Multi-Lattice Approach to Kinetic Monte Carlo...Multi-lattice kinetic Monte Carlo actually works...

Fakultät für Chemie Technische Universität München

Multi-Lattice Approachto Kinetic Monte Carlo

Max J. Hoffmann

Fritz-Haber Institut der Max-Planck-Gesellschaft

TU München

Jan 25, 2011

Outline

introduction to DFT+kMC approach for heterogeneous catalysis

multi-lattice kMC for CO oxidation on PdO(√5×√5)R27/Pd(100)1

1master thesis www.fhi-berlin.mpg.de/th/publications/Hoffmann_diploma.pdf2

Motivation

prediction of (model) catalysts performance from first principles

development of multi-scale methods

Motivation

CO+ 12O2 → CO2@RuO2(110)

2Reuter et al (2005) in Handbook of Materials Modelling, Springerhttp://www.fhi-berlin.mpg.de/th/publications/handbook_reprint.pdf

CO+ 12O2 → CO2@RuO2(110)

What is kMC?

kinetic Monte Carlo ( = dynamic Monte Carlo (DMC),Bortz-Kalos-Lebowitz (BKL-Method)3, n-fold way, Gillespie4) all referto the same basic idea

generate state-to-state trajectory from initial configuration, catalog ofelementary steps, and rate constants.

pi =∑

i kijpj − kjipi

kij : transition probability i→ j

Transition State Theory

kij =kBT

he−∆G/kBT

3BKL (1975), J Comput Phys 17, 104Gillespie (1976), J Comput Phys 22, 403

What is kMC?

kinetic Monte Carlo ( = dynamic Monte Carlo (DMC),Bortz-Kalos-Lebowitz (BKL-Method)3, n-fold way, Gillespie4) all referto the same basic ideagenerate state-to-state trajectory from initial configuration, catalog ofelementary steps, and rate constants.

pi =∑

i kijpj − kjipi

kij : transition probability i→ j

kij =kBT

he−∆G/kBT

3BKL (1975), J Comput Phys 17, 104Gillespie (1976), J Comput Phys 22, 403

More formally5

pick next process based on random number,weighted by rate constant

increase time by

t→ t− ln(u2)

ktotu2 ∈ (0, 1]

5Fichthorn, Weinberg, (1991), J Chem Phys 95, 10906

Consider PdO/Pd(100)

non-trivial surface reconstruction under reactive conditions6

6Rogal et al (2007), PRL 98, 0461017

non-trivial surface reconstruction under reactive conditions 6

6Lundgren (2006), J Phys Cond Matter 30, R4817

non-trivial but commensurable 6

6Todorova et al (2003), Surf Sci 541, 1017

Considerations for Multi-Lattice kMC

lattices need to be pairwisecommensurable (!)

integer coordinates desirable

support periodic boundaryconditions

lattices need to be pairwisecommensurable

↪→mapping approach

only some sites are active

describe processes in convenient coordinates

Reconstruction mechanism

geometrical considerations

surface slab geometryoptimizations

DFT, VASP, PW-91, J. Jelic

Reconstruction mechanism

geometrical considerations surface slab geometryoptimizations

DFT, VASP, PW-91, J. Jelic

Remove one oxygen atom

↪→no significant change10

Remove two oxygen atoms

structural changes occur

Surface Pd: Oxygen:

Pd(100) hollow: Pd(100) bridge:

Palladium atoms move tosubstrate hollow positions

Reconstruction rule

map reconstruction back to discrete positions

Refined oxide stability boundary7

7Rogal et al (2008), PRB 77, 15541013

Present statususes only guessed rates constants for new processes notconsidered in the preceding 1p-kMC from Rogal et al.considers only the onset of oxide destruction

7Rogal et al (2008), PRB 77, 15541013

Summary

DFT+kMC important tool to predict model catalyst behaviorMulti-lattice kinetic Monte Carlo actually worksPalladium surface oxide under reactive conditions

Outlookvalidation of assumed deconstruction mechanism throughfirst-principles calculationsdetailed modelling beyond the deconstruction onsetultimately: fully first-principles based kMC simulation ofsustained catalytic activity with full capability of oxideformation/reduction

Thank you for your attention

Technische Universität München

Summary

Technische Universität München

Summary

Technische Universität München14

flux-over-population

kTST =´ drdp p

m δ(r−rc)Θ(p)e−βH´ drdpe−βH

harmonic approximation

kTST = (βh)−1 e−β∆G

KMOS GUI8

8http://mhoffman.github.com/kmos16

kMC modularization

divide program parts based on degree of reusability

Oxide patching rule

reversing the initial oxide destruction

Multi-Lattice Approach to Kinetic Monte Carlo...Multi-lattice kinetic Monte Carlo actually works...

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