Self-similarity with universal property for soap film and bubble in roll-off regime

Compared to film, the volume of bubble roughly remains constant when pulled apart, as argued in Sec. IV of the Supplemental Material (SM) sm . The center of bubble surface evolves from being convex to concave in contrast to film that is always convex throughout equilibrium regime. This difference will be argued later in the theory section to give rise to a positive second derivative of $h_{\rm{min}}(L/R)$ in bubble, as opposed to a negative one in film Chen_steen_numerical ; robinson_steen_exp ; cryer_steen_exp , as shown in Fig. 3(a, b). When $\gamma$, $R$ and $V$ are fixed, $h_{\rm{min}}$ can be uniquely determined by $L$ in equilibrium regime. By use of $L^{*}-L=v_{s}(t^{*}-t)$ where $t^{*}$ denotes the time when the neck starts to collapse spontaneously, $h_{\rm{min}}$ can be alternatively expressed as a function of $v_{s}(t^{*}-t)$. But, since non-equilibrium state proceeds much faster than $v_{s}$, the time it takes is negligibly small, i.e., $t_{p}-t^{*}\approx 0$. So, we can use $\tau\equiv t_{p}-t$ to track the evolution of $h_{\rm min}$ from now on.

The necking process is expected to be size-independent for both film and bubble. Therefore we divide $h_{\rm{min}}$ and $\tau$ by $R$ and $R/v_{s}$ to render them dimensionless. When $\tau$ is fixed, $L$ is proportional to $R$ at the same $v_{s}$ for a film. A different choice of ratio of $V/R^{3}$ will shift the curve in Fig.  3(f) that still remains independent of $R$, as detailed in Sec. VII of the SM sm .

Upon entering roll-off regime, film and bubble share the following properties: First, the evolution of $h_{\rm{min}}$ is independent of (1) the pulling speed $v_{s}$ in Fig. 3 (c, d) because the shrinkage is much faster and (2) the ring or cap size if both $h_{\rm{min}}$ and $\tau$ in Fig. 3 (e, f) are properly rescaled by $R$ and the characteristic time $\sqrt{\rho R^{3}/\gamma}$ from the balance between the surface tension and inertial force. Note that the $h_{\rm min}(\tau)$ for film has been predicted to be independent of $R$ by Chen_steen_numerical . What distinguishes bubble from film is that $\alpha=1/2$ for the former instead of 2/3 Chen_steen_numerical . Although $h_{\rm{min}}$ will increase with more pumping volume, the value of $\alpha$ is checked to be independent of $V$, as shown in Fig 8. We believe this is due to the fact that there is little air in the vicinity of bubble neck. However, the constraint imposed by the volume conservation is still critical at modifying $\alpha$. Heuristically the existence of a pressure difference across the membrane impedes the collapsing of bubble and thus renders a small $\alpha$.

The second property shared by film and bubble regards the evolution of profile, $h(x)$ in Fig. 4 (a, b) and includes: (1) Both are found to exhibit self-similarity, i.e., their shapes at different time can be mapped to a master curve if both $h$ and $x$ are rescaled by $h_{\rm{min}}$, as shown in Fig. 4 (c, d). The level of similarity is quantified by cosine similarity self = 0.98 and 0.99 for film and bubble, as detailed in Sec. II of the SM sm . (2) Both master curves do not change with different ring or cap size in Fig. 4(e, f). In contrast to $h_{\rm{min}}$ that is independent of both $R$ and $V$, the contour in Fig. 4(f) is only universal with respect to $V/R^{3}$. When $V/R^{3}$ changes, the contour will become different. What distinguishes bubble from film is that $h^{\prime\prime}(x=L/2)<0$ which renders an inflection point between $x=0$ and $x=L/2$, where $h^{\prime\prime}$ denotes $d^{2}h/dx^{2}$. In contrast, the film is always concave upward. This is demonstrated theoretically in Sec. X of the SM sm .

The number of neck can be seen to double as film and bubble transits from roll-off to pinch-off regime in Fig. 5 (a, b). Furthermore, the originally different exponent $\alpha$ in roll-off becomes identical at 2/3 for film and bubble near pinch-off regime in Fig. 5 (c, d), rescaled as Fig.  3 (e, f). The evolution of profile for film looks similar to that of bubble in pinch-off regime with the shifting of minimum point from $x=$0 to $x=x_{\rm p}$ in Fig. 5 (e, f). The self-similar behavior which has been predicted by numerical and theoretical works Chen_steen_numerical ; Self_Similar_Capillary for film turns out to be also true for bubble, as shown in Fig. 5 (g, h).

Different from the flat surfaces on rings for film, two spherical bubbles survive the breakage and appear on the caps for bubble. The spherical shape is to minimize the surface energy, for which the height $b$ defined in Fig. 6 (a, b) can be calculated. In the mean time, the critical length $L^{*}$ at which the irreversible processes are initiated can also be determined theoretically by the breakdown of solution from minimizing the potential energy for equilibrium regime. Comparing these two lengths, we find that $L^{*}$ is not only bigger than $2b$, which explains the necessity for both bubbles to retract and breakage, but roughly equals $5b/2$. This is verified in our experiment.

