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Binary icosahedral clusters of hard spheres in spherical confinement

Abstract

The influence of geometry on the local and global packing of particles is important to many fundamental and applied research themes, such as the structure and stability of liquids, crystals and glasses. Here we show by experiments and simulations that a binary mixture of hard-sphere-like nanoparticles crystallizing into a MgZn2 Laves phase in bulk spontaneously forms icosahedral clusters in slowly drying droplets. Using advanced electron tomography, we are able to obtain the real-space coordinates of all the spheres in the icosahedral clusters of up to about 10,000 particles. The local structure of 70–80% of the particles became similar to that of the MgCu2 Laves phase. These observations are important for photonic applications. In addition, we observed in simulations that the icosahedral clusters nucleated away from the spherical boundary, which is distinctly different from that of the single species clusters. Our findings open the way for particle-level studies of nucleation and growth of icosahedral clusters, and of binary crystallization.

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Fig. 1: Binary Laves crystal structures.
Fig. 2: Binary SPs as obtained from experiments and computer simulations.
Fig. 3: Diffraction pattern of a simulated binary icosahedral SP.
Fig. 4: Real-space structure of the binary icosahedral cluster.
Fig. 5: Nucleation and growth of the binary icosahedral cluster in simulations.

Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

Code availability

Computer simulations were performed using HOOMD-blue, available at http://glotzerlab.engin.umich.edu/hoomd-blue/.

References

  1. 1.

    1. Turnbull, D. in Solid State Physics (eds Seitz, F. & Turnbull, D.) Vol. 3, 225–306 (Elsevier, 1956).

  2. 2.

    Frank, F. C. Supercooling of liquids. Proc. R. Soc. Lond. A 215, 43–46 (1952).

    ADS  Article  Google Scholar 

  3. 3.

    Taffs, J. & Royall, C. P. The role of fivefold symmetry in suppressing crystallization. Nat. Commun. 7, 13225 (2016).

    ADS  Article  Google Scholar 

  4. 4.

    Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784–805 (1983).

    ADS  Article  Google Scholar 

  5. 5.

    Nelson, D. R. & Spaepen, F. in Solid State Physics (eds Ehrenreich, H. & Turnbull, D.) Vol. 42, 1–90 (Elsevier, 1989).

  6. 6.

    Spaepen, F. Condensed-matter science: five-fold symmetry in liquids. Nature 408, 781–782 (2000).

    ADS  Article  Google Scholar 

  7. 7.

    Van Blaaderen, A. & Wiltzius, P. Real-space structure of colloidal hard-sphere glasses. Science 270, 1177–1179 (1995).

    ADS  Article  Google Scholar 

  8. 8.

    Gasser, U., Weeks, E. R., Schofield, A., Pusey, P. N. & Weitz, D. A. Real-space imaging of nucleation and growth in colloidal crystallization. Science 292, 258–262 (2001).

    ADS  Article  Google Scholar 

  9. 9.

    Kegel, W. K. & van Blaaderen, A. Direct observation of dynamical heterogeneities in colloidal hard-sphere suspensions. Science 287, 290–293 (2000).

    ADS  Article  Google Scholar 

  10. 10.

    Cheng, Z. Colloidal Crystallization Ch. 12, 203–248 (John Wiley & Sons, 2016).

  11. 11.

    Fernandez-Nieves, A. & Puertas, A. M. Fluids, Colloids and Soft Materials: An Introduction to Soft Matter Physics Vol. 7 (John Wiley & Sons, 2016).

  12. 12.

    Wintzheimer, S. et al. Supraparticles: functionality from uniform structural motifs. ACS Nano 12, 5093–5120 (2018).

    Article  Google Scholar 

  13. 13.

    Wang, T., LaMontagne, D., Lynch, J., Zhuang, J. & Cao, Y. C. Colloidal superparticles from nanoparticle assembly. Chem. Soc. Rev. 42, 2804–2823 (2013).

    Article  Google Scholar 

  14. 14.

    De Nijs, B. et al. Entropy-driven formation of large icosahedral colloidal clusters by spherical confinement. Nat. Mater. 14, 56–60 (2015).

    ADS  Article  Google Scholar 

  15. 15.

    Lacava, J., Born, P. & Kraus, T. Nanoparticle clusters with Lennard–Jones geometries. Nano Lett. 12, 3279–3282 (2012).

    ADS  Article  Google Scholar 

  16. 16.

