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Using molecular simulations, we show that the aperiodic growth of quasicrystals is controlled by the ability of the growing quasicrystal nucleus to incorporate kinetically trapped atoms into the solid phase with minimal rearrangement. In the system under investigation, which forms a dodecagonal quasicrystal, we show that this process occurs through the assimilation of stable icosahedral clusters by the growing quasicrystal. Our results demonstrate how local atomic interactions give rise to the long-range aperiodicity of quasicrystals.

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... In 1983 Steinhardt et al. [289] has proposed the widely used q l orientation order parameter family as 3D generalization of the ψ 6 hexatic order parameter in 2D [289]. q l and its derivates have become a fruitful instrument in identifying different crystalline phases, notably fcc, hcp and bcc [294,227,319,177,302,155,308] or icosahedral nuclei [309,158,133]. They have been used to study melting transitions [309,59,54] and fluids [59,127] too. ...

... They have been used to study melting transitions [309,59,54] and fluids [59,127] too. In the literature of glasses and supercooled fluids q 6 has arisen to the most prominent order parameter [159] searching for glass-transitions [135,220,293] and crystalline clusters [181,227,266,158,304,155]. Similar to the bond-orientation order parameter, most local parameter depend crucially on a definition of nearest neighbor sites. ...

... (1.2, 1.4)σ, used e.g. in refs. [220,231,202,85,158] or cutoff radii based on the first minimum of the two-point correlation function g(r), as in [163,54,127,1,308]. This most common definition in the literature assigns a bond between sphere k and sphere j if the distance between the sphere centers is less or equal r c . ...

This thesis is devoted to several aspects of geometry and morphology in wetting problems and hard sphere packings. First, we propose a new method to simulate wetting and slip on nanostructured substrates: a phase field model associated with a dynamical density theory approach. We showed omniphobicity, meaning repellency, no matter the chemical properties of the liquid on monovalued surfaces, i.e. surfaces without overhangs, which is in contradiction with the macroscopic Cassie-Baxter-Wenzel theory, can produce so-called We checked systematically the impact of the surface parameters on omniphobic repellency, and we show that the key ingredient are line tensions, which emerge from needle shaped surface structures. Geometrical effects have also an important influence on glassy or jammed systems, for example amorphous hard sphere systems in infinite pressure limit. Such hard sphere packings got stuck in a so-called jammed phase, and we shall demonstrate that the local structure in such systems is universal, i.e. independent of the protocol of the generation. For this, robust order parameters - so-called Minkowski tensors - are developed, which overcome robustness deficiencies of widely used order parameters. This leads to a unifying picture of local order parameters, based on geometrical principles. Furthermore, we find with the Minkowski tensor analysis crystallization in jammed sphere packs at the random closed packing point

... However, subsequent work revealed its universal character; indeed, 12-fold symmetry prevails in soft matter and other recently identified non-metallic QCs 12,25,26 . It has also been recognized that it is closely related to Frank-Kasper phases 12,16,19,23,[27][28][29][30][31] , whereby the σ-phase represents a so-called crystal approximant to ddQC, which means that the arrangement of its unit-cell constituents resembles local structures expressed in the QC state 32 . ...

... Such non-deterministic squaretriangle tilings produce a 12-fold diffraction pattern because of the necessarily inherent dodecagonal symmetry elements 38-40 . Deterministic ddQCs are likely to be stabilized energetically, whereas random-tiling configurations are likely to be entropically driven 12,17,18,28,38,41 . Thus, it is not surprising that a random-tiling phase with a ddQC signature could also be identified in molecular dynamics modelling of a monatomic liquid (formed by single atoms) during cooling treatment 42 . ...

... Thus, it is not surprising that a random-tiling phase with a ddQC signature could also be identified in molecular dynamics modelling of a monatomic liquid (formed by single atoms) during cooling treatment 42 . The underlying interaction was approximated by tailored pair potentials, similarly applicable for other condensed-matter QC systems 17,26,28,43,44 , that were recently advanced to a level such that an entire series of QC mosaic structures could be described in a unified scheme 45 . These achievements not only help to disentangle the intriguing physics and chemistry of QC evolution and establish commonalities among different classes of materials 13 , but also they could provide important assets for conceiving novel functional materials 26,46 . ...

The recognition of quasicrystals, which exhibit long-range order but lack translational symmetry, represented both the introduction of a new class of materials and a transformative breakthrough in crystallography. Concomitant with the exploration of quasicrystallinity, metal-organic architectures emerged as promising and versatile systems with significant application potential. Their building principles have been studied extensively and become manifest in a multitude of intricate amorphous and crystalline phases. To date, however, indications for quasicrystalline order have been elusive in metal-organic coordination networks (MOCNs). Here we employ rare-earth-directed assembly to construct a two-dimensional tiling with quasicrystalline characteristics at a well-defined gold substrate. By careful stoichiometry control over europium centres and functional linkers, we produced a porous network, including the simultaneous expression of four-fold, five-fold and six-fold vertices. The pertaining features were directly inspected by scanning tunnelling microscopy, and the molecule-europium reticulation was recognized as square-triangle tessellation with dodecagonal symmetry. Our findings introduce quasicrystallinity in surface-confined MOCNs with a nanoporous structure and anticipate functionalities that arise from quasicrystalline ordering of the coordinative spheres.

... Nevertheless, based on the underlying similarities in local tetrahedral particle packing, implicit in the common triangular and square tiling elements that govern the σ and DDQC phases (Fig. 2B), we expect similar overall sphericities (see below). Formation of the DDQC structure as a metastable intermediate state before the formation of the equilibrium Frank-Kasper σ phase appears quite similar to the simulated phase behavior of collections of uniformly sized spherical particles governed by the Dzugotov potential, a modified Lennard-Jones interparticle potential that contains a repulsive feature that suppresses the formation of closepacked (BCC, HCP, and FCC) crystal structures (20,30,31). There are important differences between this computational model and the experimental polymer system, most significantly the consequences of mass exchange during the formation of the σ phase, which leads to five discrete particle shapes and volumes in the tetrahedral unit cell, features that are not represented in the hard sphere simulations. ...

