2.1 Measurement of liquid–gas interfacial tension

To measure the instantaneous change of liquid–gas interfacial tension under the effects of surrounding IPA vapour, we use the pendant drop method as schematically shown in figure 1(a). A drop of liquid emerges from the lower end of a vertically situated capillary tube with inner radius $a=0.58~\text{mm}$ . As a liquid, we use deionized water and aqueous glycerine solutions of 30 wt% and 60 wt%. Table 1 lists the physical properties of the liquids used in the experiments. We make the inner wall of the glass tube highly wettable by cleaning it with the piranha solution, i.e. a 3:1 mixture of sulphuric acid and 30 % hydrogen peroxide solution. The shape of the drop is such that the hydrostatic pressure at the bottom of the tube, $\unicode[STIX]{x0394}p_{h}=\unicode[STIX]{x1D70C}gl$ with $\unicode[STIX]{x1D70C}$ being the density of liquid, $g$ the gravitational acceleration and $l$ the height of liquid column, is balanced by the Laplace pressure, i.e. a product of the drop curvature and the liquid–gas interfacial tension $\unicode[STIX]{x1D6FE}_{LG}$ . By measuring the liquid column height and the curvature, we deduce $\unicode[STIX]{x1D6FE}_{LG}$ .

Figure 1.

(a) A schematic of the experimental apparatus to measure the liquid–gas interfacial tension at different concentrations of IPA vapour. (b) Images of a swelling water drop as the IPA vapour concentration increases with time. The distance of the IPA source to the liquid drop, $s$ , and the initial liquid column height, $l$ , are 20 and 35 mm, respectively.

Table 1.

Properties of liquids used in experiments.

The measurement takes place in an acrylic vessel equipped with a shutter at the bottom under which a pool of IPA is contained. The 17 cm high vessel has a square cross-section measuring $10\times 10$ cm $^{2}$ , and is open to the atmosphere at the top. Upon placing the liquid-filled tube above the bath of IPA, we open the shutter to let the IPA vapour diffuse into the air inside the vessel. As the concentration of IPA molecules surrounding the liquid drop increases with time, it swells as shown in figure 1(b), indicating the decrease of $\unicode[STIX]{x1D6FE}_{LG}$ . The shape of the meniscus $z=z(r)$ is related to $\unicode[STIX]{x1D6FE}_{LG}$ via the Young–Laplace equation

(2.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}^{2}z}{\text{d}r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\text{d}z}{\text{d}r}}\left[1+\left({\displaystyle \frac{\text{d}z}{\text{d}r}}\right)^{2}\right]={\displaystyle \frac{\unicode[STIX]{x1D70C}g}{\unicode[STIX]{x1D6FE}_{LG}}}(l+b-l_{J}-z)\left[1+\left(\frac{\text{d}z}{\text{d}r}\right)^{2}\right]^{3/2}, & & \displaystyle\end{eqnarray}$$

where $l_{J}$ is the Jurin height, or the equilibrium capillary rise height of the uncontaminated liquid with the liquid–gas interfacial tension $\unicode[STIX]{x1D6FE}_{0}$ and zero contact angle $l_{J}=2\unicode[STIX]{x1D6FE}_{0}/(a\unicode[STIX]{x1D70C}g)$ , and $b$ is the distance between the tube bottom and the drop apex, as shown in figure 1(a). One can solve (2.1) numerically under such boundary conditions as $z=\text{d}z/\text{d}r=0$ at $r=0$ with an assumed value of $\unicode[STIX]{x1D6FE}_{LG}$ . The computational results are compared with the experimental images of the liquid drop to improve the guess of $\unicode[STIX]{x1D6FE}_{LG}$ , which eventually gives $\unicode[STIX]{x1D6FE}_{LG}$ . We see good agreement of the experimentally found liquid–gas interfacial profile and the numerical result with a correctly assumed $\unicode[STIX]{x1D6FE}_{LG}$ in figure 2(a).

Figure 2.

