polymers which only result in ﬁngers without side-branches [75] or ﬁelds of droplets left behind
[18].
A quantitative analysis shows that the mean number of ﬁngers depends only very weakly on the av-
erage concentration of the nanoparticles ρav
n ; only the mean ﬁnger width increases with increasing
concentration. However, decreasing the mobility (i.e., decreasing the diffusivity of the particles)
leads to a much denser ﬁnger pattern and also causes the front instability to appear at an earlier
stage, i.e., when the front instability is in its initial linear regime, it has a higher growth rate and a
smaller characteristic wavelength (cf. Fig. 2(c) and (d)). Decreasing the effective chemical poten-
tial (increasing its absolute value) has a similar but less strong effect. For details see [41]. These
ﬁndings lead to the conclusion that the determining factor for the front instability is the ratio of
the time-scales of the different transport processes. In particular, the front becomes more unstable
when the velocity of the dewetting front increases as compared to the mean diffusion velocity of
the nanoparticles.
If the particle diffusivity is low, the front ‘collects’ the particles, resulting in a build up of the
particles at the front that itself is slowed down. This makes the front unstable and any ﬂuctuation
along the front will trigger a transverse instability that results in an evolving ﬁngering pattern. This
happens even when the particle-liquid and particle-particle attractive interactions do not favour
clustering (i.e. demixing of the liquid and the nanoparticles). In this regime, the instability is a
purely dynamic effect and energetics plays no role in determining the number of ﬁngers. We call
this the ‘transport regime’.
To illustrate the inﬂuence of energetics (characterized by the interaction parametersεij) on ﬁnger-
ing in Fig. 3 we display the dependence of the mean ﬁnger number on particle-liquid interaction
strength εnl. For εnl ≥1.5 the mean ﬁnger number < f >is nearly constant; this is the trans-
port regime. However, on decreasing εnl below 1.5, we observe a marked increase in the value
of < f >, indicating that energy plays an important role in determining the number of ﬁngers in
this regime. In this parameter range, demixing of particles and liquid occurs at the moving front
and increases its transverse instability. In this ‘demixing regime’, the wavelength of the ﬁngering
instability is determined by the dynamics and the energetics of the system. Decreasing εnl further
(below 1.4 in Fig. 3) one ﬁrst observes in regime (iii) a slight decrease in the average ﬁnger num-
ber. This is a geometric effect resulting from our one-dimensional ﬁnger counting routine: The
ﬁngers increasingly break up and the dried-in pattern looks progressively isotropic. In regime (iv),
the measure ⟨f⟩does not represent a ﬁnger number but instead indicates a decrease in the typical
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