Multi-energy well kinetic modeling of novel PAH formation pathways in flames. Nicola Giramondi

Transcription

1 Multi-energy well kinetic modeling of novel PAH formation pathways in flames Nicola Giramondi Master of Science Thesis KTH School of Industrial Engineering and Management Energy Technology EGI MSC EKV1124 Division of Heat and Power Technology Stockholm, Sweden

2 Master of Science Thesis EGI MSC EKV1124 Multi-energy well kinetic modeling of novel PAH formation pathways in flames Nicola Giramondi Approved Examiner Supervisors 14/03/2016 Prof. Tosten Fransson Dr. Björn Waldheim CFD Engineer at Scania CV AB (Industrial Master Thesis Supervisor) Dr. Jeevan Jayasuriya (EGI supervisor) Abstract Polycyclic Aromatic Hydrocarbons (PAHs) are harmful by-products formed during combustion of hydrocarbons under locally fuel-rich conditions followed by incomplete combustion. PAHs act as precursors during the formation of soot. PAHs and soot are harmful for human health and legislation limits the emission of unburned hydrocarbons and soot. Consequently, other measures are necessary in order to limit the production of PAHs and soot in internal combustion engines applications, entailing a possible decrease of fuel efficiency and higher technical requirements for automotive manufactures. The combustion chemistry of PAHs is not fully understood, which prompts the need of further investigations. The chemical dynamics shown by novel pathways of PAH formation involving vinylacetylene addition to the phenyl radical opens up new horizons for the potential contribution to PAH formation through this class of reactions. In the present work novel pathways of the formation of naphthalene and phenanthrene are investigated for a laminar premixed benzene flame and a laminar ethylene diffusion flame. The purpose is to improve the prediction of the aromatic species concentration in the flames. A pathway chosen due the high potential aromatic yield is assessed through preliminary flame calculations relying on simplifying assumptions concerning reaction rates. Certain isomerisation steps of the pathway occur within a time-scale characteristic of thermal relaxation processes. Therefore, the solution of the energy grained master equation is necessary in order to calculate the phenomenological reaction rates resulting from a non-equilibrium kinetic modeling. Quantum chemical calculations are performed in order to calculate molecular properties of the species involved. These properties are i

3 subsequently processed to determine the rate constants of the sequence of multi-energy well reactions. Moreover, the chemical dynamics of the pathway is analyzed and the effect of temperature and pressure on the kinetic parameters is investigated. Despite of the potential yield demonstrated through the preliminary flame calculations, the computed rate constants show that the studied reactions are insignificant for the formation of naphthalene and phenanthrene in the studied flames. An effort is put on evaluating if the non-equilibrium kinetic modeling adopted for the calculation of the kinetic parameters is consistent with the kinetic modeling used in the flame calculations. The current work provides an efficient method to compute rate constants of multi-energy well reactions at different thermodynamic conditions, characteristic of flames and of combustion in commercial devices or in internal combustion engines. Pathways with a slightly different chemical dynamics should be tested applying the current methodology. Moreover, further studies should be aimed at overcoming possible limits of the kinetic modeling of multi-energy well reactions occurring in combustion environments. Key words: Polycyclic Aromatic Hydrocarbons, soot, laminar flame, flame calculation, quantum chemical calculation, non-equilibrium kinetic modeling. ii

4 Acknowledgements I am sincerely grateful to Dr. Björn Waldheim, my industrial supervisor at the NMTD department at Scania. His support, guidance and trust gave me an extraordinary motivation and allowed an outstanding professional and personal development. I could not expect a better supervision, that I will always gratefully acknowledge. I would also like to thank all the coworkers of the NMTD department, whose cordiality and helpfulness allowed a great work environment. Moreover, I consider myself privileged for the guidance and the helpfulness of Prof. Peter Lindstedt from Imperial College London. My academic supervisor at KTH, Prof. Jeevan Jayasuriya, has always supported me and my interest for the field of combustion has grown while attending his lectures at KTH. I would also like to thank my academic supervisor at Politecnico di Milano, Prof. Alessio Frassoldati, for his prompt replies despite the distance and for having demonstrated a motivating interest for this project. I am grateful to my friends at KTH and at Politecnico di Milano, for their advices and the good time spent together. Thanks to Carolina, who has never missed to support and encourage me throughout this period. My exceptional mother Gabriella has allowed me to give my best despite the sacrifices and the difficulties faced together. I am forever in her debt. My beloved father Gino gave me incomparable love and handed down his precious knowledge and experience to me. His example will guide me for life. All my efforts and results are dedicated to him. iii

5 Contents Abstract Acknowledgements Contents List of Figures List of Tables Nomenclature i iii iv viii xiv xvi 1 Introduction Background Motivation Objectives 4 3 Methodology 6 4 Literature Review PAH chemistry Introduction to PAH chemistry The formation of the first aromatic ring The HACA mechanism of aromatic growth Consumption of benzene Sectional soot modeling Modeling of general reacting flows Introduction to the modeling of general reacting flows The physics of general reacting flows Momentum equation Species conservation equation Enthalpy conservation equation Transport properties Determination of thermodynamic properties Kinetics of chemical reactions Introduction to the kinetics of chemical reactions Reaction rates Reactions that involve a third body iv

6 Contents v Reactions with pressure-dependent kinetics The estimation of reaction rates Estimations based on equilibrium considerations Estimations based on reaction class considerations Overview of the flame systems studied in this work The laminar premixed benzene flame of Bittner and Howard Flame chemistry Physics of a free-flowing laminar flat flame The laminar ethylene diffusion flame of Olten and Senkan Flame chemistry Physics of a laminar counter-flow diffusion flame Novel pathways of naphthalene formation Prediction of aromatic species concentrations in the flames studied in this work Pathways of napthalene formation involving vinylacetylene addition to the phenyl radical Previous investigations The addition of the phenyl radical to vinylacetylene triple bond The addition of the phenyl radical to vinylacetylene double bond Inspiration from the studies of astrochemical evolution of the interstellar medium Preliminary flame calculations Introduction to the preliminary flame calculations Testing a PAH formation pathway from a previous literature study Testing the novel PAH formation pathways Interpolation of the low temperature kinetic rate constants of the barrier-less pathway Testing the barrier-less pathway assuming the rate of collisions as an estimate of the pre-exponential factor Testing the pathway with a 5 kj/mol barrier at the entrance Testing the pathway with a 17 kj/mol barrier at the entrance Considerations on the preliminary flame calculations A note on the subsequent analysis and on nomenclature Quantum chemical calculations Introduction to quantum chemical calculations A note on the tools used for the quantum chemical calculations The Schrödinger equation The Hartree-Fock method Basis set Density Functional Theory Single point energies Vibrational frequencies and Intrinsic Reaction Coordinate

7 Contents vi 8 Pathway analysis for the estimation of kinetic parameters Introduction to nonequilibrium reaction pathways The Energy Grained Master Equation Introduction to the Energy Grained Master Equation Energy transfer model Energy discretization Statistical mechanics and partition functions Rotational energy levels and partition function Vibrational energy levels and partition function Microcanonical rate coefficients MESMER input specifications Introduction to MESMER input specifications Single point energies Rotational constants Symmetry numbers Vibrational Frequencies Lennard-Jones parameters Energy transfer parameters Energy discretization parameters Ensure energetic consistency Deriving reaction rates constants from the solution of the Energy Grained Master Equation Time scale separation Conservative master equations and phenomenological modeling of the chemical system Non-conservative master equation Preliminary considerations for a consistent kinetic modeling of the pathway CSEs analysis A note on well skipping Species profiles analysis Results of the computation of molecular properties and kinetic parameters Computed molecular properties Single point energies Vibrational frequencies Rotational constants Computed reaction rates Reactions implemented into the chemical mechanism of the flames Canonical reaction rates Phenomenological reaction rates Computing the species profiles by replicating the kinetic modeling of the flames Final flame calculations Introduction to the final flames calculations Flame calculations based on the canonical reaction rates currently computed120

8 Contents vii Case a: the focus on naphthalene formation Results of the calculation for the ethylene diffusion flame of Olten and Senkan Results of the calculation for the premixed benzene flame of Bittner and Howard Case b: the extension to phenanthrene formation Results of the calculation for the ethylene diffusion flame of Olten and Senkan Results of the calculation for the premixed benzene flame of Bittner and Howard Flame calculations based on the phenomenological reaction rates currently computed Case c: the application of the phenomenological reaction rate constants currently computed and the extension to phenanthrene formation Results of the calculation for the ethylene diffusion flame of Olten and Senkan Results of the calculation for the premixed benzene flame of Bittner and Howard Factors affecting the relevance of the added pathway of formation of naphthalene Conclusions and future work 133 Bibliography 136 A GAMESS calculations I A.1 GAMESS calculations settings I A.1.1 The Hartree-Fock method I A.1.2 Basis set II A.1.3 Optimization process II A.1.4 Density functional theory IV A.1.5 Single point energies IV A.1.6 Vibrational frequencies V A.2 Molecular properties resulting from GAMESS calculations VI B Additional kinetic parameters from MESMER calculations XII

9 List of Figures 2.1 A group of measured and calculated species mass fractions of the laminar ethylene diffusion flame of Olten and Senkan investigated by Waldheim [6]. The measured values are identified by circles, the computed profiles by the solid line and the ones represented by dashed and dotted lines corresponds to computation with an 100 K increase and decrease respectively. The figure is taken from the doctoral thesis of Waldheim [6] Schematic representation of the methodology adopted in the current work Major benzene oxidation pathways investigated in a jet-stirred reactor at 1000K, 10 atm and Φ = 1.5. The width of the arrows is proportional to the relevance of the reaction [1] Major benzene oxidation pathways investigated in the Princeton reactor at Φ = The width of the arrows is proportional to the relevance of the reaction [1] The NIST experimental system [17] Rate constant of the methyl recombination as a function of the pressure at a fixed temperature [21] Species involved in the reactions in Eqs. 4.64, 4.65, 4.66 and Experimental setup of the premixed benzene flame of Bittner and Howard [10] Mole fractions of relevant species, flux and mole fraction of benzene as a function of the distance from the burner in the flame of Bittner and Howard [10] Main formation pathway of Phenantrene in the flame of Bittner and Howard [6] Relevant formation pathway of Pyrene in the flame of Bittner and Howard [6] Formation of Pyrene from cyclopenta[def]phenanthrene in the flame of Bittner and Howard [6] Structure of the species relevant for the mechanisms of PAHs growth in the flame of Bittner and Howard Experimental set-up adopted by Olten and Senkan for the study of an ethylene diffusion flame [11] Temperature profile and mass fraction profiles of the major species of the laminar ethylene diffusion flame of Olten and Senkan [11] Streamlines of a counter-flow diffusion flame [24] viii

10 List of Figures ix 4.15 Measured and calculated temperature profiles of the laminar ethylene diffusion flame of Olten and Senkan. The measured temperature values are identified by circles, the radiation corrected temperature profile by squares, the computed temperature profile by the solid line and the temperature profile calculated after reducing the fuel stream velocity of 25% by dash-dotted line. The figure is taken from the doctoral thesis of Waldheim [6] Relevant measured (circles) and calculated (dash dotted lines) species profiles of the laminar ethylene diffusion flame of Olten and Senkan investigated by Waldheim [6] Relevant measured (circles) and calculated (dash dotted lines) species profiles of the laminar premixed benzene flame Bittner and Howard investigated by Waldheim [6] Pathway of naphthalene formation involving the addition of the phenyl radical to the vinylacetylene triple bond involving. Two subsequent rotations occur around the single and double bond within the side chainr [25] Pathway of naphthalene formation involving the addition of the phenyl radical to the vinylacetylene triple bond.only one rotation occurs around the single bond within the side chain [25] Planar representations of C 10 H 8 (G), C 10 H 8 (J) and C 10 H 9 (L) consistent with the study of Moriarty and Frenklach [25] Pathway of naphthalene formation involving the addition of the phenyl radical to vinylacetylene double bond studied by Moriarty and Frenklach [25] The dominant pathway of naphthalene formation subsequent to vinylacetylene addition to the phenyl radical investigated by Moriarty and Frenklach [25] Potential energy surface of the barrier-less pathway involving the phenyl radical addition to C1 of vinylacetylene [27] Potential energy surface of the pathway involving the phenyl radical addition to C4 of vinylacetylene with a 5 kj/mol barrier at the entrance [27] Potential energy surface of the pathway involving the phenyl radical addition to C2 of vinylacetylene with a 17 kj/mol barrier at the entrance [27] Hydrogen migration from the side chain to the aromatic ring [25] Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27] Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27] Molecular structures of 1-C 10 H 7, 2-C 10 H 7 and A 3 H

11 List of Figures x 6.4 Species profiles of the benzene premixed flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the barrierless pathway of Parker et al. [27] Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6]. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the barrier-less pathway of Parker et al. [27] Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 5 kj/mol barrier at the entrance Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 5 kj/mol barrier at the entrance Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 17 kj/mol barrier at the entrance Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 17 kj/mol barrier at the entrance Potential energy diagram of the pathway in analysis. The energy levels are expressed in kj/mol Spectrum of the eigenvalues of the pathway in Fig The black solid curve represent the boundary of the IEREs, the black dotted curve is the boundary of the region close to the IEREs within which the criterion adopted by MESMER is not fulfilled. The two colored curves approaching the IEREs and the non-monotonic one are the CSEs Spectrum of the eigenvalues obtained by Miller and Klippenstein [43] for an irreversible exchange reaction pathway involving 3 wells with a Potential Energy diagram qualitatively similar to the current case Potential energy diagram of the pathway with the Minimum 2 skipped. The energy levels indicated are expressed in kj/mol

12 List of Figures xi 8.5 Spectrum of the eigenvalues of the pathway with the Minimum 2 skipped. The black solid curve represent the boundary of the IEREs, the black dotted curve is the boundary of the region close to the IEREs within which the criterion adopted by MESMER in not fulfilled. The colored curves approaching the IEREs and the non-monotonic one are the CSEs Potential energy diagram of the pathway with both Minimum 2 and Minimum 3 skipped. are expressed in kj/mol Spectrum of the eigenvalues of the pathwaywith both Minimum 2 and Minimum 3 skipped. The black solid curve represent the boundary of the IEREs, the black dotted curve is the boundary of the region close to the IEREs within which the criterion adopted by MESMER in not fulfilled. The colored curve approaching the IEREs and the non-monotonic one are the CSEs Fractional species profiles as a function of time computed by MESMER for the complete pathway in Fig The subsequent diagrams correspond to the different temperatures indicated and to ambient pressure. The dashed curve corresponds to reactants, the red one with circular icons to Minimum 1, the yellow one with triangular icons to Minimum 2, the green one with squared icons to Minimum 3 and the solid curve to products Fractional species profiles as a function of time computed by MESMER for the pathway in Fig The subsequent diagrams correspond to different temperatures and to ambient pressure. The dashed curve corresponds to reactants, the red one with circular icons to Minimum 1, the green one with squared icons to Minimum 3 and the solid curve to products Fractional species profiles as a function of time computed by MESMER for the pathway in Figure 8.4 The subsequent diagrams correspond to different temperatures at a pressure of 10 6 atm. The characterization of the different curves is consistent with Fig Reaction Coordinate of the Transition States of the pathway Potential Energy Surface of the pathway with a 5 kj/mol energy barrier at the entrance investigated by Parker et al. [27]. The solid line represents the PES obtain by the authors [27], whereas the other lines are the results of the current quantum chemical calculations, applying different DFT methods and basis sets as indicated in the legend Canonical reaction rates of reactions RI, RII and RIII. The solid line corresponds to the values computed by MESMER and the dash-dotted line to the correspondent interpolations Reaction rates as a function of temperature at ambient pressure. The green solid curves with squared icons and the red dash-dotted curves with rhomboidal icons are the phenomenological reaction rates (listed in Table B.2) and the correspondent interpolations (shown in Table 9.7), respectively. When present, the black dashed curve represent the trend of the correspondent rate constants previously used into the chemical mechanism Phenomenological forward and backward reaction rate constants of the reaction in Eq. 9.1 at different temperatures and at ambient pressure. The former is represented by the blue solid line with squared icons and the latter is the red solid line with rhomboidal icons

13 List of Figures xii 9.6 Equilibrium constant of the reaction in Eq. 9.1 at different temperatures and at ambient pressure Reaction rates in m 3 /(kmol s) trends as a function of temperature at 2.67 kpa. The light blue solid curves with circular icons and the red dash-dotted curves with rhomboidal icons are the phenomenological reaction rates (listed in Table B.3) and the correspondent interpolations (shown in Table 9.8), respectively. The green solid lines with squared icons are the correspondent phenomenological reaction rates at ambient pressure of Table B.2. When present, the black dashed curve represents the trend of the correspondent rate constants previously implemented into the chemical mechanism Canonical and corresponding phenomenological reaction rates leading to products at different temperatures for reactions RI, RII and RIII. The blue curves with squared icons are the canonical reaction rates. The red curves with circular icons, the green curves with rhomboidal icons and the orange curve with triangular icons are the phenomenological reaction rates computed at 10 6, 1 atm and 2.67 kpa respectively Fractional species profiles as a function of time of the pathway in Figure 8.4. The subsequent diagrams correspond to different temperatures at ambient pressure. The dashed curve, the red one with circular icons, the green one with squared icons and the solid curve are the fractional profiles shown in Fig 8.9. The correspondent thinner curves are the fractional profiles computed as a result of the system in Eq Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the solid curves represent the measured concentrations and the currently computed species profiles for case a. The dash dotted curves represents the correspondent species profiles computed by Waldheim [6] Planar representation of C 10 H 8 (L) and of the C 10 H 9 (L) radical Formation of C 10 H 9 (T) radical through C 10 H 9 (M) isomerization Chemical structures of the species involved in the reaction in Eq Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the solid curves represent the measured concentrations and the currently computed species profiles for case a. The dash dotted curves represents the correspondent species profiles computed by Waldheim [6] Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the solid curves represent the measured concentrations and the currently computed species profiles for case b. The dash dotted curves represents the correspondent species profiles computed by Waldheim [6] Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the solid curves represent the measured concentrations and the currently computed species profiles for case b. The dash dotted curves represents the correspondent species profiles computed by Waldheim [6] Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the solid curves represent the measured concentrations and the currently computed species profiles for case c. The dash dotted curves represents the correspondent species profiles computed by Waldheim [6].. 129

14 List of Figures xiii 10.9 Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the solid curves represent the measured concentrations and the currently computed species profiles for case c. The dash dotted curves represents the correspondent species profiles computed by Waldheim [6].. 130

15 List of Tables 6.1 Significance of the reactions currently applied to the premixed benzene flame kinetic scheme based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27]. When the minus sign is indicated into brackets, the correspondent reaction turned reverse Significance of the reactions currently applied to the ethylene diffusion flame kinetic scheme based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27] Significance of the reactions currently applied to the premixed benzene flame kinetic scheme based on an approximate theoretical collision rate of the barrier-less pathway of Parker et al. [27] Significance of the reactions currently applied to the ethylene diffusion flame kinetic schem based on an approximate theoretical collision rate of the barrier-less pathway of Parker et al. [27] Significance of the reactions currently applied to the premixed benzene flame kinetic scheme, testing the pathway with a 5 kj/mol barrier at the entrance Significance of the reactions currently applied to the ethylene diffusion flame kinetic scheme, testing the pathway with a 5 kj/mol barrier at the entrance Significance of the reactions currently applied to the premixed benzene flame kinetic scheme, testing the pathway with a 17 kj/mol barrier at the entrance. When the minus sign is indicated into brackets, the correspondent reaction turned reverse Significance of the reactions currently applied to the ethylene diffusion flame kinetic scheme, testing the pathway with a 17 kj/mol barrier at the entrance Rotational symmetry numbers of all the species of the pathway Ground single point energies for the different species from B3LYP calculations adopting 6-311G basis set. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical) Ground single point energies for the different species from B3LYP calculations adopting 6-311G(d,p) basis set. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical) xiv

16 List of Tables xv 9.3 Ground single point energies for the different species according to the values of Parker et al. [27]. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical) Ground single point energies for the different species from M06-2X calculations adopting 6-311G(d,p) basis set. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical) Rotational constants [cm 1 ] of all the species of the pathway Interpolated expressions of the canonical reaction rates listed in Tab. B.1 within Appendix A. Units are m 3, kmol, s and K Interpolated expressions of the phenomenological reaction rates computed at ambient pressure and listed in Tab. B.2. Units are m 3, kmol, s and K Interpolated expressions of the phenomenological reaction rates computed at a pressure of 2.67 kpa and listed in Tab. B.3. Units are m 3, kmol, s and K A.1 Vibrational frequencies [s 1 ] of the transition sstates of the pathway.... VI A.2 Vibrational frequencies [s 1 ] of reactants, minima and products of the pathway VII A.3 Reaction coordinates [Å] of the transition states of the pathway. The data below relate to the current orientation of the transition states in the coordinate system IX B.1 Canonical reaction rates computed at different temperatures for the reactions RI, RII and RIII. Units are m 3, kmol, s XII B.2 Phenomenological reaction rates computed on the chemical system with Minimum 2 skipped (see Fig. 8.4) at different temperatures and ambient pressure. Units are m 3, kmol, s XIII B.3 Phenomenological reaction rates computed on the chemical system with Minimum 2 skipped (see Fig. 8.4) at different temperatures and 2.67 kpa. Units are m 3, kmol, s XIV

17 Nomenclature Abbreviations Abbreviation Description PAH Polycyclic Aromatic Hydrocarbon PM Particulate Matter FORTRAN FORmula TRANslation programming language DFT Density Functional Theory B3LYP Electronic density functional ab initio method M06-2X Electronic density functional ab initio method CCSD(T) Coupled Cluster ab initio method 6-311G Basis set 6-311G(d,p) Polarized basis set ZPE Zero Point Energy ME Master Equation EGME Energy Grained Master Equation CSE Chemically Significant Eigenvalue IERE Internal-Energy-Relaxation Eigenmode RRKM Rice-Ramsperger-Kassel-Marcus theory GAMESS General Atomic and Molecular Electronic Structure System MESMER Master Equation Solver for Multi-Energy Well Reactions xvi

18 Nomenclature xvii Parameters Alphabetic Designation Description SI units m Mass kg x, y Space coordinate m t Time s T Temperature K u, v Velocity m/s p Pressure P a E Energy J, Ha h Specific enthalpy J/kg c p Specific heat capacity at constant pressure J/(kg K) C v Molar specific heat capacity at constant volume J/(mol K) X Molar fraction - Y Mass fraction - f Volumetric force N/m 3 W Molar mass kg/mol Kn Dimensionless Knudsen number - Sc Dimensionless Schmidt number - V i Diffusion velocity m/s D Diffusion coefficient m 2 /s n Inverse of the molecular weight mol/kg D k Diffusion coefficient parameter m 2 /s V c Velocity correction m/s W 1,2 (1) Dimensionless function of the reduced temperature - W(2) Dimensionless function of the reduced temperature - F Dimensionless function for estimating the specific - heat capacity R Reaction rate mol/s k Reaction rate constant [m 3 ; kmol; s] E a Activation energy J/mol A Pre-exponential factor [m 3 ; kmol; s] K eq Equilibrium constant [m 3 ; kmol] V L,J Lennard-Jones potential V a Strain rate 1/s J Molecular flux kg/(m 2 s) Continue in the next page

