7th ICCK (MIT, Cambridge) July 11, 2011

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Molecular Size Dependent Molecular Size Dependent Fall-off Rate Constants for Fall-off Rate Constants for

the Recombination the Recombination Reactions of Alkyl Radicals Reactions of Alkyl Radicals

with Owith O22

Akira MiyoshiAkira Miyoshi

Department of Chemical Systems Engineering, University of TokyoDepartment of Chemical Systems Engineering, University of Tokyo

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IntroductionIntroduction

— — R (alkyl) + OR (alkyl) + O22

•• key reactions that lead to chain branching in low-temperature key reactions that lead to chain branching in low-temperature oxidation of hydrocarbonsoxidation of hydrocarbons

— — ChallengesChallenges

•• resolution of complicated pressure- and temperature- dependent resolution of complicated pressure- and temperature- dependent product specific rate constants including second Oproduct specific rate constants including second O22 addition addition

reactions to QOOHreactions to QOOH

— — ObjectivesObjectives

•• evaluation of universal fall-off rate expression for recombinationevaluation of universal fall-off rate expression for recombination•• master equation analysis for the dissociation/recombination master equation analysis for the dissociation/recombination

steady-statesteady-state

ComputationalComputational

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ComputationalComputational

— — Quantum Chemical CalculationsQuantum Chemical Calculations

•• B3B3LYP & CBS-QB3 calculations by Gaussian 03LYP & CBS-QB3 calculations by Gaussian 03•• CASPT2 calculations by MOLPRO 2008.1CASPT2 calculations by MOLPRO 2008.1

— — TST and VTST CalculationsTST and VTST Calculations

by by GPOP*GPOP* including: including:•• PPitzer-Gwinn approximation for hindered rotors, itzer-Gwinn approximation for hindered rotors, qqPGPG

(after analysis by (after analysis by BEx1D*BEx1D*))•• 1D tunneling correction (asymmetric Eckart), 1D tunneling correction (asymmetric Eckart), κκtuntun

•• rotational conformer distribution partition function, rotational conformer distribution partition function, qqRCDRCD

— — RRKM/ME CalculationsRRKM/ME Calculations

•• ρρ((EE) and ) and kk((EE) accounting for all TST feature () accounting for all TST feature (qqPGPG, , κκtuntun, and , and qqRCDRCD) ) by by

modified UNIMOL RRKM programmodified UNIMOL RRKM program•• steady-state & transient master equation calculations by steady-state & transient master equation calculations by SSUMES*SSUMES*

* http://www.frad.t.u-tokyo.ac.jp/~miyoshi/tools4kin.html

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Hindered RotorHindered Rotor (carbon-centered radical) (carbon-centered radical)

•• ppartition function calculated from eigenstate energies, artition function calculated from eigenstate energies, qqexactexact,, is well is well

approximated by approximated by qqPGPG((VV0 0 == 100100 cmcm––11) or ) or qqFRFR (free rotor) (free rotor)

— — Pitzer-Gwinn ApproximationPitzer-Gwinn Approximation

6Hindered RotorHindered Rotor (RO (RO22))

•• partition function calculated from eigenstate energies, partition function calculated from eigenstate energies, qqexactexact,, is well is well

approximated by 2approximated by 2qqHOHO++qqHOHO'' or or qqHOHOqqRCDRCD

— — Taken into Account as Rotational ConformersTaken into Account as Rotational Conformers

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Rotational ConformersRotational Conformers

tot RCD 0q q q

RCD exp ii

i

q gkT

•• rotational conformer distribution partition function, rotational conformer distribution partition function, qqRCDRCD

— — Taken into Account via Partition FunctionTaken into Account via Partition Function

tot exp ii

i

q qkT

by assuming qi q0

Molecular Size Dependent Molecular Size Dependent Fall-off Rate ConstantsFall-off Rate Constants

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Potential Energy CurvesPotential Energy Curves

•• CCASPT2(7,5)/aug-cc-pVDZ // ASPT2(7,5)/aug-cc-pVDZ // B3LYP/6-311G(d,p) potential B3LYP/6-311G(d,p) potential energy well reproduced energy well reproduced experimental experimental kk(300 K) (300 K) within within ± 25%± 25%

•• B3LYP/6-311G(d,p) potential B3LYP/6-311G(d,p) potential energy systematically energy systematically underestimated underestimated kk(300 K)(300 K)

Rate Constants for R + O2 RO2

k(300 K) / 10–11 cm3 molecule–1 s–1 R

exp. CASPT2 (%err) B3LYP (%err) C2H5 0.780 0.728 (–7) 0.411 (–47) i-C3H7 1.41 1.25 (–11) 0.829 (–41) n-C4H9 0.750 0.921 (+23) 0.354 (–53) s-C4H9 1.66 1.26 (–24) 0.426 (–74) t-C4H9 2.34 2.50 (+7) 1.73 (–26)

R (alkyl) + OR (alkyl) + O22 RO RO22

10 High-Pressure Limiting Rate Constants, High-Pressure Limiting Rate Constants, kk

same for primary R'ssame for primary R's

same for secondary R'ssame for secondary R's

•• class (class (primaryprimary, , secondarysecondary, or , or tertiarytertiary) ) determines the rate determines the rate constantconstant

— — Size-IndependentSize-Independent

— — Class-SpecificClass-Specific

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Fall-off CalculationsFall-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 Energy Transfer ModelModel

0.7

1400

1000 Kcm

T

•• experimental data for experimental data for CC22HH55 + O + O22 in fall-off in fall-off region were well region were well reproduced by the reproduced by the exponential-down exponential-down model with:model with:

12 Low-Pressure Limiting Rate Constants, Low-Pressure Limiting Rate Constants, kk00

same for three Csame for three C44 R's R's

irrespective of class (primary, secondary, or tertiary)

— — Class-IndependentClass-Independent

— — Size-DependentSize-Dependent

13 Size-Dependent Expression for Size-Dependent Expression for kk00

Parameters for modified Arrhenius Expression:Parameters for modified Arrhenius Expression:kk00 = = AA TT b b exp(exp(––EEaa / / RTRT ))

nHA = number of heavy (non-hydrogen) atoms

— — Universal Fall-off Rate Constants for R + OUniversal Fall-off Rate Constants for R + O22

•• class-specific class-specific kk + size-dependent + size-dependent kk00

Collapse of Steady-State Collapse of Steady-State Assumption?Assumption?

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?

15 Steady-State Distribution of Large ROSteady-State Distribution of Large RO22

•• ssteady-state distribution for teady-state distribution for dissociation?dissociation?

rump distribution after rump distribution after major part has gonemajor part has gone

•• ssteady-state distribution for teady-state distribution for chemical-activationchemical-activation

Boltzmann distributionBoltzmann distribution

Collapse of steady-state Collapse of steady-state assumption or Lindemann-assumption or Lindemann-Hinshelwood type mechanismHinshelwood type mechanism

2 2 2R O RO * RO 2 2 2RO RO * R O

(Miller and Klippenstein, Int. J. Chem. Kinet., 2001, 33, 654–668)

•• kk kk at high temperatures at high temperatures

Dissociation/Recombination Dissociation/Recombination Steady-StateSteady-State

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17 R + OR + O22 RO RO22 Partial Equilibrium Partial Equilibrium

indd

kt

n Jn r 0

ddis,

1( ) ( ) ( )r E k E F E

k

in

dis,( ) ( )

kn E F E

k

— — Dissociation/Recombination Steady-StateDissociation/Recombination Steady-State

CChemical activation steady statehemical activation steady state

When other channels are not present, there is trivial solutionWhen other channels are not present, there is trivial solution

wherewhere

= Boltzmann distribution= Boltzmann distribution

•• mmore general condition where near ore general condition where near FF((EE) is established) is established

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Dissociation/Recombination Steady-StateDissociation/Recombination Steady-State

— — Near Boltzmann DistributionNear Boltzmann Distribution

•• rate constants for subsequent isomerization/dissociation reactions rate constants for subsequent isomerization/dissociation reactions of ROof RO22 can be estimated to be in near high-pressure limit can be estimated to be in near high-pressure limit

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Three "Steady-States"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 HOHO22 formation in C formation in C22HH55 + O + O22 C C22HH55OO22

Experimental data by Clifford, Farrell, DeSain and Taatjes, J. Phys. Chem. A, 2000, 104, 11549–11560.

•• kk((HOHO22) ) kk((HOHO22))at moderate at moderate TTbut in partial equilibrium ofbut in partial equilibrium ofR + OR + O22 RO RO22

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Time Dependent SolutionTime Dependent Solution

indd

kt

n Jn r

Time-dependent solution forTime-dependent solution for

with with nn00 = = 00 and and kkinin = const. = const.

Build-up timeBuild-up time kkdis,FOdis,FO–1–1

•• Nearly the same with and Nearly the same with and without concerted HOwithout concerted HO22 elimination channelelimination channel

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In Autoignition ModelingIn Autoignition Modeling

1000

2000

3000T

/ K

0 1 2 310-1310-1210-1110-1010-910-810-710-610-510-410-310-2

3-C7H15

n-C7H16/air = 1,720 K, 20 atm

t / ms

mol

e fr

acti

on

OH

3-C7H15O2

near partial equilibrium

transient

23 Building-Up Transient for CBuilding-Up Transient for C88HH1717OO22

collision-free build-up of F(E) with bu

–1 kdis, >> kdis,FO

build-up of F(E) with bu

–1 kdis,FO kdis,

build-up of F(E) with• bu

–1 kdis,FO kdis, (0.01atm)

• bimodal build-up (10–6 atm)

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SummarySummary

— — Size-Dependent Fall-Off Rate Constants for R + OSize-Dependent Fall-Off Rate Constants for R + O22

•• VTST and RRKM/ME calculations forVTST and RRKM/ME calculations forR = CR = C22HH55, , ii-C-C33HH77, , nn-C-C44HH99, , ss-C-C44HH99, , tt-C-C44HH99, , nn-C-C66HH1313, and , and ii-C-C88HH1717

•• kk is class-specific but size-independentis class-specific but size-independent

•• kk00 is size-dependent but class-independentis size-dependent but class-independent

•• Universal fall-off rate expression for arbitrary R + OUniversal fall-off rate expression for arbitrary R + O22

— — Collapse of Steady-State AssumptionCollapse of Steady-State Assumption

•• For large ROFor large RO22 at high temperatures at high temperatures

— — Dissociation/Recombination Steady-StateDissociation/Recombination Steady-State

•• nnssss((EE) ) FF((EE) for RO) for RO22 in partial equilibrium with R + O in partial equilibrium with R + O22

•• HHPL(PL(kk) ) can be assumed for subsequent reactions of ROcan be assumed for subsequent reactions of RO22

•• bbuild-up timeuild-up time kkdis,FOdis,FO––11 at low at low TT

kkdis,dis,––11 at high at high TT irrespective of irrespective of PP

bbimodal build-up at midium imodal build-up at midium TT especially at low especially at low PP