Quasicrystalline structures

The publication of this paper led to a stampede of research, both experimental and theoretical, and an examination of earlier studies by eminent people like Roger Penrose and Alan Mackay in England about the possibilities of filling space by tiling with two distinct populations of tiles, as illustrated in Figure 10.8. This is the basis of quasicrystalline structure. 

Whether quasicrystalline structures are limited to alloys remains an open question. It is possible that their occurrence is much more widespread than had been previously thought. Indeed there is evidence for quasicrystallinity in both thermotropic and lyotropic liquid crystals. Diffraction patterns of decagonal symmetry have been recorded in lyotropic liquid crystals [K. Fontell, private communication], (Fig. 2.19), and there is theoretical evidence for the existence of a quasicrystalline structure within the blue phase of cholesterol (Chapters 4, 5). (The decagonal structure has quasisymmetry perpendicular to the tenfold axes, and translation symmetry along them.) Viruses crystallise in icosahedral clusters and the list continues to grow. In addition to five-fold symmetry, it has been shown that eight and ten- fold quasisymmetry is possible. ... 

Rapid solidification processes are successfully used for A1 alloys to form a dispersion particles of intermetallic phases, which resist coarsening and strengthen the alloys at elevated temperatures. It has recently been shown that metastable intermetallic phases with a quasicrystalline structure, mainly of the icosahedral type, can also be produced by rapid solidification [23], As distinct from a crystalline state, translational long-range order is absent in quasicrystals, but there is rotational symmetry with 5-, 8-, 10- or 12-fold axes, which is forbidden in crystalline materials. The absence of translational symmetry in all three orthogonal directions is characteristic of the icosahedral structure [24],... 

At low water content the quasi-crystalline motif in the distribution of inverted micelles is strong the size of the micelle is smaller than the persistence length of the bundles of backbone-chains forming the membrane skeleton. A legitimate question arises then how can one build a quasicrystalline structure of inverted micelles (aqueous droplets supported by hydrated sidechains), if the sidechains are attached to the backbones In an attempt to answer this question, a more detailed morphological model of Nafion-type ionomers was suggested [31] a quasi-crystalline arrangement of units cells as depicted in Fig. 1. 

Many nuclear phenomena can be understood better, supposing that the constituents of nuclei are organized into clusters, to be described as coherent superpositions of quasicrystalline structures. The cluster structures were studied already from the very early period of nuclear structure research. 

Bottom-up BCP SA provides a facile route to a variety of nanostructures including three-dimensional (3D) and quasicrystalline structures, usually challenging in top-down approaches. This section delineates such BCP-derived ID, 2D, 3D, and quasicrystaUine structures. The dimension of structures, D, is defined by D = 3 — n, where n stands for the number of axes that have continuous translational order. For example, the lamellar structure shown in Fig. 2 has an axis of translational order with a nonzero lattice dimension but two axes with continuous translational order. Thus, the dimension of a lamellar structure is D = with n = 2. 

Fig. 3 Quasicrystalline structures observed in BCP SA. (a) TEM image of a 2D 12-fold quasicrystal derived from a star block copolymer (the inset shows its EFT pattern), (b) Transcribed tiling pattern (reprinted with permission from [34] Copyright 2010 American Physics Society), (c) TEM image and (d, e) unit cell of the Frank-Kasper sigma phase obtained from tetra-BCP SA (c-e reprinted with permission from [33] Copyright 2010 AAAS)...

Compatibility with the model of phason-line movement by Beraha et al. [25]. This model is based on the structure solution by Boudard et al. [24]. It employs shifts in perpendicular space in a higher-dimensional crystallographic approach, as nsed for quasicrystalline structures, to model the individual atom jumps that make up a... 

Let us consider mixtures of monomers and imagine that we increase the chain-length f of the second molecular species. We may thus e q>ect a kind of periodic behaviour for the excess volume with ininiina when the length of the second species is approximately an integral multiple of that of the first one. This should provide quite a direct test for the quasicrystalline structure of liquid mixtures. At present, however, no appropriate experimental data seem to he available. 

The scheme of cluster condensation or cluster fragment condensation leads eventually to structures observed in bulk metals. Particularly through extensive condensation of tetrahedral and octahedral clusters, arrangements closely related to the hexagonal and cubic close-packed structures can be obtained. Condensation also of icosahedral five-fold symmetrical clusters may be related to crystalline and quasicrystalline metallic structures. 

Mikhailov draws attention to the fact that Persianova and Tarasov have studied compressibility of aqueous solutions of nonelectrolytes and found it necessary also to postulate the filling of cavities in a quasicrystalline lattice of water. This again agrees with our claim that solutes —both electrolytes and nonelectrolytes—do not significantly influence the temperature at which the kinks are observed and that this must be explained by assuming that there exists in such solutions elements of water structure which are unaffected by the presence of the solute. It is possible (to be discussed elsewhere) that the structured units responsible for the kinks merely possess a latent existence in pure water and that it is indeed the presence of the solute which induces the stabilization and thus furthers rather than disrupts the original structuredness of the water. 

At present, a number of quasi-crystalline alloys with icosahedral, decagonal, and octagonal symmetry are synthesized by different methods. The quasicrystalline form of the solids turned out to be widespread in a great extent. The absence of the translation symmetry and the presence of numerous interstitial sites of the different types in the structure of icosahedral quasicrystals makes some of them interesting objects for hydride chemistry. We cannot wait for any sensational discoveries here, as the general laws of M-H interaction do not depend on matrix structure. However, encouraging results were obtained for icosahedral Ti45Zr38Nii7... 

The above discussion considers a model for the stmcture of the crystalline cubic R-AlsCuTis. Now consider possible perturbations in this model to give the quasicrystalline T2-Al5CuTi3. In this coimection, the 54-vertex Mackay icosahedron (Figure 2) appears as a structural unit in certain quasicrystals. The Mackay icosahedron has a shell stmcture consisting of the following layers ... 

Motivated by the remarkable discovery of quasicrystalline ordering in solids in 1984 [1], wave propagation in deterministic non periodic media has been an area of intense research. Following the successful experimental realisation of a multitude of such structures through modem technologies, such as molecular beam epitaxy and laser ablation [2], their interest has increased ever since. The most widely known examples are quasi-periodic structures obtained by substitution rules, such as Fibonacci- or Thue-Morse-chains [3,4], Much less has been published on quasi-periodic chains constructed according to a Cantor-set algorithm, which are the subject of this note. 

---