3.1 Film deformation

Figure 2(a) shows the shape evolution of the film with time, where $\tau = t/T$, t = 0 is the instant of the impact and t = T the instant of droplet detachment. It is seen that the soap film deforms immediately to adapt to the shape of the droplet at the instant of impact. As the droplet falls, the film is continuously stretched and the contact line moves rearward in its motion towards the north pole; for convenience in description, the top and bottom extremes of the droplet are termed as the north and south poles, respectively, while the centre horizontal plane is termed the equator, as shown in the inset of figure 2(a) (see supplementary movie 1). The film shrinks and closes when the contact line crosses the equator. Subsequently, the film (1) wraps the droplet, (2) traps some air at the north pole, which eventually leads to (3) pinch-off as the wrapped droplet separates, as shown in figure 2(a). Afterwards, the film shell first coalesces with the droplet at the north pole, then it peels off and finally shoots a bubble from the south pole. In this section, we focus on the evolution of the film shape, while the peeling mechanism will be discussed later.

Figure 2. Shape evolution of the soap film upon passage of the droplet (see supplementary movie 1). (a) Sequential images and the location of the Cartesian coordinate, where $\tau = t/T$, and T denotes the period from impact to detachment of the droplet; T = 6.18 ms in this case. (b) Profile curve evolution of the film, with the embedded chart showing motion of the film evolution relative to the droplet (D ₀ = 3.0 mm, We = 126). (c) Theoretical curve of the film outline in dimensionless form; $\tilde{r} = r/{R_0}$, $\tilde{h} = h/{R_0}$, R ₀ is the droplet radius.

Figure 2(b) quantitatively shows evolution of the outline of the drop–film system in the Cartesian coordinate located at the impact point, as shown in figure 2(a), for t = 3 ms. It is seen that the film undergoes continuous and complex deformation. However, when the origin of the coordinate is moved to the south pole (see the inset in figure 2b), the outlines at various time intervals then collapse at the bottom part where the film wraps the droplet and the free surface of the film becomes a series of parallel curves. Because the film's initial perturbation at h = 0 propagates out in the r-direction in the shape of a circle and the contact line is also a circle, the film can be regarded as being attached to two circular frames with varying radii and increasing distances. The surface formed by the revolution of a catenary is well known as the catenoid (Bliss Reference Bliss1935; Taylor & Michael Reference Taylor and Michael1973; Landau & Lifshitz Reference Landau and Lifshitz2008). Assuming the film is in equilibrium and neglecting its inertia, the catenary of the film shape is $\tilde{r} = \textrm{cosh(}\tilde{h}\textrm{)}$, shown as the black curve in figure 2(c), where $\tilde{r} = r/{R_0}$, $\tilde{h} = h/{R_0}$ and R ₀ denotes the droplet radius.

Non-dimensionalizing all the data in figure 2(b), treating the point at the minimum radius of the free film as $(\tilde{r},\tilde{h}) = (1,0)$ and then replotting the experimental data in figure 2(c), the data are found to collapse onto one curve. However, the curve deviates slightly from the catenary $\tilde{r} = \textrm{cosh(}\tilde{h}\textrm{)}$, which implies that the film has not reached the equilibrium state yet but it is still under dynamic stretch. By using a modified expression for the catenary

(3.1)\begin{equation}\tilde{r} = a[\cosh (\tilde{h}/a) - 1] + 1,\end{equation}

the data then coincide with the curve with constant a = 1.15. Note that it is the equilibrium catenary when a = 1.

3.2 Droplet–film detachment

We next inquire when does the droplet detach from the film, i.e. the soap edges meet above the droplet to engulf it, as shown in figure 2(a), at t = 6 ms. Identifying this event is challenging owing to its multiple degrees of freedom, because the radii of the two rings, namely the contact ring/line on the droplet and the propagation ring of the surface wave, as well as the distance between them, are all in dynamic state. Kim & Wu (Reference Kim and Wu2010) found the detachment height of micron-sized droplets (${h_{det}}$, defined in figure 2(a) at t = 6 ms) was approximately 4R ₀, while Zou experimentally found ${h_{det}} \approx 6{R_0}$ for millimetre-sized droplets. We shall now develop a theoretical explanation of Zou's result based on the modified catenary.