19 Nomenclature xviii Continued form last page J Dimensionless molecular flux - V Dimensionless density-weighted velocity - < E d > Energy transferred on average in a collision J E vib E rot Energy contribution of the molecular vibrational J modes Energy contribution of the molecular rotational J modes I Moment of Intertia kg m 2 A, B, C Rotational constants cm 1 Concluded from last page Greek Designation Description SI units ρ Mass density kg/m 3 µ Viscosity kg/(m s) τ Viscous stress tensor N/m 2 β Rate of attachment of soot particles m 3 /s σ Collision diameter m φ m Well depth of the Lennard-Jones potential V λ Thermal conductivity W/(m K) δ j,k Kronecker delta - γ Ratio of the molar specific heat capacities - η i,j,k Parameter ensuring the conservation of mass - ψ Stream function kg/m 2 ω Cross-stream function - Φ Dimensionless velocity - η Dimensionless density-weighted space variable - µ Dimensionless density-weighted viscosity - ρ Dimensionless density - ζ Distance parameter of an atomic orbital 1/m 2 ν i Molecular vibrational frequency 1/s ω LJ Lennard-Jones frequency of collisions 1/s β T Parameter proportional to temperature m 2 kg/s 2 ɛ grain,i Average energy grain J τ j Time of relaxation s

20 Nomenclature xix Symbols Alphabetic Designation Description a i n i Z rot H V T r N c n n basis N norm l f F F x y i G i,j q i ei p M P Coefficient of the polynomial interpolation of the specific enthalpy Number of soot particles of size class i Number of molecular collisions Hamiltonian operator Potential energy operator Kinetic energy operator Particle position Number of particles Atomic oribital linear combination coefficient Number of basis functions Atomic orbital normalization constant Coordinate parameter of an atomic orbital Generic function Generic electronic density functional Force constant matrix Space coordinate vector Mass-weighted space coordinate Mass-weight factor Eigenvector of the mass-weighted force constant matrix Eigenvalue of the mass-weighted force constant matrix Vector of the grained energy distribution Grained energy distribution change matrix Collision probability q V ibrationalmode Contribution of a single mode to the vibrational partition function q vib q rot q rv g i,j EJ diatomic J n vib i W (E E 0 ) g j Vibrational partition function Rotational partition function Roto-vibrational partition function Relative speed of collision Rotational energy level of a diatomic molecule Rotational quantum number Vibrational quantum number Sum of roto-vibrational energy states Eigenvector of the EGME Continue in the next page

21 Nomenclature xx Continued form last page ˆT N chem n species n wells S c i,j (E) a i,j (E) Operator describing the time evolution of the energy distributions Number of chemically significant eigenvectors of the EGME Number of chemical species involved in a pathway Number of intermediates of a pathway Number of species with an energy distribution Energy population linear combination coefficient Molar fraction linear combination coefficient Concluded from last page Greek Designation ν A α k,i Φ i,j ξ Ψ φ χ ρ el ρ DOS σ rot Ω λ j Description Stoichiometric coefficient Third body enhancement factor Mixing coefficient Heat loss factor Wave function Molecular orbital Atomic orbital Electronic density Roto-vibrational density of state Number of rotational symmetry operations Reduced collosion integral Eigenvalue of the EGME Physical Constants Designation Description Value R Gas constant J/(mol K) k b Boltzmann constant J/K h Planck constant J s

22 Chapter 1 Introduction 1.1 Background Polycyclic aromatic hydrocarbons (PAH) are by-products formed in all the commercial combustion devices running on hydrocarbons under locally fuel-rich conditions. Even though their concentration is generally low, they act as precursors during soot formation and are harmful to human health. Aromatic species are also contained in a wide range of hydrocarbon fuels for automotive, industrial and aeronautic applications e.g. diesel, gasoline and kerosene. Despite the issues connected to PAH, the presence of aromatic species is favored in the application of spark ignition engines as there are evidences that they decrease the phenomena of knock and auto-ignition [1]. Moreover, despite the presence of aromatic compounds in the fuel, aromatic formation and growth also occurs during combustion of aliphatic fuels [2, 3]. During the last decades, one of the main topics within the research in the field of combustion has been the relation between PAH formation and oxidation in the combustion of hydrocarbons and the formation of soot. It has been demonstrated that PAH formation and subsequent growth connects the main combustion chemistry of the flames to the chemistry and the dynamics of soot formation [4]. PAH and soot emissions are harmful both for human health and for the environment. Furthermore, they may cause a reduction of combustion efficiency as well as fouling and deterioration of the combustion devices. Soot generated from automotive engines and other commercial combustion devices leads to hazardous exposure of particulate matter (PM) in populated areas. Even a short term exposure to aerosol containing PM may cause respiratory problems varying from asthma to chronic bronchitis and emphysema. There are evidences that a long term exposure to aerosols containing relevant soot concentrations may lead to cancer and heart problems [5]. 1

23 Chapter 1. Introduction 2 These critical issues made the European Union legislators introduce stricter limit values of PM and PAHs emissions with consequentially higher technical requirements for automotive manufactures [6]. The Euro VI Particulate Matter mass emission limit is 10 mg/kwh for heavy-duty compression ignited diesel engines on world harmonized stationary and transient test cycle [7]. Concerning the Particulate Matter number, the limits correspond to and particles/kwh for the world harmonized stationary and transient test cycle, respectively [6]. In order to achieve the above mentioned technical requirements the approach of the automotive companies regarding soot emissions has to prevent soot formation in the end of the combustion and/or the oxidation of soot through the exhaust aftertreatment system. The European Union emission limits legislation is coupled with local policies which have in common with the former one the aim of reducing the concentration of hazardous pollutants in densely populated areas: They are named low emission zones (LEZ) [6]. The case of the region of Lombardia in Italy and of the municipality of Milan are considered: The use of 2-stroke motorcycles below Euro 1 and public Diesel buses below Euro 3 are permanently banned. Moreover, in sub areas like the one comprising the municipality of Milan, the use of all the other petrol vehicles below Euro 1 and Diesel vehicles below Euro 3 are banned except in winter daily hours [8]. It has been demonstrated [9] that heavy duty diesel vehicles play a dominant role in the emissions of P M < 2.5, whose quantity can be one order of magnitude higher compared to the PM emissions from light duty gasoline vehicles. Moreover, diesel engines emit on average PM with a greater mass and a higher amount of ultra-fine particles, if compared to gasoline engines. Further investigations [9] have demonstrated that PAH emitted from vehicles are mainly in the PM fraction below 0.4 µm and heavy PAHs have been traced in PM fractions of a larger size. These observations represent a straightforward experimental proof of the relationship between PAH and soot formation. 1.2 Motivation The purpose of the present work is to provide an additional contribution to the doctoral thesis Modelling of soot formation and aromatic growth in laminar flames and reactor systems developed by B. B. O. Waldheim [6] from Imperial College between 2010 and 2014 in collaboration with Scania. In the doctoral work, Waldheim applied a sectional soot model to combustion processes in laboratory devices under conditions where soot measurements have been previously carried out. Moreover, he included a thorough preliminary investigation of both the aromatic and the soot chemistry involved in the systems analyzed. In the present work, the analysis will be focused on the chemistry of

24 Chapter 1. Introduction 3 the aromatic species included in the soot model mentioned above. New PAHs formation pathways will be investigated for the laminar premixed benzene flame of Bittner and Howard [10] and the laminar ethylene diffusion flame of Olten and Senkan [11] studied by Waldheim [6]. Within the process of soot formation occurring during combustion of hydrocarbons, the chemistry of the polycyclic aromatic hydrocarbons concerns the gas-phase part of the flame. More precisely, the latter represents the stage of formation of soot precursors, preliminary to soot inception and growth. Depending on the composition of the fuel and on the flame type, the pathways involved in the fuel consumption are different. As it will be shown in the course of this work, different chemical routes become relevant in the formation, growth and oxidation of PAHs [1, 3, 6]. Commercial fuels are a complex mixture of both aromatic and aliphatic hydrocarbons and depending on the relative concentration of these classes of compounds, the pathways of formations of PAHs and their subsequent growth and oxidation are different. Depending on the thermodynamic conditions of the flame, certain pathways prevail over other ones and occur in different zones of the flame [1, 6]. In order to evaluate the relevance of a particular pathway different studies can be developed. Among them, studies of laminar flames - either diffusive and premixed of aromatic or aliphatic fuels - will be quoted. The choice to investigate laminar flames is made in order to focus on the chemistry of the flames. In the kinetic mechanisms a broad range of chemical species are taken into account and measurements of the concentration of the minor species would be too technically challenging if a turbulent flame was investigated. Moreover, the implementation of comprehensive kinetic mechanisms in simulations of turbulent flames makes the computational cost too high and this is a reason why modeling of commercial combustion devices adopt reduced mechanisms. The current objective is to investigate the pathways of formation of polycyclic aromatic species under certain chemical conditions, not considering the real structure of commercial combustion devices: This clarifies the decision to investigate laminar flames modeled as one-dimensional systems. Moreover, several experiments have been conducted using different reactors for the study of the partial oxidation and combustion of fuel under different operative conditions [1]. The motivation behind the reactor studies is to focus on particular pathways occurring in different zones of the flames, with certain local equivalence ratios, temperature and pressure, as well as to validate computational results with a broader amount of experimental measurements under a wide rage of physical and chemical conditions.

25 Chapter 2 Objectives The aim of the current M.Sc. thesis project is to enhance the accuracy of the prediction of PAH and soot formation in laminar flames of aliphatic and aromatic hydrocarbons, by applying novel pathways of PAH formation to the correspondent kinetic mechanisms. This work is focused on the simulation of flames where the application of the current PAH chemistry models fails in the prediction of the amount of PAHs formed and oxidized. In particular, the main goal is to improve the prediction of PAH concentration Figure 2.1: A group of measured and calculated species mass fractions of the laminar ethylene diffusion flame of Olten and Senkan investigated by Waldheim [6]. The measured values are identified by circles, the computed profiles by the solid line and the ones represented by dashed and dotted lines corresponds to computation with an 100 K increase and decrease respectively. The figure is taken from the doctoral thesis of Waldheim [6]. 4

26 Chapter 2. Objectives 5 in the ethylene diffusion flame of Olten and Senkan [11], where the molar fractions of aromatic species such as naphthalene (C 10 H 8 ), phenanthrene (A 3 ) and pyrene (A 4 ) are underestimated. Figure 2.1 shows the underestimation of phenanthrene and pyrene in the ethylene diffusion flame investigated by Waldheim [6]. Hence, the objective will be to find reasonable agreement between the computational and the experimental PAH molar fraction profiles as a function of the distance from the burner. In order to achieve the main, general goal of this project, the following intermediate objectives have to be fulfilled: Novel pathways of PAH formation, significant in terms of aromatic growth when applied to the studied flames, have to be identified; Depending on the features of the identified PAH formation pathways, different molecular properties of the involved species have to be preliminarly calculated in order to determine the correspondent kinetic parameters; The reaction rate constants of the identified pathways have to be computed at the thermodynamic conditions of interest for the studied flames; The reliability and robustness of the method adopted for computing the reaction rate constants have to be demonstrated; The influence of the thermodinamic conditions on the computed kinetic parameters have to be clarified; The significance of the identified PAH formation pathways in terms of aromatic growth in the studied flames have to be confirmed.

27 Chapter 3 Methodology The aim of this chapter is to give the reader an overview of the workflow characterizing the current project. A schematic representation of the methdology adopted in this work is shown in Fig First, a thorough literature review is done in order to: Outline the fundamentals of the aromatic chemistry in the framework of flame and reactor studies; Introduce soot modeling, within which the current study is contextualized; Give an overview of the physics and of the chemistry of the studied flames, together with the tools adopted for performing the flame calculations. A further literature search is performed in order to identify novel PAH formation pathways potentially significant in terms of aromatic growth in the studied flames. The features of the novel pathways are outlined in comparison to analogous pathways, previously extensively investigated in the literature. In particular, the Potential Energy Surfaces of the correspondent pathways are considered. The potential significance of the novel pathways is subsequently assessed through preliminary flame calculations relying on available simplified kinetic parameters, either found in the literature or estimated. The pathway showing the highest potential in terms of aromatic yield is selected for a more refined investigation. Due to the topology of the Potential Energy Surfaces of the selected PAH formation pathway, the solution of an energy density population balance between the involved chemical species is required for computing accurate reaction rate constants. In order to 6

28 Chapter 3. Methodology 7 Figure 3.1: Schematic representation of the methodology adopted in the current work. deremine the energy distributions of the involved species, various molecular properties are calculated through quantum chemical calculations. The energy density population balance is subsequently solved for the energy distributions of the species involved in the pathway. A preliminary analysis of the chemical system represented by the single pathway is performed in order to ensure reliability for the computation of the correspondent reaction rate constants. In particular, the time-scales of the chemical reactions involved in the pathway are compared to the time-scales of the processes of achievement of thermal equilibrium within the studied chemical system. The need to introduce approximations to the pathway is outlined, since features of the correspondent Potental Energy Surface make it not suitable for the computation of reliable rate constants with the available tools. The effect of the introduced approximations on the computation of the reaction rate constants is investigated. Reaction rate constants are computed for a broad range of thermodynamic conditions: The effect of pressure and temperature on the kinetic parameters of the pathway is investigated. In particular, the rate constants of the reactions involved in the pathway are computed for the temperature range of interest and at the correspondent pressures of the studied flames. The rate constants are interpolated as a function of the temperature

29 Chapter 3. Methodology 8 fitting a modified Arrhenius expression. kinetic scheme of the flames. The latter is the expression applied to the Final flame calculations relying on the refined kinetic parameters are performed in order to confirm the significance of the identified pathway. Different assumptions are adopted and discussed in order to further investigate the influence of the thermodynamic conditions. The organization of the report is consistent with the methodology adopted in this work: Each chapter corresponds to a different phase among the ones previously depicted. The computed molecular properties and reaction rate constants are listed within a single chapter aiming at resuming the macro-phase of the computation of the kinetic parameters shown in Fig This macro-phase represents the main contribution of the current work.

30 Chapter 4 Literature Review 4.1 PAH chemistry Introduction to PAH chemistry The aim of this section is to give an overview of the complexity of the chemistry involved in the combustion of hydrocarbons: PAHs formation and oxidation pathways contribute to the chemistry of hydrocarbon flames. When considering PAHs formation pathways, it will be taken into account the formation and consumption of the first aromatic ring, its growth as well as the role of minor species in their development. The first step to consider within the chemistry of Polycyclic Aromatic Hydrocarbons is the formation of the first aromatic ring The formation of the first aromatic ring The first aromatic ring formation through the combination of aliphatic compounds often represents the rate-limiting step in PAH formation pathways [4]. During combustion of real fuels, the reactions involved in the formation of the first aromatic ring contribute to different processes within the flame, such as the consumption of the aliphatic fractions of the fuel. The first aromatic ring formation may also occur subsequently to the ring opening of the aromatic fractions of the fuel. The current discussion is mainly focused on reactions involving aliphatic radicals. Both even- and odd- carbon-atom-pathways have been investigated for a broad selection of flames and fuels at different physical conditions [4]. Among them, the role or resonance stabilized radicals like n- and iso- C 4 H 3 and C 4 H 5, as well as of the propargyl radical are relevant [4]. 9

31 Chapter 4. Literature Review 10 As an example, the role of the propargyl radical in benzene formation in a near sooting acetylene premixed flame diluted in Ar (equivalence ratio equals to 2.5) has been investigated [3]: C 3 H 3 + C 3 H 3 C 6 H 6 (4.1) There are evidences that the reaction in Eq. 4.1 represents the dominant pathway of benzene formation in the main reaction zone of the flame. However, it has to be pointed out that the reaction in Eq. 4.1 is likely to be a global mechanism, representative of a series of elementary reactions. First, a linear C 6 H 6 isomer is formed and subsequently, either isomerises to benzene or reacts with other radicals to form benzene [3]. The propargyl radical is a highly-stable hydrocarbon radical and several studies show that, once the activation of the adduct is reached, the energy barrier of the cyclization of the linear isomer of benzene is overcome [4]. The parameters that play a role in the first aromatic formation are several. The pressure of the system influences the kinetic parameters of many reactions so that, when varying the pressure, the dominant pathways may change and additional pathways may become significant. For instance, increasing the pressure, the relevance of ring-ring reactions increases and tends to prevail over the reactions involving the C 4 and propargyl radicals mentioned above. Similarly, other species besides resonantly stabilized radicals gain a more relevant role in the formation of the first aromatic ring, e.g. C 6 H x linear compounds [4] The HACA mechanism of aromatic growth Another species that plays a key role both in the formation of the first aromatic ring and in PAH growth is acetylene. Gaseous acetylene is considered the building block through which the main mechanism of aromatic growth proceeds in hydrocarbon flames [4]. Acetylene is involved in the second step of the well-known mechanism of Hydrogen- Abstraction-C 2 H 2 -Addition. First, a hydrogen atom activates an aromatic hydrocarbon molecule by abstracting one hydrogen atom from its structure: A i + H A i + H 2 (4.2) Then, gaseous acetylene is added to the activated aromatic hydrocarbon radical: A i + C 2 H 2 A i C 2 H 2 (4.3) Even though this mechanism could appear straightforward, particular kinetic and thermodynamic conditions are required to make it predominant.

32 Chapter 4. Literature Review 11 Concerning the first step of activation of the aromatic molecule, many reactions can prevent the HACA mechanism to proceed, like the reverse of the hydrogen abstraction or the recombination of the activated radical with a hydrogen atom [4]. Among these two reactions, the former one plays a more significant role with decreasing pressure and molecular size of the aromatic molecule A i [4]. Furthermore, the formation of the activated radical (A i ) may proceed through different reactions involving other types of radicals instead of the hydrogen atom. However, many investigations on shock-tubes and flames have demonstrated the H abstraction, shown in the reaction in Eq. 4.2, is predominant under several thermodynamic conditions [4]. In any case, the outcome with respect to activation of aromatic molecule precursors in the second step of the HACA machanism is the same. In order for the acetylene addition to complete the aromatic growth, two conditions have to be fulfilled [4]: The thermodynamic irreversibility of the two steps. The enhancement of the kinetics of the forward reactions in Eqs. 4.2 and 4.3. The simple acetylene addition A i + C 2 H 2 A i C 2 H 2 (4.4) has a high reversibility and even in the case of release of H, following the reaction A i + C 2 H 2 A i C 2 H + H (4.5) the reversibility may still prevail. The irreversibility of the acetylene addition, named reaction affinity (the opposite of the so-called reaction resistance ) is achieved through an entropy recovery and energy decrease of the system, that leads to the formation of highly stable hydrocarbons called stabilomers [4]. Depending on the temperature level, different kinetic regimes of the HACA mechanism establish and either the thermodynamic resistance or the rate of one or more sub-steps may control the PAH growth [4]. The sequences of reactions occurring in the process of PAHs growth terminates with irreversible steps involving ring closure which lead to the formation of stabilomers. An appropriate example is the following reaction of acetylene addition [4]: A i C 2 H 2 + C 2 H 2 A i+1 + H (4.6)

33 Chapter 4. Literature Review Consumption of benzene Depending whether the flame is a premixed or a diffusion flame as well as on the equivalence ratio and on the pressure of the system, the consumption of benzene occurs either through oxidation reactions and through pyrolysis reactions. Several experimental studies on the pyrolysis of benzene [12] both at high and low temperatures have shown the production of diacetylene (C 4 H 2 ) and acetylene as byproducts which are subsequently involved in the HACA mechanism. The pyrolysis of benzene is described by the mechanism of Bauer-Aten [13]: C 6 H 6 (+M) C 6 H 5 + H(+M) (4.7) C 6 H 6 + H C 6 H 5 + H 2 (4.8) C 6 H 5 (+M) n-c 4 H 3 + C 2 H 2 (+M) (4.9) n-c 4 H 3 (+M) C 4 H 2 + H(+M) (4.10) Acetylene is the by-product of a sequence of reactions that starts with the formation of the phenyl radical through the scission of a CH bond of the benzene ring. The latter is the rate limiting step in all conditions [14]. Through RRKM calculations [13] the formation of o-benzyne from the phenyl radical has been demonstrated to be relevant in the pyrolysis of benzene at temperatures higher than 1250 K: C 6 H 5 (+M) o-c 6 H 4 + H(+M) (4.11) The formation of acetylene and diacetylene was suggested to follow the decomposition of o-benzyne [13]: o-c 6 H 4 (+M) C 4 H 2 + C 2 H 2 (+M) (4.12) As stated above, the consumption of the fuel occur through reaction of both pyrolysis and oxidation. Relying on the investigation on the premixed benzene flame of Bittner and Howard [10] and on the subsequent study of Waldheim [6], the thermal decomposition of benzene shown in Eqs contributes 44% to benzene consumption. The second and third reaction of benzene consumption in order of importance are [6]: C 6 H 6 + OH C 6 H 5 + H 2 O (4.13) C 6 H 6 + O C 6 H 5 O + H (4.14)

34 Chapter 4. Literature Review 13 Figure 4.1: Major benzene oxidation pathways investigated in a jet-stirred reactor at 1000K, 10 atm and Φ = 1.5. The width of the arrows is proportional to the relevance of the reaction [1]. Figure 4.2: Major benzene oxidation pathways investigated in the Princeton reactor at Φ = The width of the arrows is proportional to the relevance of the reaction [1]. The three main reactions consuming the phenyl radical are [6]: C 6 H 5 + O 2 C 6 H 5 O + H 2 (4.15) C 6 H 5 O C 5 H 5 + CO (4.16) C 6 H 5 + H C 6 H 4 + H 2 (4.17) More precisely, the reaction in Eq is the main channel of formation of the phenoxy radical, that subsequently almost completely (90%) decomposes to cyclopendadienyl radical through the reaction in Eq [6]. The remaining phenoxy radical is converted to phenol [6]. The reaction in Eq shows that benzyne (C 6 H 4 ) formation contributes to the phenyl radical consumption. However, benzyne mainly undergoes hydrogenation (62%), forming the phenyl radical again [6]. As previously mentioned, in order to further investigate the mechanism of aromatic fuels consumption and to validate the computational results with a sufficient amount of experimental measurements at different physical conditions, experiments have been

35 Chapter 4. Literature Review 14 conducted using shock tubes for fuel pyrolysis, flow reactors for both fuel pyrolysis and oxidation and jet stirred reactors for fuel partial oxidation and combustion [1]. Studies on benzene oxidation conducted in the Princeton flow reactor at Φ = 1, 36 and in a pressurized jet-stirred reactor at 10 atm and 1000 K confirmed phenoxy radical to be a relevant intermediate in the pathway of benzene oxidation at different physical and chemical conditions [1]. Further investigations on benzene premixed flames with air at ambient pressure and temperature as well as on pressurized laminar premixed benzene flames over different equivalence ratios confirmed the central role of phenoxy radical in the oxidation of benzene, together with its dominant influence on the flame speed. In fact, the phenoxy recombination with H atoms to form phenol reduces the benzene flame speed [1]. In order to give to the reader an overview of the complexity of benzene oxidation in aromatic flames, Figures 4.1 and 4.2 show several major pathways under different physical and chemical conditions, investigated through reactor studies [1]. 4.2 Sectional soot modeling Even though a thorough description of the modeling of soot formation and oxidation exceeds the objectives of the current work, giving an overview of the methodology involved is important in order to properly contextualize the study of the aromatic chemistry of flames. The first soot models were approximated numerical methods for the solution of the master equations concerning particle coagulation. Among the latter ones, the method of moments relies on the assumption that knowing all the particle moments implies having the knowledge of the distribution function of the particles size [15]. The approximation necessary in order to achieve a reasonable computational effort, while determining the properties of the system with a sufficient accuracy, is to consider only the first few moments of the particle size distribution [15]. A further step has been represented by the introduction of discrete sectional models. They overcame the first approximate models since the particle size distribution is obtained by discretizing the particle sizes into classes of surrogate species referred as to bins with a fixed size relation between each other. More precisely, the classes of PAHs and soot particles are differentiated either basing on the number of carbon atoms or on the atomic mass [2]. The following digression relies on the works of Bhatt and Lindstedt [16] - concerning the sectional soot model subsequently improved by Waldheim [6] and used in the current work - and of Saggese et al. [2] - concerning the determination of particle size distribution of soot in a premixed ethylene flame applying another sectional soot model. The aerosol

