Kinetic Monte Carlo Simulation of Molecular Processes on Supported Metal Particles

Transcription

1 INFAP CONICET UNSL Kinetic Monte Carlo Simulation of Molecular Processes on Supported Metal Particles Giorgio Zgrablich Universidad Nacional de San Luis San Luis, Argentina

2 CONTENTS Motivation Non-equilibrium Statistical Mechanics Dynamic Monte Carlo Simulation Applications to Molecular Processes on Surfaces Conclusions

3 The turnover rate of a catalytic reaction often depends on the particle size. The kinetics is influenced by several factors like: geometric shape, exposed crystal planes, metal-support interactions and electronic structure.

4 EFFECT OF TEMPERATURE ON A SUPPORTED PARTICLE Initial (a) and equilibrium (b-d) shapes of a supported metal particle simulated at different temperatures: (b) 500 K, (c) 900 K, (d) 1100 K. Size of the initial particle is atoms, J mm =-10 kj/mol, J ms =-2.5 kj/mol.

5 Non-equilibrium Statistical Mechanics A rigorous basis to describe non-equilibrium statistical mechanics is provided by the theory of Stochastic Processes Molecular processes on solid surfaces usually belong to the kind of stochastic processes known as Markov Processes, i.e. processes where each transition to a new state does not depend on the history of the system The most complete information on the system is given by the probability of finding the system in a state s at time t, P s (t) In a Markov Process the time evolution of this probability is determined by the transition probability per unit time w rs through the Master Equation: t P () t = [ P() t w P() t w ] s r rs s sr r

6 Difficulties to solve the Master Equation For a surface with M adsorption sites, and in the simple case of a single adsorbed species, A, a state (configuration) of the system is represented by an M-components vector s = {s 1,s 2,,s M }, where each s i can take the values 0, 1. There are 2 M states. Therefore the Master Equation is actually a system of 2 M coupled equations. Exact solutions have been obtained only for simple reaction-diffusion processes, like the reaction A(ads) + A(ads) A 2 (gas) on one-dimensional lattices.

7 Mean field approximations A mean field approximation consists in assuming that particles are uniformly distributed in space, in that case the Master Equation can be transformed into a differential equation for the particle density, ρ( rt, ). In the simple example of the reaction-diffusion process A(ads) + A(ads) A 2 (gas) the mean field kinetic equation would be ρ( rt, ) t which can be solved easily. 2 2 = λρ ( rt, ) + ρ( rt, ) Mean field approximations may lead to incorrect predictions of the kinetics of the process (even qualitatively), when heterogeneities and fluctuations are relevant in the system.

8 Examples of relevant heterogeneities and fluctuations NO+CO/Rh(111) Adsorbed N atoms form compact islands: Zaera et al., J. Chem. Phys. 1999, 111, 8088; J. Phys. Chem. B 2001, 105, 7771 NO dissociation is inhibited by the presence of nearest-neighbors coadsorbed NO (case of the pink NOs): Niemantsverdriet et al., J. Chem. Phys. 1994, 101, 10052

9 Dynamic Monte Carlo Simulation Dynamic Monte Carlo simulation is a numerical replica of the Markov process described by the Master Equation Starting from an arbitrary state, subsequent states are generated according to the transition probabilities per unit time (rates) w = ν exp( E / k T) rs rs rs B The total transition rate for the system is R= wrs Since the time of occurrence of a transition is exponentially distributed, the real time in the process increases at each Monte Carlo step by the quantity lnξ Δ t = R where ξ is a random number uniformly distributed in (0,1) rs,

10 Application to Molecular Processes on Surfaces

11 NO-CO Catalytic Reaction on Pd Nanoclusters V. Bustos, R.O. Uñac, G. Zgrablich, C.R, Henry, PCCP 5 (2003) 2906

12 Step Edge Inhibition Effect If any of the reacting N atoms is adsorbed at a step edge site, then the reaction ocurrs with probability I < 1 I = inhibition factor is varied between 0 and 1 Other parameters: P = dissociation probability for NO; D CO and D NO, desorption probabilities for CO and NO, respectively.

13 Nanoclusters Characteristics L = side lenght in lattice units; d = side lenght in nm; r = fraction of (100) facet sites (the less active for NO dissociation).

14 Experimental Results NO reaction probability 0,6 0,5 0,4 0,3 0,2 0,1 0,0 % (d=15.6 nm) % (d=6.9 nm) % (d=2.8 nm) T( C)

15 NO reactivity versus particle size NO RP High T Low T d

16 High T Simulation Results Low T

17 Discussion Two competing factors: a) faster NO dissociation on (111) facet; b) at particle periphery lower number of NN sites, NO dissociation and reactions are depressed. Middle size particles optimize the balance between proportion of (111) facet and size of the periphery, then it is the more active. When NO desorption is high (high T) then the relative importance of a) decreases and the largest proportion of peripheral sites prevails overturning the eefficiencies of the smallest and largest particles.

18 Conclusions Simulation results show that intermediate size particles are the most active over all temperature range, while the smallest size particles are the less active at high temperature and the largest size particles are the less active at low temperature, providing simple explanations to this behavior, which is coincident with experimental observations.

19 CO + O 2 CO 2 (gas) + O(ads) on Temperature Deformed Supported particles Partícula Metálica Original Partícula Metálica a 500 ºK Partícula Metálica a 900 ºK C1 C1 C1 Simulation system: support = sites; initial metal particle = metal atoms; interaction energies, J mm = -10 kj/mol, J ms = -2.5 kj/mol.

20 Simulation Model

21 Effect of Deformation by Temperature 1,0 0,8 0,6 0,4 0,2 0,5 0,0 0,4 0,3 A ads B ads 0,2 0,1 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,0 0,8 0,6 0,4 0,2 0,5 0,0 0,4 0,3 0,2 0,1 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 undeformed AB a) b) 1,0 0,8 0,6 0,4 0,2 0,2 0,0 0,1 A ads Bads 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,0 0,8 0,6 0,4 0,2 0,2 0,0 0,1 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,0 0,8 0,6 0,4 0,2 0,0 0,2 AB a) b) c) no dif CO dif CO, M dif deformed at 500 K 0,1 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

22 Effect of CO-CO Interactions, T=500K Left, no CO difusion; right CO difusion on M and on S. a) 0, b) >0, c) < A ads B ads AB a) A ads Bads AB a) R R AB AB b) c) b) c)

23 Effect of O-O Interactions, T=500K Left, no CO difusion; right CO difusion on M and on S. a) 0, b) >0, c) < A ads Bads AB a) 1,0 0,8 0,6 0,4 0,2 0,0 0,2 0,1 A ads Bads AB a) b) c) 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,0 0,8 0,6 0,4 0,2 0,0 0,2 0,1 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,0 0,8 0,6 0,4 0,2 0,0 0,2 0,1 b) c) ,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

24 CONCLUSIONS Deformation of the particle by temperature affects considerably both the reaction windows and the overall reaction rate, depending on CO surface diffusion and adsorbate-adsorbate interactions. Deformation of the particle by temperature reduces the area of flat terraces where O 2 dissociates, thus reducing reactivity. CO diffusion enhances the reaction window and increases reactivity. CO-CO repulsive interactions increase reactivity. Repulsive O-O interactions shifts the reaction window toward the region of smaller CO molar fraction in the gas phase.