Depending on the amount of $V$, we expect two scenarios for the remnant bubble. Straightforward calculation in Sec. IV C reveals that $L^{*}/R\propto(V/R^{3})^{\beta}$ with $\beta=1$ for $V/R^{3}\ll 5.44$ and $1/3$ otherwise. This prediction is nicely verified by Fig. 6(c).

Experimentally we can stretch bubble and film horizontally or vertically. Although the sagging and non-symmetric contour due to gravity in both cases can be alleviated by minimizing $V$, it is still uncertain whether the critical behavior at pinch-off will be affected. So theoretical calculations can help us not only clarify this concern, but also give us analytic expressions for quantities of our interest, e.g., $h_{\rm min}$ and highlight the influence of volume conservation. To realize how the parameters affect the contour of bubble, we start from the minimization of total energy for bubble:

$\displaystyle\frac{U}{2}=\int_{0}^{L/2}\left[\gamma\cdot 2\pi h\sqrt{1+h^{\prime 2}}+\lambda\left(\pi h^{2}-\frac{V}{L}\right)\right]dx$

(1)

where the Lagrange multiplier $\lambda$ makes sure that the air volume is conserved. Using the second form of Euler-Lagrange equation, we can obtain

$\displaystyle\frac{h}{\sqrt{1+h^{\prime 2}}}+\frac{\lambda}{2\gamma}h^{2}=h_{\rm{min}}+\frac{\lambda}{2\gamma}h_{\rm{min}}^{2}.$

(2)

After some transpositions, Eq. (2) becomes

$\displaystyle h^{\prime}=\sqrt{\left(\frac{h}{h_{\rm{min}}+\frac{\lambda}{2\gamma}h_{\rm{min}}^{2}-\frac{\lambda}{\gamma}h^{2}}\right)^{2}-1}.$

(3)

Solving this differential equation will enable us to obtain information of the contour $h(x)$:

$\displaystyle x=\bigints_{h_{\rm{min}}}^{h}dh/\sqrt{\left(\frac{h}{h_{\rm{min}}+\frac{\lambda}{2\gamma}h_{\rm{min}}^{2}-\frac{\lambda}{\gamma}h^{2}}\right)^{2}-1}.$

(4)

where $0\leq x\leq L/2$.

By implementing the boundary condition that $h(x=L/2)=R$ and volume conservation, we get

$\displaystyle\frac{L}{2h_{\rm{min}}}=\bigints_{1}^{\frac{R}{h_{\rm{min}}}}{dy}/{\sqrt{\Big{(}\frac{y}{1+\big{(}1-y^{2}\big{)}\xi}\Big{)}^{2}-1}}$

(5)

and

$\displaystyle\frac{V}{2h_{min}^{3}}=\bigints_{1}^{\frac{R}{h_{\rm{min}}}}{\pi y^{2}dy}/{\sqrt{\Big{(}\frac{y}{1+\big{(}1-y^{2}\big{)}\xi}\Big{)}^{2}-1}}.$

(6)

where a change of variable $y=h/h_{\rm{min}}$ has been performed to render the parameters dimensionless and $\xi\equiv\frac{\lambda}{2\gamma}h_{\rm{min}}$. By setting $\lambda=0$, Eq. (5) will revert to depicting a film and give us $h_{\rm{min}}/R$ vs. $L/R$ in agreement with Fig. 3(a). During the stretching of bubble, there must be a period when $h_{\rm{min}}$ is close to $R$ and we can approximate $h_{\rm{min}}$ by $R-\Delta h$ where $\Delta h\ll R$. This allows us to Taylor expand $R/h_{\rm{min}}=R/(R-\Delta h)\approx 1+\Delta h/R$, Eqs. (5) and (6) to get

	$\displaystyle\frac{L}{2h_{\rm{min}}}$	$\displaystyle\approx$	$\displaystyle\frac{\left(\frac{R}{h_{\rm{min}}}-1\right)}{\sqrt{\big{(}\frac{1+\frac{\Delta h}{R}}{1+\big{(}1-\frac{\Delta h}{R}\big{)}^{2}\xi}\big{)}^{2}-1}}$		(7)
		$\displaystyle=$	$\displaystyle\frac{\Delta h/R}{\sqrt{\big{(}\frac{1+\Delta h/R}{1-2\xi{\Delta h}/{R}}\big{)}^{2}-1}}$

and

$\displaystyle\frac{V}{2h_{\rm{min}}^{3}}\approx\frac{\Delta h/R}{\sqrt{\big{(}\frac{1+\Delta h/R}{1-2\xi{\Delta h}/{R}}\big{)}^{2}-1}}\big{(}1+2\Delta h/R\big{)}$

(8)

where terms of order higher than ${\Delta h}/{R}$ have been neglected. By comparing the above two equations, we obtain $V/(2\pi h_{\rm min}^{3})\approx{L}/{2h_{\rm{min}}}$. Note that this result predicts a simple yet informative relation for how $h_{\rm min}$ varies with $L$:

$\displaystyle h_{\rm min}\approx\sqrt{\frac{V}{\pi L}}.$

(9)

which matches the data in early equilibrium regime for Fig. 3(b). In retrospect, this result is expected from our restricting $h_{\rm{min}}\approx R$ since the shape of bubble now mimics that of a cylinder.