36 Chapter 4. Literature Review 15 Figure 4.3: The NIST experimental system [17] dynamics modeling of the flame is subsequent to the gas phase modeling. The former is coupled with the gas-surface reactions responsible of the inception and growth of soot particles. Considering the surface chemistry involved in the soot growth, several heterogeneous reactions are taken into account, such as [2]: The HACA mechanism. The addition of resonantly stabilized radicals. The oxidation of soot particles. The first two classes in the list both occur in the gas-phase chemistry of the flame and involve solid soot particles as reactants. The boundary between the gas-phase chemistry and the chemistry of the soot solid particles is not sharp. In the modular discrete sectional approach [2], the first class of soot particles incepted is chosen to be a bin characterized by a certain mass or a certain number of carbon atoms, that is formed through the growth of PAHs in the gas-phase chemistry step. The choice of this smallest soot bin is arbitrary, however, in the soot model used by Saggese et al. [2] it is consistent with experimental measurements both concerning PAHs emission from sooting flames and concerning soot measured particle sizes. Another approach, adopted by Bhatt and Lindstedt [16], is to consider pyrene as the incepting species, i.e. as the zeroth bin. As previousely mentioned, the soot model used in the current work was developed by Bhatt and Lindstedt [16] and subsequently improved by Waldheim [6]. The model was first applied to the combustion of ethylene in the NIST reactor system in Fig.4.3. The

37 Chapter 4. Literature Review 16 latter was composed of a well-stirred-reactor (WSR) followed by a plug-flow-reactor (PFR). In the PFR, measurements of the particle size distribution of soot were available at different temperature and stoichiometric ratios [6]. As previously introduced, in this fixed sectional model representative particle sizes or bins are defined in order to avoid solving the population balance equation for obtaining the global particle size distribution [16]. The quantitative relation between the bins is fixed and given by [16]: m i+1 = f s m i (4.18) Where m i is the mass of the particle of the i-th bin and f s is the geometrical spacing factor equal to 1.5 [6]. Focusing on the aerosol dynamics, the modeling of the coagulation between soot particles relies on a simplified population balance equation. It gives the rate of variation (in particles/(m 3 s)) of the number of particles in the i-th representative particle class and it is given by [16]: dn i dt = k j i j,k m i 1 (m j +m k ) m i+1 ( 1 δ j,k 2 ) N Bin η i,j,k β j,k n j n k n i β i,k n k (4.19) where β i,k is the rate of attachment of particles of size class i with particles of size class k in m 3 /s and δ j,k is the Kronecker delta. The parameter η i,j,k ensures both the conservation of mass and of particle number. In fact, a particle of size class j +k - Newly generated from the aggregation of two particles of size classes j and k - is splitted into two adjacent bins: Either bins i 1 and i or bins i and i + 1 [16]. More precisely, η i,j,k is the fraction of the newly created particle of size class i, that the bin of size class i is assigned of, depending on the j and k size classes as follows [16]: η i,j,k = k=1 m i+1 (m j +m k ) m i+1 m i, if m i m j + m k m i+1 m i 1 (m j +m k ) m i 1 m i, if m i 1 m j + m k m i (4.20) The estimation of β i,k depends on the Knudsen number and on the geometry of the particles composing the bins [16]. The Knudsen number is given by Kn i = 2λ/d i, where λ is the mean free path and d i is the diameter of the particle i. Depending on the Knudsen number, the regime of particle motion changes, where Kn i 1, Kn i 1 and Kn i 1 correspond to the free-molecular, transition and continuum regime respectively. Concerning the surface chemistry, the addition of acetylene represented by the equation C s,i + C 2 H 2 C s,i+2 + H 2 (4.21)

38 Chapter 4. Literature Review 17 is stated to be the dominant reaction within the process of soot surface mass growth [6]. The superficial oxidation of soot particles occurs through the following reactions [6]: C s,i + OH C s,i 1 + H (4.22) C s,i + O C s,i 1 + CO (4.23) C s,i + O 2 C s,i 1 + CO + O (4.24) The soot surface chemistry of the current soot model was extended by including reactions analogous to the chemistry of naphthalene [6]. The numerical stability of sectional methods would not be ensured if the mass changes due to surface reactions was not properly taken into account [16]. More precisely, the surface reactions imply a gain or loss of mass of the soot particles and the numerical particle splitting previously described for the coagulation of soot particles is applied to take into account these mass changes as well. In the doctoral thesis of Waldheim [6] as well as in the current work, this process is simulated through a two point scheme (upwind, with a first order accuracy) modeling of the chemical movement of the particles from a bin to another depending on the extent of the superficial change of mass [6]. The formulation of the method depends whether the mass of the single particle increases (growth) or decreases (oxidation), as shown below [16]: dn i dt = I i 1n i 1 m i m i 1 I in i m i+1 m i, dn i dt = I in i m i m i 1 I i+1n i+1 m i+1 m i, in case of particle growth in case of particle oxidation (4.25) The extent of the mass change due to each surface reaction event is small if compared to the mass of the single particle since the surface kinetics involves small species compared to the dimension of soot particles. Moreover, the geometric spacing factor between bins makes the mass difference between two subsequent bins bigger if compared to the particle mass change. A particle formed after a surface reaction is splitted between the same bin (for the majority) and the previous one (in case of oxidation) or next one (in case of growth). An analogous three point scheme (with second order accuracy), could be adopted in order to avoid the need of a higher number of sections of the previous method, due to the increased numerical diffusion connected to it. However, since the three point method is less stable, it is not implemented in the model adopted in the current work [6].

39 Chapter 4. Literature Review Modeling of general reacting flows Introduction to the modeling of general reacting flows The PAH chemistry, including possible new pathways, will be investigated for two different laminar flames: a premixed benzene flame and a diffusion ethylene flame. Both flames have been recently studied by Waldheim [6]. Moreover, the study of both an aromatic and of an aliphatic flame also allows an evaluation of the influence of the fuel on the flame chemistry and structure. The added chemical pathways will be evaluated using FORTRAN codes purpose written for one-dimensional laminar premixed and diffusion flames. The codes have been developed by Lindstedt starting from the 1980s and refined by Waldheim [6]. The numerical methods applied by the FORTRAN based laminar flame codes is used to solve simplified equations governing the physics and the chemistry of the investigated flames The physics of general reacting flows An overview of the physics of general reacting flows is given in this section. More precisely, the description of the set of conservation equations that describe the laminar flames is developed. The simplifying assumptions adopted in the course of the analysis are later introduced. Both the set of conservation equation and the correspondent simplifying assumptions are thoroughly described in the doctoral thesis of Waldheim [6] Momentum equation The momentum equation is expressed below in the tensor form suitable for the analysis of a chemical system composed by several species [6]: ρu j t + ρu iu j x i = p x j + τ i,j x i n sp + ρ Y k f k,j (4.26) k=1 where u j represents the velocity field, Y k is the mass fraction of the species k, τ i,j is the viscous stress tensor and f k,j is the volumetric force acting on the species k in the x j direction. In the systems analyzed the volumetric force can be considered only due to the gravitational field [6], so that equation 4.26 becomes: ρu j t + ρu iu j x i = p x j + τ i,j x i + ρg i (4.27)

40 Chapter 4. Literature Review Species conservation equation The species conservation equation is the formulation of the conservation of mass of the species the system is composed of and, consequently, for soot particles in sooting flames. The general formulation of the species conservation equation is [6]: ρy k t + ( ρ(u i + V k,i )Y k ) x i = R k W k (4.28) where R k is the reaction rate, W k is the molar mass of the species k and V k,i is the diffusion velocity of the species k in the direction x i [6]. The latter is dependent on the binary diffusion coefficients D p,k of species p diffusing in species k and can be computed by solving the system of equations in Eq.4.29 for each time step, at each point of the computational domain and along each direction [6]: X p = n sp k=1 X p X k (V k V p ) + (Y p X p ) p D p,k p + ρ n sp p k=1 Y p Y k (f p f k ) (4.29) However, the above approach is too computationally expensive, prompting the need of a simplifying approximate treatment. Under the hypothesis of [6]: Neglecting the diffusion of the species k due to temperature gradients, Applying the Hirschfelder and Curtis approximation in order to avoid to solve the system in Eq. 4.29, which is too computationally expensive, Correcting the diffusion velocity of the species in order not to violate the conservation of mass due to the previous approximation, the final species conservation equation is [6]: Where [6]: ρy k t + ( ρ(u i + Vi c)y ) ( k = Y k (ρd k 1 x i x i x i n )) n + R k W K (4.30) x i 1. n is the inverse of the molecular weight 2. D k is a parameter introduced in the Hirschfelder and Curtis approximation, given by: D k = 1 Y k j k X j/d j,k (4.31)

41 Chapter 4. Literature Review V c i is the velocity correction mentioned above n sp ( Vi c Yk = D k 1 ) n x i n x i k=1 (4.32) Consequently, the diffusive velocity of a species k along the direction x i is given by [6]: ( 1 Y k V k,i = V c D k 1 Y k x i n ) n x i (4.33) Enthalpy conservation equation Depending on the flame structure and on the experimental set up, the accuracy of the temperature profile obtained through temperature measurments at different distances from the burner might be low. In these cases, the solution of the enthalpy conservation equation in the numerical scheme is necessary to compute a more accurate temperature profile. The implementation of the enthalpy equation in the numerical scheme depends on the coupling between the flow field and the temperature field. While investigating strained flames, inconsistencies between temperature profile and flow field might occur, for instance concerning the relative position of the peak of temperature and that of the stagnation plane. It will be shown that in the case of the ethylene diffusion flame studied by Olten and Senkan [11], the inaccuracies of the temperature measurements make the numerical implementation of the enthalpy equation necessary in order to validate subsequent temperature corrections [6]. Studying a chemical reacting system, the enthalpy variation can be expressed as the sum of the the chemical enthalpy variation - that takes into account the energy of the chemical bonds of the species composing the system - and of the sensible enthalpy variation - reflecting the system temperature change [6]. Under the hypothesis of [6]: Neglecting the viscous dissipation, Neglecting body forces, Applying the model of ideal gases ( h s = c p dt ), A suitable form for the enthalpy conservation equation for a chemically reacting system composed of several species is [6]: n sp ρh t + ρu ih = Dp x i Dt + λ h ρ Y k V k,i h k + Q. (4.34) c p x i k=1

42 Chapter 4. Literature Review 21 Under the same hypothesis, the sensible enthalpy equation is slighlty different from equation 4.34: n sp ρh t + ρu ih = Dp x i Dt + λ h s ρ Y k V k,i h s,k +. Q + h f,k R k (4.35) c p x i k=1 where the last term takes into account the enthalpy conversion from chemical to sensible. n sp k= Transport properties The transport properties of the different species can be computed based on different models, assumptions and simplifications in order to express them as a function of pressure, temperature and other physical variables of the thermodynamic system, whose measurement or estimation is feasible and reasonably technically challenging. The evaluation of the transport properties is necessary for being able to practically solve the conservation equations stated above. In other words, the equations listed above are the mathematical formulation of the system properties conservation and the correspondent advective and convective terms are dependent on the transport properties of the components of the system. The properties to be evaluated are: 1. The binary diffusion coefficients, needed for the computation of the diffusion velocity V k,i (see Eq. 4.33). The preliminary assumption made for the determination of the binary diffusion coefficients is that the model of the potential of the attractive/repulsive forces between molecules is the Lennard-Jones potential [6]: (( ) 12 ( ) 6 ) σ σ V L,J (r) = 4φ m r r (4.36) where σ and φ m are the collision diameter and the well depth, respectively, of the Lennard-Jones potential between molecules [18]. Assuming the Lennard-Jones potential model for inter-molecular forces and neglecting the exchange of rotational and vibrational energy between molecules, the binary diffusion coefficient D i,j of the species i through the species j can be express according to the Chapman and Cowling first order approximation [6]: D i,j = 3(k bt ) 3 ( 2 mi + m ) j 8pσi,j 2 W i,j(1) 2πm i m j (4.37) Where [6]:

43 Chapter 4. Literature Review 22 m i, m j are the masses of the correspondent species; k b is the Boltzmann constant; σ i,j is the average of the length scale parameters of the Lennard-Jones potential for the species i and j, given by σ i,j = (σ i + σ j )/2; W 1,2 (1) is a function of the temperature, of the Boltzmann constant and of the minimum of the Lennard-Jones potential given by φ m,i,j = (φ m,i φ m,j ) 1/2, i.e. it is a function of the reduced temperature T r = k b T/φ m,i,j. Values of this function are listed in the book of Chapman and Cowling [19]. 2. The determination of the viscosity of the gas mixture requires the estimation of the viscosity of the single gaseous components. The latter is estimated for a species k through a first order approximation, under the assumptions stated above for the evaluation of the binary diffusion coefficients [6]: µ k = 5(k bm k T ) 1 2 8σ 2 k W k(2) (4.38) Where W k (2) is again a function of the temperature, of the minimum of the Lennard-Jones potential and of the Boltzmann constant [6]. Values of this function are listed in the book of Chapman and Cowling [19]. Applying the Wilke mixing rule, it is possible to approximate the viscosity of the gas mixture with a first order accuracy [6]: µ = n sp k=1 Y i µ i nsp j=1 Y jφ i,j (4.39) Where Φ i,j is a mixing coefficient and a function of the viscosities µ k and of the molar masses of the species composing the gaseous mixture [6]: Φ i,j = (1 + (µ i/µ j ) 1/2 + (W j /W i ) 1/4 8(1 + W i /W j ) 1/2 (4.40) 3. The thermal conductivity of the gas mixture requires the estimation of the thermal conductivity of the single gaseous components. The latter is estimated for the generic species k as follows [6]: λ k = F k µ k C v,k (4.41) Where F k is a complex function of the self diffusion Schmidt number Sc k,k, the ratio of specific heat capacities γ of the species k, the contribution C v,rot of the rotational modes to the C v,k, the gas constant R and the number of collisions Z rot

44 Chapter 4. Literature Review 23 occurring before achieving the equilibrium of the transfer between rotational and kinetic energy [6]: ( ) 2 F k = (γ 1) + (5 3γ) 4 2Sc k,k 2 1 (γ 1) C v,rot (4.42) Sc k,k πrz rot Applying a mixing rule analogous to the Wilke one for viscosity shown in Eq. 4.39, it is possible to approximate the thermal conductivity of the gaseous mixture as follows: λ = n sp k=1 Y i λ i nsp j=1 Y jφ i,j (4.43) Determination of thermodynamic properties The determination and compilation of thermodynamic properties in the study of Waldheim [6] has several sources. As concerns the enthalpy of formation and the entropy of several small gaseous species within a wide temperature range, both radicals like the hydroxy radical, the oxygen and hydrogen atom as well as the major small stable species, the source is the thermodynamic properties database of Burcat and Ruscic [20]. The main aspect to take into account when determining the thermodynamic properties of a species is the evaluation of the uncertainties. Following the report of the Burcat and Ruscic concerning the correspondent database [20], the enthaply of formation at 298K of stable species up to three atoms relevant for combustion have been thoroughly evaluated and refined through calorimetric and spectroscopic methods. The level of accuracy of these species is the maximum achievable. The calorimetrical measurements are not applicable for unstable radicals: instead more difficult techniques are adopted like photoionization mass spectroscopy [20]. The consequence is a lower lever of accuracy. Regarding the more complex stable and radical species involved in the current kinetic mechanisms, the evaluation of the thermodynamic properties can be done through either quantum mechanical ab initio calculations or semi empirical molecular electronic structure calculations. A further step is the practical evaluation of the relation between the thermodynamic properties and the temperature of the system. The canonical polynomials, adopted by U.S. National Laboratories and Government agencies starting from the 1950s [20], are used in the current work. As an example, the enthalpy of a generic species at a certain temperature T is given by: h T RT = a 1 + a 2T 2 + a 3T 2 + a 4T 3 + a 5T 4 + a T (4.44)

45 Chapter 4. Literature Review 24 The enthalpy is calculated at different temperatures through the methods introduced above and the coefficients a i are subsequently obtained by fitting a polynomial expression. Moreover, the coefficients a i are evaluated for ranges of temperature of K and similar polynomials can be introduced for entropy and the Gibbs free energy, using the same coefficients and theory [20]. The polynomial formulation of c p is calculated as the temperature derivative of the enthalpy expression at constant pressure: c p R = a 1 + a 2 T + a 3 T 2 + a 4 T 3 + a 5 T 4 (4.45) The accuracy of the polynomial fit of the above mentioned polynomials is estimated in the range of % at the maximum temperature. Hence, the major source of inaccuracy derives from the evaluation of the data which the fits are based on. 4.4 Kinetics of chemical reactions Introduction to the kinetics of chemical reactions The last aspect to analyze in order to complete the introduction to the parameters involved in the modeling of the physics and of chemistry of flames is the chemical reaction kinetics. It is a broad and complex field and at this stage the aim is to give to the reader an overview of the equations adopted to model the reaction rates as a function of kinetic and thermodynamic parameters. The methods to determine the reaction rates are many and the aim of this section is to give an overview of some common ways to proceed when the (rare) direct measurements are not available Reaction rates Given a generic reaction: ν A A + ν B B ν C C + ν D D (4.46) The forward and backward reaction rates are given by: R f = k f [A] ν A [B] ν B (4.47) R r = k r [C] ν C [D] ν D (4.48) where k f and k r are respectively the forward and backward reaction rate constants, [A],[B],[C],[D], are the molar concentrations of the correspondent species and ν i are the stoichiometric coefficients. The expression generally adopted to determine the reaction

46 Chapter 4. Literature Review 25 rate constants is the modified Arrhenius equation: ( k = AT n exp E ) a RT (4.49) Where R is the gas constant, E a is the activation energy and A is the pre-exponential factor Reactions that involve a third body Several classes of reactions, e.g. many recombination reactions involves a third species as promoter of the reaction itself [21]. A possible representation of this case is given by the generic equation: A + B + M AB + M (4.50) The concentration of the third body influences the reaction rate since this type of reactions require a third body collision to proceed. In order to take it into account, a factor is added in the equations of the forward and backward reaction rates [21]: ( K ) R f = α k,i [X k ] k f [A][B] (4.51) k=1 ( K ) R r = α k,i [X k ] k r [AB] (4.52) k=1 where [X k ] is the molar fraction of the species k acting as a third body and α k,i is the correspondent third body enhancement factor. When all the components of the system make the same contribution as third bodies [21]: K K α k,i [X k ] = [X k ] = [M] (4.53) k=1 k=1 where [M] is the concentration of the gaseous mixture Reactions with pressure-dependent kinetics Using Eqs and 4.49 it is possible to express the relation between the reaction rates and the temperature of the system. However some classes of reactions show also a pressure dependence. Among them we quote the so-called fall-off unimolecular/recombination reactions.

47 Chapter 4. Literature Review 26 An appropriate recombination reaction whose rate shows a pressure dependence is the methyl recombination: CH 3 + CH 3 (+M) C 2 H 6 (+M) (4.54) The reaction above requires a third body collision to form products. However, above the high pressure limit, the concentration of the third body species is high so that the third body collisions are so frequent that this stage of the reaction does not limit the reaction rate. This is the reason why M is put into brackets [21]. Using another formulation, below and above certain levels of pressure, reaction in Eq. written respectively as: 4.54 can be CH 3 + CH 3 + M C 2 H 6 + M (4.55) CH 3 + CH 3 C 2 H 6 (4.56) The levels of pressure previously mentioned are the pressure limits above and below which the reaction rate constant of methyl recombination assumes the canonical temperature dependence: ( k = A T n exp E a, ( k 0 = A 0 T n 0 exp RT ) E a,0 RT ) (4.57) (4.58) Between the high and low pressure limits, the pressure dependence of the reaction rate constant can be modeled using the Lindemann approach [21]: ( k 0 [M] ) k = k k + k 0 [M] (4.59) Equation 4.59 takes into account the increasing third body effect while decreasing the pressure. When p, [M] and consequentially k k. Instead, when p 0, [M] 0 and consequentially k k 0 [M] [21]. In Fig. 4.4 the pressure dependence of the rate constant of methyl recombination for a fixed temperature of 1000K is shown [21]. Furthermore, it is possible to notice that also other approaches are adopted for the modeling of the pressure dependence besides the Linemann approach (for instance the Troe form, that is considered more accurate) [21].

48 Chapter 4. Literature Review 27 Figure 4.4: Rate constant of the methyl recombination as a function of the pressure at a fixed temperature [21] The estimation of reaction rates Estimations based on equilibrium considerations Consider the variation of the Gibbs free energy occurring during the generic reaction in Eq until it reaches the equilibrium: g r = ν C g C + ν D g D ν A g A ν B g B (4.60) The latter can be estimated as a function of the equilibrium temperature using the polynomials as shown for the enthalpy in the previous sections. The equilibrium constant is given by: ( ) K eq = [A]ν A[B] ν νc +ν D ν A ν B B p 0 [C] ν (4.61) C [D] ν D RT where p 0 =1 atm. The equilibrium constant is related to the variation of the Gibbs free enrgy, following the relation: ln K eq = g r RT Since at chemical equilibrium R f = R r it is straightforward to demonstrate that: (4.62) ( ) νc +ν D ν A ν B K eq = kf p 0 k r (4.63) RT

49 Chapter 4. Literature Review 28 Hence, by evaluating g r with polynomials and calculating K eq following Eq. 4.62, it is possible to calculate the forward reaction rate constant given the backward one (or vice versa) using Eq In other words, only the estimation of a single reaction rate constant (either forward or backward) is needed: The other one can be immediately calculated through Eq Estimations based on reaction class considerations Another method to evaluate the kinetic parameters of a reaction is the so-called reaction class based estimates. When reactants and products of two reactions show the same functional groups or similar structures, or in case the reactions involve common main species, it is possible to make estimations of the kinetic parameters of one reaction given the ones of the other one. Several application of this method are present in the work of Waldheim [6]. Among them, the estimation of the rate constants of the oxidation steps of the ortho-diethynylbenzene (C 10 H 6 ) is a useful example: It is equal to the kinetic parameters of the analogous oxidation steps of phenylacetylene (C 8 H 6 ) whose estimation was available [6]. Below two couples of reactions considered analogous are shown: C 10 H 6 + O C 8 H 5 + C 2 HO (4.64) C 10 H 6 + OH C 8 H 5 + C 2 H 2 O (4.65) C 8 H 6 + O C 6 H 5 + C 2 HO (4.66) C 8 H 6 + OH C 6 H 5 + C 2 H 2 O (4.67) Figure 4.5 points out the similarities between the structures of the species involved in the reactions listed above. 4.5 Overview of the flame systems studied in this work The laminar premixed benzene flame of Bittner and Howard The first flame investigated is the premixed benzene flame of Bittner and Howard [10]. The premixed reactant mixture has a molar composition of 13,5% C 6 H 6, 56,5% O 2 and 30% Ar, so that the fuel equivalence ratio is equal to 1.8. The latter value is slightly below the sooting limit of Φ = 1.9. The premixed reactants flows at a velocity of 0.5 m/s at 298 K and are stabilized on a burner with a diameter of 71 mm. The pressure of the burner chamber is 2.67 kpa. The flame composition was investigated by Bittner and Howard with a molecular beam mass spectrometer that allowed measurements of the

50 Chapter 4. Literature Review 29 Figure 4.5: Species involved in the reactions in Eqs. 4.64, 4.65, 4.66 and mole fraction profiles as a function of the distance from burner for 51 relevant chemical species. The aromatic species with the highest mass measured was pyrene and this limit was set by the sensitivity of the mass spectrometer. The calibration for the major stable species was direct whereas the absolute calibration for H and OH radicals was carried out using a stoichiometric benzene oxygen flame diluted with 30 vol% of Ar under the demonstrated condition of partial equilibrium between O 2 and H 2. The calibration for the remaining minor stable and unstable species was carried out using the ionization cross-section of similar stable species. The latter was thought to be accurate within a factor of 2 by the authors [10], an assumption confirmed by subsequent studies [14]. No attachment of the flame to the quartz probe was observed and, in order to avoid clogging, the diameter of the orifice was chosen to be 0.7 mm. However, the latter is considered by the authors being too large, so that it is a cause of uncertainty concerning the shape of the species concentration profiles even though their maxima were reproducible. Another relevant feature of the measurement system adopted by the authors concerns the temperature measurement. The thermocouple used was provided with a coating and a heating electrical resistance in order to compensate respectively for the catalytic effects and the radiation losses. The most particular aspect of this measurement system was the choice made after having obtained the temperature profile. Basing on the equilibrium of the reaction: H + H 2 O OH + H 2 (4.68) the temperature profile was uniformly decreased by 100 K and translated by 2 mm away from the burner, in order to properly take into account the perturbations of the flame due to the probe. The confidence on the accuracy of the temperature profile is enhanced by the agreement between the experimental H and OH profiles and the corresponding computed ones by Lindsted and Skevis [14].

