3 FIG. 5: (Color online) Nanoparticle density profiles from the evolu- tion of an unstable dewetting front, for the time t/t$_{l}$ = 20000 , (a) for M c l = 0 and (b) M c l = 5 , for α = 0 . 5 M nc l σ $^{4}$, µ/k$_{B}$T = − 3 . 28 and domain size is 800 σ × 800 σ ; remaining parameters and initial density profiles are as in Fig. 4. In (c) we show the dependence of the mean finger number 〈 f 〉 on the mobility coefficient for liquid diffusion, M c l . FIG. 3: (Color online) Density profiles for evaporative dewetting in the nucleation regime. (a)-(c) are the nanoparticle density profiles at times t/t$_{l}$ = 20 , 80 and 4000 , (d) and (e) are the liquid profiles for t/t$_{l}$ = 20 and 80 , for M c l = 2 and µ/k$_{B}$T = − 3 . 33 ; remaining parameters are as in Fig. 2. In (f) we plot the average density of the liquid on the substrate 〈 ρ$_{l}$ 〉 , as a function of time for this case and the case M c l = 0 . The system was initialised with the (discretised) density profiles: ρ$_{l}$ ( x, y, t = 0) = 0 . 9 + 0 . 05 χ , ρ$_{n}$ ( x, y, t = 0) = 0 . 3 + 0 . 27 χ , where χ is a random number uniformly distributed on the interval [ − 1 , 1] . It is due to this random noise that the holes are nucleated in some places and not in others. liquid density profile: FIG. 4: (Color online) Density profiles from the evolution of an unstable dewetting front. (a)–(c) are the nanoparticle density pro- files at times t/t$_{l}$ = 2000 , 20000 and 40000 , (d) and (e) are the liquid profiles for t/t$_{l}$ = 2000 and 20000 , for α = 0 . 2 M nc l σ $^{4}$, µ/k$_{B}$T = − 3 . 28 and domain size 800 σ × 800 σ ; remaining param- eters are as in Fig. 2. The initial density profiles are the same as those in Fig. 3, except for y < 0 we set ρ$_{l}$ = ρ$_{n}$ = 10 − $^{10}$, to create an ini- tially straight dewetting front at y = 0 . In (f) we plot the mean finger number 〈 f 〉 as a function of the mobility coefficient for nanoparticle diffusion α , for the case when M c l = 0 . For the liquid, the density may change either by evapora- tion/condensation from/to the substrate (non-conserved dy- namics) or may diffuse over the substrate (conserved dynam- ics). The latter dynamics is treated in a manner analogous to that for the nanoparticles. For the non-conserved dynam- ics, we assume a standard form [40], i.e., that the change of the density over time is proportional to − ( µ$_{surf}$ − µ ) = − δF [ ρ$_{n}$, ρ$_{l}$ ] /δρ$_{l}$ , where µ$_{surf}$ ( r , t ) is the local chemical po- tential of the liquid on the substrate. Combining these two contributions, we obtain the time evolution equation of the We assume that the two mobility coefficients M c l and M nc l are constants. In what follows we set k$_{B}$T = 1 and M nc l = 1 . Note that in the low density limit, when ρ$_{n}$ → 0 and ρ$_{l}$ → 0 , the conserved part in both Eq. (2) and Eq. (3) corresponds to Fickian diffusion. Before discussing results from our theory, we first make a cou- ple of comments about the status of the theory. Firstly, we note that Eq. (1) constitutes a simple ‘zeroth-order’ mean-field ap- proximation for the free energy of the system and omits (for example) terms such as ln(1 − ρ$_{n}$ − ρ$_{l}$ ) which describe the excluded area correlations between the liquid and the nano- particles. Second, due to the fact that we derive the theory from the (already) coarse-grained lattice Hamiltonian rather than by integrating over degrees of freedom (coarse-graining) in the full DDFT theory for the three-dimensional liquid film, the theory can not be regarded as a ‘fully’ microscopic theory. The reason that we have modelled the system using this sim- ple theory is because our interest is the basic question of what physics drives the behaviour displayed in the experiments. In this work we choose to start from the lattice theory, in order to compare with the KMC. However, as future work we plan to go beyond the lattice theory. Our goal here is not to construct a model describing every detail of these systems; instead we seek to examine what physics is involved in determining the observed pattern formation. Nonetheless, this DDFT does al- low us to investigate the time evolution of the postcursor film of an evaporating nanoparticle suspension under fewer restric- tions than the KMC model. To discuss the importance of liq- uid transport within the layer we compare results obtained for different liquid mobilities ( M c l > 0 ) and results without liquid transport (i.e., setting M c l = 0 ). We focus on three examples: (i) spinodal dewetting (see Fig. 2), (ii) dewetting via nucle- ation of holes in an initially flat film (see Fig. 3), and (iii) the unstable receding of an evaporative dewetting front, which ex- hibits branched fingering (see Figs. 4 and 5). Fig. 2 shows snapshots from a purely evaporative spinodal dewetting process. Panel (c) gives the evolution of mean liq- uid density with time, 〈 ρ$_{l}$ 〉 ≡ A − 1 ∫ d r ρ$_{l}$ ( r , t ) , where A is the area of the substrate, and panel (f) gives the nanoparticle Preprint – a.j.archer@lboro.ac.uk – u.thiele@lboro.ac.uk – March 27, 2019 ∂ρ$_{l}$ ∂t = ∇ · [ M c l ρ$_{l}$ ∇ δF [ ρ$_{n}$, ρ$_{l}$ ] δρ$_{l}$ ] − M nc l δF [ ρ$_{n}$, ρ$_{l}$ ] δρ$_{l}$ . (3)