122300 0.54380800 0.84374100 H 0.41709200 0.42971700 -0.91416000 H 0.32811200 -1.03819100 0.06377800 TS-1-2 C 0.00000000 0.00000000 0.00000000 C -1.46469700 0.30894200 -0.00200900 C -1.93717700 -0.97828100 -0.00195200 C -3.27674700 -1.30666600 -0.00063500 C -4.15391100 -0.22068600 -0.00001100 C -3.69003700 1.09349600 -0.00012300 C -2.32812400 1.38851400 -0.00092400 H -1.98170000 2.41554000 -0.00027100 H -4.41091200 1.90286000 0.00102800 H -5.22292400 -0.40257600 0.00070500 H -3.64171100 -2.32577800 -0.00059700 H 0.56542600 0.16159800 -0.91544500 H 0.55805200 0.15627700 0.92104500 H -0.54867800 -1.30292300 0.00291000 TS-1-3 C 0.00000000 0.00000000 0.00000000 C 1.49456600 -0.01585400 -0.09164100 C 2.26510100 -1.18708000 -0.19139200 C 3.59036100 -1.22070600 -0.14002000 C 4.39060200 -0.08253900 0.01259500 C 3.68088100 1.10081600 -0.06503300 C 2.27370500 1.13046500 -0.11874400 H 1.78056900 2.09759700 -0.14838500 H 4.21568000 2.04469100 -0.05644800 H 5.46378600 -0.10438500 0.14844900 H -0.38115100 1.01628700 0.10905000 H -0.44748400 -0.44388700 -0.89208400 H -0.33764700 -0.58692800 0.85746300 H 2.88404800 -2.22825800 0.32636100 OX6 C 0.00000000 0.00000000 0.00000000 C -1.25985800 -0.82233200 -0.00010600 C -1.18399600 -2.22868900 -0.00003300 C -2.38302100 -2.89189300 -0.00000800 C -3.63450600 -2.33682600 -0.00001600 C -3.68671500 -0.94617200 -0.00002500 C -2.50920100 -0.21041500 -0.00010800 H -2.56172900 0.87292600 -0.00001600 H -4.64496600 -0.43906800 0.00004100 H -4.53661300 -2.93692400 0.00000600 C 0.13607800 -2.95100000 -0.00002400 H 0.73046000 -2.68536200 0.87858700 H -0.01316900 -4.03001100 0.00269600 H 0.72814000 -2.68973700 -0.88154400 H 0.61516000 -0.21650800 -0.87791900 H 0.61330500 -0.21392600 0.87986800 H -0.22697800 1.06646800 -0.00173600 OX1 C 0.00000000 0.00000000 0.00000000 C -1.37370100 -0.60726700 0.00053900 C -1.54060600 -2.02675800 0.00042900 C -0.45189700 -2.90473400 0.00091500 H 0.57083600 -2.55494300 0.00185900 H -0.61417800 -3.97509900 0.00001100 C -2.86318800 -2.54205200 0.00010000 C -3.96043900 -1.71277900 0.00028000 C -3.78318800 -0.33180500 0.00070700 C -2.49604700 0.19913700 0.00067100 H -2.36765200 1.27657000 0.00076300 H -4.64101100 0.32972300 0.00086100 H -4.95885400 -2.13379800 0.00002000 OX5 C 0.00000000 0.00000000 0.00000000 C -1.18924700 0.92305500 0.00012400 C -1.00436800 2.31402100 0.00037300 C 0.37917400 2.90488200 -0.00038400 H 0.94222900 2.59421000 -0.88516800 H 0.34208400 3.99446900 0.00974500 H 0.95100000 2.57803500 0.87268700 C -2.12326200 3.15050200 0.00019600 C -3.36242800 2.56951600 -0.00017100 C -3.60241400 1.22480900 -0.00028300 C -2.47772400 0.40007100 -0.00015700 H -2.61181900 -0.67723400 -0.00037200 H -4.60210500 0.80719900 -0.00053400 325 H -2.99598900 -3.61851600 -0.00041600 H 0.57186500 -0.31044300 -0.87915800 H -0.05593300 1.08890200 -0.00237900 H 0.57099800 -0.30657300 0.88110600 H -1.99694800 4.22809000 0.00053900 H 0.62764300 0.15673000 0.88200200 H 0.63418400 0.16531000 -0.87567000 H -0.31330400 -1.04437900 -0.00615000 TS-6-1 C 0.00000000 0.00000000 0.00000000 C -1.36400100 -0.62825700 -0.00507800 C -1.55080400 -2.00280800 -0.00741200 C -0.73834200 -3.25994700 0.00223900 H -0.23255900 -3.54187100 0.92405000 H -0.22419700 -3.55332500 -0.91137600 C -2.79706500 -2.56930200 -0.00699900 C -3.97233700 -1.84620000 0.00070700 C -3.80395500 -0.46192700 0.00108200 C -2.53930700 0.12474300 -0.00268500 H -2.46842000 1.20764600 -0.00265500 H -4.67879600 0.17906800 0.00463000 H -4.95575400 -2.29781300 0.00520500 H 0.59767700 -0.35527400 -0.84321900 H -0.06175800 1.08738300 -0.06164800 H 0.54464100 -0.25669300 0.91258500 H -2.05325000 -3.78361300 0.00070400 TS-6-5 C 0.00000000 0.00000000 0.00000000 C -1.19047300 0.92304200 0.01959200 C -0.99393300 2.30559700 0.02799500 C 0.36453800 2.93117300 0.12803900 H 1.00460600 2.62421500 -0.70446700 H 0.28716700 4.01748800 0.11469000 H 0.87311400 2.63110300 1.04888800 C -2.18259700 3.03286000 -0.10405900 C -3.40304200 2.51122700 -0.05871700 C -3.63802800 1.14685800 0.12699500 C -2.49087900 0.37645800 0.07269500 H -2.58357000 -0.70520000 0.10020900 H -4.61712900 0.70816700 0.26903500 H 0.64591300 0.16927700 0.86607000 H 0.61202400 0.16480000 -0.89117900 H -0.30721100 -1.04601300 0.00831800 H -3.18126700 3.72221400 0.40922700 MX2 C 0.00000000 0.00000000 0.00000000 C -1.28803100 0.77898300 0.00791400 C -2.52082500 0.17598200 0.00755800 C -3.75365900 0.77936200 0.00446800 C -3.72680800 2.17709500 0.00086600 C -2.52031300 2.86194400 0.00110600 C -1.31412300 2.17683400 0.00503200 H -0.37693200 2.72455600 0.00853800 H -2.52014300 3.94549500 0.00112500 H -4.66383500 2.72509600 -0.00011100 C -5.04181400 0.00054400 0.00212900 H -5.13292500 -0.59947300 -0.90591800 H -5.08701200 -0.68305100 0.85224900 H -5.90146200 0.66987300 0.05556600 H 0.14662200 -0.49804100 -0.96137400 H -0.00338000 -0.77144000 0.77190200 H 0.85346400 0.65675700 0.17418600 MX1 C 0.00000000 0.00000000 0.00000000 C -1.35250500 0.66098100 0.00013500 C -2.51357400 -0.08294100 -0.00000800 C -3.79626700 0.52342200 0.00006700 C -3.85068800 1.93813200 0.00002000 C -2.69135800 2.68318500 0.00003000 C -1.44907300 2.05811700 0.00003500 H -0.54364500 2.65557200 0.00004400 H -2.74686700 3.76571100 -0.00011800 H -4.81766800 2.42851300 -0.00013100 C -4.96323900 -0.25012500 0.00001900 H -4.91511500 -1.33156300 -0.00011100 H -5.93995900 0.21646600 0.00027400 H -2.45504000 -1.16710700 -0.00013700 H 0.57788000 0.29306300 -0.88008600 H -0.08789400 -1.08718500 -0.00103600 H 0.57722200 0.29128500 0.88110700 TS-2-1 C 0.00000000 0.00000000 0.00000000 C -1.28503300 0.77483300 -0.00091100 C -2.53824700 0.18599900 -0.00095900 C -3.72467500 0.86959000 0.00036500 C -3.75696400 2.25235100 0.00013700 C -2.52019000 2.89011100 -0.00015700 C -1.32467000 2.17279600 -0.00045200 H -0.38646600 2.71934200 -0.00090300 PX2 C 0.00000000 0.00000000 0.00000000 C -1.50490400 0.02036200 -0.01373500 C -2.25193700 -1.15057600 -0.01294000 C -3.64212200 -1.12949700 -0.00899100 C -4.35242200 0.07064600 -0.00620400 C -3.56076100 1.19638200 -0.00866900 C -2.19773800 1.23937900 -0.01365200 H -1.65506400 2.17951600 -0.01864600 326 H -2.47690400 3.97323000 -0.00038800 H -4.68000100 2.81991500 -0.00001900 C -4.65047000 -0.30649700 -0.00043100 H -5.18156800 -0.54559800 0.91876900 H -5.18682800 -0.54203600 -0.91745600 H 0.05322000 -0.66542800 -0.86442800 H 0.07773500 -0.62181500 0.89476800 H 0.86334500 0.66630300 -0.02793800 H -3.39352900 -0.95395200 -0.00226700 C -5.85592300 0.12283600 -0.00306500 H -6.28015500 -0.88221800 -0.00831600 H -6.22739700 0.64492200 0.88132800 H -6.23078800 0.65571500 -0.87947300 H -4.19122200 -2.06643900 -0.01018000 H -1.73880400 -2.10615800 -0.01688300 H 0.38343000 -1.00792500 -0.16420300 H 0.38636700 0.35311200 0.95991800 H 0.41028600 0.64942500 -0.77651300 PX1 C 0.00000000 0.00000000 0.00000000 C -1.50274600 0.00953700 -0.00580100 C -2.22513200 1.20409000 -0.00534200 C -3.60289100 1.20987800 -0.00346300 C -4.34167100 0.00147900 -0.00232200 C -3.59534600 -1.20532600 -0.00346600 C -2.22137100 -1.19238500 -0.00536200 H -1.67847800 -2.13231400 -0.00713800 H -4.12845300 -2.14984700 -0.00381700 C -5.74076200 -0.00329200 -0.00101000 H -6.30079300 0.92301600 -0.00049200 H -6.29464300 -0.93328300 -0.00043700 H -4.13987400 2.15213800 -0.00377500 H -1.68845000 2.14716800 -0.00696600 H 0.40515700 1.00971800 -0.07880700 H 0.38647000 -0.44542700 0.92110200 H 0.39355200 -0.58765900 -0.83349300 TS-2-1 C 0.00000000 0.00000000 0.00000000 C -1.50238500 0.12180900 -0.01432300 C -2.29170800 -1.03704300 -0.01416500 C -3.65413400 -0.84337100 -0.00663500 C -4.25587600 0.38912900 -0.00336900 C -3.49703200 1.54193800 -0.00688900 C -2.11241000 1.37793200 -0.01230100 H -1.47833700 2.25840100 -0.01696000 H -3.93478900 2.53361400 -0.00859000 C -5.68201300 -0.06477900 -0.00414200 H -6.25850800 0.03915100 -0.92126000 H -6.25705200 0.03227000 0.91467600 H -1.83227100 -2.01881700 -0.02065900 H 0.35497100 -0.33644600 0.97794700 H 0.34401400 -0.72574900 -0.74013400 H 0.47852600 0.95603000 -0.21691700 H -5.00281300 -1.30710300 -0.00772600 TS-2-3 C 0.00000000 0.00000000 0.00000000 C 1.48962000 0.01881400 -0.15298500 C 2.24344900 1.17800700 -0.20383800 C 3.65327300 1.17831200 -0.20241900 C 4.40724600 0.01926500 -0.15045100 C 3.61146800 -1.13323500 -0.28810400 C 2.28568400 -1.13384700 -0.28934400 C 5.89715300 0.00010500 -0.00065600 H 6.36870300 -0.48187600 -0.85993100 H 6.18693300 -0.56674900 0.88761800 H 6.29960800 1.00986300 0.09343100 H 4.16428500 2.13680400 -0.21189000 H 1.73204500 2.13626900 -0.21478200 H -0.28450100 -0.49820000 0.93049700 H -0.40978500 1.01106500 0.01797000 H -0.46881300 -0.55034900 -0.81831300 H 2.94958300 -2.14214000 0.23305500 EP C 0.00000000 0.00000000 0.00000000 C -1.44344700 -0.27169300 0.00022700 C -1.99124800 -1.53843700 0.00007900 C -3.31865700 -1.85067200 0.00032700 C -4.21897000 -0.78197000 0.00077500 C -3.74080000 0.51947700 0.00092400 C -2.37601900 0.77254800 0.00063500 H -2.01502100 1.79642700 0.00071700 H -4.43788900 1.34862900 0.00125000 H -5.28559400 -0.97459400 0.00099100 H -3.66943800 -2.87621900 0.00015900 H 0.27749200 1.05067100 -0.00123200 C 0.95160300 -0.92449000 0.00125000 H 1.99884500 -0.65000500 0.00098800 H 0.71077700 -1.98222500 0.00256100 EP C 0.00000000 0.00000000 0.00000000 C -1.44344700 -0.27169300 0.00022700 C -1.99124800 -1.53843700 0.00007900 EPTS C 0.00000000 0.00000000 0.00000000 C -1.46749100 -0.18728100 -0.00041500 C -1.76594800 -1.54583700 -0.00060200 327 C -3.31865700 -1.85067200 0.00032700 C -4.21897000 -0.78197000 0.00077500 C -3.74080000 0.51947700 0.00092400 C -2.37601900 0.77254800 0.00063500 H -2.01502100 1.79642700 0.00071700 H -4.43788900 1.34862900 0.00125000 H -5.28559400 -0.97459400 0.00099100 H -3.66943800 -2.87621900 0.00015900 H 0.27749200 1.05067100 -0.00123200 C 0.95160300 -0.92449000 0.00125000 H 1.99884500 -0.65000500 0.00098800 H 0.71077700 -1.98222500 0.00256100 C -3.03707700 -2.04176600 0.00007100 C -4.08359100 -1.10633100 0.00044400 C -3.81603800 0.25132000 0.00037500 C -2.50211000 0.72695400 0.00005900 H -2.30470400 1.79348800 0.00022900 H -4.63850000 0.95678600 0.00066000 H -5.11039700 -1.45362300 0.00083200 H -3.24577800 -3.10502700 0.00025200 H 0.47457900 0.97576500 0.00013600 C 0.63292200 -1.15947900 0.00072600 H 1.67684600 -1.43844500 0.00153100 H -0.46408400 -1.99292600 -0.00017300 328 B97X-D/6-311G(2d,p) unscaled frequencies used for the RRKM/ME calculations of Chapter 4 benzyl 2mp 3mp TS-1-2 TS-1-3 202.58 30.46 10.18 -2307.03 -2161.25 366.66 205.55 210.35 190.23 98.59 395.55 337.2 352.93 377.63 178.2 483.33 422.22 422.16 394.74 336.2 506.96 490.9 488.58 424.75 363.08 538.47 523.68 524.06 494.52 441.04 633.04 638.81 634.38 566.65 475.87 692.25 699.75 683.68 643.04 546.25 713.06 750.02 766.87 716.51 586.96 781.6 803.71 800.01 755.4 687.61 840.66 862.57 870.3 835.87 778.79 842.55 945.82 901.52 887.5 804.2 911.23 991.99 987.35 951.79 910.04 981.26 995.63 1005.63 975.63 933.24 981.76 1012.55 1010.77 1003.37 987.74 997.64 1056.86 1069.08 1011.78 998.41 1003.6 1066.78 1069.47 1034.27 1065.54 1047.39 1144.6 1109.68 1058.22 1077.08 1126.98 1181.86 1186.52 1133.89 1125.89 1178.87 1231.46 1222.29 1145.16 1186.45 1191.13 1287.52 1304.47 1182.74 1273.47 1289.65 1318.31 1312.13 1218.33 1330.29 1323.04 1420.86 1420.08 1286.17 1415.35 1362.03 1458.23 1451.82 1329.96 1442.24 1491.3 1489.17 1490.98 1430.67 1477.09 1501.33 1494.34 1501.77 1479.78 1486.71 1513.63 1507.86 1506.92 1503.66 1501.68 1605.56 1609 1595.99 1626.67 1581.56 1622.52 1665.85 1668.03 1672.87 1745.18 3161.59 3054.47 3048.24 1853.17 2222.32 3181.29 3121.32 3112.13 3098.63 3048.52 3183.43 3137.37 3134.32 3182.73 3113.44 3195.62 3172.33 3179.93 3194.77 3134.07 3201.57 3185.62 3181.14 3195 3168.39 3214.91 3197.81 3195.04 3207.99 3188.2 3262.01 3211.15 3208.12 3217.94 