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Throwing Shapes

In 1993 physicists Denis Weaire and Robert Phelan discovered, using computer simulations, the lowest energy structure known of an ideal monodisperse foam in the dry limit, i.e., it is the lowest energy arrangement of packed bubbles of equal volume. entered the world of foams. This structure is now commonly known as the Weaire-Phelan structure. A sculpture representing the Weaire-Phelan structure is on display on the third floor of the SNIAM Building.

 

History 

In 1840, Belgian scientist Joseph Plateau began his investigation into the surface area of foams. It began with oil dropped into a mix of water and alcohol; Plateau noticed that the drops formed perfect spheres in the mixture. Later, using a mixture of soapy water and glycerine, he observed that the foam surfaces formed were minimal surfaces due to surface tension. Plateau calculated that three soap films are stable when meeting at 120o angles and four soap films are stable when meeting at 109.5o. The Plateau problem would emerge from these experiments. The problem is to show the existence of a surface of minimal area with a given boundary rule while satisfying Plateau’s rules (which include soap films being smooth surfaces, the curvature of a portion of film being constant, and above two angle rules). The question was pondered by scientists at the time; one scientist, Georges Buffon, would use lead shot and peas as models. Later, Belfast born scientist Lord Kelvin proposed the Kelvin problem, asking how space could be partitioned into cells of equal volume with the least surface area between them. In 1887, Lord Kelvin believed he had devised the ‘perfect’ foam. Lord Kelvin’s ‘perfect’ foam structure had truncated octahedra for its cells, with eight hexagonal faces and six square faces, all with a slight curvature to fit Plateau’s rules. The convex uniform honeycomb structure, known as the Kelvin structure, had been considered the most efficient solution to the problem for over a hundred years. In 1994, Denis Weaire and Robert Phelan, with the help of computer simulations, published a paper entitled ‘A counterexample to Kelvin’s conjecture on minimal surfaces.’  The Weaire-Phelan structure would exist solely as a mathematical concept for many years, until in 2012, when Dr. Ruggero Gabbrielli realised the concept by fabricating the Weaire-Phelan structure for the first time. More recently, in 2022, a Japanese research team developed the first polymeric Weaire-Phelan foam using facile synthetic procedures. 

 

Science and Structure 

After a century the Weaire-Phelan structure surpassed the Kelvin ‘ideal’ foam structure with 0.3% less interfacial area. In addition to having a smaller surface area, the Weaire-Phelan structure is also a lower energy arrangement than the Kelvin structure. The Weaire-Phelan structure consists of a repeating unit of eight bubbles with two kinds of polyhedral cells of equal volume. The first polyhedral cell is an irregular pentagonal dodecahedron (12 faces), which makes up two of the eight bubbles, and the second is a tetrakaidecahedron (14 faces made up of hexagons and pentagons), which makes up the remaining six bubbles. Though the Weaire-Phelan structure does not appear naturally, it has been made in a laboratory environment. In 2011, Ruggero Gabbrielli designed a container templated with the geometry of the Weaire-Phelan structure. The container was made with a translucent polymer using a 3D printer. The container was then placed into a solution of Fairy Liquid detergent. Releasing pressurised nitrogen gas from a glass capillary allowed for the production of equal-sized bubbles. Varying the flow rate of the gas meant that bubble sizes could be matched to the template. The experiment found 1,500 bubbles stacked in six layers that conformed to the Weaire-Phelan structure. 

 structure weire phelan

Beijing 2008 Olympics 

In 2003, China hosted an architectural competition for the design of its national aquatic centre in Beijing. Out of 10 proposals, the ‘Water Cube’ was chosen to be built. The Water Cube is made up of 4000 ETFE (a type of polymer) bubbles; seven bubble sizes were used for the roof and fifteen for the wall, the largest of which was nine meters in diameter. The design for the Water Cube is based on the Weaire-Phelan structure. Steel beams are used for the framework of the bubbles, while ETFE is used to cover the surface. The Water Cube was used to host the swimming, diving, and synchronised swimming events in the 2008 Beijing Olympics. In the 2022 Winter Olympics, the Water Cube was referred to as the ‘Ice Cube’ and was used to host the curling events. 

 

The Water Cube at Night 

 

Sources 

  1. Weaire and R. Phelan (1994) , A counter example to the Kelvin’s conjecture on minimal surfaces, Philosophical  magazine,  Vol 69, page 107-110, https://www.tandfonline.com/doi/abs/10.1080/09500839408241577 
  2. Ruggero Gabbrielli, Aaron J. Meagher, Denis Weaire, Kenneth A. Brakke & Stefan Hutzler (2012) An experimental realization of the Weaire–Phelan structure in monodisperse liquid foam, Philosophical Magazine Letters, 92:1, 1-6, DOI: 1080/09500839.2011.645898 
  3. Cheme Europe (Accessed July 2023), Weaire-Phelan Structure, chemeeurope.com https://www.chemeurope.com/en/encyclopedia/Weaire-Phelan_structure.html
  4. Shibaura Institute of Technology (Nov 28th 2022), Bringing the Kelvin problem solutions to life with the first ever polymeric Weaire-Phelan structure, Phys.org, https://phys.org/news/2022-11-kelvin-problem-solutions-life-first-ever.html
  5. Ball, P. (2011), Scientists make the 'perfect’ foam, Nature,  https://doi.org/10.1038/nature.2011.9504  
  6. TCD (Dec 22nd 2011), the science of bubbles – foam bubbles finally brought to order, TCD https://www.tcd.ie/news_events/articles/the-science-of-bubbles--foam-bubbles-finally-brought-to-order/ 
  7. J. O’Connor, E. F. Robertson (2006), Joseph Antoine Ferdinand Plateau, Math History, University of  St Andrews, https://mathshistory.st-andrews.ac.uk/Biographies/Plateau/ 
  8. School of mathematics and statistics (accessed July 2023), Glossary, University of St Andrews, https://mathshistory.st-andrews.ac.uk/Glossary/#plateau_problem 
  9. Kelvin structure image , By Truncated_octahedra.jpg: en:User:AndrewKepert - This file was derived from: Truncated octahedra.jpg:, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=20542549 
  10. Weaire-Phelan structure image, By Tomruen at English Wikipedia - Transferred from en.wikipedia to Commons by Sreejithk2000 using CommonsHelper., Public Domain, https://commons.wikimedia.org/w/index.php?curid=10471229 
  11. Wikipedia (Accessed July 2023) , Beijing National Aquatics centre, In Wikipedia https://en.wikipedia.org/wiki/Beijing_National_Aquatics_Centre#:~:text=Using%20the%20Weaire%E2%80%93Phelan%20geometry,of%2017%2C000%20during%20the%20games.
  12. TCD (Aug 8th 2008), Trinity inspired Water Cube Shines at the Olympic Games, TCD https://www.tcd.ie/news_events/rticles/trinity-inspired-water-cube-shines-at-the-olympic-games/
  13. Water Cube image, By Charlie fong - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=5796286
  14. Stefan Hutzler, Wiebke Drenckhan (2018), Foam Films and Foams, CH 16, The Structure of Liquid Foams, CRC Press, http://www.tara.tcd.ie/bitstream/handle/2262/83841/final-corrected-proofs-K20356_C016.pdf?sequence=1&isAllowed=y