FIG. 4: (Colour online) Density profiles for the situation where the substrate is covered by nanoparticles with average density ρ av n = 0 . 3 . The top row are the nanoparticle density profiles and the bottom row are the corresponding liquid density profiles at the times t/t$_{l}$ = 8 (left) and 80 (right), where t$_{l}$ = 1 /kT M nc l σ $^{2}$. The parameters are kT /ε$_{ll}$ = 0 . 8 , ε$_{nl}$/ε$_{ll}$ = 0 . 6 , ε$_{nn}$ = 0 , α = 0 . 4 M nc l σ $^{4}$, M c l = 0 , ρ$_{l}$ ( t = 0) = 0 . 9 ± ξ (where ξ represents white noise of amplitude 0.05) and ( µ − µ$_{coex}$ ) /kT = − 0 . 88 , where the liquid exhibits spinodal decomposition-evaporation. also diffuse over the substrate (conserved dynamics). The conserved part is treated along the lines developed above for the nanoparticles. For the non-conserved part we assume a standard form [85], i.e., the change in time of ρ$_{l}$ is proportional to − ( µ$_{surf}$ ( r , t ) − µ ) = − δF [ ρ$_{n}$, ρ$_{l}$ ] /δρ$_{l}$ ( r ) where µ$_{surf}$ ( r , t ) is the local chemical potential of the liquid at the point r on the surface at time t . This gives the evolution equation for the liquid density where we assume that the coefficients M c l and M nc l are constants. ∂ρ$_{l}$ ∂t = ∇ ·$^{[}$ M c l ρ$_{l}$ ∇ δF [ ρ$_{n}$, ρ$_{l}$ ] δρ$_{l}$ ] − M nc l δF [ ρ$_{n}$, ρ$_{l}$ ] δρ$_{l}$ , (7) 16