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How to design an icosahedral quasicrystal through directional bonding

Abstract

Icosahedral quasicrystals (IQCs) are materials that exhibit long-range order but lack periodicity in any direction. Although IQCs were the first reported quasicrystals1, they have been experimentally observed only in metallic alloys2, not in other materials. By contrast, quasicrystals with other symmetries (particularly dodecagonal) have now been found in several soft-matter systems3,4,5. Here we introduce a class of IQCs built from model patchy colloids that could be realized experimentally using DNA origami particles. Our rational design strategy leads to systems that robustly assemble in simulations into a target IQC through directional bonding. This is illustrated for both body-centred and primitive IQCs, with the simplest systems involving just two particle types. The key design feature is the geometry of the interparticle interactions favouring the propagation of an icosahedral network of bonds, despite this leading to many particles not being fully bonded. As well as furnishing model systems in which to explore the fundamental physics of IQCs, our approach provides a potential route towards functional quasicrystalline materials.

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Fig. 1: From ideal quasicrystal to patchy-particle design.
Fig. 2: Assembly of a body-centred IQC.
Fig. 3: A patchy-particle primitive IQC.
Fig. 4: Minimal quasicrystal-forming systems.

Data availability

Final configurations of all systems are available from the Oxford University Research Archive (https://doi.org/10.5287/bodleian:R8eMnrapg). Source data are provided with this paper.

Code availability

The patchy-particle Monte Carlo simulation code is available from https://github.com/evanoya/MC_GPU. The simulation code incorporating the oxDNA model is available from https://sourceforge.net/projects/oxdna/.

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Acknowledgements

We are grateful for financial support from the Agencia Estatal de Investigación and the Fondo Europeo de Desarrollo Regional (FEDER) under grant numbers FIS2015-72946-EXP(AEI) and FIS2017-89361-C3-2-P(AEI/FEDER,UE) (to E.G.N.) and the Croucher Foundation (to C.K.W.). We acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility and the Cambridge Service for Data Driven Discovery (CSD3).

Author information

Affiliations

Authors

Contributions

E.G.N. and J.P.K.D. designed the research. E.G.N. wrote the Monte Carlo code used for the simulations of systems of patchy particles and built the ideal IQC and approximants used as target structures. E.G.N., C.K.W. and P.L. performed the patchy-particle simulations and analysed the data. C.K.W. designed the DNA origami particles and performed the simulations on these systems. All the authors discussed the results. J.P.K.D., E.G.N. and C.K.W. wrote the article.

Corresponding authors

Correspondence to Eva G. Noya or Jonathan P. K. Doye.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Oleg Gang, Ron Lifshitz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Structure of the body-centred IQC.

Projections of the BCI 5P (left) and 2P (right) IQCs along the five-fold axis, three-fold axis and two-fold axis (top to bottom). On the three-fold axis, self-similar triangular motifs follow a 1:τ:τ2:τ3 scaling that is a characteristic of IQCs.

Extended Data Fig. 2 Body-centred IQC diffraction patterns.

Comparison of the diffraction patterns of the ideal body-centred IQC and its 3/2 approximant, with those of the assembled IQCs using the BCI 5P, 3P, 2P and 2P-mod models. All four assembled systems have diffraction patterns with clear five-fold symmetry confirming their IQC character. By contrast, the diffraction pattern of the approximant when viewed along the pseudo-five-fold axis shows clear deviations from fivefold symmetry. The diffraction patterns of the BCI 5P and 3P models most closely resemble those of the ideal IQC; the indexing of all peaks with a BCI indexing scheme is detailed in Supplementary Information section 4E. The diffraction patterns for the BCI 2P and 2P-mod models exhibit additional features, particularly along the two-fold and five-fold axes; these include both diffuse scattering and extra weak peaks that cannot be indexed in a BCI indexing scheme (see Supplementary Information).

