Kinetic Monte Carlo Simulation

DFT and other first-principles methods are usually used to deal with individual chemical re- actions occurring locally on certain reaction sites. However, real surface (electro-)chemical reactions, which are usually composed of several elementary steps, are much more complex

because reactions at different surface sites are strongly affect each other[122, 63], so the reaction kinetics are difficult to obtain by only applying first-principles calculations. This problem can be solved with the help of kinetic Monte Carlo (kMC) simulation, also called dynamics Monte Carlo simulation, which is a general method for numerically simulating the stochastic time evolution of many coupled events[36, 32, 121, 71].

A typical kMC algorithm for simulating the time evolution of a system, where each process i occurs with known rates ri, can be written as follows:

0. Set the time t = 0.

1. Form a list of rifor all possible event i in the system.

2. Calculate the cumulative function Ri=∑ij=1rj for i=1,. . .,N, where N is the total number of possible events, and denote R=RN

3. Get a uniform random number u′_∈ (0,1]

4. Locate the event k, for which Rk−1<u′<Rk. 5. Carry out event k.

6. Due to the above transition, total number of possible events N and all rates ri may have changed. So we need to recalculate the total number of events and their new rates to update N and the list of riaccordingly.

7. Get a new uniform random number u_∈ (0,1].

8. Update the time with t = t +∆t, where∆t is calculated by

∆t=₋log u

R . (2.22)

9. Return to step 2.

For a system of surface reactions, each individual event is a possible elementary reac- tion step at one surface site and its rican be calculated from DFT methods. Thus, we can build a multiscale model, which combines DFT results and statistic methods such as kMC

together, to study the thermodynamics, dynamics and kinetics of certain (electro-)catalytic surface reactions.

Chapter 3

Oxygen Reduction at Vacuum

Conditions–the Origin of Catalytic

Activity

The oxygen reduction by hydrogen molecule at ultra-high-vacuum (UHV) conditions, as shown in Eq. 1.3, is a famous example of catalytic reaction. As early as the 1840s, it was already found that gaseous H2 and O2, which are almost impossible to react in homoge-

neous gas phases at room temperature, react smoothly to produce water on the surface of Pt crystals at room temperature and even below[29, 9]. Being a well-defined and charac- terized “model” system, this reaction is of fundamental importance to catalysis and elec- trocatalysis. In this chapter, (111) surfaces of FCC-lattice metals are used as the model of catalyst surface to study oxygen adsorption, dissociation and reduction at UHV conditions, and several types of metals (Pt, Pt alloys, Cu and Au) are applied. The results will help us understand the origin of catalytic activity for oxygen reduction under both UHV and electrochemical conditions.

3.1

Oxygen Reduction on Pt (111) Surface

Many theoretical calculations have been performed in the framework of DFT and other first-principles methods, not only for the whole oxygen reduction as Eq. 1.3[85], but also its elementary steps such as O2adsorption and dissociation [24, 25, 55]. While these calcu-

lations have contributed a great deal to our understanding, a complete reconciliation with experiments is not yet achieved. In particular, the predicted O2dissociation rate appears to

be too slow compared to experiments, as well as the rate of H2O formation above the water

molecule desorption temperature Tdes, which is ∼170 K on Pt(111) surface under UHV

conditions.

V¨olkening et al. observed oxygen reduction reaction by hydrogen molecule in progress with scanning tunneling microscope (STM) and high-resolution electron energy loss spec-

troscopy (EELS) [143]. Two distinct temperature regimes were seen. Below T_des, the

adsorbed water has an autocatalytic effect and the reduction occurs via disproportionation reactions. Above Tdes, when water does not stay on the surface, it was postulated that water

is formed by successive additions of adsorbed H (H∗) atoms to adsorbed O (O∗) atom:

H∗+O∗_→OH∗ (3.1)

H∗+OH∗_→H2O∗ (3.2)

Michaelides and Hu studied this reaction pathway by DFT, starting from chemisorbed O∗

and H∗ atoms[85]. They found that reaction (3.1) is a highly activated process, with an activation barrier of 0.94 eV. The barrier of reaction (3.2), on the other hand, is very small, only 0.21 eV.

However, this scenario does not seem to be congruent with the experimental obser- vation [95] that Pt is still a potent catalyst above T_des. Taking T =200K [95] and trial

frequencyν =1012/s, a barrier of 0.94 eV would correspond to O∗half-life of_∼104years. So either the DFT barrier for reaction (3.1) is off, or there may be alternative reaction pathways. Recent DFT calculations for oxygen reduction reaction (ORR) under aqueous electrochemical conditions have indicated the possibility of O∗₂ reacting directly with hy- dronium H3O+(H2O)2to form OOH∗intermediates [120, 147, 148]. Experimentally, there

is also some hint for this possibility [142, 10]. It is therefore profitable for us to look at this hydroperoxyl-mediated pathway using DFT under non-aqueous condition.

H∗ and O∗ arise from the adsorption and dissociation of H2 and O2. The case of H2

is simple and has been well-studied [99]: H2 can dissociate without barrier upon contact

with Pt (111) surface. The situation with O2 is more complex. A common conclusion

of several UHV experiments [37, 73, 100, 130, 91] is that the dissociation of O2 on Pt

(111) is a thermally activated process via molecular precursor states (MPS), such as O−∗₂ (superoxo, paramagnetic) or O2₂−∗(peroxo, nonmagnetic). Using DFT, Eichler and Hafner identified two energetically nearly degenerate precursors (O−∗₂ at bridge site and O2₂−∗ at fcc site), in excellent agreement with experiments. However, they also found that the O∗₂ dissociation barriers were 0.8_∼0.9 eV[24, 25], which is definitely contradictory to the low experimental O2 dissociation temperature estimated to be∼150 K[37]. Nolan et al. used

EELS and molecular beam techniques to examine high translational energy adsorption of oxygen, and estimated the dissociation barrier to be 0.29 eV[91]. ˇSljivanˇcanin and Hammer recalculated the O∗₂dissociation barriers on flat Pt (111) surface and found the lowest barrier to be 0.6 eV[145], which is a good improvement. Recently, Hyman and Medlin obtained

the O2 dissociation barrier to be 0.44 eV from a cluster calculation[55]. However, the

disagreement of slab calculation with experiments in O∗₂dissociation still exists and needs further study.

In this section, we use DFT to study the whole reaction process of water formation, starting from the adsorption of O2and H2 on Pt (111). We resolve the two contradictions

described above by searching for different reaction paths, and checking the dependence of the energetics on adsorbate coverage. The latter could be fulfilled by changing the unit cell used in the calculation. In Sec. 3.1.1, we give details of the calculation method. In

Sec. 3.1.2, we discuss the problem of O2 adsorption and dissociation. We find that if

the unit cell is large enough ((2√3_×4) in our case), the O∗₂ dissociation barrier becomes reasonable (_∼0.3 eV). This indicates strong dependence of the O∗₂ dissociation barrier on oxygen coverage. In Sec. 3.1.3, we first verify the calculation by Michaelides and Hu for the (3.1)+(3.2) reaction by confirming that reaction barrier in Eq. 3.1 is as high as

∼1 eV, then we provide a new pathway starting from the direct protonation of O∗₂ via the

Langmuir-Hinshelwood mechanism to form OOH∗(hydroperoxyl), which only involves an

energy barrier of_∼0.4 eV. This path may be important when PO2 and PH2 are both high,

and its low activation barrier satisfies with the experimental observation of Pt as effective catalyst.