time scales for evaporation and diffusion. A large mobilityM indicates fast diffusion as compared
to evaporation. A trial move is accepted with the probabilitypacc = min[1,exp(−∆E/kT)] where
kis the Boltzmann constant, T the temperature and ∆Eis the change in energy resulting from the
potential move. Note that particles are only allowed to move into wet areas of the substrate, i.e.,
onto cells with l= 1. This models zero diffusivity of the particles on a dry substrate. The replaced
liquid ﬁlls the site left by the nanoparticle.
Without nanoparticles, the behaviour of the model is well known as it reduces to the classical
two-dimensional Ising model [74]. For kT < kTc ≈0.567 liquid and vapour coexist when µ =
µcoex = −2. For µ >−2 [µ <−2] eventually the liquid [vapour] dominates. A straight liquid-
gas interface will recede [advance] for µ <−2 [µ >−2], i.e. one ﬁnds evaporative dewetting
[wetting] fronts. If one starts, however, with a substrate covered homogeneously by the liquid,
for µ <−2 the ﬁlm will dewet via a nucleation or spinodal-like process. If the nanoparticles are
present, they form dried-in structures when all the liquid evaporates. The ﬁnal structures do not
normally change any further – at least on short time scales. However, if the liquid wets the particles
(i.e. is attracted to the particles), over long times there might be a coarsening of the structures,
facilitated by the adsorbed liquid. The dried-in patterns depend on the particular pathway taken by
the evaporative dewetting process. They range from labyrinthine to polygonal network structures
or holes in a dense particle layer. Some typical patterns are displayed in Fig. 2, for cases when
the average surface coverage of the nanoparticles ρav
n = 0 .2. Panels (a) and (b) result from a
spinodal-like and nucleation and growth process, respectively. At ﬁrst sight they look very similar
to the patterns seen for the pure solvent and one might argue that the particles solely act as passive
tracers and preserve the transient volatile dewetting structures of the solvent. This was suggested
in Refs. [26–28] for dewetting collagen solutions. However, panels (c) and (d) indicate that the
particles may at times play a rather more signiﬁcant role. When the diffusion of the particles is
slow, the evaporative dewetting fronts become transversely unstable and may result in strongly
ramiﬁed patterns. This instability is caused by the nanoparticles. The lower their mobility, the
stronger the ﬁngering effect, i.e., there are more ﬁngers in (c) than in (d) because in the latter the
mobility is larger.
The front instability is intriguing as it results in strongly branched structures. As the dewetting
front moves, new branches are continuously created and existing branches merge at the moving
contact line. However, the mean ﬁnger number in the streamwise direction of the resulting ramiﬁed
pattern is a constant. This behaviour is in contrast to the front instabilities found for dewetting
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