small holes. The competition for space results in a ﬁne-meshed polygonal network of nanoparticle
deposits. The concentration of particles is much higher at the network nodes – an effect that can
not been seen within the KMC model. As the particles attract the liquid there remains some liquid
on the substrate where the nanoparticles are.
Fig. 5 gives snapshots of the evolution of a ﬁngering instability for a retracting dewetting front.
At early times the straight front shows a rather short-wave instability, about 16 wiggles can be
seen. However, they are only a transient: the ﬁnger pattern coarsens rapidly till only about 7
ﬁngers remain. The ﬁngering then becomes stationary, i.e., just as in the KMC, the mean ﬁnger
number remains constant, although new branches are continuously created and old branches join
each other. In general, the results on ﬁngering agree well with results obtained using the KMC
model [41]. From this we conclude that jamming of discrete particles is not a necessary factor
for causing the instability, since the ﬁngering is seen here in a continuum model with a diffusion
constant that is independent of the nanoparticle concentration. The DDFT is better suited than the
KMC for investigations of the early instability stages: they are more easy to discern without the
discrete background noise of the KMC. Furthermore, one may perform a linear stability analysis of
the one-dimensional undisturbed streamwise front proﬁles with respect to transverse perturbations
(in analogy to the approach used in Refs. [19, 86, 87]).
C. Thin ﬁlm hydrodynamics
The previous two sections focused on two approaches to describe the experimentally observed
patterning dynamics in the ultrathin postcursor ﬁlm left behind by a mesoscopic receding dewet-
ting front. Although both the kinetic Monte Carlo model and the dynamical density functional
theory are able to describe well the processes in the ultrathin ﬁlm, they can not be employed to
describe mesoscale hydrodynamics. A relatively simple model for the latter can be derived in the
framework of a long-wave or lubrication equation [8, 63]. We will illustrate here the approach
by considering an isothermal situation where the nanoparticles are not surface active, i.e., they do
not act as surfactants. For a model incorporating the effects of latent heat generation and surface-
active particles resulting in thermal and solutal Marangoni stresses, see Ref. [88]. A description of
spreading particle solutions incorporating a structural disjoining pressure has also been considered
[89]. For related work on particle-laden ﬁlm ﬂow on an incline see Refs. [90, 91].
One starts from the Stokes equations, together with continuity, no-slip boundary conditions at the
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