- Physics Planet
- Wednesday September 20, 2017

Earlier this year, Paul Chaikin and his colleagues discovered a new finding; namely that oblate spheroids pack more densely than regular shaped spheres when packed randomly. That is where the M&Ms fit in – as they are oblated spheroids.

The oblated spheroids that were used in the experiment were M&Ms. The Mars Company donated 125 pounds of almond M&Ms to Chaikin, but the Mars Company had no financial investment to the research. Regular M&Ms are uniform in shape so they were used in an experiment for testing random packing theories.

A collaboration of mathematics and physics was the foundation of the experiment. The randomly packed M&Ms were found to be even denser than the perfectly packed or ordered “arrangement” of spheres.

Before Chaikin and his colleagues discovered this new finding, it was never known that randomly packed particles could be so dense. The relationships of particle shape, packing density and pouring or shaking methods have been studied in the past, but no new major findings emerged until this past February.

The National Science Foundation, the National Aeronautics and Space Administration, and the Petroleum Research Fund did support the work of Chaikin and his colleagues. The findings have significant impact for those who use granular particles.

When you change the sphere by elongating it, then you have this new surprising density packing. When put randomly into a container, spheres take up about 64% of the space of the container. However, the elongated spheres or spheroids take up approximately 68%.

Particle packing was first studied by Johannes Kepler in the 16th century. In 1998, it was proven by a mathematician that the densest arrangement of spheres would fill 74.04% of the total space. That is the densest arrangement possible and not random. Kepler had predicted it way back in the 16th century.

Chaikin is a Princeton professor, who had always had a real fascination with M&Ms. He asked a student, Evan Variaro, to measure the density of random packed M&Ms by counting the number of M&Ms in a container.

Another student, Ibrahim Cisse, further studied the effect by pouring paint in the container. After the paint dried, he counted the dots on each M&M that did not have paint on it. These dots were the contact points.

In order to counter the theory that the M&Ms somehow assume a particular order, an MRI was done. The scan of the container proved that the M&Ms were random.

Spheroids and ellipsoids (almond M&Ms) have a larger number of contact points with their surrounding particles. This, therefore, means the packing is denser.

Chaikin and his colleagues will continue to study random packing density. The findings could be helpful to any industry involving granular materials. Implications for the findings can impact ceramic materials where density of the materials forming the ceramics is important, particularly for aerospace applications.

The researchers, Chaikin and Salvatore Torquato, a Princeton chemist, worked together to understand particle packing. Torquato and his student, Aleksandar Donev worked on a computer simulation to test any shape from a sphere to spheroid to an elongated spheroid, which is a cigar-like shape. They found that when you elongated the sphere, the particles pack even more densely.

So, if you want to figure out how many M&Ms fit into a container, here is the formula. Take the volume of the container in cubic centimeters and multiply .68. Then divide the result by .636 (which is the volume of a single M&M).

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