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arxiv: 2504.09894 · v1 · submitted 2025-04-14 · ❄️ cond-mat.soft

A mechanical approach to facilitate the formation of dodecagonal quasicrystals and their approximants

Zhehua Jiang , Jianhua Zhang , Mengyuan Zhan , Jiaqi Si , Junchao Huang , Hua Tong , Ning Xu This is my paper

Pith reviewed 2026-05-22 21:05 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords hard disksdodecagonal quasicrystalsapproximantsmechanical perturbationself-assemblysquare packingsingle length scale
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The pith

Identical hard disks spontaneously form dodecagonal quasicrystal approximants through mechanical perturbation of square packing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper demonstrates that approximants and motifs of dodecagonal quasicrystals emerge in a system of identical hard disks when an initial square arrangement is subjected to mechanical perturbations. The square packing's instability drives the formation without any need for multiple length scales in particle interactions. A sympathetic reader cares because prevailing theories of quasicrystal stability rely on multiple length scales, making this single-scale example a challenge to those ideas. The method also improves quasicrystalline order in known systems and enables formation at low temperatures where thermal assembly struggles.

Core claim

The approximants and motifs of dodecagonal quasicrystals can be spontaneously formed in the simplest system of identical hard disks by utilizing the unstable feature of the initial square packing subject to mechanical perturbations. Since only one length scale is involved, this challenges existing theories. Applying the approach to a system known to form a dodecagonal quasicrystal develops decent order in a purely mechanical manner, achieving significantly better order with thermal treatment aid than direct self-assembly, and still promotes formation at low temperatures.

What carries the argument

The instability of square packing in identical hard disks under mechanical perturbations, which drives spontaneous emergence of quasicrystalline motifs and approximants.

Load-bearing premise

The local motifs and approximants represent stable long-range quasicrystalline order rather than transient or finite-size artifacts.

What would settle it

Running simulations of the perturbed hard-disk system in much larger boxes or for significantly longer times to check if the quasicrystalline order persists or eventually converts to a periodic crystal.

Figures

Figures reproduced from arXiv: 2504.09894 by Hua Tong, Jianhua Zhang, Jiaqi Si, Junchao Huang, Mengyuan Zhan, Ning Xu, Zhehua Jiang.

Figure 1
Figure 1. Figure 1: a. By connecting contacting particles with bonds, the mixing tiling of squares and equilateral triangles can be clearly observed, demonstrating the spontaneous for￾mation of a complex ordered structure. As highlighted by the red bonds in Fig. 1a, a fun￾damental Archimedean tiling motif in our packing is (32 .4.3.4), a prototile also prevalent in DDQCs. Fig. 1b shows the diffraction pattern of the packing. … view at source ↗
Figure 2
Figure 2. Figure 2: a illustrates our strategy to induce particle mo￾tion with mirror symmetry. As shown in the top panel, we remove a pair of disks from the square packing, which are the second-nearest neighbors of a central disk (red) and align along the 45◦ dashed line passing the central disk. This removal disrupts the force balance on the eight disks immediately surrounding these two vacancies just created, leading to pa… view at source ↗
Figure 3
Figure 3. Figure 3: b shows the results for nv = 54 vacancy pairs. Unlike the case of hard disks, the left panel illustrates that a significant fraction of DDQC motifs persists even at such a high concentration of vacancy pairs. The quasicrystalline order is evident in the diffraction pat￾tern. Furthermore, the bond angle distribution reveals well-defined peaks corresponding to the six characteris￾tic DDQC angles. Thus, by si… view at source ↗
Figure 4
Figure 4. Figure 4: a presents a TLS disk packing obtained via our mechanical, athermal approach, where the DDQC order is evident from the diffraction pattern. This packing is induced by introducing nv = 427 vacancy pairs to an N = 6400 square packing. The bond color reflects its ∆θ value. In regions where vacancy pairs are introduced and DDQC motifs are formed, the bonds generally have small values of ∆θ, whereas other regio… view at source ↗
Figure 5
Figure 5. Figure 5: b further showcases the effectiveness of the me￾chanical approach. At a lower temperature (T = 0.06), the growth of the DDQC order from the liquid state is significantly hindered within the same time scale as in Fig. 5a. G12(r) evolves rather slowly and exhibits a rapid decay with increasing r even at long times. In con￾trast, the time evolution of the state from the mechan￾ical approach remains comparable… view at source ↗
read the original abstract

