Evaporation-driven solutocapillary flow of thin liquid films over curved substrates

Evaporative loss of a volatile solvent can induce concentration inhomogeneities that give rise to spatial gradients in surface tension and subsequent solutocapillary Marangoni flows. This phenomenon is studied in the context of ultrathin liquid films resting atop curved convex substrates in contact with a fluid reservoir. Experiments are conducted with low-molecular-weight polydimethylsiloxane (silicone oil) mixtures composed of a volatile solvent and trace amounts of a nonvolatile solute. A theoretical model based on the thin-film approximation is developed, incorporating the effects of evaporative mass loss, gravity, capillarity, van der Waals forces, species diffusion, and Marangoni stresses. The spatiotemporal evolution of this system is studied by modulating the rate of evaporation of the volatile species and the bulk solute volume fraction in the mixture. The experiments and accompanying numerical simulations reveal that both Marangoni stresses and stabilizing van der Waals interactions between the substrate and the free surface can induce flow reversal and film regeneration. Their relative contribution is modulated by the solutocapillary Marangoni number, which is proportional to the bulk concentration of nonvolatile species in the mixture. Furthermore, it is revealed that increasing the rate of evaporation enhances the volumetric flow rate from thicker, solvent-rich areas towards thinner, solute-rich regions of the film. Although quantitative differences between the theory and the experiments are observed within certain ranges of the controlled parameters, the model qualitatively reproduces the flow regimes observed in the experiments and elucidates the complex interplay among the various physical forces.

Figure 1

(a) The dynamic fluid-film interferometer, a custom-built apparatus [37], is used to create thin liquid films between a spherical solid substrate and an initially planar free surface. Labeled components in the figure correspond to 1, Delrin chamber; 2, fused-silica substrate; 3, substrate holder; 4, motorized actuator; 5, light source; 6, top camera; and 7, DFI cover. (b) On the bottom, white-light interferometry is used to map color images recorded from the top camera to film thickness profiles. On the top, an image-processing script is used to visualize the spatiotemporal evolution of the entrained liquid film.

Figure 2

Schematic of the model geometry.

Figure 3

Comparison of the numerical computations (dashed lines) to experimental measurements (solid lines) for pure silicone fluids in the absence of evaporation. Lines and error bars in the experimental data represent the mean and standard deviation, respectively, of repeated measurements (duplicates or triplicates). The blue data are for $\nu =1$ cSt, $U=0.05$ mm/s, $t^{*}=7\mathrm{s}$, and $b=0.3$ mm; green data for $\nu =1$ cSt, $U=0.15$ mm/s, $t^{*}=2.3\mathrm{s}$, and $b=0.3$ mm; and red data for $\nu =5$ cSt, $U=0.15$ mm/s, $t^{*}=2.3\mathrm{s}$, and $b=0.3\mathrm{mm}$.

Figure 4

Time series of experimental film thickness profiles are plotted for 1.00 cSt/5.00 cSt blends ($\text{Ev}=2.6\times {10}^{-3}$) at $\text{Ma}=0.011$ (${\varphi }_{\infty }=0.01\%$), $\text{Ma}=0.107$ (${\varphi }_{\infty }=0.10\%$), and $\text{Ma}=0.320$ (${\varphi }_{\infty }=0.30\%$), as determined experimentally. Snapshots of the interferometric patterns, viewed from the top camera, are shown at the corresponding times and compositions. Experiments show the presence of three distinct flow regimes: the van der Waals regime (solid line), the intermediate regime (dashed line), and the capillary regime (dotted line).

Figure 5

Time series of the dimensionless film thickness $\overline{h}(\overline{r},\overline{t})$, solute concentration $\varphi (\overline{r},\overline{t})$, and dimensionless dynamic pressure $\overline{p}(\overline{r},\overline{t})$ profiles at three different Marangoni numbers $\text{Ma}=0.011$ (${\varphi }_{\infty }=0.01\%$), $\text{Ma}=0.107$ (${\varphi }_{\infty }=0.10\%$), and $\text{Ma}=0.320$ (${\varphi }_{\infty }=0.30\%$), as determined theoretically. The other dimensionless parameters used in the numerical calculations are $\text{Ca}=2.4\times {10}^{-6}$, $\text{Pe}={10}^3$, $\text{Bo}=4.3\times {10}^{-2}$, $\text{Ha}=1.6\times {10}^{-6}$, and $\text{Ev}=2.6\times {10}^{-3}$.

Figure 6

Plot of the experimental centerline film thickness ${\overline{h}}_0=h(0,t)/h^{*}$ against time $\overline{t}$ for a 1.00 cSt/5.00 cSt mixture ($\text{Ev}=2.6\times {10}^{-3}$) at several values of the Marangoni number $\text{Ma}=0.021\text{–}0.320$ (${\varphi }_{\infty }=0.02\%\text{–}0.30\%$). The other dimensionless parameters for a 1.00 cSt/5.00 cSt mixture are reported in Table 2. Experimental data cannot be obtained for thicknesses below the dotted line at $\overline{h}\simeq 7.5\times {10}^{-3}$ ($h\simeq 90\text{nm}$). Error bars represent the standard deviation obtained from multiple data sets.

Figure 7

Plot of the theoretical centerline film thickness ${\overline{h}}_0=h(0,t)/h^{*}$ against time $\overline{t}$ for several values of the Marangoni number $\text{Ma}=0.021\text{–}0.320$ (${\varphi }_{\infty }=0.02\%\text{–}0.30\%$). The solid lines correspond to $\mathrm{Ha}=1.6\times {10}^{-5}$, the dashed lines correspond to $\mathrm{Ha}=1.6\times {10}^{-6}$, and the region in between two curves of the same color is shaded to guide the eye. The other dimensionless parameters are $\text{Ca}=2.4\times {10}^{-6}$, $\text{Pe}={10}^3$, $\text{Bo}=4.3\times {10}^{-2}$, and $\text{Ev}=2.6\times {10}^{-3}$, corresponding to a 1.00 cSt/5.00 cSt mixture (reported in Table 2).

Figure 8

Experimental and simulation data of the dimensionless mound volume flux $d\overline{V}/d\overline{t}=(1/a{h}^{*}U)(dV/dt)$ against Ma for 1.50 cSt/10.0 cSt ($\text{Ev}=4.6\times {10}^{-4}$), 1.00 cSt/5.00 cSt ($\text{Ev}=2.6\times {10}^{-3}$), and 0.65 cSt/5.00 cSt ($\text{Ev}=2.6\times {10}^{-2}$) mixtures. Other relevant dimensionless parameters appear in Table 2. The symbols $Ⓦ$, $Ⓘ$, and $Ⓒ$ stand for the van der Waals, the intermediate, and the capillary regimes, respectively. The closed squares correspond to experimental measurements and the error bars represent the standard deviation obtained from multiple data sets. In all of the experimental measurements reported, the dimensionless mound volume $\overline{V}$ increases linearly with time. For each value of Ev in the simulations, two values of Ha are shown for comparison. Simulation data are shown by circles for low Ha and by crosses for high Ha. In the numerical computations, $\overline{V}$ typically increases nonlinearly with time; thus, the symbols represent the finite-difference approximant of $d\overline{V}/d\overline{t}$ at $\overline{t}={\overline{t}}_M$, the inception point of mound formation.