Connecting Individual Rates to Overall Dynamics

You have now learned about how to use DFT calculations to compute the rates of individual activated processes. This information is extremely useful, but it is still not enough to fully describe many interesting physical problems. In many situations, a system will evolve over time via many individual hops between local minima. For example, creation of catalytic clusters of metal atoms on metal oxide surfaces involves the hopping of multiple individual metal atoms on a surface. These clusters often nucleate at defects on the oxide surface, a process that is the net outcome from both hopping of atoms on the defect-free areas of the surface and in the neighborhood of defects. A characteristic of this problem is that it is the long time behavior of atoms as they move on a complicated energy surface defined by many different local minima that is of interest. 

There are a host of physical questions that cannot be easily answered just by knowing the rates listed in Fig. 6.13. For example, once an Ag atom is deposited on the surface, how long will it be (on average) before that Ag atom visits a site adjacent to a Pd surface atom How many different Pd surface sites will an Ag atom visit per unit time on the surface What is the net diffusion coefficient of Ag atoms on this surface To answer these questions, we need a tool to describe the evolution in time of a set of Ag atoms on the surface. 

The dynamics of systems such as this one can be followed using a method called kinetic Monte Carlo (kMC) The idea behind this method is straightforward If we know the rates for all processes that can occur given the current configuration of our atoms, we can choose an event in a random way that is consistent with these rates. By repeating this process, the system s time evolution can be simulated. 

Choose a hopping direction (up, down, left, or right) at random. Using our catalog of rates, look up the rate associated with this hop, khop. 

If s khopAmax, move the selected Ag atom in the selected direction. Otherwise, do not move the atom. 

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