Nonuniformities in miscible two-layer two-component thin liquid films

Depositing and obtaining a liquid film of uniform thickness is a problem integral to numerous applications and requires an understanding of capillary leveling, Marangoni flows, evaporation, and various other phenomena. These applications often demand multilayer films where each layer has distinct properties, and this gives rise to additional challenges. It has been experimentally demonstrated that two-layer films in which the layers are miscible can undergo dewetting, but theoretical understanding of this phenomenon is lacking. Through a lubrication-theory-based model, this work studies the mechanisms that may initiate dewetting in miscible two-layer two-component films. The model film consists of nonvolatile solvent and solute with constant density and viscosity. Two coupled fourth-order nonlinear partial differential equations describing the time evolution of the film height and solute concentration are derived and then solved with a pseudospectral method. It is found that a disparity in initial solute concentration between the film layers drives flows that lead to significant film-height nonuniformities. A parametric study is conducted to examine the influence of system parameters on this behavior and to develop several scaling relations that shed light on the underlying physical mechanisms.

Figure 1

Schematic of a two-layer liquid film resting on a solid substrate. If the liquid layers are miscible, there is no distinct interface between them.

Figure 2

Schematic of the model geometry of two miscible liquid layers resting on a solid substrate. One liquid is considered the “solvent” and the other the “solute.” Yellow indicates a higher concentration of solute, and ${\sigma }^{+}$ denotes a locally high surface tension and ${\sigma }^{-}$ denotes a locally low surface tension that arise from solute concentration gradients. The resulting surface-tension gradient causes formation of a film-height nonuniformity, through Marangoni flow, from an initially flat interface (see Fig. 1).

Figure 3

Cross sections of the initial condition $c_0(x,z)$ for three bottom layer thicknesses $h_b$ at $x=\pi /\alpha$. The top layer corresponds to layer 1 and the bottom layer to layer 2 in Fig. 1. As the bottom layer thickness $h_b$ increases, the transition between layers occurs at larger $z$ (layer 2 is thicker and layer 1 thinner in Fig. 1). The perturbation centered at $z=h_b$ cannot be seen due to its small amplitude.

Figure 4

(a) Time evolution of the film height for $\text{Pe}={10}^4$. The maximum height perturbation $\mathrm{\Delta }h$ occurs at $t_{\max }\approx 2\times {10}^3$ where the film-height profile is given by the solid black line. The perturbation slowly decays beyond this time as shown by the profile at $t={10}^4$. (b) Maximally deformed state for $\text{Pe}=1.67\times {10}^5$ with solute concentration contour. This state occurs at $t\approx 9\times {10}^2$ after which the perturbation slowly decays through lateral diffusion. The remaining parameter values are $c_p={10}^{-2}, \alpha =0.3, \epsilon =0.1$, and $h_b=0.5$.

Figure 5

Plots of (a) the maximum height perturbation $\mathrm{\Delta }h$ and (b) the time to achieve it, $t_{\max }$, at varying Pe. The plots are divided into three regions labeled I, II, and III. The blue diamond data points are predicted values from the one-layer approximation discussed in Sec. 3a, and the solid magenta line is an analytical approximation for region I derived in Sec. 3a.

Figure 6

Concentration profiles at the interface $z=h$ and $t=t_{\max }$ for three different Pe and the vertically averaged initial condition $\overline{c_0}$ [Eq. (20)]. Under pure diffusion, $\overline{c_0}$ gives the maximum possible lateral perturbation at the interface. The red line for $\text{Pe}=1.3\times {10}^3$ depicts data taken from region I of Fig. 5. Similarly, the magenta line for $\text{Pe}=7.7\times {10}^4$ and the blue line for $\text{Pe}=2.2\times {10}^5$ are data taken from regions II and III, respectively, of Fig. 5.

Figure 7

Velocity contours (black arrows) plotted over solute concentration contours (a) before ($t\approx 2.23\times {10}^2$) and (b) after ($t\approx 6.85\times {10}^2$) the development of circulatory flow for $\text{Pe}={10}^5$. The vertical velocity $v_z$ is generally an order of magnitude smaller than the horizontal velocity $v_x$ and is difficult to discern. As a visual aid, the arrows labeled “$v_z$” indicate the direction of the vertical velocity at the center of the film ($x=\pi /\alpha$) but are not to scale. At early times, there is bulk flow from the troughs to the crest. Later, circulatory flow develops while the troughs and crest continue to grow.

Figure 8

Solution (35) of convection equation (32) evaluated at $z=1$ and three different times. Laterally nonuniform vertical convection causes the rapid formation of a laterally nonuniform concentration profile, even though the initial condition is laterally uniform with $c_0^{*}(z=1)=0$.

Figure 9

Schematics depicting possible geometries of a solidified film for values of the Péclet number Pe in (a) region I, (b) region II, and (c) region III of Fig. 5.

Figure 10

(a) The maximum height perturbation $\mathrm{\Delta }h$ and (b) the time to achieve it, $t_{\max }$, at varying Pe and three perturbation magnitudes $c_p$. There are only quantitative shifts in $\mathrm{\Delta }h$, and the values of $t_{\max }$ are nearly identical for all three values of $c_p$. (c) Linear fits of the maximum height perturbation $\mathrm{\Delta }h$ versus perturbation magnitude $c_p$ at three different Pe. All lines have a slope of approximately unity, indicating a linear scaling with $c_p$. The largest value of $\text{Pe}=2.2\times {10}^5$ is taken from region III where an analytical scaling relation has not been derived.

Figure 11

(a) The maximum height perturbation $\mathrm{\Delta }h$ and (b) the time to achieve it, $t_{\max }$, at different Pe and three perturbation wavenumbers $\alpha$. (c) Linear fits of the scaled height perturbation ${\alpha }^2\mathrm{\Delta }h$ versus the scaled time ${\alpha }^2{t}_{\max }/\text{Pe}$. Data points for $\text{Pe}=1.7\times {10}^4$ in regions II and III are repeated in the zoomed-in subplot to illustrate that the scaling relation does not hold outside of region I.

Figure 12

Plots of (a) the maximum height perturbation $\mathrm{\Delta }h$ and (b) the time to achieve it, $t_{\max }$, at different Pe and three layer thicknesses $h_b$. There are only small quantitative shifts in both $\mathrm{\Delta }h$ and $t_{\max }$.

Figure 13

Plots of (a) the maximum height perturbation $\mathrm{\Delta }h$ and (b) the time to achieve, $t_{\max }$, it at varying Péclet number Pe and three different Marangoni numbers $\text{Ma}={\epsilon }^2$. There are only quantitative shifts in where region transitions occur. The inset plot in (b) shows the scaling ${\text{Pe}}_{\mathrm{crit}}\sim {\epsilon }^{-2}$.