Pressure Dependent Kinetics: Single Well Reactions · Single Well Reactions Simple Models • Lindemann-Hinshelwood • RRKM Theory • Modified Strong Collider The Master Equation

Pressure Dependent Kinetics:Single Well Reactions

Simple Models• Lindemann-Hinshelwood• RRKM Theory• Modified Strong ColliderThe Master Equation• 1-dimensional (E)• 2d Master Equation (E,J)• Energy Transfer• Troe Fitting• Product Channels CH3 + OHTheory of Unimolecular and Recombination Reactions, R. G. Gilbert and S. C.

Smith, Blackwell, 1990Unimolecular Reactions, K. A. Holbrook, M. J. Pilling, S. H. Robertson,

Wiley, 1996

Recombination KineticsRecombination is a Multistep Process - not single elementary step

A + B → AB(E) kf(E) [k(T) = ∫ k(E) P(E)]

But, E is above dissociation threshold so AB just redissociates

AB(E) → A + B kd(E)

Need some process to take away energy and stabilize ABCollisions with bath gas M (or photon emission)

AB(E) + M → AB(E') + M' kc x P(E → E')

Effective rate constant is some mix of kf(E), kd(E), kc, and P(E → E')Dissociation is related to recombination through equilibrium constant

Simple Models Lindemann-HinshelwoodAssume every collision leads to stabilizationTreat association and dissociation on canonical levelA + B → AB* kf(T)AB* → A + B kd(T)AB* + M → AB + M' kc

Steady state for [AB*] =>d[AB]/dt = keff [A] [B]keff = kf kc [M] / ( kd + kc [M] ) = kf Pstabilization

High Pressure limit ([M] → ∞)keff = kf

Low Pressure limit ([M] → 0)keff = kf kc / kd

Not accurate but good for qualitative thought

Simple Models RRKM TheoryTreat energy dependence of association and dissociation rate constantskf(E) and kd(E)keff (T,P) = dE keff (E) P(E) = dE kf(E) P(E) Pstabilization (E,P)Use transition state theory with quantum state counting to evaluate kf, kd

keff = dEN ± E( )

hρreac tan t (E)∫

ρreac tan t E( )exp(−βE)QAQB

kc M[ ]kd (E) + kc M[ ]

keff =1

hQAQB

dEN ± E( )exp(−βE) kc M[ ]kd (E) + kc M[ ]∫

keff∞ =

1hQAQB

dEN ± E( )exp(−βE) = kBT

hQAQB

dEρ± E( )exp(−βE)∫∫

keff∞ =

kBT

h

Q±

QAQB

Consider High Pressure Limit; [M] → ∞

Simple Models Modified Strong ColliderAssume only a fraction βc of collisions lead to stabilization

keff =1

hQAQB

dEN ± E( )exp(−βE) βckc M[ ]kd (E) + βckc M[ ]∫

Consider low pressure limit; [M] → 0

keff0 =

1

hQAQB

dEN ± E( )exp(−βE) βckc M[ ]kd (E)

∫

keff0 =

βckc[M]

QAQB

dEρAB E( )exp(−βE)0

∞

∫keff

0 does not depend on transition state! Only the threshold Ematters

βc is a fitting parameter -typical value ~ 0.1

Master EquationConsider n(E,t) = time-dependent population of AB molecule at

energy EMaster equation Irreversible Formulation

dn(E)

dt= kc[M] dE ' P E,E '( )n E ', t( ) − P(E ',E)n E,t( )[ ]∫ − kd E( )n E,t( )

Replace n(E,t) with normalized population x(E,t) = n(E,t)/ dEn(E,t)

Steady state for x =>

−k(T, p)x(E) = kc[M] dE 'P E,E '( )x E '( ) − kc[M]x E( )∫ − kd E( )x E( )

Master equation Reversible Formulation

dn(E)

dt= kc[M] dE ' P E,E '( )n E ', t( ) − P(E ',E)n E,t( )[ ]∫ − kd E( )n E,t( ) +

k f (E)ρreac tan t E( )exp(−βE)

