What's KMC All About and Why Bother

Traditional kinetic models

Langmuir-Hinshelwood-type models can describe the kinetics of this reaction mechanism by invoking a number of assumptions:

1. Each site can be occupied by at most one adsorbate. This is sometimes referred to as the "exclusion principle". It is usually true, although there are some notable exceptions e.g. two CO molecules can bind to a cobalt site. One can circumvent this problem by defining a new species e.g. C₂O₂*.
2. All catalytic sites are equivalent. This means that CO*, for instance, has the same "stability" (quantified by the binding energy) at any site on the surface. Consequently the probability of finding any site occupied by CO* is equal to the coverage of CO*. This assumption is not true in general: catalytic nanoparticles expose a variety of sites with different coordination numbers, onto which adsorbates bind with different strengths. Still though, one can formulate a more complicated Langmuir-Hinshelwood model taking into account the various binding states for the same species.
3. Adsorption/desorption steps are quasi-equilibrated. This assumption is typically employed in Langmuir-Hinshelwood models and makes things easy by providing analytical expressions of the coverage of each surface reactant with respect to the gas phase composition, pressure and temperature. Of course one can easily formulate more complicated models in which the coverages are variables of a differential equation system to be solved for.
4. No spatial correlations are observed between adsorbates. To visualize this, imagine a pair of sites A and B and let us only consider CO adsorption. The assumption can be restated as: the probability of finding a CO* on site B, does not depend on the occupancy of site A. Thus, the probability of finding a pair of CO* adsorbates next to each other is θ_CO·θ_CO = θ_CO². This is a rather limiting assumption: imagine what would happen is CO adsorbates exerted repulsive interactions between each other. Then the real probability of finding a pair of COs could be much lower than θ_CO². Likewise, imagine what would happen (in the case of pure O₂ adsorption) if O* diffusion was not very fast: since O₂ adsorbs dissociatively, one would find pairs of sites occupied by O* more frequently than θ_O².
5. Kinetic constants are independent of coverage. Thus, a CO oxidation elementary event (CO* + O* → CO₂) is assumed to have the same kinetic rate constant, k_oxi, irrespective of whether the two reactants are in low versus high coverages. The reaction rate will be given by k_oxi·θ_CO·θ_O, where the product of the two coverages is essentially the probability of finding a reactive CO*-O* pair and k_oxi is indeed constant. However, it is known that rate "constants" are coverage-dependent: the activation energy of the CO oxidation elementary event drops at high coverages, and the reaction tends to proceed faster. This makes sense intuitively: when the catalytic surface is cramped with molecules, the repulsions between the reacting CO* and O* and the (several) CO* and/or O* spectators destabilize the reactant configuration, thereby lowering the reaction barrier.

We therefore see already that some of the assumptions just noted pose limitations on the validity of the Langmuir-Hinshelwood model. Under these assumptions, the CO oxidation rate expression is:

Rate_LH = k_oxi·θ_CO·θ_O

where

θ_CO = K_CO·P_CO/(1+K_CO·P_CO+√(K_O2·P_O))

θ_O = √(K_O2·P_O)/(1+K_CO·P_CO+√(K_O2·P_O))

K_CO = k_ads,CO/k_des,CO

K_O2 = k_ads,O2/k_des,O2