An interesting aquarelle on display in the Napoleon’s Last Headquarters Museum, near Waterloo, depicts the Belle Alliance inn. The roof and walls are seen shot by cannonballs; an officer of the Chasseurs Imperiales, with the uniform represented in full details, walks on his horse; some remains of the battle can be seen on the ground, hats, swords, jackets, soldiers carrying wounded companions, an abandoned cannon still at aim, with the dead gunner laying on the side. This dramatic document represents one of the rare accounts by an eye-witness, a young kid of age 13, who on the day of June 19, 1815, took a walk on the bloody Waterloo battlefield. The kid was Joseph Plateau, the son of a painter decorator from Bruxelles who had died in April of that same year. He had been taken in foster care with his two sisters by uncle Leopold, who decided to move away from the big city to the small village of Ohain, just on June 17, the day before the great battle. It is certainly amazing that the young Plateau, then student in the school of arts, could be so daring to take a walk among that horror, and start drawing with such a detachment. However, his qualities would have amazed even more in the following years when, after obtaining a degree in law at the university of Liege, he also started a doctorate in mathematics and physics, which he completed at the age of 28. Merely four years later, Plateau was nominated professor of experimental physics at the Royal Academy of Bruxelles, and then in Gand, where he continued his original studies on the physics of human vision, a subject that had interested him since his youngest age, and that unfortunately would also fail him miserably, becoming totally blind in 1843. In the meantime, he had contributed a number of interesting studies on stroboscopic vision, capillarity, surface tension, and problems of minimal surfaces, which he had the idea of studying with the help of soap bubbles. He continued working and writing, thanks to his wife Fanny Clavareau, an amateur scientist herself, who transcribed Plateau’s notes and lectures collected in a precious book, Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires, published in 1873.

As every kid knows, soap bubbles are made by mixing some soap with water (in case you missed this part of childhood experiences, detailed instructions are provided in PRL 116 (2016) 077801). If we blow air with a straw directly into pure water, we would form some bubbles that however are immediately destroyed by the energy cost of making extra water free surface in contact with air. But when soap is added, its amphiphilic molecules arrange in two parallel monolayers, sandwiching a thin layer of water inbetween; the resulting triple-layer membrane has a finite value of surface tension that can resist a (small) pressure, giving rise to our beautiful spherical bubbles. Without knowing, children are exploring the properties of surface tension and minimal surfaces.  

Minimal surfaces have been largely studied over the years because they are one of the most complex domains of mathematical physics, as well as having a lot of practical applications in biology, engineering or architecture, to name a few. One of the most important empirical discoveries by Joseph Plateau is the fact that the angles among joining soap bubbles can have only two possible values, that is 120o in two dimensions, and 109o5 in three dimensions. It took more than 100 years before these experimental observations could be rigorously proved by the American mathematician Jean Taylor [The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. Math. (2nd ser.) 103, 489–539 (1976)]. Empirically, these two special values correspond to the angles delimiting equal-area surfaces in two dimensions, or equal-volume regions in three dimensions, respectively. Therefore, they are connected with the practical realisation of minimal surfaces and their limiting shapes.

Limiting shapes originate from the state of curvature of a surface, synthetically described by the Laplace equation. This law can be derived for example for a soap bubble with a given surface tension 𝛴, inflated by air at pressure P (actually, a pressure difference with respect to the environment); by elementary observations, it is easily obtained that the radius R of the bubble is related to 𝛴 and P as: (Pin–Pout)=2𝛴/R. The radius of the bubble represents a local state of curvature, which can be defined in a general way on both the inner and outer side of the surface. Actually, the factor 2 in Laplace’s equation comes just from the definition of local curvature, that is the sum of curvatures on both sides: for a sphere both curvatures are equal to 1/R, hence the 2/R in the equation. To appreciate the difference between “inner” and “outer” curvature, try to make a soap “tube” by catching the bubble between two parallel rings held at a short distance: in this case you obtain a saddle-shaped soapy surface with equal and opposite curvatures on the inner and outer side, their sum is zero, coherently with the fact that such an open surface has the same pressure on both sides.

I renewed my interest in soap bubbles in adult age, both because my little nephew Niccolò loves them, and in connection with the kinetics of phase coarsening when I was studying solid-state grain growth [FC, A stochastic grain growth model based on a variational principle for dissipative systems, Physica A282, (2000) 339]. Interesting observations about minimal surfaces can be made also at home, by playing with two transparent rectangles (such as two pieces of glass) joined by four thin (2-3mm) biadhesive pads at the corners, and dipping the ensemble in soapy water. Similar to a microstructure of polycrystalline grains, a quasi-2D dense network of squeezed soap bubbles forms in the thin space between the plates, and one can observe how bigger bubbles grow progressively “eating up” smaller ones. Doing it in 3D is even simpler, since you just need to blow with a straw in a dish containing the soap-water mixture (the results will be better if replacing a small part of the soap by glycerine, to increase viscosity); however, observing angles at joining bubbles is much easier in 2D. The Laplace equation above tells us that the smaller the radius of the bubble, the larger the pressure (P is inversely proportional to R), therefore when a small bubble touches a large one, the gas inside diffuses to the direction of lower pressure, and the small bubble will merge into the larger.

