structure factor S ( k ) ≡ 〈 ρ$_{n}$ ( k ) $^{2}$〉 , where ρ$_{n}$ ( k ) is the Fourier transform of ρ$_{n}$ ( r ) . For small times, the unstable film de- velops a typical spinodal labyrinthine pattern with a typical wavelength 2 π/k$_{max}$ (note that the symmetry breaking in the density profiles in all our calculations is due to the addition of a small amplitude random noise to the density profiles at time t = 0 and no noise is added at later times). The nanoparticles concentrate where the remaining liquid is situated. However, they are ‘slow’ in their reaction: when ρ$_{l}$ already takes val- ues in the range 0.08 to 0.83, the nanoparticle concentration has only deviated by about 25% from its mean value. The film thins rapidly forming many small holes. The competi- tion for space results in a fine-meshed network of nanoparti- cle deposits with a much higher concentration of particles at the network nodes – an effect that can not be seen within the KMC model. Because the liquid wets the nanoparticles, some liquid always remains on the substrate. Accounting for sol- vent diffusion, the rate of the dewetting process is increased [see Fig. 2(c)] and leads to a more strongly modulated final pattern – i.e. the peaks in S ( k ) are higher for M c l > 0 than for M c l = 0 [2(f)]. 

Fig. 3 shows snapshots from a dewetting process triggered by nucleation events. The holes nucleate at several arbitrary places due to the random initial noise in the density profiles, and grow to form a random polygonal network of rims of highly concentrated solution. On a very long time scale the network coarsens into an array of drops. The influence of liq- uid transport can be seen in the final panel of Fig. 3, where we display a plot of the average density of the liquid on the substrate as a function of (log) time. We see that the liquid is able to evaporate from the substrate faster and that for times t/t$_{l}$ ∼ 10 $^{2}$− 10 $^{4}$, the total amount of liquid on the substrate is less when M c l = 2 , than when M c l = 0 . However, over very long times, the drops on the substrate slowly move and ‘eat- up’ the network pattern. This process is faster with solvent diffusion, leading to a faster increase of 〈 ρ$_{l}$ 〉 at long times. Since the liquid wets the nanoparticles, some liquid also re- condenses back onto the substrate. 

The final example in Figs. 4 and 5 is the evolution of the fin- gering instability for a receding dewetting front. The fingering instability is caused by a build up of the nanoparticles at the receding front, which collects the nanoparticles due to their attraction to the liquid. In Fig. 4(a) we see that at early times the initially straight front shows a rather short-wave instabil- ity; about 20 short ‘fingers’ can be seen. However, the fin- ger pattern coarsens rapidly to a stationary pattern containing only about half the initial number of fingers. Intriguingly, the mean finger number remains constant although at the mov- ing contact line new branches are created and old branches merge continuously. The occurrence of this phenomenon in 

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the present continuum model (DDFT) is similar to results of the KMC [32], and proves that jamming of discrete particles is not a necessary mechanism for causing the instability. In Fig. 4(f) we show how the average number of fingers per unit length, 〈 f 〉 , varies as a function of α , the mobility coefficient of the nanoparticles on the wet substrate, for the case when M c l = 0 . We see that as α is decreased, the number of fin- gers increases. This increase in 〈 f 〉 occurs because when the mobility of the nanoparticles is decreased the front ‘collects’ more particles (less of them diffuse further from the front). The resulting region of high concentration solution at the front may be ‘dynamically unstable’: As the front velocity depends non-linearly on the amount of particles collected, any fluctua- tion along the front may trigger a transverse instability. 

Fig. 5 shows that the finger number 〈 f 〉 depends non- monotonically on the mobility coefficient for liquid diffusion, M c l . Although the overall trend is an increase of 〈 f 〉 with in- creasing M c l , there exists an intermediate region ( 5 ≲ M c l ≲ 7 ) where 〈 f 〉 slightly decreases. The overall trend results from an increase in front velocity (due to the increase in M c l ) at fixed particle diffusivity. However, we currently have no ex- planation for the intermediate slight decrease. 

Note also that in all cases the instability may be strongly am- plified when the particle interactions favour the clustering of the nanoparticles (this is not the case for the parameters used here), where the higher concentration at the receding front leads to a local demixing of nanoparticles and liquid, that it- self enforces the deposition of a highly branched finger pat- tern. In this ‘demixing regime’ the instability is determined by the dynamics and the energetics of the system whereas for the case studied in Fig. 4 it mainly depends on the dynamics. 

We note finally, that the fingering process can be seen as a self-optimisation of the front motion, so that the average front velocity is kept constant by expelling particles into the fin- gers. A similar effect exists for dewetting polymer films [22], where surplus liquid is expelled from the growing moving rim which collects the dewetted polymer. However, front instabil- ities found for dewetting polymers only result in fingers with- out side-branches [41] or fields of droplets left behind [22]. 

In this Letter we have developed a versatile DDFT that is ca- pable of describing the pattern formation observed in evapo- rating dewetting thin films of suspensions. Since our DDFT takes account of all the basic solvent and solute transport and phase change processes in a consistent manner, we believe that it will be the basis for many successful future studies of the behaviour of suspensions and solutions at interfaces. 

AJA and MJR gratefully acknowledge financial support by RCUK and EPSRC, respectively. We acknowledge support by the EU via grant PITN-GA-2008-214919 (MULTIFLOW). 

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