Extended Data Fig. 3 An ideal primitive IQC.

a, Primitive IQC model with a dodecahedral occupation domain viewed along the five-, three- and two-fold rotational axes (left to right). In the top row, only red and those blue particles that are bonded to red particles are shown; these views clearly show that not all the clusters are perfect. In the bottom row, all particles are shown. The yellow particles form an additional shell around the icosahedral cluster depicted in Fig. 3a and all other matrix particles are coloured pink. b, Views of the 8/5 rational approximant along the psuedo-five-, three- and two-fold rotational axes (left to right).

Extended Data Fig. 4 Structure of the primitive IQC.

Projections of the primitive icosahedral 3P WT (left) and 2P WT (right) models along the five-fold axis, three-fold axis and two-fold axis (top to bottom). On the three-fold axis, self-similar triangular motifs follow a 1:τ:τ2:τ3 scaling that is a characteristic of IQCs.

Extended Data Fig. 5 Primitive IQC diffraction patterns.

Comparison of the diffraction patterns of the ideal primitive IQC and its 8/5 approximant with those of the IQCs obtained from simulations. All four assembled systems have diffraction patterns with clear five-fold symmetry confirming their IQC character. By contrast, the diffraction pattern of the approximant when viewed along the pseudo-five-fold axis shows clear deviations from five-fold symmetry. Despite the assembled systems being simpler than the ideal IQC in terms of the number of environments, the diffraction patterns are very similar to that for the ideal IQC with the patterns along the two-fold axis showing the most differences. The primitive icosahedral 2P diffraction patterns could be satisfactorily indexed with a primitive icosahedral indexing scheme, but the better-resolved primitive icosahedral 3P patterns have additional weak peaks that require a face-centred icosahedral indexing scheme with a hypercubic lattice parameter that is double that of the primitive icosahedral indexing scheme, indicating some superstructural ordering (see Supplementary Information section 4E). The diffraction patterns for the systems with and without torsions show no clear differences, confirming their structural similarity.

Extended Data Fig. 6 Structural analysis of the body-centred IQCs.

The properties of the body-centred IQCs obtained with the 5P, 3P and 2P models are compared with those of the ideal body-centred IQC and its 3/2 rational approximant. a, PDFs considering only those particles within a sphere of radius 15σLJ centred at the centre of mass of the solid cluster. There is good coincidence between the broadened (due to thermal motion) peaks for the assembled IQCs and the distances present in the ideal IQC consistent with their similar structure. The PDFs for the BCI 5P and 3P systems are essentially the same, with the differences from the BCI 2P system still being relatively small. b, Radial density. The bulk densities of the ideal IQC, the BCI 5P and 3P systems and the 3/2 approximant are very similar, but the bulk density for the BCI 2P system is lower, as expected, due to the absence of the red particles. The radial density of the assembled systems decreases from its bulk value to zero over 10σLJ–15σLJ, reflecting the irregular nature of the surface of the assemblies. c, Mole fraction of each particle type in the different structures. The incorporation of the yellow and purple particles into the BCI 5P assembly is much lower than for the ideal IQC. The fraction of red and blue particles in the BCI 5P and 3P assemblies is also closer to that for the approximant than the ideal IQC. The BCI 2P assemblies have a substantially increased fraction of green particles. d–i, Probability of the number of dangling bonds for each particle type (d–h) and for all particle types (i). The assembled IQCs show a greater number of dangling bonds for the blue matrix particles than the ideal IQC, whereas the ideal IQC has a greater number of dangling bonds for the red and green particles that form the triacontahedral clusters.

Extended Data Fig. 7 Structural analysis of the primitive IQCs.

The properties of the primitive IQCs obtained with the 3P and 2P models with and without torsions are compared with those of the ideal primitive IQC. a, PDF considering only those particles within a sphere of radius 15σLJ centred at the centre of mass of the solid cluster. The peaks in the PDFs are consistent with the distances in the ideal IQC, even though the assembled systems do not have particles corresponding to many of the environments in the ideal IQC. The 2P system does not have the peak at ~0.8σLJ that is due to the red–red bonds. b, Radial density. The bulk densities of the 3P systems are a bit lower than the ideal IQC with the 2P systems having even lower densities due to the absence of red particles. c, Mole fraction of each particle type in the different structures. Note that in the ideal structure the mole fractions do not add to 1, as there are additional local environments that are not included in this plot. Particularly noticeable is the much smaller number of red particles in the ideal IQC. d–g, Probability of the number of dangling bonds for each particle type (d–f) and for all particle types (g). The blue particles in the 3P systems have a particularly low probability of having no missing bonds, because when forming part of the inter-cluster matrix, they do not use their two patches that can only bond to the red particles that form the dodecahedral clusters; such particles contribute to the relatively large number of blue particles with two dangling bonds.

