dewetted liquid. The front recedes until all liquid is collected in a central drop. Since no liquid
evaporates [Qnc = 0 in Eq. (1)], the particle concentration does not change during the process.
The situation changes when allowing for evaporation ( Qnc > 0). Now the front may retract
by convection and/or evaporation. Evaporation leads to the possibility of a strong increase in
the particle concentration at the contact line as evaporation is strongest there. Due to the strong
nonlinear dependence of the viscosity on the particle concentration, this may lead to a dramatic
decrease of the convective contribution to the front velocity. For moderate evaporation rates, this
may result in a (temporary) self-pinning of the front. Within the present basic model, the process
can (after complete dry-in) result in three different basic deposition patterns: (i) for very fast
evaporation rates, all other processes occur over time scales that are much larger. In particular, the
effects of convective redistribution of the liquid are neglectable. As a result one ﬁnds that a nearly
homogeneous ﬁlm of nanoparticles of thicknesshp = φ0h0 is deposited (see Fig. 6(a)). Convection
only results in the small heap of material visible at the left hand side of Fig. 6(a). The decrease
in hp on the right side of Fig. 6(a) arises due to the diffusion of particles to the right of the initial
front position; (ii) for very low evaporation rates, the ﬁlm dynamics is dominated by convective
dewetting as this process acts on a much shorter time scale than evaporation. As a result, all the
liquid is collected into a drop before evaporation slowly removes the remaining solvent. Under
these conditions most of the nanoparticles are deposited in a single heap (see Fig. 6(c)). Depending
on the diffusivity, the heap might be highest at the centre or show a depression there; (iii) at
intermediate evaporation rates, one may observe the deposition of a nanoparticle ring around a
region with a nanoparticle ﬁlm of much lower height. At the centre deposition might increase
again (see Fig. 6(b)).
The most intriguing feature is the ring formation that has been observed experimentally for sus-
pensions of very different particle sizes ranging from nanometers [32, 36, 46, 47] to hundreds of
micrometers. Pinning of the contact line and thermal Marangoni effects are often mentioned as
necessary conditions for the ring formation. The contact line pinning is often assumed to result
from substrate heterogeneities. Film height and concentration proﬁles at various instants during
the dewetting process are displayed in Fig. 7. The proﬁles are from before, at and after self-pinning
of the contact line. In Fig. 8 we display a space-time plot for the complete process. At ﬁrst, the
front recedes in the same manner as when there is no evaporation, but now driven by convection
and evaporation. A small capillary rim forms that collects all the dewetted liquid that does not
evaporate. The particle concentration slowly increases at the contact line (Fig. 7(a) and regime
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