FIG. 4: (Colour online) Density proﬁles for the situation where the substrate is covered by nanoparticles
with average density ρav
n = 0.3. The top row are the nanoparticle density proﬁles and the bottom row are
the corresponding liquid density proﬁles at the timest/tl = 8 (left) and 80 (right), where tl = 1/kTMnc
l σ2.
The parameters are kT/εll = 0.8, εnl/εll = 0.6, εnn = 0, α= 0.4Mnc
l σ4, Mc
l = 0, ρl(t= 0) = 0.9 ±ξ
(where ξrepresents white noise of amplitude 0.05) and (µ−µcoex)/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 ρl is proportional to −(µsurf(r,t) −µ) = −δF[ρn,ρl]/δρl(r)
where µsurf(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
∂ρl
∂t = ∇·
[
Mc
l ρl∇δF[ρn,ρl]
δρl
]
−Mnc
l
δF[ρn,ρl]
δρl
, (7)
where we assume that the coefﬁcients Mc
l and Mnc
l are constants.
16