is similar to the size of the nanoparticles. At a certain distance from the macroscopic front, the
ultrathin ﬁlm starts to evolve a locally isotropic pattern of holes. The holes themselves grow in an
unstable manner resulting in an array of isotropically branched structures as shown, e.g., above in
Fig. 1. This indicates that at least some of the patterns described in the literature may have arisen
from processes in similar ultrathin ‘postcursor’ ﬁlms.
The existence of the ultrathin ‘postcursor’ ﬁlm is an experimental ﬁnding that can be drawn on
when choosing a theoretical approach to account for the pattern formation (see below). Note how-
ever, that at the moment there exists no explanation for its existence. A possible hypothesis is
that the substrate strongly attracts the nanoparticles. As a result they form a dense suspension
layer having a thickness roughly equal to the diameter of the nanoparticles. The observed meso-
scopic dewetting front then actually correspond to an autophobic dewetting of a low concentration
suspension from the higher concentration suspension on the surface of the substrate.
III. MODELLING APPROACHES
Models of dewetting thin ﬁlms of pure liquids or polymers are often based on thin ﬁlm hydro-
dynamics. Starting from the Stokes equations, together with continuity and boundary conditions
at the substrate and free surface, one applies a long-wave approximation (assuming small surface
slopes and contact angles) [8, 63] and obtains a non-linear evolution equation for the ﬁlm thickness
proﬁle h(x,y,t ). In the case of volatile liquids one ﬁnds [55–58, 64]
∂th = ∇·
[
Qc∇δF
δh
]
−Qe
δF
δh, (1)
with the mobility functions Qc(h) = h3/3η ≥0 (assuming Poiseuille ﬂow in the ﬁlm and no slip
at the substrate; η is the dynamic viscosity) and Qe ≥0 for the convective and evaporative part
of the dynamics, respectively. Qe is a rate constant that can be obtained from gas kinetic theory
or from experiment [57]. Note that Eq. (1) only applies if the pressure in the vapour above the
ﬁlm is close to the saturation pressure. For alternative expressions that are used to describe the
non-conserved evaporative dynamics see, e.g., Refs. [56, 57, 65–69]. Finally, ∇= (∂x,∂y), and
∂t, ∂x and ∂y denote partial derivatives w.r.t. time and the coordinates.
Focusing on the inﬂuence of capillarity and wettability only, the energy functional F[h] is given
by
F[h] =
∫
dx
∫
dy
[γ
2 (∇h)2 + f(h) −µh
]
(2)
7