Figures 2(a) and 2(b) show that the droplet–film interaction process consists of two stages, defined by three instantaneous locations of P_ff,min, namely the point of minimum radius on the free film (R_ff,min), as shown in figure 3(a). Here Position I is the onset of the impact when the droplet contacts the film, Position II is the instant when the free film is tangent to the droplet at the equator and Position III refers to the pinch-off of the film when the free film-surface downstream of the droplet contracts to meet at the axis. The contact angles $\alpha $ (defined in figure 3a) are 0°, 90° and a critical value at Positions I, II and III, respectively.

Figure 3. (a) Three critical positions of the drop–film interaction process. (b) Trajectory of the point P_ff,min with the minimum radius of the free film (R_ff,min).

The detachment height, thus, can be divided into two parts by the pinch-off point B, as shown in figure 3(a): the lower height (nondimensionalized by R ₀) is ${\tilde{h}_3}$; the upper height is ${\tilde{h}_1} + {\tilde{h}_2}$, where ${\tilde{h}_2}$ is the height during shrinkage of the film tunnel, and ${\tilde{h}_1}$ is the height between the wire ring and the equator at Position II. It is noted that ${\tilde{h}_1}$, ${\tilde{h}_2}$, ${\tilde{h}_3}$ and $\; {\tilde{h}_{det}}$ are the absolute distances between the corresponding locations and, as such, they are always positive.

In Stage I (from Position I to II), R_ff,min increases until R_ff,min = R ₀ at the equator when $\alpha = 90\mathrm{^\circ }$. However, owing to deformation of the droplet at impact, this change occurs at the early stage of impact and R_ff,min almost reaches R ₀ (or $\alpha $ becomes 90°) immediately when the equator is visible (figure 2a), and ${\tilde{R}_{ff,min}} = {R_{ff,min}}/{R_0}$ remains at unity till Position II is reached, shown as the vertical part of the curves in figure 3(b). Here we note that for an ideal case with a soap film supported by the two coaxial rings with equal radius R, the maximum attainable distance between the two rings is 2 × 0.663R = 1.33R (Taylor & Michael Reference Taylor and Michael1973; Isenberg Reference Isenberg1992; Landau & Lifshitz Reference Landau and Lifshitz2008; Salkin et al. Reference Salkin, Schmit, Panizza and Courbin2014) as shown in figure 4, beyond which the catenoid collapses to form two separate planar circular films attached to the two rings, respectively. Thus, the critical values of $\tilde{r}$ and $\tilde{h}$ in figure 2(c) should be located on the line $\tilde{h} = 0.663\tilde{r}$. This line is the tangent of the series of a catenary family with the same ratio of the ring distance to the ring radius (Bliss Reference Bliss1935). Because it intersects with the modified catenary applicable for the current problem at $\tilde{h} = 1.91$ (shown in figure 2c), the theoretically expected value for ${\tilde{h}_1}$ is 1.91.

Figure 4. Schematic of the shrinkage of a soap film along the critical line with decreasing radius of the rings.

In Stage II (from Position II to III), although the distance between the two rings grows and the radius of the upper ring R ₁ also increases owing to propagation of the perturbation in the film, the radius of the lower ring (attached to the north hemisphere of the droplet) R ₂ (segment CD in figure 3a) decreases through ${R_2} = {R_0}\sin \alpha $. Thus, the soap cavity above the droplet may shrink stably rather than collapses, the point P_ff,min moves to point O (see figure 4), where the film surface shrinks and the droplet detaches from the film. The shrinkage of the cavity through R ₂ leads to an interesting phenomenon, in that point P_ff,min moves theoretically along the surface of the south hemisphere of the droplet at position II, depicted as the quarter circle ${(\tilde{h} + 2)^2} + {\tilde{r}^2} = 1$ in figure 3(b). Thus, the shrinkage height (from point A to B in figure 3a) is ${\tilde{h}_2} \approx 1$, while the theoretically expected upper detachment height is ${\tilde{h}_1} + {\tilde{h}_2} \approx 2.91$. This is indeed close to the experimentally measured values irrespective of the film/frame radii (R_f), as compared in figure 5(a). It is noted that the dimensionless propagation diameter of the soap film surface at the instant of the detachment is 4.58, which is the maximum film diameter that can influence the detachment height. Because this value is almost half of the minimum diameter of the copper ring tested in this study, ${\tilde{D}_{f,min}} = {D_{f,min}}/{D_0} = 8.33$, (see the legends in figure 5a), the film size in this study has no detectable effect on the threshold distance.

