The research group led by Prof. Ning Xu from School of Physical Sciences of University of Science and Technology of China (USTC) obtained quasicrystals in a very simple model systems, in collaboration with Prof. Peng Tan from Fudan University. Their findings raise new questions to the development of theories of quasicrystals. The results was published online in Nature Communications on December 12, 2017.

Fig. 1: a Solid phase diagram in the α-ρ plane of two-dimensional systems of soft-core particles interacting via a repulsion U(r)=(1-r)^α/α where σ=1. The inset shows examples of the interaction. b,c Configurations of an octagonal quasicrystal (OQC, b) and a dodecagonal quasicrystal (DDQC, c) and the polygon tilings based on centers of pentagon clusters. The insets show the diffraction patterns. The particle diameter shown here is 20% of the range of interaction σ.

When cooled or compressed slowly, a liquid undergoes symmetry breaking and turns into a solid with a specific rotational symmetry. Required by periodicity, normal crystals usually exhibit 2-, 3-, 4-, or 6-fold rotational symmetry. Other symmetries are unfavorable due to the constraint of periodicity. In 1982, Israel scientist D. Shechtman observed a metallic phase in the Al-Mn alloy with 10-fold symmetry, which violated the allowed crystal symmetries. Shechtman and coworker published their work in 1984 after two year’s further study and understanding, which opened the field of quasicrystals. Shechtman was also awarded the Nobel Prize in Chemistry in 2011 for his discovery of quasicrystals. By definition, quasicrystals are ordered solids exhibiting rotational symmetries prohibited in normal crystals and without translational symmetry and periodicity. Within twenty years after the discovery, quasicrystals were mostly observed in alloys. Quasicrystals was first observed in soft matter until 2004. Because of the advantages in visualization and synthesis of soft matter systems such as colloids, soft quasicrystals have inspired great interest since then.

For particulate systems forming quasicrystals, previous studies have shown that in order to have quasicrystals particles should be bidisperse in size (e.g., alloys), anisotropic in shape (e.g., tetrahedral particles or patchy particles), or interact via a decorated potential with multiple length scales to stabilize quasicrystals. There seems to be a consensus that in order to form quasicrystals two length scales have to be prepared in advance. No one has ever seen the formation of quasicrystals by monodisperse and isotropic particles interacting via a smooth potential which does not explicitly contain multiple length scales. Moreover, octagonal quasicrystals are rare. There has been no report of convincing observations of soft octagonal quasicrystals.

The research group of USTC and Fudan University found the self assembly of both octagonal and dodecagonal quasicrystals in high-density two-dimensional systems consisting of monodisperse and isotropic particles interacting via a very simple soft-core repulsion:U(r)=(1-r/σ)α/α ,where r∈[0,σ] is the particle separation, σ is the range of interaction, and α is a tunable parameter to determine the softness of the interaction. When α=2, the interaction is the well-known harmonic (linear spring) potential. Seen from inset of Fig. 1a, there are no explicit multiple length scales in such an interaction. According to the two-length framework of quasicrystals, it would be a surprise if quasicrystals can be formed in such simple systems. The formation of such quasicrystals will be a challenge to current theories. Another exciting finding is the first convincing observation of octagonal quasicrystals in soft matter systems.

A necessary condition for quasicrystals to self assemble in such simple systems is high density (the density is calculated assuming  is the particle diameter). As shown in Fig. 1a, with increasing density, multiple types of solids occur in sequence. Quasicrystals can be formed in certain density regimes relying on the interaction exponent . Interestingly, in quasicrystals particles tend to form pentagon clusters. The quasicrystal order is developed based on the pentagon clusters. As illustrated by the polygon tessellations in Fig. 1b and c, when centers of non-edge-adjacent pentagons are connected, the octagonal (dodecagonal) quasicrystals show beautiful square-rhombus (square-triangle) tilings. Moreover, it can be seen that each pentagon is surrounded by an octagon (dodecagon) formed by eight (twelve) particles for octagonal (dodecagonal) quasicrystals. This complex structure proposes a possible motif for the design of n-fold quasicrystals.

The first author of this paper is Dr. Mengjie Zu, who graduated from USTC this summer. She also observed novel phenomena of two-dimensional melting of the same soft-core systems, which was published as a letter in Physical Review Letters in 2016. This work was supported by National Natural Science Foundation of China and Fundamental Research Funds for the Central Universities.

Link to the paper：http://www.nature.com/articles/s41467-017-02316-3