QAQB

nAnB

Master Equation Symmetrized Formf2(E) = ρ(E) exp(-βE)=F(E)Q(T)y(E) = x(E)/f(E)Discretize master equation

d y

dt= G' y

Gij' = kc[M]P Ei,E j( ) f E j( )

f Ei( ) δE − 1+kd E( )kc[M]

⎡

⎣ ⎢

⎤

⎦ ⎥ δ ij

Diagonalize

Eigenvalues are all negativeOne with smallest magnitude defines the rate coefficientk(T,p) = -ξ1Others are related to rate of energy transfer - form continuum

y t( ) = expj=1

N

∑ ξ j t( ) g j' g j

' y 0( )

Master Equation Problems at Low T

numerical difficulties with diagonalization due tolarge dynamic rangeVarious Solutions

1. Integrate in time2. Quadruple Precision3. Reformulate with sink for complex => Matrix

inversion

Master Equation Problems at high TDissociation occurs on same time scale as energy

relaxation

Nonequilibrium factor fne

fne =dEc E( )∫( )2

dEc 2 E( )F(E)∫

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2 c(E) = steady state distribution

Deviation of fne from unity indicates how muchdissociation happens before relaxation

Detailed balance is still satisfied for fraction thathappens after relaxation

Boltzmann Distributions CH4

Boltzmann Distributions C2H5O2

Nonequilibrium Factor

Master Equation 2-DimensionalTotal Angular Momentum J - conserved between

collisionsMaster equation in E and J

n(E,J,t) or x(E,J,t)P(E,J,E',J')k(E,J)

Numerical solution timeconsumingNeed more information on energy transfer than we

have

Approximate Reduction from 2D to 1DE model

P(E,J,E',J') = P(E,E') ϕ(E,J)Rotational energy transfer like vibrationalJ distribution given by phase space volumeϕ(E,J)=(2J+1) ρ(E,J) / ρ(E)ρ(E) = ∑J (2J+1) ρ(E,J)k(E) = ∑J (2J+1) N‡(E,J) / hρ(E)Use k(E) and P(E,E’) in 1D Master Eqn

Does not resolve J dependent thresholdsAll rotational degrees of freedom are activeIncorrect low pressure limit

2D Master Equation E,J ModelE,J model

like E model, but treat k(E,J) properlyk(E) = ∑J k(E,J) y(E,J) / ∑J y(E,J)y(E,J) = ϕ(E,J) / { kc[M] + k(E,J) }x(E) = ∑J x(E,J)

x(E,J) =kc[M]ϕ(E,J)Z + k(E,J)

dE 'P E,E '( )x E '( )∫Proper treatment of J dependent thresholdsProper zero-pressure limitProper high-pressure limit Consistent with detailed balance

2D Master Equation ε,J Modelε,J model

Active energy - does not include overall rotation ε = E - EJEJ = BJ(J+1)P(ε,J,ε',J') = P(ε,ε')Φ(ε,J)

Φ(ε,J) = (2J+1)ρ(ε,J)exp(-βEJ)/∑J(2J+1)ρ(ε,J)exp(-βEJ) ρ(ε,J) = density of states for active degrees of

freedomThermally equilibrated J distributionSatisfies Detailed balance

Steady State Distribution CH4

E model E,J model

Low Pressure Limit CH4

Low Pressure Limit

Reduced Falloff Curves

Collision RatesHard Sphere

kcHS =

8kBT

πμπd2

Lennard-Jones

kcLJ = kc

HSΩ2,2*

Underestimates collision rate Correct with larger average energy transferredDipole Corrections

Ω2,2* =

1.16145

T*( )0.14874+

0.52487

exp 0.7732T*( ) +2.16178

exp 2.437887T*( ) T* = kBT/ε

Energy Transfer FormsExponential Down

P E,E '( ) = 1

CN E '( ) exp −ΔE /α( )

P E,E '( ) = 1

CN E '( ) exp − ΔE /α( )2[ ]

P E,E '( ) = 1

CN E '( ) 1− f( )exp −ΔE /α1( ) + f exp −ΔE /α2( )[ ]Double Exponential Down

Gaussian Down

α = α0(T /298)n

α0 ~ 50-400 cm-1

n ~ 0.85Fit to experiment

Energy Transfer MomentsAverage Energy Transferred

ΔE = dE E '−E( )∫ P E,E '( )

for exponential down

Average Downwards Energy Transferred

ΔE d = dE ' E '−E( )0

E '

