Chemical reactions as network of rare events: Kinetic MonteCarlo

Transcription

1 Chemical reactions as network of rare events: Kinetic MonteCarlo

2 Extending the scale Length (m) Potential Energy Surface: {Ri} 10 6 (3N+1) dimensional 10 9 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 {Ri} 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).

3 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

4 Reminder: Nudged Elastic Band Harmonic Transition State Theory: Elastic band: normalized local tangent at i Climbing image:

5 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:

6 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)

7 Kinetic MonteCarlo: theory Many independent (loss of memory assumption)poisson processes: : number of overall events in the ensemble after time t Invoking independence Overall Poisson process:

8 Simple adsorption/desorption (Langmuir) fractional surface coverage desorption rate adsorption rate Initial condition: Solution:

9 (ab initio) kmc for surface catalysis: implementation

10 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 (DFT PBE+vdW) Spin forbidden! K. Reuter and M. Scheffler, Phys. Rev. Lett. 90, (2003); Phys. Rev. B 68, (2003)

11 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 DFT ZPE y ECO ECO CO ]

12 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 DFT ZPE y ECO ECO CO Obr / COcus CObr / COcus Gad (mev/å2) Obr / Ocus ]

13 Obr / Obr / COcus ΔμCO (ev) Obr / Ocus CObr / COcus ΔμO (ev)

14 (ab initio) kmc for surface catalysis: implementation Escape prob:

15 (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

16 (ab initio) kmc for surface catalysis: implementation Desorption Detailed balance: Diffusion

17 (ab initio) kmc for surface catalysis: showcase Model lattice One CO+O CO2 reaction (both cus)

18 (ab initio) kmc for surface catalysis: fitting from DFT From ab initio calculations: Reactions: Adsorption/desorption: Diffusion:

19 (ab initio) kmc for surface catalysis: fitting from DFT

20 (ab initio) kmc for surface catalysis: results No reaction: Same as thermodynamics, but with time evolution information: From fully O covered surface (red), induction time is 0.1 s (!)

21 (ab initio) kmc for surface catalysis: global vs local? 13% 86% 1% Bridge (gray): only 6% CO Cus: 73% CO

22 (ab initio) kmc for surface catalysis: global vs local

23 (ab initio) kmc for surface catalysis: global vs local 0.5% 99.5% Bridge (gray): 0% CO

24 (ab initio) kmc for surface catalysis: global vs local

25 Surface catalysis: What about gas supply and heat transfer?

26 Surface catalysis: What about gas supply and heat transfer? Stagnation flow geometry Combination of 1p kmc 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)

27 Surface catalysis: What about gas supply and heat transfer? Adiabatic limit Adiabatic limit O poisoned CO poisoned

28 Surface catalysis: What about gas supply and heat transfer? At high intrinsic TOF, mass transports reduces actual TOF

29 kmc for surface catalysis: sensitivity to accuracy of calculations? Degree of rate control: (DRC)

30 kmc for surface catalysis: sensitivity to accuracy of calculations? The myth of the Rate limiting step

31 kmc for surface catalysis: The myth of the Rate limiting step

32 kmc for surface catalysis: apparent (global) activation energy??? 2.85 ev Assuming: weak dependence on β

33 kmc for surface catalysis: apparent (global) activation energy??? 2.85 ev x2

34 kmc for surface catalysis: adjusting an empirical approach

35 kmc for surface catalysis: self consistent accuracy control?