canonical ensemble. The free energy functional is ﬁrst deﬁned on the original KMC lattice. How-
ever, after re-writing the interaction terms employing gradient operators [78] one ﬁnally obtains
the free energy functional for a continuous system
F[ρl,ρn] =
∫
dr
[
f(ρl,ρn) + εll
2 (∇ρl)2 + εnn
2 (∇ρn)2 + εnl(∇ρn) ·(∇ρl) −µρl
]
, (4)
where
f(ρl,ρn) = kT[ρl ln ρl + (1 −ρl) ln(1−ρl)]
+ kT[ρn ln ρn + (1 −ρn) ln(1−ρn)]
−2εllρ2
l −2εnnρ2
n −4εnlρnρl. (5)
Since the liquid may evaporate from the surface into the vapour above the surface, µis the (true)
chemical potential of this reservoir and determines the rate of evaporation [condensation] from
[to] the surface. Note that normally a free energy of the form in Eq. (4) is obtained by making a
gradient expansion of the free energy functional of a continuous system [84]. However, here we
have made the mapping from the free energy of the lattice KMC system.
The chemical potential for the nanoparticles may be determined from the functional derivative
µn = δF[ρn,ρl]/δρn(r). In equilibrium it is constant throughout the system, but it may vary
spatially in a non-equilibrium system, i.e., µn = µn(r,t). We assume that the dynamics of the
nanoparticles is governed by the thermodynamic force ∇µn – i.e. that the nanoparticle current
is j = −Mnρn∇µn, where Mn(ρl) is a mobility coefﬁcient that depends on the local density of
the liquid. Combining this expression for the current with the continuity equation, we obtain the
following evolution equation for the nanoparticle density proﬁle
∂ρn
∂t = ∇·
[
Mnρn∇δF[ρn,ρl]
δρn
]
. (6)
Note that this equation of motion may also be obtained by assuming that the nanoparticles have
over-damped stochastic equations of motion [80–83]. Here, we assume that Mn(ρl) = αΘs(ρl −
0.5), where Θs(x) is a continuous function that switches smoothly from the value 0 to the value
1 at x = 0 (i.e. it is essentially a smooth analogue of the Heaviside function). This ensures that
the nanoparticles are immobile when the local liquid density is small (dry substrate) and have a
mobility coefﬁcient αwhen ρl is high (wet substrate).
For the evolution of the liquid density distribution we assume that the liquid is able to evaporate
from the surface into the vapour (reservoir) above the surface (non-conserved dynamics) and may
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