Extended Data Fig. 8 Bond orientational order.

ad, Contributions to the BOOD for the BCI 5P (a), BCI 2P (b), primitive icosahedral 3P WT (c) and primitive icosahedral 2P WT (d) quasicrystals from the different patch–patch bonds types in these systems. For the BCI systems, the green–green and purple–purple bonds are directed along the two-fold axes, the blue–green and blue–purple along the three-fold axes, the blue–red, red–green, blue–yellow and yellow–purple along the five-fold axes and the blue–blue can be directed along five-fold (bonds formed through patches 1 to 3) and three-fold (bonds formed by patch 4) axes. For the primitive icosahedral systems, the red–red bonds are directed along the two-fold axes, and the green–blue and red–blue along the three-fold axes. Thus, for the primitive icosahedral 2P WT system the total BOOD only exhibits spots directed along the three-fold axes, whereas for the primitive icosahedral 3P WT system there are also spots along the two-fold axes. Contributions to the BOODs calculated using a distance cutoff exhibit additional spots but always directed along the rotational axes of the Ih point group (Supplementary Fig. 7). e, f, The relationship between the symmetry of the patchy particles and the Ih point group. All particles can be oriented so that their patch vectors point exclusively along the rotational axes of Ih. This is illustrated for the patchy particles in the BCI 5P (e) and primitive icosahedral 3P (f) models by depicting the direction of the patch vectors Pi by circles (with the colours representing (one of) the particles with which they interact) on the surface of an icosahedron. The projections chosen are along the highest rotational axis of the particles. Edges on the back faces of the icosahedra are dashed and cyan. Similarly, for patch vectors on the back faces, the colour shade is lighter and ringed in cyan rather than black. The BOODs in a–d are fully consistent with the particles being oriented in this way with respect to the global icosahedral order of the IQCs. We also note that the preferred torsional angles of each patch–patch interaction are those that ensure the propagation of this global orientational order.

Extended Data Fig. 9 Dynamics of the body-centred IQCs.

Van Hove autocorrelation functions for the simulated body-centred IQCs, evaluated after one million Monte Carlo cycles and considering only those particles within a radius of 20σLJ, which corresponds to the interior of the cluster where the radial density is constant (Extended Data Fig. 6b). The majority of the particle mobility is associated with the blue matrix particles, whereas the red and green particles that form the inner shell of the icosahedral clusters have very limited mobility. The yellow particles in the 5P system are the most mobile but are only present at very low mole fraction.

Extended Data Fig. 10 Dynamics of the primitive IQCs.

Van Hove autocorrelation function for the simulated primitive IQCs, evaluated after one million Monte Carlo cycles and considering only those particles within a radius of 20σLJ, which corresponds to the interior of the cluster where the radial density is constant (Extended Data Fig. 7b). Although the overall pattern shows clear five-fold symmetry, the peaks merge into each other more than for the BCI systems. This is partly because the size of the allowed hops is shorter. The green particles are the most mobile, and the red particles, which form the inner shell of the icosahedral clusters, are least mobile. It is also noticeable that without torsions, the 3P system has greater mobility, and that the 2P patterns are less well defined, with the motion of the blue particles close to isotropic.

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This file contains supplementary text, supplementary equations s1 – s17, supplementary figures s1 – s15 and supplementary references.

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Noya, E.G., Wong, C.K., Llombart, P. et al. How to design an icosahedral quasicrystal through directional bonding. Nature 596, 367–371 (2021). https://doi.org/10.1038/s41586-021-03700-2

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