# Spreading of soap bubbles on dry and wet surfaces We report on the spreading of soap bubbles made from the three solutions (cf. Table 1) on four different glass plates. Figure 2 depicts exemplary three snapshots of the spreading dynamics once the soap bubble makes contact with the substrate. For this particular example the surface is coated with indium tin oxide (ITO). At time of contact, (t=0), between the soap bubble and the substrate we see a small spot. This spot at (t=0.25,hbox {ms}) has expanded into a ring of (approx 0.2,hbox {mm}) in diameter and keeps its nearly perfect round shape when growing to (approx 2.2,hbox {mm}) at (t=0.86,hbox {ms}). Within the ring the image is brighter indicating a thinning of the liquid film, while the dark contrast of the rim is caused by accumulating liquid from the film into a highly curved region. Additionally, there is a second annular ring traveling with about twice the velocity of the rim indicated with an arrow in Fig. 2 at (t=0.25,hbox {ms}). We identify this as a capillary wave originating from the initial point of contact. Interestingly, the capillary wave is traveling faster upwards than downwards. This behavior is explained with a thickness dependent capillary speed: due to gravity driven drainage the soap film is thinner above the initial point of contact than below, see Fig. 1.

Next we focus on the velocity of the rim spreading on the glass substrate. Figure 3 is an artificial streak image generated from horizontal lines taken from consecutive images of the high-speed recordings. The pixel values are taken from the line indicated in Fig. 2 ((t=0.25,hbox {ms})) just before contact until (t=11,hbox {ms}). The initial radius of the soap bubble is (5.0pm 0.1) mm. The tangent of the (x-t) line is the velocity of the spreading rim. After contact at (t=0) the rim grows with a constant velocity (v=1.6pm 0.2) m/s, which is indicated by the red line. After about (t_0approx 0.2pm 0.05,hbox {ms}) the rim velocity decelerates and eventually approaches 0. That is the moment when the bubble adopts a hemispherical shape on the glass slide, i. e. (2^{1/3}) times the diameter of the original bubble. We call this time (t_0) the cross-over time.

The velocity of the capillary wave (v_c) can be determined from the slope of the dashed line in the (x-t) plot (Fig. 3). The point of contact between bubble and glass plate was selected and another one on the line produced by the capillary wave, where it appears still linear. Between these points a line was calculated to obtain the spatial positions of the capillary wave over time, from which the velocity can be calculated. To verify this method, the position of the capillary wave is tracked in each frame and the velocity is determined from the position over time. Since we only observe a projection of the wave on a plane, the curvature of the bubble needs to be taken into account. The spatial distance y needs to be corrected: (y_{corrected}=r(pi /2-text {arccos}(y/r))), r is the initial radius of the bubble. The horizontal speed of the capillary wave is (3.6pm 0.2) m/s. The knowledge of the velocity of the capillary wave allows us to calculate the film thickness of the bubble. The wave speed of an anti-symmetric capillary wave (v_c) can be calculated as19, 20

begin{aligned} v_c=sqrt{frac{2 sigma }{rho h}}, end{aligned}

(1)

where (sigma ) is the surface tension, (rho ) is the density of the soap solution, and h is the film thickness. Thus, the film thickness of the soap bubble can be calculated from the velocity of the capillary wave in the horizontal direction, which is (5.7pm 0.4,upmu hbox {m}).

From the space–time plot we extract the spatial coordinate of the rim. Let us first compare the measurements with a power law of for the early spreading of a droplet on a dry surface, i. e. (rsim t^{1/2})12. In Fig. 4 the rim radius r is plotted in (a) as a function of t and in (b) as a function of (sqrt{t}). Comparing both graphs, we see that the spreading follows very accurately a (rsim t^{1/2}) dependency, except during the very early time: here (t< 0.8,hbox {ms}).The cross-over time (t_0) is determined from (b), where the first part (blue curve) is fitted with (r=a,t) and the remaining data are fitted with (r=b+csqrt{t}) (a, b and c are fitting parameters). The intersection of both fits gives (t_0).

Next the effect of viscosity is investigated by varying the amount of glycerol added to the solution (cf. Table 1). In Fig. 5 the location of the rim r as a function of time for three different solutions are shown. Additionally, four different glass substrates are used. The soap solutions show different equilibrium contact angles on the different substrates. The values are presented in Table 2. However, these different contact angles have no significant effect on the spreading of the soap bubble. Small deviations in the spreading can be attributed to a varying film thickness of the soap bubble. The soap solution S1 without glycerol and lowest viscosity is spreading the fastest. For S2 and S3 the spreading event lasts longer, which can be seen in Fig. 5. Additionally, the cross-over time when deceleration starts is affected by the viscosity: solution S2 has an cross-over time at (t_0=0.30pm 0.05) ms ((v=0.62pm 0.05) m/s) and solution S3 of (t_0=0.6pm 0.1) ms ((v=0.32pm 0.05) m/s).

