You might think that how to pack spheres into as small a space as possible would be exactly known by now — but you’d be surprised. The most efficient packing of an infinite number of identical spheres was only definitively proved in the twenty-first century; if you have a finite number of spheres everything gets more complicated, especially if you have a practical problem to solve. Mathematicians have proven how best to pack spheres in 42 dimensions (or more!) but in the ordinary 3-dimensional world, the situation is murkier. For fewer than 55 spheres, there is good evidence that the most efficient packing puts the spheres in a 1D line, in what mathematicians call a sausage packing (think tennis balls in a tube). For more than 65 spheres it has been proved that there are 3D clusters of spheres that are more space-efficient than the sausage packing, although the actual structure of these clusters isn’t always known. And for packing some numbers between 55 and 65 spheres, sometimes the most efficient known packing is linear, whereas for other numbers, it is a cluster. The sudden unexplained transition from the optimal packing being a sausage to a cluster is known as the ‘sausage catastrophe’.
The experiments involve putting micrometre-scale spherical particles inside giant unilamellar vesicles (GUVs) — essentially, elastic containers — of different volumes, and then imaging the conformations using fast confocal scanning microscopy. Packing large numbers of particles into GUVs is technically challenging, so the experiments are limited to 3–9 particles. The particles are small enough that they undergo Brownian motion, which also leads to fluctuations in the GUV membranes. For some combinations of sphere number and vesicle volume, there is one stable conformation of the particles, which can be 1D (sausages), 2D (particles in a plane) or 3D (clusters). For other combinations, the conformations exhibit bistability; that is, they fluctuate between 1D and 2D arrangements, or 2D and 3D. Studying the stability of packings doesn’t tell you what the most efficient packing is, but it does give you an idea of how real-life Brownian spheres behave, which is useful for understanding problems such as self-assembly.
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