x z y (a) (b) FIG. 4: Evolution of unstable fronts in substrates of different wettabilities that originate from a slightly perturbed front at t/t$_{c}$ = 0. (a) Saturated sawtooth: θ$_{e}$ = 0 ◦ and W/x$_{c}$ = 13. (b) Growing finger: θ$_{e}$ = 90 ◦ and W/x$_{c}$ = 8. The width of each system is chosen to match the most unstable wavelength as measured from figure 3, depending on the wetting properties of the substrate. The plot shows the contact line position at time intervals of Δ t/t$_{c}$ = 13 . 4, starting at t/t$_{c}$ = 0. For the last time, t/t$_{c}$ = 134, the full three dimensional interface is shown. ponential dependence on time to extract the growth rate of the perturbation. The resulting dispersion relations for each front are shown in figure 3. The band of modes over which the front is unstable is much narrower for θ$_{e}$ = 0 ◦ than for θ$_{e}$ = 90 ◦ . This is due to the thin film structure near the contact line, which tends to accumulate more mass when the fluid is in contact with hydrophobic substrates than with hydrophilic ones. As a result, fronts on hydrophilic substrates are stable to longer wavelengths and have smaller growth rates. In a recent work [21], we have validated our numerical method by comparing the linear stability results of the diffuse interface model to those of lubrication theory. A direct comparison is only possible for small dynamic angles and small capillary number, Ca . Given that our model allows for slip at the contact line, it is natural to compare with the results obtained by Spaid and Homsy [16], for which the contact line is allowed to slip by imposing a Navier boundary condition at the solid. As explained in the aforementioned paper, our results approach the lubrication theory results for small Ca . B. Sawtooth and Finger formation Figure 3 shows that the most unstable wavelengths are Λ$_{max}$( θ$_{e}$ = 0 ◦ ) /x$_{c}$ ≈ 14 and Λ$_{max}$( θ$_{e}$ = 90 ◦ ) /x$_{c}$ ≈ 8. These wavelengths set the typical length scale of the contact line 15