and present the conclusions of this work. 

## II. DIFFUSE INTERFACE MODEL FOR DRIVEN THIN FILMS 

In this section we present the diffuse interface framework on which we rely to study thin film dynamics on chemically heterogeneous substrates. In a recent work [21], we have studied the dynamics of thin films on homogeneous substrates in the aforementioned framework, which considers the hydrodynamics of two immiscible liquid phases coupled by a diffuse interface. Such a model allows for contact line dynamics and has been validated against molecular dynamics simulations [22]. Contact line dynamics emerges as a consequence of a diffusive mechanism at the contact line, making the use of explicit boundary conditions at the contact line unnecessary [23, 24]. This model has been used to study the dynamics of the contact line between parallel plates on hydrophilic-hydrophobic domains by Wang et al. [25]. Lattice-Boltzmann simulations, the powerful computational fluid dynamics algorithm that we shall use in this work, have been used to apply this model on a variety of situations. Briant and Yeomans [23] studied the motion of contact lines between shearing parallel plates. The development of thin films in Hele-Shaw cells was studied by Ledesma- Aguilar et al. [24]. Three-dimensional effects on the Saffman-Taylor finger was studied by Ledesma-Aguilar et al. [26]. In Ref. [21] we have performed lattice-Boltzmann simulations of thin films flowing on homogeneous substrates of arbitrary wetting properties. Our results for the linear stability of the front show that the contact line is unstable to a wider band of perturbation modes as the equilibrium contact angle increases. To validate our model, we have shown that the previous lubrication theory results are approached for sufficiently small driving velocities. In addition, quantitative agreement has been obtained between the lattice-Boltzmann thin film profiles and those obtained by Spaid and Homsy [16] for the slip-velocity lubrication model in the same small-velocity regime. Long time dynamics show that a sawtooth saturates when the fluid wets the substrate, while fingers grow when forcing non-wetting fluids. In agreement with previous results [13, 18], we have observed that sawtooth shaped fronts are obtained by decreasing the inclination angle of the substrate. 

