Packing, geometry & entropy: Crystallization of spheres in a sphere

When twelve equally sized spheres are packed around a central sphere, the smallest volume of the resulting cluster and thus the highest packing 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.

However, it has been proven only very recently that a close-packed arrangement of equally sized spheres, composed of stackings of 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 this close-packed arrangement, the local packing is less dense than the one with the icosahedral symmetry. A high local density is incompatible with a high global density.


In this talk, I will show that if spheres crystallize in a spherical confinement in experiments and simulations, clusters with icosahedral symmetry will be formed up to about 100.000 spheres [1]. Using simulations, I will show that the free energy of these icosahedral clusters is lower than that of the close-packed crystal for as many as 100.000 spheres. With larger numbers of spheres, the system crystallizes in the close-packed bulk crystal.

In the case of a mixture of two sizes of hard spheres, with a size ratio around 0.8, I will show that such a mixture of spheres would not form the bulk equilibrium MgZn2 phase, but instead a so-called binary icosahedral cluster consisting of MgCu2 crystalline domains [2]. The resulting clusters, a.k.a. supraparticles, can be used for a next self-assembly step in order to structure matter over multiple length scales.

References:

[1] Entropy-driven formation of large icosahedral colloidal clusters by spherical confinement
B. de Nijs, S. Dussi, F. Smallenburg, J.D. Meeldijk, D.J. Groenendijk, L. Filion, A. Imhof, A. van Blaaderen, and M. Dijkstra, Nature Materials 14, 56-60 (2015).

[2] Binary icosahedral clusters of hard spheres in spherical confinement
D. Wang, T. Dasgupta, E.B. van der Wee, D. Zanaga, T. Altantzis, Y. Wu, G.M. Coli, C.B. Murray, S. Bals, M. Dijkstra and A. van Blaaderen, Nature Physics 17, 128–134 (2021).

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