∫ P E,E '( ) / dE '0

E '

∫ P E,E '( )

ΔE 2 = dE E − E '( )∫2P E,E '( )

ΔE d ≈α

Average squared energy transfer

Fits to Experiment H + C2H2 Addition

Fits to Experiment C2H3 Dissociation

Fits to Experiment T dependent ΔEd

Energy Transfer from Trajectories

Collisional energy transfer in unimolecularreactions: Direct classical trajectoriesfor CH4=CH3+H in Helium

A. W. Jasper, J. A. Miller, J. Phys. Chem. A 113,5612 (2009).

α0=110 cm-1 n=0.81

Barker is studying P(E,J,E’,J’)

Troe FittingNeed to represent k(T,P) for Global ModelsStandard is Troe Fitting

k(T, p) =k0[M]k

∞

k∞ + k0[M]F log10 F =

log10 Fcent

1+log10(p*) + c

N − d log10 p*( ) + c( )⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2

c = −0.4 − 0.67log10 Fcent N = 0.75 −1.27log10 Fcent

d = 0.14p* = k0[M]/k∞

Fcent = 1− a( )exp −T /T***( ) + aexp −T /T*( ) + exp −T** /T( )

Fit k0 & k∞ to modified Arrhenius k0 = A0Tn 0 exp(−E0 /T)

Fit Fcent to:

Troe Fitting ProblemsLimited AccuracyTypical Errors ~ 10 to 20%Improved Fitting Formulas

A Fitting Formula for the Falloff Curves of UnimolecularReactions, P. Zhang, C. K. Law, Int. J. Chem. Kinet. 41,727 (2009)

Still problems for tunnelingMultiple channels - actual P dependence is dramatically

different from Troe Form

Use Log Interpolation

logk = logki + logki+1 − logki( ) log p − log pi( )log pi+1 − log pi( )

Part of Current ChemKin

CH3 + OH

Potential EnergySurface

G2M MRCI//CASQCISD(T)/CBS

Reactions with Products: CH3 + OHExperiment:Triangles - De AvillezPereira, Baulch, Pilling,Robertson, and Zeng, 1997Circles - Deters, Otting,Wagner, Temps, László,Dóbé, Bérces, 1998Theory: Master EquationsDotted - De Avillez Pereiraet al.Solid & Dashed - PresentWork

<ΔEd> = 133 (T/298)0.8 cm-1

± 25%

CH3 + OH: Higher T and P ~1 atm

Shock tube studies• 1991, Bott and

Cohen (1 atm)• 2004, Krasnoperov

and Michael (100−1100 torr)

• 2006, Srinivasan,Su, and Michael(200−750 torr)

Methanol decomposition: Low pressure limit

CH3OH → CH3 + OH Experimental• 2004, Krasnoperov and Michael• 2006, Srinivasan, Su, and Michael• 1981−2000, Many others• k independent of P (100−1000 torr)• 60−90% CH3 + OHPrevious theory• 2001, Xia, Zhu, Lin, and Mebel

(shown at 1 atm)• Falloff below 1 atm• ~33% CH3 + OH

~52% CH2 + H2O~15% H2 + HCOH

Present theory• Low-P limit at 1 atm• ~75% CH3 + OH

~20% CH2 + H2O< 5% H2 + HCOH

Methanol decomposition: Product branching

Secondary kinetics of methanol decompositionShock tube OH absorption profilesensitivities for CH3OH → CH3 + OH

Srinivasan, Su, and Michael, JPCA, 2007

Well characterizedOH + OH → O + H2OH + OH → O + H2

Not well characterized3CH2 + OH → CH2O + H

Ambiguous experiments3CH2 + 3CH2 → C2H2 + 2HCH3 + 3CH2 → C2H4 + H

Secondary kinetics: OH Time Traces

Michael et al.