... There are important differences between this computational model and the experimental polymer system, most significantly the consequences of mass exchange during the formation of the σ phase, which leads to five discrete particle shapes and volumes in the tetrahedral unit cell, features that are not represented in the hard sphere simulations. Nevertheless, the simulations demonstrate the metastable nature of the DDQC and identify icosahedral clusters as the seeds that nucleate the quasicrystals that grow to fill space before transforming to the σ phase, which is dominated by lattice sites configured as distorted icosahedral motifs (20). ...

... Unfortunately, there is insufficient detail in the overall SAXS pattern to extract quantitative information regarding the detailed radial distribution function. Perhaps these reflect icosahedral clusters as suggested by molecular simulations (20,45) and as implicated in many investigations of glass-forming systems (46,47). ...

We report the discovery of a dodecagonal quasicrystalline state (DDQC) in a sphere (micelle) forming poly(isoprene-b-lactide) (IL) diblock copolymer melt, investigated as a function of time following rapid cooling from above the order-disorder transition temperature (TODT = 66 °C) using small-angle X-ray scattering (SAXS) measurements. Between TODT and the order-order transition temperature TOOT = 42 °C, an equilibrium body-centered cubic (BCC) structure forms, whereas below TOOT the Frank-Kasper σ phase is the stable morphology. At T < 40 °C the supercooled disordered state evolves into a metastable DDQC that transforms with time to the σ phase. The times required to form the DDQC and σ phases are strongly temperature dependent, requiring several hours and about 2 d at 35 °C and more than 10 and 200 d at 25 °C, respectively. Remarkably, the DDQC forms only from the supercooled disordered state, whereas the σ phase grows directly when the BCC phase is cooled below TOOT and vice versa upon heating. A transition in the rapidly supercooled disordered material, from an ergodic liquid-like arrangement of particles to a nonergodic soft glassy-like solid, occurs below ∼40 °C, coincident with the temperature associated with the formation of the DDQC. We speculate that this stiffening reflects the development of particle clusters with local tetrahedral or icosahedral symmetry that seed growth of the temporally transient DDQC state. This work highlights extraordinary opportunities to uncover the origins and stability of aperiodic order in condensed matter using model block polymers.

... Other variations of the free energy functional have also been considered, e.g. a mean field density functional theory has been employed to reveal the formation of complex structures [110,374] and fundamental measure theory has been used in order to study the structure and the growth dynamics on incommensurate and/or quasicrystalline substrates [264,344,369]. Additionally, the growth behavior can be studied, e.g. with molecular dynamics simulations that have been employed to investigate atomic dodecagonal quasicrystals growing from clusters that form in the melt [375] or to determine the formation of quasicrystals [92,102,[376][377][378]. ...

... In our study of the growth of a quasicrystalline structure we aim to prevent homogeneous nucleation that occurs, e.g. in an undercooled liquid (see [375]). Therefore, we model the growth at a solid-gas interface where the grown structure is at the bottom of the simulation box and particles are inserted at the top (see Figure 6.2). ...

... [387,388]). Finally, while we only study the growth obtained by depositing one particle after the other on the surface, it might be interesting to have a closer look at the growth that occurs if small clusters can form in the melt and such clusters are deposited on the surface of the quasicrystal (see [375]). ...

Quasicrystals are remarkable ordered structures without periodic translational symmetry. They can possess any discrete rotational symmetry including those that must not be present in periodic crystals. An important characteristic of quasicrystals is the existence of additional degrees of freedom whose excitations lead to rearrangements of particles. Such rearrangements are called phasonic flips. Several material properties like the elasticity of quasicrystals are affected by phasonic excitations. By now, quasicrystals have been synthesized in experiments and simulations and have even been found in nature. The aim of this thesis is to contribute to the understanding of the amazing order and properties of quasicrystals. For this purpose we employ computational simulations of two-dimensional colloidal model systems where mesoscopic particles are suspended in a liquid. By implementing appropriate external or internal interactions quasicrystalline order of the colloids can be induced. Significant new insights into the structural and dynamical complexity of quasicrystals are gained. Our results are essentially different from what is known from periodic crystals. In particular, investigations of the phase behavior of quasicrystals reveal a surprisingly rich phase diagram. Even in the solid, positional order is short-ranged due to excited phasonic degrees of freedom, and the transition to liquid is of first order. Furthermore, we illustrate how the growth of quasicrystals is affected by thermodynamic parameters and phasonic flips. Especially the growth of nearly defect-free quasicrystals is presented. In addition, we focus on the dynamics of quasicrystals. The stability of quasicrystals against phasonic perturbations is investigated. Particles are identified which easily perform phasonic flips, while other ones are rather stable. Phasonic drifts lead to complex trajectories of the particles. Even in intrinsic quasicrystals, which form under internal interactions alone, correlated phasonic flips are found and analyzed. Our work provides significant progress in theory and simulations of quasicrystals and our results obtained from colloidal model systems are also relevant for other fields in physics, chemistry and material science. We expect that our work motivates further theoretical and experimental research on quasicrystals and might also advance the design of novel applications based on quasicrystals.

... But interestingly, icosahedral clusters consisting of a small amount of atoms which show fivefold rotations have low free energy, and can exist in liquid metals [10][11][12] . And quasicrystals can form based on icosahedral clusters present in undercooled liquid during solidification [13][14][15][16][17][18] . It is also known that fivefold rings of atoms may be produced at dislocation cores in crystals 1,19 . ...

... It is also known that fivefold rings of atoms may be produced at dislocation cores in crystals 1,19 . This knowledge about quasicrystals and dislocations 1,[13][14][15][16][17][18]20 implies that dislocations in crystals might serve as nucleation sites for icosahedral clusters. However, it has been unclear whether icosahedral clusters form and interact with the broken symmetry along dislocations in crystalline materials at the earliest stage of precipitation [2][3][4][5] . ...

Dislocations in crystals naturally break the symmetry of the bulk, introducing local atomic configurations with symmetries such as fivefold rings. But dislocations do not usually nucleate aperiodic structure along their length. Here we demonstrate the formation of extended binary quasicrystalline precipitates with Penrose-like random-tiling structures, beginning with chemical ordering within the pentagonal structure at cores of prismatic dislocations in Mg-Zn alloys. Atomic resolution observations indicate that icosahedral chains centered along [0001] pillars of Zn interstitial atoms are formed templated by the fivefold rings at dislocation cores. They subsequently form columns of rhombic and elongated hexagonal tiles parallel to the dislocation lines. Quasicrystalline precipitates are formed by random tiling of these rhombic and hexagonal tiles. Such precipitation may impact dislocation glide and alloy strength.

