  In many areas of research it is interesting how lattices can be filled with
particles that have no nearest neighbors, or they are in limited quantities.
Examples may be found in statistical physics, chemistry, materials science,
discrete mathematics, etc. Using Monte Carlo (MC) simulation we study the
kinetics of filling of square lattice (2D). Two complementary rules are used to
fill the lattice. We study their influence on the kinetics of the process as
well as on the properties of the obtained systems. According to the first rule
the occupied sites may not share edges (nearest neighbors occupations are not
permitted). Under this condition, the maximum possible concentration is 0.5,
forming a checkerboard type structure. However, we found that if the deposition
is done by random selection of sites the concentration of 0.5 is inaccessible
and the maximum concentration is Cmax(2D)=0.3638 (0.0003) for 2D lattice. If
the lattice is 3D we find that the maximal concentration is even lower
Cmax(3D)=0.326 (0.001). The second rule establishes permission to break the
first one with certain probability 0<=p<=1, thus the occupied sites can start
to share edges when p>0. In this case higher then 0.3638 concentrations are
accessible, even up to C=1. Therefore the percolation threshold Pc can be
reached. Its value depends on the value of the probability p. Our model
describes the kinetics of formation of thin films of particles attracted by the
substrate but repulsing each other.
