Overview. kmcand DFT. An example to start with. An example to start with 03/09/2014

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1 An Introduction of the Use of Monte Carlo Simulations in Catalysis. Tonek Jansen verview Kinetics and kinetic Monte Carlo The Master Equation Getting the Rate Constants Algorithms Examples Bangalore, 5 September 014 kmcand DFT densityfunctional theory adsorption sites other quantum chemical methods molecular mechanics experiments V ( R) reaction mechanisms rate constants molecular dynamics thermodynamics Kinetics and kinetic Monte Carlo fantasy/imagination/insight kinetics An example to start with C+ adlayer Pt(110) substrate An example to start with adlayer substrate 0.4 µm 0.4 µm time: length: seconds meter C 1x 1x1 Salazar et al., PRE, 69 (004) C 1x 1x1 Salazar et al., PRE, 69 (004)

2 Typical kmc simulations Length and time scales grid size: 18x18 104x104 system size: 5 nm 0. µm second coarsegrained kmc rate equations real time simulated: sec processor time used: 1 sec 1 hour number of reactions: number of reaction types: 5 memory used: 8 18 Mb quantum chemistry kmc MD meter Chemical kinetics an example: [ C] d C(gas) C(ads) (gas) (ads) C(ads) + (ads) C(gas) [ ] [ C][ ] = k ads * k form hat s wrong with... macroscopic rate equations, mass balance equations, reactiondiffusion equations, These equations implicitly assume that the reactants are homogeneously/randomly mixed; i.e., there is no correlation between the position of the reactants. surface science: d C = kads* k form C coverages Shortrange order The rate of bimolecular reactions depends on the number of pairs of adsorbates that react. The number of pairs are assumed to be equal to a product like C. hen there is shortrange order this is not correct. Shortrange order arises from diffusion limitations, lateral interactions, and defects. ith lateral interactions it is not even possible to talk about one rate constant for a reaction; i.e., k form C is not even correct if there is no shortrange order. Phenomenology an example: d C(gas) C(ads) (gas) (ads) C(ads) + (ads) C(gas) k C = ads* k form C Rate equations can be used to fit results of kinetic experiments. This is often successful. The drawbacks are that this only works for a limited range of reaction conditions, and the rate constants have no physical meaning. ne may also wonder why these particular equations are used.

3 The (111) surface of an fccmetal bridge/fold sites fcc/fold top/1fold sites sites hcp/fold sites The Master Equation Key concept Atoms and molecules adsorb at welldefined sites. These sites form a regular grid or lattice. This grid of sites with a description of what is adsorbed at each site is called a latticegas model. Each grid or lattice point has a label that specifies the occupation of the corresponding site and possibly other properties of the site. Phase space N particles: phase space: q, q,, q 1 K N coordinates p,, p, 1 K p N conjugate momenta ( q q, K, q, p, p, K, p ) ( q, p) 1, N 1 N = p q Evolution of a system in phase space p Time scales vibrations: subpicoseconds reactions: seconds q

4 Evolution of a system in phase space Key concept p q If you only specify the sites and their occupation, then the times when reactions take place can not be determined exactly. Instead we get probability distributions for the reaction times. If more reactions are possible, then we also can not say which reaction will take place, but can only give probabilities. Reactions in phase space dp dqdp ρ ( t) = p h t = R dqdp P ( t) = ρ p h ( q, t) D, D R dqdp h D i= 1 R ( q, t) D, ρ H ρ H pi qi qi pi The master equation dp ( TS ) kbt Q = ( IS ) h Q = [ P P ] ( IS ) dp Q = dq exp D h R B ( ) dp dp dp + dp TS 1K j 1 j 1K D Q = ds exp D 1 h S B T H k T H k The master equation dp = [ P P ] configuration probability time rate constant Key concept The evolution of a latticegas model of a surface and its adsorbates can be described by the Master Equation dp = [ P P ] 4

5 The role of the master equation ab initio kinetics (quantum chemistry) dp = [ P P ] Getting the Rate Constants numerical analytical analytical kinetic Monte Carlo rate equations Calculating rate constants dp = [ P P ] ( TS ) k = BT Q E ( ) bar exp IS h Q kbt Differences between transition probabilities and rate constants are a matter of definition. e do not distinguish between them. Rate constants from experiments Property X has value X in configuration. Statistical average: Rate equation: X d X = P X = dp X = [ P P ] X = P [ X X ] = P [ X X ] change in X average over initial configurations rate constant of change Key concept The evolution of a property can be described by the macroscopic or phenomenological equation d X = P [ X X ] Simple desorption e assume that we have just one type of adsorbate that desorbs one atom/molecule at a time, and suppose that a surface area with S sites has N adsorbates. Summation over : d N = P [ N N ] = 0, des, if is a desorption otherwise If is a desorption then N N = 1 There are N desorptions in = #nonvanishing terms. d N = des P N = des N 5

6 Simple desorption d N d = des N divide by S: = des macroscopic rate equation: d = des = k des k des The macroscopic rate equation is exact. determined from experiments coverage Bimolecular reaction e assume that we have two types of adsorbate that react with each other; A + B K ( A) d N ( A) ( A) = P [ N N ] Summation over : = 0, rx, if is a reaction otherwise ( A ) ( A ) If is a reaction then N N = 1 There are ( AB) N #AB pairs reactions in = #nonvanishing terms. ( A) d N ( AB) ( AB) = rx P N = rx N Bimolecular reaction ( A) d N ( AB) ( AB) = rx P N = rx N Two approximations need to be introduced. The adsorbatesare randomly distributed. Fluctuations can be neglected. macroscopic rate equation: rx = k rx Z d A d B = = krx A B determined from experiments da d B = = Zrx A B The macroscopic rate equations are not exact. Algorithms Kinetic Monte Carlo KMC generates an instantiation of the probabilities of the master equation. This means that it generates a configuration at time t with probability P (t), which is a solution of the master equation. t = 0 : 1 0 t 1 t L t n n Variable Step Size Method (VSSM) Initialize 1 Reaction time t = ln r, t t + t Reaction p( ) = Continuation stop, 6