... Simulation studies indicate that the growth of quasicrystals is controlled by the ability of a growing quasicrystal nucleus to incorporate building spherical motifs into the solid phase with minimal rearrangement of these motifs; i.e., during the bulk quasicrystal phase formation, the "growth rule" is the tendency to retain configurations of building motifs rather than copying a nucleus surface template to form a traditional crystal. 35 The σ phase is a periodic approximant structure to the DQC phase. They share similar local tetrahedral closed packing features. ...

... The less ordered DQC could be able to reach a "structural compromise" with the surrounding motifs in a more rapid rate as a kinetic favorable phase when the rearrangement ability of the building motifs is limited, compared with the σ phase which requires relative decent rearrangements of the building motifs into the crystal lattice. 14 The rearrangement of motifs during phase formation would be a combination of configuration rearrangements of multiple sphere clusters 35 and internal rearrangement of the spheres themselves. 14 However, to better understand the real subunit cell information on these complex structures, high fidelity 3D reconstruction studies in the future are necessary. ...

... Theoretically, the quasicrystalline ordering in softmatter is addressed using pair potentials with hard core and a soft repulsive shoulder interactions [16][17][18][19]. The growth of quasicrystalline long-range ordering from simple periodic approximants interacting via short-range potentials were also studied using molecular simulations [20][21][22]. Schematic representation of the spherical supramolecules formed from the cone-shaped dendrimers and the quasicrystalline structures normal to the rotation axis. (a,b) Cone shaped dendrimers assemble into spherical supramolecules. ...

Quasicrystalline ordering was first observed in synthetic multi-component metallic alloys. These solid state materials exhibit quasicrystalline atomic ordering at nanometer length scales. Softmatter systems are another class of versatile materials that can exhibit quasicrystalline ordering across supra-nanometer (>10 nm) to supra-micrometer (>10 μm) length scales as recently observed in materials like-supramolecular dendritic molecules, ABC star polymers, binary nanoparticle systems and block co-polymers in condensed matter systems. The underlying mechanism in most of these soft quasicrystals seems to be the presence of two or more length scales in the system. Another class of development in self-assembled quasicrystals in softmatter is being observed in low molecular weight chiral and achiral nematic liquid crystals. Liquid crystal forms an efficient matrix for self- and directed-assemblies of colloidal structures where surface and geometry-tuning the particles in nematic liquid crystals gives rise to complex inter-particle interactions while the long-range order results in self-assembled structures of higher order rotational symmetries. Furthermore, there has also been attempts to generate colloidal quasicrystalline defect structures by directing the assemblies using multiple and single beam lasing techniques. In the present article, we will review self- and assisted-assembly of quasicrystalline structures in nematic liquid crystals (both chiral and achiral) and discuss the underlying mechanisms.

... The bond-orientational order metric Q 6 [22] is an important order parameter when describing glass transitions [46][47][48] and crystalline clusters [49][50][51]. For each particle, a set of bonds connect its center to the centers of its neighbor particles. ...

... Amman tiles were also used to simulate phase transitions in quasicrystals (see, e.g.,Leuzzi and Parisi, 2000;Koch and Radin, 2010;Aristoff and Radin, 2011). There are a number of quasicrystal models (see, e.g.,Onoda et al., 1988;The Toner, 1990;Marcia, 2006;Keys andGlotzer, 2007, Steinhardt, 2008). These models are based on different assumptions about the mechanisms of interaction of elementary structures, which form a quasicrystal. ...

The current protein folding literature is reviewed. Two main approaches to the problem of folding were selected for this review: geometrical and biophysical. The geometrical approach allows the formulation of topological restrictions on folding, that are usually not taken into account in the construction of physical models. In particular, the topological constraints do not allow the known funnel-like energy landscape modeling, although most common methods of resolving the paradox are based on this method. The very paradox is based on the fact that complex molecules must reach their native conformations (complexes that result from reactions) in an exponentially long time, which clearly contradicts the observed experimental data. In this respect we considered the complexity of the reactions between ligands and proteins. On this general basis, the folding-reaction paradox was reformulated and generalized. We conclude that prospects for solving the paradox should be associated with incorporating a topology aspect in biophysical models of protein folding, through the construction of hybrid models. However, such models should explicitly include long-range force fields and local cell biological conditions, such as structured water complexes and photon/phonon/soliton waves, ordered in discrete frequency bands. In this framework, collective and coherent oscillations in, and between, macromolecules are instrumental in inducing intra- and intercellular resonance, serving as an integral guiding network of life communication: the electrome aspect of the cell. Yet, to identify the actual mechanisms underlying the bonds between molecules (atoms), it will be necessary to perform dedicated experiments to more definitely solve the particular time paradox.

... The normalized local bond-orientational order parameterQ 6local is similar to the original bond-orientational order parameter Q 6 [49], which is an important order parameter when describing glass transitions [50][51][52] and crystalline clusters [53][54][55]. The Q 6 is calculated as follows [49,56]. ...

The superellipsoid model is a rich geometric model and is convenient to study the particle shape effects on random packings. The particle shape significantly influences the macroscopic and microscopic structure properties of random packings. In this work, we find uniform and decoupled shape effects on the maximally dense random packings (MDRPs) of hard superellipsoids. Slightly changing the surface shape or elongating (compressing) the particles may influence the random packing density significantly. The influences of surface shape parameter p and aspect ratio w on the random packing densities are decoupled. For the aspect ratio effects, all the packing density curves show “M” type with various p. Meanwhile, the aspect ratio effects are applicable to all the symmetric particles with three equal main cross sections when w = 1.0. For the surface shape effects, the packing density curve is also in “M” type with various w. The maximum of the random packing density is obtained at p ≈ 0.7, 2.0 and w ≈ 0.7, 1.5. Moreover, we obtain the MDRPs of hard superellipsoids via the inverse Monte Carlo packing method with a wide range of the surface shape parameter. The normalized local cubatic order parameter and a new normalized local bond-orientational order parameter are used to evaluate the order degrees of orientations and bond-orientations in random packings, respectively. The local analyses of the MDRPs of superellipsoids are carried out via the Voronoi tessellation. Two linear relationships between the mean and standard deviation of the reduced Voronoi cell volumes are obtained. Our findings should lead to a better understanding of random packings and are helpful in guiding the granular material design.

