Sofia Deloudi

Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry

Micelles are the simplest example of self-assembly found in nature. As many other colloids, they can self-assemble in aqueous solution to form ordered periodic structures. These structures so far all exhibited classical crystallographic symmetries. Here we report that micelles in solution can self-assemble into quasicrystalline phases. We observe phases with 12-fold and 18-fold diffraction symmetry. Colloidal water-based quasicrystals are physically and chemically very simple systems. Macroscopic monodomain samples of centimeter dimension can be easily prepared. Phase transitions between the fcc phase and the two quasicrystalline phases can be easily followed in situ by time-resolved diffraction experiments. The discovery of quasicrystalline colloidal solutions advances the theoretical understanding of quasicrystals considerably, as for these systems the stability of quasicrystalline states has been theoretically predicted for the concentration and temperature range, where they are experimentally observed. Also for the use of quasicrystals in advanced materials this discovery is of particular importance, as it opens the route to quasicrystalline photonic band gap materials via established water-based colloidal self-assembly techniques.

Reciprocal-space imaging of a real quasicrystal. A feasibility study with PILATUS 6M

How many of the theoretically densely distributed Bragg reflections of a quasicrystal can be observed employing an area detector and synchrotron radiation? How does the reflection density of a real quasicrystal change as a function of exposure time, and what is the minimum distance between reflections? What does the distribution of diffuse scattering look like? To answer these questions, the Bragg reflection density of a perfect icosahedral quasicrystal with composition Al64Cu23Fe13 was measured employing a novel type of single-photon-counting X-ray pixel detector, PILATUS 6M, which allows noise-free data collection with the extraordinarily large dynamic range of 20 bit. The reflection density was found to be two orders of magnitude lower than expected for a strictly quasiperiodic structure. Moreover, diffuse scattering reflects significant structural disorder, breaking six-dimensional F-lattice symmetry. These findings have some implications for the interpretation of physical properties.

Unifying cluster-based structure models of decagonal Al–Co–Ni, Al–Co–Cu and Al–Fe–Ni

The geometrical building principles of Al-based decagonal quasicrystals and their approximants are discussed from a cluster-based approach. Our investigations cover 11 modifications with two- or four-layer periodicity in the systems Al-Co-Ni, Al-Co-Cu and Al-Fe-Ni. We identified a cluster that leads to a unifying view of all these phases. This unit cluster has ~20 Å diameter, four-layer periodicity along its tenfold axis and rod symmetry group p102m. The models obtained are in agreement with all the electron-density maps and electron-microscopy images available.

Cluster Packing from a Higher Dimensional Perspective

The way to find the optimum packing of quasicrystal-constituting clusters is discussed based on the projected-cell approach. We illustrate why the quasiperiodic arrangement of partially overlapping clusters with decagonal or icosahedral symmetry is the most efficient one, by relating it to the packing of unit cells in hypercubic lattices.

Bulk and surface structure of the clean and adsorbate-covered decagonal Al-Co-Ni quasicrystal

We review our Al adsorption experiments on the tenfold-symmetry surface of the decagonal Al–Co–Ni quasicrystal and present computational simulations of adsorption on a structural model based on a fundamental Al–Co cluster with 20 A diameter, symmetry , and 8 A periodicity. This cluster is the building unit of τ2-Al13Co4, from which, by a sequence of minor changes, the structures of the phases in the stability region of decagonal Al–Co–Ni can be derived. The model used for the decagonal Al70Co15Ni15 is an idealized model with a two-layer periodicity (4 A) and no chemical or structural disorder. We find that the bulk and surface properties of this model are in good agreement with experiments. Our molecular-dynamics simulations of Al adsorption reproduce the experimental results and show that by varying the thermal relaxation rates of the adsorbed layer, a variety of different surface morphologies can be achieved. We also present our recent experiments on dissociative adsorption of oxygen on the decagonal surface.