51 Chapter 4. Literature Review 30 Figure 4.6: Experimental setup of the premixed benzene flame of Bittner and Howard [10] Further experimental investigations have been carried out, among them it is possible to quote the one of Yang et al. [1] for a premixed benzene flame diluted in Argon with Φ = 1.78 whose description exceedes the scope of the current work. However, the measurement system set by Bittner and Howard can be considered relevant still today. In Fig. 4.6 it is shown the experimental set up used by the authors [10]. A relevant aspect discussed by Bittner and Howard concerns the benzene decomposition. Observing figure 4.7 [10], the benzene flux appears to be almost constant until about 6 mm from the burner at which it starts to decrease rapidly. Comparing the latter to the mole fraction of benzene, the authors [10] concluded that the benzene chemical consumption starts at a certain distance from the burner. The decrease in the mole fraction of benzene observed within 6 mm from the burner is mainly due to its diffusion along the flame. There is a connection between the two phenomena: the diffusion of benzene occurring in the region close to the burner is driven by its chemical consumption starting at about 6 mm from the burner. The latter causes a steep decrease of concentration, dampened by the diffusion itself until it is reached a steady state. This study developed at the beginning of the 1980s by Bittner and Howard and the correspondent experimental results have been investigated in subsequent studies [6, 14, 22].

52 Chapter 4. Literature Review 31 Figure 4.7: Mole fractions of relevant species, flux and mole fraction of benzene as a function of the distance from the burner in the flame of Bittner and Howard [10] Flame chemistry A review of the kinetic mechanisms implemented used to simulate the flame of Bittner and Howard [10] is made in this section. This kinetic mechanism was developed in the studies of Lindstedt and Skevis [14], Lindstedt et al. [22] and Waldheim [6]. The aim is to outline the subsequent refinements introduced in terms of submechanisms and species took into consideration. Starting from the work of Lindstedt and Skevis [14], a relevant feature pointed out is the difference between the benzene consumption pathways comparing the low-temperature and high-temperature region of the flame. The former is the zone of the flame close to the burner, with an oxidant environment and a maximum temperature of about 1000 K. Reactions involving addition with low activation energy prevail, among which the addition of O to benzene leading to phenol predominates: C 6 H 6 + O C 6 H 5 OH (4.69) The latter region is the main reaction zone with temperatures above 1500 K. Reactions involving decomposition, ring opening and H abstraction prevail, among which the abstraction by H and OH predominates, leading to phenyl radical: C 6 H 6 + H C 6 H 5 + H 2 (4.70) C 6 H 6 + OH C 6 H 5 + H 2 O (4.71)

53 Chapter 4. Literature Review 32 The chemistry of C 4 species is stated to be central: resonantly stabilized n-c 4 H 5 and n-c 4 H 3 radicals are the products of the cyclopentadienyl radical (c-c 5 H 5 ) ring rupture in the low temperature zone and from the decomposition of the phenyl radical in the main reaction zone, respectively. The main C 4 products measured in the flame are diacetylene and vinylacetylene C 4 H 4, which are major sources for the production of acetylene. A relevant pathway for the production of vinylacetylene is the linearisation of the phenyl radical, the subsequent decomposition into n-c 4 H 3 and C 2 H 2 followed by hydrogen addition to n-c 4 H 3. The introduction of a C 5 submechanism is one of the main improvements provided by the work of Lindstedt and Skevis [14]. It starts with the decomposition of phenoxy radical: C 6 H 5 O c-c 5 H 5 + CO (4.72) and it is followed by the postulated opening of the carbon ring through O attack leading to n-c 4 H 5 and CO. This mechanism was introduced to provide a solution to the discrepancies between the computed and measured ratio CO/CO 2 in the postflame region. Robinson and Lindstedt [23] took an important step forward towards the understanding of C 5 submecanisms involved in the chemistry of aromatic fuels oxidation. Quantum mechanical G4/G4MP2 together with RRKM/ME methods were applied to determine the thermodynamic properties, the potential energy surfaces and the rates of several cyclopentadienyl consumption pathways. Among the latter, the pathways initiated by [23]: the formation of O through the reaction: C 5 H 5 + O 2 C 5 H 5 O + O (4.73) the reaction between C 5 H 5 and OH: C 5 H 5 + OH C 5 H 5 OH (4.74) are relevant. Another relevant pathway investigated was initiated by the reaction between C 5 H 5 and O, with alternative channels of formation of 1,3-butadienyl radical and carbon monoxide (Eq. 4.75) or of cyclopentadienone (Eq. 4.76) [23]: C 5 H 5 + O C=CC=C. + CO (4.75) C 5 H 5 + O C 5 H 4 O + H (4.76)

54 Chapter 4. Literature Review 33 Figure 4.8: Main formation pathway of Phenantrene in the flame of Bittner and Howard [6] Figure 4.9: Relevant formation pathway of Pyrene in the flame of Bittner and Howard [6] Moreover, the channel of decomposition of both cyclopentadiene and the cyclopentadienyl radical have been thoroughly studied [23]. The study of Lindstedt et al. [22] introduced further relevant improvements in the understanding of the chemistry of the flame of Bittner and Howard, like the introduction of kinetic schemes for the formation and oxidation of naphthalene and indene. A relevant pathway that will be further investigated also in the present work is the vinylacetylene addition to the phenyl radical, leading to the formation of naphthalene. The recent study of Waldheim [6] develops a thorough review of the whole reaction mechanism with relevant updates concerning the estimation of the kinetic parameters of several reactions. The main step further made in the study of Waldheim concerns PAHs growth. In the study of Lindstedt et al. [22] the kinetic modeling was developed for PAHs up to naphthalene and indene. Waldheim [6] improved the kinetic modeling and evaluated the implementation against the experimental data of the flame [10] for species up to pyrene. Figure 4.8 and 4.9 show two relevant kinetic routes [6] for the formation of phenanthrene and pyrene that involves two subsequent HACA steps. The combination of two phenyl radicals leads to biphenyl, that undergoes H abstraction releasing the ortho-phenyl-phenyl radical. The subsequent acetylene addition (that completes the

55 Chapter 4. Literature Review 34 Figure 4.10: Formation of Pyrene from cyclopenta[def]phenanthrene [6] Figure 4.11: Structure of the species relevant for the mechanisms of PAHs growth in the flame of Bittner and Howard HACA mechanism) leads to phenanthrene. A further HACA step occurs starting from phenanthrene leading to to pyrene, as depicted in Fig The possible alternative pathway for pyrene formation starts with the addition of triplet carbene to phenanthrene releasing 4-phenantrenylmethyl radical. The former undergoes ring-closure forming cyclopenta[def]phenanthrene. The final sequence that leads to pyrene is depicted in Fig Figure 4.11 gives an overview of the structures of the species relevant for the mechanisms of PAHs growth depicted above. In this section the author considers not appropriate to develop a short description of the complete kinetic mechanism since it would only narrow the analysis of Waldheim [6] Physics of a free-flowing laminar flat flame In this section the assumptions applied in the analysis by Waldheim [6] and made for simplifying the modeling of the physics of the current flame are shown. The main hypotheses is that the flame is one-dimensional and that the viscous effects are negligible since the axial velocity is low. Moreover, the temperature profile is not computed but measured: the reliability of the temperature measurements is considered sufficient and the physics of the flame implies a straightforward coupling between the velocity and temperature fields - v = f ( ρ(t, Y k ) ) -, so that no inconsistencies between the temperature and velocity fields are possible. For these reasons, the enthalpy equation

56 Chapter 4. Literature Review 35 is not implemented for the investigation of the flame of Bittner and Howard developed by Waldheim [6]. By modeling the system under these assumptions, the expressions of the continuity equation (Eq. 4.77) and of the species conservation equation (Eq. 4.78) are simplified as follows: ρy k t + ( ρ(v + V c )Y k ) y ρ t + ρv y = 0 (4.77) ( )) = (ρd Y k k y y 1 n + R k W K (4.78) n y The further step is the coordinate transformation introduced by fixing the constant mass flux along the coordinate system. defined as follows: More precisely, the stream function is introduced, ψ y = ρ (4.79) ψ t = ρv (4.80) Since the coordinate transformation implies a constant mass flux m, the mass conservation equation (Eq. 4.77) becomes implicit. ψ I and ψ E are the values of the stream function at the extremes of the domain and they are set in accordance to the initial conditions of the system. The conservation of mass is ensured since the computational domain changes its size following a change of density so that the constraint for the physical domain to accommodate the same mass is not violated. More precisely, the mass fraction profiles of the different species are computed in an equally-spaced, normalized ψ domain (as explained below) and a local change of density is instantaneously translated into a proportional change of the correspondent cell size in the y domain, following Eq The normalization of the ψ domain mentioned above is pursued through the following coordinate transformation: ω = ψ ψ I ψ E ψ I (4.81) where ω is the cross-stream function whose values are included in the range 0 ω 1. So, the final and unique equation to be implemented in order to compute the mass fraction profiles of the species in the flame is: ρy k t + ( 1 m + ρv c) Y k ψ E ψ I ω = ( 1 Y k (ρd (ψ E ψ I ) 2 k ω ω 1 n )) n + R k W K ω (4.82)

57 Chapter 4. Literature Review The laminar ethylene diffusion flame of Olten and Senkan The second flame investigated is the laminar counter-flow ethylene diffusion flame with a strain rate of 56.6 s 1 studied by Olten and Senkan [11]. As observed by the authors [11], this type of flame allows a technical advantage concerning the investigation of its structure due to its wider spatial extension if compared to the correspondent laminar flame. The additional parameter to be taken into account in a diffusion flame is the strain rate, which is the gradient of the velocity of the flame along the axial direction, calculated following Eq [11]: a = 2v 0 L ( 1 + v f v 0 ( ρf ρ 0 ) 0.5 ) (4.83) The strain rate a depends on the distance between the jet burners L, the oxidizer outlet velocity v 0, the fuel outlet velocity v f and the correspondent densities ρ 0, ρ f. It influences significantly the chemistry of the flame since both the residence time and the temperatures in the pre-flame zone decreases with the increasing strain rate. The experimental set-up adopted and described by Olten and Senkan is shown in Fig The counter-flow flame was stabilized between two nozzles with a diameter of 2.54 cm. The upper one injected the oxidizer flow composed by 22 vol% O 2 and 78 vol% Ar. The bottom burner injected the fuel together with Argon, the latter having a molar concentration of 75 vol% C 2 H 4 and 25 vol% Ar. Argon was also used to separate the Figure 4.12: Experimental set-up adopted by Olten and Senkan for the study of an ethylene diffusion flame [11]

58 Chapter 4. Literature Review 37 flame from the surrounding air through an annular co-flow established around the flame itself. The sampling device was composed by a microprobe made of quartz with an orifice diameter of mm. The measured species concentrations were transmitted from the quartz probe to an on-line gas chromatograph/mass spectrometer through a silicacoated stainless steel sampling line. In case calibration standards were available, species identification was made by comparing retention times and mass spectral fingerprints to standard mass spectra libraries. As for the concentration, calibration mixtures were used, with an estimated accuracy within 15%. In the other cases, the relative ionization cross section method was used, with and estimated accuracy within a factor of 2. The chemistry of the flame was not significantly influenced by the probe as investigated by the authors [11] both for pyrolysis within the probe and for reaction quenching. The contribution of radical conversion to stable species was numerically estimated to be between 1% and 5%. Temperature was measured using a type-r, Omega Pt/Pt+13%Rh thermocouple with mm wire, mm head coated with silica. In the sooting region of the flame, a soot deposit tended to form on the thermocouple and was subsequently burnt off at the end of each series of measurement. The accuracy of the position with the respect to the burner both of the temperature and of the species concentration profiles was estimated by the authors to be ±0.25 mm. The measurement devices were kept above 300 C to avoid PAH condensation. No radiation corrections were applied to the temperature measurements by the authors [11]. Hence, Waldheim [6] applied a temperature correction to the expermental measurements in order to take into account the radiation losses. For the same purpose, another temperature correction was applied to the temperature profile calculations, since the enthalpy equation solved for computing the temperature profile (see Eq ) does not account for radiation losses. The latter temperature correction is given by the empirical expression below: ( ( T T c = T 1 ξ T ad ) 2 ) (4.84) where ξ is the heat loss factor and T ad is the adiabatic temperature of the flame, set to and the maximum temperature of the flame without radiation losses, respectively Flame chemistry The laminar ethylene diffusion flame shows three zones, characterized by different colors. A blue zone extends for 2 mm towards the oxidizer nozzle starting at 7.5 mm from the fuel burner. A yellow zone develops just behind the previous one, closer to the fuel nozzle, ending in the orange zone. The maximum temperature of 1577 C was measured at about the boundary between the yellow and the blue zone. The value of maximum

59 Chapter 4. Literature Review 38 Figure 4.13: Temperature profile and mass fraction profiles of the major species of the laminar ethylene diffusion flame of Olten and Senkan [11] temperature is the one measured by Olten and Senkan [11] and therefore not subjected to the temperature corrections mentioned in the previous section. Observing Fig it is possible to characterize the zones depending on the class of reactions that occur. In the region in front of the fuel burner including the orange zone, the pyrolysis of the fuel occurs, leading mainly to vinyl radical (C 2 H 3 ) through hydrogen abstraction. Vinyl radical subsequently decomposes to acetylene [11]: C 2 H 4 + H C 2 H 3 + H (4.85) C 2 H 3 (+M) C 2 H 2 + H(+M) (4.86) This mechanism justifies the increasing trend of acetylene concentration that starts at about 2 mm from the fuel nozzle, as it can be observed in Fig In parallel to the increase in the acetylene concentration also CO concentration increases. This corresponds to the gradual transition from the pyrolysis zone to the partial oxidation zone of the flame, moving from the orange towards the yellow zone. According to Olten and Senkan [11], the partial oxidation of acetylene leading to carbene and carbon monoxide is the main reaction occurring in this zone: C 2 H 2 + O CH 2 + CO (4.87)

60 Chapter 4. Literature Review 39 In the study of Waldheim [6], the latter reaction is demonstrated to contribute only 7% to the acetylene consumption. Moreover, the main product of the oxygen attack to acetylene is demonstrated to be the ethynyloxy radical (11%), shown in the reaction in Eq The oxidation of acetylene through OH attack shown in the reaction in Eq is indicated to be the predominant channel of consumption of acetylene (30%). It is followed by hydrogen abstraction in the reactions with OH (24%) and H (19%) forming the ethenyl radical, shown in the reactions in Eqs and 4.91, respectively [6]. C 2 H 2 + O C 2 HO + H (4.88) C 2 H 2 + OH C 2 H 2 O + H (4.89) C 2 H 2 + OH C 2 H + H 2 O (4.90) C 2 H 2 + H C 2 H + H 2 (4.91) The decrease of CO concentration is connected to the beginning of the total oxidation zone, the blue one. The two main reactions that occur in this zone are the OH attack on hydrogen and on carbon monoxide, leading to the formation of the main combustion products, water and carbon dioxide [11]: H 2 + OH H 2 O + H (4.92) CO + OH CO 2 + H (4.93) Concerning the formation of the first aromatic ring, benzene concentration has its peak at about 4.5 mm from the fuel nozzle in the pyrolysis zone. As the majority of the products of the pyrolysis of ethylene have an even number of carbon atoms it is reasonable to suppose that even carbon mechanisms could be predominant in the formation of the first aromatic ring [11]. In fact, the numerical simulations done by Waldheim [6] confirms that the 40% of the total formation rate of benzene is due to the addition of vinyl radical to vinylacetylene: C 4 H 4 + C 2 H 3 C 6 H 6 + H (4.94) However, Waldheim [6] states that also propargyl recombination plays a relevant role in the formation of benzene, contributing 31% through the following mechanism involving the isomerisation of different linear C 6 H 6 species: C 3 H 3 + C 3 H 3 l-c 6 H 6 (4.95) l-c 6 H 6 C 6 H 6 (4.96)

61 Chapter 4. Literature Review 40 In order to avoid to make just a narrow resumé of the thorough chemical mechanisms implemented by Waldheim [6] concerning the formation of PAHs containing up to 4 aromatic rings, we leave the description of them to the sections concerning the flame computations. In the following section we focus on the modeling of the physics of the current flame in order to compare its features to the ones of the premixed benzene flame Physics of a laminar counter-flow diffusion flame The counter-flow diffusion flame streamlines can be presented as in Fig [24]. Under the hypothesis of inviscid and incompressible flow with an infinite stagnation plane, the 2-dimensional flow field for the current system is given by: u e = ax, v e = 2ay (4.97) where a is the strain rate, u e is the free stream tangential velocity along the radial direction and v e is the free stream transverse velocity along the axial direction [24]. The coordinate system is axisymmetric, so the continuity equation along the central Figure 4.14: Streamlines of a counter-flow diffusion flame [24]

62 Chapter 4. Literature Review 41 streamline is [24]: ρux x + ρvx y = 0 (4.98) Whereas the momentum and species transport equations along the central streamline are [24]: ρu u u + ρv x y + p x = ( µ u ) y y J k represents the molecular flux, expressed below [6]: (4.99) ρv Y k y = J k y + R kw k (4.100) ( ) Yk J k = ρd k y Y 1 n w k ρv c Y k (4.101) n w y The symbols adopted in the previous sections concerning the governing conservation equations are consistent to the symbols in Eqs. 4.99, 4.100, and Assuming the flow field in Eq. 4.97, the pressure gradient along the x-direction can be derived from Eq as: p x = aρ eu e (4.102) where ρ e is the density at the nozzles outlet [6]. A coordinate transformation is applied to the conservation equations listed above. First, the following dimensionless variables are introduced: Φ = u u e (4.103) V = ρv ( ρe µ e a ) 1/2 (4.104) ( ) a 1/2 y η = ρ dx (4.105) ρ e µ e 0 µ = ρµ ρ e µ e (4.106) ρ = ρ ρ e (4.107) Applying the previous substitutions, Eqs. 4.98, 4.99 and become: 2Φ + V η = 0 (4.108) V Φ η = ) (µ Φ Φ η η ρ (4.109) respectively. V Y k η = J k η + R kw k ρa (4.110)

63 Chapter 4. Literature Review 42 In Eq , J k is given by: ( J k = D k Yk ρ2 ρ e µ e η Y k 1 n w n w η ) ρv c Y k ( ρe µ e a ) 1/2 (4.111) Given the temperature measurements inaccuracy previousely outlined, the temperature profile is computed by solving the enthalpy conservation equation. Considering the first assumptions made concerning the physics of the flame, the enthalpy equation for the current flame is given by: ρv h y = ( ) λ h + [ n sp ( h k J k y c p y y y λ )] Y k c p y k=1 (4.112) Substituting the dimensionless variables listed above, Eq becomes: V h η = ( ) ρλ h + [ n sp η ρ e µ e c p η η k=1 h k ( J k η ρλ )] Y k ρ e µ e c p η (4.113) At the nozzles exits, the velocities have only the transverse component v e respectively equals to cm/s and cm/s for the oxidizer nozzle and for the fuel nozzle (with opposite signs). At the nozzle exist Φ is set to 1 and the temperatures are set to the ones measured closest to both sides. The boundary condition applied for species at the Figure 4.15: Measured and calculated temperature profiles of the laminar ethylene diffusion flame ofolten and Senkan. The measured temperature values are identified by circles, the radiation corrected temperature profile by squares, the computed temperature profile by the solid line and the temperature profile calculated after reducing the fuel stream velocity of 25% by the dash-dotted line. The figure is taken from the doctoral thesis of Waldheim [6].

64 Chapter 4. Literature Review 43 boundaries of the computational domain is: ( V Yk + J k) e = V Y k,e (4.114) As previously mentioned, the temperature profile of the flame computed through Eq is validated against the corrected temperature profile. Figure 4.15 shows the measured temperature values (circles), the radiation corrected temperature values (squares), the computed temperature profile (solid line) and the temperature profile calculated after reducing the fuel stream velocity of 25% (dash-dotted line) [6]. It is evident that the computed temperature profile is shifted towards the fuel nozzle if compared to the measurements. Due to this temperature uncertainty, the species molar fraction profiles have been computed by Waldheim [6] for three different cases: using the computed temperature profile itself (solid line) and uniformly increasing and decreasing the computed temperature profile of ±100K.

65 Chapter 5 Novel pathways of naphthalene formation 5.1 Prediction of aromatic species concentrations in the flames studied in this work In order to improve the prediction of PAHs concentrations in the flames studied in this work, novel pathways of PAH formation have to be investigated. Figure 5.1 shows that the concentrations of naphthalene (C 10 H 8 ), phenanthrene (A 3 ) and pyrene (A 4 ) are underpredicted by one/two orders of magnitude in the laminar ethylene diffusion flame of Olten and Senkan [6]. Concerning the premixed benzene flame of Bittner and Howard, Fig. 5.2 shows that the calculated naphthalene concentration profile [6] agrees with the experimental measurements, since they differ by less than a factor of two for a broad range of distances from the burner. This factor is indicated to be the maximum level of inaccuracy of the measurements performed by Bittner and Howard [10]. Moreover, the modeled submechanism of pyrene oxidation underestimates the consumption of pyrene in the flame [6]. As shown in Figures 5.1 and 5.2, the peak of the fractional profile of vinylacetylene occurs at a distance from the burner close to the region of maximum concentration of naphthalene, phenanthrene and pyrene. Pathways involving the addition of the phenyl radical (C 6 H 5 ) to vinylacetylene leading to naphthalene formation have been the subject of several investigations [25, 26]. Novel pathways of formation of naphthalene [27] involving vinylacetylene addition to the phenyl radical open up new horizons for the potential contribution to PAH formation of this class of reactions. The features of the 44

66 Chapter 5. Novel pathways of naphthalene formation 45 Figure 5.1: Relevant measured (circles) and calculated (dash dotted lines) species profiles of the laminar ethylene diffusion flame of Olten and Senkan investigated by Waldheim [6] Figure 5.2: Relevant measured (circles) and calculated (dash dotted lines) species profiles of the laminar premixed benzene flame Bittner and Howard investigated by Waldheim [6].