3218.84 329 ox1 TS-6-1 ox6 ox5 TS-6-5 mx2 TS-2-1 mx1 155.31 -2340.07 126.23 133.88 -2148.53 25.93 -2294.41 46.06 191.18 54.16 175.82 177.84 114.64 41.17 34.01 197.85 258.15 185.11 184.41 195.96 159.8 190.92 177.97 211.67 342.37 203.15 252.35 269.55 161.38 219.98 217.32 288.91 420.04 266.03 306.81 324.16 242.82 264.54 283.3 419.53 422.32 386.91 402.88 421.26 303.74 399.13 383.6 426.51 492.25 475.47 473.62 439.18 372.06 476.43 465.99 507.19 512.45 485.69 506.12 515.74 411.41 514.14 480.79 522.38 540.96 501.37 516.65 516.75 459.27 520 507.93 542.51 597.4 517.73 595.65 584.95 516.5 550.84 531.35 553.57 710.12 629.45 709.41 707.95 526.95 704.33 595.99 691.23 733.46 721.34 755.46 752.99 570.9 746.75 715.09 723.07 756.09 752.63 766.45 796.91 701.05 785.82 735.66 752.27 779.66 777.19 833.21 822.39 726.05 903.37 787.44 797.88 866.98 859.52 899.11 865.87 797.36 924.95 912.24 881.51 872.34 915.04 979.13 953.28 824.54 983.61 951.09 903.11 949.47 972.43 997.95 1013.44 962.34 987.89 975.56 945.85 985.36 983.1 1007.68 1019.48 979.8 1016.99 988.45 985.12 986.32 1000.27 1045.6 1052.13 1001.37 1056.14 1006.68 990.66 1011.76 1044.11 1077.76 1079.18 1042.96 1062.94 1038.45 1000.12 1065.4 1074.76 1096.92 1082.22 1076.05 1067.65 1056.74 1043.48 1073.48 1087.55 1150.14 1141.65 1112.37 1123.76 1065.34 1070.42 1157.86 1140.65 1189.18 1195.39 1140.49 1192.59 1105.87 1128.99 1180.88 1154.73 1228.8 1243.86 1188.7 1263.63 1139.39 1187.04 1206.72 1188.36 1280.61 1296.3 1258.48 1288.72 1190.25 1206.28 1288.23 1220.77 1305.45 1303.91 1330.89 1318.1 1272.81 1285.82 1321.32 1287.5 1414.54 1410.12 1394.9 1418.94 1283.42 1323.87 1324.69 1329.08 1425.77 1422.99 1421.96 1420.84 1326.6 1351.52 1421.95 1421.89 1452.91 1430.2 1430.24 1433.95 1420.06 1422.48 1465.35 1431.46 1484.96 1484.24 1474.11 1489.7 1426.76 1455.84 1489.88 1465.45 1496.22 1495.74 1484.6 1490.46 1458.49 1491.21 1502.38 1489.6 1501.77 1500.1 1494.44 1493.58 1488.85 1501.1 1511.38 1499.72 1503.26 1503.03 1500.36 1506.49 1504.42 1504.83 1519.16 1518.24 1506.08 1510.48 1505.16 1511.31 1515.46 1519.7 1596.81 1613.31 1593.57 1611.14 1575.75 1614.54 1622.56 1603.05 1630.89 1686.8 1675.36 1656.62 1739.49 1670.66 1686.94 1635.81 3039.72 1850.48 3044.4 3040.07 2215.99 3052.37 1853.69 3047.08 3096.78 3044.35 3045.38 3041.27 3040.43 3053.38 3051.88 3109.66 3133.5 3092.47 3101.18 3095.01 3043.15 3119.03 3095.4 3133.1 3172 3104.6 3104.72 3099.08 3094.92 3119.76 3116.5 3161.51 3178.59 3132.56 3135.44 3133.32 3101.52 3135.83 3136.21 3176.19 3183.72 3177.54 3144.61 3135.25 3135.71 3136.85 3170.45 3180.66 3200.32 3187.06 3180.32 3174.1 3146.82 3172.27 3191.33 3189.58 3215.99 3195.9 3194.29 3180.75 3172.97 3177.58 3192.13 3206.42 3271.46 3218.18 3207.13 3203.29 3215.39 3206.13 3208.18 3262.04 330 mxTS-6-1* mx5* px2,px3 pxTS-2-1 px1 TS-2-3 -2320.97 22.46 17.18 -2337.05 16.05 -2110.88 26.49 31.66 28.29 22.29 133.6 88.45 168.23 195.03 134.96 134.92 301.89 95.14 227.73 230.7 284.6 270.6 304.9 119.94 313.1 272.54 304.45 319.89 397.48 278.93 379.04 408.94 390.57 386.23 400.04 315.73 425.61 432.27 424.89 400.5 471.21 368.61 452.51 523.6 460.86 441.14 497.46 377.76 525.29 534.41 519.99 497.29 510.81 443.99 534 546.04 659.23 524.05 655.22 485.11 583.02 697.7 696.91 651.56 702.23 508.42 712.55 733.16 738.44 718.29 736.89 600.44 765.84 803.43 828.31 771.35 757.18 687.61 820.06 893.33 840.87 830.71 834.39 736.78 903.45 918.91 870.33 858.26 838.45 830.5 922.22 953.7 962.44 901.32 853.43 845.64 965.45 1004.65 989.07 968.4 967.47 953.84 982.72 1009.98 1025.73 977.08 976.25 966.22 1010.75 1035.6 1033.78 1015.8 978.12 974.09 1021.49 1066.35 1065.55 1026.67 1019.08 1033.28 1058.02 1071.47 1070.25 1057.42 1028.39 1065.42 1070.61 1152.68 1167.89 1070.2 1063.42 1066.01 1139.38 1185.73 1219.44 1143.52 1155.69 1162.98 1156.09 1259.4 1241.85 1153.17 1187.91 1214.82 1163.3 1302.48 1290.81 1215.91 1248.31 1289.64 1248.29 1315.82 1307.91 1225.79 1293.38 1329.14 1298.82 1419.12 1420.15 1287.83 1309.06 1408.81 1313.6 1421.05 1421.08 1325.53 1348.37 1420.98 1421.5 1428.35 1428.88 1420.2 1421.15 1441.67 1428.72 1472.9 1488.65 1421.57 1462.28 1478.04 1452.07 1489.38 1489.28 1435.48 1489.81 1485.51 1495.52 1495.8 1492.48 1492.53 1500.3 1486.54 1498.73 1502.17 1505.86 1501.88 1501.45 1497.6 1504.77 1512.1 1522.45 1527.19 1527.15 1502.81 1638.98 1631.98 1598.61 1628.52 1587.36 1591.98 1670.32 1657.94 1676.95 1675.87 1636.89 1752.19 1855.73 3046.59 3047.41 1849.37 3041.57 2230.38 3047.94 3052.77 3053.55 3046.37 3101.14 3049.11 3095.21 3110.53 3110.76 3096.11 3130.9 3049.73 3112.09 3119.28 3119.75 3110.6 3161.53 3114.03 3133.62 3132.42 3133.8 3131.24 3172.6 3114.23 3176.64 3135.85 3136.11 3182.83 3175.52 3133.14 3182.66 3160.37 3168.28 3191.43 3192.15 3135.78 3190.79 3176.42 3174.09 3194.34 3195.55 3160.61 3212.87 3197.43 3191.09 3202.02 3261.65 3178.27 * data for these columns not used since the system was assumed to be the same as that for 2MP 331 B3LYP/6-311++G(2d,p) parameters for species in Chapter 5 cC6H11 C 0.241693 0.711507 1.265886 C -0.266058 1.410662 0.000000 C 0.241693 0.711507 -1.265886 C -0.157063 -0.776818 -1.285625 C 0.163280 -1.461636 0.000000 C -0.157063 -0.776818 1.285625 H -0.144448 1.210970 -2.159333 H -1.363777 1.408023 0.000000 H 0.041955 2.460559 0.000000 H 1.333848 0.788653 1.306008 H -0.144448 1.210970 2.159333 H -1.244969 -0.834161 -1.474375 H 0.316267 -1.289253 -2.127933 H 0.401665 -2.519071 0.000000 H -1.244969 -0.834161 1.474375 H 0.316267 -1.289253 2.127933 H 1.333848 0.788653 -1.306008 Nuclear Repulsion Energy = 244.9082704979 hartrees 5-hexen-1-yl (pxc6h11) C -0.532932 0.326996 -0.090678 C 0.700036 -0.582411 -0.220193 H 0.780430 -1.224550 0.662926 H 0.552842 -1.247101 -1.080797 C 1.974905 0.187312 -0.400686 C 3.001028 0.181182 0.442679 H 3.892542 0.769234 0.259443 H 2.985983 -0.415907 1.349017 H 2.035347 0.804367 -1.295744 H -0.605622 0.978108 -0.968084 H -0.410432 0.985123 0.774330 C -1.848857 -0.476807 0.055107 C -3.057605 0.380524 0.172347 H -3.384960 0.755667 1.133752 H -3.556267 0.762233 -0.709857 H -1.758418 -1.122086 0.936231 H -1.941168 -1.139461 -0.813175 Nuclear Repulsion Energy = 219.0286186765 hartrees TS-1-2: (cC6H11-> 5-hexen-1-yl) C 1.485574 0.025431 -0.266593 C 0.991541 1.334804 0.275483 H 0.903905 1.436066 1.352249 H 1.278140 2.243154 -0.244936 C 0.729130 -1.217299 0.236663 C -0.749178 -1.261248 -0.222757 C -1.478288 -0.050505 0.274184 C -1.251103 1.168555 -0.276108 H -0.909269 1.259075 -1.301127 H -1.674013 2.066683 0.158351 H -1.957357 -0.112523 1.247313 H -0.773097 -1.297106 -1.317965 H -1.213855 -2.182561 0.138635 H 1.239237 -2.117662 -0.119653 H 0.767499 -1.247936 1.331483 H 1.452539 0.044042 -1.363309 H 2.550069 -0.099015 -0.011550 Nuclear Repulsion Energy = 238.1193889818 hartrees TS-1-3 (cC6H11-> cC6H10+H) C 1.301570 -0.770045 -0.107511 C 1.212788 0.680466 0.382443 C 0.053959 1.413351 -0.299990 C -1.289417 0.745947 0.024243 C -1.206717 -0.762314 -0.039354 C -0.045183 -1.426362 -0.191063 H -0.067129 -2.502994 -0.331723 H -2.139086 -1.304239 -0.156792 H -1.639152 1.056807 1.014313 H -2.056144 1.085176 -0.680269 H 0.031567 2.464469 -0.000557 H 0.212978 1.397983 -1.384095 H 1.053178 0.686285 1.466044 H 2.157813 1.198885 0.200250 H 1.781295 -0.806920 -1.095606 H 1.952515 -1.356571 0.551144 H -1.511146 -1.225129 1.881442 Nuclear Repulsion Energy = 244.7564952112 hartrees c5h9ch2 TS-2-6 (5-hexen-1-yl -> c5h9ch2)   332 C -0.678868 0.134736 -0.579497 C -0.709818 0.107344 0.974994 C 0.843254 0.158833 -0.851898 C 0.472739 -0.799109 1.388745 C 1.433365 -0.832844 0.165458 H -1.076850 -0.820104 -0.943369 H -1.670852 -0.233773 1.363852 H -0.557810 1.127135 1.341864 H 1.215141 1.171630 -0.663011 H 1.088150 -0.088775 -1.886585 H 0.123465 -1.805500 1.629217 H 0.966020 -0.417708 2.284713 H 2.460560 -0.580320 0.435108 H 1.459317 -1.837105 -0.264868 C -1.439513 1.248611 -1.193293 H -2.335717 1.082750 -1.775866 H -1.131703 2.274345 -1.021726 Nuclear Repulsion Energy = 242.8774119615 hartrees C 1.099415 0.033861 -0.480131 C -0.588607 1.443818 0.001919 C 0.216935 -1.201823 -0.382239 C -1.654069 0.393957 -0.027522 C -1.049107 -0.934965 0.445133 H 1.157890 0.508303 -1.454205 H -0.532764 2.175967 -0.795900 H -0.270082 1.809528 0.971651 H 0.784513 -2.028570 0.052178 H -0.074255 -1.514279 -1.389948 H -2.046493 0.277735 -1.042859 H -2.505320 0.667613 0.610452 H -0.780433 -0.849607 1.503659 H -1.763435 -1.759285 0.360944 C 2.131499 0.260193 0.386323 H 2.832990 1.070247 0.229713 H 2.234838 -0.309876 1.303564 Nuclear Repulsion Energy = 236.9111401082 hartrees cC6H10 C 1.493877 -0.111406 0.047122 C 0.696706 0.318222 -1.190887 C -0.696706 -0.318222 -1.190887 C -1.493877 0.111406 0.047122 C -0.663132 0.057792 1.302303 C 0.663132 -0.057792 1.302303 H -1.242107 -0.056186 -2.101623 H 0.590431 1.408852 -1.189242 H 1.242107 0.056186 -2.101623 H 1.881770 -1.129974 -0.088225 H 2.378710 0.523462 0.166138 H -1.881770 1.129974 -0.088225 H -2.378710 -0.523462 0.166138 H -1.192880 0.115306 2.249087 H 1.192880 -0.115306 2.249087 H -0.590431 -1.408852 -1.189242 Nuclear Repulsion Energy = 236.8098580505 hartrees TS-6-8 (ch2cC5H9 -> ch2cC5H8+H) C 0.795619 0.008980 -0.103749 C -0.054484 1.214014 0.194992 C -1.461380 0.758628 -0.224246 C -1.497604 -0.732625 0.164107 C -0.044927 -1.247652 -0.022947 H 0.265570 -1.880384 0.819954 H 0.053906 -1.869172 -0.919582 H -1.786642 -0.830004 1.214161 H -2.224358 -1.300891 -0.419221 H -1.574648 0.869523 -1.306788 H -2.258254 1.335000 0.250379 H 0.277599 2.121084 -0.317526 H -0.029450 1.436398 1.275499 H 1.610240 0.110522 -1.104914 C 2.271296 0.003434 0.024228 H 2.788959 0.917846 0.280171 H 2.796562 -0.938181 0.107633 Nuclear Repulsion Energy = 243.2245854809 hartrees ch2cC5H9 C 0.046270 -1.217244 0.145405 C -0.847762 0.009270 -0.016189 C 0.042418 1.250761 -0.076553 C 1.456071 0.738326 0.269760 C 1.454786 -0.719769 -0.216752 H 1.591468 -0.751480 -1.301718 H 2.246305 -1.321154 0.233852 H 1.607770 0.765080 1.353317 H 2.244924 1.340691 -0.183781 H -0.308363 2.029064 0.603809 H 0.030035 1.673206 -1.083924 cC6H9 C 1.058792 0.894402 -0.322362 C -0.286426 1.445977 0.171554 C -1.402661 0.473238 -0.070304 C -1.151882 -0.884424 -0.180999 C 0.119780 -1.418013 -0.054510 H 0.264762 -2.490811 -0.106482 C 1.307519 -0.533629 0.185701 H 2.198591 -0.955606 -0.290067 H 1.534457 -0.507647 1.262635 H -1.985964 -1.557067 -0.356414 H -2.418857 0.844503 -0.135239   333 C -2.179352 0.026055 0.184622 H -2.744786 -0.891816 0.297574 H -2.742068 0.952322 0.171741 H -0.286056 -2.067155 -0.450608 H 0.022930 -1.525829 1.196992 H -1.034885 -0.243826 -1.924306 Nuclear Repulsion Energy = 242.8774119615 hartrees H -0.502344 2.402817 -0.314055 H -0.219937 1.669654 1.247357 H 1.875647 1.554017 -0.018154 H 1.050199 0.880527 -1.417032 Nuclear Repulsion Energy = 227.4191598855 hartrees ts-1-2(cC6H9 -> pxc6h9) C 1.521236 -0.046611 0.194145 C 1.026070 1.207848 -0.457896 H 1.358194 2.166086 -0.077605 H 0.762844 1.175484 -1.507405 C 0.611666 -1.255608 0.116264 C -0.717454 -1.253354 -0.095307 C -1.519203 -0.052356 -0.196481 C -1.185101 1.099785 0.441885 H -1.724894 2.019134 0.247159 H -0.577322 1.089346 1.335654 H -2.340484 -0.050108 -0.908311 H -1.213089 -2.207855 -0.252171 H 1.114337 -2.215296 0.191833 H 2.476531 -0.335538 -0.270741 H 1.773410 0.152518 1.244020 Nuclear Repulsion Energy = 228.6458753554 hartrees cC6H8 C 1.198176 -0.733741 -0.224781 C 1.197981 0.733919 0.225080 C -0.106457 1.418078 -0.099740 C -1.247880 0.720695 -0.126198 C -1.247768 -0.721029 0.125835 C -0.106193 -1.418175 0.099736 H -0.110123 -2.492389 0.249896 H -2.193796 -1.217604 0.311860 H -2.193956 1.217083 -0.312465 H -0.110585 2.492307 -0.249798 H 2.038353 1.265259 -0.228086 H 1.354404 0.786852 1.313122 H 2.038568 -1.264922 0.228530 H 1.354821 -0.786591 -1.312795 Nuclear Repulsion Energy = 219.2915753589 hartrees pxc6h9 C 1.877132 -0.292194 0.487597 C 0.553633 0.365729 0.218728 C -0.622599 -0.267973 0.159562 H -0.652830 -1.344746 0.317285 C -1.893460 0.385512 -0.099311 C -3.071649 -0.242088 -0.147724 H -3.992067 0.293388 -0.342736 H -3.149363 -1.312838 0.008919 H -1.859175 1.460582 -0.260728 H 0.581477 1.441141 0.054179 C 2.873480 -0.100118 -0.606690 H 2.554789 -0.061595 -1.640299 H 3.935250 -0.106873 -0.397444 H 1.710146 -1.366380 0.669297 H 2.295971 0.097618 1.424920 Nuclear Repulsion Energy = 201.7686499665 hartrees ch2cC5H7 C 0.00000000 0.00000000 0.00000000 C -0.87776800 1.20287600 0.31593900 C -2.14261300 0.86591400 0.54818300 C -2.36507700 -0.62133400 0.43406000 C -0.92633300 -1.18591300 0.43976200 H -0.65523500 -1.47826100 1.45696700 H -0.79898500 -2.06194500 -0.19751500 H -2.97483500 -1.02614500 1.24667300 H -2.89317300 -0.85863900 -0.49793000 H -2.94257400 1.56828800 0.75143700 H -0.48967700 2.21428500 0.30356900 H 0.15252500 -0.04341600 -1.08653900 C 1.33140500 0.00100400 0.66205100 H 2.24120600 -0.21438400 0.11768000 H 1.40350500 0.15749000 1.73239200 Nuclear Repulsion Energy = 225.2191173547 hartrees ch2cC5H6 C 0.143615 -1.185436 -0.001403 C -0.840223 -0.019180 -0.000319 C -0.044537 1.205766 -0.000776 C 1.266815 0.942951 0.000103   334 C 1.553561 -0.535715 0.001638 H 2.146332 -0.827825 -0.871471 H 2.141667 -0.826345 0.878443 H 2.049692 1.691858 0.000213 H -0.485402 2.195102 -0.001374 C -2.173190 -0.101263 0.000916 H -2.791896 0.788457 0.001371 H -2.686172 -1.055918 0.001521 H -0.004981 -1.815763 -0.880541 H -0.007438 -1.820481 0.873853 Nuclear Repulsion Energy = 216.8905920030 hartrees ts-2-6 C 1.610251 -0.365376 -0.308418 C 1.080913 1.032948 -0.096438 C -0.187390 1.201111 0.260720 H -0.591157 2.198318 0.413133 C -1.115214 0.050282 0.442598 C -2.080000 -0.254530 -0.473210 H -2.850345 -0.983985 -0.255008 H -2.060067 0.165258 -1.472353 H -1.277999 -0.283238 1.464465 H 1.743876 1.879293 -0.249476 C 0.641855 -1.368931 0.243578 H 0.268515 -2.161712 -0.391211 H 0.722476 -1.621388 1.294575 H 1.790229 -0.530620 -1.377103 H 2.590387 -0.459548 0.178309 Nuclear Repulsion Energy = 218.2066957307 hartrees ts-6-8 C 1.546784 -0.608036 0.058068 C 1.333986 0.879269 -0.059149 C 0.038924 1.204933 -0.041510 H -0.361924 2.210277 -0.067412 C -0.811136 0.008636 0.034374 C -2.161863 -0.014046 -0.085387 H -2.711517 -0.946621 -0.115633 H -2.739121 0.902347 -0.078095 H -0.834652 -0.057548 1.928533 C 0.123632 -1.189471 -0.124067 H 0.008939 -1.583556 -1.137699 H -0.105482 -1.996474 0.570468 H 2.151443 1.587977 -0.108882 H 2.254570 -0.992042 -0.681517 H 1.960465 -0.854152 1.043266 Nuclear Repulsion Energy = 225.5280002993 hartrees c6h6 (Benzene) C 1.391778 0.000000 0.000000 C 0.695890 1.205439 0.000000 C -0.695890 1.205439 0.000000 C -1.391778 0.000000 0.000000 C -0.695890 -1.205439 0.000000 C 0.695890 -1.205439 0.000000 H 2.475596 0.000000 0.000000 H 1.237839 2.143995 0.000000 H -1.237839 2.143995 0.000000 H -2.475596 0.000000 0.000000 H -1.237839 -2.143995 0.000000 H 1.237839 -2.143995 0.000000 Nuclear Repulsion Energy = 203.8684480609 hartrees cC6H7 C 1.409284 -0.056659 0.001990 C 0.678282 1.250272 0.067237 C -0.680496 1.318804 0.043690 C -1.478667 0.152957 -0.044811 C -0.854867 -1.115627 -0.114413 C 0.499708 -1.243744 -0.096494 H 0.955234 -2.226092 -0.152343 H -1.473761 -2.003851 -0.184173 H -2.557825 0.231578 -0.061410 H -1.166399 2.287388 0.094357 H 1.269611 2.156405 0.137380 H 2.117095 -0.051558 -0.845245 H 2.070906 -0.162174 0.879578 Nuclear Repulsion Energy = 210.1310960156 hartrees nc6h7 C 1.788048 -0.432844 0.000087 ts-1-2 (cC6H7 -> nc6h7) C 1.499403 1.293804 0.041837   335 C 0.530585 0.283375 0.000050 C -0.677262 -0.309817 0.000017 H -0.737912 -1.394727 0.000016 C -1.937059 0.422830 -0.000017 C -3.134457 -0.123395 -0.000031 H -3.570590 -1.111393 -0.000028 H -1.857506 1.511017 -0.000031 H 0.584495 1.369974 0.000050 C 2.990290 0.153286 0.000125 H 3.092665 1.233091 0.000131 H 3.903980 -0.426974 0.000152 H 1.727661 -1.518636 0.000082 Nuclear Repulsion Energy = 186.1101988679 hartrees C -2.066268 0.856886 0.347525 C -1.483403 -0.039347 -0.401475 C -0.585386 -1.129714 0.062462 C 0.747588 -1.106196 0.173712 C 1.691011 -0.018894 -0.101756 H 2.671639 -0.354448 -0.432702 H 1.217550 -2.042632 0.464663 H -1.089857 -2.068074 0.280714 H -1.719059 -0.045944 -1.474987 H -2.759699 1.673930 0.214306 H 2.291783 1.993286 -0.196657 H 0.569980 1.704654 0.410822 Nuclear Repulsion Energy = 193.5494686568 hartrees ts-1-3 (cC6H7 -> c6h6+H) C 1.354234 -0.000180 -0.081710 C 0.642683 1.213501 -0.094042 C -0.739808 1.208467 -0.000356 C -1.434642 0.000172 0.057652 C -0.740109 -1.208282 -0.000310 C 0.642389 -1.213683 -0.094019 H 1.182796 -2.151185 -0.144453 H -1.284382 -2.145232 0.017281 H -2.515640 0.000282 0.131212 H -1.283879 2.145533 0.017247 H 1.183319 2.150873 -0.144393 H 2.416150 -0.000260 -0.293041 H 1.953151 0.000018 1.692864 Nuclear Repulsion Energy = 211.3218032762 hartrees mch(1,equitorial) C 1.885967 -0.003977 -0.307495 C 1.179998 1.240350 0.239525 C -0.307118 1.246947 -0.130277 C -1.039439 -0.018189 0.365551 C -0.276968 -1.250259 -0.001624 C 1.202171 -1.292405 0.185281 H 1.437518 -1.411827 1.258889 H 1.629516 -2.168518 -0.310427 H -0.819734 -2.173759 -0.174891 C -2.492786 -0.061114 -0.114499 H -2.539227 -0.106087 -1.206194 H -3.012773 -0.938286 0.279241 H -3.042129 0.826685 0.209352 H -1.057889 0.050254 1.470604 H -0.797193 2.135404 0.280783 H -0.408792 1.306604 -1.220670 H 1.284116 1.268933 1.331471 H 1.662940 2.145295 -0.141220 H 1.857275 0.019769 -1.402615 mch(axial) C 1.824097 -0.041608 0.094223 C 0.968561 1.194904 0.383484 C -0.247701 1.254514 -0.548228 C -1.133332 -0.010428 -0.469085 C -0.300341 -1.256214 -0.505108 C 0.998271 -1.332568 0.228568 H 0.812893 -1.509461 1.304030 H 1.577537 -2.196290 -0.110856 H -0.762499 -2.176672 -0.847385 H -1.794205 -0.013624 -1.343922 C -2.057388 0.016795 0.773998 H -2.699677 -0.866291 0.800962 H -1.483885 0.036561 1.703472 H -2.699609 0.902243 0.754845 H -0.853481 2.140070 -0.330759 H 0.109539 1.365989 -1.577939 H 0.642249 1.176771 1.429716 H 1.566805 2.103344 0.265186 H 2.220493 0.026902 -0.925092 336 H 2.940681 -0.006387 -0.017216 Nuclear Repulsion Energy = 316.1590684741 hartrees H 2.685208 -0.077926 0.767671 Nuclear Repulsion Energy = 318.7322102670 hartrees 3ch3cC6H10 (3) C 1.825779 -0.062035 0.045677 C 0.932524 1.158135 0.292648 C -0.227950 1.196320 -0.706147 C -1.122847 -0.054464 -0.601501 C -2.165283 0.053725 0.526251 H -2.837201 0.898592 0.352625 H -2.773738 -0.852284 0.587639 H -1.686849 0.192222 1.498886 C -0.287650 -1.303697 -0.455312 C 1.015717 -1.310021 -0.183399 H 1.540170 -2.259326 -0.116867 H -0.807121 -2.250266 -0.584039 H -1.683186 -0.145621 -1.540789 H -0.830598 2.097662 -0.559567 H 0.182539 1.254756 -1.720192 H 0.536611 1.112112 1.312339 H 1.519880 2.078019 0.227505 H 2.475293 0.114329 -0.822506 H 2.503729 -0.213805 0.892135 Nuclear Repulsion Energy = 309.5211923759 hartrees 2ch3cC6H10 (4) C 1.847676 0.010692 -0.346513 C 1.157547 1.211435 0.303949 C -0.325258 1.263567 -0.081434 C -1.008238 -0.082970 -0.004074 C -0.312583 -1.217486 0.090509 C 1.191269 -1.298762 0.103487 H 1.534307 -1.562497 1.113542 H 1.517778 -2.124008 -0.538691 H -0.852605 -2.158777 0.160818 C -2.511531 -0.062833 -0.049858 H -2.868782 0.435525 -0.958173 H -2.930221 -1.070120 -0.027953 H -2.923978 0.499217 0.795406 H -0.855334 1.973955 0.563644 H -0.435689 1.659100 -1.100708 H 1.244444 1.126599 1.393026 H 1.653170 2.143873 0.020074 H 1.767436 0.098481 -1.435992 H 2.914838 0.001621 -0.108290 Nuclear Repulsion Energy = 306.3974711262 hartrees pxch251c6h11 (6) C 0.615120 -0.242092 0.515879 C 1.794817 0.697981 0.496728 C 2.956645 0.525762 -0.124494 H 3.732759 1.280264 -0.074337 H 3.184008 -0.362886 -0.701335 H 1.643315 1.621142 1.054388 C -0.632998 0.499273 -0.018839 C -1.971166 -0.250244 0.202572 C -3.156156 0.541814 -0.221828 H -3.514040 1.365508 0.383821 H -3.607414 0.403488 -1.195617 H -2.046871 -0.492713 1.270673 H -1.948792 -1.199839 -0.337476 H -0.500635 0.700076 -1.087120 H -0.702016 1.475948 0.470535 H 0.414969 -0.469925 1.573445 C 0.869376 -1.565479 -0.208425 H 0.022367 -2.242938 -0.094669 H 1.029549 -1.406691 -1.278637 H 1.750443 -2.070506 0.192762 Nuclear Repulsion Energy = 294.2672590476 hartrees px7-2c7h13 (5,bad geometry, force calc resulted in an imag freq, therefore this structure was not used for CCSD(T)/cc-pVDZ calculations) C 0.006332 -0.757694 0.108602 C -1.339789 -0.182751 0.439413 C -2.384693 -0.146792 -0.382764 C -3.723560 0.444127 -0.058012 H -3.743452 0.857678 0.952405 H -4.517757 -0.305798 -0.134156 H -3.982031 1.246167 -0.757420 H -2.278068 -0.577231 -1.377632 H -1.444112 0.249181 1.434671 C 1.133188 0.287341 0.169045 C 2.524207 -0.316885 -0.142823 C 3.630399 0.674395 -0.081117 H 3.879040 1.286591 -0.938877 H 4.120640 0.904711 0.856534 H 2.482865 -0.771006 -1.139429 H 2.710327 -1.131417 0.566809 H 1.155397 0.746526 1.162714 H 0.919321 1.093456 -0.539115 H -0.021660 -1.210577 -0.887806 H 0.244421 -1.565953 0.812241 Nuclear Repulsion Energy = 279.4204271724 hartrees 337 pxch231c6h11 (9) C 0.753569 -0.467981 0.431990 C 2.078478 0.277957 0.377177 C 2.314339 1.453017 -0.197994 H 3.308945 1.883322 -0.195478 H 1.538036 2.030899 -0.684764 H 2.908018 -0.249896 0.841807 C -0.449803 0.358036 -0.049152 C -1.798100 -0.318731 0.211295 C -2.983217 0.493366 -0.311892 H -3.021606 1.483851 0.150628 H -3.932118 -0.006602 -0.103474 H -2.915972 0.637903 -1.393980 H -1.914953 -0.483301 1.288664 H -1.802319 -1.312084 -0.248915 H -0.345162 0.556118 -1.122497 H -0.435543 1.333717 0.447636 H 0.586864 -0.729637 1.485974 C 0.916873 -1.740441 -0.343434 H 0.836189 -1.720021 -1.424064 H 1.316744 -2.634522 0.117493 Nuclear Repulsion Energy = 294.7353966106 hartrees sxc2h4cC5H8 (7) C 0.317107 0.081392 -0.181780 C -0.744321 1.176279 0.059633 C -2.111528 0.490922 -0.165614 C -1.848774 -1.037769 -0.041454 C -0.384534 -1.171837 0.413560 H -0.314909 -1.128168 1.505364 H 0.080621 -2.106314 0.093388 H -2.538919 -1.523910 0.650778 H -1.983405 -1.523293 -1.011430 H -2.509336 0.731585 -1.153591 H -2.851074 0.835056 0.560115 H -0.593515 2.047569 -0.580711 H -0.657779 1.523736 1.094481 H 0.411461 -0.083452 -1.264274 C 1.662002 0.381345 0.375715 H 1.709972 0.795515 1.379034 C 2.917163 -0.108773 -0.257204 H 2.916023 0.063674 -1.339818 H 3.801715 0.376223 0.161354 H 3.055653 -1.194239 -0.122865 Nuclear Repulsion Energy = 308.8050128953 hartrees ch3pxch212cC5H8 (8) C 0.019582 -0.847674 0.307329 C -1.473442 -0.741164 -0.078308 C -1.818833 0.765188 0.002758 C -0.457746 1.510286 -0.007932 C 0.609737 0.448813 -0.330663 H 0.628953 0.288339 -1.416383 C 2.016638 0.804759 0.134897 H 2.044839 0.945030 1.220257 H 2.731289 0.018196 -0.120036 H 2.365748 1.732126 -0.327669 H -0.437494 2.339051 -0.719398 H -0.251955 1.937229 0.978597 H -2.375793 0.990952 0.914562 H -2.454401 1.067686 -0.831748 H -2.108749 -1.360621 0.557347 H -1.595515 -1.106545 -1.102714 H 0.111337 -0.735333 1.395480 C 0.669957 -2.110844 -0.112361 H 1.350104 -2.651251 0.532205 H 0.520952 -2.488917 -1.117905 Nuclear Repulsion Energy = 316.5533901767 hartrees px1-2c7h13 (10) C -0.141865 -0.740128 0.069057 C -1.464447 -0.124163 0.387178 C -2.570091 -0.154378 -0.444001 C -3.798520 0.410868 -0.165073 H -4.620100 0.345300 -0.865542 H -3.975651 0.939146 0.764611 H -2.459812 -0.669994 -1.396748 H -1.554155 0.387221 1.343681 C 1.030308 0.257550 0.126676 C 2.385911 -0.386207 -0.170551 C 3.555461 0.593393 -0.066691 H 3.440726 1.425939 -0.766628 H 4.506339 0.103181 -0.288938 H 3.627627 1.016649 0.939050 H 2.364652 -0.825030 -1.174396 H 2.548399 -1.220960 0.521027 H 1.057299 0.720560 1.119919 H 0.844619 1.070625 -0.583708 H -0.179677 -1.206680 -0.920897 H 0.067769 -1.549936 0.783139 Nuclear Repulsion Energy = 279.9989140702 hartrees ts12 (axial) C 1.518479 -0.133041 -0.796424 C 