The kinetics of nucleation and growth of thin films is investigated using phenomenological rate equations modified from the equations by Venables (Ref. 6). These consist of coupled first-order differential equations for single adatom density, stable-cluster density, and the fractional coverage of a layer. The rate of evolution of single adatom density on a substrate is governed by the flux of adatoms, the rates of desorption and adsorption of a single adatom, and incorporation of adatoms with other adatoms or with a stable cluster. This approach serves as a basis for a rate-equation model to describe multilayer growth. The effect of impurities is also discussed.
Nieminen, J. A., & Kaski, K. (1989). Rate-equation study of nucleation and growth of thin films. I. Growth of one monolayer. Physical Review A, 40(4), 2088-2095. https://doi.org/10.1103/PhysRevA.40.2088