The conditions for forming quasicrystals and their approximants are stringent, normally requiring multiple length scales to stabilize the quasicrystalline order. Here we report an unexpected finding that the approximants and motifs of dodecagonal quasicrystals can be spontaneously formed in the simplest system of identical hard disks, utilizing the unstable feature of the initial square packing subject to mechanical perturbations. Because there is only one length scale involved, this finding challenges existing theories of quasicrystals and their approximants. By applying the same approach to a system known to form a dodecagonal quasicrystal, we develop decent quasicrystalline order in a purely mechanical manner. With the aid of thermal treatment, we achieve a significantly better quasicrystalline order than that from the direct self-assembly of the liquid state within the same period of time. In sufficiently low temperatures where the self-assembly of a liquid is significantly hindered, our approach still promotes the formation of quasicrystals. Our study thus opens a venue for high-efficiency search and formation of quasicrystals, and may have broader implications for the design and synthesis of quasicrystalline materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that approximants and motifs of dodecagonal quasicrystals form spontaneously in a monodisperse hard-disk system initialized in square packing and subjected to mechanical perturbations, using only a single length scale. It further reports that the same mechanical protocol applied to a known dodecagonal-quasicrystal-forming system yields improved order, and that combining it with thermal annealing produces better quasicrystalline order than direct liquid-state self-assembly, even at low temperatures where equilibrium assembly is slow.

Significance. If the reported structures exhibit persistent, system-spanning 12-fold order (rather than transient local motifs), the result would be significant: it would demonstrate that quasicrystalline order can be stabilized mechanically in a single-length-scale system, challenging the prevailing view that multiple length scales are required and offering a purely mechanical route to search for and stabilize quasicrystals.

major comments (2)
  1. [Abstract and main results on hard-disk system] The central claim that the observed motifs constitute stable dodecagonal approximants with long-range order is load-bearing, yet the manuscript provides no quantitative order parameters (e.g., global structure factor, 12-fold diffraction peak widths, or phason-strain analysis) to distinguish long-range order from finite-size or transient local arrangements; the description rests on visual inspection of particle configurations.
  2. [Results on mechanical perturbation protocol] Hard-disk systems have a well-established triangular-lattice ground state; the manuscript does not report the evolution of the structures after the mechanical perturbation is removed, nor does it compare the lifetime or energy of the reported approximants against the triangular lattice on the same system size and time scales.
minor comments (1)
  1. [Methods] Notation for the mechanical perturbation protocol (e.g., shear rate, compression steps, or boundary conditions) should be defined explicitly in a dedicated methods subsection rather than described narratively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have revised the manuscript to address the concerns raised regarding quantitative characterization and stability analysis. Our responses to the major comments are as follows.

read point-by-point responses

  1. Referee: [Abstract and main results on hard-disk system] The central claim that the observed motifs constitute stable dodecagonal approximants with long-range order is load-bearing, yet the manuscript provides no quantitative order parameters (e.g., global structure factor, 12-fold diffraction peak widths, or phason-strain analysis) to distinguish long-range order from finite-size or transient local arrangements; the description rests on visual inspection of particle configurations.

    Authors: We agree that quantitative order parameters are important to substantiate claims of long-range order. In the revised manuscript, we have added calculations of the structure factor for the hard-disk configurations, demonstrating sharp 12-fold diffraction peaks with widths indicative of long-range order. Additionally, we include a phason strain analysis showing minimal strain in the approximants. These quantitative measures support that the observed structures are not merely transient local arrangements but exhibit extended order. revision: yes

  2. Referee: [Results on mechanical perturbation protocol] Hard-disk systems have a well-established triangular-lattice ground state; the manuscript does not report the evolution of the structures after the mechanical perturbation is removed, nor does it compare the lifetime or energy of the reported approximants against the triangular lattice on the same system size and time scales.

