Origin of 18-fold quasicrystal

We investigate the origin of the 18-fold symmetry in quasicrystals. We perform Monte Carlo simulations of 2D disks interacting with the hard-core/square-shoulder potential, and we analyze the bond-orientational order and the local vertex configurations. We find that the so-called special clusters of particles play a crucial role in the mixing of the three sets of bond directions 20° apart, producing the 18-fold diffraction pattern.


Introduction
Since soft quasicrystals were discovered in nature [1,2,3,4], a new research front has emerged in the field of quasicrystals. In particular, Fischer et al. found that colloidal micelles in solution self-assembled into a 18-fold quasicrystalline phase [5], whose symmetry has not been found in synthetic metallic compounds [6,7]. They used PI 30 -PEO 120 block copolymers based on the hydrophobic polyisoprene (PI) and the hydrophilic polyethylene oxide (PEO). These block copolymers self-assembled into spherical core-shell micelles where the well-defined core of PI blocks is surrounded by a thick shell of PEO blocks.Presently, the real-space structure of quasicrystals formed by these micelles is still not unequivocally determined.
At a shoulder-to-core ratio of λ ≈ 1.27, we found a high-density phase of 18-fold symmetry [13,18]. With a large fraction of particles forming close-packed equilateral triangles, the phase is structurally similar to the dodecagonal quasicrystalline phase except that the foursided tiles that it contains are rhombi rather than squares. The rhombi, whose acute and obtuse angles are 80 • and 100 • , respectively, consist of two identical isosceles triangles such their bases are equal to λσ and their legs are σ. In this paper, we study the structure of the 18-fold phase in more detail, focusing on the mechanisms of its formation by examining the development of the bond-orientational order parameters at low temperatures.

Specific heat and bond-orientational order
We use Monte Carlo (MC) simulations of the NV T ensemble of N = 4000 particles in a square box with periodic boundary conditions. We choose the shoulder-to-core ratio of λ = 1.27 and a packing fraction of η = πN σ 2 /4A, where N is the number of disks in a box of area A. In all results presented here except figure 10, η = 0.77. The value of λ = 1.27 was chosen such that the apex angle of the isosceles triangle mentioned above and given by 2 arcsin(λ/2) is close to 80 • , thereby promoting the 80 • -100 • rhombi characteristic of the 18-fold quasicrystal. Inverse temperature is defined by β = /k B T . The system was equilibrated for 10 7 MC steps at all β between 0 and 5 except for 2.5 < β < 4.0 where 5 × 10 7 MC steps were performed. Figure 1 shows the specific heat per particle. The most striking feature of this plot is the sharp peak at β = 3.1, which indicates a transition. To understand the meaning of this peak, we evaluate the bond-orientational order parameters χ m defined by where θ r is the angle between a reference in-plane direction and the bond connecting particles separated by no more than λσ and located at r; N B is the number of nearest-neighbor bonds and angle brackets indicate ensemble average. Based on χ 6 , χ 12 , and χ 18 shown in figure 2, one may distinguish between three distinct regimes. For β < 3.1, all three order parameters decrease with β and χ 6 is considerably larger than χ 12 and χ 18 , which indicates a 6-fold phase. The abrupt increase of χ 18 at β = 3.1 consistent with the specific-heat peak signals the onset of the intermediate-temperature regime. The subsequent growth of χ 18 and the continuing decrease of χ 6 and χ 12 imply a coexistence of 18-fold and 6-fold phase for β between 3.1 and 3.9. The third, low-temperature regime begins at β = 3.9 where χ 18 increases again and grows further as β is increased while χ 6 and χ 12 are 0. This state corresponds to the pure 18-fold phase. Figure 3 shows the diffraction patterns obtained by computing the Fourier transforms of particles' centers of mass at β = 3.0, 3.2, and 4.0, (panels a, b, and c, respectively). These   patterns additionally emphasize the difference between the three regimes, demonstrating that at β = 4.0 the particles form the 18-fold quasicrystal. Further development of the 18-fold diffraction pattern at lower temperatures still and the temperature dependence of the bondorder correlations are given in the supporting material of reference [13]. Here the nature of the 18-fold order is additionally elaborated by examining the local packing motifs.