1.331238 1.137302 0.036433 C -0.153046 1.369231 0.353630 C -0.860598 0.089685 0.776485 ts13 C 1.850307 -0.059999 -0.363780 C 1.207593 1.190039 0.245896 C -0.273466 1.285000 -0.131011 C -1.068658 0.071466 0.374078 338 C -0.215332 -1.113925 0.753679 C 1.052109 -1.364369 -0.009015 H 1.838191 -1.675336 0.694240 H 0.919683 -2.220113 -0.683255 H -0.674553 -1.967134 1.244376 C -2.473619 -0.094924 -0.799300 H -1.881391 -0.139525 -1.704422 H -2.981779 -1.007944 -0.516565 H -3.013985 0.828193 -0.624635 H -1.680198 0.204381 1.476300 H -0.248055 2.103860 1.159951 H -0.645672 1.816999 -0.514402 H 1.895319 1.037684 0.970507 H 1.740010 2.006435 -0.486285 H 0.936651 -0.043364 -1.720605 H 2.563349 -0.251511 -1.095293 Nuclear Repulsion Energy = 310.4133745226 hartrees C -0.326311 -1.211519 0.106952 C 0.982297 -1.278563 -0.183562 H 1.424264 -2.243141 -0.412510 H -0.894485 -2.133875 0.195725 C -2.486154 0.031928 -0.211988 H -2.457756 -0.072141 -1.300353 H -3.058385 -0.808228 0.189958 H -3.031581 0.949657 0.023321 H -1.173074 0.165254 1.465909 H -0.713312 2.204139 0.267636 H -0.367410 1.341508 -1.222497 H 1.299420 1.141990 1.336366 H 1.743976 2.086042 -0.077634 H 2.024376 0.090721 -1.437609 H 2.835314 -0.237818 0.076608 H 1.581767 -1.720593 1.824382 Nuclear Repulsion Energy = 316.2433663531 hartrees ts14 C 1.871893 0.009406 -0.355991 C 1.177132 1.219131 0.273907 C -0.301859 1.266659 -0.120629 C -0.998225 -0.070639 0.017597 C -0.293227 -1.222207 0.043355 C 1.205312 -1.294544 0.096737 H 1.512466 -1.533169 1.125287 H 1.552917 -2.134521 -0.513797 H -0.836415 -2.162270 0.090645 C -2.495105 -0.060467 -0.161014 H -2.750386 0.273339 -1.172954 H -2.924808 -1.051850 -0.011407 H -2.974164 0.631159 0.536042 H -1.034117 -0.030884 2.026415 H -0.831112 2.016343 0.475170 H -0.400472 1.591737 -1.165633 H 1.251968 1.149995 1.364399 H 1.673753 2.147159 -0.021662 H 1.809064 0.086414 -1.447416 H 2.935150 -0.002574 -0.101877 Nuclear Repulsion Energy = 316.1687046508 hartrees ts15 C 1.859390 -0.311845 -0.303406 C 1.469918 1.078350 0.229383 C 0.114851 1.556407 -0.201608 C -1.226400 -0.200156 0.389055 C -0.484732 -1.176847 -0.203918 C 0.930222 -1.437187 0.211986 H 1.006622 -1.478768 1.304613 H 1.282119 -2.400124 -0.168851 H -0.808812 -1.571643 -1.164110 C -2.613206 0.173707 -0.060594 H -2.776647 -0.093252 -1.106634 H -3.368273 -0.343783 0.540535 H -2.794884 1.245213 0.048399 H -0.975960 0.077619 1.410036 H -0.343425 2.360202 0.366826 H -0.093863 1.581623 -1.266744 H 1.527551 1.070171 1.325018 H 2.238686 1.797197 -0.095933 H 1.837678 -0.300470 -1.399029 H 2.891124 -0.534583 -0.013140 Nuclear Repulsion Energy = 307.3362623182 hartrees ts16 C -1.278448 -1.215580 0.254346 C -2.106604 -0.114067 -0.337935 H -2.091501 0.008274 -1.415813 H -3.052128 0.128249 0.136369 C 0.200067 -1.229664 -0.175410 C 1.015247 -0.005701 0.317800 C 0.378172 1.243161 -0.218671 H 0.734948 1.607927 -1.179469 C -0.800391 1.704555 0.267171 ts68 C 1.277087 -0.896328 -0.227805 C 1.011935 0.403492 0.471552 H 1.074151 0.355698 1.556992 C 1.485709 1.687935 -0.130903 H 2.583407 1.750700 -0.119954 H 1.167138 1.781370 -1.172721 H 1.106519 2.555402 0.414003 C 0.129635 -1.868725 0.068307 C -1.177172 -1.137181 -0.255090 339 H -1.114383 1.474622 1.279157 H -1.311569 2.535512 -0.204563 H 0.941212 0.007450 1.413047 C 2.491510 -0.130965 -0.070245 H 2.931270 -1.045774 0.334861 H 2.606591 -0.160601 -1.157967 H 3.070436 0.716941 0.303370 H 0.673521 -2.140165 0.206701 H 0.262527 -1.281810 -1.269271 H -1.333159 -1.171427 1.348984 H -1.715079 -2.189625 -0.017749 Nuclear Repulsion Energy = 309.2056305515 hartrees C -1.204390 0.241802 0.368873 C -1.724870 1.326624 -0.285665 H -1.918602 2.259949 0.228432 H -1.878648 1.315503 -1.359175 H -1.240399 0.259232 1.455127 H -2.039183 -1.715598 0.094231 H -1.283348 -1.038405 -1.340820 H 0.148722 -2.143067 1.129293 H 0.223242 -2.797156 -0.501637 H 1.343715 -0.726936 -1.308600 H 2.240400 -1.325653 0.081723 Nuclear Repulsion Energy = 309.2669618657 hartrees ts611 C 0.905261 -0.451550 0.265805 C 0.945020 0.942821 -0.300244 C 0.976685 2.064109 0.409085 H 1.022014 3.036329 -0.067396 H 0.958409 2.052337 1.494474 H 0.962888 1.003542 -1.387766 C -0.251101 -1.248878 -0.274717 C -2.222929 -0.319093 0.439887 C -2.763034 0.478944 -0.515988 H -2.484372 1.521622 -0.605492 H -3.433231 0.082788 -1.269920 H -1.678937 0.115606 1.269036 H -2.616858 -1.315166 0.599801 H -0.415675 -2.226376 0.169167 H -0.428750 -1.206807 -1.344194 H 0.810250 -0.378977 1.355903 C 2.243024 -1.181085 -0.026836 H 2.231959 -2.190190 0.391445 H 2.413697 -1.264481 -1.103107 H 3.081315 -0.632547 0.410058 Nuclear Repulsion Energy = 288.0618190171 hartrees ch3 (methyl radical) C -0.000022 -0.000003 0.000000 H -0.391530 -1.006154 0.000000 H 1.067197 0.163975 0.000000 H -0.675532 0.842200 0.000000 Nuclear Repulsion Energy = 9.6712154831 hartrees 340 Vibrational Frequencies used in the calculations for Table 5.5 ts12 90.63 146.36 176.25 231.01 288.29 378.9 455.87 490.5 523.81 567.8 686.61 717.61 799.75 825.72 852.62 879.82 905.38 922.65 975.41 1021.65 1051.95 1065.9 1094.1 1157.14 1168.45 1229.74 1269.89 1295.06 1355.88 1367.47 1368.34 1382.89 1416.69 1427.26 1430.31 1468.34 1489.51 1498.43 1508.79 1553.69 2971.68 2996.13 3011.04 3020.2 3023.89 3043.82 3057.98 3062.62 3078.81 3134.85 3155.97 3227.94 3237.3 ts14 132.54 160.07 231.1 306.88 317.94 380.57 435.1 453.25 474.3 519.64 620.16 762.72 803.7 831.66 862.31 895.69 932.31 975.71 1010.07 1041.98 1058.96 1101.07 1102.34 1162.45 1170.31 1199.07 1267.56 1297.98 1341.97 1369.43 1374.18 1383.34 1396.77 1407.7 1469.39 1473.44 1476.83 1490 1494.45 1502.97 1648.56 2975.47 2986.25 3010.73 3013.19 3019.4 3033.02 3043.54 3057.4 3062.39 3065.53 3101.6 3132.39 341 Appendix F: Sample RRKM/Master Equation Input and Source Code Sample RRKM/Master Equation input file for the cyclohexyl system studied in Chapter 5. Here, cyclohexyl is assigned as the reactant. This calculation is for T = 1500 K, P = 7.6 torr: cyclohexyl , thermal activation ! Title 2 ! seed number .97 ! multiplication factor for vibrational frequencies 250.0 25.0 260.0 ! Max. energy (kcal/mol), Energy spacing (/cm) and - ! <Edown>(/cm)=298(T/298)^.45, built in code (value of 260 will be neglected) 1 1 ! No. of pressures and no of temperatures 7.6 ! Pressures(torr) 1500 ! Temperatures (K) 20000 ! No. of molecules 39.95 3.405 120 ! MW, LJ collision diameter, Well depth of the bath gas, Ar 3 2 4 ! No. of stable isomers, No. of products, No. of TS 1 ! Index of initial isomer 1 ! Index of reactant cC6H11 ! Name of reactant 0.0 ! Heat of formation of isomer at 0K 2 ! No. of channels associated with reactant 4 5 ! Index of TSs associated with reactant 2 3 ! Index of other isomer associated with TS, enter '0' if it is a product 2 ! Index of isomer pxc6h11 ! Name of isomer 21.6 ! Heat of formation of isomer at 0K 2 ! No. of channels associated with isomer 4 7 ! Index of TS associated with isomer 1 6 ! Index of other isomer associated with TS 6 ! Index of isomer c5h9ch2 ! Name of isomer 5.6 ! Heat of formation of isomer at 0K 2 ! No. of channels associated with isomer 7 9 ! Index of TS associated with isomer 2 8 ! Index of other isomer associated with TS 3 ! Index of product cC6H10+H ! Name of product 8 ! Index of TS c5h8ch2+H ! Name of product 4 ! Index of TS TS-1-2 ! Name of TS 30.9 ! Heat of formation of fake TS at 0K 1 2 ! Indices of reactant and product associated with TS 2 1 ! Reaction path degeneracy 5 ! Index of TS TS-1-3 ! Name of TS 34.8 ! Heat of formation of fake TS at 0K 1 3 ! Indices of reactant and product associated with TS 2 2 ! Reaction path degeneracy 7 ! Index of TS 342 TS-2-6 ! Name of TS 28.8 ! Heat of formation of fake TS at 0K 2 6 ! Indices of reactant and product associated with TS 1 1 ! Reaction path degeneracy 9 ! Index of TS TS-6-8 ! Name of TS 41.1 ! Heat of formation of fake TS at 0K 6 8 ! Indices of reactant and product associated with TS 1 1 ! Reaction path degeneracy Sample RRKM/Master Equation input data file for the cyclohexyl system, cyclohexyl isomer cC6H11 ! name g2mp2 energy from knepp, geo from sir ! comments 83.15 5.29 464.8 ! Molecular wt., LJ parameters 0.1477 1 2 ! External inactive rotational constant,Symmetry number,Dimension 1 ! Number of active rotors 0.0826 1 1 ! External active rotational constant,Symmetry number,Dimension 0 ! No. internal hindered rotors 45 ! Number of vibrational unscaled frequencies followed by frequencies 184.63 216.55 324.77 398.05 442.21 460.03 611.51 785.96 803.57 856.76 870.95 876.31 928.33 1016.94 1032.47 1054.81 1097.39 1114.06 1128.68 1151.67 1255.97 1283.08 1295.69 1342.19 1349.18 1354.75 1377.27 1391.61 1395.88 1469.58 1475.31 1489.58 1492.94 343 1502.71 2895.92 2900.68 2993.7 3011.97 3012.06 3045.12 3047.75 3049.1 3050.59 3054.38 3164.23 RRKM/Master Equation source code RRKM/Master Equation code for unimolecular thermal isomerization reactions: C*********************************************************************** C Altered to specifically address multi-channel unimolecular C thermal activation reactions. Further altered to take channel C specific Eckart tunneling into account through the simple C approximation that k_new(E,T)=F_eckart(T)*k(E). Edown is also C now a function of temperature. C C Enoch Dames 2011 C*********************************************************************** C Program to determine rate constants for multi-channel reaction C networks using a Monte Carlo approach. C C Xiaoqing You, Ameya Joshi, Hai Wang (2007) C C*********************************************************************** PROGRAM MASTEREQUATION C BOLTZ is in units (/cm/K) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION PRES(50),TEMP(50),ID(20),E00(20),Emin(20) DIMENSION WT(20),SIG(20),EPS(20),WTAVE(20),SIGMA(20),EPSLON(20) DIMENSION DELHF0(20),NCHAN(30),IDTS(20,10),TSINT(20,20) DIMENSION ROTX(20),SYMX(20),DIMX(20), EQ(50),VIMAG(20) DIMENSION ATOT(20),HIGHP(20),ISID(20,10) DIMENSION NACROT(20),ROTACT(20,20),SYMACT(20,20) DIMENSION DIMACT(20,20),NVIB(20),VIBX(20,100) DIMENSION REAC(20),PROD(20),IDEG(20,20),BARRIER(20),MGRAINS(20) DIMENSION QRSP(20),QVSP(20),QTOT(20) DIMENSION AKE(20) DIMENSION HIGH(20,50),ICOUNTER(20),ENTRK(50),Z(20) 344 DIMENSION IFIT(20),FIT(20,15) DIMENSION NAHROT(20),ROTAH(20,20),SYMAH(20,20),VH(20,20) DIMENSION ROTH(20),SYMH(20),V(20) ALLOCATABLE P(:), PKE(:,:), DENS(:), GINIT(:) ALLOCATABLE PACT(:), PDEACT(:), PE(:) ALLOCATABLE CIJ(:), PIJACT(:,:), PIJDEACT(:,:) ALLOCATABLE P2D(:,:), EINIT(:) C ALLOCATABLE PROB(:,:,:) REAL*8 KT,NTRIALS CHARACTER TITLE*50,NAME(20)*15 CHARACTER ANAME*15, CSTART, CSEED INTEGER ISEED(3) DATA AVGD/6.0222E+23/,BOLTZ/0.695045/,PLANCK/3.3362E-11/, 1 PI/3.1415926/,SMALL/1.0D-78/,SPEED/2.9979E+10/ COMMON NARRAY, NTOL COMMON /DENSITY/ RO(0:120000,20) COMMON /REACT/ VIB(100),ROT(20),SYM(20),DIM(20),T,NV,NROT COMMON /RANDOM/ISEED DATA ISEED(1),ISEED(2),ISEED(3)/1,10000,3000/ OPEN (UNIT=15, FILE='mc.inp') C WRITE(*,*) 'ENTER SEED (1-9)' C READ(15,*) ISEED(1) CALL TIME(time_start) READ(15,500) TITLE READ(15,123) ISEED(1) WRITE(*,*) 'SEED NUMBER IS', ISEED(1) IF(ISEED(1) .EQ. 1) CSEED = '1' IF(ISEED(1) .EQ. 2) CSEED = '2' IF(ISEED(1) .EQ. 3) CSEED = '3' IF(ISEED(1) .EQ. 4) CSEED = '4' IF(ISEED(1) .EQ. 5) CSEED = '5' IF(ISEED(1) .EQ. 6) CSEED = '6' IF(ISEED(1) .EQ. 7) CSEED = '7' IF(ISEED(1) .EQ. 8) CSEED = '8' IF(ISEED(1) .EQ. 9) CSEED = '9' C OPEN (UNIT=15, FILE='mc.inp') OPEN (UNIT=16, FILE=CSEED//'.out') OPEN (UNIT=17, FILE='energy.out') OPEN (UNIT=18, FILE='mcrc.out') OPEN (UNIT=36, FILE='temp.out') OPEN (UNIT=19, FILE='time.out') C OPEN (UNIT=37, FILE='P3mp.out') C OPEN (UNIT=38, FILE='Pbenzyl.out') c OPEN (UNIT=39, FILE='Preactant.out') C EMAX = Maximum energy (input in kcal/mol) 345 C DELTAE = Energy spacing between successive grains (/cm) C EDOWN = -<Edown>(/cm) C NPRES = Number