... Furthermore, although this may be the stable phase, the presence of an underlying FCC crystal cannot be ruled out. Figure 4d shows that under strong bias, our model may provide a suitable means to investigate the nucleation process in quasicrystals. We note that related behaviour has been observed in fivefold symmetric patchy particles 39 , the Dzugutov model 40 , which is also designed to suppress FCC symmetry 41 and indeed in hard tetraheda, although in that case the local ordering was not fivefold symmetric 42 . These observations lead us to propose the phase diagram shown in Fig. 1c. ...

Although long assumed to have an important role in the suppression of crystallization and the development of glassformers, the effect of local fivefold symmetry has never been directly tested. Here we consider whether such suppression of crystallization has a kinetic or thermodynamic nature and investigate its mechanism. We introduce a model in which the degree of fivefold symmetry can be tuned by favouring arrangements of particles in pentagonal bipyramids. We thus show that fivefold symmetry has both kinetic and thermodynamic effects on the mechanism of crystallization to a face-centred cubic crystal. Our results suggest that the mechanism of crystallization suppression is related to the surface tension between fluid and crystal. Interestingly, the degree of fivefold symmetry has little effect on crystal growth rate, suggesting that growth may be only weakly coupled to fluid structure in hard sphere like systems. Upon increasing the fivefold symmetry, we find a first-order transition to an alternative icosahedra-rich phase. At intermediate bias strengths we find a one-component glassformer.

... In the system studied stable icosahedral clusters are assimilated. For small nuclei and long-range ordering it appears that different growth mechanisms are at work (Keys and Glotzer, 2007). Tsai-type clusters of the quasicrystal approximant CaCd6 emerge from an atomic-size-driven transformation from planar arrangements to spherical clusters (Berns and Fredrickson, 2013), perhaps reminiscent of the formation of clathrin cages from triskelia. ...

Since the discovery of metallic quasicrystals, which lack translational symmetry, much work has been done on their characterisation. In particular mathematical aperiodic tilings have been invoked in an effort to explain their existence. It appears that in three-dimensional quasicrystals non-local quantum effects may not be required. However, here I present some instances - ribosome organisation in embryonic plants, neurotransmitter receptor complexes in animal nervous systems, the cross-section of microtubule bundles and nucleic acids - where biological two-dimensional quasicrystals may occur. Two-dimensional quasicrystals do require non-local quantum effects. This has possible ramifications for both plant development and consciousness. The golden mean is integral to the structure of aperiodic tilings and is found widely in nature, including in plant development, brain EEG waves, and in the consciousness of beauty. I suggest that somehow the encoding of the golden mean in sub-cellular quasicrystals leads to its expression in these macro systems.

... Both simple and elaborated two-body interaction potentials have been used to model the effective interactions between mesoscopic entities composing the soft materials. By using Monte Carlo (MC) and molecular dynamics (MD) simulations, equilibrium structures, phase diagrams and kinetic routes for the growth of quasicrystals have been explored [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Most of these theoretical and simulation works have focused on single-component systems, which already display an impressive palette of structural arrangements. ...

Hierarchical self-assembly of soft matter provides a powerful route to create complex materials with enhanced physical properties. The understanding of the fundamental processes leading to such organization can provide design rules to create new functional materials. In this work, we use a simple model of polymer-grafted nanoparticles to explore the self-assembly of binary mixtures. By using Monte Carlo simulations we study the interplay of composition, density and particle sizes on the self-organization of such nanoparticle systems. It is found that complex hierarchical organization can take place for conditions where one-component systems form simple lattices. In particular, a mixture where one component forms a structure with 18-fold symmetry in a sea of an apparent disordered phase of the second component is observed to emerge for a certain parameter combinations.

... Despite a strong potential for numerous technical applications, the conditions under which quasicrystals form are still poorly understood [106,4,107,108,109,110]. Currently, it is not clear why all two-dimensional quasicrystals are of a rank ∆ = 4 but no single example with ∆ ≥ 6 has ever been observed [111,112]. ...