(a) Meniscus profile of an IPA-contaminated water drop with $\unicode[STIX]{x1D6FE}_{LG}=0.050$  N m $^{-1}$ . The circles and solid line correspond to the experimental results and the solution of the Young–Laplace equation (2.1), respectively. (b) The scaled liquid–gas interfacial tension $\unicode[STIX]{x1D6FE}_{LG}/\unicode[STIX]{x1D6FE}_{0}$ versus time. (c) Re-plotting the data of $\unicode[STIX]{x1D6FE}_{LG}/\unicode[STIX]{x1D6FE}_{0}$ , which correspond to the instantaneous response of surface tension to the IPA vapour in phase I, versus $c/c_{s}$ .

Figure 2(b) shows the measurement results of the temporal evolution of the surface tension coefficient of the liquid drops at different distances from the source, $s$ . The scaled surface tension $\unicode[STIX]{x1D6FE}_{LG}/\unicode[STIX]{x1D6FE}_{0}$ decreases for initial ${\sim}$ 20 s (phase I) and then it plateaus (phase II). The change of the surface tension is related to the IPA vapour concentration in the air, $c$ , which obeys the one-dimensional diffusion equation with the boundary conditions being such that $c=c_{s}$ at the surface of IPA pool ( $s=0$ ) and $c=0$ as $s\rightarrow \infty$ as follows

(2.2) $$\begin{eqnarray}\displaystyle c=c_{s}\text{erfc}\left({\displaystyle \frac{s}{2\sqrt{Dt}}}\right). & & \displaystyle\end{eqnarray}$$

Here, $c_{s}=142~\text{g}~\text{m}^{-3}$ and $D=1\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ (Lugg Reference Lugg1968) are the saturation concentration and the diffusion coefficient of IPA in the air at 25  $^{\circ }$ C, respectively. The surface tension coefficient of water with volumetric mixing of liquid IPA was measured (Hernández-Sánchez et al. Reference Hernández-Sánchez, Eddi and Snoeijer2015), which showed monotonic decrease of the surface tension with the increase of the volumetric composition of IPA. However, considering that $c$ increases monotonically with time in the current experiments, the surface tension data of the volumetric mixture of water and liquid IPA are not likely to explain the plateau in figure 2(b).

We carry out additional experiments to scrutinize the effects of the IPA vapour on the surface tension, in which we suddenly remove the IPA vapour in air by closing the shutter and supplying fresh air in the vessel. When the removal occurs in phase I (at 10 s), the pendant drop almost recovers its original shape but at a lower rate than when $\unicode[STIX]{x1D6FE}_{LG}$ is being lowered, as shown in figure 9(a). This indicates a profound effect of the IPA vapour concentration in the surrounding air on the surface tension of the liquid in phase I.

As discussed in appendix A, in the early stages of the exposure of an aqueous liquid surface to IPA vapour, the adsorption of IPA vapour molecules to the liquid surface is supposed to play a dominant role in altering the surface tension of the liquid. Therefore, it is the IPA vapour concentration in the surrounding air, not the volumetric composition of IPA in the water-IPA mixture, that determines the surface tension of liquid in phase I. Here, phase I refers to the initial period when the surface tension is decreased under a low concentration of IPA vapour in the surrounding air ( $c/c_{s}<0.2$ ). We see that the rather scattered data of $\unicode[STIX]{x1D6FE}_{LG}/\unicode[STIX]{x1D6FE}_{0}$ are collapsed onto a single line when plotted versus $c/c_{s}$ in figure 2(c), implying a dominant effect of the IPA vapour concentration on surface tension in phase I. In the following experiments of dewetting films, fresh surfaces are continually generated, whose instantaneous surface tension under the effects of IPA vapour is assumed to follow the behaviour exhibited in phase I.

Figure 3.

IPA vapour-induced dewetting of water films with the original thickness, $h_{0}$ , of (a) 30  $\unicode[STIX]{x03BC}$ m and (b) $50~\unicode[STIX]{x03BC}\text{m}$ .

