Instability and dynamics of volatile thin films

Volatile viscous fluids on partially wetting solid substrates can exhibit interesting interfacial instabilities and pattern formation. We study the dynamics of vapor condensation and fluid evaporation governed by a one-sided model in a low-Reynolds-number lubrication approximation incorporating surface tension, intermolecular effects, and evaporative fluxes. Parameter ranges for evaporation-dominated and condensation-dominated regimes and a critical case are identified. Interfacial instabilities driven by the competition between the disjoining pressure and evaporative effects are studied via linear stability analysis. Transient pattern formation in nearly flat evolving films in the critical case is investigated. In the weak evaporation limit unstable modes of finite-amplitude nonuniform steady states lead to rich droplet dynamics, including flattening, symmetry breaking, and droplet merging. Numerical simulations show that long-time behaviors leading to evaporation or condensation are sensitive to transitions between filmwise and dropwise dynamics.

Figure 1

Schematic representation of a two-dimensional thin fluid film on a uniformly heated (cooled) solid substrate where $J$ represents the evaporative (condensing) mass flux.

Figure 2

Plot of $\tilde{\mathrm{\Pi }}\left(h\right)$ given by (1e) with $\epsilon =0.1$, with the dashed lines representing the thresholds of the parameter $P_0$, which determines the multiplicity of spatially uniform solutions in the system. The thresholds separate the undersaturated, critical case, and supersaturated regimes.

Figure 3

Bifurcation diagram of the uniform steady states $H_c$ and $H_m$ as functions of $P_0$. Arrows indicate the dynamics for the mean thickness $\overline{h}\left(t\right)$ in each of the regimes US, CC, and SS: condensation with $\overline{h}\left(t\right)$ increasing or evaporation with $\overline{h}\left(t\right)$ decreasing, and $H_m$ being stable while $H_c$ is an unstable steady state.

Figure 4

(a) Plot of $d\sigma /dt$ vs $\overline{h}$ at several values of $P_0$ (with $k=6,\beta =0.001$, and $L=20$ fixed). Here $P_c\approx 8.455$ is a threshold for growth of spatial instabilities for $P_0<P_c$. (b) Critical film heights ${\overline{h}}_{-},{\overline{h}}_{+}$ marking the range where spatial perturbations grow for $P_0=0.5$. This range of heights is different than the ranges for evaporation or condensation of uniform films set by the equilibria $H_m,H_c$.

Figure 5

(a) Evaporation with transient pattern formation starting from the initial data $h_0\left(x\right)=0.2+{10}^{-4}cos(12\pi x/L)$ with the system parameters $P_0=0.5,\beta =0.001$, and $L=20$. (b) Dots represent the effective rate of evaporation $d\langle h\rangle /dt$ plotted against the decreasing average film thickness in the PDE simulation, compared against the prediction $-\beta \tilde{J}\left(\langle h\rangle \right)$ from (8a) (solid curve).

Figure 6

Dispersion relation plot of $d\sigma /dt$ against the wave number $k$ at the values $\overline{h}=0.127,0.133,0.2,0.3$. The parameter $P_0=0.5$ is in the critical case range $0<P_0<{\mathcal{P}}_{\epsilon }$ and the other parameters are identical to those in Fig. 4.

Figure 7

(a) Condensation dynamics with transient pattern formation starting from the initial data $h_0\left(x\right)=0.3+0.001{\sum }_{k=1}^{6}cos(2k\pi x/L)$ with $L=40$ and other parameters identical to Fig. 5. (b) Contour plot of the growth rate $d\sigma /dt$ with the thick line representing the computed most unstable wave number $k^{*}$ as the condensation occurs.

Figure 8

Phase plane for the second-order steady-state solutions satisfying (19) in the critical case with $0<P_0<{\mathcal{P}}_{\epsilon }$. The homoclinic orbit (solid line) corresponds to the droplet solution in Fig. 9 and the dashed line represents the periodic solution in Fig. 9.

Figure 9

(a) Typical steady-state droplet solution and (b) a periodic solution satisfying the ODE (19).