Quasicrystals are somewhat paradoxical structures which exhibit many amazing properties distinguishing them from ordinary crystals. Although the atoms are not localized at periodic positions, quasicrystals posses perfect long-range order. Until the early 1980s it was unanimously established that ordered matter is always periodic. Accordingly, the rotational symmetry in real space was thought to be limited to n=2,3,4 and 6. However more than a hundred complex metal alloys, for instance the discretely diffracting icosahedral AlPdMn or decagonal AlNiCo, have defied these crystallographic rules and self-organized into quasicrystals. Although the majority of the identified quasicrystals are complex metal alloys synthesized in the laboratory, recent experimental results proved that quasiperiodic order is not limited to metals. Matter also organizes itself aperiodically at larger length scales where thermal fluctuations play an important role. Recent experiments have shown that quasiperiodic order is also oberved in soft matter systems, such as micellars, polymers, and binary nanoparticles. Quasicrystals show many interesting properties which are quite different from that of periodic crystals. Accordingly, they are considered as materials with high technological potential e.g. as surface coatings, thermal barriers, catalysts or photonic materials.
Quasicrystalline structures have been theoretically predicted also in systems with a single type of particles. Nevertheless, experimentally their spontaneous formation has been only observed in binary, ternary or even more complex alloys. Accordingly, their surfaces exhibit a high degree of structural and chemical complexity and show intriguing properties. In order to understand the origin of those characteristics it would be helpful to disentangle structural and chemical aspects which can be achieved by growing single-element monolayers to quasicrystalline surfaces. Apart from understanding how quasicrystalline properties can be transferred to such monolayers, this approach might allow fabrication of materials with novel properties. First heteroepitatic growth experiments on decagonal and icosahedral surfaces indeed demonstrate the formation of Pb, Bi and Sb monolayers with a high degree of quasicrystalline order as determined by low-energy electron diffraction and elastic helium atom scattering experiments. Compared to reciprocal space studies, only recently atomically resolved scanning tunneling microscopy investigations of the adsorbate morphology became possible. Even then, however, it is difficult to relate the structure of the adsorbate to that of the underlying substrate.
In that respect, the study of the phase behaviour of colloidal particles interacting with quasiperiodic laser fields can throw new light on fundamental problems of broad interest in the physics of quasicrystals and in condensed matter physics. In fact colloidal systems are meanwhile established as excellent model for atomic systems and colloidal physics have demonstrated that such systems can give answers to many basic physics questions. Depending on the pair-interaction and the concentration, colloidal systems show analogues of all the states of atomic systems: gas, liquid and solid states. The mesoscopic size (nm-µm), the time scales (ms-s) and the tunability of the pair interaction in colloidal systems make them a convenient model system for experimental and theoretical studies. As a consequence, real space analysis by means of video microscopy allows tracking the trajectories of the individual particles and makes the time evolution of the system accessible in detail. Such information is inaccessible in systems investigated by diffraction experiments, as the scattering information is available only averaged over the scattering area. Because in a colloidal system there is direct access to real space information, the strength and nature of the different interactions, the origins of the complex phase behavior could be in different examples identified. In conclusion, the study of the rich phase behavior of colloidal suspensions provides ideal conditions for experimental and theoretical studies.
In this Thesis, we report on a real-space investigation of the phase behaviour of charged colloidal monolayers interacting with quasicrystalline decagonal or tetradecagonal substrates created by interfering five or seven laser beams. Different starting configurations, such as dense fluid and triangular crystals with different densities, are prepared. At low intensities and high particle densities, the electrostatic colloidal repulsion dominates over the colloid-substrate interaction and the crystalline structure remains mainly intact. As expected, at very high intensities the colloid-substrate interaction dominates and a quasiperiodic ordering is observed. Interestingly, at intermediate intensities we observe the alignment of crystalline domains along the 5 directions of the quasicrystalline substrate. This is in agreement with observations of Xenon atoms adsorbed on the ten-fold decagonal Al-Ni-Co surface and numerical simulations of weakly adsorbed atomic systems. Intermediate phases are observed for colloid-substrate interactions strong enough to produce defects in the crystal. These defects adapt the form of rows of quadratic tiles. Surprisingly, for specific particle densities (at which the colloid-substrate interaction is minimized) we identify a novel pseudomorphic ordering. This intermediate phase which exhibits likewise crystalline and quasicrystalline structural properties can be described by an Archimedean-like tiling consisting of alternating rows of quadratic and triangular tiles. The calculated diffraction pattern of this phase is in agreement with recent observations of copper adsorbed on icosahedral AlPdMn surfaces. Interestingly, we also observe the formation of the same phase on tetradecagonal substrates also at densities for which the potential energy of the colloidal system is minimized. Although the structure can also be described by rows of triangles and rows of squares, a closer analysis reveals substantial differences. Here, large domains with almost periodic ordering are found. We show that this behavior is closely related to the low density of highly symmetric local motifs in the substrate potential.
In the second part of this Thesis the conditions under which quasicrystals form are investigated. Currently, it is not clear why most quasicrystals hold 5- or 10-fold symmetry but no single example with 7 or 9-fold symmetry has ever been observed. Since the properties of quasicrystals are strongly connected to their atomic structure, a better understanding of their growth mechanisms is of great importance. In contrast to crystals which are periodic in all three dimensions, quasiperiodicity is always (except for icosahedral quasicrystals) restricted to two dimensions. Accordingly, three-dimensional quasicrystals are comprised of a periodic stacking of quasiperiodic layers and any hurdle in the formation of quasiperiodic order within a single layer will eventually prohibit their growth along the periodic direction. In this Thesis, we also report on geometrical constraints which impede the formation of quasicrystals with certain symmetries in a colloidal model system. This is achieved by subjecting a colloidal monolayer to N=5- and 7-beam quasiperiodic potential landscapes. Our results clearly demonstrate that quasicrystalline order is much easier established for N = 5 compared to N = 7. With increasing laser intensity we observe that the colloids first adopt quasiperiodic order at local areas which then laterally grow until an extended quasicrystalline layer forms. As nucleation sites where quasiperiodicity originates, we identify highly symmetric motifs in the laser pattern. We find that their density strongly varies with n and surprisingly is smallest exactly for those quasicrystalline symmetries which have never been observed in atomic systems. Since such high symmetry motifs also exist in atomic quasicrystals where they act as preferential adsorption sites, this suggests that it is indeed the deficiency of such motifs which accounts for the absence of e.g. materials with 7-fold symmetry.
In addition to the fundamental aspects, we report in this Thesis on the fabrication of large colloidal quasiperiodic layers incorporated in a polymer hydrogel matrix. Because quasicrystals have higher point group symmetry than ordinary crystals, micrometer-scale quasicrystalline materials are expected to exhibit large and isotropic photonic bandgaps in the visible range. In our case, the quasiperiodic symmetries are induced using extended light fields. The reported gelled colloidal quasicrystals are unique in that they have large sizes as well as good optical uniformity. With laser diffraction the in situ variable length scale of such materials is demonstrated.
In conclusion, we have studied the phase behavior of charged colloidal particles interacting with quasiperiodic laser fields. We showed that novel pseudomorphic growth can lead to the formation of a phase which exhibits likewise crystalline and quasicrystalline structural properties. We also performed unconventional measurements in order to understand why the formation of quasicrystals is limited to specific rotational symmetries. We have found that geometrical hurdles play a crucial role in the proliferation of quasiperiodicity and that such hurdles can hindered or even prohibited the formation of e.g. 7- or 9-fold symmetry. And finally, we have shown that the combination of extended light fields and hydrogel matrices leads to the formation of large quasiperiodically ordered colloidal materials.

... This finding reflects a higher interfacial free energy at a certain normalized temperature, due to relatively less similarity of local orders between the liquid and crystal. A recent simulation result showed that an increasing contribution of five-fold symmetry in liquid, leading to an increase in crystal-liquid interfacial free energies, which consequently suppressed the formation of crystal nucleation [49,50]. ...