of pressures C NTEMP = Number of temperatures C PRES = Pressures (torr) C TEMP = Temperatures (K) C NGRAINS = Total no. of grains C NTRIALS = No. of stochastic runs, in the input files, this may be referred to as 'No. molecules' READ(15,*) FACTOR READ(15,*) EMAX,DELTAE,EDOWN READ(15,*) NPRES,NTEMP WRITE(16,520) WRITE(16,530) TITLE WRITE(16,540) EMAX,DELTAE IF(NPRES.GT.50) THEN WRITE(*,*) ' Limit number of pressures is 50 -- abort' STOP ENDIF IF(NTEMP.GT.50) THEN WRITE(*,*) ' Limit number of temperature is 50 - abort' STOP ENDIF READ(15,*) (PRES(I),I=1,NPRES) READ(15,*) (TEMP(I),I=1,NTEMP) EMAX=EMAX*3.4964E2 ! /cm units NGRAINS=NINT(EMAX/DELTAE) PTOL=1.0E-3 IF(NGRAINS.GT.120000) THEN WRITE(*,*) ' Maximum number of energy spacing exceeded' WRITE(*,*) ' Reduce EMAX or increase DELT' STOP ENDIF READ(15,*) NTRIALS C WTBATH = Molecular weight C SBATH = Lennard-Jones collision diameter C EBATH = Well depth of the bath gas READ(15,*) WTBATH,SBATH,EBATH C NISOMERS = No. of stable isomers, excluding products C NPRODUCTS = No. of products in reaction network C NTS = No. of Transition states C NSPECIES = Total no. of species READ(15,*) NISOMERS, NPRODUCTS, NTS 346 NSPECIES = NISOMERS + NPRODUCTS IF(NSPECIES.GT.20) THEN WRITE(*,*) ' Limit number of species is 20 -- abort' STOP ENDIF WRITE(16,550) NTRIALS WRITE(16,560) WTBATH,SBATH,EBATH NMOLEC=NTRIALS NARRAY=NGRAINS+1 NTOL = NISOMERS+NPRODUCTS+NTS C Section for allocating working arrays: C P: C PKE: C DENS: C allocate (P(0:NARRAY), PKE(NTOL,0:NARRAY), DENS(0:NARRAY)) allocate (GINIT(0:NARRAY), CIJ(0:NARRAY), EINIT(NMOLEC)) allocate (PACT(0:NARRAY), PDEACT(0:NARRAY)) allocate (PIJACT(NTOL,0:NARRAY),PIJDEACT(NTOL,0:NARRAY)) allocate (P2D(0:NARRAY,0:NARRAY)) C allocate (PROB(0:NARRAY,0:NARRAY,NISOMERS)) C Read in data for reactants, isomers and products C C ISINIT = ID of initial isomer C TSINIT = ID of initial transition state which means...? C ID = Index of species C NAME = Name of species C NCHAN = No. of channels associated with isomer C IDTS, TEMPID = Indices of TS associated with isomer C ISID = indeces of isomers or products associated with IDTS C TSINT = Internal index associated with isomer-channel pair C INCR = Integer serving as above index C WT = Molecular weight C SIG = Lennard-Jones collision diameter of species C EPS = Well depth of the species C DELHF0 = Heat of formation at 0K C C I = 1 -- Reactant isomer C = 1, NISOMERS -- Isomers and reactant isomer C = NISOMERS, NSPECIES -- Products C = NSPECIES, NSPECIES+NTS -- Transition states C INCR = 1 READ(15,*) ISINIT ! Data for reactant isomer C READ(15,*) ID(1) ! Data for reactant C READ(15,510) NAME(ID(1)) C READ(15,*) NCHAN(ID(1)) C READ(15,*) (IDTS(ID(1),J),J=1,NCHAN(ID(1))) 347 WRITE(16,565) DO 10 I = 1, NISOMERS ! Data for isomers READ(15,*) ID(I) READ(15,510) ANAME READ(15,*) DELHF0(ID(I)) READ(15,*) NCHAN(ID(I)) READ(15,*) (IDTS(ID(I),J), J = 1, NCHAN(ID(I))) DO J = 1, NCHAN(ID(I)) TEMPID = IDTS(ID(I),J) ! set array for TS(s) for each isomer TSINT(ID(I),TEMPID) = INCR INCR = INCR + 1 END DO READ(15,*) (ISID(ID(I),J), J = 1, NCHAN(ID(I))) C ICH = Integer locating first non-blank character in string C LCH = Integer locating last non-blank character in string C ANAME = array of isomers names ICH = IFIRCH(ANAME) LCH = ILASCH(ANAME) C Beginning of isomer card OPEN (UNIT=20, FILE=ANAME(ICH:LCH)//'.dat') READ(20,*) NAME(ID(I)) READ(20,*) READ(20,*) WT(ID(I)),SIG(ID(I)),EPS(ID(I)) IFIT(ID(I))=0 WTAVE(ID(I)) = WT(ID(I))*WTBATH/(WT(ID(I))+WTBATH) SIGMA(ID(I)) = (SIG(ID(I))+SBATH)/2.0 EPSLON(ID(I))= SQRT(EPS(ID(I))*EBATH) C ROTX = External inactive rotational constant C SYMX = Symmetry number C DIMX = Dimension of external rotor (2 or 3) READ(20,*) ROTX(ID(I)),SYMX(ID(I)),DIMX(ID(I)) C NACT, NACROT = Number of active rotors READ(20,*) NACT NACROT(ID(I)) = NACT C ROTACT = Rotational constant of active rotor C SYMACT = Symmetry number of active rotor C DIMACT = Dimension of active rotor DO J=1,NACT READ(20,*) ROTACT(ID(I),J),SYMACT(ID(I),J),DIMACT(ID(I),J) END DO 348 C NAHROT = No of hindered rotors READ(20,*) NHROT NAHROT(ID(I))=NHROT C ROTH = Rotational constant of hindered rotor C SYMH = Symmetry number of hindered rotor C VH = Barrier of hindered rotor DO J=1,NHROT READ(20,*) ROTAH(ID(I),J),SYMAH(ID(I),J),VH(ID(I),J) VH(ID(I),J)=VH(ID(I),J)*3.4964E2 END DO C NV, NVIB = Number of vibrational frequencies C VIBX = Vibrational frequencies READ(20,*) NV NVIB(ID(I)) = NV READ(20,*) (VIBX(ID(I),J),J=1,NV) C C Scale vibrational frequencies due to anharmonicity C DO J = 1, NV VIBX(ID(I),J) = VIBX(ID(I),J) * FACTOR END DO CLOSE(20) C End of isomer card C*********************************************************** C C Begin computing density of states for each isomer C C*********************************************************** DO J=1,NV VIB(J)=VIBX(ID(I),J) END DO DO J=1,NACT ROT(J)=ROTACT(ID(I),J) SYM(J)=SYMACT(ID(I),J) DIM(J)=DIMACT(ID(I),J) END DO DO J=1,NHROT ROTH(J)=ROTAH(ID(I),J) SYMH(J)=SYMAH(ID(I),J) V(J)=VH(ID(I),J) END DO 349 C INDEX = 1 --- Returns density of states C INDEX = 2 --- Returns sum of states INDEX = 1 c IF(INDEX.EQ.1) THEN c WRITE(*,*) 'Computing density of states of ', ANAME c ELSE c WRITE(*,*) 'Computing sum of states of ', ANAME c ENDIF CALL BSCOUNT(VIB,NV,ROT,NACT,SYM,DIM,EMAX,DELTAE,P,NGRAINS 1 ,NHROT,ROTH,SYMH,V,INDEX) OPEN (UNIT=2, FILE=ANAME(ICH:LCH)//'.DEN.txt') WRITE(2,*) ' No. (cm-1) Density' DO J=0,NGRAINS RO(J,ID(I))=P(J) WRITE(2,*) J, J*DELTAE, P(J) END DO CLOSE(2) WRITE(16,570) NAME(ID(I)) WRITE(16,580) DELHF0(ID(I)) WRITE(16,590) WT(ID(I)),SIG(ID(I)),EPS(ID(I)), 1 WTAVE(ID(I)),SIGMA(ID(I)),EPSLON(ID(I)) WRITE(16,600) ROTX(ID(I)) WRITE(16,610) SYMX(ID(I)) WRITE(16,620) NINT(DIMX(ID(I))) WRITE(16,630) NACROT(ID(I)) WRITE(16,640) (ROTACT(ID(I),J),J=1,NACT) WRITE(16,650) (SYMACT(ID(I),J),J=1,NACT) WRITE(16,660) (DIMACT(ID(I),J),J=1,NACT) WRITE(16,665) NHROT WRITE(16,666) (ROTAH(ID(I),J),J=1,NHROT) WRITE(16,667) (SYMAH(ID(I),J),J=1,NHROT) WRITE(16,668) (VH(ID(I),J)/3.4964E2,J=1,NHROT) WRITE(16,670) WRITE(16,680) (VIBX(ID(I),J),J=1,NVIB(ID(I))) DELHF0(ID(I)) = DELHF0(ID(I))*3.4964E2 C*********************************************************** C C End computing density of states for each isomer C C*********************************************************** 10 CONTINUE DO I = (NISOMERS+1), NSPECIES ! Data for products READ(15,*) ID(I) READ(15,510) NAME(ID(I)) write(*,*) 'product ID and name are', ID(I),NAME(ID(I)) END DO 350 WRITE(16,685) DO 70 I = NSPECIES+1, NSPECIES+NTS ! Data for Transition States READ(15,*) ID(I) READ(15,510) ANAME READ(15,*) DELHF0(ID(I)) C REAC = Reactant ID for TS C PROD = Product ID for TS READ(15,*) REAC(ID(I)), PROD(ID(I)) C IDEG = Reaction path degeneracy READ(15,*) IDEG(REAC(ID(I)),ID(I)), IDEG(PROD(ID(I)),ID(I)) ICH = IFIRCH(ANAME) LCH = ILASCH(ANAME) C Beginning of TS card OPEN (UNIT=20, FILE=ANAME(ICH:LCH)//'.dat') READ(20,510) NAME(ID(I)) READ(20,*) READ(20,*) VIMAG(ID(I)) !read in imaginary freq for tunneling calc, READ(20,*) ROTX(ID(I)),SYMX(ID(I)),DIMX(ID(I)) IF (ROTX(ID(I)) .LT. 0.0) THEN IFIT(ID(I))=1 READ(20,*) (FIT(ID(I),NFIT),NFIT=1,10) ENDIF READ(20,*) NACT NACROT(ID(I)) = NACT DO J=1,NACT READ(20,*) ROTACT(ID(I),J),SYMACT(ID(I),J),DIMACT(ID(I),J) END DO C NAHROT = No of hindered rotors READ(20,*) NHROT NAHROT(ID(I))=NHROT C ROTH = Rotational constant of hindered rotor C SYMH = Symmetry number of hindered rotor C VH = Barrier of hindered rotor DO J=1,NHROT READ(20,*) ROTAH(ID(I),J),SYMAH(ID(I),J),VH(ID(I),J) VH(ID(I),J)=VH(ID(I),J)*3.4964E2 END DO 351 READ(20,*) NV NVIB(ID(I)) = NV READ(20,*) (VIBX(ID(I),J),J=1,NV) C C Scale vibrational frequencies C DO J=1,NV VIBX(ID(I),J)=VIBX(ID(I),J) * FACTOR END DO CLOSE(20) C*********************************************************** C C Begin computing sum of states for TS C C*********************************************************** DO J=1,NV VIB(J)=VIBX(ID(I),J) END DO DO J=1,NACT ROT(J)=ROTACT(ID(I),J) SYM(J)=SYMACT(ID(I),J) DIM(J)=DIMACT(ID(I),J) END DO DO J=1,NHROT ROTH(J)=ROTAH(ID(I),J) SYMH(J)=SYMAH(ID(I),J) V(J)=VH(ID(I),J) END DO c VIMG=VIMAG(ID(I)) C INDEX = 2 --- Returns sum of states C INDEX = 1 --- Returns density of states INDEX = 2 c IF(INDEX.EQ.1) THEN c WRITE(*,*) 'Computing density of states of ', ANAME c ELSE c WRITE(*,*) 'Computing sum of states of ', ANAME c ENDIF CALL BSCOUNT(VIB,NV,ROT,NACT,SYM,DIM,EMAX,DELTAE,P,NGRAINS 1 ,NHROT,ROTH,SYMH,V,INDEX) OPEN (UNIT=2, FILE=ANAME(ICH:LCH)//'.SUM.txt') WRITE(2,*) ' No. (cm-1) Sum of states' DO J=0,NGRAINS RO(J,ID(I))=P(J) WRITE(2,*) J, J*DELTAE, P(J) END DO 352 CLOSE(2) C*********************************************************** C C End computing sum of states for each TS C C*********************************************************** WRITE(16,570) NAME(ID(I)) WRITE(16,580) DELHF0(ID(I)) WRITE(16,600) ROTX(ID(I)) WRITE(16,610) SYMX(ID(I)) WRITE(16,620) NINT(DIMX(ID(I))) WRITE(16,630) NACROT(ID(I)) WRITE(16,640) (ROTACT(ID(I),J),J=1,NACT) WRITE(16,650) (SYMACT(ID(I),J),J=1,NACT) WRITE(16,660) (DIMACT(ID(I),J),J=1,NACT) WRITE(16,665) NHROT WRITE(16,666) (ROTAH(ID(I),J),J=1,NHROT) WRITE(16,667) (SYMAH(ID(I),J),J=1,NHROT) WRITE(16,668) (VH(ID(I),J)/3.4964E2,J=1,NHROT) WRITE(16,670) WRITE(16,680) (VIBX(ID(I),J),J=1,NVIB(ID(I))) WRITE(16,690) IDEG(REAC(ID(I)),ID(I)), IDEG(PROD(ID(I)),ID(I)) DELHF0(ID(I)) = DELHF0(ID(I))*3.4964E2 70 CONTINUE C End of TS card C Scan for first well C TSINIT = TS connecting reactants and isomer C ISINIT = First isomer formed from reactants C M1 = Index of first channel C MGR1 = Grain no. at critical energy of first channel (BUT SINCE NO FIRST CHAN., SET TO ZERO, ed) 162 CONTINUE C M1 = TSINT(ISINIT,TSINIT) C write(*,*) 'the first channel is ', M1 ISINK = 0 WRITE(16,735) NTRIALS DO 120 IT=1,NTEMP T = TEMP(IT) KT = BOLTZ*T BETA = 1/KT EDOWN = 250.0*(T/298.0)**0.450 C EQ(IT) = AEQ*T**BEQ*EXP(-EEQ*1E3/1.9872/T) 353 WRITE(*,*) 'TEMP = ', T WRITE(*,*) 'EDOWN= ', EDOWN WRITE(16,700) T C********************************************************** C C Compute total partition function for each isomer C C********************************************************** C IDIS= Temporary storage of ID of isomer WRITE(16,705) DO 130 I=1,NSPECIES+NTS IF (I .GE. NISOMERS+1 .AND. I .LE. NSPECIES) GOTO 130 IDIS = ID(I) IF(IFIT(ID(I)) .EQ. 1) THEN ROTX(ID(I))=0.0 RX=0.0 DO NFIT=1,10 RX = FIT(ID(I),NFIT)*T**(NFIT-1) ROTX(ID(I))=ROTX(ID(I))+RX END DO IF(ROTX(ID(I)) .LE. 0) ROTX(ID(I))=1.0E-6 ENDIF CALL QTOTAL(IDIS,T,NACROT,ROTX,SYMX,DIMX,ROTACT,SYMACT, 1 DIMACT,NAHROT,ROTAH,SYMAH,VH,NVIB,VIBX,QTOT) WRITE(16,710) NAME(ID(I)),QTOT(ID(I)) ! write(*,*), 'Qtotal is ', QTOT 130 CONTINUE C***************************************************************** C C Begin computing collision probability matrix for each isomer C C***************************************************************** DO 201 I = 1,NISOMERS DO J=0,NGRAINS DO K=0,NGRAINS P2D(J,K)=0.0 END DO END DO ANAME = NAME(ID(I)) WRITE(*,*) 'Computing collision probability matrix for ',ANAME 354 DO J=0,NGRAINS DENS(J)=RO(J,ID(I)) END DO CALL NORM(NGRAINS,DELTAE,DENS,EDOWN,BETA,CIJ) DO J = 0, NGRAINS DO K = 0, NGRAINS IF(J .GE. K) THEN P2D(K,J) = CIJ(J)*EXP(-DELTAE*(J-K)/EDOWN) ELSE P2D(K,J) = CIJ(K)*DENS(K)/DENS(J)*EXP(-(BETA+1/EDOWN)* 1 (DELTAE*(K-J))) ENDIF END DO END DO IF (NISOMERS .NE. 1) THEN ICH = IFIRCH(ANAME) LCH = ILASCH(ANAME) OPEN (UNIT=35,FORM='UNFORMATTED',FILE=ANAME(ICH:LCH)//'.prob') WRITE(*,*)'Writing collision probability matrix to scratch file' DO 214 J=0,NGRAINS ! write Pjj diag WRITE(35) P2D(J,J) C PROB(J,J,ID(I))=P2D(J,J) ! write upper diag within tolerance (no loss of accuracy, A.J thesis) DO K1=(J+1),NGRAINS WRITE(35) P2D(K1,J) C PROB(K1,J,ID(I))=P2D(K1,J) IF( P2D(K1,J)/P2D(J,J) .LT. PTOL) GOTO 211 END DO ! write lower diag within tolerance (no loss of