    Yang, Y. et al. Scalable assembly of crystalline binary nanocrystal superparticles and their enhanced magnetic and electrochemical properties. J. Am. Chem. Soc. 140, 15038–15047 (2018).

    Article  Google Scholar 

  17. 17.

    Wang, J. et al. Magic number colloidal clusters as minimum free energy structures. Nat. Commun. 9, 5259 (2018).

    ADS  Article  Google Scholar 

  18. 18.

    Wang, D. et al. Interplay between spherical confinement and particle shape on the self-assembly of rounded cubes. Nat. Commun. 9, 2228 (2018).

    ADS  Article  Google Scholar 

  19. 19.

    Wang, J. et al. Free energy landscape of colloidal clusters in spherical confinement. ACS Nano 13, 9005–9015 (2019).

    Article  Google Scholar 

  20. 20.

    Laves, F. & Witte, H. Der einfluß von valenzelektronen auf die kristallstruktur ternärer magnesiumlegierungen. Metallwirtschaft 15, 840–842 (1936).

    Google Scholar 

  21. 21.

    Berry, R. L. & Raynor, G. V. The crystal chemistry of the Laves phases. Acta Cryst. 6, 178–186 (1953).

    Article  Google Scholar 

  22. 22.

    Hynninen, A.-P., Filion, L. & Dijkstra, M. Stability of LS and LS2 crystal structures in binary mixtures of hard and charged spheres. J. Chem. Phys. 131, 064902 (2009).

    ADS  Article  Google Scholar 

  23. 23.

    Hynninen, A.-P., Thijssen, J. H. J., Vermolen, E. C. M., Dijkstra, M. & van Blaaderen, A. Self-assembly route for photonic crystals with a bandgap in the visible region. Nat. Mater. 6, 202–205 (2007).

    ADS  Article  Google Scholar 

  24. 24.

    Dong, A., Chen, J., Vora, P. M., Kikkawa, J. M. & Murray, C. B. Binary nanocrystal superlattice membranes self-assembled at the liquid–air interface. Nature 466, 474–477 (2010).

    ADS  Article  Google Scholar 

  25. 25.

    Saghi, Z. & Midgley, P. A. Electron tomography in the (S)TEM: from nanoscale morphological analysis to 3D atomic imaging. Annu. Rev. Mater. Res. 42, 59–79 (2012).

    ADS  Article  Google Scholar 

  26. 26.

    Bals, S., Goris, B., Liz-Marzán, L. M. & Van Tendeloo, G. Three-dimensional characterization of noble-metal nanoparticles and their assemblies by electron tomography. Angew. Chem. Int. Ed. 53, 10600–10610 (2014).

    Article  Google Scholar 

  27. 27.

    Zanaga, D. et al. Quantitative 3D analysis of huge nanoparticle assemblies. Nanoscale 8, 292–299 (2016).

    ADS  Article  Google Scholar 

  28. 28.

    Wang, P.-p, Qiao, Q., Zhu, Y. & Ouyang, M. Colloidal binary supracrystals with tunable structural lattices. J. Am. Chem. Soc. 140, 9095–9098 (2018).

    Article  Google Scholar 

  29. 29.

    Chen, O. et al. Magneto-fluorescent core-shell supernanoparticles. Nat. Commun. 5, 5093 (2014).

    ADS  Article  Google Scholar 

  30. 30.

    Kister, T., Mravlak, M., Schilling, T. & Kraus, T. Pressure-controlled formation of crystalline, Janus, and core–shell supraparticles. Nanoscale 8, 13377–13384 (2016).

    ADS  Article  Google Scholar 

  31. 31.

    Yang, Z. et al. Supracrystalline colloidal eggs: epitaxial growth and free standing three-dimensional supracrystals in nanoscaled colloidosomes. J. Am. Chem. Soc. 138, 3493–3500 (2016).

    Article  Google Scholar 

  32. 32.

    Anderson, J. A., Lorenz, C. D. & Travesset, A. General purpose molecular dynamics simulations fully implemented on graphics processing units. J. Comput. Phys. 227, 5342 – 5359 (2008).

    MATH  Article  Google Scholar 

  33. 33.

    Glaser, J. et al. Strong scaling of general-purpose molecular dynamics simulations on GPUs. Comput. Phys. Commun. 192, 97 – 107 (2015).

    Article  Google Scholar 

  34. 34.