Coarsening in extended two-dimensional networks of random “bubbles” can be simulated by computer models, which nicely confirm the theoretical predictions for single bubbles (the equivalent for condensed phases is the Ostwald ripening). In particular, simple geometrical considerations lead to the Von Neumann growth law, dA/dt=k(n–6), stating that the area of polygonal bubbles in two dimensions grows or shrinks in time for bubbles with more or less than 6 sides, respectively, while it remains constant for a 6-sided bubble. This result together with minimization of surface tension gives the Plateau-like network of perfectly hexagonal bubbles with angles at 120o, which is indefinitely stable in two dimensions, as can be confirmed by computer simulations and by honey bees. In 3D the situation is more complicated, and the equivalent of a polyhedral growth law can only be obtained by numerical approximations (e.g., MacPherson & Srolovitz, The von Neumann’s relation generalized to coarsening of three-dimensional microstructures, Nature 446 (2007) 1053). As far as I know, nobody has yet proved what the absolutely stable state should be in 3D… maybe (my guess) a honeycomb of tetrahedra and octahedra, both sharing vertex angles of 109o47 (tetrahedra alone cannot exactly fill the space, despite this was believed for more than 2,000 years, based on a wrong construction by Aristotle).

Have you ever wondered what the thermodynamic properties of a such a material without a bulk phase would be? The thickness of the water layer in a soap bubble can be as small as 10 nanometers, that is a length of about 40-50 water molecules: an almost ideal 2D system, with no bulk to exchange heat with. As we know from years of work with nano-confined systems, the correlation length is much larger in a nanoscale 2D layer than in the corresponding 3D bulk, and temperature fluctuations die off as N–2  compared to N–3. Therefore, we expect very large temperature fluctuations in the thin shell of water when, for example, it is brought close to freezing temperature. Such a peculiar condition gives rise to multi-site heterogeneous nucleation of snow crystals, which are free to move inside the water layer, with wonderful phenomena such as the “snow globe effect” that you can admire in the title image, and in this YouTube video .

The singular freezing kinetics of soap bubbles has been characterized in a recent paper by the group of J. Boreyko at Virginia Tech (Nature Comm. 10 (2019) 2531), which also suggests funny experiments to try at home (I did, details below). They studied two regimes: for bubbles deposited on an icy substrate contained within an isothermal freezer, the freeze front induced local heating at the bottom of the bubble; this results in a Marangoni-like flow (that is, mass transfer along an interface subject to a gradient of surface tension), strong enough to detach and entrain several growing ice crystals, such that the water layer in the bubble freezes from multiple fronts. On the other hand, when bubbles are deposited on a chilled, icy substrate in a room-temperature environment, the freeze front uniformly grows bottom-up, before stopping entirely at a critical height. It is interesting to note that since the temperature at the contact is quite lower than that of the water, there is latent heat generated at the growing freeze front.

I performed the two experiments in the kitchen; the results are funny, and you may repeat them with your kids, as a family-made Christmas event. The second experiment is easier, just take a few ice cubes from the freezer and blow bubbles above them. The quick freezing arrested up to about mid-height of the bubble is easily observed, but no ice crystals can be seen, even if looking with a magnifying glass. The effect of latent heat release is manifested by micro-melting of the ice cube surface all around the perimeter of the contact: the bubble finds itself sitting on a ring of water, capillarity sucks water molecules into the bubble, until it breaks by gravity. The first experiment however is more complicated. I tried to attain nearly isothermal conditions by emptying the bottom freezer compartment of my refrigerator. Soap bubbles can be deposited on the freezer trays, locally at –18oC, as close as possible to the side walls (clean metal trays with no ice deposited do a better job, plastic trays instead are too hydrophobic). Put a flashlight inside, beaming the bubble from the back (I used the LED lamp of the iPhone). With some luck, beautiful ice-crystal dendrites will be seen to grow in the bubble over a time of 10-20 seconds, similar to the YouTube video. However, the best isothermal conditions are to be met in a freezing outdoors, if you can find one. I suggested my wife Olga we should try this in Saint Petersburg, by going to blow bubbles on the frozen surface of the Neva river… a very romantic Christmas decoration!

One last question: I also tried to blow a bubble on top of an ice cube, which then I pushed in and out the freezer compartment; the experiment kinda worked, but when I took it back out of the freezer, the bubble immediately deflated like a punctured balloon, I don’t understand why. Could it be that the soap molecules in the frozen state have a higher permeability, letting air molecules to zip out once the temperature is quickly raised…? Or maybe the air enclosed in the bubble at lower temperature has a lower pressure, which is crushed by the pressure of air at room temperature…? Do you have any better suggestions…?

Christmas soap-snowglass

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