2.2 Observations of film dewetting

To observe the dynamics of a liquid film under the effects of IPA vapour, we prepare a highly hydrophilic surface in the following procedure. We immerse a 10 cm diameter silicon wafer in the piranha solution for 1 h. Upon withdrawing the wafer from the solution, we rinse it with deionized water. The wafer is dried with nitrogen gas and then treated with oxygen plasma for 5 min. To obtain a thin film on the circular wafer, we place liquid drops on the wafer and horizontally vibrate it until the entire circular area is covered with a thin film. The resulting film is almost free from interference patterns owing to its flatness as shown in the first images (0 s) of figures 3(a) and 3(b). The film thickness, $h_{0}$ , is deduced from its weight measured by a precision balance (Mettler Toledo XS205). The resolution of the balance is 0.1 mg, which corresponds to the uncertainty range of $\pm 0.01~\unicode[STIX]{x03BC}\text{m}$ in deducing the film thickness. Special care is taken to keep the wafer surface horizontal and free from contamination, failure of which results in notably abnormal film dewetting behaviours including elliptic dry holes and locally pinned contact lines.

A capillary tube is filled with IPA to its Jurin height, $l_{J}\approx 1~\text{cm}$ with $a=0.58~\text{mm}$ . This forms a meniscus of zero curvature at the lower end of the tube, allowing us to neglect the effects of interface curvature on the vapour pressure of IPA (Butt, Graf & Kappl Reference Butt, Graf and Kappl2006). We slowly bring the capillary tube from ${\sim}3~\text{cm}$ to a distance $d$ above the film, which takes approximately $\unicode[STIX]{x1D70F}_{0}=10~\text{s}$ . The time $t$ is measured since the tube reaches the final position at $d$ . The dynamics of the liquid film is imaged by a CMOS camera (Photron SA 1.1) at a frame rate of $60~\text{s}^{-1}$ with a resolution of $896\times 896$ pixels. Figure 3 shows the responses of liquid films to IPA vapour emitted from the capillary tube.

For a film of $h_{0}=30~\unicode[STIX]{x03BC}\text{m}$ , figure 3(a), we observe the reduction of film thickness below the IPA source, indicating the recession of bulk from the centre. The growing region of reduced thickness is enclosed by a dotted line ( $t=0.5$ and 1.4 s). The physical origin of the bulk recession is simple – the high concentration of IPA molecules adsorbed on the water surface in the centre lowers the liquid–gas interfacial tension locally. Then the peripheral area with a relatively high surface tension draws the liquid away from the region of a high IPA concentration, causing the bulk to recede from the centre. The continual decrease of the thickness eventually punctures the film ( $t>2.3~\text{s}$ ). The retraction of a fringe film right next to the dry area is unstable and thus forms fingers ( $t=2.3$ and 4.9 s), which are diminished as the dry hole keeps expanding ( $t>7.6~\text{s}$ ).

Figure 3(b) for a thicker film with $h_{0}=50~\unicode[STIX]{x03BC}\text{m}$ exhibits a narrower area of reduced thickness compared with figure 3(a), which allows us to see a thickness profile of the film from the central dry area to the peripheral bulk area in one image (1.3 and 2.3 s). The image at $t=1.3~\text{s}$ clearly shows that the surrounding bulk area and an inner area developing fingers is separated by an annulus of very small thickness. It is indicated as a thinning region, being thinner than both its inner and outer regions.

Therefore, we schematically draw the thickness profile of the film from the centre to periphery as in figure 4(a). The dry region meets the fringe film at a distance $r_{f}$ from the centre, and the fringe film is surrounded by the thinning region at $r_{t}$ . Thus, $r_{f}$ and $r_{t}$ correspond to the inner radii of the fringe film and the thinning region, respectively. We note that the average radius of the dry hole was used to obtain the value of $r_{f}$ when fingers were developed. The thinning region ends at $r_{b}$ , where the bulk begins. The film thickness reaches $h_{0}$ after bulging to form a ridge of characteristic length $\unicode[STIX]{x1D6EC}$ . Figure 4(b) plots the growth of $r_{f}$ , $r_{t}$ , and $r_{b}$ with the continual supply of IPA vapour from the source.

Figure 4.

(a) Thickness profile of a dewetting film. (b) Temporal evolution of the inner radii of bulk film $r_{b}$ , thinning region $r_{t}$ , and fringe film $r_{f}$ . The original thickness of the water film was $h_{0}=50~\unicode[STIX]{x03BC}\text{m}$ , and the IPA source was $d=2~\text{mm}$ above the film.