Figure 5. Comparison of experimental data with theoretical predictions of (a) total, (b) lower, (c) upper detachment heights and (d) current work and work of Zou et al (Reference Zou, Wang, Ji and Pan2017). Each dot in panel (d) consists of at least 8 cases and the error bars are ±5 %. Here, ${D_f}$ is the diameter of the frame (copper wire ring). Note that ${\tilde{h}_1}$, ${\tilde{h}_2}$, ${\tilde{h}_3}$ and ${\tilde{h}_{det}}$ are absolute distances and hence they are always positive.

Furthermore, when ${U_0} < 2.0\;\textrm{m}\;{\textrm{s}^{ - 1}}$, the trajectory of P_ff,min departs marginally from the theoretical quarter circle owing to oscillation of the droplet location. In particular, at ${U_0} = 0.75\;\textrm{m}\;{\textrm{s}^{ - 1}}$, large droplet oscillation causes P_ff,min to drop to the position $\tilde{h} ={-} 3.7$ and then it increases as the droplet rebounds to create a figure-of-eight-shaped trajectory, as shown in the inset of figure 3(b). Smaller U ₀ leads to larger departure owing to the longer contact time, which will be discussed in the next section.

Figure 3(a) shows the lower detachment height ${\tilde{h}_3} = {\widetilde {BO}_{III}} + 1 = \widetilde {BD} - \cos \alpha + 1$, where $\widetilde {BD}$ can be obtained by the evolution equations of the tangency of the catenary and the unit circle. At the point of detachment, the three geometric conditions listed below must be satisfied:

(3.2) \begin{equation}\left. {\begin{gathered} {\widetilde {CD} = a\left[ {\cosh \left( {\dfrac{{\widetilde {BD}}}{a}} \right) - 1} \right]\; \; \; \; \; }\\ 1\\ {\widetilde {CD} = {R_o}\sin \alpha \; \; \; \; \; \; \; \; \; \; \; \; \; \; }\\ {\; \; \dfrac{{\textrm{d}\tilde{r}}}{{\textrm{d}\tilde{h}}} = a \sinh \left( {\dfrac{{\widetilde {BD}}}{a}} \right) = \tan ({\rm \pi} - \alpha )} \end{gathered}} \right\}.\end{equation}

By solving this equation set, one obtains $\widetilde {CD} = 0.6520$, $\widetilde {BD} = 1.1351$ and $\alpha = 130.4\mathrm{^\circ }$. Thus, ${\tilde{h}_3} = \widetilde {BD} - \cos \alpha + 1 = 2.89$. Consequently, the theoretical detachment height is given by ${\tilde{h}_{det}} = {\tilde{h}_1} + {\tilde{h}_2} + {\tilde{h}_3} = 5.80$. The experimental data, including those in Zou, confirm these predictions, as illustrated in figure 5(b–d), which demonstrate that the droplet–film detachment is determined by the droplet geometry rather than the impact Weber number of the droplet and, as such, the dynamic process can be regarded as quasi-static.

Because the stretched section of the soap film is fed by the free section, the modified catenoid moves down almost in parallel (see the embedded figure in figure 2b). Consequently, stretch of the film can be neglected, at least in the vertical direction. The maximum stretch occurs in the horizontal direction at pinch-off.

Here, we note that the dynamics of detachment for very low- or high-We impacts are different and require special treatments. Although detailed analysis of such regimes is not a focus of this work, the Appendix provides a generic description of the low- and high-We regimes.

