on the model (see above). The purely two-dimensional character of the KMC was extended to
a ‘pseudo three-dimensional’ one by making the effective chemical potential dependent on the
mean liquid coverage [38]. As the latter is related to a mean ﬁlm thickness, this corresponds to
the introduction of a ‘global’ thickness-dependent disjoining pressure into the evaporation term
without an explicit consideration of a ﬁlm thickness. The amended model can reproduce bimodal
structures that are beyond the scope of the purely two-dimensional model [38, 39]. Fully three-
dimensional models are also discussed in the literature [76, 77].
B. Dynamical Density Functional theory
The limitations of the kinetic Monte Carlo model introduced in the previous Section are related
to its character as a two-dimensional lattice gas with only three states: gas, liquid or particle.
This implies that (i) no liquid can be transported to a site on the surface already ﬁlled with liquid,
i.e., diffusion of the liquid can not be incorporated in a sensible way and (ii) one is not able to
distinguish between the inﬂuence of the short- and the long-range parts of the interactions with the
substrate, as all such interactions are absorbed into the effective chemical potential.
However, using dynamical density functional theory (DDFT) [78–83] one can develop a model
for the processes in the ultrathin postcursor ﬁlm without these limitations, although here we limit
ourselves to developing the theory at the level of the KMC and solely discuss how to extend it to
incorporate the inﬂuence of the liquid diffusion over the surface. Such a DDFT model describes
the coupled dynamics of the density ﬁelds of the liquid ρl and the nanoparticles ρn. The densities
ρl and ρn are deﬁned as the probabilities of ﬁnding a given lattice site on the surface to be occupied
by a ﬁlm of liquid or by a nanoparticle, respectively. Note that the probability densities correspond
to number densities as we use the lattice spacing σ= 1 as our unit of length.
To develop the DDFT, one must ﬁrst derive the underlying free energy functional F[ρl,ρn], and
secondly, devise dynamical equations for both density ﬁelds that account for the conserved and the
non-conserved aspects of their dynamics, i.e., transport and phase change processes, respectively.
For a system governed by the hamiltonian (3), we may construct a mean-ﬁeld (Bragg-Williams)
approximation for the free energy of the system [78, 84] which contains an entropic contribution
and contributions from the interactions between the different species (nanoparticles and liquid).
The free energy is a semi-grand free energy, since the liquid is treated grand canonically (it is
coupled to a reservoir with chemical potential µ), whereas the nanoparticles are treated in the
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