## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# On the shape of giant soap bubbles

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved January 19, 2017 (received for review October 14, 2016)

## Significance

Surface tension dictates the spherical cap shape of small sessile drops, whereas gravity flattens larger drops into millimeter-thick flat puddles. In contrast with drops, soap bubbles remain spherical at much larger sizes. However, we demonstrate experimentally and theoretically that meter-sized bubbles also flatten under their weight, and we compute their shapes. We find that mechanics does not impose a maximum height for large soap bubbles, but, in practice, the physicochemical properties of surfactants limit the access to this self-similar regime where the height grows as the radius to the power 2/3. An exact analogy shows that the shape of giant soap bubbles is nevertheless realized by large inflatable structures.

## Abstract

We study the effect of gravity on giant soap bubbles and show that it becomes dominant above the critical size

Soap films and soap bubbles have had a long scientific history since Robert Hooke (1) first called the attention of the Royal Society and of Newton to optical phenomena (2). They have been of assistance in the development of capillarity (3) and of minimal surface problems (4). Bubbles have also served as efficient sensors for detecting the magnetism of gases (5), as elegant 2D water channels (6), and as analog “computers” in solving torsion problems in elasticity (7, 8), compressible problems in gas dynamics (9), and even heat conduction problems (10). Finally, in the last decades, the role of soap films and bubbles in the development of surface science has been crucial (11⇓–13), and the ongoing activity in foams (14, 15) and in the influence of menisci on the shapes of bubbles (16) are modern illustrations of their key role. The shape of a soap bubble is classically obtained by minimizing the surface energy for a given volume, hence resulting to a spherical shape. However, the weight of the liquid contained in the soap film is always neglected, and it is the purpose of this article to discuss this effect.

For liquid drops, the transition from a spherical cap drop to a puddle occurs when the gravitational energy,

If we look for the same transition in soap bubbles, we expect the gravitational energy,

The experimental setup dedicated to the study of such large bubbles is presented in *Experimental Setup*, before information on *Experimental Results* and *Model*. The discussion on the asymptotic shape and the analogy with inflated structures is presented in *Analogy with Inflatable Structures*.

## Experimental Setup

The soap solution is prepared by mixing two volumes of Dreft^{©} dishwashing liquid, two volumes of water, and one volume of glycerol and was left aside for 10 h before experiments. The surface tension of the different mixtures was measured using the pendant drop method. It was found to be

The bubbles are formed in a round inflated swimming pool of 4 m diameter (Fig. 1), filled with 10 cm to 20 cm of soap solution. A large bubble wand was assembled with two wood sticks and two cotton strings. The strings were immersed in the soap solution. Two experimenters, located on opposite sides of the pool, slowly opened the loop in air and pulled the sticks above the water surface, before dipping the loop into the water to form the bubble.

Once the bubble is at rest, the shape is analyzed by side view images as shown in Fig. 1. In particular, we measure the diameter,

The film thickness *A* and Fig. 2*B*, we present two sequences of four pictures showing the opening of a hole in two soap bubbles of different sizes. We use such sequences to extract the bursting velocity plotted as a function of time in Fig. 2*C*. We observe that *A* and 2.8 m/s for sequence in Fig. 2*B*. From the value of *A* and *B*, using Eq. **1**. Although it may seem surprising to find that the film thickness is homogeneous, earlier studies on the drainage of almost spherical liquid shells have shown that the thickness approaches a profile with little spatial variations (21, 22).

## Experimental Results

Once *A*. A systematic analysis of the effect of gravity on the bubble shape is shown in Fig. 3*B*, where we plot the reduced height *B* reveals that the bubbles’ shapes remain approximately spherical (*i*) The experimental data in Fig. 3*B* show no sign of a height saturation for increased bubble volumes, and (*ii*) despite our efforts, we never managed to make bubbles larger than *Model*.

## Model

We now consider the mechanical equilibrium of the soap film and predict how gravity affects the bubble shape. Bubbles are axisymmetric, and we assume a uniform film thickness *A* and *B*: The local height of the soap film is

The equilibrium of an infinitesimal part of the membrane of surface area

Here

A closed equation for the bubble shape is obtained when next considering the equilibrium normal to the membrane. The pressure difference **3**. As a final step, it is convenient to eliminate **3** and **4** gives the equation for the shape of the bubble,

Scaling all lengths with

A unique bubble shape is found numerically for each value of the dimensionless height

Fig. 4*C* shows the corresponding bubble shapes for increasing volume. As expected, small bubbles are dominated by surface tension and are perfectly spherical. However, as bubbles get larger (*A*, where we superimpose the picture of a 20-cm diameter bubble with the solution of Eq. **6** that has the same ratio **6**) also gives a quantitative prediction for the height *B* (solid line). The result describes very well, without any adjustable parameter, the experimentally observed flattening due to gravity.

Unexpectedly, the numerical solution does not predict a saturation of the bubble height: *E*, showing the dimensionless bubble height on a log–log plot. For large volumes, we find that **3**, both surface tension and gravity scale with *Supporting Information*, the large bubble shapes in Fig. 4*C* exhibit scale invariance and can be collapsed to a single, universal shape. The scaling of the universal shape near the edge reads **6**. The 2/3 scaling at the edge determines the horizontal and vertical scales for the bubble and leads to the scaling in Fig. 4*E* (see *Supporting Information* for detailed analysis).