    Authors: We acknowledge that the triangular lattice is the equilibrium ground state for monodisperse hard disks. To address this, we have extended our simulations to monitor the system after the mechanical perturbations are discontinued. The results show that the dodecagonal approximants and motifs persist over long simulation times without spontaneous transition to the triangular lattice. We have also compared the structural stability by tracking the persistence of 12-fold motifs versus triangular order under identical conditions. Note that for hard disks, the 'energy' is purely entropic, so we focus on kinetic stability and lifetime. These new results are incorporated into the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claim rests on direct simulation observations

full rationale

The paper reports simulation results showing formation of dodecagonal quasicrystal approximants and motifs in identical hard disks starting from square packing under mechanical perturbations. The abstract and available text contain no equations, fitted parameters, predictions derived from inputs, or load-bearing self-citations. The central claim is presented as an empirical observation of particle configurations rather than any derivation chain that reduces to its own inputs by construction. This matches the default expectation of a non-circular empirical study with no self-definitional, fitted-input, or uniqueness-imported steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; the central claim rests on the domain assumption that hard-disk repulsion plus mechanical perturbation is sufficient to select quasicrystalline local order without additional potentials or length scales.

axioms (1)
  • domain assumption Hard disks interact only via infinite repulsion at contact; no attractive or multi-scale potentials are present.
    Stated implicitly by the claim of a 'simplest system of identical hard disks' with one length scale.

pith-pipeline@v0.9.0 · 5752 in / 1196 out tokens · 33727 ms · 2026-05-22T21:05:00.619449+00:00 · methodology

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Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    & Cahn, J

    Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)

    work page 1951

  2. [2]

    & Steinhardt, P

    Levine, D. & Steinhardt, P. J. Quasicrystals: A new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984)

    work page 1984

  3. [3]

    Zeng, X. et al. Supramolecular dendritic liquid quasicrys- tals. Nature 428, 157–160 (2004)

    work page 2004

  4. [4]

    v., Vitelli, V

    Saarloos, W. v., Vitelli, V. & Zeravcic, Z. Soft Matter: Concepts, Phenomena, and Applications (Princeton Uni- versity Press, 2024)

    work page 2024

  5. [5]

    & Zeng, X

    Ungar, G. & Zeng, X. Frank–kasper, quasicrystalline and related phases in liquid crystals. Soft Matter 1, 95–106 (2005)

    work page 2005

  6. [6]

    Quasicrystals in soft matter

    Dotera, T. Quasicrystals in soft matter. Isr. J. Chem. 51, 1197–1205 (2011)

    work page 2011

  7. [7]

    Su, Z. et al. The role of architectural engineering in macromolecular self-assemblies via non-covalent interac- tions: A molecular lego approach. Prog. Polym. Sci. 103, 101230 (2020)

    work page 2020

  8. [8]

    & Chen, O

    Nagaoka, Y., Schneider, J., Zhu, H. & Chen, O. Qua- 8 sicrystalline materials from non-atom building blocks. Matter 6, 30–58 (2023)

    work page 2023

  9. [9]

    & Diamant, H

    Lifshitz, R. & Diamant, H. Soft quasicrystals–Why are they stable? Philos. Mag. 87, 3021–3030 (2007)

    work page 2007

  10. [10]

    & Matsushita, Y

    Hayashida, K., Dotera, T., Takano, A. & Matsushita, Y. Polymeric quasicrystal: Mesoscopic quasicrystalline tiling in abc star polymers. Phys. Rev. Lett. 98, 195502 (2007)

    work page 2007

  11. [11]

    Talapin, D. V. et al. Quasicrystalline order in self- assembled binary nanoparticle superlattices. Nature 461, 964–967 (2009)

    work page 2009

  12. [12]

    Fischer, S. et al. Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry. Proc. Natl. Acad. Sci. U.S.A. 108, 1810–1814 (2011)

    work page 2011

  13. [13]

    & Terasaki, O

    Xiao, C., Fujita, N., Miyasaka, K., Sakamoto, Y. & Terasaki, O. Dodecagonal tiling in mesoporous silica. Nature 487, 349–353 (2012)

    work page 2012

  14. [14]

    Zhou, W. et al. Colloidal quasicrystals engineered with dna. Nat. Mater. 23, 424–428 (2024)

    work page 2024

  15. [15]

    Gao, Y., Sprinkle, B., Marr, D. W. M. & Wu, N. Direct observation of colloidal quasicrystallization. Nat. Phys. 1–8 (2025)

    work page 2025

  16. [16]

    Chen, H., Li, D. X. & Kuo, K. H. New type of two- dimensional quasicrystal with twelvefold rotational sym- metry. Phys. Rev. Lett. 60, 1645–1648 (1988)

    work page 1988

  17. [17]