Good agreementat long timesusing ourpredicted ratesfor

3CH2 + OH3CH2 + 3CH2

CH3 + 3CH2

CH3 + OH

OH

con

cent

ratio

n

Multiple-Well Multiple-ChannelTime Dependent Master Equation

1. The Kinetic Model2. Collisionless Limit3. CH + N24. Time Dependent Populations5. Kinetic Phenomenology6. C2H5 + O27. Reduction in Species at High Pressure8. C3H3 + H9. Radical Oxidation10.C3H3+C3H3

Multiple-Well Multiple-Channel Master Equation

M Wells Np ProductsM+1 Chemical SpeciesnB>>nm>>nR B=Bath, m=Molecule, R=RadicalLinear Master Equation

dni(E)

dt= kcnB dE 'Pi E,E '( )ni E '( ) − kcnBni E( )∫ − kdi E( )ni E( ) − kpi E( )

p=1

N p

∑ ni E( ) −

k jiisom E( )

j≠ i

M

∑ ni E( ) + kijisom

j≠ i

M

∑ E( )n j E( ) + Keqikdi E( )Fi E( )nRnm

dnRdt

= dEkdi E( )ni E( )∫i=1

M

∑ − nRnm KeqidEkdi E( )Fi E( )∫

i=1

M

∑

Collisionless LimitConsider Z→ 0

d n(E,J)

dt= −K E,J( ) n E,J( ) + nRnm b(E,J) ρRm (E,J)exp −βE( ) /QRm

Steady State for n(E,J)

d P(E,J)

dt= D E,J( ) n E,J( )

d P(E,J)

dt= D E,J( )K −1 E,J( ) b E,J( ) nRnmρRm E,J( )exp −βE( ) /QRm (T)

k0(T) =1

QRm (T)2J +1( )

J∑ dED E,J( )∫ K −1 E,J( ) b E,J( ) ρRm E,J( )exp −βE( )

Flux coefficients

CH + N2 Prompt NO• 1971 Fenimore 2CH + N2 → HCN + 4N• 1991 Dean, Hanson & Bowman -- shock tube measurements of

rate for 2CH + N2 → Products• 1991 Manaa & Yarkony -- located minimum crossing point for

doublet to quartet transition• 1996 Miller & Walch -- found maximum on spin forbidden path

corresponds to dissociation of the quartet complex; not thedoublet-quartet crossing; presume rapid ISC and fit experimentaldata

• 1999 Qui, Morokuma, Bowman & Klippenstein -- predicted spin-forbidden reaction to be less than observed rate by at least 102

• 2000 Moskaleva, Xia & Lin -- predicted new spin allowedmechanism,

2CH + N2 → HNCN → 2H + 3NCN• 2007 Szpunar, Faulhaber, Kautzman, Crider & Neumark --

observed the photodissociation of DNCN to CD+N2 and D+NCNwith 1:1 branching ratio

Recent Modeling• Williams, Fleming

Proc. Comb. Inst. 31, 1109-1117, 2007NO severely underpredicted in CH4 and C3H8 flames

• El Bakali, Pillier, Desgroux, Lefort, Gasnot, Pauwels, da Costa,Fuel 85, 896, 2006Increasing CH + N2 rate by 1-2 orders of magnitude over the1000 to 1500 K range yields good predictions for NO in naturalgas flames

• Sutton, Williams, Fleming,Comb. Flame, 2008, in press.Improved modeling for CH4/O2/N2 flames with rates of ElBakali et al.

Moskaleva, Xia and Lin (2000)

Present

HN C N

N N N N

H H

N N

H

CAS+1+2+QC/aug-cc-pvtz

Contour Increments: Thick- 5 kcal/mol, Thin- 1 kcal/mol

CASPT2 CAS+1+2+QC

CCSD(T)B3LYP

656, 1029, 1062,1424, 2985, 361i

CASPT2:

B3LYP:507, 540, 1059,

1494, 2505, 1225i

Discretize Energy Levels

Transition Matrix; Renormalize → real, symmetric; G

Diagonalize

Time-Dependent Populations

( ) ( )d w t G w tdt

=

( ).