3.1 Receding of bulk

We first consider the rate of bulk receding, or the growth of $r_{b}$ . The Reynolds number based on the characteristic receding speed ${\dot{r}}_{b}$ , and the film height $h_{0}$ , $\mathit{Re}=\unicode[STIX]{x1D70C}{\dot{r}}_{b}h_{0}/\unicode[STIX]{x1D707}$ , is less than $1$ for all the cases. Furthermore, the product of Re and the aspect ratio $h_{0}/r_{b}$ is less than $10^{-1}$ , and thus we neglect inertia as compared with viscous forces (Batchelor Reference Batchelor1967). The flow is driven by the difference in the liquid–gas interfacial tensions across the ridge of the characteristic width $\unicode[STIX]{x1D6EC}$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}_{LG}\sim 10^{-2}~\text{N}~\text{m}^{-1}$ . The gravitational effects are negligible because the Bond number $\mathit{Bo}=\unicode[STIX]{x1D70C}gh_{0}^{2}/\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}_{LG}\sim 10^{-2}$ . The balance of the driving interfacial forces and the resisting viscous forces gives the receding rate of the bulk, or $r_{b}(t)$ .

Because $h_{0}\ll r_{b}$ for most of the dewetting process, we ignore the curvature effects and consider a simple force balance for a dotted control volume in figure 4(a). The liquid–gas interfacial tension in the periphery beyond the ridge is assumed to be close to that of the uncontaminated liquid, $\unicode[STIX]{x1D6FE}_{0}$ , because of continual replenishing flow in the bulk. The liquid–gas interfacial tension at the outer rim of the thinning region, $r=r_{b}$ , is lower than $\unicode[STIX]{x1D6FE}_{0}$ because the IPA molecules adsorbed on the thinning region are hardly advected to the surrounding for the small thickness of the region. The surface tension coefficient $\unicode[STIX]{x1D6FE}_{LG}$ varies with the distance from the IPA source and time, and thus the driving force $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}-\unicode[STIX]{x1D6FE}_{LG}$ changes as the bulk recedes, which we evaluate in the following paragraph.

As $\unicode[STIX]{x1D6FE}_{LG}$ is a function of IPA concentration $c$ as shown in figure 2(b), we first consider $c$ on a film at a distance $(r_{b}^{2}+d^{2})^{1/2}$ from the IPA source. Adopting the solution for diffusion in the spherical coordinates from a source of radius $a$ (Carslaw & Jaeger Reference Carslaw and Jaeger1959), we estimate $c/c_{s}\approx (a/r_{b})\text{erfc}(1/2r_{b}/\sqrt{D\unicode[STIX]{x1D70F}})$ , where we used $(r_{b}^{2}+d^{2})^{1/2}\approx r_{b}$ for $d\ll r_{b}$ . Here $\unicode[STIX]{x1D70F}\approx \unicode[STIX]{x1D70F}_{0}+t$ is the duration the water film is exposed to IPA, as explained in § 2.2. We have $\unicode[STIX]{x1D70F}\gg 1/4r_{b}^{2}/D\sim 1$  s, and thus $c/c_{s}\approx a/r_{b}$ . When $c/c_{s}$ is small for $a/r_{b}\ll 1$ , the data in figure 2(c) allows us to approximate $\unicode[STIX]{x1D6FE}_{LG}/\unicode[STIX]{x1D6FE}_{0}\approx 1-\unicode[STIX]{x1D709}c/c_{s}$ with $\unicode[STIX]{x1D709}\approx 1$ . Therefore, the liquid–gas interfacial tension at $r_{b}$ is scaled as $\unicode[STIX]{x1D6FE}_{LG}/\unicode[STIX]{x1D6FE}_{0}\sim 1-a/r_{b}$ , which gives the driving force per unit depth $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}\sim \unicode[STIX]{x1D6FE}_{0}a/r_{b}$ .

