  We present a simple non-equilibrium model of mass condensation with
Lennard-Jones interactions between particles and the substrate. We show that
when some number of particles is deposited onto the surface and the system is
left to equilibrate, particles condense into an island if the density of
particles becomes higher than some critical density. We illustrate this with
numerically obtained phase diagrams for three-dimensional systems. We also
solve a two-dimensional counterpart of this model analytically and show that
not only the phase diagram but also the shape of the cross-sections of
three-dimensional condensates qualitatively matches the two-dimensional
predictions. Lastly, we show that when particles are being deposited with a
constant rate, the system has two phases: a single condensate for low
deposition rates, and multiple condensates for fast deposition. The behaviour
of our model is thus similar to that of thin film growth processes, and in
particular to Stranski-Krastanov growth.
