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librium contact angle, θ$_{e}$ . In order to explore the effect of substrate heterogeneity on the dynamics of the film, we must first explore the dynamics on homogeneous substrates, to characterize ˙ L and Λ$_{max}$. In this section we describe the dynamics of thin films on two separate homogeneous substrates with equilibrium contact angles θ$_{e}$ = 0 ◦ and θ$_{e}$ = 90 ◦ , respectively. Our aim is to characterize the linear instability by measuring the dispersion relation on each substrate, and to characterize the base structures that emerge at long times. 

## A. Linear Stability 

A numerical linear stability analysis of the front can be performed by following the evo- lution of an initially flat contact line that is perturbed transversely with a single mode k/k$_{c}$ , where k$_{c}$ = 2 π/x$_{c}$ . At short times, the perturbation either grows or decays exponentially with a growth rate, ω ( k ). We follow the evolution of single mode perturbations of small initial amplitude A$_{0}$/W ≈ 10 − $^{2}$, and focus on timescales where the amplitude shows an ex- 

