Fully nonlinear dynamics of stochastic thin-film dewetting

The spontaneous formation of droplets via dewetting of a thin fluid film from a solid substrate allows materials nanostructuring. Often, it is crucial to be able to control the evolution, and to produce patterns characterized by regularly spaced droplets. While thermal fluctuations are expected to play a role in the dewetting process, their relevance has remained poorly understood, particularly during the nonlinear stages of evolution that involve droplet formation. Within a stochastic lubrication framework, we show that thermal noise substantially influences the process of droplets formation. Stochastic systems feature a smaller number of droplets with a larger variability in size and space distribution, when compared to their deterministic counterparts. Finally, we discuss the influence of stochasticity on droplet coarsening for asymptotically long times.

Figure 1

Space-time plot of droplet formation and evolution as predicted by Eq. (1) for $\sigma =0$ (a) and $\sigma ={10}^{-2}$ (b), for the same parameter values and initial conditions, see main text. Brighter (darker) color corresponds to larger (smaller) values of the film thickness $h(x,t)$ according to the color scale on the right of each panel.

Figure 2

Time evolution of the position of the main maximum, $q_{\mathrm{m}}$, of the structure factor. The horizontal dashed black line indicates the deterministic linear prediction, $q_0$. Black circles (squares) correspond to predictions from Eq. (2) for $\sigma ={10}^{-2}$ ($\sigma =0$). Red up (blue down) triangles provide the position of $q_{\mathrm{m},\mathrm{sto}}\left(t\right) \left[q_{\mathrm{m},\det }\left(t\right)\right]$ as obtained in numerical simulations of Eq. (1) for $\sigma ={10}^{-2}$ ($\sigma =0$). Rupture times are signaled by arrows. All results are obtained by averaging over 200 noise realizations. Inset: Number of droplets for different noise amplitudes at $t=220$. All lines are guides to the eye.

Figure 3

(a), (c): Surface morphologies from simulations of Eq. (1) for $\sigma =0$ (thin blue line) and $\sigma ={10}^{-2}$ (single realization, thick red line) at $t=120$ (a) and 220 (c). (b), (d): Structure factor averaged over 200 noise realizations, at $t=120$ (b) and 220 (d) for $\sigma =0$ (blue squares) and $\sigma ={10}^{-2}$ (red circles). The thick black dashed (solid) line in (b) provides the analytical prediction from Eq. (2) for $\sigma ={10}^{-2}$ ($\sigma =0$). Thin solid blue and red lines in (b) and (d) are guides to the eye.

Figure 4

Distribution functions of drop heights (a) and interdrop distances (b) at time $t=220$ for $\sigma =0$ (blue squares) and $\sigma ={10}^{-2}$ (red circles). Inset: width of the main peak of $S_q$ at $t=220$, as a function of noise amplitude. Solid black, blue, and red lines are guides to the eye. (c) Number of droplets vs time, $N\left(t\right)$, for very long times up to $t=10000$, averaged over 40 noise realizations. We have set ${\widehat{h}}_{*}=0.04$ for computational feasibility. Blue squares (red circles) correspond to $\sigma =0$ ($\sigma ={10}^{-5/2}$). The dashed lines correspond to the power-law decay $N\left(t\right)\sim t^{-2/5}$ of very large deterministic systems [29]. Solid blue and red lines are guides to the eye.