    Bergman, G., Waugh, J. L. & Pauling, L. The crystal structure of the metallic phase Mg32(Al, Zn)49. Acta Cryst. 10, 254–259 (1957).

    Article  Google Scholar 

  35. 35.

    Lechner, W. & Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. J. Chem. Phys. 129, 114707 (2008).

    ADS  Article  Google Scholar 

  36. 36.

    Yang, Z. et al. Precipitation of binary quasicrystals along dislocations. Nat. Commun. 9, 809 (2018).

    ADS  Article  Google Scholar 

  37. 37.

    Ducrot, É., He, M., Yi, G.-R. & Pine, D. J. Colloidal alloys with preassembled clusters and spheres. Nat. Mater. 16, 652–657 (2017).

    ADS  Article  Google Scholar 

  38. 38.

    Wang, Y., Jenkins, I. C., McGinley, J. T., Sinno, T. & Crocker, J. C. Colloidal crystals with diamond symmetry at optical lengthscales. Nat. Commun. 8, 14173 (2017).

    ADS  Article  Google Scholar 

  39. 39.

    Pietryga, J. M. et al. Utilizing the lability of lead selenide to produce heterostructured nanocrystals with bright, stable infrared emission. J. Am. Chem. Soc. 130, 4879–4885 (2008).

    Article  Google Scholar 

  40. 40.

    Steckel, J. S., Yen, B. K. H., Oertel, D. C. & Bawendi, M. G. On the mechanism of lead chalcogenide nanocrystal formation. J. Am. Chem. Soc. 128, 13032–13033 (2006).

    Article  Google Scholar 

  41. 41.

    Evers, W. H. et al. Entropy-driven formation of binary semiconductor-nanocrystal superlattices. Nano Lett. 10, 4235–4241 (2010).

    ADS  Article  Google Scholar 

  42. 42.

    Mason, T. G. & Bibette, J. Shear rupturing of droplets in complex fluids. Langmuir 13, 4600–4613 (1997).

    Article  Google Scholar 

  43. 43.

    Guizar-Sicairos, M., Thurman, S. T. & Fienup, J. R. Efficient subpixel image registration algorithms. Opt. Lett. 33, 156–158 (2008).

    ADS  Article  Google Scholar 

  44. 44.

    Gilbert, P. Iterative methods for the three-dimensional reconstruction of an object from projections. J. Theor. Biol. 36, 105–117 (1972).

    Article  Google Scholar 

  45. 45.

    Martyna, G. J., Tobias, D. J. & Klein, M. L. Constant pressure molecular dynamics algorithms. J. Chem. Phys. 101, 4177–4189 (1994).

    ADS  Article  Google Scholar 

  46. 46.

    Nosé, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81, 511–519 (1984).

    ADS  Article  Google Scholar 

  47. 47.

    Hoover, W. G. Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

D.W., E.B.v.d.W. and A.v.B. acknowledge partial financial support from the European Research Council under the European Union’s Seventh Framework Programme (FP-2007-2013)/ERC Advanced Grant Agreement 291667 HierarSACol. T.D. and M.D. acknowledge financial support from the Industrial Partnership Programme, ‘Computational Sciences for Energy Research’ (grant number 13CSER025), of the Netherlands Organization for Scientific Research (NWO), which was co-financed by Shell Global Solutions International BV G.M.C. was also financially supported by NWO. S.B. acknowledges financial support from ERC Consolidator Grant Number 815128 REALNANO. T.A. acknowledges a post-doctoral grant from the Research Foundation Flanders (FWO, Belgium). C.B.M. and Y.W. acknowledge support for materials synthesis from the Office of Naval Research Multidisciplinary University Research Initiative Award ONR N00014-18-1-2497. G. A. Blab is gratefully acknowledged for 3D printing numerous truncated tetrahedra, which increased our understanding of the connection between the binary icosahedral cluster and Laves phase structures. N. Tasios is sincerely thanked for providing the code for the diffraction pattern calculation. M. Hermes is sincerely thanked for providing interactive views of the structures in this work. We thank G. van Tendeloo, M. Engel, J. Wang, S. Dussi, L. Filion, E. Boattini, S. Paliwal, N. Tasios, B. van der Meer, I. Lobato, J. Wu and L. Laurens for fruitful discussions. We acknowledge the EM Square centre at Utrecht University for the access to the microscopes.

