Physicists solve 150-year-old mystery of equation governing sandcastle physics
December 10, 2020
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The secret to a stable sandcastle primarily lies in the right proportion of water to sand. Mathematically, the forces at play are described by the
Enlarge / The secret to a stable sandcastle primarily lies in the right proportion of water to sand. Mathematically, the forces at play are described by the “Kelvin equation,” first referenced in 1871.

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Building sandcastles at the beach is a time-honored tradition around the world, elevated into an art form in recent years thanks to hundreds of annual competitions. While the basic underlying physics is well-known, physicists have continued to gain new insights into this fascinating granular material over the last decade or so. The latest breakthrough comes from Nobel Laureate Andre Geim’s laboratory at the University of Manchester in England, where Geim and his colleagues have solved a mathematical puzzle—the “Kelvin equation“—dating back 150 years, according to a new paper just published in Nature.

All you really need to make a sandcastle is sand and water; the water acts as a kind of glue holding the grains of sand together via capillary forces. Studies have shown that the ideal ratio for building a structurally sound sandcastle is one pail of water for every eight pails of sand, although it’s still possible to build a decent structure with varying water content. But if you want to build the kind of elaborate, towering sandcastles that win competitions, you’d be wise to stick with that ideal ratio.

Back in 2008, physicists decided to delve a little deeper into why sand becomes sticky when it gets wet. Using X-ray microtomography, they took 3D images of wet glass beads of similar shape and size as grains of sand. When they added liquid to dry beads, they observed liquid “capillary bridges” forming between individual beads. Adding more liquid caused the bridges to grow larger, and as that happened, the bead surfaces came into contact with more water, further increasing the binding effect. However, the increased binding effect was canceled out by a corresponding decrease of the capillary forces as the bridge structures grew bigger. The team concluded that even if the moisture content changes, the forces binding the beads together do not change.

It’s similar to how soap bubbles will tend to be spherical because that’s the shape that minimizes the total surface area, thereby using the least energy, according to Daniel Bonn, a physicist at the University of Amsterdam who has conducted several experiments with sand over the years. Bonn has become something of an expert on what’s involved with building the perfect sandcastle. “Likewise, a small amount of water between two sand grains forms a small liquid bridge that minimizes the surface area between the water and the air,” he told Vice in 2015. “If one then moves one grain with respect to the other, one automatically creates surface area. This costs energy, and therefore there will be a resistance to deformation.”

Mathematically, this kind of capillary condensation—i.e., how water vapor from ambient air will condense spontaneously inside porous materials or between touching surfaces—is typically described by an equation devised by Sir William Thompson (later Lord Kelvin) and first referenced in an 1871 paper. It’s a macroscopic equation that nonetheless has proven to be remarkably accurate down to the 10-nanometer scale, but the lack of a complete description that can account for even tinier scales has long frustrated physicists.

Typical humidity for this kind of condensation is between 30 and 50 percent, but at molecular scales of 1 nanometer or less (a water molecule is about 0.3nm in diameter), only one or two molecular layers of water would be able to fit inside 1nm-thick capillaries. At that scale, the Kelvin equation did not appear to make sense. That might not matter for building sandcastles, but capillary condensation is also relevant to many microelectronic, pharmaceutical, and food processing industries. Geim and his colleagues found a way to overcome the longstanding experimental challenges of studying capillaries at the molecular scale.

Geim won the 2010 Nobel Prize in Physics for his groundbreaking experiments on graphene, a thin flake of ordinary carbon just one atom thick, giving the material unusual properties. Physicists struggled to isolate graphene from graphite (just like that found in pencils), but Geim and his Manchester colleague Konstantin Novoselov developed a novel method using used Scotch tape to collect the atom-thick flakes from graphite. He also won an Ig Nobel prize for his discovery of direct diamagnetic levitation of water—work that famously involved using magnets to levitate a frog in the lab. And he once created a gecko-inspired sticky tape strong enough to suspend a Spider-Man action figure from the ceiling indefinitely.

For this latest work, Geim’s team painstakingly constructed molecular-scale capillaries by layering atom-thin crystals of mica and graphite on top of each other, with narrow strips of graphene in between each layer to serve as spacers. With this method, the team built capillaries of varying height, including capillaries that were just one atom high—just enough to fit one layer of water molecules, the smallest such structure possible.

Geim et al. found that the Kelvin equation is still an excellent qualitative description of capillary condensation at the molecular scale—contradicting expectations, since the properties of water are expected to become more discrete and layered at the 1nm scale. Apparently in that regime, there are microscopic adjustments to the capillaries, which suppress any additional effects that might otherwise cause the equation to break down as expected.

“This came as a big surprise. I expected a complete breakdown of conventional physics,” said co-author Qian Yang. “The old equation turned out to work well. A bit disappointing but also exciting to finally solve the century old mystery. So we can relax, all those numerous condensation effects and related properties are now backed by hard evidence rather than a hunch that ‘it seems to work so therefore it should be OK to use the equation.'”

“Good theory often works beyond its applicability limits,” said Geim. “Lord Kelvin was a remarkable scientist, making many discoveries but even he would surely be surprised to find that his theory—originally considering millimeter-sized tubes—holds even at the one-atom scale. In fact, in his seminal paper Kelvin commented about exactly this impossibility. So our work has proved him both right and wrong, at the same time.”

DOI: Nature, 2020. 10.1038/s41586-020-2978-1  (About DOIs).

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