Icosahedral symmetry, which is not compatible with truly long-range order, can be found in many systems, such as liquids, glasses, atomic clusters, quasicrystals and virus-capsids1,2,3,4,5,6,7,8,9,10,11,12. To obtain arrangements with a high degree of icosahedral order from tens of particles or more, interparticle attractive interactions are considered to be essential1,3,6,7,8,9,10,11,12. Here, we report that entropy and spherical confinement suffice for the formation of icosahedral clusters consisting of up to 100,000 particles. Specifically, by using real-space measurements on nanometre- and micrometre-sized colloids, as well as computer simulations, we show that tens of thousands of hard spheres compressed under spherical confinement spontaneously crystallize into icosahedral clusters that are entropically favoured over the bulk face-centred cubic crystal structure13,14. Our findings provide insights into the interplay between confinement and crystallization and into how these are connected to the formation of icosahedral structures.
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Nelson, D. R. & Halperin, B. I. Pentagonal and icosahedral order in rapidly cooled metals. Science 229, 233–238 (1985).
Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784–805 (1983).
Doye, J. P. K. & Wales, D. J. The structure and stability of atomic liquids: From clusters to bulk. Science 271, 484–487 (1996).
Karayiannis, N. C., Malshe, R., Kröger, M., de Pablo, J. J. & Laso, M. Evolution of fivefold local symmetry during crystal nucleation and growth in dense hard-sphere packings. Soft Matter 8, 844–858 (2012).
Bernal, J. D. & Finney, J. L. Random close-packed hard-sphere model. II. Geometry of random packing of hard spheres. Discuss. Faraday Soc. 43, 62–69 (1967).
Mackay, A. L. A dense non-crystallographic packing of equal spheres. Acta Crystallogr. 15, 916–918 (1962).
Doye, J. P. K. & Wales, D. J. Thermally-induced surface reconstructions of Mackay icosahedra. Z. Phys. D 40, 466–468 (1997).
Meng, G., Arkus, N., Brenner, M. P. & Manoharan, V. N. The free-energy landscape of clusters of attractive hard spheres. Science 327, 560–563 (2010).
Hendy, S. C. & Doye, J. P. K. Surface-reconstructed icosahedral structures for lead clusters. Phys. Rev. B 66, 235402 (2002).
Wang, Y., Teitel, S. & Dellago, C. Melting of icosahedral gold nanoclusters from molecular dynamics simulations. J. Chem. Phys. 122, 214722 (2005).
Lacava, J., Born, P. & Kraus, T. Nanoparticle clusters with Lennard-Jones geometries. Nano Lett. 12, 3279–3282 (2012).
Frank, F. C. Supercooling of liquids. Proc. R. Soc. Lond. A 215, 43–46 (1952).
Pusey, P. N. & Van Megen, W. Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. Nature 320, 340–342 (1986).
Bolhuis, P. G., Frenkel, D., Mau, S-C. & Huse, D. A. Entropy difference between crystal phases. Nature 388, 235–236 (1997).
Conway, J. H. & Sloane, N. J. A. Sphere Packings, Lattices and Groups (Springer, 1998).
McGinley, J. T., Jenkins, I., Sinno, T. & Crocker, J. C. Assembling colloidal clusters using crystalline templates and reprogrammable DNA interactions. Soft Matter 9, 9119–9128 (2013).
O’Malley, B. & Snook, I. Crystal nucleation in the hard sphere system. Phys. Rev. Lett. 90, 085702 (2003).
Hubert, H. et al. Icosahedral packing of B12 icosahedra in boron suboxide (B6O). Nature 391, 376–378 (1998).
Doye, J. P. K. & Calvo, F. Entropic effects on the structure of Lennard-Jones clusters. J. Chem. Phys. 116, 8307–8317 (2002).
Hofmeister, H. Forty years study of fivefold twinned structures in small particles and thin films. Cryst. Res. Technol. 33, 3–25 (1998).
Langille, M. R., Zhang, J., Personick, M. L., Li, S. & Mirkin, C. A. Stepwise evolution of spherical seeds into 20-fold twinned icosahedra. Science 337, 954–957 (2012).
Tang, J. et al. Hard-sphere packing and icosahedral assembly in the formation of mesoporous materials. J. Am. Chem. Soc. 129, 9044–9048 (2007).
Wales, D. J. Surveying a complex potential energy landscape: Overcoming broken ergodicity using basin-sampling. Chem. Phys. Lett. 584, 1–9 (2013).
Fortini, A. & Dijkstra, M. Phase behaviour of hard spheres confined between parallel hard plates: Manipulation of colloidal crystal structures by confinement. J. Phys. Condens. Matter 18, L371–L378 (2006).
Löwen, H., Oguz, E. C., Assoud, L. & Messina, R. Colloidal crystallization between two and three dimensions. Adv. Chem. Phys. 148, 225–249 (2012).
Bodnarchuk, M. I. et al. Exchange-coupled bimagnetic wüstite/metal ferrite core/shell nanocrystals: Size, shape, and compositional control. Small 5, 2247–2252 (2009).
Van Blaaderen, A. & Vrij, A. Synthesis and characterization of colloidal dispersions of fluorescent, monodisperse silica spheres. Langmuir 8, 2921–2931 (1992).
Farges, J., De Feraudy, M. F., Raoult, B. & Torchet, G. Noncrystalline structure of argon clusters. II. Multilayer icosahedral structure of ArN clusters 50 < N < 750. J. Chem. Phys. 84, 3491–3501 (1986).
Friedrich, H. et al. Quantitative structural analysis of binary nanocrystal superlattices by electron tomography. Nano Lett. 9, 2719–2724 (2009).
Van Blaaderen, A. & Wiltzius, P. Real-space structure of colloidal hard-sphere glasses. Science 270, 1177–1179 (1995).
Schilling, T. & Schmid, F. Computing absolute free energies of disordered structures by molecular simulation. J. Chem. Phys. 131, 231102 (2009).
Cacciuto, A., Auer, S. & Frenkel, D. Onset of heterogeneous crystal nucleation in colloidal suspensions. Nature 428, 404–406 (2004).
Manoharan, V. N., Elsesser, M. T. & Pine, D. J. Dense packing and symmetry in small clusters of microspheres. Science 301, 483–487 (2003).
Lauga, E. & Brenner, M. P. Evaporation-driven assembly of colloidal particles. Phys. Rev. Lett. 93, 238301 (2004).
Bai, F. et al. A versatile bottom-up assembly approach to colloidal spheres from nanocrystals. Angew. Chem. Int. Ed. 46, 6650–6653 (2007).
We thank R. J. A. Moes (who is funded by the FOM programme Control over Functional Nanoparticle Solids (FNPS)) for synthesis of the semiconductor particles, A. Kuijk for the synthesis of the silica colloids, and T. H. Besseling for the two-dimensional tracking. We thank J. R. Edison, W. Vlug and R. v. Roij for critical reading of the manuscript. B.d.N. acknowledges financial support from a ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek’ (NWO) CW grant. M.D. and F.S. acknowledge financial support from an NWO-VICI grant. S.D. and M.D. acknowledge financial support from an NWO-CW-Echo grant.
The authors declare no competing financial interests.
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de Nijs, B., Dussi, S., Smallenburg, F. et al. Entropy-driven formation of large icosahedral colloidal clusters by spherical confinement. Nature Mater 14, 56–60 (2015) doi:10.1038/nmat4072
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