Fig. 1. Numerical simulations of the lubrication model for a dewetting thin film: (left) the evolution of a film at successive times showing droplet collision and collapse events (both circled), (right) the long-time scaling behavior for coarsening of the number of drops N(t)in a large system as a function of time.

Fig. 2. A stable steady-state droplet solution showing the three regions in the asymptotic structure of the solution for →0: (i) droplet core, (ii) contact line, (iii) outer ultra-thin film. The dashed curve shows the leading order asymptotic solution for the droplet core, the parabola (2.7), with width .

Fig. 3. Dynamics of two typical adjacent droplets in a dewetting film. The interaction of the droplets generally yields mass exchange and spatial motion.

Fig. 4. Fundamental modes of evolution for a single droplet: (left) translation and (right) change of mass.

Fig. 5. The C_X(P)drift coefficient function (2.13) and its asymptotic behaviors for large drops (P→0)(2.22) and small drops (P→p_max).

Fig. 6. Numerical simulation of two interacting drops in the lubrication PDE (2.1). As predicted by the ODE model (3.4), collision cannot occur.

Fig. 7. A schematic figure of a system of four drops used to study collision interactions.

Fig. 8. Contour lines of the asymptotic collision distance ratio, r=D_2,3/D_1,2(3.12), as a function of P₁,P₃for a fixed value of P₂. This structure of the solution is generic for all values of P₂.

Fig. 9. Plots of droplet histories in the xtplane for two different versions of the collision events suggested by (3.6): (a) a nearly symmetric 3-drop interaction, and (b) a more asymmetric quasi-2-drop collision. The solid stripes indicate the positions of droplet cores, with supports given on for k=1,2,3,4.

Fig. 10. Droplet histories in the xt plane (corresponding to Fig. 1) from numerical simulations of the full PDE, notice that the collision event (left) resembles the asymmetric “quasi-2-drop” collision in Fig. 9 b, while the collapse event (right) is more symmetric.

Fig. 11. Details of the collision and merging process: (a) two near-equilibrium droplet evolving toward collision, (b) the formation of the collided “two-drop complex” and its rapid convexification, (c) symmetrization and contraction to yield a single merged near-equilibrium droplet.

Fig. 12. Evolution of the energy for the numerical solution of the PDE, corresponding to the dynamics shown in Fig. 1 and Fig. 10 including a collision and a collapse event.

Fig. 13. Schematic of the fastest growing global instability modes: (a) the primary instability, connected to coarsening by collapse (4.10), and (b) the dominant secondary instability, connected to coarsening by collision (4.13).

Fig. 14. Droplet evolution tracks for a system of 100 drops with =0.01 and at three different values of : (a) , coarsening dominated by collapse events, (b) , a balance between collapses and collisions, (c) , collision dominated coarsening.

Fig. 15. Transition between collision- and collapse-dominated coarsening: the fraction of collisions as a function of the coarsening number and sampled from 1000 runs with 100 drops each, on a range of and =10^−j for j=1 –4.

Fig. 16. (Left) Plot of N(t)from simulations of (2.15) at several values of , for j=0,1,2,…,5, and (right) the same data rescaled according to (4.19).

Fig. 17. N(t) for the three simulations shown in Fig. 14 for values in the collision, collapse and mixed coarsening regimes (solid dots) compared with the scaling law predicted by (4.22) (dotted lines).

Fig. 18. The phase plane for Eq. (2.6) at a value of in the range . The droplet solution is given by the homoclinic orbit to (solid curve).