The formation mechanism of metastable crystals from metastable liquids still remains elusive, although controlling the metastability of crystals and liquids already plays a crucial role in designing new materials in physics, chemistry, biology, and materials science. This review article describes how metastable phases can be obtained by controlling temperature, concentration, and pressure. In particular, I show the role of crystal-liquid interfacial free energy in the formation of metastable crystals from metastable liquids at a given driving force. In a microscopic viewpoint, local structure similarity between the metastable crystals and liquid determines the crystal-liquid interfacial free energy, and thus the nucleation barrier for the metastable crystals. The effect of the interfacial free energy on the formation of metastable crystals from supercooled, supersaturated, and supercompressed liquids will be demonstrated with metallic liquids, aqueous solutions, and water.

... FK phases are topologically close-packed and contain exclusively tetrahedral interstices composed of a polyhedron with coordination number CN = 12 and at least one polyhedron with higher CN (CN = 14, 15 or 16) (Fig. 1a). Often, FK phases can be viewed as ordered approximates of quasicrystal phases due to some shared construction rules from sphere packing [5][6][7] . So far, 27 different types of FK phases have been found experimentally 8 . ...

Frank–Kasper phases, a family of ordered structures formed from particles with spherical motifs, are found in a host of materials, such as metal alloys, inorganic colloids and various types of soft matter. All the experimentally observed Frank–Kasper phases can be constructed from the basic units of three fundamental structures called the A15, C15 and Z phases. The Z phase, typically observed in metal alloys, is associated with a relatively large volume ratio between its constituents, and this constraint inhibits its formation in most self-assembled single-component soft-matter systems. We have assembled a series of nanosized shape amphiphiles that comprise a triphenylene core and six polyhedral oligomeric silsesquioxane cages grafted onto it through linkers to give a variety of unconventional structures, which include the Z phase. This structure was obtained through fine tuning of the linker lengths between the core and the peripheral polyhedral oligomeric silsesquioxane cages, and exhibits a relatively large volume asymmetry between its constituent polyhedral particle motifs.

... The relative importance of the order parameters for bulk system classification is shown in Fig. 4 and most important parameters are Q 12 , Q 4 , LSI, and d 5 . The highest weights applied in the bond orientational orders for classifying bulk phases coincides with previous studies showing that Q 12 is sensitive to the symmetry of crystal systems 50,51 and Q 4 can be used to distinguish liquid and ice phases with Q 6 48 . ...

Understanding phases of water molecules based on local structure is essential for understanding their anomalous properties. However, due to complicated structural motifs formed via hydrogen bonds, conventional order parameters represent the water molecules incompletely. In this paper, we develop a GCIceNet, which automatically generates machine-based order parameters for classifying the phases of the water molecules via supervised and unsupervised learning. Multiple graph convolutional layers in the GCIceNet can learn topological informations of the complex hydrogen bond networks. It shows a substantial improvement of accuracy for predicting the phase of water molecules in the bulk system and the ice/vapor interface system. A relative importance analysis shows that the GCIceNet can capture the structural features of the given system hidden in the input data. Augmented with the vast amount of data provided by molecular dynamics simulations, the GCIceNet is expected to serve as a powerful tool for the fields of glassy liquids and hydration layers around biomolecules.

... Notably, dodecagonal quasicrystal (DQC) phase usually appears as the metastable state of phase, since they share some constructing rules and FK phases can be viewed as periodic approximants of quasicrystal phase [159,166]. A DQC phase only has twelve-fold rotational symmetry without translational symmetry on the ab plane, and has translational symmetry on c axis. ...

Macromolecular self-assembly has made explosive development in the last several decades, are being extensively explored in the fields of drug delivery, lithography, catalysis, molecular electronics, sensors, and so many others. In this review, we summarize the self-assembly of macromolecules such as polymers, dendrimers, molecular nanoparticles, colloids, DNA and proteins, from the aspect of architectural engineering and mainly focus on the periodic and quasi-periodic assembled structures. In particular, simple building blocks can be conjugated together to construct complex macromolecular motifs with different architectures and physical interactions. We first introduce the concept of architectural engineering, then present a brief overview of supramolecular interactions and five main categories of building blocks, including polymer coils, dendrons, rods, discs, and polyhedra. We further discuss the detailed self-assembly behaviors of fifteen types of specific macromolecular motifs involving different building blocks. Special attention is paid to the architectural effect of macromolecular self-assembly. In the end, future perspective on architectural engineering is briefly mentioned.

... High temperature kinetics seems to favor the quasicrystal over related complex crystals. Due to the wealth of possible local configurations in quasicrystals, their formation is much easier than that of a periodic crystal, where each particle has its fixed position [92]. This might explain, why quasicrystals are often found in rapidly solidified intermetallic compounds. ...

Komplexe Kristalle und Quasikristalle sind geordnete Festkörperstrukturen mit sehr großen beziehungsweise unendlichen großen Einheitszellen. Ihre Teilchendynamik unterscheidet sich grundlegend von derjenigen der einfachen Kristalle: Aufgrund der strukturellen Komplexität treten lokale Umordnungen, sogenannte Phasonflips, und neueartige Versetzungen auf. Um das Verhalten auf einer elementaren Ebene zu verstehen, werden drei ein- und zweidimensionale Modellsysteme eingeführt. Die Systeme werden sowohl analytisch, als auch mit numerischen Simulationen untersucht.
(i) Die Strukturfaktoren der dynamischen Fibonacci-Kette wurden in hoher Auflösung berechnet. Sie zeigen eine charakteristische Verbreiterung der Phonondispersionsrelation. Die Teilchendynamik im Realraum zeichnet sich durch Soliton- und Breathermoden aus, welche zusammen mit Phasonflips auftreten.
(ii) Mit Hilfe eines Tilingmodells wurde die experimentelle Beobachtung von Metaversetzungen im intermetallischen System AlPdMn und die damit verbundene kollektive strukturelle Umordnung erklärt. Die Burgersvektoren der Versetzungen können aus Energieüberlegungen abgeleitet werden.
(iii) Im Lennard-Jones-Gauß-System treten eine überaschende Vielzahl an zweidimensionalen komplexen Kristallen, ein dekagonaler und zwei dodekagonale Quasikristalle auf. Bei Temperaturen nahe des Schmelzpunktes ordnen sich die Teilchen per Phasonflips um. Während des Abkühlens transformieren sich die entropisch stabilisierten Quasikristalle reversibel in komplexe Kristalle. Mit dem Lennard-Jones-Gauß-System ist es zum ersten Mal möglich, das Wachstum, die Gleichgewichtsdynamik und die Defekte von Quasikristallen und komplexen Kristallen in Simulationen zu untersuchen.