The viscous resisting force per unit depth, $F_{r}$ , associated with flows in the ridge can be estimated using the well known inertialess wedge flow assumption (de Gennes Reference de Gennes1985; Redon et al. Reference Redon, Brochard-Wyart and Rondelez1991; Kim, Lee & Kang Reference Kim, Lee and Kang2002; Kim Reference Kim2007). By solving the Stokes equation and integrating the wall shear stress in the ridge, one obtains $F_{r}\sim \unicode[STIX]{x1D707}{\dot{r}}_{b}\ln (\unicode[STIX]{x1D6EC}/h_{t})/\unicode[STIX]{x1D703}_{b}$ , where $h_{t}$ is the thickness of the thinning region and $\unicode[STIX]{x1D703}_{b}$ is taken to be a constant of 0.1 as optically measured in separate experiments. The interference patterns in figure 3 allow us to estimate the film thickness in the thinning region (Isenberg Reference Isenberg1992; van der Veen et al. Reference van der Veen, Tran, Lohse and Sun2012), which ranges from 50 to 200 nm. Thus, we take $10^{2}~\text{nm}$ as a characteristic scale of $h_{t}$ , a value consistent with the thickness of a thinning region that appears when a surfactant spreads on a liquid film (Jensen & Grotberg Reference Jensen and Grotberg1992).

Balancing the driving force $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}$ and the resisting force $F_{r}$ leads to a scaling law for $r_{b}$ given by

(3.1) $$\begin{eqnarray}\displaystyle r_{b}\sim \sqrt{\frac{\unicode[STIX]{x1D6FE}_{0}a\unicode[STIX]{x1D703}_{b}}{\unicode[STIX]{x1D707}\ln (\unicode[STIX]{x1D6EC}/h_{t})}}t^{1/2}. & & \displaystyle\end{eqnarray}$$

We compare the theory with our experimental measurement results of $r_{b}$ for different liquids, original film thickness and source locations in figure 5. Here we take $\unicode[STIX]{x1D6EC}=3~\text{mm}$ , an empirical value which hardly changes in the course of dewetting regardless of experimental conditions. Slight variations of $\unicode[STIX]{x1D6EC}/h_{t}$ have negligible effects on its logarithmic value and the scaling result. We see that the raw data, which are scattered in figure 5(a), are collapsed onto a single line when plotted according to (3.1) in figure 5(b). The slope of the best fitting line is 0.36. We note that the bulk rim was also reported to recede with its inner radius growing like $t^{1/2}$ when a liquid alcohol, instead of vapour, was continuously supplied at a localized spot on a water film (Hernández-Sánchez et al. Reference Hernández-Sánchez, Eddi and Snoeijer2015).

Figure 5.

(a) Experimentally measured inner radius of bulk film $r_{b}$ versus time $t$ . (b) $r_{b}$ plotted according to the scaling law (3.1). (c) Experimental conditions for the symbols.

3.2 Receding of fringe film

The inner radius of the fringe film, $r_{f}$ , matters when the centre is completely dried to result in hole nucleation. The annular fringe spacings of interference patterns, which are evident in figure 3, allow us to construct the thickness profile near the contact line as shown in figure 6. Figure 6(a) shows that the locations of the contact line, or $r_{f}$ , differ even at the same $t$ depending on the distance of the source $d$ . It is because of a delay time, $t_{d}$ , before the hole nucleates, which is a function of $d$ : $t_{d}=0.2$ and 1.0 s for $d=1.6$ and 3.3 mm, respectively. When we plot the thickness profiles under different source distances $d$ at the same time after nucleation, $t-t_{d}$ , they coincide as shown in figure 6(b). The slopes of the trend lines in figures 6(a) and 6(b) give the receding contact angles of the fringe film, $\unicode[STIX]{x1D703}_{f}$ , as a function of $r_{f}$ . Figure 6(c) shows that $\unicode[STIX]{x1D703}_{f}$ is almost constant except for the extremely early stages of the hole nucleation.

Figure 6.

(a) Experimentally measured thickness profiles of the fringe film at $t=1.8~\text{s}$ under different source distances, $d=1.6$ (circles) and 3.3 mm (squares). (b) Thickness profiles of the dewetting fringe film. Circles, squares and triangles correspond to the measured height at $t-t_{d}=2.2$ , 5.0 and 7.4 s, respectively. The filled and empty symbols are for $d=1.6$ and 3.3 mm, respectively. (c) The receding contact angle of the fringe film $\unicode[STIX]{x1D703}_{f}$ versus $r_{f}$ . The liquid is water and the original film thickness $h_{0}=30~\unicode[STIX]{x03BC}\text{m}$ .

