Lattice Boltzmann method for thin-liquid-film hydrodynamics

We propose an approach to the numerical simulation of thin-film flows based on the lattice Boltzmann method. We outline the basic features of the method, show in which limits the expected thin-film equations are recovered, and perform validation tests. The numerical scheme is applied to the viscous Rayleigh-Taylor instability of a thin film and to the spreading of a sessile drop toward its equilibrium contact angle configuration. We show that the Cox-Voinov law is satisfied and that the effect of a tunable slip length on the substrate is correctly captured. We address, then, the problem of a droplet sliding on an inclined plane, finding that the Capillary number scales linearly with the Bond number, in agreement with experimental results. At last, we demonstrate the ability of the method to handle heterogenous and complex systems by showcasing the controlled dewetting of a thin film on a chemically structured substrate.

Figure 1

Schematic sketch of a model system: a thin liquid film deposited on a flat substrate. The air-liquid interface is represented by the height $h(x,y,t)$. The characteristic thickness of the film is given by $H$.

Figure 2

Time evolution of the free surface for the Rayleigh-Taylor instability at ${\tau }_{\text{cap}}\approx 50,75,100,188$, corresponding to the lattice Boltzmann time steps [9000, 14000, 19000, 35000] $\mathrm{\Delta }t$ in (a)–(d). For a more clear visualization we only show a small patch of size $256\times 256$ centered in the middle of the $2048\times 2048$ domain. The fluctuations of earlier states still follow the linear stability analysis (see Fig. 3 for the the power spectrum of the height fluctuations versus wave number).

Figure 3

Power spectrum of the height fluctuations versus wave number. The different colors and symbols belong to different values of gravitational acceleration; $g=4\times {10}^{-5}$ is given by blue circles ($•$), $g=6\times {10}^{-5}$ by orange triangles ($▴$), and $g=8\times {10}^{-5}$ is given by green squares ($\blacksquare$). Same colored lines are taken at different time steps. In the inset we show the growth rate $\sigma \left(k\right)$ for the largest value of $g$ (symbols) and the theoretical growth rate according to Eq. (20) (solid line).

Figure 4

Relaxation of an out-of-equilibrium droplet. The initial state (a) is shown with a contact angle of $\theta =\pi /6$. Toward the end of the simulation (b) the droplet relaxes to the equilibrium contact angle.

Figure 5

Difference of cubed instantaneous and equilibrium contact angles, ${\theta }_{\mathrm{num}}^3-{\theta }_{\mathrm{eq}}^3$, vs. capillary number Ca for a spreading droplet; the dashed line shows a linear dependence (consistent with the Cox-Voinov law). The different symbols represent different viscosities, while the dashed line is a linear function of the capillary number.

Figure 6

Time evolution of the droplet base radius of a spreading droplet; the dashed red line shows a ${\tilde{t}}^{1/10}$ power law (consistent with Tanner's law). As in Fig. 5 different symbols refer to different viscosities. The radius clearly grows with the predicted power law until it saturates. On rescaling the time with ${\tau }_{\text{cap}}$ the curves of all three viscosities collapse into a single one.

Figure 7

Ca vs. Bo for a sliding droplet: Notice that a finite minimum forcing (corresponding to ${\mathrm{Bo}}_c$) is needed to actuate the droplet. For $\mathrm{Bo}>{\mathrm{Bo}}_c$ a linear relation, $\mathrm{Ca}\sim \mathrm{Bo}$, is observed. In the insets we show the shape of the droplet for both the pinned (upper left) as well as the sliding (lower right) case.

Figure 8

Time evolution of the free surface on a chemically patterned substrate on a $512\times 512\mathrm{\Delta }{x}^2$ domain. Varying the contact angle between the letters and the rest of the substrate yields the shown dewetting pattern. The letters are more wettable then the rest. To emphasis the process we use a color gradient raging from dark blue to light blue. Starting from a randomly perturbed film height, the fluid starts to dewet the pattern and after $2400\mathrm{\Delta }t$ the letters and a surrounding rim structure are clearly visible (a). After $16800\mathrm{\Delta }t$ a depletion around the letters is already very prominent (b). Toward the end of the simulation at $97200\mathrm{\Delta }t$, the instability of the thin film also leads to film rupture. Holes form among the letters E, R, and N (c).

Figure 9

Time evolution of the ${\mathbf{L}}_2$ norm [as defined by Eq. (A1)] of the $x$ component of the terms appearing on the right-hand side of the second of Eqs. (15). The plot is in log-lin scale. The panels refer to three different numerical experiments: (a) spreading droplet, (b) sliding droplet, and (c) thin-film dewetting. The symbols correspond to film pressure gradient, $-h{\partial }_x{p}_{\text{film}}$ ($★$); friction, $-\nu \alpha \left(h\right)u_x$ ($•$); and longitudinal dissipation terms $\nu {\nabla }^2\left(h{u}_x\right)+2\nu {\partial }_x[\mathbf{\nabla }·\left(h\mathbf{u}\right)]$, ($▴$).