Systematic cluster-based modelling of the phases in the stability region of decagonal Al–Co–Ni

Systematic cluster-based structural modelling of decagonal Al–Co–Ni is performed, based on a fundamental Al–Co cluster with 20 Å diameter, symmetry and 8 Å periodicity. This cluster is the building unit of τ2-Al13Co4, from which, via minor changes, the structures of the W-approximant, the basic Co-rich phase, the superstructures of type I, II, and S1, and the basic Ni-rich phase are derived. Good agreement with available experimental data based on X-ray-diffraction is achieved.

Ab initio structure solution by iterative phase-retrieval methods: performance tests on charge flipping and low-density elimination

Comprehensive tests on the density-modification methods charge flipping [Oszlanyi & Suto (2004). Acta Cryst. A60, 134-141] and low-density elimination [Shiono & Woolfson (1992). Acta Cryst. A48, 451-456] for solving crystal structures are performed on simulated diffraction data of periodic structures and quasicrystals. A novel model-independent figure of merit, which characterizes the reliability of the retrieved phase of each reflection, is introduced and tested. The results of the performance tests show that the quality of the phase retrieval highly depends on the presence or absence of an inversion center and on the algorithm used for solving the structure. Charge flipping has a higher success rate for solving structures, while low-density elimination leads to a higher accuracy in phase retrieval. The best results can be obtained by combining the two methods, i.e. by solving a structure with charge flipping followed by a few cycles of low-density elimination. It is shown that these additional cycles dramatically improve the phases not only of the weak reflections but also of the strong ones. The results can be improved further by averaging the results of several runs and by applying a correction term that compensates for a reduction of the structure-factor amplitudes by averaging of inconsistently observed reflections. It is further shown that in most cases the retrieved phases converge to the best solution obtainable with a given method.

Periodic average structures in phononic quasicrystals

Periodic average structures for the pentagonal, heptagonal, octagonal and dodecagonal tilings were constructed and used for the interpretation of transmission spectra of corresponding two-dimensional phononic quasicrystals (QPNCs). In the hard scattering regime of steel rods in water, the transmission behaviour of QPNCs resembles that of disordered periodic phononic crystals. The average structures allow prediction of position and width of the first bandgap. Additionally, the type of deviation of the quasiperiodic from its periodic average structure is shown to support the understanding of the shape of the transmission spectra and may assist the choice of an optimal structure for specific phononic crystal applications.

5D Modelling of decagonal Al–Co–Ni based on the W-approximant

A structure model for decagonal Al–Co–Ni with 8 Å periodicity along the decagonal axis is proposed. The model agrees well with available experimental information, such as electron microscopic images and three-dimensional (3D) Patterson maps calculated from X-ray-diffraction data. The model is based on a novel columnar cluster with 20 Å diameter and symmetry 5/mm building the W-approximant, Al72.5Co20Ni7.5. The proposed cluster also allows modelling of the various types of disorder and superstructures found in decagonal Al–Co–Ni.

Higher-dimensional crystallography of N-fold quasiperiodic tilings.

Crystallography and periodic average structures (PASs) of two-dimensional (2D) quasiperiodic tilings with N-fold symmetry (N-QPTs with N = 7, 8, 9, 10, 11, 12, 13, 15) were studied using the higher-dimensional approach. By identifying the best (most representative) PASs for each case, it was found that the complexity of the PASs and the degree of average periodicity (DAP) strongly depend on the dimensionality and topology of the hypersurfaces (HSs) carrying the structural information. The distribution of deviations from periodicity is given by the HSs projected upon physical space. The 8-, 10- and 12-QPTs with their 2D HSs have the highest DAP. In the case of the 7-, 9-, 11-, 13- and 15-QPTs, the dimensionality of the HSs is greater than two, and is therefore reduced in the projection upon 2D physical space. This results in a non-homogeneous distribution of deviations from the periodic average lattice, and therefore in a higher complexity of the PASs. Contrary to the 7- and 9-QPTs, which still have representative PASs and DAPs, the 11-, 13- and 15-QPTs have a very low DAP.