Long-time evolution of interfacial structure of partial wetting

When a solid plate is withdrawn from a partially wetting liquid, a liquid layer dewets the moving substrate. High-speed imaging reveals alternating thin and thick regions in the entrained layer in the transverse direction at steady state. This paper systematically compares this situation to the reversed process, forced wetting, where a solid entrains an air layer along its surface as it is pushed into a liquid. To quantify the absolute thickness of these steady-state structures precisely, I have developed an optical technique, taking advantage of the angle dependence of interference, combined with a method based on a maximum-likelihood estimation. The data show that the thicknesses of both regions of the film scale with the capillary number, $\mathrm{Ca}$. Further, an additional region is observed during onset, the quantitative explanation of which requires future investigation.

Figure 1

(a) Schematic of forced wetting. A film of air is entrained into a liquid bath by the surface of a moving substrate. Inset A: Front view of the steady state of an entrained air film in forced wetting. A mylar tape of width $w=12.7$ mm plunges vertically into a water-glycerin mixture of viscosity ${\eta }_{\mathrm{out}}=226$ cP at $U=130$ mm/s. The entrained air layer assumes a V shape with two thin, flat sections at the upper corners. (b) Schematic of dewetting. A film of liquid is dragged out from a liquid bath by the surface of a moving substrate. Inset B: Front view of the steady state of a liquid film in dewetting. An acrylic plate of width $w=20.3$ mm is pulled vertically out of a water-glycerin bath of viscosity $\eta =100$ cP at $U=4.4$ mm/s. An entrained liquid layer forms an upside-down V shape with two thin sections at the lower corners.

Figure 2

(a) Measured liquid-air interfacial tension $\gamma$ versus glycerin mass fraction $M_g$ of water-glycerin mixtures. Solid line, prediction of Eq. (1). (b) Receding and advancing contact angles on an acrylic substrate (Rain-${\mathrm{X}}^{\text{®}}$ coated) versus $M_g$. At each value of $M_g$, the data are displaced slightly in the $x$ direction to make overlapping symbols visible. Dashed line, linear fit for receding contact angles.

Figure 3

Images of a typical evolution of a dewetting liquid layer, for a duration of $\approx 40$ s. An acrylic plate of width $w=20.3$ mm is pulled vertically out of a water-glycerin bath of viscosity $\eta =100$ cP at $U=4.4$ mm/s. The fluid layer reaches its steady state in (d). (The images in (a)–(d) are not evenly spaced in time in order to show the different stages clearly.)

Figure 4

Increasing substrate width $w$ increases the number of thin-thick alternations in film thickness. (a) A 25.4-mm-wide recording tape entrains an air film with three thick sections. The upper portion of the thin-thick boundary can be approximated by a circle of radius $R$. (b) A 35.3-mm-wide acrylic plate entrains a water-glycerin film with two thick sections with width $W$. In (a) and (b) the substrate widths $w$ are twice what they were in the corresponding images shown in Figs. 1 and 1. Scale bar: 5 mm.

Figure 5

Geometry of dewetting films. (a) Width of a thick section, $W$, versus substrate width, $w$, for water dewetting a mylar surface. (b) Number of thick parts, $N$, versus $w$, for water dewetting a mylar surface. Error bars indicate different thin-thick configurations observed, for a given $w$. Dashed line, linear fit to the mean observation.

Figure 6

Geometry of forced-wetting films. Radius of curvature, $R$, of the upper portion of the thick part is plotted against liquid viscosity ${\eta }_{\mathrm{out}}$. Dashed line, capillary length $l_c=\sqrt{\gamma /\mathrm{\Delta }\rho g}$. Inset: Superimposed edge-detected images of two different air films, entrained into different liquids of viscosities ${\eta }_{\mathrm{out}}=160$ and 290 cP, respectively, at $U=150$ mm/s.

Figure 7

Contact-line inclination and contact-line velocity in dewetting. (a) Evolution of the inclination angle $\varphi$ (relative to the horizontal direction) of the lateral contact lines versus time $t$ for seven different runs under the same condition. Plate velocity $U=4.4$ mm/s. Liquid viscosity $\eta =100$ cP. ${(cos\varphi )}^{-1}$ versus $U$ for (b) onset (${\varphi }_{\mathrm{on}}$, circles) and (c) steady state (${\varphi }_{\mathrm{ss}}$, squares). Solid lines, linear fits to each data set ($\text{slope}=1$). (d) Fitted normal relative velocity, $U_{n,\mathrm{fit}}$, versus liquid viscosity $\eta$ for both onset (circles) and steady state (squares). Top solid line, $U_{n,\mathrm{fit}}\propto {\eta }^{-0.55}$; bottom solid line, $U_{n,\mathrm{fit}}\propto {\eta }^{-0.59}$. Legend is consistent for (b), (c), and (d).