... Some interparticle interactions, which are negligibly weak at larger scales, such as the hydrophobic interaction of surface ligands, can have a significant role in nanoparticle assembly 156,157 . As a consequence of the complexity of interparticle interactions, nanoparticle self-assembly often leads to unpredictable results, as in the case of quasicrystalline spherical nanoparticle superlattices 158,159 . Usually, the self-assembly of spherical particles is an entropy-driven process that maximizes space-filling efficiency, which leads to the formation of Bravais lattice structures with translational symmetry. ...

The synthesis of nanoparticles with particular compositions and structures can lead to nanoparticles with notable physicochemical properties, thus promoting their use in various applications. In this area of nanoscience, the focus is shifting from size- and shape-uniform single-component nanoparticles to multicomponent nanoparticles with enhanced performance and/or multifunctionality. With the increasing complexity of synthetic reactions, an understanding of the formation mechanisms of the nanoparticles is needed to enable a systematic synthetic approach. This Review highlights mechanistic studies underlying the synthesis of nanoparticles, with an emphasis on nucleation and growth behaviours that are not expected from classical theories. We discuss the structural properties of nanoclusters that are of a size that bridges molecules and solids. We then describe the role of nanoclusters in the prenucleation process as well as in nonclassical nucleation models. The growth of nanoparticles via the assembly and merging of primary particles is also overviewed. Finally, we present the heterogeneous nucleation mechanisms behind the synthesis of multicomponent nanoparticles.

... Early molecular dynamics simulations considered hard spheres [64] and configurations of model protein molecules [65]. Recent simulations that apply statistical mechanics to problems in self-assembly have elucidated control over tethered nanoparticle structure [66][67][68][69][70], quasicrystal formation [71,72], and how interaction anisotropy can be used to control structure [3,73,74]. Exploiting new computational hardware such as graphics processing units (GPUs) [75] and implementing new techniques on them [76,77] will advance our ability to engineer new materials using self-assembly. ...

We develop advanced Monte Carlo sampling schemes and new methods of calculating thermodynamic partition functions that are used to study the self-assembly of complicated ``patchy '' particles. Patchy particles are characterized by their strong anisotropic interactions, which can cause critical slowing down in Monte Carlo simulations of their self-assembly. We prove that detailed balance is maintained for our implementation of Monte Carlo cluster moves that ameliorate critical slowing down and use these simulations to predict the structures self-assembled by patchy tetrominoes. We compare structures predicted from our simulations with those generated by an alternative learning-augmented Monte Carlo approach and show that the learning-augmented approach fails to sample thermodynamic ensembles. We prove one way to maintain detailed balance when parallelizing Monte Carlo using the checkerboard domain decomposition scheme by enumerating the state-to-state transitions for a simple model with general applicability. Our implementation of checkerboard Monte Carlo on graphics processing units enables accelerated sampling of thermodynamic properties and we use it to confirm the fluid-hexatic transition observed at high packing fractions of hard disks. We develop a new method, bottom-up building block assembly, which generates partition functions hierarchically. Bottom-up building block assembly provides a means to answer the question of which structures are favored at a given temperature and allows accelerated prediction of potential energy minimizing structures, which are difficult to determine with Monte Carlo methods. We show how the sequences of clusters generated by bottom-up building block assembly can be used to inform ``assembly pathway engineering'', the design of patchy particles whose assembly propensity is optimized for a target structure. The utility of bottom-up building block assembly is demonstrated for systems of CdTe/CdS tetrahedra, DNA-tethered nanospheres, colloidal analogues of patchy tetrominoes and shape-shifting particles.

Three-dimensional morphology and formation process of icosahedral quasicrystal phase have been investigated in a melt-spun Al–18Mn alloy (in wt%). Three distinct layers corresponding to varying temperature gradient have been observed on the cross section of the ribbons. 3D morphologies of cellular and dendritic icosahedral phase have been obtained through electro-etching. A model has been proposed to describe the formation process of the icosahedral phase and α-Al during the rapid solidification. The icosahedral phases are primarily precipitated from the melt into fine cellular and dendritic particles, and subsequently engulfed by the α-Al which propagates in a planar morphology.

As crystallography merges with materials science and engineering, mathematical crystallography is growing in new directions, including: Characterizing new materials with unusual properties; Imaging, including but not limited to diffraction; Exploring and exploiting superspaces; Mapping the aperiodic landscape, from chaos to classical periodicity and beyond; Re-modeling the structures of real crystals, both periodic and aperiodic; Modeling self-assembly and self-reorganization on the nanoscale. In short, it’s not (just) about space groups and tilings anymore.

We propose a novel, highly-efficient approach for the evaluation of bond-orientational order parameters (BOPs). Our approach exploits the properties of spherical harmonics and Wigner 3j-symbols to reduce the number of terms in the expressions for BOPs, and employs simultaneous interpolation of normalised associated Legendre polynomials and trigonometric functions to dramatically reduce the total number of arithmetic operations. Using realistic test cases, we show how the above, combined with a CPU-cache-friendly data structure, leads to a 10 to 50-fold performance increase over approaches currently in use, depending on the size of the interpolation grids and the machine used. As the proposed approach is an approximation, we demonstrate that the errors it introduces are well-behaved, controllable and essentially negligible for practical grid sizes. We benchmark our approach against other structure identification methods (centro-symmetry parameter (CSP), common neighbour analysis (CNA), common neighbourhood parameter (CNP) and Voronoi analysis), generally regarded as much faster than BOPs, and demonstrate that our formulation is able to outperform them for all studied systems.