Author information

Affiliations

Authors

Contributions

A.v.B. initiated the investigation of icosahedral order in binary crystals under spherical confinement and supervised D.W. and E.B.v.d.W. M.D. initiated the simulation study on the binary icosahedral clusters in spherical confinement and supervised T.D. and G.M.C. T.D. performed computer simulations of the spontaneous nucleation of binary icosahedral clusters. D.W. synthesized the SPs and obtained, together with T.A., the electron tomography. D.Z. performed SSR tomographic reconstruction and T.A. performed manual segmentations, under the supervision of S.B. Y.W. synthesized NCs under the supervision of C.B.M. E.B.v.d.W., T.D. and G.M.C. developed the BOP analysis of the Laves phases, the binary icosahedral clusters and the cluster criterion to track the nucleation of the binary icosahedral clusters. D.W., T.D., E.B.v.d.W., M.D. and A.v.B. analysed the results. A.v.B., M.D., D.W., T.D. and E.B.v.d.W. co-wrote the manuscript. All authors discussed the text and interpretation of the results.

Corresponding authors

Correspondence to Marjolein Dijkstra or Alfons van Blaaderen.

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The authors declare no competing interests.

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Supplementary information

Supplementary Information

Supplementary methods, Tables 1 and 2, Figs. 1–13 and refs. 1–12.

Supplementary Data 1

Crystal structure of the MgZn2 Laves phase.

Supplementary Data 2

Crystal structure of the MgCu2 Laves phase.

Supplementary Data 3

Crystal structure of the MgNi2 Laves phase.

Supplementary Data 4

An experimental binary icosahedral cluster supraparticle composed of 2,609 nanocrystals, with a size ratio of 0.78.

Supplementary Data 5

A simulated binary icosahedral cluster supraparticle composed of 5,001 HS-like particles with a size ratio of 0.78, after compression.

Supplementary Data 6

Cluster made up of the first eight shells from the centre of symmetry, of a simulated binary icosahedral cluster supraparticle composed of 5,001 HS-like particles with a size ratio of 0.78, after compression.

Supplementary Data 7

Cluster made up of the first eight shells from the centre of symmetry, of an experimental binary icosahedral cluster supraparticle composed of 2,609 nanocrystals.

Supplementary Data 8

Pentagonal tubes composed of alternating pentagons of L and S particles, in a simulated binary icosahedral cluster supraparticle composed of 5,001 HS-like particles with a size ratio of 0.78, after compression.

Supplementary Data 9

Pentagonal tubes composed of alternating pentagons of L and S particles, in an experimental binary icosahedral cluster supraparticle composed of 2,609 nanocrystals.

Supplementary Video 1

HAADF-STEM tilt series of an experimental binary icosahedral cluster supraparticle with a diameter of 110 nm, consisting of 2,609 nanocrystals.

Supplementary Video 2

Three-dimensional representation and orthoslices views of the reconstructed binary icosahedral cluster supraparticle with a diameter of 110 nm by SSR algorithm. Large PbSe nanocrystals and small CdSe nanocrystals are coloured in cyan and red, respectively.

Supplementary Video 3

HAADF-STEM tilt series of an experimental binary icosahedral cluster supraparticle with a diameter of 187 nm, consisting of 9,898 nanocrystals.

Supplementary Video 4

Three-dimensional representation and orthoslices views of the reconstructed binary icosahedral cluster supraparticle with a diameter of 187 nm by SSR algorithm. Large PbSe nanocrystals and small CdSe nanocrystals are coloured in cyan and red, respectively.

Supplementary Video 5

A video showing the nucleation of a binary icosahedral cluster in computer simulation (Ntot = 5,001). Different crystalline domains are shown in different colours. Particles not belonging to the binary icosahedral cluster are reduced in size for visual clarity.

Source data

Source Data Fig. 2

Coordinates of six experimental supraparticles and two simulated supraparticles composed of Ntot = 3,501 and Ntot = 5,001 HS-like particles.

Source Data Supplementary Table 2

Coordinates of a simulated supraparticles composed of Ntot = 10,002 HS-like particles.

Source Data Fig. 5

Plots of the fraction of crystalline particles as a function of the radial distance from the centre of the supraparticle (Ntot = 5,001) at different molecular dynamics times.

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Wang, D., Dasgupta, T., van der Wee, E.B. et al. Binary icosahedral clusters of hard spheres in spherical confinement. Nat. Phys. 17, 128–134 (2021). https://doi.org/10.1038/s41567-020-1003-9

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