Because we have theoretically evaluated the geometry of the catenoid at the point of detachment, the volume of the trapped air in the pocket, V_a, can also be calculated. Noting that the droplet's free surface at the north pole is flat (see CC’ in figure 8) when the pocket is closed in figure 2(a) at t = 5.5 ms to 6.25 ms, V_a is theoretically equal to the volume generated by rotation of the area BCD around the axis O_IIO_III in figure 3(a). Thus,

(3.3)\begin{equation}{V_a} = \int_D^B {{\rm \pi} {{\tilde{r}}^2}\,\textrm{d}\tilde{h}} = {\rm \pi}\int_{ - 1.1351}^0 {{a^2}{{\left[ {\cosh \left( {\frac{{\tilde{h}}}{a}} \right) - 1} \right]}^2}\,\textrm{d}\tilde{h}} = 0.09584{\rm \pi} .\end{equation}

It is shown from the following section that the trapped air is the filler of the ejected bubble. Thus, the estimated inner radius of the bubble is ${\tilde{r}_{b,in}} = {({V_a}/{V_d})^{1/3}} = 0.4158$, where ${V_d} = (4/3){\rm \pi} $ is the volume of the unit sphere. Furthermore, because the relative thickness of the original film is $\tilde{\delta } = \delta /{R_0} = 0.02$, while the film is totally peeled off from the droplet, to be demonstrated later, the film thickness of the ejected bubble $({\tilde{\delta }_b})$ should increase to ${\tilde{\delta }_{bf}} = \tilde{\delta } (4{\rm \pi} \tilde{R}_0^2)/(4{\rm \pi} \tilde{r}_{b,in}^2) = 0.1157$. Consequently, the bubble diameter is ${\tilde{D}_b} = {D_b}/{D_0} = {\tilde{r}_b} = {\tilde{r}_{b,in}} + {\tilde{\delta }_{bf}} = 0.53$, which is consistent with the experimental data, see figure 6. One can also see that the volume ratio of the trapped air in the bubble to the original droplet is ${\tilde{V}_b} = \tilde{D}_{b,in}^3 = 7.2\,\%$.

Figure 6. Comparison of experimental data with theoretical predictions of the diameter of the ejected bubble/trapped air.

3.3. Piercing of air pocket

After the droplet detaches from the soap film, the free film air pocket retracts to the north pole. Meanwhile, a jet is formed at the north pole by the convergence of the capillary wave. It rises, then collides and finally punctures the film. Subsequently, the film shell contracts and swipes down along the droplet surface from the north pole to the south pole. Finally, it is peeled off from the droplet and ejects a bubble as depicted by Zou. However, as shown in figure 7(a) (see supplementary movie 1), the bubble in the present study is not resorbed again by the droplet, as reported in the experiments by Zou. The trajectories of the jet tip and pocket joint in figure 7(b) show that the rising jet tip actually punctures the film when We < 184. Take We = 252 for instance: the tip cannot meet the pocket joint and thus the intact shell is maintained. The experimental result from the present study agrees with the value provided by Zou, where they reported the upper limit of the Weber number for the collision as 220. Here, we provide the theoretical explanation for this critical value.

Figure 7. Peeling of the film shell off the droplet (D ₀ = 3.0 mm, We = 126): (a) sequential evolution images of the pass-through and bubble ejection process (see supplementary movie 1); (b) trajectories of the droplet tip and the pocket joint, $\tilde{t} = t/{t_0}$, ${t_0} = {R_0}/{U_0}$; (c) theoretical value of the dimensionless jet tip height and pocket height.

3.3.1. Height of the jet tip

As shown in figure 7(b), collision of the film juncture B and the jet tip F is triggered by detachment of the droplet from the film at ${\tilde{t}_{det}} = 5.80$, when the trajectory of point B intersects that of point F. The dimensionless height of the jet tip $({\tilde{r}_{tip}} = {r_{tip}}/{R_0})$ decreases with increasing Weber number and ${\tilde{r}_{tip}}$ becomes too low to meet point B when We > 184. Let us assume that the travelling wave on the droplet surface meets at the north pole of the droplet to form a short jet and attains a kinetic energy $\mathrm{(\Delta }{E_k})$. Whether this jet is able to penetrate the soap film, to cause the peel-off of the film shell (at a distance of ${\tilde{h}_{det}}$), depends on this kinetic energy. The impulse that the film imposes on the droplet can be calculated from the following equation:

(3.4)\begin{equation}I = \int_0^{{t_{det}}} {2{\sigma _f} 2{\rm \pi} {R_{ff,min}} \textrm{d}t} \approx 23.2{\rm \pi} {\sigma _f}{\bar{R}_{ff,min}}{t_0},\end{equation}

where ${t_0} = {R_0}/{U_0}$ and the mean radius is

(3.5)\begin{equation}{\bar{R}_{ff,min}} = \frac{1}{{{h_{det}}}}\int_0^{{h_{det}}} {{R_{ff,min}}\,\textrm{d}h} \approx \frac{1}{{{h_{det}}}}\left[ {({h_1} + {h_2}) {R_0} + {h_3} \frac{{\sqrt 2 }}{2}{R_0}} \right] = 0.8536{R_0}.\end{equation}