67 Chapter 5. Novel pathways of naphthalene formation 46 novel pathways oulined in the recent study of Parker et al. [27] are described in the following sections, with respect to analogous pathways previously studied [25, 26]. 5.2 Pathways of napthalene formation involving vinylacetylene addition to the phenyl radical Previous investigations In order to give the reader an overview of relevant pathways of napthalene formation involving vinylacetylene addition to the phenyl radical, the study of Moriarty and Frenklach [25] is examined. This study outlines different chemical dynamics of this class of reactions, depending on the initial relative orientation of the reactants. More precisely, the addition of the phenyl radical to either the vinylacetylene double and triple bond was investigated [25] The addition of the phenyl radical to vinylacetylene triple bond Figures 5.3 and 5.4 [25] show two different dynamics of the addition of the phenyl radical to vinylacetylene triple bond, leading to two rotational isomers of the phenyl-1,3- butadien-2-yl radical (A 1 CHCCHCH 2 or C 10 H 9 (B)). In the pathway shown in Fig. 5.3, Figure 5.3: Pathway of naphthalene formation involving the addition of the phenyl radical to the vinylacetylene triple bond. Two subsequent rotations occur around the single and double bond within the side chain [25].

68 Chapter 5. Novel pathways of naphthalene formation 47 Figure 5.4: Pathway of naphthalene formation involving the addition of the phenyl radical to the vinylacetylene triple bond. Only one rotation occurs around the single bond within the side chain [25]. Figure 5.5: Planar representations of C 10 H 8 (G), C 10 H 8 (J) and C 10 H 9 (L) consistent with the study of Moriarty and Frenklach [25]. the C 10 H 9 (B) radical that is formed after the first addition step with an energy barrier of about 8 kj/mol may undergo hydrogen removal forming either phenylbutatriene (A 1 CHCCCH 2 or C 10 H 8 (G)) or phenyl-buta-1-yn-3-ene (A 1 CCCHCH 2 or C 10 H 8 (J)). These branching reactions have higher energy barriers than the step of hydrogen migration from the aromatic ring of the C 10 H 9 (B) radical to the side chain leading to the 1,3-butadienyl-benzen-2-yl radical (C 10 H 9 (L)). In Fig. 5.5 the structures of C 10 H 9 (G), C 10 H 8 (J) and C 10 H 9 (L) [25] are shown. The highest energy barrier of the pathway in Fig. 5.3 corresponds to the step involving the rotation about the double bond within the side chain of the C 10 H 9 (L) radical, leading to the second aromatic ring formation [25]. Since this step may be rate limiting, the second pathway shown in Fig 5.4 was investigated [25]. In this pathway the intermediate formed after the first reaction (with a

69 Chapter 5. Novel pathways of naphthalene formation 48 Figure 5.6: Pathway of naphthalene formation involving the addition of the phenyl radical to vinylacetylene double bond studied by Moriarty and Frenklach [25] similar energy barrier if compared to the previous one) requires only one rotation about a single bound within the side chain before the formation of the second aromatic ring. However, in this case the rate limiting step may be the hydrogen migration within the side chain. In both pathways, the last step required for the formation of naphthalene is the hydrogen removal from the sp 3 hybridized carbon of the 1-hydro-naphtalene-2-ylradical [27] The addition of the phenyl radical to vinylacetylene double bond Figure 5.6 shows the mechanism of naphthalene formation involving the addition of the phenyl radical to vinylacetylene double bond studied by Moriarty and Frenklach [25]. The high energy barriers of the second and the last steps may slow down the global reaction rate [25]. However, the computed reaction rate of this pathway is comparable to the one of the pathway involving the addition of phenyl radical to vinylacetylene triple bond [25]. The global reaction rate constants of all the above mentioned pathways are relatively low, i.e. the dominant pathways of naphthalene formation involve the production of phenyl-c 4 H 3 species, like phenylbutatriene as shown in Fig The mechanism in Fig. 5.7 was implemented in the kinetic scheme adopted in the flame calculations of Waldheim [25]. The only channel of phenylbutatriene formation is the addition of the phenyl radical to vinylacetylene. The hydrogen addition to phenylbutatriene accounts

70 Chapter 5. Novel pathways of naphthalene formation 49 Figure 5.7: The most probable pathway of formation of naphthalene subsequent to vinylacetylene addition to the phenyl radical investigated by Moriarty and Frenklach [25] for 30.6% of C 10 H 9 (T) formation and the subsequent H removal accounts for 43.65% of naphthalene formation in the ethylene diffusion flame of Olten and Senkan Inspiration from the studies of astrochemical evolution of the interstellar medium A broad literature review was performed with the aim to find pathways napthalene formation involving vinylacetylene addition to the phenyl radical with a different chemical dynamics from the one observed in the pathways decribed in the previous section. The inspiration of the current study derives from a recent investigation on naphthalene formation at low temperatures occurring in carbon-rich stars done by Parker et al. [27]. In fact, PAHs are not only harmful intermediates formed during the combustion of hydrocarbons: they are also key species involved in the evolution of the interstellar medium [27]. More precisely, pathways of PAH formation were found to be relevant in the processes of molecular growth in the circumstellar envelopes of carbon rich stars [27]. Consequentially, the investigations of the pathways of PAH formation in the interstellar medium open up new horizons in the understanding of the kinetics of PAH formation in flames. Since the temperature ranges of the studied interstellar systems are extremely low (around 10 K), the pathways investigated either show low or zero energy barriers at the entrance. The investigation of the mechanism of naphthalene formation was performed through electronic molecular structure calculations [27]. Three alternative relative orientations of the reactants were considered: the addition of

71 Chapter 5. Novel pathways of naphthalene formation 50 Figure 5.8: Potential energy surface of the barrier-less pathway involving the the phenyl radical addition to C1 of vinylacetylene [27] the phenyl radical to different carbon atoms of vinylacetylene - named C1, C2 and C4 - was investigated [27]. The addition of the phenyl radical to C1 of vinylacetylene - the vinyl group characterized by a carbon-carbon double bond - is shown in Fig It has a zero energy barrier at the entrance due to the formation of Van der Waals interactions between the reactants leading to the first transition state. The addition to C4 of vinylacetylene - the acetylene group characterized by a carbon-carbon triple bond - is depicted in Fig. 5.9 and it has an energy barrier at the entrance of 5 kj/mol. In both cases the first transition state has the highest energy level of the pathway, i.e. both reactions show submerged barriers [27]. The last intermediate before the formation of naphtalene is the 1-hydro-naphtalene- 2-yl radical: the same species of the pathways investigated in the study of Moriarty and Frenklach [25]. Similarly, the 1-hydro-naphtalene-2-yl radical undergoes hydrogen dissociationl to form naphthalene. The pathway in Fig is initiated by the phenyl radical addition to the C2 of vinylacetylene. Due to the high number of isomerisation steps and due to the higher barrier at the entrance, this pathway is the least likely one to give a relevant contribution to PAH formation among the three investigated by Parker et al. [27]. The definition of the energy barries and the description of the methodology adopted to compute the energy levels of the pathways is given in Chapter 7.

72 Chapter 5. Novel pathways of naphthalene formation 51 Figure 5.9: Potential energy surface of the pathway involving the phenyl radical addition to C4 of vinylacetylene with a 5 kj/mol barrier at the entrance [27] Figure 5.10: Potential energy surface of the pathway involving the phenyl radical addition to C2 of vinylacetylene with a 17 kj/mol barrier at the entrance [27] Moriarty and Frenklach [25] outlined two key features of the pathways of naphthalene formation involving vinylacetylene addition to the phenyl radical: The hydrogen migration from the side chain to the aromatic ring shown in Fig. 5.11

73 Chapter 5. Novel pathways of naphthalene formation 52 Figure 5.11: Hydrogen migration from the side chain to the aromatic ring [25] was demonstrated to occur at a rate sufficient for making this step relevant in hightemperature aromatics chemistry. The rate limiting steps of the pathways involving the phenyl radical addition to the vinylacetylene triple and double bonds were demonstrated to be either the first addition step or the trans-cis isomerisation occurring within the side chain of the intermediate species. The study of Parker et al. [27], instead, introduces the possibility that only the first addition step is rate limiting, i.e. the trans-cis isomerisation within the side chain would not represent a kinetic bottleneck since the energy barriers involved in this step is submerged below. The activated conglomerates with a sufficient energy to overcome the barrier at the entrance of the pathway isomerise through the configurations correspondent to the wells of the potential energy surfaces in Figures 5.8, 5.9 and 5.10 (depending on the relative orientation of the reactants) within a time scale faster if compared either to the one characteristic of chemical changes or to the one of the initial addition step.

74 Chapter 6 Preliminary flame calculations 6.1 Introduction to the preliminary flame calculations In the previous chapter, various relevant PAH formation pathways were described. In order to assess their significance in terms of aromatic yield, preliminary flame calculations are performed. Approximated kinetic parameters are used in order to assess which pathway among the ones preliminary tested is the most significant for the aromatic growth in the studied flames. The first pathway to be tested is taken from the study of Moriarty and Frenklach [25]. Then, the significance of the novel PAH formation pathways of Parker et. al [27] is investigated. 6.2 Testing a PAH formation pathway from a previous literature study Relying on the study of Moriarty and Frenklach [25] mentioned in chapter 5, the rate constants of 1-hydro-napthalen-2-yl (C 10 H 9 (T)) formation computed at amospheric pressure at 1500 K and 2000 K are equal to and cm3 mol s, respectively. As a first attempt, these values were interpolated in the form of the modified Arrhenius equation in order to model the correspondent temperature dependence: k = e RT (6.1) Where A is expressed in m3 kmol s and E a in kj kmol. Due to the lack of further data concerning the kinetics of the intermediate steps leading to the 1-hydro-naphthalen-2-yl radical 53

75 Chapter 6. Preliminary flame calculations 54 (C 10 H 9 (T)) formation, the rate constant in Eq. 6.1 was assumed for the global reaction: C 6 H 5 + C 4 H 4 C 10 H 9 (T) (6.2) The reaction was applied to the flames kinetic mechanism, representing a new channel of formation of C 10 H 9 (T). No significant improvements in the prediction of the fractional species profiles of aromatic species resulted after this change, since the reaction in Eq. 6.1 accounted for 0.02% to 1-hydro-napthalen-2-yl radical C 10 H 9 (T) formation both for the benzene premixed and for the ethylene diffusion flame. 6.3 Testing the novel PAH formation pathways Interpolation of the low temperature kinetic rate constants of the barrier-less pathway Relying on the study of Parker et al. [27], the rate constants of the first step of the barrier-less pathway in Fig. 5.8 are equal to and s 1 at 298 K and 100 K, respectively. Therefore, the first barrier-less step is favored at low temperatures. The first two isomers of the pathway were not modeled in the flame codes used by Waldheim [6], i.e. no channel of formation of naphthalene starting from the configuration of the first well of the PES in Fig. 5.8 can be applied to the flames kinetic mechanism Figure 6.1: Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27].

76 Chapter 6. Preliminary flame calculations 55 Figure 6.2: Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27]. without preliminary calculations of molecular properties. Assuming the first step to be rate limiting, the two rate constants given by Parker et al. [27] were interpolated in the form of the modified Arrhenius equation and associated with the global reaction: C 6 H 5 + C 4 H 4 C 10 H 9 (T) (6.3) The last step of the barrier-less pathway (C 10 H 9 (T) C 10 H 8 + H) was already implemented in the flame kinetic scheme, i.e. a new channel of formation of napthalene has been implemented following this methodology. reaction in Eq. 6.3 is: The rate constant expression for the k = e RT (6.4) Where A is expressed in m3 kmol s and E a in kj kmol. Moreover, the dynamics of formation of the C 10 H 9 (T) radical can be extended to an homologous mechanism involving the formation of hydro-phenanthrene (A 3 H) initiated by the vinylacetylene addition to the naphthyl radical. For this reason the expression in Eq. 6.4 obtained for the reaction in Eq. 6.3 was extended to the reaction of hydro-phenanthrene formation involving vinylacetylene addition to the 1- and 2- naphthyl radicals: 1-C 10 H 7 + C 4 H 4 A 3 H (6.5) 2-C 10 H 7 + C 4 H 4 A 3 H (6.6)

77 Chapter 6. Preliminary flame calculations 56 Figure 6.3: Molecular structures of 1-C 10 H 7, 2-C 10 H 7 and A 3 H Table 6.1: Significance of the reactions currently applied to the premixed benzene flame kinetic scheme based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27]. When the minus sign is indicated into brackets, the correspondent reaction turned reverse. Reaction Prod./Cons. Integral contribution Peak contribution C 6 H 5 + C 4 H 4 C 10 H 9 (T) C 10 H 9 (T) Prod % 38.16% C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 36.79% C 10 H 9 (T) C 10 H 8 + H C 10 H 9 (T) Cons % 98.77% 1-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 48.56% 2-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod. (-)0.05% (-)15.25% A 3 H A 3 + H A 3 Prod. 2.67% 1.54% Table 6.2: Significance of the reactions currently applied to the ethylene diffusion flame kinetic scheme based on the low temperature rate constants interpolation of the barrier-less pathway of Parker et al. [27]. Reaction Prod./Cons. Integral contribution Peak contribution C 6 H 5 + C 4 H 4 C 10 H 9 (T) C 10 H 9 (T) Prod % 32.75% C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 55.77% C 10 H 9 (T) C 10 H 8 + H C 10 H 9 (T) Cons % 98.90% 1-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 27.01% 2-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 70.26% A 3 H A 3 + H A 3 Prod % 49.89% The species involved in the reactions listed above are shown in Fig In Table 6.1 the integral and peak value contributions to the production and consumption of the species involved in the reactions in Eqs. 6.3, 6.5 and 6.6 are listed for the laminar premixed benzene flame. In Table 6.1, the relevant contribution to C 10 H 8 and A 3 production of the reactions currently applied to the correspondent flame kinetic scheme is assessed. Figure 6.1 shows a comparison between the species profiles relevant for the current pathway computed by Waldheim [6] and resulting from the changes in the

78 Chapter 6. Preliminary flame calculations 57 kinetic scheme of the flames described in this subsection. No significant changes with respect to the concentration profiles computed by Waldheim [6] are observable. The same procedure was implemented for the ethylene diffusion flame of Olten and Senkan and the correspondent results are shown in Table 6.2 and in Fig The improvement in the prediction of naphthalene concentration is insignificant, whereas the currently computed phenanthrene and pyrene mole fraction profiles are one order of magnitude higer around the maxima if compared to the ones computed by Waldheim [6]. Further comparisons or explanations are avoided since they will not be sufficiently robust due to the low reliability of the low-temperature interpolation, in reality an extrapolation for the high temperature of interest for combustion Testing the barrier-less pathway assuming the rate of collisions as an estimate of the pre-exponential factor A simplifying assumption was formulated in order to perform further flame calculations with the aim to assess the possible improvements in the prediction of PAHs concentrations subsequent to the application of the barrier-less pathway investigated by Parker et al. [27] to the kinetic model of the studied flames. In this subsection, the results of the flame calculations performed assuming the approximate theoretical rate of collisions as an estimate of the rate of the reactions in Eq. 6.3, 6.5 and 6.6 are presented. The Figure 6.4: Species profiles of the benzene premixed flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the barrier-less pathway of Parker et al. [27].

79 Chapter 6. Preliminary flame calculations 58 Figure 6.5: Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6]. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the barrier-less pathway of Parker et al. [27]. Table 6.3: Significance of the reactions currently applied to the premixed benzene flame kinetic scheme based on an approximate theoretical collision rate of the barrierless pathway of Parker et al. [27]. Reaction Prod./Cons. Integral contribution Peak contribution C 6 H 5 + C 4 H 4 C 10 H 9 (T) C 10 H 9 (T) Prod % 68.03% C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 51.00% C 10 H 9 (T) C 10 H 8 + H C 10 H 9 (T) Cons % 98.33% 1-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 67.80% 2-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod. 4.95% 16.79% A 3 H A 3 + H A 3 Prod. 5.63% 7.10% reaction rate adopted was k = m 3 /(kmol K) [28]. In Tables 6.3 and 6.4 the integral and peak value contributions to the production and consumption of the species involved in the reactions in Eqs. 6.3, 6.5 and 6.6 are listed for the the premixed benzene and the ethylene diffusion flame, respectively. Moreover, Figures 6.4 and 6.5 show the comparison between the species profiles relevant for the current pathway computed by Waldheim [6] and those resulting from the changes described in this subsection concerning the kinetic scheme of the premixed benzene and the ethylene diffusion flame. After the current changes the computed naphthalene concentration in the benzene premixed flame almost triples the one obtained by Waldheim [6]. The changes in the computed phenanthrene and pyrene concentration profiles are instead insignificant. Concerning the

80 Chapter 6. Preliminary flame calculations 59 Table 6.4: Significance of the reactions currently applied to the ethylene diffusion flame kinetic schem based on an approximate theoretical collision rate of the barrierless pathway of Parker et al. [27]. Reaction Prod./Cons. Integral contribution Peak contribution C 6 H 5 + C 4 H 4 C 10 H 9 (T) C 10 H 9 (T) Prod % 63.16% C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 70.84% C 10 H 9 (T) C 10 H 8 + H C 10 H 9 (T) Cons % 98.48% 1-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 28.03% 2-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 71.08% A 3 H A 3 + H A 3 Prod % 65.85% ethylene diffusion flame, the computed naphthalene concentration profile almost doubles the one obtained by Waldheim [6]. The currently computed phenanthrene and pyrene mole fraction profiles are more than one order of magnitude higer around the maxima if compared to the ones computed by Waldheim [6] Testing the pathway with a 5 kj/mol barrier at the entrance In the current subsection the results of the test performed for the pathway with a 5 kj/mol barrier at the entrance (see Fig. 5.9) are shown. The pre-exponential factor was again estimated basing on an approximate theoretical rate of collisions equal to Figure 6.6: Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], repectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 5 kj/mol barrier at the entrance.

81 Chapter 6. Preliminary flame calculations 60 Figure 6.7: Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 5 kj/mol barrier at the entrance. Table 6.5: Significance of the reactions currently applied to the premixed benzene flame kinetic scheme, testing the pathway with a 5 kj/mol barrier at the entrance Reaction Prod./Cons. Integral contribution Peak contribution C 6 H 5 + C 4 H 4 C 10 H 9 (T) C 10 H 9 (T) Prod % 59.85% C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 46.38% C 10 H 9 (T) C 10 H 8 + H C 10 H 9 (T) Cons % 98.54% 1-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 62.19% 2-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod. 2.31% 15.49% A 3 H A 3 + H A 3 Prod. 4.11% 4.48% m 3 /(kmol K) [28]. The activation energy was roughly estimated to be equal to the barrier at the entrance of the pathway [28]. Consequentially, the kinetic rate constant adopted for the reactions in Eqs. 6.3, 6.5 and 6.6 is equal to: k = e RT (6.7) Where A is expressed in m3 kmol s and E a in kj kmol. Similarly to the previous subsection, in Tables 6.5 and 6.6 the integral and peak value contributions to the production and consumption of the species involved in the reactions in Eqs. 6.3, 6.5 and 6.6 are listed for the the premixed benzene and the ethylene diffusion flame, respectively. Moreover, Figures 6.6 and 6.7 show the comparison between the species profiles relevant for the current pathway computed by Waldheim [6] and resulting from the changes described in

82 Chapter 6. Preliminary flame calculations 61 Table 6.6: Significance of the reactions currently applied to the ethylene diffusion flame kinetic scheme, testing the pathway with a 5 kj/mol barrier at the entrance Reaction Prod./Cons. Integral contribution Peak contribution C 6 H 5 + C 4 H 4 C 10 H 9 (T) C 10 H 9 (T) Prod % 54.41% C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 66.90% C 10 H 9 (T) C 10 H 8 + H C 10 H 9 (T) Cons % 98.58% 1-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 27.89% 2-C 10 H 7 + C 4 H 4 A 3 H A 3 H Prod % 70.90% A 3 H A 3 + H A 3 Prod % 62.52% this subsection concerning the kinetic scheme of the premixed benzene and the ethylene diffusion flame. The same observations done for the flames simulations testing the barrier-less pathway assuming an approximate rate of collision as an estimate of the pre-exponential factor can be replicated for the current cases Testing the pathway with a 17 kj/mol barrier at the entrance The same procedure described in the previous subsection was replicated in order to estimate the potential contribution of the pathway with a 17 kj/mol barrier at the entrance (see Fig. 5.10) to naphthalene and phenanthrene formation. However, this pathway is Figure 6.8: Species profiles of the premixed benzene flame of Bittner and Howard. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 17 kj/mol barrier at the entrance.

83 Chapter 6. Preliminary flame calculations 62 Figure 6.9: Species profiles of the ethylene diffusion flame of Olten and Senkan. The circles and the dash dotted curve represent the measured species concentrations and the species profiles computed by Waldheim [6], respectively. The solid curve represents the currently computed species profiles based on an approximate theoretical collision rate of the reaction pathway with a 17 kj/mol barrier at the entrance. Table 6.7: Significance of the reactions currently applied to the premixed benzene flame kinetic scheme, testing the pathway with a 17 kj/mol barrier at the entrance. When the minus sign is indicated into brackets, the correspondent reaction turned reverse. Reaction Prod./Cons. Integral contribution Peak contribution C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 16.19% 1-C 10 H 7 + C 4 H 4 A 3 + H A 3 Prod. 0.52% 1.17% 2-C 10 H 7 + C 4 H 4 A 3 + H A 3 Prod. (-)0.03% (-)0.35% Table 6.8: Significance of the reactions currently applied to the ethylene diffusion flame kinetic scheme, testing the pathway with a 17 kj/mol barrier at the entrance. Reaction Prod./Cons. Integral contribution Peak contribution C 10 H 9 (T) C 10 H 8 + H C 10 H 8 Prod % 18.48% 1-C 10 H 7 + C 4 H 4 A 3 + H A 3 Prod % 14.15% 2-C 10 H 7 + C 4 H 4 A 3 + H A 3 Prod % 35.13% not terminated by the hydrogen removal from the C 10 H 9 (T) radical. Therefore, assuming the first step of the pathway to be rate limiting, the reactions currently integrated in the flame kinetic schemes involve the direct formation of naphthalene and pyrene: C 6 H 5 + C 4 H 4 C 10 H 8 + H (6.8) 1-C 10 H 7 + C 4 H 4 A 3 + H (6.9)

84 Chapter 6. Preliminary flame calculations 63 2-C 10 H 7 + C 4 H 4 A 3 + H (6.10) The expression of kinetic rate constant adopted for the reactions in Eqs. 6.8, 6.9 and 6.10 is: k = e RT (6.11) Where A is expressed in m3 kmol s and E a in kj kmol. The integral and peak value contributions to the production and consumption of the species involved in the reactions in Eqs. 6.8, 6.9 and 6.10 for the the premixed benzene and the ethylene diffusion flame are listed in Tables 6.7 and 6.8, respectively. Furthermore, Figures 6.8 and 6.9 show the comparison between the species profiles relevant for the current pathway computed by Waldheim [6] and resulting from the changes described in this subsection concerning the kinetic scheme of the premixed benzene and the ethylene diffusion flame. No significant improvement in the prediction of napthtalene concentration is observed for both flames. The same applies to phenanthrene and pyrene concentrations for the premixed benzene flame. However, a significant improvement in the prediction of phenanthrene concentration - Of about one order of magnitude around the maximum if compared to the profile obtained by Waldheim [6] - and of pyrene concentration is observed for the ethylene diffusion flame. 6.4 Considerations on the preliminary flame calculations Relying on the preliminary calculations performed, the pathways of formation of naphthalene investigated in the astrochemical study of Parker et al. [27] were confirmed to have a promising potential contribution to naphthalene production also when tested in the chemical environment of premixed and diffusion flames. The tests performed in order to evaluate the relevance of the barrier-less pathway in Fig. 5.8 showed that the introduced channel of formation of the hydro-naphthalene radical is relevant for both flames. However, the extension of the barrier-less pathway of formation of naphthalene to phenanthrene through a reaction class based estimate is significant only for the ethylene diffusion flame. In both cases the roles of the 1-C 10 H 7 and 2-C 10 H 7 radicals in the formation of A 3 H are swapped between the two flames: In the premixed benzene flame the 2-C 10 H 7 radical is not relevant for the formation of A 3 H, but in the ethylene diffusion flame the step involving 2-C 10 H 7 becomes the dominant one for the formation of A 3 H. The different behavior of the two radicals might be related to that different sub-mechanisms are applied to the isomers in the kinetic scheme of the flames. This issue is due to that the modeling of the 1-C 10 H 7 radical was introduced at a later stage if compared to the 2-C 10 H 7 radical [28]. The conclusions regarding the barrier-less pathway of formation of naphthalene are valid also for the pathway involving a barrier at the

85 Chapter 6. Preliminary flame calculations 64 entrance of 5 kj/mol (see Fig. 5.9). In fact, the two pathways are not properly distinguished without refined estimates of the rate constants and without a detailed modeling of the correspondent intermediate steps. Concerning the pathway involving a barrier at the entrance of 17 kj/mol (see Fig. 5.10), the decreased relevance of this channel of formation of naphthalene compared to the previous cases was expected due to the higher energy barrier at the entrance. The new channel of formation of phenanthrene is insignificant for the premixed benzene flame, whereas it is relevant for the ethylene diffusion flame. 6.5 A note on the subsequent analysis and on nomenclature In this chapter, the relevance of novel PAH formation pathways in terms of aromatic yield have been assessed for two laminar flames. The study of Parker et al. [27] is the starting point of the subsequent analysis of this work. In fact, the next chapters describe the calculation of the rate constants of a pathway involving vinylacetylene addition to the phenyl radical studied by Parker et al. [27]. The calculation of the refined rate constants is motivated by the low accuracy of the simplifications adopted in this preliminary phase of the study concerning the kinetic parameters of the tested reactions. Due to the high technical challenge required in order to study the barrier-less pathway and since the pathway involving a barrier at the entrance of 17 kj/mol shows the least potential contribution to aromatic formation, the focus is put on the pathway involving a barrier at the entrance of 5 kj/mol (see Fig. 5.9). Parker et al. [27] provided only the ground energy levels and the atomic coordinates of both the stable species and the transition states of the studied pathways. Hence, quantum chemical calculations are performed in order to obtain the missing molecular properties of the species involved in the investigated pathway, necessary for the rate constants refinement. These properties are subsequently processed for the solution of an energy population balance - the Energy Grained Master Equation - in order to determine the rate constants of the sequence of multi-energy well reactions. The species involved in the pathway in Fig. 5.9 were already modeled in the flame codes used in this work. Concerning the nomenclature adopted in this work, the molecular configurations of the wells of the pathway are generically named Minimum i. They are differentiated form the reactants and the products of the pathway, consistently with the kinetic modeling adopted.