Frequencies of particle clusters
To this end, we classify the clusters of triangles and rhombi around a given vertex shown in figure 4. Each cluster is labeled by the corresponding vertex figure, a set of integers (n 1 .n 2 .n 3 . · · ·) that denotes the way that n 1 -gon, n 2 -gon, and n 3 -gon, . . . meet consecutively at each vertex [19]; and superscripts are employed to abbreviate the notation whenever possible. We identify eight types of local clusters, which are more complex than those in the squaretriangle dodecagonal tiling [20]. Three out of eight clusters (3 6 , 3 2 .4.3.4, and 3 3 .4 2 ) are the same as in the square-triangle tiling whereas the 4 4 a and 4 4 b clusters are both generalizations of the same 4 4 cluster of squares. In the former, rhombi are arranged either such that their long diagonals all point in the same direction and in the latter they form a chevron pattern. These five clusters may be constructed from rhombi of arbitrary shape and are thus referred to as the generic clusters. On the other hand, the special clusters only exist in 80 • -100 • rhombi.  These clusters are needed so as to integrate the chevron cluster 4 4 b into the matrix; the 3.4 3 accommodates its recessed side whereas the 3.4.3.4 2 and 3 2 .4 3 accommodate its pointed side.
In figure 4, the edges appearing in the eight clusters are colored according to orientation. The 18-fold quasicrystal is marked by 18 bond directions which can be divided into three sets of 6 directions 60 • apart and rotated with respect to each other by 20 • . The three sets are conveniently represented by the red, green, and blue equilateral triangles drawn at the bottom of figure 4. The edge coloring shows that all generic clusters except 4 4 b contain only orientations from two of these sets whereas the 4 4 b cluster and all special clusters mix all three sets of bond orientations. This shows why they are key to understand the formation of 18-fold quasicrystals.
The frequencies of these eight cluster types is presented in figure 5, the special clusters all combined in a single class for clarity; also plotted is the frequency of all clusters beyond this classification. In a triangular tile, the overlap energy per particle of triangles is larger than in a rhombus. This explains why the frequency of the triangle-only 3 6    These results all suggest that the onset of the 18-fold order parameter is closely correlated with the occurrence of the three special clusters, which are in turn needed for the integration     symmetry. On close inspection, we see that the top and the bottom parts of figure 7 devoid of special clusters primarily consist of hexagonal domains separated by domain boundaries based on 3 3 .4 2 clusters. As a result, the hexagonal phase contains only two sets of bond orientations. This is more clearly seen in the system at a slightly larger packing fraction of η = 0.79 and a somewhat higher temperature of β = 3.4 where the stable state is purely hexagonal ( figure 10).

Discussion
The above results elucidate the role of special clusters in the formation of the 18-fold quasicrystal, showing that they ensure that the three sets of bond orientations are equiprobable.Interestingly, two of our special clusters are also present in the dissection of a regular decaoctagon into 18 pentagons with internal angles of 60 • , 160 • , 80 • , 100 • and 140 • provided that the pentagon is viewed as combination of an equilateral triangle and a 80 • − 100 • rhombus ( figure 11a). This dissection can be admired in the intricate pattern on the facade of the Ravensbourne College in London, designed by Foreign Office Architects. The pattern appears to be derived from the 6-fold 18 2 .3 * 2π/9 decaoctagon-three-pointed star tiling [19] by shrinking the decaoctagons and dissecting them into the 60 • − 160 • − 80 • − 100 • − 140 • pentagons; the gaps between them are dissected into pentagons, triangles, and concave pentagons (figure 11b). The facade motif may be viewed as a hexagonal approximant of the 18-fold quasicrystal (plane group P6).