Figure 8

Data collapsing for dewetting using Eq. (5). Only data from the linear regimes of Fig. 7 are used. Inset: Fitted $\overline{\ln }$ using Eq. (5) versus the glycerin mass fraction $M_g$.

Figure 9

Data collapsing for dewetting using Eq. (6). Legend consistent with Fig. 8. Dotted line, unstable branch of Eq. (6); solid line, stable branch.

Figure 10

Forced wetting data plotted as ${\mathrm{Ca}}_n$ versus $ln{\eta }_{\mathrm{out}}$. Solid line: a least-squares fit shows a good linear relation.

Figure 11

Normal relative velocity, $U_n$, of contact line during formation of a V. Color scale: $U_n$ deduced from Eq. (9). (a) Contact-line positions at intervals of 100 ms, for an acrylic moving out of a water-glycerin mixture of viscosity $\eta =18$ cP at $U=15.8$ mm/s. (b) Contact-line positions at intervals of 50 ms, for a mylar moving into a water-glycerin mixture of ${\eta }_{\mathrm{out}}=226$ cP at $U=130$ mm/s.

Figure 12

Thickness fitting by likelihood maximization. An acrylic substrate travels out of a liquid of viscosity $\eta =18\mathrm{cP}$ at $U=17$ mm/s. Left column is an image of the entrained fluid. The arrows pointing to focusing spots indicate location of the measurement. Middle column: Corresponding interference fringes (Haidinger's fringes; see Appendix pp1 and Ref. [49]) captured by high-speed camera. Right column: Reconstructions of fringes using maximizing likelihood (see Appendix pp2), giving thicknesses of $h=84.2$ $\mu \mathrm{m}$ for the thin region (top row) and $285.0\mu \mathrm{m}$ for the thick region (bottom row).

Figure 13

(a) Thickness $h$ for steady state of dewetting, versus substrate width $w$. Markers connected by solid lines, thin part; markers connected by dashed lines, thick part. Liquid viscosity $\eta =41$ cP. (b) A typical configuration of the entrained liquid layer on a wide substrate. (c) Zoomed-in view of the indicated region (red square) in (b). Sample illuminated by a sodium-vapor lamp (wavelength $\lambda =589$ nm). Interference stripes in the thin regions indicate the flatness.

Figure 14

Thickness for dewetting layers normalized by capillary length, $h/l_c$, versus capillary number $\mathrm{Ca}$, for both the thin parts and the thick parts. Top solid line, $h/l_c\propto {\mathrm{Ca}}^{0.46}$; bottom solid line, $h/l_c\propto {\mathrm{Ca}}^{0.48}$; substrate width, $w=17.1$ mm.

Figure 15

Left: In forced wetting, the thickness variation of the entrained layer is a perturbation to the dominant bulk liquid, resulting in a roughly equal interface velocity for the thin part $U_{I;\mathrm{thin}}$ and the thick part $U_{I;\mathrm{thick}}$. Right: In dewetting, the thickness variation of the entrained layer is expected to break the uniformity of the interface velocity, making $U_{I;\mathrm{thin}}\ne U_{I;\mathrm{thick}}$ in general.

Figure 16

(a) Fringe fitting using likelihood maximization in forced wetting. Left: Interference fringes of equal height produced by part of an air film (Newton's fringes). Middle: Fringes reconstructed by maximum-likelihood fitting. Frame partitioned into four smaller parts for separate fitting. Right: A typical local edge detection algorithm for comparison. (b) Topography of an air film reconstructed. Peak thickness, $87\mu \mathrm{m}$; thin flat part, $3\mu \mathrm{m}$. Absolute thickness obtained by using multiwavelength interference discussed in Ref. [1].

Figure 17

(a) Schematic showing the formation of a film of intermediate thickness (red) on a wide plate. (b) Measured thickness scaled by capillary length, $h/l_c$, versus $\mathrm{Ca}$. Top solid line, $h/l_c\propto {\mathrm{Ca}}^{0.63}$; bottom solid line, $h/l_c\propto {\mathrm{Ca}}^{0.59}$. Substrate width $w=37.7$ mm. Inset: Typical image of onset film, showing the ridge, the intermediate region, and the thin region.

Figure 18

Schematic of interference produced by a beam with angle of incidence ${\theta }_i$ onto a sample of thickness $h$. Light reflected from top surface at ${\theta }_i$ and bottom surface at ${\theta }_j$ interferes at the focal plane of a lens placed above.