86 Chapter 7 Quantum chemical calculations 7.1 Introduction to quantum chemical calculations The aim of this chapter is to give the reader an overview of the theory and the methodology behind the quantum chemical calculations performed in order to compute the molecular properties of the species involved in the pathway with a 5 kj/mol barrier at the entrance studied by Parker et al. [27]. Specifically, the molecular properties to be computed are the vibrational frequencies and the moments of inertia of each configuration of the pathway. The ground state energy levels were instead given by the authors [27]. These properties will be processed in the subsequent analysis of this work, with the aim to refine the kinetic parameters of the pathway. Parker et al. [27] used B3LYP Density Functional Theory coupled with 6-311G(d,p) basis set for performing the molecular geometry optimizations, i.e. to locate the molecular configurations of both the stable species and the transition states of the pathway. The same methodology was adopted for the calculation of the obtained molecular configurations vibrational frequencies and moments of intertia. In order to achieve consistency, this methodology is replicated in the present work for the calculation of the same molecular properties, since Parker et al. [27] did not publish them. The correspondent ground state energy levels were computed by the authors [27] using a highly accurate composite energy method. Since reproducing the exact resaults of Parker et al [27] is out of scope and since the above mentioned energy composite method is too computational expensive in the context of the current scope, the ground energy levels are computed with the same methdology adopted for the others molecular properties. They are used to validate the currently performed quantum chemical calculations against the results of Parker et al. [27]. 65

87 Chapter 7. Quantum chemical calculations A note on the tools used for the quantum chemical calculations The computational chemistry software used for performing the quantum chemical calculations in this work is GAMESS, an acronym which stands for General Atomic and Molecular Electronic Structure System [29]. The results of the quantum chemical calculations will be presented in chapter The Schrödinger equation A quantum chemical calculation consists of the numerical solution of the Schrödinger equation [30] under certain simplifying assumptions. The chemical systems studied in this work are the molecular configurations of the pathway involving vinylacetylene addition to the phenyl radical with a barrier at the entrance of 5 kj/mol investigated in the pathway of Parker et al. [27]. More precisely, they represent stationary quantum mechanical systems [30], described by the time-independent Schröedinger equation. The expression of the time-independent Schröedinger equation for a single particle is [31]: H(r)Ψ(r) = E(r)Ψ(r) (7.1) Where Ψ is the wave function of the particle, representing the square root of the probability-density to find the particle at a certain position r [31]. H is the Hamiltonian operator, given by the sum of the potential (V) and kinetic energy (T) operators [31]: H = T + V = 1 2m 2 + V (7.2) For a system composed by N-particles, the Hamiltonian operator assumes the form [31]: H = T + V = N i=1 1 2m i 2 i + N V ij (7.3) i>j In the current work the aim is to evaluate the energy and the energy-dependent variables of stationary configurations of bounded atoms. Considering a molecule, the Hamiltonian operator in Eq. 7.2 is represented by the summation of different contributions [31]: H tot = H e + T n = (T e + V ne + V ee + V nn ) + T n (7.4) Where the subscripts e and n indicates electrons and nuclei, respectively.

88 Chapter 7. Quantum chemical calculations The Hartree-Fock method The Born-Oppenheimer approximation relies on the observation that the mass of the electrons is negligible if compared to the one of the nuclei [30]. As a consequence, electrons respond instantaneously to a change in the configuration of the nuclei. Subsequently, the wave functions for the nucleus and for the electrons can be decoupled and the spatial configuration of the nuclei considered a parameter independent from the position of the electrons. More precisely, the electrons are considered moving in the field of fixed nucleus [30]. Concerning the Schrödinger equation, this assumption allows to decouple the electronic wave function from the momentum of the nuclei [31]: T n = 0 (7.5) H e (r)ψ e (R,r) = E e (R)Ψ e (R,r) (7.6) Where R and r represents the coordinates of nuclei and electrons, respectively. The second simplifying approximation is the independent particle approximation for which molecular orbitals are approximated as a linear combination of atomic orbitals [30]: φ = n c n χ n (7.7) where φ and χ n are the molecular and atomic orbital functions, respectively. This assumption implies that the dynamics of one electron can be considered independently from the others. Hence, the interaction between electrons is accounted as an average interaction on the single electrons due to the motion of all the others [31]. Due to this approximation the correlation energy between single electrons is not taken into account. The correlation energy pertains to the tendency of electrons to correlate their motions in order to be at the maximum distance in order to to minimize the mutual repulsion. The energy of the exact wave function is therefore lower than the so-called Hartree-Fock energy [30]. Wave functions of polyelectronic systems are approximated accounting for the electronic correlation as an average effect through the Slater determinants. Slater determinants were formulated in order to express wave functions as a combination of single-electron orbitals and to ensure the antisymmetric property stated by the Pauli Principle [31].

89 Chapter 7. Quantum chemical calculations 68 The Slater determinant to approximate a generic wave function Ψ is given by [31]: Ψ Sl = 1 N! φ 1 (1) φ 2 (1)... φ N (1) φ 1 (2) φ 2 (2)... φ N (2) φ 1 (N) φ 2 (N)... φ N (N) (7.8) Each element of the determinant is a one-electron function φ i (j) for an electron j in a quantum state i charcterized by a certain spatial orbital and a certain spin. This approach represents the base of the Hartree-Fock method for the determination of the energy of a stationary quantum system [31]. Relying on Eq. 7.6, the Hartree-Fock energy can be expressed as [31]: E HF = Ψ Sl H e Ψ Sl dr Ψ Sl Ψ Sl dr (7.9) The Slater determinant is not unique and the one correspondent to the wave function minimizes the Hartree-Fock energy (E HF ) in Eq. 7.9, according to the variational principle [31]. The variational principle states that expectaction energy expressed in Eq. 7.9 is alway higher that the exact energy of the correspondent wave function [30]. The more energy interaction terms are accounted for in the Hamiltonian H e, the lower obtained expectation energy value is, i.e. a more accurate modeling of the wave function is implemented [30]. In the electronic Hamiltonian operator (H e ) (see Eq. 7.4), the nuclear potential operator V nn is independent from the positions of the electrons as a consequence of the Born- Oppenheimer approximation. Therefore, the nuclear repulsive energy can be included a posteriori in order to calculate the ground energy of the molecules [31]. A thorough description of the Hartree-Fock method goes beyond the scope of the current work and can be found elsewhere [30, 31]. The Fock operator computes the energy of a single electron accounting for the kinetic, the nuclear attractive and the average electronic repulsive contributions [31]. In order to account for the last contribution, the complete electronic configuration of the molecule is needed [31]. This condition leads to an iterative procedure which starts from an initial guess of the molecular orbitals, according to the self-consistent field theory [31]. Moreover, either the unrestricted or the restricted Hartree-Fock model has to be specified for the electronic configuration of the molecules. The latter assumes that each spatial orbital of the molecule is occupied by an electron pair with opposite spins [31]. This is not possible for a radical due to the odd number of electrons. Moreover, the

90 Chapter 7. Quantum chemical calculations 69 energy of an unrestricted Hartree-Fock wave function is always lower or equal to the one corresponding to the restricted modeling of the electronic configuration (when both are applicable) [31]. 7.4 Basis set As a consequence of the introduction of the independent particle approximation, molecular orbitals have been expressed in terms of linear combination of atomic orbitals. In this section, a mathematical formulation of the functions describing the atomic orbitals (see Eq. 7.7) has to be specified. A basis set is a collection of basis functions representative of the atomic orbitals [31]. The computational requirement of quantum mechanical calculations involving ab initio methods increases with the increasing number of basis function (n basis ) according to the following expression: Computational cost n α basis (7.10) Where α is a constant dependent on the method adopted [30]. Hence, a trade-off between the level of approximation of the molecular orbital functions and the computational requirements has to be considered [31]. Gaussian basis functions compose the basis set adopted in this work. Gaussian type atomic orbitals are represented by the following expression [31]: χ(x, y, z) = N norm x lx y ly z lz e ζr2 (7.11) Where N norm is a normalization constant, l x, l y and l z are the parameters determining the type of orbital (s,p,d or f) and the dependence on the distance from the nuclei is expressed through the term e ζr2. Depending on the number of electrons of the molecule, a minimum number of basis function is required for the formation of the socalled minimum basis set [31]. In order to refine the modeling of the distribution of the electrons around the nuclei, the number of basis function adopted can be doubled or tripled. Since the electrons closer to the nuclei are less dependent on the chemical environment of the species, only the valence orbitals are double (or tripled) to avoid a too high computational requirement. As a result, the double (or triple) split valence basis sets [31] are obtained. In order to decrease the computational requirement for the description of the inner orbitals of the molecules, the so-called primitive basis set can be contracted into a smaller set of basis functions resulting from a linear combination of the initial functions: the contracted basis set [31]. Basis functions with higher angular momentum have to be introduced to better model the electronic distribution within certain chemical bonds. For instance, since the species involved in the current pathway are composed only by carbon and hydrogen, a minimum basis set would consist only of s

91 Chapter 7. Quantum chemical calculations 70 and p basis functions. However, the integration of p and d basis functions for hydrogen and carbon atoms, respectively, has been demonstrated to increase the accuracy of the modeling of the molecular orbitals of the C-H bond. The introduced basis functions with higher angular momentum are called polarization functions [31]. Another type of basis functions that can be adopted increasing the accuracy of the molecular orbital modeling are the diffusive functions. The main feature of these functions is the low magnitude of their exponents, suitable for representing the valence orbitals of species with electrons weakly bounded to the nuclei (anions) [31]. Since this is not the case for the species involved in the studied pathway, these functions are not added to the basis set. As stated at the beginning of this chapter, in order to achieve consistency with the work of Parker et al. [27], the adopted basis set is the Pople s 6-311G(d,p) basis set [32]. This basis set is composed by a core resulting from the contraction of 6 primitive Gaussian type orbitals. The valence orbitals are triple-split and the correspondent three functions are the contraction of 3 primitive Gaussian type orbitals and two single additional ones, respectively. P and d polarization functions are added to hydrogen and carbon atoms [31], respectively. A sensitivity analysis was performed concerning the effect of polarization functions removal from the adopted basis set, i.e. calculations using the 6-311G basis set were performed. 7.5 Density Functional Theory Density Functional Theory (DFT) allows the determination of the global ground energy of a system of electrons as a function of the electronic coordinates. The electronic density is a measurable quantity of particular interest within Density Functional Theory, given by the following integral [30]: ρ el (r) = N Ψ(x i ) 2 dx i ds 1 (7.12) Relying on Eq. 7.12, the electronic density is the integral of the square of the wave function over all the spatial variables but three and over the spin coordinates defining the states of the polyelectronic system. The central feature of DFT is that the properties of the ground state of electronic systems can be expressed in terms of functionals of the electronic density. However, the exact expressions of the DFT functionals are unknown [30]. Assuming the formalism of Ramachandran et. al. [30]: σ el = ρ el ρ el, ˆσ el = ρ el ˆρ el, ˆσ el = ˆρ el ˆρ el, ν el = 2 σ el, ˆν el = 2ˆσ el.

92 Chapter 7. Quantum chemical calculations 71 Where ρ el = ρ α el +ρβ el is the global electronic density given by the sum of the contributions of the electrons with opposite spins and ˆρ el = ρ α el ρβ el. The most general form of the approximate functionals adopted in DFT is [30]: F = ) f (ρ el, ˆρ el, σ el, ˆσ el, ˆσ el, ν el, ˆν el, τ el, ˆτ el d 3 r (7.13) Where τ el and ˆτ el are functions representing the kinetic energy density [30]. The density functionals adopted in different DFT methods can be found elsewhere[30, 31]. Two DFT methods are adopted in this work: the B3LYP method in order to achieve consistency with the work of Parker et al. [27] and the M06-2X method. The latter was demonstrated to have enhanced accuracy and efficient computational cost in the study of Robinson and Lindstedt [33]. 7.6 Single point energies The energy barriers related to each step of the pathway are given by the difference between the ground energy of the well (or the reactants) and the subsequent transition state. The ground energy at 0 K for each species is given by the sum of different contributions that can be expressed as follows: E ground = ZP E + E nuclear + E electronic (7.14) The physical meaning of each contribution is clarified below: ZP E is the Zero Point Energy of the molecule, representing the contribution of the rotovibrational modes. E nuclear is the nuclear repulsive potential energy. E electronic is the sum of the potential energy correspondent to the interaction between electrons and the kinetic energy of the electrons. Two observations are now of central importance: The rotovibrational energy contributions were included in the computation of the ground state energy of the molecules to be consistent with the methodology adopted by Parker et al. [27]. Moreover this is necessary for achieving consistency with the subsequent solution of the energy grained master equation. In fact, the kinetic modeling of the current pathway is based on exchanges of rovibrational energies between the species involved.

93 Chapter 7. Quantum chemical calculations 72 The other energy contributions to the ground state energy are formulated based on the Born-Oppenheimer approximation: the state of the molecules at 0 K is a function of the nuclear configuration, i.e. of the position of the nuclei of the atoms [30]. Therefore, the energy of electrons - both kinetic and potential - is a function of the nuclear configuration. This statement is equivalent to a theoretical imposition of the nuclear potential followed by the adjustment of the kinetic and potential electronic energy for the sake of reaching the minimum energy of the global system [30]. 7.7 Vibrational frequencies and Intrinsic Reaction Coordinate As a result of the DFT calculations the PES has been obtained as a function of 3xN atoms dimensions for each species of the pathway. The vibrational frequencies needed for the subsequent rate constants refinement are energy-dependent properties derived after computing the ground state energies. The derivation of the vibrational frequencies is presented in this section. The Taylor expansion of the potential energy of a single species is expressed below [31]: E(x) = E(x 0 ) + ( de dx ) t x 0 (x x 0 ) (x x 0) t ( d 2 E dx 2 ) x 0 (x x 0 ) +... (7.15) Considering that each species involved in the pathway of Parker et al. [27] represent a configuration correspondent to a stationary point in the PES and setting E(x 0 ) = 0, the expansion in Eq is truncated after the first term different from zero as follows [31]: ( ) E(x) 1 2 (x x 0) t d 2 E dx 2 (x x 0 ) = 1 2 xt F x (7.16) x 0 Where F is the 3 N atoms x 3 N atoms matrix of the second derivatives of the potential energy with respect to the 3 N atoms coordinates, named force constant matrix [31]. Subsequently, the expression of the nuclear Schrödinger equation for a generic polyatomic species becomes [31]: [ ( ) ] 1 2 2m i x i 2 xt F x Ψ nuc = E nuc Ψ nuc (7.17) N atom i=1

94 Chapter 7. Quantum chemical calculations 73 Introducing the following atomic mass-weighted coordinate system [31] y i = m i x i (7.18) G i,j = the nuclear Schrödinger equation becomes [31]: [ ( 1 2 N atom i=1 2 y 2 i ) 1 mi m j (7.19) yt (F G)y ] Ψ nuc = E nuc Ψ nuc (7.20) By introducing a proper coordinate transformation, the matrix (F G) is diagonalized and a set of 3 N atoms eigenvalues e i with the correspondent eigenvectors q i [31] are obtained: [ ( 1 2 N atom i=1 2 q 2 i ) qt (U(F G)U t )q ] Ψ nuc = E nuc Ψ nuc (7.21) The key feature for the analysis that follows in Chapter 8 concerning the energy related to the vibrational modes of the species of the pathway is that in the q-coordinate system - named vibrational normal coordinates - there is no coupling between the vibrational degrees of freedom of the species [31]. The relation between the eigenvalues e i and the vibrational frequencies ν i is [31]: ν i = ei 2π (7.22) Among the 3 N atoms vibrational frequencies, a total of 6 frequencies is equal to zero since they correspond to rigid rotations and translations of the entire molecule. Due to numerical precision and tolerances, it is appropriate to outline that a group of 6 computed vibrational frequencies (out of the total of 3 N atoms ) have magnitudes below 10 3 s 1. If the stationary point of the PES is a Transition State, it represents a minimum of the PES with respect to 3 N atoms 1 coordinates but a maximum for the last coordinate. That means that one eigenvalue e i will be negative, and the correspondent vibrational frequency imaginary. The correspondent eigenvector q i is the mode of internal vibration of the species that leads to the following minimum of the energy surface. In other words, perturbing the molecule along q i leads to a displacement on the PES towards the minimum that follows the TS. Subsequently, q i represents the so-called reaction coordinate, named Intrinsic Reaction Coordinate (IRC) in atomic mass-weighted coordinates.

95 Chapter 8 Pathway analysis for the estimation of kinetic parameters 8.1 Introduction to nonequilibrium reaction pathways The studied pathway consists of a bimolecular association followed by two subsequent unimolecular isomerizations and the final hydrogen dissociation, modeled as an irreversible step. The kinetics of this reaction pathway cannot be modeled relying on the semi-classical Rice-Ramsperger-Kassel-Marcus (RRKM) theory since it assumes that all the stable species and the transition states are in thermal equilibrium with the bath gas [31]. The last feature applies to a reaction pathway only if the time scale of thermal relaxation of the stable species involved in the pathway is sharply shorter if compared to the time scale of the reactions determining chemically relevant changes between the species [34]. The unimolecular isomerizations occurring as intermediate steps in the process of formation of naphthalene and hydrogen previously depicted in Chapter 5 are characterized by non-boltzmann energy distributions since the kinetic time scales of the intermediate steps are comparable to the time scales of thermalization with the bath gas [34]. More precisely, it will be shown that certain isomerisation steps are classifiable as thermal relaxation processes rather than as chemically significant changes. Under these conditions another approach has to be adopted in order to make a prediction of such a nonequilibrium kinetics: The solution of the energy grained master equation using MESMER. MESMER is a computer software that implements the numerical solution of the energy grained master equation and it is oriented to provide the phenomenological description of pathways showing a multi-well potential energy topology. 74

96 Chapter 8. Pathway analysis for the estimation of kinetic parameters The Energy Grained Master Equation Introduction to the Energy Grained Master Equation The Master Equation (ME) describes the time evolution of the energy distributions of all the species (reactants, products and wells) involved in the pathway. The compact matrix formulation of the energy grained master equation (EGME) [35] is given below: p t = Mp (8.1) Where p is the vector of the grained energy distribution as a function of the ground state energy of each species and M is a matrix modeling the configuration changes between different species of the pathway as well as the collisional energy transfer between molecules of single species and the bath gas. The continuum formulation of the system in Eq. 8.1 allows a better understanding of the modeling of nonquilibrium chemical dynamics [34]. Moreover, a further overview of the studied chemical process is needed in order to better contextualize the current mathematical formulation. The Potential Energy Surface in Fig 5.9 is representative of a reaction pathway initiated by a bimolecular association between vinylacetylene and the phenyl radical. The first step of the sequence of reactions leads to the C 10 H 9 (B) radical formation. This species corresponds to a well in the PES and therefore to a meta-stable molecular configuration [34]. It undergoes two subsequent unimolecular isomerizations towards the formation of the C 10 H 9 (T) radical. Hence, three wells are interconnected through unimolecular Transitions States in the pathway, which are projections of saddle points of the 3 N atom -dimensional Potential Energy Surfaces along the reaction coordinate. From an energetic prospective, each Transition State is an energy barrier to be overcome, which means that there is a need of a mean of energy supply for each isomer in order for it to be converted to the following chemical configuration present in the pathway [34]. The energy transfer is modeled to occur through interactions with a thermal bath of Argon atoms, which is supposed to be in large excess if compared with the reacting species. This choice is consistent with the inert gas diluent used in the flames studied in this work [10, 11]. The formulation of a sub-group of equation composing the system in Eq. 8.1 is given below [34]: p m t = ω LJ P (E E )p m (E ) de ω LJ p m (E)+ E 0m + k mn (E)p n (E) k nm (E)p m (E) k Sm (E)p m (E)+ n m n m + K eq Rm k Rm(E) ρdos m (E)e β T E n A p B k Rm (E)p m (E) q rv,m (β T ) (8.2)

97 Chapter 8. Pathway analysis for the estimation of kinetic parameters 76 Equation 8.2 outlines the seven ways through which the energy population p m of isomer m with energy E is modeled to change as a function of time. The first term is representative of an energy change from a grain with energy E to the grain with energy E within the same energy distribution of isomer m due to a collisional energy transfer: ω LJ is the Lennard-Jones frequency of collisions and P (E E ) is the probability for the collisions to cause an energy population change from E to E. The second term represents a population loss of the grain with energy E of isomer m due to collisional energy transfer. The third and fourth terms are representative of reversible population gains and losses of the energy grain E due to population transfers between isomer m and the others isomers n of the pathway. The rate of the exchange is estimated through RRKM microcanonical reaction rates k mn and k nm. Since the products are modeled as sinks - which means that only populations gains can occur for the products - the fifth term models the irreversible population loss due to a collisional transfer from the energy distribution of isomer m to the bimolecular products energy distributions. Since the reaction modeled is a bimolecular irreversible exchange, the last two terms of equation 8.2 represent the reversible population gain and loss due to bimolecular association of the reactants towards the grain E of isomer m and the reversed process, respectively [34]. q rv,m (β T ) is the roto-vibrational partition function of isomer m - where β T = (k b T ) 1 -, ρ DOS m (E) is the density of rotovibrational states, n A is the number density of the excess reactant A and p B is the energy distribution of the deficient reactant B. Concerning the last two terms of Eq. 8.2, the following two assumptions are introduced [34]: The energy distributions of the reactants are Boltzmann distributions, i.e. reactants are in thermal equilibrium with the bath gas. the One of the two reactants is in large excess if compared with the other one: [A] >> [B] (8.3) The first assumption is coherent with the other one concerning the large amount of the bath gas if compared to the reacting species. Moreover it appears to be reasonable since vinylacetylene was measured across a wide range of distances from the burner in the current flames [10, 11]. Therefore the time scale of residence of the species between the formation and the subsequent consumption processes in the flames studied in this work overcomes the time scale of thermalization. The second assumption allows the linearity of the master equation (Eq. 8.1) since the concentration of the excess reactant can be assumed to be constant [34]. In the case of irreversible exchange reactions (A + B C + B) as well as in the case of bimolecular association reactions (A + B AB) the system in Eq. 8.2 has to be coupled