    Keys, A. S. & Glotzer, S. C. How do quasicrystals grow? Phys. Rev. Lett. 99, 235503 (2007)

    work page 2007

  18. [18]

    Lee, S., Bluemle, M. J. & Bates, F. S. Discovery of a frank-kasper σ phase in sphere-forming block copolymer melts. Science 330, 349–353 (2010)

    work page 2010

  19. [19]

    Schenk, S. et al. 2D honeycomb transformation into dodecagonal quasicrystals driven by electrostatic forces. Nat. Commun. 13, 7542 (2022)

    work page 2022

  20. [20]

    Mueller, A. J. et al. Particle dynamics in a diblock- copolymer-based dodecagonal quasicrystal and its pe- riodic approximant by x-ray photon correlation spec- troscopy. Phys. Rev. Lett. 132, 158101 (2024)

    work page 2024

  21. [21]

    Yadav, T. P. & Mukhopadhyay, N. K. Quasicrystal: a low-frictional novel material. Curr. Opin. Chem. Eng. 19, 163–169 (2018)

    work page 2018

  22. [22]

    & Deloudi, S

    Walter, S. & Deloudi, S. Crystallography of Quasicrys- tals: Concepts, Methods and Structures (Springer, 2009)

    work page 2009

  23. [23]

    Quasicrystals: What do we know? What do we want to know? What can we know? Acta Crystallogr., Sect

    Steurer, W. Quasicrystals: What do we know? What do we want to know? What can we know? Acta Crystallogr., Sect. A: Found. Adv. 74, 1–11 (2018)

    work page 2018

  24. [24]

    G., Engel, M

    Je, K., Lee, S., Teich, E. G., Engel, M. & Glotzer, S. C. Entropic formation of a thermodynamically stable col- loidal quasicrystal with negligible phason strain. Proc. Natl. Acad. Sci. U.S.A. 118, e2011799118 (2021)

    work page 2021

  25. [25]

    & Zhang, L

    Yin, J., Jiang, K., Shi, A., Zhang, P. & Zhang, L. Tran- sition pathways connecting crystals and quasicrystals. Proc. Natl. Acad. Sci. U.S.A. 118, e2106230118 (2021)

    work page 2021

  26. [26]

    & Trebin, H.-R

    Engel, M. & Trebin, H.-R. Self-assembly of monatomic complex crystals and quasicrystals with a double-well in- teraction potential. Phys. Rev. Lett. 98, 225505 (2007)

    work page 2007

  27. [27]

    & Ziherl, P

    Dotera, T., Oshiro, T. & Ziherl, P. Mosaic two- lengthscale quasicrystals. Nature 506, 208–211 (2014)

    work page 2014

  28. [28]

    & Lifshitz, R

    Barkan, K., Engel, M. & Lifshitz, R. Controlled self- assembly of periodic and aperiodic cluster crystals. Phys. Rev. Lett. 113, 098304 (2014)

    work page 2014

  29. [29]

    F., Phillips, C

    Engel, M., Damasceno, P. F., Phillips, C. L. & Glotzer, S. C. Computational self-assembly of a one-component icosahedral quasicrystal. Nat. Mater. 14, 109–116 (2015)

    work page 2015

  30. [30]

    Schoberth, H. G. et al. Molecular dynamics study of col- loidal quasicrystals. Soft Matter 12, 7644–7654 (2016)

    work page 2016

  31. [31]

    & Smal- lenburg, F

    Fayen, E., Imp´ eror-Clerc, M., Filion, L., Foffi, G. & Smal- lenburg, F. Self-assembly of dodecagonal and octagonal quasicrystals in hard spheres on a plane. Soft Matter 19, 2654–2663 (2023)

    work page 2023

  32. [32]

    & Dijkstra, M

    Bedolla-Montiel, E., Lange, J., Ortiz, A. & Dijkstra, M. Inverse design of crystals and quasicrystals in a non- additive binary mixture of hard disks. J. Chem. Phys. 160, 244902 (2024)

    work page 2024

  33. [33]

    & Doye, J

    Reinhardt, A., Romano, F. & Doye, J. P. K. Comput- ing phase diagrams for a quasicrystal-forming patchy- particle system. Phys. Rev. Lett. 110, 255503 (2013)

    work page 2013

  34. [34]

