text: substrate and force equilibria at the free surface, and applies a long-wave approximation. Under 
  
  the assumption that concentrations equilibrate rapidly over the ﬁlm thickness, we obtain coupled 
  
  non-linear evolution equations for the ﬁlm thickness proﬁle h(x, t) and the amount of nanoparticles 
  
  per unit length hp = φh, where φ is the volume concentration of the nanoparticles. Note, that hp 
  
  corresponds to the local thickness of the nanoparticle layer when all the solvent is evaporated. The 
  
  resulting evolution equation for the ﬁlm thickness is Eq. (1) above and focusing on the inﬂuence 
  
  of particle-independent capillarity and wettability only, the energy functional F [h] is given by 
  
  Eq. (2) above. Note that the viscosity η depends on the particle concentration. Following Refs. 
  
  [88, 89, 91, 92] we use the Quemada law for dense suspensions [93–95] 
  
  (cid:18)  (cid:19)−2 
  
  φ 
  φc 
  η(φ) = η0 
  1 −  (8) 
  
  where φc = 0.64 corresponds to random close packing of spherical particles. For the nanoparticle 
  
  volume per length hp = φh one obtains the following evolution equation: 
  
  (cid:20)  (cid:21) 
  
  δF 
  δh 
  ∂t(φh) = ∇ ·  φQc∇ 
  + ∇ · [D(φ)h∇φ] ,  (9) 