Stability of slowly evaporating thin liquid films of binary mixtures

We consider the evaporation of a thin liquid layer which consists of a binary mixture of volatile liquids. The mixture is on top of a heated substrate and in contact with the gas phase that consists of the same vapor as the binary mixture. The effects of thermocapillarity, solutocapillarity, and the van der Waals interactions are considered. We derive the long-wave evolution equations for the free interface and the volume fraction that govern the two-dimensional stability of the layer subject to the above coupled mechanisms and perform a linear stability analysis. Our results demonstrate two modes of instabilities, a monotonic instability mode and an oscillatory instability mode. We supplement our results from stability analysis with transient simulations to examine the dynamics in the nonlinear regime and analyze how these instabilities evolve with time. More precisely we discuss how the effect of relative volatility along with the competition between thermal and solutal Marangoni effect define the mode of instability that develops during the evaporation of the liquid layer due to preferential evaporation of one of the components.

Figure 1

Schematic of the physical system describing an evaporating thin liquid film on top of a horizontal heated solid substrate in a periodic domain, with periodic boundary condition at $\tilde{x}=0$ and ${\tilde{L}}_o$.

Figure 2

Time evolution of (a) the height of the flat interface, (b) the evaporation flux of components A and B, and (c) the temperature difference between the solid substrate and the interface of the liquid layer, for $K=0$ and $K\ne 0$. Here $E={10}^{-5}$ and the remaining parameters are shown in Table 2.

Figure 3

The growth rates $r_{\pm }$ vs wave number $k$, for $K=0$ and $E=0$. The remaining parameters are shown in Table 2.

Figure 4

The growth rates $r_{\pm }$ vs wave number $k$, for $K=0$ and $E={10}^{-5}$. The remaining parameters are shown in Table 2.

Figure 5

The growth rates $r_{+}$ and $r_{-}$ vs wave number $k$, for $E=0$. The remaining parameters are shown in Table 2.

Figure 6

The growth rates $r_{+}$ and $r_{-}$ vs wave number $k$, for $E={10}^{-5}$. The remaining parameters are shown in Table 2.

Figure 7

Dependence on $M_T$: Growth rates $r_{+}$ and $r_{-}$ vs wave number $k$ for $K\ne 0$ and $E={10}^{-5}$. (a) Oscillatory instability mode for $M_T=100$. (b) Monotonic instability mode for $M_T=1000$. The remaining parameters are shown in Table 2.

Figure 8

Dependence on $M_c$: Growth rates $r_{+}$ and $r_{-}$ vs wave number $k$ for $K\ne 0$ and $E={10}^{-5}$. The monotonic instability mode for $M_c=10$ and oscillatory instability mode for $M_c=100$ are also shown. The remaining parameters are shown in Table 2.

Figure 9

Dependence on volatility: Growth rates $r_{+}$ and $r_{-}$ vs wave number $k$ for $K\ne 0$ and $E={10}^{-5}$. (a) Monotonic instability mode for $\alpha =0.5$. (b) Oscillatory instability mode for $\alpha =2$. The remaining parameters are shown in Table 2.

Figure 10

(a) Neutral curves for two different values of relative volatility leading to either a monotonic or an oscillatory mode of instability. (b) Imaginary part of the most unstable eigenvalue for the oscillatory mode.

Figure 11

Maps showing the regions of the monotonic instability mode (crosses) and oscillatory instability mode (squares) in the parameter space of $M_c/M_T$ vs $\alpha$. (a) Contour lines of the growth rate of the most unstable wave number. (b) Contour lines of the most unstable wave number. (c) Contour lines of the frequency of the oscillations. Here $E={10}^{-5}$ and the remaining parameters are shown in Table 2. The experimental points for water-ethanol (parameters from Table 2) and water-butanol are plotted in green and blue bullets, respectively. The parameters used for butanol are $p^o=22\times {10}^3$ ($\alpha =0.46$) and $\sigma =19.45\times {10}^{-3}$.

Figure 12

(a) Schematic of the oscillatory instability for the case where component A is less volatile than B and has higher surface tension. (b) Schematic of the monotonic instability for the case where component A is more volatile than B and has higher surface tension. Here $F_{\text{TC}}$ is the thermocapillary force and $F_{\text{SC}}$ is the solutocapillary force.

Figure 13

Growth rate of the amplitude of the initial perturbation over time derived from the transient simulation. Inset: Growth rate vs wave number derived from the linear stability analysis for the case of the monotonic instability mode with $\alpha =0.5$. The solid and dashed line, respectively, show $r_{+}$ and $r_{-}$. Here $E={10}^{-5}$ and the remaining parameters are shown in Table 2.

Figure 14

Growth rate of the maximum amplitude of the oscillations over time derived from the transient simulation. Inset: Growth rate vs wave number derived from the linear stability analysis for the case of the oscillatory instability mode with $\alpha =2.28$. The solid and dashed line, respectively, show $r_{+}$ and $r_{-}$. Here $E={10}^{-5}$ and the remaining parameters are shown in Table 2.

Figure 15

Time evolution of the interface for $\alpha =0.5$ and $E={10}^{-5}$ showing the monotonic instability mode. The remaining parameters are shown in Table 2.

Figure 16

Time evolution of the interface for $\alpha =2.28$ and $E={10}^{-5}$ showing the oscillatory instability mode. The remaining parameters are shown in Table 2.

Figure 17

Evolution of higher-order Fourier modes for transient simulations presented for oscillatory instability in Fig. 16: (a) the interface ($s_1=2.21\times {10}^{-4}$, $s_2=7.20\times {10}^{-4}$, $s_3=9.92\times {10}^{-4}$), (b) the temperature ($s_1=2.44\times {10}^{-4}$, $s_2=7.36\times {10}^{-4}$, $s_3=1.04\times {10}^{-3}$), and (c) the volume fraction ($s_1=2.57\times {10}^{-4}$, $s_2=7.40\times {10}^{-4}$, $s_3=1.07\times {10}^{-3}$). (d) First mode of the Fourier transform of the interface (H), temperature (T), and volume fraction (C).