accuracy, A.J thesis) 211 DO K2=(J-1),0,-1 WRITE(35) P2D(K2,J) C PROB(K2,J,ID(I))=P2D(K2,J) IF( P2D(K2,J)/P2D(J,J) .LT. PTOL) GOTO 214 END DO 214 CONTINUE CLOSE(35) END IF C Scan the probability matrix and define total probability of activation/deactivaton C (Pact, Pdeact) for each energy grain: DO J = 0, NGRAINS PACT(J) = 0 DO K = J+1, NGRAINS 355 PACT(J) = PACT(J) + P2D(K,J) IF (PACT(J).GT.1D+00) THEN PACT(J) = 1D+00 ENDIF END DO END DO DO J = 0, NGRAINS PDEACT(J) = 0 DO K = 0, J PDEACT(J) = PDEACT(J) + P2D(K,J) IF (PDEACT(J).LT.0) THEN PDEACT(J) = 0 ENDIF END DO END DO C Save the energy-grained total Pact and Pdeact for each isomer: DO J=0,NGRAINS PIJACT(ID(I),J)=PACT(J) PIJDEACT(ID(I),J)=PDEACT(J) END DO 201 CONTINUE C***************************************************************** C C End computing collision probability matrix for each isomer C C***************************************************************** C****************************************************************** C C Begin loop for microcanonical and high pressure rate constants C for each channel C C****************************************************************** C IDIS = Temporary storage of index of isomer C TEMPID = Temporary storage of index of TS associated with isomer C TEMPIS = Temporary storage of index of other isomer associated with TS C needed for calculating the Eckart tunneling coefficient C M = Temporary storage of internal index of isomer-channel pair C MGRAINS = Grain no. at critical energy WRITE(*,*) 'Computing microcanonical rate constants' WRITE(16,720) WRITE(16,725) DO 140 I=1,(NISOMERS) DO 150 J=1,NCHAN(ID(I)) IDIS = ID(I) 356 TEMPID = IDTS(ID(I),J) TEMPIS = ISID(ID(I),J) DEGEN = IDEG(IDIS,TEMPID) M = TSINT(IDIS,TEMPID) E0 = DELHF0(TEMPID)-DELHF0(IDIS) BARRIER(M)=E0 E00(J)= DELHF0(TEMPID)-DELHF0(IDIS) IF (J .GE. 2) THEN If (E00(J) .LE. Emin(I)) THEN Emin(I)=E00(J) END IF ELSE Emin(I)=E00(1) END IF V1=EO V2=DELHF0(TEMPID)-DELHF0(TEMPIS) VIMG=VIMAG(TEMPID) write(*,*) 'VIMG IS ', VIMG C Call MCRC subroutine to calculate MGRAINS, PKE for all channels CALL MCRC(T,IDIS,TEMPID,TEMPIS,M,DELTAE,V2,VIMG,ROTX, 1 DIMX,SYMX,DEGEN,NGRAINS,BARRIER,MGRAINS,PKE,TUN) C Calculate the high pressure rate constant for all channels HIGH(M,IT) = DEGEN*KT/PLANCK*QTOT(TEMPID) 1 /QTOT(IDIS)*EXP(-BARRIER(M)/KT)*TUN MGR1 = 0 ! set to zero because this code is for mutual isomerization reactions only M1 = 1 ! set to one since the reactant and first isomer are the same WRITE(*,*) TEMPID,DELHF0(TEMPID), DELHF0(IDIS),IDIS, E0,M WRITE(16,730) NAME(IDIS),NAME(TEMPID),HIGH(M,IT) WRITE(18,*) NAME(IDIS),NAME(TEMPID) DO K=0,NGRAINS WRITE(18,*) K, PKE(M,K) ! write to mcrc.out the mcrc as a func. of grain no. K, for every chan. M END DO 150 CONTINUE 140 CONTINUE C****************************************************************** C C End of loop for microcanonical and high pressure rate constants C for each channel C C****************************************************************** C Calculate the initial distribution of reactant for C chemically or thermally activated system as a function of energy C GINIT = Array containing above distribution CALL INDIST (T,M1,ISINIT,PKE,MGR1,NGRAINS,DELTAE,GINIT) C Begin the stochastic run C T = array of temperatures C P5 = Pressure in torr 357 C C5 = Concentration in mol/cm^3 C COUNTER = No. of molecules of particular species at end of simulation DO 180 IP = 1,NPRES P5=PRES(IP) C5=P5/760.0*1.013E-1/8.314/T WRITE(*,*) 'Stochastic run begun for pressure ', P5, ' torr' WRITE(16,740) P5 DO I = 1,NSPECIES ICOUNTER(ID(I)) = 0 END DO C Calculate the collision frequency for each isomer with the bath gas: DO I = 1, NISOMERS Z(ID(I))=ZLJ(T,WTAVE(ID(I)),SIGMA(ID(I)),EPSLON(ID(I)))*C5 ! write(*,*) Z(ID(I)) END DO C Initialize timer to track evolution of system: TIMER2=0 TIMER=0 TAU=0 C Start the kinetic Monte Carlo sampling for chosen number of trials: DO 190 NMOL=1,NTRIALS C******************************************************* C Determine initial energy of chosen molecule C******************************************************* C EINIT = Initial energy of chosen molecule !a random sampling of Boltzmann distribution, ed. C TIMER = Time of stochastic run for each molecule C TIMER2 = Time evolution of system over all trials FRAC2 = ICOUNTER(ID(2)) !trace appearance of isomer with index 2 WRITE(19,*) TIMER2,FRAC2 TIMER2=TIMER2+TIMER !TAU TIMER=0.0 CALL RAND(U) R1=U DO I =0,NGRAINS IF(R1 .LE. GINIT(I)) THEN EINIT(NMOL)=DELTAE*(I)-(GINIT(I)-R1) 1 /(GINIT(I)-GINIT(I-1))*DELTAE WRITE(17,*) NAME(ISINIT), ' ', NMOL, EINIT(NMOL) ! write to energy.out for initial isomer GOTO 200 ENDIF END DO C************************************************** C Initial energy of chosen molecule determined C************************************************** 358 C*************************************************************** C C Begin loop to determine C C (a) Time before molecule changes state C (b) Next process (activation/deactivation/isomerization) C (c) Energy at end of process if activation/deactivation C C*************************************************************** C IDSPECIES = ID of isomer C E = Energy of molecule C INEW = 0 -- molecule is same isomer as in previous time step C = 1 -- molecule has undergone isomerization in previous time step C Z = Collision frequency at P and T C ATOTAL = Sum of microcanonical rate constants and C collision frequency for isomer at energy E (/sec) C TAU = Time before molecule changes state C ISINK = the energy grain for which the total probability of C activation is equal to that for deactivation 200 IDSPECIES = ISINIT ! start with another reactant, ISINIT IDOLD = ISINIT E = EINIT(NMOL) INEW=1 210 CONTINUE IF (INEW .EQ. 1) THEN E = E + DELHF0(IDOLD) - DELHF0(IDSPECIES) IDOLD = IDSPECIES NUM = NMOL/100 IF ((NUM*100) .EQ. NMOL) WRITE (*,*) NMOL IF (NISOMERS .NE. 1) THEN ! for a mutual isom. system, NISOMERS always greater than 1 ANAME = NAME(IDSPECIES) ICH = IFIRCH(ANAME) LCH = ILASCH(ANAME) OPEN (UNIT=35,FORM='UNFORMATTED',FILE=ANAME(ICH:LCH)//'.prob') ! open and read Pij for chosen isomer DO 215 J=0,NGRAINS READ (35) P2D(J,J) DO K1=(J+1),NGRAINS READ(35) P2D(K1,J) IF( P2D(K1,J)/P2D(J,J) .LT. PTOL) GOTO 212 END DO 212 DO K2=(J-1),0,-1 359 READ(35) P2D(K2,J) IF( P2D(K2,J)/P2D(J,J) .LT. PTOL) GOTO 215 END DO 215 CONTINUE ! done reading in Pij for chosen isomer CLOSE(35) END IF ! Find where in the energy grains total Pdeact = Pact, mark by ISINK DO ITER = 0,NGRAINS IF (PIJACT(IDSPECIES,ITER) .GT. PIJDEACT(IDSPECIES,ITER) 1 .AND. PIJACT(IDSPECIES,ITER+1) .LT. PIJDEACT(IDSPECIES, 1 ITER+1)) THEN ISINK = ITER+1 ENDIF END DO ENDIF CALL RAND(U) R2 = U IF ( R2 .EQ. 0.0 ) R2 = SMALL ATOTAL = 0.0 DO 220 J=1,NCHAN(IDSPECIES) TEMPID = IDTS(IDSPECIES,J) M = TSINT(IDSPECIES,TEMPID) C Interpolation to find k(E) from array PKE for each channel assc. w/ isomer C AKE = Microcanonical rate constant at energy E, channel M C IGRAIN = Grain in which E lies DO IM = MGRAINS(M),NGRAINS IF(E .LT. DELTAE*(MGRAINS(M)-1)) THEN AKE(M) = 0.0 GOTO 230 ENDIF IF (E .LE. (DELTAE*IM)) THEN AKE(M) = PKE(M,IM-1)+(PKE(M,IM)-PKE(M,IM-1))* 1 (E-DELTAE*(IM-1))/DELTAE WRITE(36,*) AKE(M) GOTO 230 ENDIF END DO !Sum the rate constants for every channel associated w/ a particular isomer: 230 ATOT(J)=ATOTAL+AKE(M) ATOTAL = ATOT(J) 220 CONTINUE ATOTAL = ATOTAL + Z(IDSPECIES) TAU = -LOG(R2)/ATOTAL TIMER = TIMER + TAU 360 CALL RAND(U) R3 = U C Check if process is activation/deactivation/isomerization C ICHECK = 0 -- activation/deactivation C ICHECK = 1 -- isomerization ICHECK = 0 DO J=1,NCHAN(IDSPECIES) ATOTP=R3*ATOTAL IF(ATOT(J) .GE. (R3*ATOTAL)) THEN TEMPID = IDTS(IDSPECIES,J) ! TS and respective channel selected IF(IDSPECIES .EQ. REAC(TEMPID)) THEN IDSPECIES = PROD(TEMPID) ICHECK = 1 INEW = 1 GOTO 240 ELSE IDSPECIES = REAC(TEMPID) ICHECK = 1 INEW = 1 GOTO 240 ENDIF ENDIF END DO C Check if species formed on isomerization is reactant / product C Scan the isomers to see which one was chosen and then mark in counter: 240 DO 250 IPROD = (NISOMERS+1), NSPECIES IF(IDSPECIES .EQ. ID(IPROD)) THEN ICOUNTER(ID(IPROD))=ICOUNTER(ID(IPROD)) + 1 GOTO 190 ENDIF 250 CONTINUE C Process is activating/deactivating collision, call step to find new energy IF(ICHECK .EQ. 0) THEN CALL STEP(IDSPECIES,DELTAE,E,NGRAINS, 1 P2D,PIJACT,PIJDEACT,ENEW) E = ENEW INEW=0 IF ( E .LE. DELTAE*ISINK ) THEN EISINK = DELTAE*ISINK ICOUNTER(IDSPECIES)=ICOUNTER(IDSPECIES) + 1 361 GOTO 190 ENDIF ENDIF GOTO 210 C*********************************************************** C Stochastic run complete for single molecule C*********************************************************** 190 CONTINUE WRITE(16,750) DO I=1,NSPECIES FRAC = ICOUNTER(ID(I))/NTRIALS RES = FRAC WRITE(16,760) NAME(ID(I)),ICOUNTER(ID(I)),FRAC,RES END DO C*********************************************************** C Calculation complete for single pressure C*********************************************************** 180 CONTINUE C*********************************************************** C Calculation complete for single temperature C*********************************************************** 120 CONTINUE CALL TIME(time_end) WRITE(16,770) time_start WRITE(16,780) time_end C Format statements 500 FORMAT(A50) 123 FORMAT(I1) 510 FORMAT(A15) 520 FORMAT(10X,'Master equation modeling using Monte Carlo approach' 1/15X,'Xiaoqing You, Ameya Joshi, Hai Wang (2007)',//80('*')) 530 FORMAT(//1X,A50) 540 FORMAT(//,80('-')//,'Maximum energy counted: ',1PE10.3,0P, 1 'kcal/mol'/'Energy spacing:',9X,1PE10.3,0P,'cm-1') 550 FORMAT('Number of molecules:',E10.2,//70('-')) 560 FORMAT(//,'Bath gas data',//45X,'Lennard-Jones',/15X, 1'Molecular Weight',4X,'Coll. Dia. (A)',4X,'Well Depth (K)'// 2T12,3(5X,F8.2,7X),//70('-')) 565 FORMAT(//,'Data for isomers') 570 FORMAT(//,A15) 580 FORMAT(//,'Relative Energy (kcal/mol)',10X,F6.2) 590 FORMAT(//,45X,'Lennard-Jones',/15X,'Molecular Weight',4X, 362 1'Coll. Dia. (A)',4X,'Well Depth (K)'//,'Isomer' 2T12,3(5X,F8.2,7X),/70('.')/'Average',T12,3(5X,F8.2,7X)) 600 FORMAT(//,'External rotational constant (/cm)',T41,2X,F6.2,2X) 610 FORMAT('Symmetry number',T41,4X,F6.2,5X) 620 FORMAT('Dimension of the external rotor',T41,4X,I1,5X) 630 FORMAT(/'No. of active rotors:',10X,I1) 640 FORMAT( 'Rotational constant (/cm)',T41,8(F8.4,2X)) 650 FORMAT( 'Symmetry number',T41,8(2X,F6.4,2X)) 660 FORMAT( 'Dimension',T41,8(2X,F6.4,2X)) 665 FORMAT(/'No. of hindered rotors:',10X,I1) 666 FORMAT( 'Rotational constant (/cm)',T41,8(F8.4,2X)) 667 FORMAT( 'Symmetry number',T41,8(2X,F6.4,2X)) 668 FORMAT( 'Barrier to rotation (kcal/mol)',T41,8(F8.4,2X)) 670 FORMAT(/'Vibrational frequencies (/cm):') 680 FORMAT(T41,F6.0) 685 FORMAT(//80('-'),//,'Data for Transition States') 690 FORMAT(/'Reaction path degeneracies:',T44,I2,2X,I2,2X/) 700 FORMAT(/,80('-'),//,25('*'),/2X,'Temperature =',F7.1,' K'/25('*')) 705 FORMAT(//,2X,'Total Partition Functions',/) 710 FORMAT(2X,'Partition function for',1X,A15,':',1PE10.3,0P) 720 FORMAT(//2X,'High pressure limit rate constants') 725 FORMAT(/,2X,'ISOMER ',5X,'TRANSITION STATE',5X,'Kinf') 730 FORMAT(2X,A15,5X,A15,5X,E10.3) 735 FORMAT(/,80('-'),//,'Number of molecules =',1PE10.3) 736 FORMAT(/,'k-High pressure for entrance channel',1PE10.3) 740 FORMAT(/1X,'Pressure = ',1PE10.3) 750 FORMAT(/,5X,'Channel',8X,'No. of molecules',5X,'Fraction',9X,'k'/) 760 FORMAT(A20,I8,15X,F5.3,5X,1PE10.3) 770 FORMAT(/,5X,'Time at start of program:',5X,A10) 780 FORMAT(/,5X,'Time at end of program:',7X,A10) STOP END FUNCTION IFIRCH (STRING) C BEGIN PROLOGUE IFIRCH C DATE WRITTEN 850626 C REVISION DATE 850626 C CATEGORY NO. M4. C KEYWORDS CHARACTER STRINGS,SIGNIFICANT CHARACTERS C AUTHOR CLARK,G.L.,GROUP C-3 LOS ALAMOS NAT'L LAB C PURPOSE Determines first significant (non-blank) character C in character variable C DESCRIPTION