The Cheerios Effect

August 25, 2016

If mothers told their young physicists not to play with their food, the youngsters weren't listening. Quite a few discoveries have been made through observations of food items. I've written about several on this blog, including the packing of M&M candy (Packing, November 30, 2010), the Brazil nut effect (Rivers of Sand, December 22, 2014), and wine "legs" (Oenodynamics, November 29, 2011).

The unexpected discovery about the lens-shaped M&M candies is that their random packing fills space to 68%, while a random packing of spheres achieves just 64%. Ellipsoids of a different aspect ratio than the M&M candies were found to fill space 78% in random packing.[1]

The well-known Brazil nut effect is the tendency for large nuts to move to the top of a container of mixed nuts after shaking. While noticed in cans of mixed nuts, this phenomenon happens in any vertically-shaken, dry, granular mixture of large and small particles.[2-3]

Screenshots of a simulation of the Brazil nut effect. Vibration in the vertical direction causes the large object to rise in a sea of smaller objects. (Used with permission from the Granular Dynamics Update Web Site of Prof. Derek C. Richardson of the University of Maryland - College Park. For further information, see ref. 9)

One interesting food property that young physicist are likely to have seen is the Cheerios effect, named after a popular breakfast cereal.[4] The Cheerios effect describes the aggregation of objects floating on the surface of a liquid. The individual Cheerios sit on a meniscus that's lower than the liquid level, so neighbor Cheerios will "fall" towards each other; that is, they attract each other.

A demonstration of the Cheerios effect. The presence of the central hole is irrelevant to the attraction, but it aids in keeping the individual pieces afloat. (Photo by author.)

A decade ago, a pair of Harvard University (Cambridge, Massachusetts) physicists calculated the attractive force between floating objects for the idealized case of spheres.[4] Not surprisingly, the mathematics of this calculation is quite complex, involving a Bessel function, something I encountered just once in a permeability calculation. The force depends on the density, separation, and radius of the spheres, the surface tension, and the contact angle. It's easier to eat your Cheerios than analyze them.

Cheerios were always made from whole-grain oats, but wheat is a common contaminant of oats. General Mills, manufacturer of Cheerios, now certifies it to be gluten-free. (Photo by Conrad Irwin, via Wikimedia Commons.)

In further research on the Cheerios effect published earlier this year, physicists from Montclair State University (Montclair, New Jersey) and Harvard University found that aggregation is enhanced when the liquid temperature is increased. The size of the aggregate increased with temperature until all the floating particles are joined. This happens because the viscosity and surface tension of the liquid changes, and convection causes liquid flow.[5]

When you have many small particles floating on a large surface area, they don't aggregate into a single large island. Instead, they aggregate into smaller islands separated by large voids.[6] Our universe has a similar structure, with stars aggregating into galaxies separated by voids. In one case, it's surface tension that drives the attraction, while in the other it's gravitation.[6]

While the Cheerios effect is demonstrated as solid particles floating on a liquid, what's important isn't the liquid or solid, but the mechanical properties of the materials. In fact, the solids and liquids can be interchanged to have liquid droplets showing the Cheerios effect on a solid surface. This was demonstrated by scientists at the University of Twente (Enschede, the Netherlands), the University of Leiden (Leiden, The Netherlands), the Université Claude Bernard (Lyon, France), Queen Mary University of London (London, United Kingdom), the University of Maryland (College Park), the Université Paris–Diderot (Paris, France), and Eindhoven University of Technology (Eindhoven, The Netherlands).[7-8]

A similar interaction to the solid-on-liquid Cheerios effect can happen in the inverse case of liquid droplets on a solid surface. The trick to this is that the solid surface must be soft enough to mimic how the solid particles of the regular Cheerios effect sink into the liquid.[7] This inverse effect depends on a combination of two physical effects, capillarity and bulk elasticity.[7]

Still image from a video of water droplets moving on a gel substrate. (Queen Mary University of London image by S. Karpitschka.)

Interestingly, the interaction between droplets can be attractive, leading to drop–drop coalescence, or repulsive, preventing the merger of droplets, depending on the thickness of the solid layer, a fact that might be exploited technologically.[7] Says co-author Lorenzo Botto, a lecturer in the School of Engineering and Materials Science at Queen Mary University of London,

"...The physical phenomena we have highlighted in this paper suggest ways to design surfaces that prevent fogging or control heat transfer; for instance to create car windows that are always transparent despite high humidity or surfaces that improve heat management in conditioners or boilers. By making surfaces softer or harder, and changing the thickness of the soft layer, we will be able to control how the drops coalesce and spread on the substrate."[8]

References:

1. Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato and P. M. Chaikin, "Improving the Density of Jammed Disordered Packings Using Ellipsoids," Science, vol. 303, no. 5660 (February 13, 2004), pp. 990-993.
2. Heinrich M. Jaeger, "Why does shaking a can of coffee cause the larger grains to move to the surface?" Scientific American, March 24, 2003.
3. M. Bose, U. U. Kumar, P. R. Nott, and V. Kumaran, "Brazil nut effect and excluded volume attraction in vibrofluidized granular mixtures," Phys. Rev. E, vol. 72, Article No. 021305, August 25, 2005.
4. D. Vella and L. Mahadevan, "The 'Cheerios effect'," Am. J. Physics, vol. 73, p. 817, http://dx.doi.org/10.1119/1.1898523. Also at arXiv.
5. Eric Forgostona, Leo Hentschkerb, Siobhan Soltaua, Patrick Truitta, and Ashwin Vaidya, "Thermally induced aggregation of rigid spheres on a liquid surface," Physics Letters A, vol. 380, nos. 1–2 (January 8, 2016), pp. 227-231.
6. Eric Betz, "A well-known effect in breakfast cereal helps physicists understand the universe," Phys.org, September 9, 2010.
7. Stefan Karpitschka, Anupam Pandey, Luuk A. Lubbers, Joost H. Weijs, Lorenzo Botto, Siddhartha Das, Bruno Andreotti, and Jacco H. Snoeijer, "Liquid drops attract or repel by the inverted cheerios effect," Proceedings of the National Academy of Sciences, Early Edition, doi: 10.1073/pnas.1601411113. A PDF file is available here, and a version is available on arXiv.
8. Cereal science: how scientists inverted the Cheerios effect, Queen Mary University of London Press Release, June 14, 2016.
9. Soko Matsumura, Derek C. Richardson, Patrick Michel, Stephen R. Schwartz, and Ronald-Louis Ballouz, "The Brazil nut effect and its application to asteroids," Mon. Not. R. Astron. Soc., vol. 443, no. 4 (October 1, 2014), pp. 3368-3380. A PDF file is available here, and also at arXiv.