substrate and force equilibria at the free surface, and applies a long-wave approximation. Under
the assumption that concentrations equilibrate rapidly over the ﬁlm thickness, we obtain coupled
non-linear evolution equations for the ﬁlm thickness proﬁleh(x,t) and the amount of nanoparticles
per unit length hp = φh, where φis the volume concentration of the nanoparticles. Note, that hp
corresponds to the local thickness of the nanoparticle layer when all the solvent is evaporated. The
resulting evolution equation for the ﬁlm thickness is Eq. (1) above and focusing on the inﬂuence
of particle-independent capillarity and wettability only, the energy functional F[h] is given by
Eq. (2) above. Note that the viscosity η depends on the particle concentration. Following Refs.
[88, 89, 91, 92] we use the Quemada law for dense suspensions [93–95]
η(φ) = η0
(
1 −φ
φc
)−2
(8)
where φc = 0.64 corresponds to random close packing of spherical particles. For the nanoparticle
volume per length hp = φhone obtains the following evolution equation:
∂t(φh) = ∇·
[
φQc∇δF
δh
]
+ ∇·[D(φ)h∇φ] , (9)
where the particle concentration dependent diffusion coefﬁcientD(φ) is related to the viscosity by
the Einstein relation D(φ) = kT/6πRη(φ), where Ris the radius of the nanoparticles [96].
We illustrate results obtained employing this thin ﬁlm theory using the single example of a re-
ceding dewetting front for a partially wetting ﬁlm. We use the disjoining pressure and material
constants for the liquid considered in Ref. [57], where the evaporative and convective dewetting
of a ﬁlm of volatile liquid is studied. We add, however, the nanoparticles to the system. The
expression that we employ for the local free energy term in Eq. (2) is:
f(h) = SLW d2
0
h2 + SP exp
(d0 −h
l0
)
, (10)
where the parameters characterising the interaction between the liquid ﬁlm and the surface are
the apolar and polar spreading coefﬁcients SLW and SP , respectively, the Debye length l0 and the
Born repulsion length d0 [57]. The resulting disjoining pressure Π = −∂hf(h) allows for a stable
precursor ﬁlm (thickness hprecursor) and also has a second (larger) thickness (h0) that corresponds
to a secondary minimum of the underlying energy functional. See Refs. [11, 97] for studies of
ﬁlm and drop states for similar disjoining pressures. Our results are calculated for a system where
the proﬁles only vary in one Cartesian direction ( x), corresponding to a straight dewetting front.
However, our results may also be interpreted as applying to a circular ﬂat drop whose front remains
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