C C----------------------------------------------------------------------- C IFIRCH locates the first non-blank character in a string of C arbitrary length. If no characters are found, IFIRCH is set = 0. C When used with the companion routine ILASCH, the length of a string C can be determined, and/or a concatenated substring containing the C significant characters produced. C----------------------------------------------------------------------- C C REFERENCES (NONE) 363 C ROUTINES CALLED (NONE) C END PROLOGUE IFIRCH IMPLICIT DOUBLE PRECISION (A-H,O-Z), INTEGER (I-N) C*****END precision > double C*****precision > single C IMPLICIT REAL (A-H,O-Z), INTEGER (I-N) C*****END precision > single C CHARACTER* (*)STRING C C FIRST EXECUTABLE STATEMENT IFIRCH NLOOP = LEN(STRING) C IF (NLOOP.EQ.0 .OR. STRING.EQ.' ') THEN IFIRCH = 0 RETURN ENDIF C DO I = 1, NLOOP IF (STRING(I:I) .NE. ' ') GO TO 120 END DO C IFIRCH = 0 RETURN 120 CONTINUE IFIRCH = I END FUNCTION ILASCH (STRING) C BEGIN PROLOGUE ILASCH C DATE WRITTEN 850626 C REVISION DATE 850626 C CATEGORY NO. M4. C KEYWORDS CHARACTER STRINGS,SIGNIFICANT CHARACTERS C AUTHOR CLARK,G.L.,GROUP C-3 LOS ALAMOS NAT'L LAB C PURPOSE Determines last significant (non-blank) character C in character variable C DESCRIPTION C C----------------------------------------------------------------------- C IFIRCH locates the last non-blank character in a string of C arbitrary length. If no characters are found, ILASCH is set = 0. C When used with the companion routine IFIRCH, the length of a string C can be determined, and/or a concatenated substring containing the C significant characters produced. C Note that the FORTRAN intrinsic function LEN returns the length C of a character string as declared, rather than as filled. The C declared length includes leading and trailing blanks, and thus is C not useful in generating 'significant' substrings. C----------------------------------------------------------------------- C C REFERENCES (NONE) C ROUTINES CALLED (NONE) 364 C END PROLOGUE IFIRCH C*****precision > double IMPLICIT DOUBLE PRECISION (A-H,O-Z), INTEGER (I-N) C*****END precision > double C*****precision > single C IMPLICIT REAL (A-H,O-Z), INTEGER (I-N) C*****END precision > single C CHARACTER*(*) STRING C C FIRST EXECUTABLE STATEMENT ILASCH NLOOP = LEN(STRING) IF (NLOOP.EQ.0 .OR. STRING.EQ.' ') THEN ILASCH = 0 RETURN ENDIF C DO I = NLOOP, 1, -1 ILASCH = I IF (STRING(I:I) .NE. ' ') RETURN END DO C END C Subroutine to calculate the initial distribution of reactant for C chemically or thermally activated system as a function of energy C SUBROUTINE INDIST (T,M,ISINIT,PKE,MGR,NGRAINS,DELTAE,GINIT) C M = Index of first channel ! for mutual iso. there is no 'first chan', just reactant isomer, ed C MGR = Grain no. at critical energy of first channel ! set to 1, see above, ed IMPLICIT REAL*8 (A-H,O-Z) REAL*8 KT DATA AVGD/6.0222E+23/,BOLTZ/0.695045/,PLANCK/3.3362E-11/, 1 PI/3.1415926/,SMALL/1.0D-78/,SPEED/2.9979E+10/ COMMON NARRAY, NTOL COMMON /DENSITY/ RO(0:120000,20) DIMENSION GINIT(0:NARRAY), PKE(NTOL,0:NARRAY), TOTAL(0:NARRAY) KT = BOLTZ*T G = 0.0 DO I=0,NGRAINS GINIT(I) = 0.0 TOTAL(I) = 0.0 END DO 365 C Integration of Rho(E)*exp(-E/kT)dE (multiplied by k(E) if C.A. rxn) DO ITER=MGR,NGRAINS G = RO(ITER,ISINIT)*EXP(-DELTAE*FLOAT(ITER)/KT) IF((ITER-MGR) .EQ. 0) THEN TOTAL(ITER)=TOTAL(ITER)+G/2.0 ! TOTAL(ITER)=TOTAL(ITER)+G*PKE(M,ITER)/2.0 !turn on for C.A. rxn ELSE TOTAL(ITER)=TOTAL(ITER-1)+G ! TOTAL(ITER)=TOTAL(ITER-1)+G*PKE(M,ITER) !turn on for C.A. rxn ENDIF END DO C End of integration DO ITER=MGR,NGRAINS GINIT(ITER) = TOTAL(ITER)/TOTAL(NGRAINS) END DO RETURN END C Subroutine to find the normalization constants associated with C the energy transfer probability matrix. The exponential down C model is used. C Based on the algorithm described in "Unimolecular Reactions" by C Holbrook, K.A., Pilling, M.J. and Robertson S.H. C C NGRAINS = No. of energy grains C DELTAE = Energy spacing between successive grains (/cm) C DENS(NGRAINS) = Array containing density of states in each grains C EDOWN = Average energy transferred per collision <Edown> C BETA = 1/kT C CIJ(NGRAINS) = Array containing normalization constants C SUBROUTINE NORM(NGRAINS, DELTAE, DENS, EDOWN, BETA, CIJ) IMPLICIT REAL*8 (A-H, O-Z) COMMON NARRAY, NTOL DIMENSION DENS(0:NARRAY),CIJ(0:NARRAY) F = -(BETA+1/EDOWN)*DELTAE ! THRES=INT(500/DELTAE) ! DO J = 0,THRES ! CIJ(J) = 0.01 ! END DO DO 10 J = NGRAINS, 0, -1 ! DO 10 J = NGRAINS, 0, THRES+1 366 SUM = 1.0 TOT = 0.0 IF (J .NE. NGRAINS) THEN DO I = NGRAINS, J+1, -1 ! DO I = NGRAINS, J+1, THRES+1 SUM = SUM - CIJ(I) * DENS(I)/DENS(J) * EXP(F * (I-J)) END DO ENDIF DO K = 0, J TOT = TOT + EXP(DELTAE*(K-J)/EDOWN) END DO CIJ(J) = SUM/TOT 10 CONTINUE RETURN END C Function QR computes rotational partition function. C C B = rotational constants in cm-1. C N = number of rotors. C SIGMA = symmetry numbers. C D = dimension of the rotor C T = temperature, K. C DOUBLE PRECISION FUNCTION QR(B,N,SIGMA,D,T) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION B(N),SIGMA(N),D(N) DATA BOLTZ/0.695045/,PI/3.1415926/ C BOLTZ here is in unit of (/cm/K) QR=1.0 C DO 10 I=1,N ID=NINT(D(I)) GOTO (20,30,40), ID C C one-dimensional rotor C 20 QR=QR*SQRT(PI*BOLTZ*T/B(I))/SIGMA(I) GOTO 10 C C two-dimensional rotor 367 C 30 QR=QR*(BOLTZ*T/SIGMA(I)/B(I)) GOTO 10 C C Three dimensional rotor C 40 QR=QR*SQRT(PI)*(SQRT(BOLTZ*T/B(I)))**3/SIGMA(I) 10 CONTINUE RETURN END C Function QT computes the translational partition function C per unit volume (1/cm^3) C C WT = molecular weight C T = temperature, K C DOUBLE PRECISION FUNCTION QT(WT,T) IMPLICIT REAL*8 (A-H,O-Z) DATA AVGD/6.0222E+23/,BOLTZ/1.3806E-16/,PLANCK/6.6262E-27/, 1 PI/3.1415926/ WMASS=WT/AVGD QT=(2.0*PI*WMASS*BOLTZ*T)**(1.5)/PLANCK**3 RETURN END C C Subroutine to find the total partition function (including both C active and inactive rotations) of the desired species. C SUBROUTINE QTOTAL(IDIS,T,NACROT,ROTX,SYMX,DIMX,ROTACT,SYMACT, 1 DIMACT,NAHROT,ROTAH,SYMAH,VH,NVIB,VIBX,QTOT) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION ROT(20),SYM(20),D(20),VIB(100) DIMENSION ROTX(20),SYMX(20),DIMX(20) DIMENSION NACROT(20),ROTACT(20,20),SYMACT(20,20) DIMENSION DIMACT(20,20),NVIB(20),VIBX(20,100) DIMENSION QRSP(20),QVSP(20),QTOT(20),VH(20,20) DIMENSION NAHROT(20),ROTAH(20,20),SYMAH(20,20) DATA BOLTZ/0.695045/,PI/3.1415926/ C BOLTZ here is in unit of (/cm/K) C*********************************** C C Rotational partition function C 368 C*********************************** QRH = 1.0 NROT=1+NACROT(IDIS) ROT(NROT)=ROTX(IDIS) SYM(NROT)=SYMX(IDIS) D(NROT)=DIMX(IDIS) DO J=1,NACROT(IDIS) ROT(J)=ROTACT(IDIS,J) SYM(J)=SYMACT(IDIS,J) D(J)=DIMACT(IDIS,J) END DO QRSP(IDIS)=QR(ROT,NROT,SYM,D,T) IF(NAHROT(IDIS) .GT. 0) THEN DO J=1,NAHROT(IDIS) ROTH=ROTAH(IDIS,J) SYMH=SYMAH(IDIS,J) V=VH(IDIS,J) ARG=V/2/BOLTZ/T QRH=QRH*QR(ROTH,1,SYMH,1.0D+0,T)*EXP(-ARG)*BESSEL(ARG) END DO END IF QRSP(IDIS)=QRSP(IDIS)*QRH C************************************* C C Vibrational partition function C C************************************* NV=NVIB(IDIS) DO J=1,NV VIB(J)=VIBX(IDIS,J) END DO QVSP(IDIS)=QV(VIB,NV,T) QTOT(IDIS)=QVSP(IDIS)*QRSP(IDIS) RETURN END C C Function to calculate the modified Bessel function of the first C kind of order zero I0(x) C DOUBLE PRECISION FUNCTION BESSEL(R) IMPLICIT REAL*8 (A-H,O-Z) DATA PI/3.1415926/ DE=1.D-7 369 N=NINT(1.0D0/DE) BESSEL=0.0D0 DO I=0,N-1 T=DE*I BESSEL=BESSEL+DE*COSH(R*T)/SQRT(1.0D0-T**2) END DO BESSEL=BESSEL*2.0D0/PI RETURN END C Function QV computes the vibrational partitional function C C VIB = vibrational frequencies, in /cm C NVIB = number of vibrational frequencies C DOUBLE PRECISION FUNCTION QV(VIB,NVIB,T) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION VIB(NVIB) DATA BOLTZ/0.695045/ QV=1.0 DO I=1,NVIB QV=QV/(1.0-EXP(-VIB(I)/BOLTZ/T)) END DO RETURN END C======================================================================= C Portable pseudo-random integer generator, especially for C microcomputers with a limitation of 16 bit integers. Translated from C original Pascal version(1) to Fortran 77 by H. D. Knoble, PSU. C C The supporting paper is: C (1) B. Wichmann & D. Hill, "Building a Random-Number Generator", C BYTE, March, 1987, 12(3):127-128. C C Also see the following related works: C (2) Wichmann, B.A. and Hill, I.D., "An Efficient and Portable", C Pseudo-random Number Generator", Applied Statistics, C Volume 31, 1982, pages 188-190. C (3) Haas, Alexander, "The Multiple Prime Random Number Generator", C ACM Transactions on Mathematical Software; December, C 1987, 13(4):368-381. C (4) L'Ecuyer, Pierre, "Efficient and Portable Combined Random Number C Generators", Communications of the ACM; June, 1988, C 31(6):742-749,774. C C Use... C CALL RAND(U,N) C To generate a sequence, U, of N Uniform(0,1) numbers. 370 C Cycle length is ((30269-1)*(30307-1)*(30323-1))/4 or C 6953607871644 > 6.95E+12. C C To access the SEED vector in the calling program use statements: C INTEGER SEED(3) C COMMON/RANDOM/SEED C C The common variable SEED is the array of three current seeds. c INTEGER SEED(3) c COMMON/RANDOM/SEED c DATA SEED(1),SEED(2),SEED(3)/1,10000,3000/ c END C======================================================================= SUBROUTINE RAND(U) INTEGER X,Y,Z REAL*8 U,V COMMON /RANDOM/X,Y,Z X=171*MOD(X,177)-2*(X/177) IF(X.LT.0) X=X+30269 Y=172*MOD(Y,176)-35*(Y/176) IF(Y.LT.0) Y=Y+30307 Z=170*MOD(Z,178)-63*(Z/178) IF(Z.LT.0) Z=Z+30323 V=X/30269.0 + Y/30308.0 + Z/30323.0 U=V-INT(V) RETURN END SUBROUTINE MCRC(T,IDIS,TEMPID,TEMPIS,M,DELTAE,V2,VIMG,ROTX, 1 DIMX,SYMX,DEGEN,NGRAINS,BARRIER,MGRAINS,PKE,TUN) C IDIS = Index of isomer C TEMPID = Index of TS associated with isomer C TEMPIS = Index of other isomer or product assoc with tempid C M = Temporary storage of internal index of isomer-channel pair C ROTX = External inactive rotational constant C SYMX = Symmetry number C DIMX = Dimension of external rotor (2 or 3) C MGRAINS = No. of grains in energy barrier C PKE = Microcanonical rate constants IMPLICIT REAL*8 (A-H,O-Z) COMMON NARRAY, NTOL DIMENSION ROTX(20),SYMX(20),DIMX(20),PE(NTOL) DIMENSION BARRIER(20),MGRAINS(20),PKE(NTOL,0:NARRAY) COMMON /DENSITY/ RO(0:120000,20) 371 DATA AVGD/6.0222E+23/,BOLTZ/0.695045/,PLANCK/3.3362E-11/, 1 PI/3.1415926/,SMALL/1.0D-78/,SPEED/2.9979E+10/ V1=BARRIER(M) WRITE(*,*) 'V1,V2,VIMG', V1,V2,VIMG CALL ECKART(T,V1,V2,VIMG,M,PE) TUN=PE(M) WRITE(*,*) 'tunnel factor is', PE(M) E0 = BARRIER(M) MGRAINS(M) = NINT(E0/DELTAE) ROTIN=ROTX(IDIS) DIMIN=DIMX(IDIS) SYMIN=SYMX(IDIS) ROTTIN=ROTX(TEMPID) DIMTIN=DIMX(TEMPID) SYMTIN=SYMX(TEMPID) QRR=QR(ROTIN,1,SYMIN,DIMIN,T) QRT=QR(ROTTIN,1,SYMTIN,DIMTIN,T) OTHER=DEGEN*(QRT/QRR)/PLANCK DO I = 0,NGRAINS PKE(M,I) = 0.0 END DO DO ITER = MGRAINS(M),NGRAINS PKE(M,ITER) = OTHER*RO(ITER-MGRAINS(M),TEMPID) 1 /RO(ITER,IDIS)*PE(M) END DO RETURN END C Subroutine to determine : C (a) If the collision is activating / deactivating C (b) The new energy level after collision C C R = Random number between 0 and 1 C DELTAE = Width of energy grain C E = Exact energy of molecule before collision C N1 = Energy grain before collision C NGRAINS = No. of energy grains C PIJACT(20,NARRAY) Summation of activation probabilities C PIJDEACT(20,NARRAY) Summation of deactivation probabilities C ENEW = New energy level after collision C SUBROUTINE STEP(ID,DELTAE,E,NGRAINS, 1 PIJ,PIJACT,PIJDEACT,ENEW) IMPLICIT REAL*8 (A-H, O-Z) 372 COMMON NARRAY, NTOL DIMENSION PIJACT(NTOL,0:NARRAY), PIJDEACT(NTOL,0:NARRAY) DIMENSION PIJ(0:NARRAY,0:NARRAY) ENEW = E CALL RAND(R) R4=R DO IM = 0, NGRAINS IF (E .LE. (DELTAE*IM)) THEN N1 = IM GOTO 5 ENDIF END DO 5 IF (R4 .LT. PIJACT(ID,N1)) THEN INDEX = 1 ! Collision is activating ELSE INDEX = 2 ! Collision is deactivating ENDIF IF (INDEX .EQ. 1) THEN ! if activating