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Shake 'N Pack: How the M&M Got Its Shape

Article written by Kun-Chun Lee, an indie scientist who has worked in various fields of basic science, including polymer physics, colloids, cellular bio-mechanics, and numerical algorithms. When he is not working, he can be found drinking coffee at Roos' Cafe.

Title:  "Shake 'n Pack:  How the M&M Got Its Shapes"

-By Kun-Chun Lee

Here is a simple experiment to try at home. Take two gallon-sized empty pretzel jars (like the ones you get from Costco) and fill one with spherically shaped marbles and the other with M&M candy-like flattened marbles. Shake the jars a few times to create more space, add more of the same shaped marble to each, and repeat until you have completely fill the jars. Now fill each jar with water, pour the water out into a clean graduate cylinder, and measure how much volume of water was in each jar to determine how much volume was occupied by each type of marble.

This may look like an odd experiment but it is exactly what a group of Princeton scientists did using more sophisticated methods [1]. The scientists were studying a concept called random close packing (RCP), which has a very long tradition dating back to the time of Kepler and Newton. It is a fairly loose concept that roughly means the maximum volume density obtainable when one randomly dump stuffs into a box. Surprisingly, the study of how things pack randomly is important for many industrial applications, like the design of grain silos, liquid crystal displays (LCD), crystallization of salts, size of bubble-gum machines, mechanical stability of shaving creams, development of robotic hand, fragility of glass and of course the storage of M&M candy. In fact, the group studied a series of spheroidal and ellipsoidal shaped objects and concluded that M&M candies (both with, ellipsoid-shaped, and without peanut, oblate-shaped) are closest to the most densely packed possible in their study, a most intriguing result. The M&M company neither admitted nor refuted the claim that the M&M engineers specifically designed the candies that way although one might speculate that they did in order to save space on storage for their mega-sized bags.

One of the main obstacle to studying RCP is making the random samples, how to be sure that a sample packing is random.  One definition might be that a different result is obtained with each experiment, assuming one follows the protocol exactly. However, this is true with all practical experiments anyway, you always get a distribution of results that you hope with repeated experiment will average to some well defined number. Many natural phenomenon have this property called self-averaging, meaning that not only is the average defined, so are higher moments of deviations from the average so that the error bar (standard error) decreases as more samples are studied.  RCP, as is typical of highly disordered system, is an exception, one of its defining feature is that it is none self-averaging, namely the error bar does not vanish as more samples are studied, making it one of the toughest problems to crack. At the same time, it provides a way to define RCP if not exactly and it suggests that RCP can be studied using methods from other none self-averaging systems. Further studies have concluded that RCP is in fact a very special condition (the technical term is that it is a jamming critical point) that gives rise to glass-like mechanical property to the whole structure because of its random geometry: The resulting material is fragile and hard like glass, but small changes in density and/or (shear) stress can lead to dramatic mechanical transformations [2].

The idea of putting a bunch of stuffs inside a jar and studying their packing behavior is probably as old as the invention of a jar. Toddlers seem to be the experts at this game  as they enjoy putting everything inside  boxes and shake, including themselves. If you enjoy packing and shaking things too, I suggest putting a bunch of toothpicks inside a jar and shake it to see what happens as you add more toothpicks.

suggested additional readings:

1.  Candy-coated Research Leads to Discovery in Physics, Steven Schultz (https://www.princeton.edu/pr/pwb/04/0223/)

2.  Robotic Gripper Made from Coffee-Filled Balloon Picks Up Anything, Erico Guizzo (http://spectrum.ieee.org/automaton/robotics/industrial-robots/universal-jamming-gripper)

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