The growth of the hole can be driven either by the surface tension effects or the evaporation. We first check whether the Marangoni effects that worked for the aforementioned bulk receding play an important role. Balancing the difference of the liquid–gas interfacial tension across the fringe film (from $r_{f}$ to $r_{t}$ ), $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}\sim 10^{-2}~\text{N}~\text{m}^{-1}$ , with the viscous resisting force in the wedge-like fringe film per unit depth, $F_{r}\sim \unicode[STIX]{x1D707}{\dot{r}}_{f}\ln [(r_{t}-r_{f})/\unicode[STIX]{x1D706}]/\unicode[STIX]{x1D703}_{f}$ , gives ${\dot{r}}_{f}\sim \unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D703}_{f}/\{\unicode[STIX]{x1D707}\ln [(r_{t}-r_{f})/\unicode[STIX]{x1D706}]\}$ . Here, $\unicode[STIX]{x1D706}$ is the cut-off distance of molecular length scale, which is introduced to relieve the singularity at the contact line (de Gennes Reference de Gennes1985). Considering that $\unicode[STIX]{x1D706}$ ranges from 1 to 10 nm, typical for highly wettable smooth surfaces (Bonaccurso, Kappl & Butt Reference Bonaccurso, Kappl and Butt2002; Cheng & Giordano Reference Cheng and Giordano2002; Joseph & Tabeling Reference Joseph and Tabeling2005), and that $r_{t}-r_{f}\sim 1~\text{mm}$ , we find ${\dot{r}}_{f}\sim 10^{-5}~\text{m}~\text{s}^{-1}$ , which is much lower than the observed dewetting velocity of the fringe film ( ${\sim}10^{-3}~\text{m}~\text{s}^{-1}$ ).

We thus turn to the effects of evaporation on the dewetting of fringe film. The steady diffusion equation from the liquid interface to the air gives the volume evaporation rate per unit area near the contact line, $J=J_{0}x^{-(\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D703}_{f})/(\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{f})}$ , where $x$ is the distance from the contact line, $x=r-r_{f}$ (Deegan et al. Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten1997). For $\unicode[STIX]{x1D703}_{f}\ll 1$ , we get $J\approx J_{0}/\sqrt{x}$ with $J_{0}$ , a function of the diffusivity and the saturation concentration of the liquid molecules in the air and a typical length scale of the fringe film, being estimated to be $J_{0}\sim 10^{-9}~\text{m}^{3/2}~\text{s}^{-1}$ (Berteloot et al. Reference Berteloot, Pham, Daerr, Lequeux and Limat2008). Balancing the rate of volume evaporation per unit depth with the rate of wedge disappearance, we write $\int _{0}^{r_{f}}J_{0}/\sqrt{x}\,\text{d}x\sim \text{d}(r_{f}^{2}\unicode[STIX]{x1D703}_{f})/\text{d}t$ , which leads to $\text{d}r_{f}/\text{d}t\sim (J_{0}/\unicode[STIX]{x1D703}_{f})r_{f}^{-1/2}$ . Because the radius of a hole increases from zero to $r_{f}$ for the duration of $t-t_{d}$ , we finally obtain

(3.2) $$\begin{eqnarray}\displaystyle r_{f}\sim \left({\displaystyle \frac{J_{0}}{\unicode[STIX]{x1D703}_{f}}}\right)^{2/3}(t-t_{d})^{2/3}. & & \displaystyle\end{eqnarray}$$

With typical measured values of $\unicode[STIX]{x1D703}_{f}\sim 10^{-4}$ and $r_{f}\sim 10^{-3}~\text{m}$ , ${\dot{r}}_{f}$ is estimated to be of the order of $10^{-3}~\text{m}~\text{s}^{-1}$ , consistent with the experimental results shown in figure 7(a). When plotting the scattered raw data obtained for different original film thickness and source distances according to the scaling law (3.2), they are collapsed onto a single line as shown in figure 7(b). Therefore, we see that the dewetting of fringe film is caused primarily by evaporation rather than the Marangoni effects. A similar power law for the receding distance of the contact line, which depends on $t^{2/3}$ , was reported previously when a sessile drop dries while leaving a band of suspension particles at its periphery (Rio et al. Reference Rio, Daerr, Lequeux and Limat2006; Berteloot et al. Reference Berteloot, Hoang, Daerr, Kavehpour, Lequeux and Limat2012).