Thus, the momentum is $I = mv$ and $\mathrm{\Delta }{E_k} = (1/2)m{v^2}$, where m is the droplet mass and v the translational velocity equivalent to the oscillatory velocity of the droplet. One can show that $\mathrm{\Delta }{E_k} = {I^2}/(2m) = 73.5 \cdot {E_{ds}} \cdot ({\gamma ^2}/We)$. Here, ${E_{ds}} = 4{\rm \pi} {\sigma _d}R_0^2$ is the surface energy of the droplet and $\gamma $ is the surface tension ratio, $\gamma = ({\sigma _f}/{\sigma _d}) = 1/3$. Thus,

(3.6)\begin{equation}\mathrm{\Delta }{\tilde{E}_k} = \frac{{\mathrm{\Delta }{E_k}}}{{{E_{ds}}}} = 73.5\frac{{{\gamma ^2}}}{{We}}.\end{equation}

For the capillary wave, the wave energy density is $\bar{e} = (1/2)\rho h_w^2{f^2}$, where $f = {[(8{\sigma _d})/({\rho _d}R_0^3)]^{1/2}}$ is the smallest oscillation frequency of the droplet (Landau & Lifshitz Reference Landau and Lifshitz2008) and h_w is the wave amplitude. Thus, the kinetic energy of the capillary wave is ${E_{osc}} = \bar{e} {\textstyle{4 \over 3}}{\rm \pi} R_0^3 = (4/3){E_{ds}} \tilde{h}_w^2$, where ${\tilde{h}_w} = ({h_w}/{R_0})$. The normalized kinetic energy of the oscillation is

(3.7)\begin{equation}{\tilde{E}_{osc}} = \frac{{{E_{osc}}}}{{{E_{ds}}}} = \frac{4}{3}\tilde{h}_w^2.\end{equation}

Consequently, the maximum crest height of the wave can be obtained from $\mathrm{\Delta }{\tilde{E}_k} = {\tilde{E}_{osc}}$ such that ${\tilde{h}_w} = 2.46/\sqrt {We} $. Because the wave is converged at the north pole above the chord CC’ in figure 7(c), the height of the jet tip should be ${t_{tip}} = D{O_{III}} + 2{h_w}$, which leads to

(3.8)\begin{equation}{\tilde{r}_{tip}} = \frac{{4.92}}{{\sqrt {We} }} + 0.7583.\end{equation}

This theoretical ${\tilde{r}_{tip}}$ is consistent with the experimental data, as shown in figure 7(c).

3.3.2. Height of the air pocket

Because it is the rising jet tip that punctures the pocket, the necessary condition for the drop–film collision is the pocket height ${r_p} \le {r_{tip}}$. The solid red and yellow arcs in figure 8 (zoomed in from the image at t = 6.25 ms in figure 2a) show the actual geometry of the droplet and the air pocket shortly after the detachment, respectively. The dashed curve CFC’ infers the cavity arising from the tip oscillation. The air pocket (B), which has sharp-pointed edge at the instant of drop–film detachment (shown by the blue dotted line) quickly collapses and, as such, point B retracts rapidly to B ₁ to form a bubble with the air trapped inside the segment above the chord CC’. The chord height ${B_1}D({h_{ch}})$ then decreases slowly with the oscillation of the entire droplet. These fast and slow retractions of point B are reflected by the cliffy and mild trajectories in figure 7(b). The chord height ${h_{ch}}$ is the least height that the jet tip must reach to pierce the air pocket.

Figure 8. Geometry of the droplet and air pocket soon after the detachment.

Neglecting the curvature difference of the air pocket with the droplet and accounting for the volume of the trapped air, ${\tilde{h}_{ch}}$ can be described by the equation ${V_a} = {\rm \pi}\tilde{h}_{ch}^2(1 - {\tilde{h}_{ch}}/3) = 0.09583{\rm \pi} $ and thus, ${\tilde{h}_{ch}} = 0.3280$. The pocket height is, then, ${\tilde{r}_{poc}} = {\tilde{h}_{ch}} + 0.7583 = 1.0863$.