1

. . 1

( ) 0I M

jN N

tj j

jw t e g g wλ

+

=

+

= ∑

w(t) = yI (E0 I),L,yI (Emax ),L,yi(E0 i

),L,yi(Emax ),L,nm

QRmδE⎛ ⎝ ⎜

⎞ ⎠ ⎟ 1/ 2

XR ,L⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

T

yi E,t( ) = xi E,t( ) / fi E( )

Kinetic PhenomenologyExperimental Viewpoint

• Find regimes of single exponential decay (λ)– λ implies total rate coefficient– Eigenvector corresponding to λ implies branching– Branching implies individual rate coefficient (ktot)

• When is decay close enough to single exponential?– Suppose 2nd eigenvector contributes to only 1% of

the initial decay but that λ2/λ1 = 100– Rate coefficient will differ by a factor of two from

apparent exponential decay– Branching similarly incorrect

• Difficult to find single exponential decay regimes inmultiple well situations

Eigenvalues C3H3 + C3H3

A Simple Solution: Separation of Timescales• M+1 modes corresponding to chemical change have

least negative eigenvalues.• λ’s for chemical modes well separated from remaining

λ’s for energy transfer• After energy relaxation can treat populations as

•

•

• Eigenpairs (λi, ΔXi) correspond to Normal modes ofchemical relaxation

( ) jtj jj

1

=1

M+ ët e g g (0)Aw w= Σ ll

( ) ( ) ( )1

1; (0)j A

ij

M t Ai jj ij A i jj i

d XX e X g w E f E gdt

λ δλ+

= ∈= − Σ Δ = −Δ Σ l l

l

λ

Method 1 t=0 Limit and Start in Well A

• Phenomenology

• Master Equation

•

• Similarly, consider dXi/dt implies

•

•

( ) ( )0 0ATA A

dX k Xdt

= −

( ) ( )1

10

MAA

j Ajj

dX Xdt

λ+

== − Σ Δ

( )1

1

MA

TA j Ajjk Xλ

+

== Σ Δ

( )1

1

MA

Ai j ijjk Xλ

+

== − Σ Δ

( )1/ 2

1

1; (0)

MA Rm

AR j Rj Rj j j Ajm

Q Ek X X g g wnδλ

+

=

⎛ ⎞= − Σ Δ Δ = −⎜ ⎟

⎝ ⎠l

( )1

1; 0

M MA

Ap j pj R p ij i Ij

k X X X Xλ+

= =

⎛ ⎞= − Σ Δ Δ + Δ + Σ Δ =⎜ ⎟⎝ ⎠

Method 2 Long time limit

•

•

•

•

( )1 1

0 0

jM Mt

i ij ij jj jX t a e a vλ+ +

= == Σ ≡ Σ X A v= v B X=

1 2

0 1

M Mi

j ij jj

dX a b Xdt

λ+ +

= == Σ Σ l l

l

ii i ii i

dX k X k Xdt ≠ ≠

= Σ − Σl l ll l

1

0

M

i j ij jjk a b iλ

+

== Σ ≠l l l

C2H5 + O2 Potential Energy Surface

C2H5 + O2

C2H5 + O2 T Dependent Rate Coefficients

C2H5 + O2 P Dependent Rate Coefficients

C2H5O2 → C2H4 + HO2 P Dependent Rate Coefficients

C3H7 + O2 Formally Direct Pathways; QOOH

C3H4 Potential Energy Surface

C3H4 Rate Coefficients

C3H3 + C3H3 Eigenvalues

C3H3 + C3H3 Rate Coefficients

C3H3 + C3H3 Isomerization Rate Coefficients

C3H3 + C3H3 Product Branching in 1,5-Hexadiyne Pyrolysis

C3H3 + C3H3 Product Branching

C3H3 + C3H3 Product Branching

C3H3 + C3H3 Rate Coefficients

C6H6 Dissociation Rates

Master Equation CodesEigenvalue Eigenvector Methods

VariFlex KlippensteinResearch Code - Not usable without personal training

MESMER Pilling (Leeds)http://sourceforge.net/projects/mesmer/

Stochastic Master Equation SolversExperimental Perspective onlyMultiwell Barker(Michigan)

http://esse.engin.umich.edu/multiwell/MultiWell/MultiWell%20Home/MultiWell%20Home.html

Vereecken and Peeters (Leuven)Steady State Solvers

ChemRate Tsang (NIST)http://www.mokrushin.com/ChemRate/chemrate.html