text: by a ﬁlm of liquid or by a nanoparticle, respectively. Note that the probability densities correspond 
  
  to number densities as we use the lattice spacing σ = 1 as our unit of length. 
  
  To develop the DDFT, one must ﬁrst derive the underlying free energy functional F [ρl, ρn], and 
  
  secondly, devise dynamical equations for both density ﬁelds that account for the conserved and the 
  
  non-conserved aspects of their dynamics, i.e., transport and phase change processes, respectively. 
  
  For a system governed by the hamiltonian (3), we may construct a mean-ﬁeld (Bragg-Williams) 
  
  approximation for the free energy of the system [78, 84] which contains an entropic contribution 
  
  and contributions from the interactions between the different species (nanoparticles and liquid). 
  
  The free energy is a semi-grand free energy, since the liquid is treated grand canonically (it is 
  
  coupled to a reservoir with chemical potential µ), whereas the nanoparticles are treated in the 