Lattice Boltzmann simulations of stochastic thin film dewetting

We study numerically the effect of thermal fluctuations and of variable fluid-substrate interactions on the spontaneous dewetting of thin liquid films. To this aim, we use a recently developed lattice Boltzmann method for thin liquid film flows, equipped with a properly devised stochastic term. While it is known that thermal fluctuations yield shorter rupture times, we show that this is a general feature of hydrophilic substrates, irrespective of the contact angle $\theta$. The ratio between deterministic and stochastic rupture times, though, decreases with $\theta$. Finally, we discuss the case of fluctuating thin film dewetting on chemically patterned substrates and its dependence on the form of the wettability gradients.

Figure 1

Height fluctuations spectra from deterministic (circles) and stochastic (triangles) simulations at $t=0.2t_0$ (filled symbols) and $t=0.7t_0$ (empty symbols), on a substrate with $\theta =\pi /9$. The theoretical predictions, Eq. (29), are reported with solid ($\sigma =0$) and dashed ($\sigma ={\sigma }_0$) lines.

Figure 2

Histograms of the droplet height distributions from the deterministic (blue, line-patterned) and stochastic (orange, dot-patterned) simulations with $\theta =\pi /9$, at $t\approx 20t_0$; the corresponding Gaussian kernel density estimations are also plotted, as a guide to the eye, with solid blue (dashed orange) curve for the deterministic (stochastic) data.

Figure 3

Time evolution of $\mathrm{\Delta }h\left(t\right)={max}_x\left\{h(x,t)\right\}-{min}_x\left\{h(x,t)\right\}$ for the athermal (bullets) and fluctuating (triangles) systems with $\theta =\pi /9$. The blue dashed-dotted line depicts an exponential fit of the data for the athermal case ($\sigma =0$). The vertical lines mark the rupture times for the deteministic (solid) and stochastic (dotted) simulations.

Figure 4

Ratio of rupture times from athermal and fluctuating dewetting, ${\chi }_{{\sigma }_0}\left(\theta \right)$ [Eq. (32)], as a function of the contact angle $\theta$. The dashed line is Eq. (36) for $\sigma ={\sigma }_0$. Inset: Rupture times ${\tau }_r$ normalized by $t_0$ (see Table 1) vs $\theta$, for the deterministic (orange bullets $•$) and stochastic (green triangles $▴$) simulations.

Figure 5

Time dependence of the wavenumber, $q_m$, at which the spectrum has the global maximum for various contact angles $\theta$ and $\sigma ={\sigma }_0$, using $\delta =0$. The full blue, orange and green lines correspond to theoretical curve $q_m\left(t\right)$ derived from Eq. (29) for $\theta =\pi /9,5\pi /36,\pi /6$. The dashed lines show the corresponding $q_0\left(\theta \right)$. In the inset we show that the data collapse onto a single curve upon rescaling each $q_m$ by its respective $q_0$.

Figure 6

Space-time plot of the height field $h(x,t)$ evolution over a sinusoidally patterned substrate undergoing athermal (a) and fluctuating (b) dewetting, respectively. In panel (c) we report the contact angle profile $\theta \left(x\right)$ [Eq. (37)].

Figure 7

Height profiles from deterministic dewetting ($\sigma =0$) over the sinusoidally patterned substrate, Eq. (37) at $t=0.26t_0$ (a) and $t=0.66t_0$ (b) and corresponding spectra (c). In panel (c) also the initial spectrum, $S_0\left(q\right)$, is reported; the vertical lines indicate the pattern wave mode $q_{\theta }$ (solid) and its multiples $2q_{\theta }$ (dashed) and $3q_{\theta }$ (dashed dotted). As explained in the text, the convention $t_0=t_0(\pi /6)$ and $q_0=q_0(\pi /6)$ applies.

Figure 8

Height profiles from stochastic dewetting ($\sigma ={\sigma }_0$) over the sinusoidally patterned substrate, Eq. (37) at $t=0.26t_0$ (a) and $t=0.66t_0$ (b) and corresponding spectra (c). In panel (c) also the initial spectrum, $S_0\left(q\right)$, is reported; the vertical lines indicate the pattern wave mode $q_{\theta }$ (solid) and its multiples $2q_{\theta }$ (dashed) and $3q_{\theta }$ (dashed dotted).

Figure 9

Space-time plot of the height field $h(x,t)$ evolution over a square-wave patterned substrate undergoing athermal (a) and fluctuating (b) dewetting, respectively. In panel (c) we report the contact angle profile $\theta \left(x\right)$ [Eq. (38)].

Figure 10

(a), (b) Space-time plots showing the evolution of the height field for athermal (a) and fluctuating (b) dewetting over the square-wave patterned substrate, in a neighbourhood of the instant of time at which the first rupture event (in the athermal case) occurred; for the sake of visualization we mark the rupture events with red bullets ($•$). (c) Distribution of times of occurrence of rupture events for the athermal ($\sigma =0$, blue, line-patterned) and fluctuating ($\sigma ={\sigma }_0$, orange, dot-patterned) dewetting.

Figure 11

Spectra of the dewetting film on the ${\theta }^{\left(2\right)}\left(x\right)$ pattern, Eq. (38), at $t=0.1t_0(\pi /6)$, with ($\sigma ={\sigma }_0, ▴$) and without ($\sigma =0, •$) thermal fluctuations. The vertical lines indicate the substrate wave mode $q_{\theta }$ (solid) and its multiples $2q_{\theta }$ (dashed), $3q_{\theta }$ (thin dashed-dotted), and $7q_{\theta }$ (thick dashed-dotted).