CALL RAND(U) R=U ASUM1 = 0 DO 10 I = N1, NGRAINS ASUM2 = ASUM1 + PIJ(I,N1) IF (ASUM2 .GE. R) THEN ENEW = (R-ASUM1)/(ASUM2-ASUM1)*DELTAE + DELTAE*(I-1) GOTO 15 ENDIF ASUM1=ASUM2 10 CONTINUE ELSE CALL RAND(U) R=U DSUM1 = 0 DO 20 I = N1, 0, -1 DSUM2 = DSUM1 + PIJ(I,N1) IF (DSUM2 .GE. R) THEN ENEW = (R-DSUM2)/(DSUM1-DSUM2)*DELTAE + DELTAE*I GOTO 15 ENDIF DSUM1=DSUM2 20 CONTINUE ENDIF 15 RETURN 373 END C C<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><> C C Subroutine BSCOUNT performed ro-vibrational states counting. C The vibrational degrees of freedom are treated as hamonic oscillators. C C The counting of rotational states are performed by using C the method described by Gilbert and Smith C (R. O. Gilbert and S. C. Smith, C Theory of Unimolecular and Recombination Reactions, Blackwell, C Oxford, 1990, Chapter 3). C SUBROUTINE BSCOUNT(VIB,IVIB,ROT,IROT,SIGMA,DIM,EMAX,DELTAE,P,NMAX, 1 NHROT,ROTH,SIGMAH,V,INDEX) C C IVIB = number of vibrational degrees of freedom, input; C VIB(I) = vibrational frequencies (cmˆ-1), input; C IROT = number of active (rigid) rotors, input; C (The maximum number of rotor is 4 for direct count.) C ROT(J) = rotational constant (cmˆ-1), input; C SIGMA(J) = symmetric number of the Jth rotor, input; C DIM(J) = dimension of the Jth rotor, input; C EMAX = maximum energy (Evr, cmˆ-1), input; C DELTE = energy spacing; input (DELTE*NMAX=EMAX) C P = array containing density of states in DELTAE energy spacing. C INDEX = 1 --- returns density of state C = 2 --- returns sum of state C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION VIB(IVIB),ROT(IROT),SIGMA(IROT),DIM(IROT),P(0:NMAX) DIMENSION NVIB(200) DIMENSION ROTH(NHROT),SIGMAH(NHROT),V(NHROT) ALLOCATABLE PD(:),PS(:),PP(:),S(:) IF(NHROT.EQ.0) THEN DE=DELTAE ELSE DE=1.0 ENDIF NN=NINT(DELTAE/DE) allocate (PD(0:NMAX*NN),PS(0:NMAX*NN),PP(0:NMAX*NN),S(0:NMAX*NN)) C DO 10 I=1,IVIB NVIB(I)=NINT(VIB(I)/DE) 10 CONTINUE C C Initialize P array. C IF(IROT.EQ.0) THEN C 374 C No active rotational degrees of freedom C PD(0)=1.0 DO 20 I=1,NMAX*NN PD(I)=0.0 20 CONTINUE ELSE C C with active rotational degress of freedom C PD(0)=0.0 U=0.0 P2=0.0 C C Count number if 1-diemnsional free rotors, U, and number of C 2-dimensional rotors, P2, and calculate the products C PROD=1.0 DO 30 J=1,IROT IF(NINT(DIM(J)).EQ.1) THEN U=U+1 PROD=PROD/(SIGMA(J)*SQRT(ROT(J))) ENDIF IF(NINT(DIM(J)).EQ.2) THEN P2=P2+1 PROD=PROD/(SIGMA(J)*ROT(J)) ENDIF 30 CONTINUE C C Assign intial value of rotational densitity of states to array C P(I) C PROD=PROD*3.1415926**(U/2.0)/DGAMMA(P2+U/2.0) EXPON=P2+U/2.0-1.0 DO 40 I=1,NMAX*NN ESUM=FLOAT(I)*DE PD(I)=PROD*ESUM**EXPON 40 CONTINUE ENDIF C C Perform vibrational state counting. P(J),J=1,2,...,M contains C the number of energy states on level E(J)/DELTAE. C CALL COUNT(NVIB,IVIB,NMAX*NN,PD) C IF(IROT.EQ.0) THEN DO 45 I=1,NMAX*NN PD(I)=PD(I)/DE 45 CONTINUE ENDIF C IF(INDEX.NE.1) THEN 375 DO 50 I=1,NMAX*NN PS(I)=PS(I-1)+PD(I)*DE 50 CONTINUE END IF C IF(NHROT .EQ. 0) GOTO 75 C C Convolute with density of states of hindered rotor, if any C DO IHROT=1,NHROT CALL HINCOUNT(ROTH(IHROT),SIGMAH(IHROT),V(IHROT),DE,NMAX*NN,S) DO I=0,NMAX*NN RHS=0.0 DO K=0,I RHS=RHS+S(K)*PD(I-K)*DE END DO PP(I)=RHS END DO DO I=0,NMAX*NN PD(I)=PP(I) END DO END DO C IF(INDEX.NE.1) THEN DO 60 I=1,NMAX*NN PS(I)=PS(I-1)+PD(I)*DE 60 CONTINUE END IF 75 DO I=1,NMAX IF(INDEX.EQ.1) THEN P(I)=PD(I*NN) ELSE P(I)=PS(I*NN) ENDIF END DO RETURN END C Subroutine HINCOUNT performed states counting for hindered rotors C C The counting of rotational states are performed by using C the method described by Knyazev et al. (JPC,1994,98,5279) C SUBROUTINE HINCOUNT(ROT,SIGMA,V,DE,N,DEN) C C ROT = rotational constant (cm-1), input; C SIGMA = symmetric number of the Jth rotor, input; C V = Barrier height of rotor (cm-1), input; 376 C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION DEN(0:N) DATA PI/3.1415926/ C DO I=0,N E=FLOAT(I)*DE ARG=V/E IF(E .GT. V) THEN ARG=SQRT(ARG) DEN(I)=2.0*EKP(ARG)/PI/SIGMA/SQRT(ROT*E) ELSE ARG=SQRT(1.0/ARG) DEN(I)=2.0*EKP(ARG)/PI/SIGMA/SQRT(ROT*V) ENDIF END DO RETURN END C C C Function to find the elliptic integral of the first kind C DOUBLE PRECISION FUNCTION EKP(ARG) IMPLICIT REAL*8 (A-H,O-Z) ARG2=ARG*ARG IF(ARG.LT.0.80) THEN EKP=3.1415926*(16.0-5.0*ARG2)/(2.0*(16.0-9.0*ARG2)) ELSE EKP=(5.0-ARG2)*log(16.0/(1.0-ARG2))/8.0-(1.0-ARG2)/4.0 ENDIF RETURN END C C Subroutine HINCOUNT2 performed states counting for hindered rotors C C The counting of rotational states are performed by using C the method described by Knyazev et al. (JPC,1994,98,5279) C SUBROUTINE HINCOUNT2(ROT,SIGMA,V1,V2,DE,N,DEN) C C ROT = rotational constant (cm-1), input; C SIGMA = symmetric number of the Jth rotor, input; C V = Barrier height of rotor (cm-1), input; C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION DEN(0:N) DATA PI/3.1415926/ C DO I=0,N E=FLOAT(I)*DE 377 VV1=MIN(V1,V2) VV2=MAX(V1,V2) ARG1=VV1/E ARG2=VV2/E IF (E .GT. VV2) THEN ARG11=SQRT(ARG1) ARG22=SQRT(ARG2) DEN(I)=1.0*(EKP(ARG11)+EKP(ARG22))/PI/SIGMA/SQRT(ROT*E) ELSEIF((E .GE. VV1) .AND. (E.LT.VV2)) THEN ARG11=SQRT(ARG1) ARG22=SQRT(1.0/ARG2) DEN(I)=1.0/SIGMA/PI*(EKP(ARG11)/SQRT(ROT*E)+EKP(ARG22)/ 1 SQRT(ROT*VV2)) ELSE ARG11=SQRT(1.0/ARG1) ARG22=SQRT(1.0/ARG2) DEN(I)=1.0/PI/SIGMA*(EKP(ARG11)/SQRT(ROT*VV1)+EKP(ARG22)/ 1 SQRT(ROT*VV2)) ENDIF END DO RETURN END C C<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><> C C The subroutine COUNT is taken from C T.Beyer and D.F.Swinehart, Commun. Assoc. Comput. Machin. 16, 379 C (1973). C SUBROUTINE COUNT(IC,N,MCOUNT,P) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION IC(N),P(0:MCOUNT) C C Perform state counting. C DO 10 I=1,N J=IC(I) JP1=J+1 P(J)=P(J)+1.0 DO 20 M=JP1,MCOUNT MJ=M-J P(M)=P(M)+P(MJ) 20 CONTINUE 10 CONTINUE C RETURN END C Function ZLJ evaluates Lennard-Jones collision frequency C factor DOUBLE PRECISION FUNCTION ZLJ(T,W,SIGMA,EPSLON) 378 IMPLICIT REAL*8 (A-H,O-Z) DATA BOLTZ/1.3806E-16/,PI/3.1415926/,AVGD/6.0222E+23/ IF(EPSLON.LE.0.0001D0) THEN OMEGA=1.0 ELSE TS=T/EPSLON OMEGA=1.16145/TS**0.14874 OMEGA=OMEGA+0.52487/EXP(0.7732*TS) OMEGA=OMEGA+2.16178/EXP(2.437887*TS) ENDIF WT=W/AVGD ZLJ=(SIGMA*1.0E-8)**2*SQRT(8.0*PI*BOLTZ*T/WT)*OMEGA*AVGD C This has units : cm^3/mol/sec RETURN END C C This subroutine was adopted from R.L. Brown, 'A Method of Calculating Tunneling Corrections C For Eckart Potential Barriers, J. Res. of NIST (86) 4, 1961. The subroutine devised by the C author is accurate to with in +/- a few percent. Thus, in some cases, unphysical results C are obtained. To avoid this, the tranmission probablity output has two flags: C 1) If it is calculated to be less than 1.0, it is set to 1.0 C 2) If k(T2)>k(T1), where T2>T1, k(T2) is set to k(T1) C C Created by Enoch Dames, 2011 C C ************************************************************************************** ** SUBROUTINE ECKART(T,V1,V2,VIMG,M,PE) ! IMPLICIT REAL*8 (A-H,O-Z) COMMON NTOL DIMENSION X(6), W(6), PE(NTOL) DATA X/-.9324695,-.6612094,-.2386192,.2386192,.6612094,.9324695/ DATA W/.1713245,.3607616,.4679139,.4679139,.3607616,.1713245/ DATA PI/3.1415926/,PI2/6.2831853/,PISQ/9.8696044/ DATA PLANCK/6.6262E-34/,BOLTZ/1.3806E-23/,SPEED/2.9979E+10/ DATA AVGD/6.0222E+23/ REAL*8 KT C OPEN (UNIT=15, FILE='eckart.inp') c OPEN (UNIT=16, FILE='eckart.out') C V1,V2 = BARRIERS IN KCAL/MOL C V = IMAGINARY FREQUENCY IN CM-1 C PE = Eckart tunneling correction factor C ALPH1 = 2*pi*V1/(hv) C ALPH2 = 2*pi*V2/(hv) C U = hv/KT C KT = Boltzmann constant * T 379 C WRITE(16,200) C CONVERT cm-1 to J/MOL C WRITE(*,*) 'V1,V2,VIMG', V1,V2,VIMG V1=V1*4184/394.75 V2=V2*4184/394.75 ALPH1 = 2*PI*V1/(PLANCK*VIMG*SPEED)/AVGD ALPH2 = 2*PI*V2/(PLANCK*VIMG*SPEED)/AVGD C WRITE(*,*) 'Alphas are', ALPH1, ALPH2 C READ(15,*) (TEMP(I),I=1,NTEMP) C DO 120 IT=1,MCOUNT C Convert cm-1 to K: C T = DELTAE*IT*0.6947 !0.6947=k/(hc) C T = TEMP(IT) KT = BOLTZ*T U = PLANCK*VIMG*SPEED/KT C CONVERT KCAL/MOL J/MOL UPI2=U/PI2 C=.125*PI*U*(1./SQRT(ALPH1)+1./SQRT(ALPH2))**2 V1=UPI2*ALPH1 V2=UPI2*ALPH2 D=4.*ALPH1*ALPH2-PISQ IF(D.LT.0) GOTO 10 DF=COSH(SQRT(D)) GOTO 11 10 DF=COS(SQRT(-D)) 11 IF(V2.GE.V1) EZ=-V1 IF(V1.GT.V2) EZ=-V2 EB=MIN((C*(ALOG(2.*(1.+DF)/.014)/PI2)**2-.5*(V1+V2)),3.2) EM=.5*(EB-EZ) EP=.5*(EB+EZ) G=0 DO 20 N=1,6 E=EM*X(N)+EP A1=PI*SQRT((E+V1)/C) A2=PI*SQRT((E+V2)/C) FP=COSH(A1+A2) FM=COSH(A1-A2) 20 G=G+W(N)*EXP(-E)*(FP-FM)/(FP+DF) G=EM*G+EXP(-EB) IF (G.LT.1.0) G=1.0 PE(M)=G C WRITE(16,100) T, G IF(V1.GE.V2) G=1.0 write(*,*) T, G PE(M)=G c200 FORMAT(2X,'T',10X,'k',5x) 100 FORMAT(F5.0,F15.2) 120 CONTINUE RETURN END

Abstract (if available)

Abstract Future internal combustion engine design will rely on accurate kinetic models of surrogate fuels that mimic their fossil fuel derived counterparts, having hundreds or thousands of different chemical species. At the same time, technological advances in alternative fuel production will enable tailor made surrogate fuels for use in pre-existing internal combustion engines. For these reasons, a large collaborative effort has been underway to identify and understand key species and their reaction kinetics. This work focuses on two such compound classes - one-ringed aromatics and alkanes, as they make up a significant portion of almost all transportation fuels. ❧ Attempts at understanding the combustion characteristics of new compounds frequently results in the identification of missing kinetic pathways and/or key intermediate species. For a kineticist, there is never a lack of new and interesting reactions to explore. This thesis work presents mechanisms seeking to explain observed behavior in the three different premixed flame regions: the postflame region, the flame sheet, and the preflame region. ❧ Three topics will be addressed: 1) persistent free radicals in large aromatic compounds, which serve as analogs to systems in nascent and mature soot that may help explain the continued growth of soot in the postflame region where hydrogen radicals are scarce. 2) mutual isomerization through H-atom shifts in benzylic and methylphenyl type radicals for toluene, and o-, m-, and p-xylene isomers. The relative structure of xylene isomers and specifically the number of H atoms immediately adjacent to the methyl groups can have a direct impact on their high-temperature oxidation and appear to explain the observed differences in xylene oxidation behind reflected shock waves and in laminar premixed flames. 3) the isomerization and dissociation of cyclohexyl radicals, which marks a necessary step towards a detailed understanding of dehydrogenation from cyclohexane and methylcyclohexane to benzene and toluene. Focus lies on branching fractions of cyclohexyl isomerization/dissociation for various temperatures and pressures, and its relationship to differing observations on the role of early dehydrogenation in benzene formation for various flame configurations.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Dames, Enoch Edward (author)

Core Title Kinetic modeling of high-temperature oxidation and pyrolysis of one-ringed aromatic and alkane compounds

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 07/05/2012

Defense Date 07/05/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag ab initio,combustion,gas-phase chemistry,kinetics,OAI-PMH Harvest,reaction rate theory,soot

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Wang, Hai (committee chair), Egolfopoulos, Fokion N. (committee member), Krylov, Anna I. (committee member)

Creator Email enoch.dames@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-51809

Unique identifier UC11290364

Identifier usctheses-c3-51809 (legacy record id)

Legacy Identifier etd-DamesEnoch-912.pdf

Dmrecord 51809

Document Type Dissertation

Rights Dames, Enoch Edward

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA