2 FIG. 1: (Color online) (a) Phase diagram of the pure liquid in the plane spanned by the density ρ$_{l}$ and temperature T . (b) Liquid and nanoparticle densities ρ$_{l}$ and ρ$_{n}$ at coexistence as a function of the average nanoparticle density ¯ ρ$_{n}$ , for ε$_{ll}$/k$_{B}$T = 1 . 25 , ε$_{nl}$/k$_{B}$T = 0 . 6 , ε$_{nn}$ = 0 . FIG. 2: (Color online) Results for spinodal evaporative dewetting of a nanoparticle suspension. (a) and (b) are typical nanoparticle density profiles, for times t/t$_{l}$ = 6 and 20 , and (d) and (e) are the corresponding liquid density profiles, for M c l = 0 . The domain size is 200 σ × 200 σ . (c) and (f) give the corresponding time evolution of the mean liquid density 〈 ρ$_{l}$ 〉 and the structure factor 〈 ρ$_{n}$ ( k ) $^{2}$〉 for various values of M c l . The remaining parameters are ε$_{ll}$/k$_{B}$T = 1 . 25 , ε$_{nl}$/k$_{B}$T = 0 . 6 , ε$_{nn}$ = 0 , α = 0 . 4 M nc l σ $^{4}$, µ/k$_{B}$T = − 3 . 4 , and 〈 ρ$_{n}$ 〉 = 0 . 3 . dynamics of the solute diffusion and the solvent evaporation [30–32]. The neglect of convective solvent transport can be justified based on estimates comparing its influence with the one of evaporative solvent transport [32]. However, so far the 2d KMC models have not incorporated diffusive solvent trans- port that might be important in the postcursor layer. In this Letter we present an alternative description for the structure formation in the postcursor film that does not have the limitations of the previous approaches. We develop a 2d dynamical density functional theory (DDFT) [33–36] to de- scribe the coupled dynamics of the density fields of the liq- uid ρ$_{l}$ ( r , t ) and the nanoparticles ρ$_{n}$ ( r , t ) . In this approach, diffusive liquid transport can be incorporated in a straight- forward manner enabling us to go beyond previous 2d KMC studies and examine its influence. To construct the DDFT model, we (i) develop via coarse-graining an approximation for the underlying free energy functional of the system and (ii) form equations governing the dynamics of the two den- sity fields. These are able to account for the non-conserved and conserved aspects of the dynamics, i.e., phase change and diffusive transport processes, respectively. In order to compare results with the established KMC lattice model [30–32], we start from the lattice Hamiltonian to de- velop a mean-field (Bragg-Williams) approximation for the free energy functional [35, 37]. Expressing the interaction terms (sums over neighbouring lattice sites) as gradient opera- tors [35], the following semi-grand [38] free energy functional is obtained: where f ( ρ$_{l}$, ρ$_{n}$ ) = k$_{B}$T [ ρ$_{l}$ ln ρ$_{l}$ + (1 − ρ$_{l}$ ) ln(1 − ρ$_{l}$ )] + k$_{B}$T [ ρ$_{n}$ ln ρ$_{n}$ + (1 − ρ$_{n}$ ) ln(1 − ρ$_{n}$ )] − 2 ε$_{ll}$ρ 2 l − 2 ε$_{nn}$ρ 2 n − 4 ε$_{nl}$ρ$_{n}$ρ$_{l}$, includes entropic contributions and various inter- action terms – the parameters ε$_{ij}$ , where i, j = n, l , are the energies for having neighbouring pairs of lattice sites occu- pied by species i and j , respectively, T is the temperature, k$_{B}$ is Boltzmann’s constant and we have set the lattice spacing σ = 1 . Note that F in Eq. (1) can also be obtained by making a gradient expansion of the (non-local) free energy functional of a continuous system [37]. The rate of evaporation of the liquid from the substrate is determined by the chemical poten- tial µ in the reservoir (i.e. in the vapour above the substrate). When the temperature is sufficiently low, and the chemical potential µ = µ$_{coex}$ , we observe coexistence between a thick (high density) and a thin (low density) liquid film. In Fig. 1(a) we display the limit of linear stability (spinodal) and the equi- librium coexistence curve (binodal) for the pure liquid (i.e. with ρ$_{n}$ = 0 ). To indicate the influence of the solute we plot in Fig. 1(b) the densities ρ$_{l}$ and ρ$_{n}$ at coexistence for a fixed temperature k$_{B}$T /ε$_{ll}$ = 0 . 8 as a function of the average con- centration ¯ ρ$_{n}$ = 1 $_{2}$( ρ a $_{n}$+ ρ b $_{n}$) , where ρ a n and ρ b n are the densities of the nanoparticles in the coexisting a and b phases. For fur- ther details concerning the phase diagram and its topology see Ref. [39]. At equilibrium, the derivative µ$_{n}$ ≡ δF [ ρ$_{n}$, ρ$_{l}$ ] /δρ$_{n}$ is a con- stant, corresponding to the chemical potential of the nanopar- ticles. However, when the system is out of equilibrium, µ$_{n}$ may vary along the substrate. We assume that the thermo- dynamic force ∇ µ$_{n}$ drives the dynamics of the nanoparticles and that the nanoparticle current is j = − M$_{n}$ρ$_{n}$ ∇ µ$_{n}$ , where M$_{n}$ ( ρ$_{l}$ ) is a mobility coefficient. This expression for the cur- rent, together with the continuity equation, yields the time evolution equation for the nanoparticle density profile: This equation may also be obtained by assuming over-damped stochastic equations of motion for the nanoparticles [33, 34]. To model the fact that nanoparticles do not diffuse over the dry substrate (when ρ$_{l}$ is small) we set the mobility M$_{n}$ ( ρ$_{l}$ ) to switch at ρ$_{l}$ = 0 . 5 (smoothly) from zero for the dry substrate (low ρ$_{l}$ ) to α for the wet substrate (high ρ$_{l}$ ). Note that our results are not sensitive to the precise form of M$_{n}$ ( ρ$_{l}$ ) . Preprint – a.j.archer@lboro.ac.uk – u.thiele@lboro.ac.uk – March 27, 2019 ∂ρ$_{n}$ ∂t = ∇ · [ M$_{n}$ρ$_{n}$ ∇ δF [ ρ$_{n}$, ρ$_{l}$ ] δρ$_{n}$ ] . (2) F [ ρ$_{l}$, ρ$_{n}$ ] = ∫ d r$^{[}$ f ( ρ$_{l}$, ρ$_{n}$ ) + ε$_{ll}$ 2 ( ∇ ρ$_{l}$ ) 2 + ε$_{nn}$ 2 ( ∇ ρ$_{n}$ ) 2 + ε$_{nl}$ ( ∇ ρ$_{n}$ ) · ( ∇ ρ$_{l}$ ) − µρ$_{l}$$^{]}$, (1)