Download Chemical reactions as network of rare events: Kinetic MonteCarlo

Chemical reactions as network of rare events: Kinetic MonteCarlo
Extending the scale
Length
(m)
1
10­3
Potential Energy Surface: {Ri}
10­6
(3N+1)­dimensional
10­9
E
Thermodynamics:
p, T, V, N
continuum
ls
Macroscopic
i
a
t
e
regime
d
e
average over
or
m
all processes
many atoms
es
Mesoscopic
s
s
e
c
regime
ro
p
many processes
e few atoms
r
o
m
Microscopic
regime few processes
10­15
{Ri}
10­9
10­3
1
Time (s)
Essentials of computational chemistry: theories and models. 2nd edition.
C. J. Cramer, JohnWiley and Sons Ltd (West Sussex, 2004).
Ab initio atomistic thermodynamics and statistical mechanics
of surface properties and functions
K. Reuter, C. Stampfl, and M. Scheffler, in: Handbook of Materials Modeling Vol. 1,
(Ed.) S. Yip, Springer (Berlin, 2005). http://www.fhi-berlin.mpg.de/th/paper.html
Chemical energy conversion: catalysis
Reactant(s)
Non-catalytic free-energy barrier
Free energy
ΔFnon-cat
Reaction
Product(s)
ΔFcat
Adsorption
Desorption
Reaction coordinate
●
●
Issues:
Reaction rate: proportional to exp (­ ΔF / kT) Selectivity: eliminate or at least reduce the undesired products
Reminder: Nudged Elastic Band
Harmonic Transition State Theory:
Elastic band:
normalized local tangent at i Climbing image:
Kinetic MonteCarlo: theory
Master equation:
Probability to be in σ at time t
State
Steady state:
Transition probability per unit time
0
Detailed balance (sufficient, not necessary!):
Canonical distribution:
Define a set of events:
With (so called, elementary) transition rates:
Kinetic MonteCarlo: theory
# of time intervals of length Task: propagate system from time 0 to t
δ containing events
­ subdivide t in smaller times:
­ the average rate is: ­ in the limit there is at most one event per time interval (each time interval has equal probability to contain an event
­ Probability that events occur within time t:
(binomial distribution)
(Poisson distribution)
­ Probability density of times between two subsequent events:
(Poisson process)
Kinetic MonteCarlo: theory
Many independent (loss­of­memory assumption)Poisson processes:
: number of overall events in the ensemble after time t Invoking independence
Overall Poisson process:
Simple adsorption/desorption (Langmuir)
fractional surface coverage
desorption rate
adsorption rate
Initial condition:
Solution:
(ab initio) kMC for surface catalysis: implementation
Adsorption of O2 and CO on RuO2(110)
CO oxidation over a Ru catalyst:
O2
CO2
CO
RuO2
What are the preferred structures for RuO2·(O)x·(CO)y?
This reaction does not take place in the gas phase:
Singlet + Triplet
Singlet
E= ­3.27 eV (DFT­PBE+vdW)
Spin forbidden!
K. Reuter and M. Scheffler, Phys. Rev. Lett. 90, 046103 (2003); Phys. Rev. B 68, 045407 (2003)
Constrained equilibrium
X
μO (T, p)
2
μCO(T, p)
G(T, p) ≈ Etot
RuO2(110) + x O + y CO ↔RuO2·(O)x·(CO)y
1
G = [
G
−G RuO  110−x  O− y CO]]
A [ RuO ·O · CO
ad
2
x
1
 G ≈ [E RuO ·O
A
y
ad
2
x
2

1 DFT 1 ZPE
E O  E O   O
·CO  −E RuO 110 −x
2
2
y
2
2
2
DFT
ZPE
− y  ECO
ECO
  CO  ]

Obr / ­
 Gad ≈
1
E RuO ·O
A
[
2
x
·CO  −E RuO 110 −x
y
2

1 DFT 1 ZPE
E O  E O   O
2
2
2
2
DFT
ZPE
− y  ECO
ECO
  CO 
Obr / COcus
CObr / COcus
∆Gad (meV/Å2)
Obr / Ocus
]

Obr / ­
Obr / COcus
ΔμCO (eV)
Obr / Ocus
CObr / COcus
ΔμO (eV)
(ab initio) kMC for surface catalysis: implementation
Escape prob:
(ab initio) kMC for surface catalysis: implementation
Adsorption
Local sticking coefficient
Area unit cell
Rate of adsorption of species i on given site type st area of site type
internal DoF
gas particle
(ab initio) kMC for surface catalysis: implementation
Desorption
Detailed balance:
Diffusion
(ab initio) kMC for surface catalysis: showcase
Model lattice
One CO+O → CO2 reaction (both cus)
(ab initio) kMC for surface catalysis: fitting from DFT
From ab initio calculations:
Reactions:
Adsorption/desorption:
Diffusion:
(ab initio) kMC for surface catalysis: fitting from DFT
(ab initio) kMC for surface catalysis: results
No reaction: Same as thermodynamics, but with time evolution information: From fully O­covered surface (red), induction time is 0.1 s (!)
(ab initio) kMC for surface catalysis: global vs local
?
13%
86%
1%
Bridge (gray): only 6% CO
Cus: 73% CO
(ab initio) kMC for surface catalysis: global vs local
(ab initio) kMC for surface catalysis: global vs local
0.5%
99.5%
Bridge (gray): 0% CO
(ab initio) kMC for surface catalysis: global vs local
Surface catalysis: What about gas supply and heat transfer?
Surface catalysis: What about gas supply and heat transfer?
Stagnation flow geometry
Combination of 1p­kMC and computational fluid dynamics for TOF
Boundary conditions:
­ species conversion rates (DFT)
­ chemical reactions heat release (DFT)
­ heat flux into the solid, limiting cases,
a) isothermal (perfect heat conductor)
b) adiabatic (perfect heat insulator)
Surface catalysis: What about gas supply and heat transfer?
Adiabatic limit
Adiabatic limit
O poisoned
CO poisoned
Surface catalysis: What about gas supply and heat transfer?
At high intrinsic TOF, mass transports reduces actual TOF
kMC for surface catalysis: sensitivity to accuracy of calculations?
Degree of rate control:
(DRC)
kMC for surface catalysis: sensitivity to accuracy of calculations?
The myth of the Rate limiting step
kMC for surface catalysis: The myth of the Rate limiting step
kMC for surface catalysis: apparent (global) activation energy
??? 2.85 eV
Assuming:
weak dependence on β
kMC for surface catalysis: apparent (global) activation energy
??? 2.85 eV
x2
kMC for surface catalysis: adjusting an empirical approach
kMC for surface catalysis: self consistent accuracy control?