Packing, geometry & entropy: unexpected crystallization of mixtures of spheres of two sizes in spherical confinement

How particles pack under different constraints is not only of fundamental interest to physics, chemistry and materials science, but also to mathematics. A team of researchers has now shown that mixing particles of two different sizes inside a sphere results in unexpected, cubic structures, as they describe today in a publication in Nature Physics. These findings do not only lead to new fundamental insights into packing and crystallization under frustration, but may also form a basis for the construction of photonic crystals with new properties.

Packing thirteen spheres

When packing twelve equally sized spheres around a central sphere, the smallest volume of the resulting cluster and thus the highest packing fraction is obtained for a regular icosahedron, a shape with twenty equilateral triangles as faces and a five-fold symmetry. This five-fold symmetry makes this packing interesting, because it can be proven that clusters with such a symmetry cannot fully pack 3D space. This has been known for a long time.

However, only twenty years ago, it has been proven that a close packed arrangement of equally sized spheres, composed of stacked hexagonal layers of spheres, is the closest packed arrangement in 3D of any single sized sphere packing. However, if the twelve spheres are arranged in the closest packing that fills all of the available space, the local packing is less dense than the one with the icosahedral symmetry. In short: a high local density is incompatible with a high global density.

Icosahedral symmetry

The current publication is based on earlier research supervised by Prof. Alfons van Blaaderen and Prof. Marjolein Dijkstra, published in Nature Materials in 2015. These experiments showed that if one lets hard spheres crystallize in a spherical confinement (in the experiments realized through a slowly drying emulsion droplet with nanoparticles inside) one forms crystals with icosahedral symmetry until roughly 100.000 spheres.

With larger numbers of spheres, the system crystallizes in the close packed bulk crystal. For particles with hard interactions it is only entropy that determines if a spontaneous phase transition like crystallization can take place. The experiments and simulations proved that the entropy of an icosahedral crystal in a spherical confinement is higher than that of a close packed crystal even for as many of 100.000 spheres.

As explained, this was expected to be the case for 13 particles, but it amazed many in the field that this would extend to so many spheres. This behaviour is caused by the fact that icosahedral crystals cannot pack in ordinary 3D space, but can do so better if one adds the frustration of  a spherical confinement.

Mixture of two sizes

These results have now been extended to a mixture of two sizes of hard spheres, with a size ratio around 0.8. The researchers found that such a mixture of spheres would also not form the bulk equilibrium phase (which is a crystal analogous to that of MgZn₂), but instead a so-called binary icosahedral crystal.

New photonic crystals

These extended findings of icosahedral crystal formation induced by the frustration of being confined to crystallize inside a sphere lead to new fundamental insights into packing and crystallization under frustration. Additionally, the results may be used to construct so-called photonic crystals with new properties, if the crystals are made with particles with a size of several hundred nanometres instead of the nanoparticles used in this study. Recently, the Soft Condensed Matter group, together with collaborating groups from both the Debye Institute and the University of Twente, obtained funding to pursue exactly this exciting research direction.

Researchers

The experiments were carried out by Da Wang and Ernest van der Wee, in close collaboration with computer simulator Tonnishtha Dasgupta, under supervision of Profs. Van Blaaderen and Dijkstra and in collaboration with the groups of Prof. Sara Bals (University of Antwerp), who is an expert in electron microscopy tomography, and experts on binary crystallization from the group of Prof. Chris Murray (University of Pennsylvania).

Publication

Binary icosahedral clusters of hard spheres in spherical confinement
Da Wang*, Tonnishtha Dasgupta*, Ernest B. van der Wee*, Daniele Zanaga, Thomas Altantzis, Yaoting Wu, Gabriele M. Coli*, Christopher B. Murray, Sara Bals, Marjolein Dijkstra* and Alfons van Blaaderen*
Nature Physics, 31 August 2020, DOI 10.1038/s41567-020-1003-9
* researchers affiliated with Utrecht Universit