The derivations from Eq. (2) to (8) serve several purposes: First, to illustrate the important role of the Lagrange multiplier $\lambda$ and how it mathematically prohibits the bubble from adopting the catenoid profile as the film. Second, $h^{\prime\prime}$ near the edge can be easily shown to exhibit different signs for film and bubble - positive and negative from the derivative of Eq. (2), respectively, as detailed in Sec. X of the SM sm . Third, they highlight the importance of $\lambda$ and subsequent parameter $\xi\approx[-2+(\Delta h/R)]/4$, without which the right-hand side of Eq. (7) will be of order $\mathcal{O}(\sqrt{\Delta h/R})$ - meaning the cylindrical shape is only possible when the two rings are very close for film.

Some experience can be borrowed from the analysis of film. After setting $\xi=0$, we plot both sides of Eq. (5) as a function of $R/h_{\rm{min}}$ in Fig. 7 (a).

When $L>L^{*}$, there is no intersection - meaning that the starting point of minimizing the surface energy is problematic. This is consistent with our expectation that film will collapse automatically at large $L$ when we need to resort to minimizing the action. When $L=L^{*}$, both sides of the equation become tangent. The same is expected for bubble, i.e., we should differentiate both sides of Eq. (5) with respect to $h_{\rm{min}}$ to locate $L^{*}$. There are two intersections for $L<L^{*}$ with one being unphysical as explained in Fig. 7 (a). When we plot $h_{\rm{min}}$ vs. $L$, the solution should be double-valued until $L$ reaches $L^{*}$ in Fig. 7(b). Therefore, we expect $L(h_{\rm{min}})\approx-\chi(h_{\rm{min}}-h^{*}_{\rm{min}})^{2}+L^{*}$ where $\chi$ is a constant when $h_{\rm{min}}$ is close to the critical neck radius $h^{*}_{\rm{min}}$ beyond which the neck collapses spontaneously. Simple rearrangement gives

$\displaystyle h_{\rm{min}}\approx h^{*}_{\rm{min}}+\chi^{-1/2}\sqrt{L^{*}-L}.$

(10)

Our confidence on Eqs. (9) and (10) is supported by its rightful prediction of an inflection point in Fig. 3(b) due to the fact that their curvatures are of opposite sign.

The collapsing speed of neck is very fast. As will be delineated later, there will be two complications before the final breakage. First, two necks will be developed in pinch-off regime. Second, this is followed by the breaking stage when a satellite bubble is formed in the middle of these two necks after the hollow thin tube connecting them becomes a liquid string. There is no gas leakage throughout breaking and relaxing stages and the volume of satellite bubble can be neglected. Therefore, $V$ should equal to the combined volume of the two remnant bubbles after breakage. We can directly obtain the relation between $V$ and $L^{*}$ which can be estimated from Fig. 6(b) where the chord length equals $2R$ and the radius of partial sphere is denoted by $a$. From the geometry in Fig. 6(b), we can write down

$$a=\frac{R^{2}+b^{2}}{2b}$$

(11)

and

$$\cos\theta=\frac{R^{2}-b^{2}}{R^{2}+b^{2}}.$$

(12)

The volume of each partial sphere can be calculated as

$\displaystyle\frac{V}{2}=\frac{\pi L^{*}}{15}\left(3R^{2}+b^{2}\right)$

(13)

Rearranging both sides to make them dimensionless, we found

$\displaystyle\frac{V}{R^{3}}=\frac{\pi}{3}\frac{b}{R}\Big{[}3+\Big{(}\frac{b}{R}\Big{)}^{2}\Big{]}$

(14)

By inserting the experimental result $b=2L^{*}/5$, there are two limiting cases to Eq. (14). When ${L^{*}}/{R}\ll\frac{5}{2}\sqrt{3}$, the cubic term in Eq. (14) can be neglected and

$\displaystyle\frac{V}{R^{3}}\cong\frac{\pi}{2}\frac{L^{*}}{R}$

(15)

In the other extreme ${L^{*}}/{R}\gg\frac{5}{2}\sqrt{3}$, the linear term becomes negligible and

$\displaystyle\frac{V}{R^{3}}\cong\frac{8\pi}{375}\left(\frac{L^{*}}{R}\right)^{3}.$

(16)

The two regions in Eqs. (15) and (16) are vindicated by Fig. 6(c).