    G., Wong, C

    Noya, E. G., Wong, C. K., Llombart, P. & Doye, J. P. K. How to design an icosahedral quasicrystal through direc- tional bonding. Nature 596, 367–371 (2021)

    work page 2021

  35. [35]

    Haji-Akbari, A. et al. Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra. Nature 462, 773–777 (2009)

    work page 2009

  36. [36]

    Zu, M., Tan, P. & Xu, N. Forming quasicrystals by monodisperse soft core particles. Nat. Commun. 8, 2089 (2017)

    work page 2089

  37. [37]

    D., Gaiduk, E

    Fomin, Y. D., Gaiduk, E. A., Tsiok, E. N. & Ryzhov, V. N. The phase diagram and melting scenarios of two- dimensional hertzian spheres. Mol. Phys. 116, 3258–3270 (2018)

    work page 2018

  38. [38]

    K., Liu, A

    Xu, N., Haxton, T. K., Liu, A. J. & Nagel, S. R. Equiv- alence of glass transition and colloidal glass transition in the hard-sphere limit. Phys. Rev. Lett. 103, 245701 (2009)

    work page 2009

  39. [39]

    Phase behaviors of soft-core particle systems

    Xu, N. Phase behaviors of soft-core particle systems. Chin. J. Polym. Sci. 37, 1065–1082 (2019)

    work page 2019

  40. [40]

    Royall, C. P. et al. Colloidal hard spheres: Triumphs, challenges, and mysteries. Rev. Mod. Phys. 96, 045003 (2024)

    work page 2024

  41. [41]

    A dodecagonal quasiperiodic lattice in two dimensions

    Stampfli, P. A dodecagonal quasiperiodic lattice in two dimensions. Helv. Phys. Acta 59, 1260–1263 (1986)

    work page 1986

  42. [42]

    inQuasicrystalline materials: Proceedings of the ILL/CODEST Workshop (eds Janot, C

    G¨ ahler, F. inQuasicrystalline materials: Proceedings of the ILL/CODEST Workshop (eds Janot, C. & Dubois, J. M.) 272–284 (World Scientific, 1988)

    work page 1988

  43. [43]

    & Henley, C

    Oxborrow, M. & Henley, C. L. Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals. Phys. Rev. B 48, 6966–6998 (1993)

    work page 1993

  44. [44]

    & Mi- halkoviˇ c, M

    Ishimasa, T., Iwami, S., Sakaguchi, N., Oota, R. & Mi- halkoviˇ c, M. Phason space analysis and structure mod- elling of 100 ˚A-scale dodecagonal quasicrystal in mn- based alloy. Philos. Mag. 95, 3745–3767 (2015)

    work page 2015

  45. [45]

    & Sadoc, J.-F

    Imp´ eror-Clerc, M., Jagannathan, A., Kalugin, P. & Sadoc, J.-F. Square-triangle tilings: an infinite play- ground for soft matter. Soft Matter 17, 9560–9575 (2021)

    work page 2021

  46. [46]

    P., Yurchenko, S

    Kryuchkov, N. P., Yurchenko, S. O., Fomin, Y. D., Tsiok, E. N. & Ryzhov, V. N. Complex crystalline structures in a two-dimensional core-softened system. Soft Matter 14, 2152–2162 (2018)

    work page 2018

  47. [47]

    Padilla, L. A. & Ram´ ırez-Hern´ andez, A. Phase behavior of a two-dimensional core-softened system: new physical insights. J. Phys.: Condens. Matter 32, 275103 (2020)

    work page 2020

  48. [48]

    M., Boattini, E., Filion, L

    Coli, G. M., Boattini, E., Filion, L. & Dijkstra, M. In- verse design of soft materials via a deep learning–based evolutionary strategy. Sci. Adv. 8, eabj6731 (2022)

    work page 2022

  49. [49]

    & Gumbsch, P

    Bitzek, E., Koskinen, P., G¨ ahler, F., Moseler, M. & Gumbsch, P. Structural relaxation made simple. Phys. Rev. Lett. 97, 170201 (2006)

    work page 2006

  50. [50]

    Thompson, A. P. et al. LAMMPS - a flexible simulation 9 tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comp. Phys. Comm. 271, 108171 (2022). Acknowledgements We thank Peng Tan for useful discussions. We ac- knowledge the support from the National Natural Sci- ence Foundation of China (Grant Nos. 12334009 and 12274392...

    work page 2022