The Ancient Physics Quest of ‘Perfect Packing’ Has a New Solution

Physicists can observe and measure individual subatomic particles, build self-assembling nanomachines inside of living organisms, and trap antimatter in tiny cages—and yet there are macroscale, everyday phenomena that seem impossible to physically explain. One frustrating example is the behavior of granular materials under pressure: sand under a foot, the seismic waves of an earthquake traveling through soil, a precariously unstable slab of snow. All mysteries, in some large part.

"If you walk on sand, it supports your weight," write a team of Duke researchers in a new study. "How do the disordered forces between particles in sand organize, to keep you from sinking? This simple question is surprisingly difficult to answer experimentally: measuring forces in three dimensions, between deeply buried grains, is challenging."

As described in this week's Nature Communications, the Duke team has devised a new method of imaging the effects of forces—pressure in particular—on granular materials. The newfound ability comes in part thanks to a new computer algorithm, but also an advanced array of sensors, lasers, and digital imaging.

Part of the challenge lies in the unusual coupling between very macroscale physical forces (the weight of a human, for example) with relatively microscale objects (sand, packed snow crystals). So it's a translation of very big things to very small things, which then again have very big effects (supporting a human weight, transmitting seismic waves).

What Duke's Nicolas Brodu and company came up with is a calculation linking the two worlds, a technology-enabled breakthrough after centuries of research that began with a 17th century clergyman and chemist, Stephen Hales, puzzling over the expansion of dried peas soaking in water. In his volume Vegetable Staticks, Hales wrote:

I compressed several fresh parcels of Pease in the same Pot, with a force equal to 1600, 800, and 400 pounds; in which Experiments, tho' the Pease dilated, yet they did not raise the lever, because what they increased in bulk was, by the great incumbent weight, pressed into the interstices of the Pease, which they adequately filled up, being thereby formed into pretty regular Dodecahedrons.

The general conundrum is known in math as a packing problem—in which empty space is minimized—and the ultimate example is sand. How does sand, a seemingly fluid collection of material, become firm and resilient under pressure? The question is the source of deeply challenging "mathematical instabilities," Brodu and team explain.

"The motivation for such studies is clear," the Duke team writes. "Packings of particles surround us: coffee beans and rice; soils, embankments, many industrial processes, and geophysical processes from earthquakes to landslides involve granular materials."

Part of the difficulty is in achieving full access to the granular material's microstructure as it evolves in time in response to various surface deformations. How does one single grain affect another single grain and how does that all come together into a unified picture?

The answer comes courtesy of an imaging method known as refractive index matching tomography, which is a bit like an MRI scan. The granular material is imaged in 2D slices, one by one, until they can all be put together into an extremely detailed cross-sectional picture of the material as a whole with all of its constituent grains. The researchers then used these images to observe the 3D structure of those grains and the deformations that result of the material being squeezed and packed (by feet or whatever).

"Our work crucially adds the ability to measure contact forces in vectorial detail while straining the sample, enabling us to track the system-scale stress tensor properties over small strain steps," Brodu and his group write. "This feature gives access to the complete micro-macro range of mechanical details of the packing, including ingredients for constitutive modeling and its underlying physical mechanisms."

Hales would be proud, as would, uh, Jesus, who in the actual Bible (via the actual physics textbook The Pursuit of Perfect Packing) describes a good measure as being, "packed down, shaken together, running over."

Finally, Brodu offers in a statement, "This gives us hope of understanding what happens in disasters like a landslide, when packed soil and rocks on a mountain become loose and slide down. First it acts like a solid, and then for reasons physicists don't completely understand, all of a sudden it destabilizes and starts to flow like a liquid. This transition from solid to liquid can only be understood if you know what's going on inside the soil."

An open-access preprint version of the study can be accessed at arXiv.