98 Chapter 8. Pathway analysis for the estimation of kinetic parameters 77 with the equation below [34]: dp B t = M m=1 E 0m k Rm (E)p m (E) de n A p B M m=1 Where the notation is coherent with respect to Eq K eq Rm k Rm (E) ρdos m (E)e β T E E 0i q rv,m (β T ) de (8.4) The step taken to move from the continuum formulation of Eq. 8.2 coupled with Eq. 8.4 to the matrix formulation in Eq. 8.1 is the discretization of the energy axes into grains. Consequently, the energetic states with similar energy of each isomer are bundled together to form discretized energy distributions p [34] Energy transfer model In Eq. 8.2, the term P (E E ) represents the probability of energy change from a grain with energy E to a highr level E due to nonreactive, activating unelastic collisions with the bath gas [34]. With the decreasing difference between E and E, the energy transfer that causes the energy change becomes more frequent [36]. The model adopted for the probability of energy change stated above is [36]: ( P (E E ) = A(E) exp (E E) ) < E d > (8.5) Where A(E) is the normalization coefficient so that the probability fulfill the normalization constraint [34]: P (E E) = 1 (8.6) and < E d > is the energy that is transferred on average in a deactivating collision [36]. In Eq. 8.2, ω LJ represents the Lennard-Jones collision frequency between the bath gas and the isomer m, given by [18]: ω LJ = πσ 2 i,jω g i,j (8.7) Where: σ i,j is the Lennard-Jones length scale parameter (see Eq. 4.36) related to to the bath gas molecules and the isomer m; Ω is a reduced collision integral dependent on the intermolecular potential defined in Eq. 4.36;

99 Chapter 8. Pathway analysis for the estimation of kinetic parameters 78 g i,j is the average relative speed of collision between the two species, given by [18]: g i,j = ( 8k b T π mi + m j m i m j ) 1/2 (8.8) Where m i, m j are the masses of the two species and k b is the Boltzmann s constant Energy discretization The choice of the dimension of the energy grains is a result of a trade-off between: A lower limit necessary for the achievement of a sufficient resolution of the microscopic nonequilibrium energy distributions of the wells. An upper limit in order not to overcome the energy transferred on average during inelastic collisions. The aim of avoiding excessive computational requirements. A reasonable range for the dimension of the energy grains according to Glowacki et al. [34] is between 0.5 and 3 kj/mol. The energy units required in MESMER input files for the energy grains dimension corresponds to the energy of a photon with a wavelength of 1 cm and is indicated by cm 1. The above mentioned range for the dimension of the energy grains corresponds to cm 1 [34]. The algorithm of energy discretization in MESMER consists of a coarsening subsequent to the initial definition of a finer grid of grains with a dimension of 1 cm 1, named cells. The algorithm to allocate the quantum states within the initial cells can be found in the literature [34] and goes beyond the scope of the current work. In the subsequent step, the average energy of a grain - parameter used to evaluate the canonical partition functions - is calculated as [34]: ɛ grain,i = 1 N C,tot i Ej C Nj C (8.9) Where N C j is the number of cells which have an energy E C j within the grain i and N C,tot i = j i N C j is the total amount of cells within the grain i. j i Statistical mechanics and partition functions Statistical mechanics relates the properties of a microscopic system to the ones of a correspondent macroscopic sample [31]. The PES showed in Fig. 5.9 corresponds to the energy ground state at 0 K of the different species. At finite temperatures, a distribution

100 Chapter 8. Pathway analysis for the estimation of kinetic parameters 79 of molecules in all the allowed quantum states establishes. The probability of finding a molecule in a state with energy E in equilibrium with the surrounding system at temperature T is given by the Boltzmann distribution [31]: P (E) = e E k b T (8.10) The energy distributions representative of the thermal nonequilibrium of the wells differs from the Boltzmann distribution for energy levels above the threshold required for the reaction to proceed, or in other words to overcome the energy barriers of the transition states [37]. The higher the threshold, the smaller is the population deviating from the Boltzmann distribution: In this case the near-boltzmann approximation can be introduced [37]. The last feature is of central importance in the current chemical system: the time-scale of the population thermalization of each isomer is close to the time-scale of the chemical changes involved in the sequence of reactions of the pathway. For this reason, the changes of energy states involving an exchange between two different isomers populations - modeled through all the terms of Eq. 8.2 besides the first two - causes the deviation of the single wells energy distributions from the Boltzmann distribution. The deviation is mainly confined within the high-energy tail of the Boltzmann distribution. Assuming thermal equilibrium, the partition function of a species is given by the sum of the Boltzmann probabilities for a molecule of that species to have an energy E at temperature T, considering all the allowed quantum states [31]: q = i=states e E k b T (8.11) In Eq. 8.2 q rv,m (β T ) is the roto-vibrational partition function of isomer m. Subsequently to the discretization of the master equation the average energy of the grains - ɛ grain,i in Eq substitutes E in Eq Following the Born-Oppenheimer approximation, within a molecule the inertia of the nuclei sharply overcomes the electronic one, i.e electrons almost instantaneously respond to a change in the nuclear configuration [30]. Hence, the motion of the nuclei and of the electrons is decoupled and the electronic Schrödinger equation can be solved as a function of the nuclear positions, resulting in the Potential Energy Surface [31]. Therefore the chemical dynamics of a generic pathway can be described as a function of nuclear configurations along the intrinsic reaction coordinate. In other words, a sequence of reactions can be described in terms of nuclear dynamics. Therefore, the characterization of molecular quantum states relies on the analysis of the vibrational and rotational modes of the nuclei [31].

101 Chapter 8. Pathway analysis for the estimation of kinetic parameters 80 Recalling the rovibrational partition function (q rv,m (β T )) in Eq. 8.2, the energy E of the allowed molecular quantum states is calculated as the sum of the vibrational and rotational modes contributions [31]: E tot = E rot + E vib (8.12) Subsequently the rovibrational partition function is calculated as the product between the rotational and vibrational partition functions [31]: q rv = q rot q vib (8.13) In the following sections an overview of the expressions of the rotational and vibrational energy contributions - as well as of the correspondent partition functions - is given. A thorough review of the theory behind the following equations is too extensive to be performed in the current work and the aim of the following sections is to give the reader an understanding of input specifications required by MESMER [34] Rotational energy levels and partition function In order to evaluate the rotational energy contribution of a molecule, the latter is assumed to be a rigid collection of nuclei. In order to achieve higher accuracies, the centrifugal stretching effect of molecular rotation on the nuclear configuration should be taken into account [31]. The energy levels correspondent to the rotation of a rigid diatomic molecule derives from the solution of the Schrödinger equation and can be expressed as follows [31]: E diatomic J = J(J + 1) h2 8π 2 I (8.14) Where J is a quantum number varying from 0 to infinite, h is the Planck s constant and I is the moment of inertia of the nuclei with respect to the center of mass of the molecule. If the molecule is polyatomic, a 3x3 inertia matrix has to be computed instead of a single moment of inertia. By making the coordinate system match the principal axes of inertia of the molecule, the inertia matrix is diagonalized and the diagonal elements are the three moments of inertia I 1, I 2, I 3 of the molecule [31]. An approximate expression of the rotational partition function becomes [31]: ( ) q rot = π1/2 8π 2 k b T σ rot h 2 (I 1 I 2 I 3 ) 1/2 (8.15) Where σ rot is the number of rotational symmetry operations allowed for the single molecular configurations of the pathway. MESMER input files require three rotational

102 Chapter 8. Pathway analysis for the estimation of kinetic parameters 81 constants A, B and C directly proportional to the moments of inertia and listed in GAMESS output files Vibrational energy levels and partition function Recalling the theoretical analysis developed in Chapter 7, the vibrational frequencies were derived from the force constant matrix at stationary points of the PES. The vibrational frequencies of each molecular configuration of the pathway are calculated using GAMESS. Subsequently, the correspondent vibrational energies are computed assuming that molecular vibrational modes can be approximated to harmonic oscillators [31]. As stated in Chapter 7, the vibrational degrees of freedom are decoupled in the vibrational normal coordinates and the total energy correspondent to molecular vibrational modes can be express as a sum of single contributions [31]: E vib = 3N atom 6(7) i=1 ( n vib i ) hν i (8.16) Where n vib i is a quantum number ranging between zero and infinite. The single contributions are the energy levels derived from the Schrödinger equation for a 1D harmonic oscillator of a diatomic system. The modes are in total 3N atom 6 except for the Transition States in which one mode is the reaction coordinate, correspondent to internal molecular vibration leading to the chemical change towards the following well of the pathway [31]. The partition function is the sum over the infinite quantum numbers of the single probabilities of finding the molecule in a quantum state as expressed in Eq Concerning the contributions of each vibrational mode, it can be expressed in a closed form [31]: q V ibrationalmode = e hν/2k bt 1 e hν/k bt (8.17) Relying on the principle expressed in Eq and on the uncoupling between the vibrational degrees of freedom, the global vibrational partition function is given by [31]: q vib = 3N atom 6(7) i=1 e hν i/2k b T 1 e hν i/k b T (8.18) To conclude this section, the definition of ground energy adopted in this work and described in Chapter 7 is recalled. No traslational partition function was considered since

103 Chapter 8. Pathway analysis for the estimation of kinetic parameters 82 MESMER does not account for it, due to the constrained unimolecolarity of the microcanonical reaction rates. It has been previously outlined that the forward bimolecular reaction rate constants (when present) are computed based on detailed balance, i.e. through the correspondent unimolecular backward reaction rate constants. Crossing the first transition state, the translational degrees of freedom of the bimolecular reactants become internal vibrational degrees of freedom. This is the chemical reason that no parameters regarding molecular translational degrees of freedom are required in the MESMER input files Microcanonical rate coefficients The microcanonical rate coefficients in Eq. 8.2 are computed basing on Rice-Ramsperger- Kassel-Marcus (RRKM) theory [34]. The main hypotheses behind this theory is the socalled ergodicity assumption: The time scale of thermalization of the species is sharply smaller than the time scale of a chemical changes [34]. The consequence of this assumption on the modeling of energy exchange through activating inelastic collisions is that molecules are energized and the additional energy is distributed between the rotovibrational modes in the thermalization process [31]. If the additional energy is accumulated in the vibrational mode corresponding to the reaction coordinate, the molecule reaches the Transition State. This is the mechanism of overcoming the energy barriers of the pathway. However, the ergodicity assumption is not exactly fulfilled in the current pathway due to the near-equilibrium that establishes between the wells [31], which will be discussed further in this chapter. The RRKM unimolecular rate coefficient as a function of the energy of the grains k(e) is given by the ratio between the sum of rotovibrational energy states above the activation energy threshold of the activated molecule at the Transition State and the product between the Planck s constant and the rotovibrational energy of the reactants [34]: k(e) = W (E E 0) hρ DOS (E) (8.19) Where E 0 is the threshold given by the activation energy of the unimolecular reaction considered, W (E E 0 ) is the sum of rotovibrational energy states above the activation energy threshold and ρ DOS (E) is the density of rotovibrational states of the reactants. Only unimolecular microcanonical reaction rates are computed in MESMER calculations (see Eq. 8.2): The forward reaction rate related of bimolecular reactions leading to a single molecule is estimated through a detailed balance as previously described.

104 Chapter 8. Pathway analysis for the estimation of kinetic parameters 83 Table 8.1: Rotational symmetry numbers of all the species of the pathway C 6 H 5 C 4 H 4 TS1 Min1 TS2 Min2 TS3 Min3 TS4 C 10 H MESMER input specifications Introduction to MESMER input specifications This section aims to clarify all the input specifications defined for MESMER calculations. The base case for the current analysis is considered. Many parameters were varied in order to perform a sensitivity analysis and this will be the object of a further discussion Single point energies The energy barriers are set in order to achieve consistency with the values given by Parker et al. [27] listed in Tab MESMER input files require a single point energy value for each species and the energy barriers of each step of the pathway are computed as differences between the correspondent values of the Transition States and of the contiguous wells Rotational constants The rotational constants A, B and C required as an input by MESMER are parameters proportional to the moments of inertia of the molecules necessary to compute the rotational partition functions of the species. They are given as outputs of the ab initio calculations and they are listed in Table Symmetry numbers The symmetry number required as an input by MESMER is the parameter σ in Eq Since it concerns the rotational symmetry operations allowed for each molecular configuration, it represents a subgroup of the symmetry operations correspondent to the molecular point group [31]. The values of the rotational symmetry number for each species are listed in Table 8.1. Minimum 3 has a symmetry operation represented by the reflection plane in which the two aromatic rings lay: Since it is not a rotational symmetry operation the correspondent symmetry number is 1.

105 Chapter 8. Pathway analysis for the estimation of kinetic parameters Vibrational Frequencies The vibrational frequencies have to be specified in MESMER in order to compute the vibrational partition functions of the species. They were given as output from GAMESS and are listed in Tables A.1 and A.2 in Appendix A. The input of the imaginary eigenfrequencies of the Transition States is separate from the other ones. The former ones are listed at the beginning of Table A Lennard-Jones parameters The Lennard-Jones σ and ψ m parameters have to be specified in MESMER input files for each molecular configuration corresponding to a well. The parameters are shown in Eq and mentioned in Section concerning the modeling of non-elastic collisions with the bath gas. The computation of the forward rate constants from the bimolecular fragments (reactants) towards the wells is outlined in Eq. 8.2 and relies on a detailed balance. In other words, the forward rate constants of the reactions involving the bimolecular reactants of the pathway are computed through the correspondent equilibrium and backward rate constants. This is the reason why MESMER requires only the Lennard-Jones parameters of the modeled wells of the pathway, whereas they are not required for the bimolecular reactants. Moreover, since the products are modeled as infinite sinks the backward rate constants of the reactions going from the products towards the wells are forced to be zero. Hence, only the forward rate constants of the reactions of products formation are computed and Lennard-Jones parameters are not required for the products themselves. In order to estimate the Lennard-Jones parameter for the wells of the pathway a brief literature review was performed, leading to the following observations: Observing the previous estimates of the Lennard-Jones parameters in the flame code transport property library, the values of σ and φ m for the radicals C 10 H 9 (B) and C 10 H 9 (T) - Minimum 1 and 3 of the current pathway - were estimated basing on the correspondent values for naphthalene, i.e. the stable species with the most similar structure with respect to the above mentioned radicals. The author is aware of the border line significance of the last statement due to the evident difference between the side chain of minimum 1 and the second carbon ring of Minimum 3 and naphthalene. This may be a valid estimation in the context of flame simulations, but at the current level of detail a more accurate value is required. However, the following findings were considered a better way to proceed.

106 Chapter 8. Pathway analysis for the estimation of kinetic parameters 85 The models adopted for the estimate of naphthalene Lennard-Jones parameters were taken from Poling et al. [38]: φ m k b = T c (8.20) σ = 0.809V 1/3 c (8.21) Where k b is the Boltzmann constant, T c is the critical temperature in K and V c is the critical volume in [cm 3 /mol]. Naphthalene has T c = 748 K and V c = 409 cm 3 /mol [39]. The resulting values adopted in the MESMER input files for the modeled wells of the pathway are σ = Å and φm k b = K. These values are comparable with the ones previously estimated in the flame code, respectively equal to Å (σ) and K ( φm k b ). Moreover, through a simple MESMER test run, it was observed that even using the last couple of values the sensitivity of the MESMER calculations in terms of canonical and phenomenological rate constants was negligible. Furthermore, the last observation was an implicit confirmation of the reliability of the models adopted and of the calculations performed Energy transfer parameters This section is intended to complete the overview of the other parameters required by MESMER in order to model the energy transfer between the molecules of the species of the pathway and the bath gas. Recalling Eq. 8.5, MESMER requires a value for the energy that is transferred on average in a deactivating collision (< E d >). The choice of this parameter was particularly delicate. Since the value < E d >= 230 cm 1 was used by Robertson et al. [35] for an irreversible exchange reaction involving the 1-, 2-pentyl radical in Ar as bath gas, this value was adopted for the current calculations. This choice was taken by the following considerations: In order to make a reliable estimate, < E d > should be calibrated based on experimental data-fitting [34]. Relying on previous experimental estimates made with Helium [34] a reliable range for < E d > is stated to be between cm 1. However, Helium has a mass almost 20 times lower that Argon and an eventual similarity with the collisional behaviour of Argon itself requires further investigations.

107 Chapter 8. Pathway analysis for the estimation of kinetic parameters 86 Concerning the temperature dependence, a possible correlation for < E d > when either Ar or Kr are the bath gas is [40]: ( ) 0.85 T < E d >= 133 [cm 1 ] (8.22) 300K The dependency of < E d > on temperature is not investigated in the current work. However, the expression in Eq was used to validate the value chosen for < E d >. As shown in Eq. 8.8, the molar mass of each species of the pathway is needed in input by MESMER. Moreover, the species composing the bath gas is indicated in the input files Energy discretization parameters Two inputs are required by MESMER in order to discretize Eqs. 8.2 and 8.4: According to Glowacki et al. [34], the dimension of the energy grains was set to 40 cm 1. This was considered a consistent choice to achieve a reasonable computational cost without excessively decrease the resolution of the energy distributions of the species. The value of the grain at the highest energy in the system, i.e. the upper energy boundary. This value was initially set to be automatically computed by MESMER in order to exclude the highest-energy tails of the energy distributions within a certain threshold. Since the significance of this threshold was not completely clarified, the location of the highest energy grain was set to be 40 k b T above the stationary point with the highest energy, i.e. above Transition State 1. A reliable value for the energy difference between the highest energy grain and the stationary point with the greatest energy is 20 k b T, according to Glowacki et al. [34]. However, bigger values should be used for heavier molecules [34] Ensure energetic consistency A further input required by MESMER is the frequency scaling factor, to be specified considering the knowing deficiencies of B3LYP DFT calculations coupled with 6-311G(d,p) basis set under the harmonic oscillator assumption [23]. According to Andersson and Uvdal [41], the value was chosen to be equal to Parker et al. [27] did not mention

108 Chapter 8. Pathway analysis for the estimation of kinetic parameters 87 any particular correction applied to the vibrational frequencies, therefore the consistency between this empirical correction and the methodology adopted by Parker et al. [27] for the electronic structure calculations is not ensured. A correction applied to vibrational frequencies of the molecules influences the computation of the energy distribution of the species. Hence, it should be consistent with the energy barriers computed as the difference between the ground state energies of two subsequent configurations of the pathway. However, considering that the barriers were set to be consistent with the energy levels of Parker et al. [27] without seperately specifying the vibrational contribution, this could not be done in pratice. 8.4 Deriving reaction rates constants from the solution of the Energy Grained Master Equation Time scale separation The formulation of the Energy Grained Master Equation [35] is shown in Eq The solution of Eq. 8.1, i.e. the explicit functional time dependence of the energy distribution of each species, is given by [40, 42]: ( N p(t) = e λjt g j g j )p(0) = ˆT p(0) (8.23) j=0 Where N is the number of eigenvalues λ j of the system in Eq. 8.1, g j are the correspondent eigenvectors, stands for the vector product and p(0) is the population of the species at t = 0. ˆT is the operator describing the time evolution of the system and it can be applied to any arbitrary initial condition p(0). Analyzing the eigenvectors g j of the system in Eq. 8.1 is fundamental in order to achieve an understanding of the extensive description of the current chemical system and enables a mapping of reaction rate constants to a system where the energy distributions are not resolved, i.e. in the flame simulations of laminar flames. If none of the species of the pathway is modeled as an infinite sink the system reaches an equilibrium at t and the eigenvector g j correspondent to λ j = 0 represents the evolution of the initial system towards the relative thermal and chemical equilibrium between the species of the pathway and with the bath gas. In the current case, since the bimolecular products are modeled as infinite sinks, there is no eigenvalue λ j that is equal to 0.

109 Chapter 8. Pathway analysis for the estimation of kinetic parameters 88 The number of modes (eigenvectors) describing a chemically significant evolution of the system (a chemical change between two stables configurations of the system) is a function of the number of configurations that reaches thermal and chemical equilibrium at t. Defining S as the number of the species with an energy distribution that evolves towards thermal and chemical equilibrium, the number of chemically significant eigenvectors is equal to N chem = S 1 [42]. In the current case, the bimolecular products are modeled as infinite sinks. This means that the equilibrium that establishes between the products and the other stable molecular configurations within the pathway is completely unbalanced towards the former ones. The bimolecular products are accounted as a single additional configuration within S. The N chem eigenvectors mentioned above describe the normal modes of relaxation of the system. A parallel can be drawn between the normal modes of relaxation of the system and the Intrinsic Reaction Coordinate (IRC) introduced in Chapter 7. In order to achieve a chemically significant evolution of the energy populations of the system, the molecular vibrational modes to be excited correspond the IRC. When other vibrational modes are excited, the energy distributions of the species diverges from the Boltzmann distribution, representative of the thermal and chemical equilibrium of the correspondent species. The normal modes of relaxation of the system describe the energetic evolution of the energy population of the species at the stationary points of the PES of the pathway, resulting in a chemical change. The latter ones can be seen as the excitation of molecules of the different configurations of the pathway along the IRC, resulting in a rearrangement of the species energy populations towards the thermal and chemical equilibrium, starting from an arbitrary initial condition. Therefore, the eigenvalues λ j correspondent to the N chem eigenvectors are named chemically significant eigenpairs (CSEs) [42] in contrast with the residual eigenvalues λ j, named fast internal-energy-relaxation eigenmodes (IEREs). The latter are representative of the internal rearrangements within single species populations, subsequent to the excitation of vibrational modes different from the IRC. More generally, they are representative of an energetic evolution of the system towards the equilibrium, not involving a chemically significant change between the different chemical configurations. At this stage a crucial condition of the current analysis has to be pointed out: the time scale of the IEREs has to be sharply lower than the one of the CSEs. Since λ j 0 for each j, the last statement translates into the condition: λ IEREs << λ CSEs (8.24)

110 Chapter 8. Pathway analysis for the estimation of kinetic parameters 89 In other words, the CSEs are the N chem least negative eigenvalues among λ j, where j [(0)1, N] [42]. The importance of this condition will be pointed out in the following digression Conservative master equations and phenomenological modeling of the chemical system In order to develop a phenomenological (macroscopic) description of the chemical dynamics of the pathway, the condition in Eq has to be fulfilled. The phenomenological description of the pathway derives from the so-called Bartis and Widom analysis [34]. At this stage, it is helpful to focus on the conservative case of a bimolecular association pathway and to derive the correspondent phenomenological description. The products are not modeled as infinite sinks since the system is conservative [43]. In other words, the reversibility of the reactions leading to the formation of the products of the pathway has to be allowed. Relying on Eq. 8.23, the energy population of the n wells of the system is given by [43]: N p i (E, t) = c ij (E)e λ jt j=0 (8.25) Where i [1; n wells ] and N [1; CSEs + IEREs]. To make the energy dependency implicit, Eq is integrated over E. Subsequently, the molar fraction of the n species of the pathway is given by [43]: N X i (t) = a ij e λ jt j=0 (8.26) Where [43]: a i,j = E 0i c i,j (E) de (8.27) In Eqs and 8.27: i [1; n species ] and N [1; CSEs + IEREs]. The difference between n species and n wells is that the former includes the deficient component of the bimolecular reactants. This difference rises from the system modeling (see Eq. 8.2): The energy integration is not necessary for the bimolecular fragments [43]. If λ IEREs << λ CSEs, Eq can be truncated after the Nchem th term [43]: X i (t) N chem j=0 a ij e λ jt (8.28)

111 Chapter 8. Pathway analysis for the estimation of kinetic parameters 90 It can be demonstrated [43] that the first derivative of Eq with respect to time can be written as: dx i dt = n species ( Nchem l=1 j=0 a ij λ j a 1 jl ) X l (8.29) The left-hand-side of Eq can be expressed in terms of phenomenological temporal evolution of the chemical system as follows [43]: dx i dt = l i k il X l X i k li (8.30) l i Where i [1; n species ] and k il is the rate coefficient of the forward reaction starting from the molecular configuration l and leading to the molecular configuration i of the pathway. Comparing Eq with Eq. 8.29, the phenomenological reaction rate coefficients k il (with i l) can be expressed as follows [43]: k il = N chem j=0 a ij λ j a 1 jl (8.31) Even though the parameters a i,j are dependent on the initial condition p(0), it can be demonstrated that the phenomenological rate constants are not [43]. This is an important point, since in MESMER calculations only the bimolecular reactants are assumed to be present at t=0, with a Boltzmann energy distribution. The phenomenological rate coefficients obtained through the above theoretical digression are the Bartis and Widom rate coefficients computed by MESMER. According to Miller and Klippenstein [42], the mathematical derivation adopted by Bartis and Widom is oriented to demonstrate that the coefficients k il satisfies detailed balance for λ IEREs << λ CSEs [42]. However, the approach of Miller and Klippenstein [42] allows a clearer focus on the phenomelogical characterization of the pathway. It is appropriate to recall that the crucial approximation made in the above theoretical digression is the truncation of Eq. 8.26, leading to Eq The approximation is robust if λ IEREs << λ CSEs and if the N chem CSEs does not tend to merge. MESMER adopts a quantitative criterion for considering whether the condition in Eq is fulfilled: the N chem CSEs should differ at least of one order of magnitude between each other and from the first IERE [34]. If the last criterion is not fulfilled, the phenomenological rate constants computed by MESMER are considered not strictly reliable and the software gives an output warning message.

