The diffusion, "explosion" and "evaporation" of dimers and the subsequent coalescence are treated in a formal way by identifying and solving the differential equations deduced from the respective behaviors of dimers in the different cases. This study leads to analytic formulas allowing to calculate, in a simple and fast way, the size statistics obtained after the coalescence of the dimers or their constituents once the dimers have completely disappeared. These formulas are of capital interest to characterize systems in which the dimers initially present disappear.
Study assumptions.Six assumptions will be made for this study: 1. Initial probabilities are known. G.S. -28-4-2021 -02:33 5 DIMER DIFFUSION 3 2. Only dimers disappear.3. The diffusion of dimers or constituents can be decomposed in two of tree steps according the scenario:(a) If the dimer diffuses: i. A dimer is removed from the set of dimers. ii. The taken dimer is then placed randomly at the surface (i.e. on a nucleation site) (b) If the dimer explode: i. A dimer is removed from the set of dimers ii. The first monomer is then placed randomly at the surface on a site holding a cluster. iii. The second single atom is then also placed randomly on a site holding a cluster.(c) If the dimer evaporates: i. A dimer is removed from the set of dimers and a monomer is added to the set of monomers. ii. A single atom is placed on a site on the surface.4. The number N of nucleation sites is very large : 1/N ≪ 1 5. The mean free path of a diffusing atom is large in comparison of the distance of nucleation centers.6. Finally, the size of a cluster is negligible in regard of the distance of nucleation center. This means that the capture probability for an atom does not depend of the size of the cluster already present on the nucleation center, and that condition 5 will be fulfilled.
Dimer DiffusionWe will suppose here that the dimers leave their position and that, without breaking up, they diffuse until they reach a nucleation site (already occupied or not).Let us consider a starting situation in which we have the probability P 0 of having an empty site, the probability P 1 of having a site with a monomer, P 2 of having a site with a dimer... and P n of having a site with a cluster of n atoms. A priori, P 1 should be equal to 0, but, as said above, we are going to keep it, it won't make the calculations more complicated, and it could allow us to "solve" the improbable but not impossible problem where monomers, because of their enhanced interaction with the substrate, could be more efficiently trapped on the surface defects than the dimers.