Pressure Dependent Kinetics: Single Well Reactions · Single Well Reactions Simple Models •...
Transcript of Pressure Dependent Kinetics: Single Well Reactions · Single Well Reactions Simple Models •...
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