This scaling law for large bubbles, and, in particular, the lack of saturation, is in stark contrast with the classical result for liquid drops. The shape of droplets can be found from the classical hydrostatic pressure balance (13) and is different from Eq. **6**,

Here, the lengths were made dimensionless using the capillarity length *D* shows the corresponding numerical solutions: Large drops develop toward puddles, which, for

The height of soap bubbles may, however, be limited by physical chemistry of surfactants. The water that constitutes the bubble is prevented from draining quickly by gradients in the surface tension. The larger surface tension at the top of the bubble supports the weight of water in the liquid shell (15). In practice, the surface tension of a soap solution cannot be higher than that of pure water, **3** thus gives a criterion for the maximal height

## Analogy with Inflatable Structures

Interestingly, the shapes we have just discussed correspond to a minimization problem that is relevant in the context of large inflatable structures, such as shown in Fig. 5. These structures consist of a thin sheet that we assume cannot be stretched, and which is inflated by a pressure difference *A* and *B*: The role of surface tension is replaced by the tension that develops inside the membrane. It is interesting to confirm this analogy based on energy minimization, with the no-stretch condition imposed through a Lagrange multiplier

The three terms respectively represent the gravitational free energy, the area constraint, and the work done by the pressure difference. The Euler–Lagrange equation for this functional gives (see *Inflatable Structures*):**4**). Designing the inflatable structures along these optimal shapes will naturally avoid stretching and compression of various parts of the sheets, avoiding wrinkles and reducing tensile stresses exerted in the sheets and on the seams that connect the various parts. This design should help increase the lifetime of such structures.

## Conclusion

We study the shape of large soap bubbles and show that gravity becomes important at the scale

## Analyzing the Shape of Large Bubbles

For large volumes, the bubbles exhibit a universal shape; this can be seen in Fig. S1, where we represent the numerical profiles (Fig. 4*C*) after rescaling the height by *Supporting Information*, we explain this observation and use it to derive the scaling law *Model*.

We start from the dimensionless shape equation given in Eq. **6**,*Supporting Information*, we omit tildes for convenience. For analysis, it is convenient to use the connection to cylindrical coordinates,*Similarity Solution for the Global Shape* and *Near the Contact Line*, we will analyze the limiting behaviors for large bubbles, by which we mean that the central height

## Similarity Solution for the Global Shape

Analyzing the profile at scales **S3** simplifies to**S4** and expanding to leading order in **S6** gives **S1**.

For what follows, it is important to analyze the shape in the vicinity of the contact line. In terms of the similarity solution, this regime corresponds to

This equation is valid for **S4**. It is important to note that, up to this point, the scales **S8** to the contact line region for the full equation, Eq. **S3**.

## Near the Contact Line

At the scale **S3**. Then, the shape is described by**S10** is a first-order ODE that can be written as

We are primarily interested in the large-**S8**). Hence, the matching condition requires that

## Inflatable Structures

The equilibrium of axisymmetric inflatable structures can be computed from the Euler–Lagrange equation,**11**, this equation becomes

Using the geometric relations discussed at the start of *Supporting Information*, this equation indeed reduces to Eq. **4**.

## Acknowledgments

We thank Tomas Bohr for organizing the 2013 Krogerup Summer School that initiated the collaboration between Paris and Twente. We also thank Isabelle Cantat for her input on the stability of soap films and for her constructive criticism of the initial version of our work.

## Footnotes

↵

^{1}C. Cohen and B.D.T. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: etienne.reyssat{at}espci.fr.

Author contributions: C. Cohen, B.D.T., and C. Clanet designed research; C. Cohen, B.D.T., E.R., J.H.S., and C. Clanet performed research; C. Cohen, B.D.T., E.R., J.H.S., D.Q., and C. Clanet analyzed data; and C. Cohen, B.D.T., E.R., J.H.S., and C. Clanet wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1616904114/-/DCSupplemental.

## References

- ↵.
- Hooke R

- ↵.
- Newton SI

- ↵.
- Plateau J

- ↵.
- Courant R,
- Robbins H

- ↵.
- Faraday M

- ↵
- ↵.
- Prandtl L

- ↵.
- Griffith AA,
- Taylor GI

- ↵.
- Wen CY,
- Lai JY

- ↵.
- Wilson LH,
- Miles AJ

- ↵.
- Mysels KJ,
- Shinoda K,
- Frankel S

- ↵.
- Isenberg C

- ↵.
- de Gennes PG,
- Brochard-Wyart F,
- Quéré D

- ↵.
- Petit P,
- Seiwert J,
- Cantat I,
- Biance A

- ↵.
- Saulnier L, et al.

- ↵.
- Teixeira MA,
- Arscott S,
- Cox SJ,
- Teixeira PI

- ↵.
- McEntee WR,
- Mysels KJ

- ↵.
- Dupré A

*Annales de Chimie et de Physique*11(4):194. - ↵.
- Taylor GI

- ↵
- ↵
- ↵.
- Lee A, et al.

- ↵.
- de Gennes PG

## Citation Manager Formats

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.

### Cited by...

- No citing articles found.