1. 7th ICCK (MIT, Cambridge) July 11, 2011 1 Molecular Size Dependent Fall-off Rate Constants for the Recombination Reactions of Alkyl Radicals with O 2 Akira Miyoshi Department of Chemical Systems Engineering, University of Tokyo

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

3. Computational 3

4. 4 Computational — Quantum Chemical Calculations B3LYP & CBS-QB3 calculations by Gaussian 03B3LYP & CBS-QB3 calculations by Gaussian 03 CASPT2 calculations by MOLPRO 2008.1CASPT2 calculations by MOLPRO 2008.1 — TST and VTST Calculations by GPOP* including: by GPOP* including: Pitzer-Gwinn approximation for hindered rotors, q PG (after analysis by BEx1D*)Pitzer-Gwinn approximation for hindered rotors, q PG (after analysis by BEx1D*) 1D tunneling correction (asymmetric Eckart), κ tun1D tunneling correction (asymmetric Eckart), κ tun rotational conformer distribution partition function, q RCDrotational conformer distribution partition function, q RCD — RRKM/ME Calculations ρ(E) and k(E) accounting for all TST feature (q PG, κ tun, and q RCD ) by modified UNIMOL RRKM programρ(E) and k(E) accounting for all TST feature (q PG, κ tun, and q RCD ) by modified UNIMOL RRKM program steady-state & transient master equation calculations by SSUMES*steady-state & transient master equation calculations by SSUMES* * http://www.frad.t.u-tokyo.ac.jp/~miyoshi/tools4kin.html

5. 5 Hindered Rotor (carbon-centered radical) partition function calculated from eigenstate energies, q exact, is well approximated by q PG (V 0 = 100 cm –1 ) or q FR (free rotor)partition function calculated from eigenstate energies, q exact, is well approximated by q PG (V 0 = 100 cm –1 ) or q FR (free rotor) — Pitzer-Gwinn Approximation

6. 6 Hindered Rotor (RO 2 ) partition function calculated from eigenstate energies, q exact, is well approximated by 2q HO +q HO ' or q HO q RCDpartition function calculated from eigenstate energies, q exact, is well approximated by 2q HO +q HO ' or q HO q RCD — Taken into Account as Rotational Conformers

7. 7 Rotational Conformers rotational conformer distribution partition function, q RCDrotational conformer distribution partition function, q RCD — Taken into Account via Partition Function by assuming q i  q 0

8. Molecular Size Dependent Fall-off Rate Constants 8

9. 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%CASPT2(7,5)/aug-cc-pVDZ // B3LYP/6-311G(d,p) potential energy well reproduced experimental k(300 K) within ± 25% B3LYP/6-311G(d,p) potential energy systematically underestimated k(300 K)B3LYP/6-311G(d,p) potential energy systematically underestimated k(300 K) R (alkyl) + O 2  RO 2

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

11. 11 Fall-off Calculations 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. — Energy Transfer Model experimental data for C 2 H 5 + O 2 in fall-off region were well reproduced by the exponential-down model with:experimental data for C 2 H 5 + O 2 in fall-off region were well reproduced by the exponential-down model with:

12. 12 Low-Pressure Limiting Rate Constants, k 0  same for three C 4 R's irrespective of class (primary, secondary, or tertiary) — Class-Independent — Size-Dependent

13. 13 Size-Dependent Expression for k 0 Parameters for modified Arrhenius Expression: k 0 = A T b exp(–E a / RT ) n HA = number of heavy (non-hydrogen) atoms — Universal Fall-off Rate Constants for R + O 2 class-specific k  + size-dependent k 0class-specific k  + size-dependent k 0

14. Collapse of Steady-State Assumption? 14 ?

15. 15 Steady-State Distribution of Large RO 2 steady-state distribution for dissociation?steady-state distribution for dissociation?  rump distribution after major part has gone steady-state distribution for chemical-activationsteady-state distribution for chemical-activation  Boltzmann distribution Collapse of steady-state assumption or Lindemann- Hinshelwood type mechanism (Miller and Klippenstein, Int. J. Chem. Kinet., 2001, 33, 654–668) k  k  at high temperaturesk  k  at high temperatures

16. Dissociation/Recombination Steady-State 16

17. 17 R + O 2 RO 2 Partial Equilibrium — Dissociation/Recombination Steady-State Chemical activation steady state When other channels are not present, there is trivial solution where = Boltzmann distribution more general condition where near F(E) is establishedmore general condition where near F(E) is established

18. 18 Dissociation/Recombination Steady-State — Near Boltzmann Distribution rate constants for subsequent isomerization/dissociation reactions of RO 2 can be estimated to be in near high-pressure limitrate constants for subsequent isomerization/dissociation reactions of RO 2 can be estimated to be in near high-pressure limit

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

20. 20 HO 2 formation in C 2 H 5 + O 2 C 2 H 5 O 2 Experimental data by Clifford, Farrell, DeSain and Taatjes, J. Phys. Chem. A, 2000, 104, 11549–11560. k(HO 2 )  k  (HO 2 ) at moderate T but in partial equilibrium of R + O 2  RO 2k(HO 2 )  k  (HO 2 ) at moderate T but in partial equilibrium of R + O 2  RO 2

21. 21 Time Dependent Solution Time-dependent solution for with n 0 = 0 and k in = const. Build-up time  k dis,FO –1 Nearly the same with and without concerted HO 2 elimination channelNearly the same with and without concerted HO 2 elimination channel

22. 22 In Autoignition Modeling near partial equilibrium transient

23. 23 Building-Up Transient for C 8 H 17 O 2 collision-free build-up of F(E) with  bu –1  k dis,  >> k dis,FO build-up of F(E) with  bu –1  k dis,FO  k dis,  build-up of F(E) with  bu –1  k dis,FO  k dis,  (0.01atm) bimodal build-up (10 –6 atm)

24. 24 Summary — Size-Dependent Fall-Off Rate Constants for R + O 2 VTST and RRKM/ME calculations for R = C 2 H 5, i-C 3 H 7, n-C 4 H 9, s-C 4 H 9, t-C 4 H 9, n-C 6 H 13, and i-C 8 H 17VTST and RRKM/ME calculations for R = C 2 H 5, i-C 3 H 7, n-C 4 H 9, s-C 4 H 9, t-C 4 H 9, n-C 6 H 13, and i-C 8 H 17 k  is class-specific but size-independentk  is class-specific but size-independent k 0 is size-dependent but class-independentk 0 is size-dependent but class-independent Universal fall-off rate expression for arbitrary R + O 2Universal fall-off rate expression for arbitrary R + O 2 — Collapse of Steady-State Assumption For large RO 2 at high temperaturesFor large RO 2 at high temperatures — Dissociation/Recombination Steady-State n ss (E)  F(E) for RO 2 in partial equilibrium with R + O 2n ss (E)  F(E) for RO 2 in partial equilibrium with R + O 2 HPL(k  ) can be assumed for subsequent reactions of RO 2HPL(k  ) can be assumed for subsequent reactions of RO 2 build-up time  k dis,FO –1 at low T  k dis,  –1 at high T irrespective of P bimodal build-up at midium T especially at low Pbuild-up time  k dis,FO –1 at low T  k dis,  –1 at high T irrespective of P bimodal build-up at midium T especially at low P