Recognizing that at the critical condition, the ${\tilde{r}_{tip}}$ (given by (1.8)) must be equal to the pocket height, ${\tilde{r}_{poc}}$, we can set ${\tilde{r}_{tip}} = {\tilde{r}_{poc}}$ to obtain the critical We. As shown in figure 7(c), this leads to $W{e_{cri}} = 225$, which is close to the experimentally observed value in our study and in the analysis by Zou et al. (Reference Zou, Wang, Ji and Pan2017) $(We \approx 220)$.

3.4. Dynamics of the peel-off

After collision of the jet tip and the film juncture, the film shell coalesces with the droplet, contracts and propagates down to the south pole. We now extend the work of Zou to investigate the dominant role that controls the propagation. Here we assume that the ‘peel-off’ process removes the entire soap layer that is initially wrapped around the droplet. The section of the film which remains separated from the north pole P to point M accumulates at M, as shown in figure 9(a). It is powered by the surface energy $({E_s})$ of the crown $\widehat {PM}$ to move down along the droplet surface. Note that it loses two surfaces of equal area from the droplet and the film, respectively. The energy balance of this motion is ${E_s} \approx {E_k}$, i.e. $({\sigma _d} + {\sigma _f}) S \approx {\textstyle{1 \over 2}} \rho \delta S {v^2}$, so $v \approx {[(2({\sigma _d} + {\sigma _f}))/(\rho \delta )]^{1/2}}$, where S is the surface of the spherical crown $\widehat {PM}$. For systems with similar surface tension between the droplet and the film (i.e. replacing ${\sigma _d}$ with ${\sigma _f}$) we find

(3.9)\begin{equation}v \approx \sqrt {\frac{{4{\sigma _f}}}{{\rho \delta }}} = \sqrt 2 c = 1.79\;\textrm{m}\;{\textrm{s}^{ - 1}}.\end{equation}

This value matches well with the experimental results, as shown in figure 9(b–d), where the angular displacement $\theta \sim (vt/{R_0})(180\mathrm{^\circ }/{\rm \pi} )$, and x and y coordinates of point B are $x = vt \sin \theta $ and $y = vt \cos \theta $, respectively. Note that the theoretical curves are moved right to avoid the effects of incoherent shape caused by the rising tip at the early times. The substitution of surface tension infers that the soap solvent diffuses faster than the movement of the contact point and the thickness of the diffused layer is negligibly thin relative to that of the soap film. Thus, the underlying physics is that the film shell peels off the droplet and simultaneously coats the droplet surface with solvents, which has been validated by the oscillation period of the detached droplets in our previous work (Yang et al. Reference Yang, Mei, Zhang, Wei and Liu2019; Wei et al. Reference Wei, Zhang, Yang, Zhang and Liu2020). This unique feature has application potential to produce composite droplets with transplanted surfaces, say in the fields of chemistry, metallurgy, biology and medicine.

Figure 9. (a) Schematic illustration of the geometry of the air ring and evolution of the (b) angular displacement $\theta $, (c) horizontal coordinate x and (d) vertical coordinate y, of the contact point.

It is noted that we observed the occasional coalescence of a pocketless film shell that also leads to bubble shooting when the Weber number We = 20.1, as shown in figure 10 (see supplementary movie 2). Because no air is trapped at pinch-off, the air in the ejected bubble is from the interfacial air layer between the droplet and the film. Thus, the mean layer thickness can be estimated to be 264 nm, which is similar to the typical gas layer thickness (approximately 200 nm) of a droplet impacting a pool surface just before merging (Couder et al. Reference Couder, Protiere, Fort and Boudaoud2005; Hicks & Purvis Reference Hicks and Purvis2011; Thoroddsen et al. Reference Thoroddsen, Thoraval, Takehara and Etoh2012; De Gennes, Brochard-Wyart & Quéré Reference De Gennes, Brochard-Wyart and Quéré2013; Tang et al. Reference Tang, Saha, Law and Sun2019).

Figure 10. Occasional bubble shooting from cushion break without air pocket, at We = 20.1 (see supplementary movie 2).