Figure 10

Bifurcation diagram for $\langle h\rangle$ parametrized by the domain size $L$ with the parameter $P_0=0.5$.

Figure 11

(a) Bifurcation diagram of steady states to (19) with the domain size $L=3.05$ showing that a family of nonuniform steady states coexists with the constant steady states $H_m$ and $H_c$ in a specific range of $P_0$. The solid lines represent the stable steady states and the dashed lines correspond to the unstable ones. (b) Coexisting equilibria of (19) with the average film thickness $\langle h\rangle =0.24$ and the domain size $L=3.05$.

Figure 12

Dependence of dominant eigenvalues of $H_s\left(x\right)$ on the period $\ell$ for the eigenproblem (29) with $P_0=0.5$, where the critical period is given by ${\ell }^{*}=3.005$.

Figure 13

Plots of the largest three eigenvalues of (23) over a range of $\beta$ with the period $\ell$ satisfying (a) $\ell =2.9<{\ell }^{*}$, (b) $\ell =3.005={\ell }^{*}$, and (c) $\ell =3.2>{\ell }^{*}$. The critical period is ${\ell }^{*}\approx 3.005$ for $P_0=0.5$.

Figure 14

(a) Early-stage evolution $\left(0\le t<250\right)$ and (b) the later stage $\left(t>250\right)$ of evaporation with the initial condition close to the nonuniform steady-state solution $H_s$ for a PDE simulation starting from the initial data $h_0\left(x\right)=H_s-0.0001{\mathrm{\Psi }}^V$ with domain size $L=\ell =3$.

Figure 15

Growth and decay rates of the $L^2$-norms between the PDE solution shown in Fig. 14 and the steady states $H_s\left(x\right)$ and $H_m$, showing that the solution evolves from the $H_s\left(x\right)$ state towards $H_m$.

Figure 16

Unstable eigenmodes of the nonuniform steady state $H_{s,2}\left(x\right)$ with two zero-mean modes ${\mathrm{\Psi }}^B$ and ${\mathrm{\Psi }}^M$ and a nonzero-mean mode ${\mathrm{\Psi }}^P$ where the domain size $L=2\ell =7$ and the other system parameters are identical to those in Fig. 14.

Figure 17

The PDE simulations of two droplets on $0\le x\le 7$ starting from $h_0\left(x\right)=H_{s,2}\left(x\right)+0.1\mathrm{\Psi }$ showing (a) symmetry breaking with $\mathrm{\Psi }={\mathrm{\Psi }}^B$, (b) droplet merging with $\mathrm{\Psi }={\mathrm{\Psi }}^M$, and (c) flattening with $\mathrm{\Psi }={\mathrm{\Psi }}^P$. (d) Corresponding changes of $\langle h\rangle$ in time, where the long-time behavior of the solutions perturbed by the ${\mathrm{\Psi }}^B$ mode is identical to that of the ${\mathrm{\Psi }}^M$ mode shifted by $\mathrm{\Delta }t=217$.

Figure 18

Transition-state diagram connecting linearized dynamics in the small-domain case with $\ell <{\ell }^{*}$.

Figure 19

(a) Numerical simulation of dewetting of a volatile thin film starting from typical initial conditions: early-stage dewetting yielding three metastable droplets, subsequent collapse of the left and right droplets occurring while the center droplet gradually grows, followed by filmwise condensation after the substrate is flooded. (b) Plot showing the average thickness $\langle h\rangle$ decreasing in the early regime for $0\le t<t^{*}\approx 440$ corresponding to dewetting and evaporation, with later condensation yielding increasing average thickness for $t>t^{*}$.

Figure 20

(a) The PDE simulation starting from initial data $h_0\left(x\right)=0.197-0.03sin(6\pi x/L)+0.01sin(4\pi x/L)$ for the model (1) and (b) the corresponding monotonically decreasing average thickness in time.

Figure 21

Plot showing the influences of $\beta$ and the initial film height to evaporation (denoted by open circles) or condensation (denoted by closed circles) dynamics. The $(\beta ,{\overline{h}}_0)$ values used in Figs. 19 and 20 are labeled by closed and open squares, respectively.