Growth model of dodecagonal quasicrystal based on correlated tiling of squares and equilateral triangles

Formation pathways of mesoporous silica nanoparticles with dodecagonal tiling

Considerable progress in the fabrication of quasicrystals demonstrates that they can be realized in a broad range of materials. However, the development of chemistries enabling direct experimental observation of early quasicrystal growth pathways remains challenging. Here, we report the synthesis of four surfactant-directed mesoporous silica nanoparticle structures, including dodecagonal quasicrystalline nanoparticles, as a function of micelle pore expander concentration or stirring rate. We demonstrate that the early formation stages of dodecagonal quasicrystalline mesoporous silica nanoparticles can be preserved, where precise control of mesoporous silica nanoparticle size down to <30 nm facilitates comparison between mesoporous silica nanoparticles and simulated single-particle growth trajectories beginning with a single tiling unit. Our results reveal details of the building block size distributions during early growth and how they promote quasicrystal formation. This work identifies simple synthetic parameters, such as stirring rate, that may be exploited to design other quasicrystal-forming self-assembly chemistries and processes.Probing the growth pathways of quasicrystalline materials, where tiling units arrange with local but no long-range order, remains challenging. Here, the authors demonstrate that dodecagonal tiling of mesoporous silica nanoparticles occurs via irreversible packing of micelles with non-uniform size distribution.

Effect of shape on the self-assembly of faceted patchy nanoplates with irregular shape into tiling patterns

Recent reports of the synthesis and assembly of faceted nanoplates with a wide range of shapes and composition motivates the possibility of a new class of two-dimensional materials with specific patterns targeted for a host of exciting properties. Yet, studies of how nanoplate shape controls their assembly - knowledge necessary for their inverse design from target structures - has been performed for only a handful of systems. By constructing a general framework in which many known faceted nanoplates may be described in terms of four anisotropy dimensions, we discover design rules to guide future synthesis and assembly. We study via Monte Carlo simulations attractive polygons whose shape is altered systematically under the following four transformations: faceting, pinching, elongation and truncation. We report that (i) faceting leads to regular porous structures (ii) pinching stabilizes complex structures such as dodecagonal quasicrystals (iii) elongation leads to asymmetric phase behavior, where low and high aspect ratio nanoplates self-assemble completely different structures and (iv) low and high degrees of truncation transform a complex self-assembler into a disk-like assembler, providing design ideas that could lead to switchable structures. We provide important insight into how the shape and attractive interactions of a nanoplate can be exploited or designed to target specific classes of structures, including space-filling, porous, and complex tilings.

The first find of dodecagonal quasiperiodic tiling in historical Islamic architecture

The tympanum of the entrance of the Zaouïa Moulay Idriss II in Fez contains the only known example of a dodecagonal cartwheel quasiperiodic pattern in Islamic art, dating possibly from the Merinid epoch. This pattern, carved in a marble plate, is based on a type of Ammann quasilattice known also from modern mathematical literature. The central portions of this pattern were used as elements in a periodic pattern on the walls of the Saadian mausoleum in Marrakech.

The structure of quasicrystals

The aim of the present article is to give a review of the state-of-the-art on quasicrystal structure analysis. After a short discussion of the term “crystal” in chapter 1, the geometrical generation of quasilattices is touched in chapter 2. In the following the higher-dimensional description of 1d, 2d and 3d quasi-crystals is demonstrated in detail as well as the derivation of structure factor equations and symmetry relationships in the higher-dimensional space. Chapter 4 shows the experimental techniques and structure determination methods for the study of quasicrystals. The experimental results of structural studies performed with different tech-niques are critically reviewed in chapter 5. Some of the results of the literature research are that five years after the detection of the first quasicrystal not a single quantitative (in terms of a regular structure determination) analysis of its structure has been carried out, and that the famous Mackay-icosahedra do not play the important role as the basic structural building elements as one supposed before.

Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals

Is quasicrystal structure analysis a never-ending story? Why is still not a single quasicrystal structure known with the same precision and reliability as structures of regular periodic crystals? What is the state-of-the-art of structure analysis of axial quasicrystals? The present comprehensive review summarizes the results of almost twenty years of structure analysis of axial quasicrystals and tries to answer these questions as far as possible. More than 2000 references have been screened for the most reliable structural models of pentagonal, octagonal, decagonal and dodecagonal quasicrystals. These models, mainly based on diffraction data and/or on bulk and surface microscopic images are critically discussed together with the limits and potentialities of the respective methods employed.

Discrete elastic model for two-dimensional melting

We present a network model for the study of melting and liquid structure in two dimensions, the first in which the presence and energy of topological defects (dislocations and disclinations) and of geometrical defects (elemental voids) can be independently controlled. Interparticle interaction is via harmonic springs and control is achieved by Monte Carlo moves which springs can either be orientationally "flipped" between particles to generate topological defects, or can be "popped" in force-free shape, to generate geometrical defects. With the geometrical defects suppressed the transition to the liquid phase occurs via disclination unbinding, as described by the Kosterlitz-Thouless-Halperin-Nelson-Young model and found in soft potential two-dimensional (2D) systems, such as the dipole-dipole potential [H. H. von Grünberg, Phys. Rev. Lett. 93, 255703 (2004)]. By contrast, with topological defects suppressed, a disordering transition, the Glaser-Clark condensation of geometrical defects [M. A. Glaser and N. A. Clark, Adv. Chem. Phys. 83, 543 (1993); M. A. Glaser, (Springer-Verlag, Berlin, 1990), Vol. 52, p. 141], produces a state that accurately characterizes the local liquid structure and first-order melting observed in hard-potential 2D systems, such as hard disk and the Weeks-Chandler-Andersen (WCA) potentials (M. A. Glaser and co-workers, see above). Thus both the geometrical and topological defect systems play a role in melting. The present work introduces a system in which the relative roles of topological and geometrical defects and their interactions can be explored. We perform Monte Carlo simulations of this model in the isobaric-isothermal ensemble, and present the phase diagram as well as various thermodynamic, statistical, and structural quantities as a function of the relative populations of geometrical and topological defects. The model exhibits a rich phase behavior including hexagonal and square crystals, expanded crystal, dodecagonal quasicrystal, and isotropic liquid phases. In this system the geometrical defects effectively control the melting, reducing the solid-liquid transition temperature by a factor of relative to the topological-only case. The local structure of the dense liquid has been investigated and the results are compared to that from simulations of WCA systems.