7 First Reaction Method (FRM) Initialize Generate configuration, set time t, make a list of all reactions, set stop conditions, determine: 1 t = t ln r Reaction Take the first reaction to occur. Performance FRM VSSM ZGB TPD w. repulsion C+H/PtRu 60 RSM 800 FRM +RSM VSSM +RSM #sites sulphate/cu(111) C+/Pt(100) N/Rh(111) 8 7 Continuation stop NH /Pt(111) 1 ) modified model ) approximate algorithm in 10 processes per CPU second A short history of Carlos Pizzazz: a predecessor of Carlos (1994). generalpurpose code using userspecified labels firstreaction method to solve the Master Equation including linear timedependence of the temperature Carlos ( ) first version (1996): improved version of Pizzazz version.0 (1998): three algorithms, improved userfriendliness, graphical output version 4.0 (004): lateral interactions, more general timedependent rate constants version 5.0 (009): more flexible user interface Kinetix (010 ) commercial version; part of Accelrys s Materials Studio Examples Reduction of NH on Pt(111) Reduction of NH on Pt(111) NH (gas) NH (ads) NH (ads) NH (ads) + H(ads) NH (ads) NH(ads) + H(ads) NH(ads) N(ads) + H(ads) N(ads) N (gas) H(ads) H (gas) 0 58 NH (ads) 9 NH (ads) NH(ads) N(ads) 7 H (gas) 9 6 N (gas) ffermans et al., Surf. Sci., 600, 1714 (006) 7

8 kmc snapshot Site blocking N NH vacant NH Coverages Coverages Coverages (T = 1000 K, P = 1.18 atm): kmc microkinetics Boltzmann NH NH NH N H /T Sulphate on Cu(111) with steps Sulphate on Cu(111) S + 4 +** S4 e adsorption sites blocked sites 8

9 Sulphate on Cu(111) with steps Sulphate on Cu(111) with steps current current potential (V) potential (V) Sulphate on Cu(111) with steps Sulphate on Cu(111) with steps This voltammogram is called the butterfly. Surface with defects! C.G.M. Hermse et al., Surf. Sci., 57, 4760 (004). Sulphate on Cu(111) with steps Bitartrate on Cu(110) current Steps suppress the phase transition. C HCH HCH C potential (V) Lorenzo, Baddeley, Muryn, Raval, Nature,, 76 (000) 9

10 Nonlinear response of LEED STM of mixtures 50:50 60:40 Haq, Liu, Humblot, Jansen, Raval, Nature Chemistry, 1, 409 (009) kmc simulations Domain formation of bisuccinate 50:50 60:40 time Chiral induction by bitartrate TPD of N from Ru(0001) Timedependent temperature and defects! 10

11 TPD of N from Ru(0001) TPD of N from Ru(0001) increased step density Temperature (K) Temperature (K) step terrace TPD of N from Ru(0001) Tmax (K) Step density increased step density C electrooxidation on PtRu H(ads) + H H (l) + * H(ads) + H + + e H (l) + * C(ads) + H(ads) C (gas) + * + H * H H * C + H * + + e + + e Initially the electrode is at low potential and almost completely covered by C. In the experiment the potential is increased and the current is measured. All reactions take place on Pt and Ru, but with different rate constants. C electrooxidation on PtRu The rate constants depend on the electrode potential. M rx transfer coefficient M ads M des = k = k M ads M des Me0E exp kbt exp ( 1 ) k 1,M M1,M M1 = krx exp k e T M B,M B electrode potential T e E E 0 0 C electrooxidation on PtRu more reactive n Pt the C formation is fast, but the water adsorption is slow. n Ru the C formation is slow, but the water adsorption is fast. If the C formation is fast at the PtRu boundary we get the observed synergic effect. 11

12 C electrooxidation on PtRu TPD of C/Rh(100) C H Pt Ru C prefers top sites. There is a strong NN repulsion. Ru Pt 0.50 ML C Rh M.T.M. Koper et al., J. Phys. Chem. B, 10, (1999). TPD of C/Rh(100) TPD of C/Rh(100) Some C molecules move to bridge sites to reduce repulsion. Some C molecules move to bridge sites to reduce repulsion ML 0.8 ML TPD of C/Rh(100) TPD of C/Rh(100) desorption prefactor adsorption energy top sites adsorption bridge sites NN toptop interaction NNN topbridge interaction geminal bridgebridge interaction vicinal bridgebridge interaction desorption prefactor /s adsorption energy top sites 106 kj/mol adsorption bridge sites 94 kj/mol NN toptop interaction 14. kj/mol NNN topbridge interaction.6 kj/mol e used Differential Evolution to get these parameters by fitting kmc results to experimental TPD spectra. geminal bridgebridge interaction 1.8 kj/mol vicinal bridgebridge interaction 0.6 kj/mol 1

13 TPD of C/Rh(100) TPD of C/Rh(100) C on bridge site C on top site M.M.M. Jansen et al., PCCP, 1, (010). For more information see Thank you for your attention 1