The Process Syntax

In KMC language a process is uniquely defined by a configuration before the process is executed, a configuration after the process is executed, and a rate constant. Here, this model is used to define a process by giving it a:

  • condition_list

  • action_list

  • rate_constant

As you might guess, each condition corresponds to one before, and each action corresponds to one after. In fact, conditions and actions are actually of the same class or data type: each condition and action consists of a coordinate and a species which has to be or will be at the coordinate. This model of process definition also means that each process in one unit cell has to be defined explicitly. Typically, on a single crystal surface one will have not only one diffusion per species, but as many as there are equivalent directions:

  • species_diffusion_right

  • species_diffusion_up

  • species_diffusion_left

  • species_diffusion_down

While it may seem like a lot of work to define that many processes, it allows for a very clean and straightforward definition of a process itself. Later you can use geometric measures to abstract these cases as you will see further down.

Adsorption

Let’s start with a very simple and basic process: molecular adsorption of a gas phase species, let’s call it A, on a surface site. For this we need a species

from kmos3.types import *
kmc_model = kmos3.create_kmc_model()

A = Species(name='A')
kmc_model.add_species(A)

empty = Species(name='empty')
kmc_model.add_species(empty)

and the coordinates of a surface site

layer = Layer(name='default')
kmc_model.add_layer(layer)
layer.sites.append(Site(name='a'))
coord = kmc_model.lattice.generate_coord('a.(0,0,0).default')

which is for now all we need to define an adsorption process

adsorption = Process(
    name='adsorption_A_a',
    condition_list=[Condition(coord=coord, species='empty')],
    action_list=[Action(coord=coord, species='A')]
    )
kmc_model.add_process(adsorption)

Now this wasn’t hard, was it?

Diffusion

Let’s move to another example, namely the diffusion of a particle to the next unit cell in the y-direction. Initially, you need the coordinate of the final site

final = kmc_model.lattice.generate_coord('a.(0,1,0).default')

and you are good to go

diffusion_up = Process(
    name='diffusion_A_up',
    condition_list=[Condition(coord=coord, species='A'),
                    Condition(coord=final, species='empty')],
    condition_list=[Condition(coord=coord, species='empty'),
                    Condition(coord=final, species='A')],
    )
kmc_model.add_process(diffusion_up)

You can complicate this ad infinitum but you already know all the elements necessary to define processes.

Avoid Double Counting

Finally a word of warning: double counting is a phenomenon that can arise for certain processes where there are more than one equivalent directions and coordinates. Consider, for example, the dissociative oxygen adsorption. Novices typically collect all possible directions (e.g., right, up, left, down) and then define this process for each direction. However, this approach may be incorrect depending on how the rate constant value was defined. If the rate constant represents a net rate for adsorption/desorption from a pair of sites (from an average or from a single transition state that bridges two sites), then the right, up, left, down procedures will result in double counting. This is because, for example, adsorption_up is the same process as adsorption_down, just executed from one site above or below. To address this issue, one can compensate by dividing each adsorption and desorption rate constant by 2. Alternatively, one can avoid double counting by only defining geometrically equivalent sites once per unit cell: a simple trick to achieve this is to only consider processes in the positive directions, for example.

However, it’s crucial to recognize that in cases where there are two transition state possibilities (one above each site) due to a heterolytic cleavage transition state, it’s essential to acknowledge and include both processes. These processes should not be dismissed as double counting because they represent distinct pathways with unique transition states sharing the same reactants and products.

Most rate constants in the literature for dissociative adsorption are defined as an average for the two sites, or involve a single homolytic transition state, and thus most cases of dissociative adsorption should have a single process between up and down as well as between left and right (yielding two processes, not four, for a simple cubic system).

Taking It Home

  • A process consists of conditions, actions and a rate constant

  • double counting is best avoided from the beginning