text: 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 
  
  (cid:20) 
  Mnρn∇ 
  
  (cid:21) 
  
  ∂ρn 
  ∂t 
  δF [ρn, ρl] 
  δρn 
  = ∇ ·  .  (6) 
  
  Note that this equation of motion may also be obtained by assuming that the nanoparticles have 