Figure 7.

(a) Experimentally measured inner radius of the fringe film $r_{f}$ versus time $t$ . (b) $r_{f}$ plotted according to the scaling law (3.2). The slope of the best fitting line is 3.4. (c) Experimental conditions for the water films.

3.3 Creation of thinning region

The thinning region, located between the IPA-rich fringe film and the IPA-poor bulk film, has a very small thickness as shown in figure 4(a). Figure 8(a) shows that the experimentally measured onset radius of the thinning region, $r_{t,0}$ , decreases with $h_{0}$ . That is, the larger the original film thickness, the earlier the thinning region appears. Furthermore, we see in figure 8(a) that it emerges earlier for a less viscous liquid (liquid A, filled symbols). The film thickness is reduced in the thinning region because too much liquid is drawn into the bulk. We attribute this flow to excessively large Marangoni forces along the same line as Gaver & Grotberg (Reference Gaver and Grotberg1990, Reference Gaver and Grotberg1992) and Warner et al. (Reference Warner, Craster and Matar2004), which considered a thinning region arising from spreading of a surfactant drop on a thin liquid film.

Figure 8.

(a) Experimentally measured inner radius of thinning region at its onset, $r_{t,0}$ , versus the original film thickness $h_{0}$ . (b) Measurement values of $r_{t,0}$ plotted according to the scaling law (3.3). The slope of the best fitting line is 1200. Filled and empty symbols correspond to liquid A and B, respectively.

When the thinning region emerges, the imbalance of IPA concentrations of the liquid surface within and beyond the radius $r_{t,0}$ occurs, which allows us to obtain a scaling law for $r_{t,0}$ as a function of $h_{0}$ and the material properties. IPA molecules are supplied from the source by diffusion in the air, and the IPA concentration on the liquid surface at the radius $r$ is estimated as $c/c_{s}\approx a/r$ in § 3.1. The supply rate of IPA to the liquid surface by diffusion, or the diffusion velocity $U_{d}$ is scaled as $U_{d}\sim -D\unicode[STIX]{x2202}(c/c_{s})/\unicode[STIX]{x2202}r\sim Da/r^{2}$ . Those IPA molecules are rapidly carried away by advection into the bulk region, whose velocity $U_{a}$ is given by the balance between the Marangoni stress and the viscous shear stress, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6EC}\sim \unicode[STIX]{x1D707}U_{a}/h_{0}$ . Here, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}$ corresponds to the difference of the liquid–gas interfacial tension across the ridge over which the driving force for advection acts: $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}\sim \unicode[STIX]{x1D6FE}_{0}a/r$ as derived in § 3.1. Thus, $U_{a}$ is scaled as $U_{a}\sim \unicode[STIX]{x1D6FE}_{0}ah_{0}/(\unicode[STIX]{x1D707}r\unicode[STIX]{x1D6EC})$ .

The thinning region occurs where $U_{a}$ overwhelms $U_{d}$ , so that we scale the onset radius of thinning region as

(3.3) $$\begin{eqnarray}\displaystyle r_{t,0}\sim {\displaystyle \frac{D\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D6FE}_{0}}}{\displaystyle \frac{\unicode[STIX]{x1D6EC}}{h_{0}}}. & & \displaystyle\end{eqnarray}$$

Figure 8 shows that the scattered raw data of the onset radius of the thinning region in (a) are collapsed onto a single line in (b) when plotted according to the scaling law (3.3). It implies that for greater $h_{0}$ and lower $\unicode[STIX]{x1D707}$ , advection of IPA becomes stronger as compared with diffusion, resulting in an earlier appearance of the thinning region. The slope of the best fitting line in figure 8(b) corresponds to the Péclet number, $\mathit{Pe}=U_{a}/U_{d}=1200$ , a ratio of IPA advection rate in liquid to IPA diffusion rate in air.