112 Chapter 8. Pathway analysis for the estimation of kinetic parameters 91 Recalling the study of Miller and Klippenstein [43], assuming that the relaxation processes in the chemical system occur infinitely faster than the chemical changes, the parameters a ij in Eq express the global change (from the initial condition to t ) in the population of the i-th species of the pathway due to j-th eigenmode g j : a ij = E 0i c ij (E) de = X ij (8.32) In the derivation of Eq only one chemical configuration is considered to be present at t=0. This condition is adopted also in MESMER calculations, considering only the bimolecular reactants to be present at t=0 with a mole fraction arbitrarily set to 1. Under this condition, Eqs and 8.30 are evaluated at t=0 [43]: Nchem dx i dt (0) = a ij λ j (8.33) j=1 respectively. dx i dt (0) = k ireactantsx Reactants (0) = k ireactants (8.34) In Eq. 8.34, j = Reactants indicates that at t = 0 the only species present are the reactants (more preciselythe deficient reactant is considered). with the initial condition adopted by MESMER: X Reactants (0) = 1. Comparing Eqs and 8.34 Miller and Klippenstein [43] conclude that: k i,reactants = N chem j=1 Eq is consistent a ij λ j (8.35) The observation translated into Eq becomes more clear and the final step towards the understanding of the chemical meaning of λ j Eq [43]: λ j can be artificially defined as [43]: j=1 is achieved inserting Eq into N chem k i,reactants = λ j X ij (8.36) λ j = 1 τ j (8.37) Where τ j represent the time of relaxation [43] of the correspondent eigenmode g j. Hence, the phenomenological rate coefficients of Bartis and Widom represent the sum over the N chem CSEs of the chemically significant changes in the population of a species

113 Chapter 8. Pathway analysis for the estimation of kinetic parameters 92 (the reactants in the previous digression) due to the correspondent eigenmodes g j divided by τ j : k i,reactants = N chem j=1 X i,j τ j (8.38) Non-conservative master equation In the modeling of the pathway of Parker et al. [27] the products are modeled as infinite sinks. This constraint was imposed by MESMER for the type of reactions involved in the current pathway. As a consequence, the reactions that lead to product formation are irreversible. As already explained at the beginning of this section, no eigenvalues of the diagonalized transition matrix is equal to 0. However, another difference arises with respect to the conservative case [43]. The quantity X ij, that was derived as a function of the parameters of the transition matrix in the conservative case (see Eq. 8.32), has to be evaluated as a residual quantity for the infinite sink (products) in the current case. More precisely, X ij has to be computed for all the species besides the products and subsequently the following conservation equation is applied to compute X P j for the products [43]: Where j [1; N chem ]. ( X R + X P + n wells i=1 X i ) j = 0 (8.39) 8.5 Preliminary considerations for a consistent kinetic modeling of the pathway The previous section showed that the separation of the time scales between the internal relaxations and the chemical changes within the pathway is the necessary condition for the derivation of a macroscopic (phenomenological) rate constants. It is appropriate to recall the criterion adopted by MESMER for which the CSEs must differ at least of one order of magnitude both from the IEREs and between each other in order to achieve a reliable phenomenological description of the pathway. After running the first simulation, a critical overlapping between CSEs was observed. Subsequently, different pathway configurations were tested in order to investigate the issue. Miller and Klippenstein [43] developed a methodology for solving this issue without

114 Chapter 8. Pathway analysis for the estimation of kinetic parameters 93 Figure 8.1: Potential energy diagram of the pathway in analysis. The energy levels are expressed in kj/mol. introducing further approximations. CSEs are representative of a chemical equilibration [43] between different stable species within the pathway. If one CSE approaches the IEREs, it means that the chemical change that it represents starts to occur with a time scale of internal relaxations within the single energy distributions of each species. Therefore, according to Miller and Klippenstein [43] in order to overcome the issue the energy distributions of the equilibrating species have to be merged after solving the Energy Grained Master Equation for the system. However, the implementation of this method is not available in MESMER [43] CSEs analysis All the parameters required by MESMER in order to perform a phenomenological analysis were set as previously described in this chapter. As a result, the Potential Energy Diagram of Figure 8.1 was obtained: as already explained the energy levels indicated agree with the values listed in Table 9.3. The number of species present in the pathway are 5, therefore there are 4 CSEs and the fifth eigenvalue represents the first IERE. More precisely, the first IERE represents the boundary of the region where the eigenvalues are representative of internal energy relaxations. The eigenvalues spectrum obtained by Miller and Klippenstein [43] for an irreversible exchange reaction pathway involving 3 wells and with a qualitative Potential Energy diagram similar to the current case is shown in Fig The consistency between the spectra of eigenvalues shown in Figure 8.2 and 8.3 is evident. There is one species whose chemical equilibration [43] with another one in the pathway occurs with a time-scale comparable to an internal relaxation. In fact, there is one eigenvalue that merges with the IEREs at temperatures below 400 K as it can be observed in Figure 8.2. The mathematical investigation for

115 Chapter 8. Pathway analysis for the estimation of kinetic parameters 94 Figure 8.2: Spectrum of the eigenvalues of the pathway in Fig The black solid curve represent the boundary of the IEREs, the black dotted curve is the boundary of the region close to the IEREs within which the criterion adopted by MESMER is not fulfilled. The two colored curves approaching the IEREs and the non-monotonic one are the CSEs. Figure 8.3: Spectrum of the eigenvalues obtained by Miller and Klippenstein [43] for an irreversible exchange reaction pathway involving 3 wells with a Potential Energy diagram qualitatively similar to the current case depicted in Fig the identification of that species is not straightforward. Nevertheless, Miller and Klippenstein [43] observe that the equilibrating species usually corresponds to the shallower well of the pathway. The shallower well of the current pathway corresponds to Minimum 2 in Fig It is likely that a large fraction of the molecules with a sufficient energy to overcome the barrier correspondent to Transition State 2 would not be retained [43] within the second well of the pathway. The isomerization towards Minimum 3 will occur with a time scale characteristic not of isomerization reactions but of internal relaxations.

116 Chapter 8. Pathway analysis for the estimation of kinetic parameters 95 Figure 8.4: Potential energy diagram of the pathway with the Minimum 2 skipped. The energy levels indicated are expressed in kj/mol. Figure 8.5: Spectrum of the eigenvalues of the pathway with the Minimum 2 skipped. The black solid curve represent the boundary of the IEREs, the black dotted curve is the boundary of the region close to the IEREs within which the criterion adopted by MESMER in not fulfilled. The colored curves approaching the IEREs and the nonmonotonic one are the CSEs. In order to confirm this hypothesis, since the methodology of Miller and Klippenstein [43] is not available in MESMER, an approximation in the system was introduced. Instead of merging energy distributions subsequently to the energy master equation solution, Minimum 2 was removed from the pathway, as depicted in Fig The correspondent eigenvalues spectrum (shown in Fig. 8.5) confirmed the hypothesis above: the CSEs that merged with the IEREs below 400 K is not present any more. However the merging issue is still evident for temperatures above 1000K.

117 Chapter 8. Pathway analysis for the estimation of kinetic parameters 96 Figure 8.6: Potential energy diagram of the pathway with both Minimum 2 and Minimum 3 skipped. The energy levels indicated are expressed in kj/mol. Figure 8.7: Spectrum of the eigenvalues of the pathway with both Minimum 2 and Minimum 3 skipped. The black solid curve represent the boundary of the IEREs, the black dotted curve is the boundary of the region close to the IEREs within which the criterion adopted by MESMER in not fulfilled. The colored curve approaching the IEREs and the non-monotonic one are the CSEs A note on well skipping Removing another well from the pathway would sharply detach the current analysis from the pathway depicted by Parker et al. [27] and it was not considered a reliable option. The following considerations support this choice: Concerning the pathway depicted in Fig. 8.4 the CSEs that tend to merge to the IEREs are two. The third CSE - the one that starts to decrease after about 1000 K - is likely to be representative of a chemical equilibration involving the products

118 Chapter 8. Pathway analysis for the estimation of kinetic parameters 97 due to the infinite sink approximation and the subsequent completely unbalanced equilibrium. The last statement is consistent to the observation of Miller and Klippenstein [43]: The eigenvalue that is far distant from the IEREs is usually related to equlibrations involving products or reactants. The point stated to be the the critical intersection by Miller and Klippenstein [43] shown in Figure 8.3 occurs at about 1800 K in the pathway in Fig. 8.4 as it can be observed in the eigenvalue spectrum of Fig Hence, the merging of eigenvalues does not occur for a broad range of temperatures of interest for this pathways when applied to the studied flames. Moreover, according to Miller and Klippenstein [43] the phenomenological rate coefficients can be considered an acceptable first approximation even when the merging of eigenvalues occurs. Furthermore, the criterion adopted by MESMER might not be interpreted as a strict definition of a quantitative condition that if not fulfilled make the macroscopic analysis of the system unreliable. Concerning the temperature range of interest for the studied pathway, the profile of vinylacetylene in the flame of Olten and Senkan has a peak at a temperature of about 1500 K, i.e. at a temperature in which the critical intersection has not occurred yet. In order to investigate the effect of skipping another well on the correspondent phenomenological description of the current chemical system, the pathway in Fig. 8.6 was studied. In this pathway both Minimum 2 and Minimum 3 are skipped (the reason of this choice will be explained in the following section). The correspondent eigenvalues spectrum in Fig. 8.7 shows that the merging issue is still present, even though the critical intersection of eigenvalues does not occur within the range of temperatures considered. The following analysis concerning the species profiles computed by MESMER confirms that the skipping of a further well represents an inconsistent approximation in the phenomenological description of the pathway Species profiles analysis The expressions of the species profiles as a function of time are obtained by integrating the species energy density distributions over E (see Eq. 8.25). Hence, the resulting species profiles expressions (see Eq. 8.26) do not involve any approximation related to the time-scale separation betweene CSEs and IEREs and are for the non-conservative

119 Chapter 8. Pathway analysis for the estimation of kinetic parameters 98 case [43]: X R (t) = X i (t) = N chem j=0 N chem j=0 a Rj e λ jt a ij e λ jt (8.40) (8.41) With i [1; n wells ]. X P (t) = N chem j=0 a P j e λ jt (8.42) In Fig. 8.8 the fractional species profiles as a function of time computed by MESMER for the complete pathway in Fig. 8.1 are shown. The subsequent diagrams correspond to the indicated different temperatures at ambient pressure (the pressure of the ethylene diffusion flame of Olten and Senkan [11]). Numerical instability leading to nonphysical results was observed at low temperatures: For this reason the first temperature considered is 500 K. This pathway showed a critical intersection of eigenvalues starting from 400 K. Observing Fig. 8.8 it is possibile to understand that the choice to skip Minimum 2 was consistent. In fact the fractional profile of Minimum 2 lays below 10 7 : As soon as Minimum 2 is formed, it is converted to another species of the pathway. Minimum 3 fractional profile tends to approach the correspondent one of Minimum 2 with increasing temperature. Even though the same qualitative trend is shown by the fractional profile of Minimum 1, the correspondent mole fraction is sharply higher than the one of the other intermediates of the pathway and comparable to the mole fraction of the products at low temperatures. A parallel can be drawn between the dynamics of the fractional species profiles described above and the eigenvalue spectra analyzed in the previous section. With increasing temperature the conversion of the wells of the pathway occurs within decreasing time-scales. At a certain temperature the conversion starts to occur within a time scale characteristic of thermal relaxations. In order to investigate the effect of skipping Minimum 2, the species profile analysis was performed for the pathway in Fig. 8.4 (see Fig. 8.9). The species profiles are almost unchanged if compared to the fractional profiles in Fig The only change is represented by the disappearance of the fractional profile of Minimum 2. This observation leads to a relevant conclusion: This well skipping does not introduce any distortion in the parameters used by MESMER for calculating the remaining species profiles (i.e. a ij and λ j in Eq. 8.41). The same parameters are used by MESMER for computing the phenomenological rates (see Eq. 8.31). Hence, the phenomenological rates of the reactions involving the other stable species of the pathway should not be affected by the skipping of Minimum 2. Moreover, since the merging issues does not occur for temperatures below 1000 K, both Minimum 1 and Minimum 3 are relevant species for the

120 Chapter 8. Pathway analysis for the estimation of kinetic parameters 99 phenomenological description of the pathway of Parker et al. [27] for a broad range of temperatures relevant in the studied flames. Therefore, the skipping of Minimum 3 - and even more the skipping of Minimum 1 - would be inconsistent choices within the current analysis. Considering the observations above, it is possible to understand that the CSE that is closest to the IEREs in the eigenvalue spectrum of Fig. 8.5 corresponds to an equilibration involving Minimum 3. At high temperatures, the same equilibration occurs for Minimum 1, as shown by the dampening of the correspondent factional profile in Fig. 8.9 (analogous to the one occurring for Minimum 3). This similar dynamics ends up in a pattern for which the approaching of the second CSE in Fig. 8.5 to the IEREs Figure 8.8: Fractional species profiles as a function of time computed by MESMER for the complete pathway in Fig The subsequent diagrams correspond to the different temperatures indicated and to ambient pressure. The dashed curve corresponds to reactants, the red one with circular icons to Minimum 1, the yellow one with triangular icons to Minimum 2, the green one with squared icons to Minimum 3 and the solid curve to products.

121 Chapter 8. Pathway analysis for the estimation of kinetic parameters 100 Figure 8.9: Fractional species profiles as a function of time computed by MESMER for the pathway in Fig The subsequent diagrams correspond to different temperatures and to ambient pressure. The dashed curve corresponds to reactants, the red one with circular icons to Minimum 1, the green one with squared icons to Minimum 3 and the solid curve to products. boundary is expected. A final note on Figures 8.8 and 8.9 concerns the numerical instability observed in low temperatures calculations. Relying on the plot correspondent to T=500 K in Figures 8.8 and 8.9, a nonphysical trend of species profiles is observed. The effect of the pressure on the current chemical system was investigated performing a simulation for the pathway in Fig. 8.4 at a pressure of 10 6 atm. The qualitative trend of the fractional profiles is the same as the one observed for the atmospheric pressure case (see Fig. 8.9). However both Minimum 1 and 3 are stabilized since the correspondent fractional profiles appears to be translated upwards. In other words, the dampening of the the species profiles of Minimum 1 and 3 was observed at higher temperatures if compared to the case at atmospheric pressure. Moreover, the eigenvalue merging issue does not occur for this case within the investigated temperature range. Another

122 Chapter 8. Pathway analysis for the estimation of kinetic parameters 101 Figure 8.10: Fractional species profiles as a function of time computed by MESMER for the pathway in Figure 8.4 The subsequent diagrams correspond to different temperatures at a pressure of 10 6 atm. The characterization of the different curves is consistent with Fig simulation was performed for the same chemical system in Fig. 8.4 at a pressure equal to 2.67 kpa (the pressure of the benzene premixed flame of Bittner and Howard [10]) and similar considerations could be done for the correspondent factional species profiles. These two tests represented the proof that pressure tends to stabilize the wells of the current pathway. As stated above, with decreasing pressure the intermediates are less stabilized into the wells. It will be shown in the next sections that with the decreasing stabilization of the wells, the rate constant of the reaction involving the direct formation of the products from the reactants increases. This is due to the equilibrium of the first step of the pathway, that moves towards the reactants with decreasing pressure (as it could be expected simply based on Le Chatelier s principles) and with increasing temperature. In fact, an increase of the backward reaction rate constant for the first step of the pathway was observed with increasing temperature. In other words, with decreasing pressure, or with increasing temperature, the grains with enough energy to overcame the highest barrier of the pathway corresponding to Transition State 1 tends to stabilize directly into the products and not into the intermediate wells.

123 Chapter 9 Results of the computation of molecular properties and kinetic parameters 9.1 Computed molecular properties Single point energies In this section, the values of single point energies obtained from GAMESS B3LYP calculations are outlined and commented. The values are shown in Tab. 9.1 and 9.2 for the 6-311G and the 6-311G(d,p) basis set, respectively. The motivation behind this comparison is to evaluate the sensitivity of the energy levels computed with respect to the polarization functions of the basis set. The non-polarized basis set is nowadays considered obsolete [32], whereas the 6-311G(d,p) basis set was adopted by Parker et al. [27] for obtaining the vibrational frequencies, Zero Point Eenergies, and the other molecular properties needed in the current study. The series of single point energies obtained through GAMESS calculations using B3LYP Density Functional Theory and either 6-311G or 6-311G(d,p) basis sets qualitatively agree but quantitatively differ from the ones obtained by Parker et al. [27]. One of the most relevant discrepancies is the barrier at the entrance of the pathway: the barriers obtained as a result of GAMESS calculations are kj/mol and kj/mol adopting the 6-311G and 6-311G(d,p) basis sets, respectively. The values differ almost by a factor of two from the barrier at the entrance of the pathway computed by Parker et al. [27]. The reason of this discrepancy is the CCSD(T) based composite energy method adopted by Parker et al. [27] for the single point energy calculations. As explained at 102

124 Chapter 9. Results of the computations of molecular properties and kinetic parameters 103 Table 9.1: Ground single point energies for the different species from B3LYP calculations adopting 6-311G basis set. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical). Species E 0 [kj/mol] E 0 [kj/mol] Phenyl Radical Vinylacetylene (Reactants sum) Transition State Minimum Transition State Minimum Transition State Minimum Transition State Naphtalene Hydrogen (Products sum) Table 9.2: Ground single point energies for the different species from B3LYP calculations adopting 6-311G(d,p) basis set. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical). Species E 0 [kj/mol] E 0 [kj/mol] Phenyl Radical Vinylacetylene (Reactants sum) Transition State Minimum Transition State Minimum Transition State Minimum Transition State Naphtalene Hydrogen (Products sum) the beginning of this chapter, the replication of this computationally expensive method was out of scope since the reliability of the analysis developed by Parker et al. [27] is not doubted. Since the potential energy surface computed by Parker et al. [27] was considered the most accurate due to the composite energy method adopted, the correspondent energy levels with respect to the ground energy of the reactants were used in the current study. The resulting single point ground energies are listed in Tab Another set of ground energies was obtained through calculations using M06-2X DFT method, coupled with the same polarized 6-311G(d,p) basis set. This was considered a test of reliability of this more recent functional. The corresponding computed energy

125 Chapter 9. Results of the computations of molecular properties and kinetic parameters 104 Table 9.3: Ground single point energies for the different species according to the values of Parker et al. [27]. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical). Species E 0 [kj/mol] E 0 [kj/mol] Phenyl Radical Vinylacetylene (Reactants sum) Transition State Minimum Transition State Minimum Transition State Minimum Transition State Naphtalene Hydrogen (Products sum) Table 9.4: Ground single point energies for the different species from M06-2X calculations adopting 6-311G(d,p) basis set. E 0 is the difference between E 0 of each species and the sum of E 0 of the reactants (vinylacetylene + the phenyl radical). Species E 0 [kj/mol] E 0 [kj/mol] Phenyl Radical Vinylacetylene (Reactants sum) Transition State Minimum Transition State Minimum Transition State Minimum Transition State Naphtalene Hydrogen (Products sum) levels are listed in Tab The convergence of the GAMESS optimization algorithm was never reached for the molecular configuration related to Transition State 1. The author believes that neither the most refined mesh set used in these calculations was able to resolve some components the potential energy gradient of this configuration within an acceptable tolerance of 10 5 Hartrees during the optimization runs for the exact location of the saddle point. This statement is based on the continuous oscillations observed between several values of the energy gradient matrix after each single geometry search iteration. Their magnitude never fell below the set tolerance threshold. Even if the tolerance threshold was relaxed down to Hartrees - value stated by the GAMESS documentation [29] to be the lowest acceptable - the issue was not solved. Moreover, the ground energy of the hydrogen atom was kept equal to the one calculated through

126 Chapter 9. Results of the computations of molecular properties and kinetic parameters 105 the B3LYP DFT method coupled with the current basis set. This was due to numerical divergence encoutered during the computation of the M06-2X single point energy. The author believes that the numerical algorithm implemented in GAMESS leads to numerical instability for M06-2X calculations on species with a single electron. A test calculation on a single carbon atom succeeded in the computation of the ground energy with the same settings as the hydrogen atom calculation and without encountering any numerical instability. The four different Potential Energy Surfaces correspondent to the ground energy values of Tables 9.1, 9.2, 9.3 and 9.4 are plotted in Fig Considering that further refinements could increase the accuracy of the computed profiles, it can be concluded that M06-2X functional brings to the mostly accurate profile with respect to the one obtained by Parker et al. [27] using a CCSD(T) based composite energy method. Concerning the comparison between the ground state energies obtained using B3LYP with 6-311G and 6-311G(d,p) basis sets, they followed the preliminary expectations for which by adding polarization functions to the basis set the accuracy is enhanced. This is observable from Fig. 9.2 since the energy profile corresponding to B3LYP/6-311G(d,p) is closer to the profile coherent with Parker et al. [27]. (a) Transition State 1 (b) Transition State 2 (c) Transition State 3 (d) Transition State 4 Figure 9.1: Reaction Coordinate of the Transition States of the pathway.