The final radius the rim attains is determined by the radius of the bubble before contact. Typically, the rim grows to about the size of the soap bubble just prior contact. As this initial size varies between experimental runs we re-scale the radius of the rim with the initial radius of the bubble (R_0). Accounting for balance of inertia and capillary forces, the time is scaled by the capillary-inertial time (t’=sqrt{rho R_0^3/sigma })  8. Interestingly Fig. 6 reveals that after this rescaling of time and space the (r-t) curves for an particular soap solution collapse into a single curve. With higher viscosity the curves are shifted to the right; thus their rims are slower than that of the pure soap solution (S1). Yet, we observe a considerable spread in the velocity of the radius during the early times. We attribute this to a variation of the film thicknesses of the soap bubbles which is not controlled in the experiments. Here, a thicker film reduces the initial acceleration due to inertia of the forming rim5. Thus thicker films are initially slower than thinner films explaining the spread of velocities.

Next we report and discuss the spreading of soap bubbles on wetted surfaces.Note that only the macroscopic wetting is studied here. The wetting according to the molecular-kinetic theory21 cannot be resolved in the current setup. The wetting is realized with hydrophilic glass slides that are wetted with a glycerol containing solution (S2 or S3). This coating is obtained by spreading a bubble on the dry surface prior to the present experiment. Solution S1 shows dewetting and is therefore not used as a wet coating. Solutions S2 and S3 form a thin stable fluid layer with an almost uniform thickness of approx. 50 (upmu hbox {m}). When a soap bubble is approaching those wetted glass slides, the spreading characteristics are altered considerably as compared to the dry surface. We now observe a rim that grows linear with time.

Figure 7 compares the rim growth for solutions S2 and S3 on a dry and on a wetted hydrophilic surface. The spreading of solutions S2 and S3 on a dry surface are colored in green and yellow, respectively, and the spreading on the wetted surface is colored in red and blue for S2 and S3. Interestingly, for S2 the difference with the non-wetted surface is small until (t=2.5,hbox {ms}). The linear slope of these curves is almost identical for all surfaces: (approx 0.62pm 0.2,hbox {m/s}). The rim growth on the wetted surface remains linear with time before its speed decreases once the radius becomes comparable to the equilibrium radius of the hemispherical bubble. For solution S3, the spreading also remains constant on the wetted surface, the spreading velocity is (0.45pm 0.05) m/s. This, however, is considerably faster than the spreading on a dry surface (i. e. (0.32pm 0.05) m/s) due to the lubricating effect of the liquid layer.

Observing the spreading in a top view, the formation of a second rim can be seen (see arrow in Fig. 8d at (t=0.02)  ms). This rim is thinner and moves a little slower than the leading rim. The latter displays undulations, which occur approx. 1 ms after the formation of the liquid bridge (see inset). In order to understand the origin of the second rim, experiments are performed from the side view (Fig. 8a). Here it can be seen that behind the leading rim a wedge shaped soap film follows. Its tip is marked with a dashed line in (a). Between this line and the leading rim the inner rim appears in the interferometric measurements. Since the outer rim collects more and more fluid from the substrate, the volume of the wedged shaped fluid film increases, such that it becomes wider (the distance between the leading rim and the dashed line in (a) increases). The inclined side view (Fig. 8b) helps to understand the geometry. Here, the outer rim builds up a crest of fluid upstream. These thickness variations are visible under monochromatic illumination, since interference fringes occur in front of the rim (dashed circle in (d)). The entire geometry is sketched in (c) and the positions of the inner and the outer rim are marked. Note that in the side view the glass slide is now oriented horizontally, such that the thin film coating does not run off the glass slide. Hence, the thickness of the fluid film is thicker than in the top view experiments. In the latter, the film thickness on the glass slide lies around 50 (upmu hbox {m}), whereas it is around 200 (upmu hbox {m}) for the side view experiments.

Additionally to the formation of the inner and the outer rim, an instability of the rim was observed in some of the experiments, which is characterized by equally spaced indentations of the outer rim (cf. Fig. 8d). The wavelength of these indentations grow with time due to the radial expansion of the rim. Furthermore, radial bright and dark stripes can be seen downstream of the rim. We hypothesize that these may be the result of an instability during the formation and early expansion of the rim upon splitting.

The rim expands slower in regions with a thicker film. This is upon close inspection where brighter radial wrinkles connect to the rim, see Fig. 8d at (t=1.02) ms. At the endpoints of the darker radial rays the rim has already split into two. If we further assume that the surface curvature of the film refracts the collimated light, we expect brighter regions close to the crests and darker regions at the troughs.

For the troughs the film profile looks like as sketched in Fig. 8c, where the inner rim appears at a certain curvature at the wedge shaped area behind the leading rim. However, when crests are present, the curvature changes which reduces the contrast of the inner rim.