Akira Miyoshi

1. Molecular Size Dependent Fall-off Rate Constants for the Recombination Reactions of Alkyl Radicals with O2 Akira Miyoshi Department of Chemical Systems Engineering, University of Tokyo 7th ICCK (MIT, Cambridge) July 11, 2011

2. Introduction — R (alkyl) + O2 • key reactions that lead to chain branching in low-temperature oxidation of hydrocarbons — Challenges • resolution of complicated pressure- and temperature- dependent product specific rate constants including second O2 addition reactions to QOOH — Objectives •evaluation of universal fall-off rate expression for recombination •master equation analysis for the dissociation/recombination steady-state

3. 3 Computational

4. Computational — Quantum Chemical Calculations • B3LYP & CBS-QB3 calculations by Gaussian 03 •CASPT2 calculations by MOLPRO 2008.1 — TST and VTST Calculations by GPOP* including: • Pitzer-Gwinn approximation for hindered rotors, qPG(after analysis by BEx1D*) •1D tunneling correction (asymmetric Eckart), κtun •rotational conformer distribution partition function, qRCD — RRKM/ME Calculations •ρ(E) and k(E) accounting for all TST feature (qPG, κtun, and qRCD) by modified UNIMOL RRKM program •steady-state & transient master equation calculations by SSUMES* * http://www.frad.t.u-tokyo.ac.jp/~miyoshi/tools4kin.html

5. Hindered Rotor (carbon-centered radical) — Pitzer-Gwinn Approximation • partition function calculated from eigenstate energies, qexact, is well approximated by qPG(V0 =100cm–1) or qFR (free rotor)

6. Hindered Rotor (RO2) — Taken into Account as Rotational Conformers •partition function calculated from eigenstate energies, qexact, is well approximated by 2qHO+qHO' or qHOqRCD

7. Rotational Conformers — Taken into Account via Partition Function • rotational conformer distribution partition function, qRCD by assuming qi q0

8. 8 Molecular Size Dependent Fall-off Rate Constants

9. Potential Energy Curves • CASPT2(7,5)/aug-cc-pVDZ // B3LYP/6-311G(d,p) potential energy well reproduced experimental k(300 K) within ± 25% R (alkyl) + O2 RO2 •B3LYP/6-311G(d,p) potential energy systematically underestimated k(300 K)

10. High-Pressure Limiting Rate Constants, k — Size-Independent  same for secondary R's  same for primary R's — Class-Specific •class (primary, secondary, or tertiary) determines the rate constant

11. Fall-off Calculations — Energy Transfer Model •experimental data for C2H5 + O2 in fall-off region were well reproduced by the exponential-down model with: Plumb & Ryan, Int. J. Chem. Kinet., 1981, 13, 1011;Slagle et al., J. Phys. Chem., 1984, 88, 3648; Wagner et al., J. Phys. Chem., 1990, 94, 1853.

12. Low-Pressure Limiting Rate Constants, k0 — Size-Dependent — Class-Independent  same for three C4 R's irrespective of class (primary, secondary, or tertiary)

13. Size-Dependent Expression for k0 Parameters for modified Arrhenius Expression:k0 = AT bexp(–Ea / RT) nHA = number of heavy (non-hydrogen) atoms — Universal Fall-off Rate Constants for R + O2 •class-specific k + size-dependent k0

14. 14 Collapse of Steady-State Assumption? ?

15. Steady-State Distribution of Large RO2 •steady-state distribution for dissociation?  rump distribution after major part has gone • steady-state distribution for chemical-activation  Boltzmann distribution Collapse of steady-state assumption or Lindemann-Hinshelwood type mechanism •kk at high temperatures (Miller and Klippenstein, Int. J. Chem. Kinet., 2001, 33, 654–668)

16. 16 Dissociation/Recombination Steady-State

17.   R + O2 RO2 Partial Equilibrium — Dissociation/Recombination Steady-State • more general condition where near F(E) is established Chemical activation steady state where When other channels are not present, there is trivial solution = Boltzmann distribution

18. Dissociation/Recombination Steady-State — Near Boltzmann Distribution •rate constants for subsequent isomerization/dissociation reactions of RO2 can be estimated to be in near high-pressure limit

19. Three "Steady-States" "delayed" "prompt" Miller and Klippenstein, Int. J. Chem. Kinet., 2001, 33, 654–668. Clifford, Farrell, DeSain and Taatjes, J. Phys. Chem. A, 2000, 104, 11549–11560.

20.   HO2 formation in C2H5 + O2 C2H5O2 •k(HO2) k(HO2)at moderate Tbut in partial equilibrium ofR + O2 RO2 Experimental data by Clifford, Farrell, DeSain and Taatjes, J. Phys. Chem. A, 2000, 104, 11549–11560.

21. Time Dependent Solution Time-dependent solution for with n0 = 0 and kin = const. • Nearly the same with and without concerted HO2 elimination channel Build-up time kdis,FO–1

22. In Autoignition Modeling near partial equilibrium transient

23. Building-Up Transient for C8H17O2 build-up of F(E) with • bu–1  kdis,FO  kdis, (0.01atm) • bimodal build-up (10–6 atm) build-up of F(E) with bu–1  kdis,FO  kdis, collision-free build-up of F(E) with bu–1  kdis, >> kdis,FO

24. Summary — Size-Dependent Fall-Off Rate Constants for R + O2 • VTST and RRKM/ME calculations for R = C2H5, i-C3H7, n-C4H9, s-C4H9, t-C4H9, n-C6H13, and i-C8H17 •kis class-specific but size-independent •k0is size-dependent but class-independent •Universal fall-off rate expression for arbitrary R + O2 — Collapse of Steady-State Assumption •For large RO2 at high temperatures — Dissociation/Recombination Steady-State •nss(E)  F(E) for RO2 in partial equilibrium with R + O2 •HPL(k) can be assumed for subsequent reactions of RO2 •build-up time  kdis,FO–1 at low T  kdis,–1 at high T irrespective of Pbimodal build-up at midium T especially at low P