Using molecular dynamics simulations we show that a one-component system can be driven to a three-dimensional decagonal (10-fold) quasicrystalline state just by purely repulsive, isotropic and monotonic interaction pair potential with two characteristic length scales; no attraction is needed. We found that self-assembly of a decagonal quasicrystal from a fluid can be predicted by two dimensionless effective parameters describing the fluid structure. We demonstrate stability of the results under changes of the potential by obtaining the decagonal order for three particle systems with different interaction potentials, both purely repulsive and attractive, but with the same values of the effective parameters. Our results suggest that soft matter quasicrystals with decagonal symmetry can be experimentally observed for the same systems demonstrating the dodecagonal order for an appropriate tuning of the effective parameters.

Systems of soft-core particles interacting via a two-scale potential are
studied. The potential is responsible for peaks in the structure factor of the
liquid state at two different but comparable length scales, and a similar
bimodal structure is evident in the dispersion relation. Dynamical density
functional theory in two dimensions is used to identify two novel states of
this system, the crystal-liquid state, in which the majority of the particles
are located on lattice sites but a minority remains free and so behaves like a
liquid, and a 12-fold quasicrystalline state. Both are present even for deeply
quenched liquids and are found in a regime in which the liquid is unstable with
respect to modulations on the smaller scale only. As a result the system
initially evolves towards a small scale crystal state; this state is not a
minimum of the free energy, however, and so the system subsequently attempts to
reorganize to generate the lower energy larger scale crystals. This dynamical
process generates a disordered state with quasicrystalline domains, and takes
place even when this large scale is linearly stable, i.e., it is a nonlinear
process. With controlled initial conditions a perfect quasicrystal can form.
The results are corroborated using Brownian dynamics simulations.

Phase transformation of quasicrystals is of interest in various fields of science and technology. Interestingly, we directly observed unexpected solid-state epitaxial nucleation and growth of Zn6Mg3Y icosahedral quasicrystals in a Mg alloy at about 573 K which is about 300 K below the melting point of Zn6Mg3Y, in contrast to formation of quasicrystals through solidification that was usually found in many alloys. Maximizing local packing density of atoms associated with segregation of Y and Zn in Mg adjacent to Mg/Zn3MgY interfaces triggered atomic rearrangement in Mg to form icosahedra coupled epitaxially with surface distorted icosahedra of Zn 3 MgY, which plays a critical role in the nucleation of icosahedral clusters. A local Zn:Mg:Y ratio close to 6:3:1, corresponding to a valence electron concentration of about 2.15, should have been reached to trigger the formation of quasicrystals at Mg/Zn3MgY interfaces. The solid-state icosahedral ordering in crystals opens a new window for growing quasicrystals and understanding their atomic origin mechanisms. Epitaxial growth of quasicrystals onto crystals can modify the surface/interface structures and properties of crystalline materials.

The relative stability of two-dimensional soft quasicrystals is examined
using a recently developed projection method which provides a unified numerical
framework to compute the free energy of periodic crystal and quasicrystals.
Accurate free energies of numerous ordered phases, including dodecagonal,
decagonal and octagonal quasicrystals, are obtained for a simple model, i.e.
the Lifshitz-Petrich free energy functional, of soft quasicrystals with two
length-scales. The availability of the free energy allows us to construct phase
diagrams of the system, demonstrating that, for the Lifshitz-Petrich model, the
dodecagonal and decagonal quasicrystals can become stable phases, whereas the
octagonal quasicrystal stays as a metastable phase.

We use Monte Carlo simulations and free-energy techniques to show that binary solutions of penta- and hexavalent two-dimensional patchy particles can form thermodynamically stable quasicrystals even at very narrow patch widths, provided their patch interactions are chosen in an appropriate way. Such patchy particles can be thought of as a coarse-grained representation of DNA multi-arm `star' motifs, which can be chosen to bond with one another very specifically by tuning the DNA sequences of the protruding arms. We explore several possible design strategies and conclude that DNA star tiles that are designed to interact with one another in a specific but not overly constrained way could potentially be used to construct soft quasicrystals in experiment. We verify that such star tiles can form stable dodecagonal motifs using oxDNA, a realistic coarse-grained model of DNA.

We report on the results of a high-energy x-ray diffraction study of Al–Pd–Mn to investigate the solidification products obtained during free-cooling using an electrostatic levitation furnace. The primary solidification product from the melt is i-Al–Pd–Mn which coexists with a significant remaining liquid component. As the sample cools further, we find that the solidification pathway is consistent with the liquidus projection and pseudo-binary cut through the ternary phase diagram reported previously. At ambient temperature we have identified the major phase to be the ξ′-phase orthorhombic approximant, along with minor phases identified as Al and, most likely, the R-phase orthorhombic approximant. We have also observed a distinct prepeak in the liquid at high temperature, signifying the presence of extended atomic order. Interestingly, this prepeak was not observed in previous neutron diffraction measurements on the Al–Pd–Mn system. No undercooling was observed preceding the solidification of the i-Al–Pd–Mn phase from the melt which may signal the close similarity of the short-range order in the solid and liquid. However, this can not be clearly determined because of the potential for heterogenous nucleation associated with the presence of an Al2O3 impurity at the surface of the sample.

We use Monte Carlo simulations and free-energy techniques to show that binary solutions of penta- and hexavalent two-dimensional patchy particles can form thermodynamically stable quasicrystals even at very narrow patch widths, provided their patch interactions are chosen in an appropriate way. Such patchy particles can be thought of as a coarse-grained representation of DNA multi-arm 'star' motifs, which can be chosen to bond with one another very specifically by tuning the DNA sequences of the protruding arms. We explore several possible design strategies and conclude that DNA star tiles that are designed to interact with one another in a specific but not overly constrained way could potentially be used to construct soft quasicrystals in experiment. We verify that such star tiles can form stable dodecagonal motifs using oxDNA, a realistic coarse-grained model of DNA.

We explore the growth of colloidal quasicrystals with dodecagonal symmetry in two dimensions by employing Brownian dynamics simulations. On the one hand, we study the growth behavior of structures obtained in a system of particles that interact according to an isotropic pair potential with two typical length scales. On the other hand, we consider patchy colloids that possess only one typical interaction length scale but prefer given binding angles. In case of the isotropic particles, we show that an imbalance in the competition between the two distances might lead to defects with wrong nearest-neighbor distances in the resulting structure. In contrast, during the growth of quasicrystals with patchy colloids such defects do not occur due to the lack of a second interaction length scale. However, as a downside, the diffusion of patchy particles along a surface typically is slower such that domains occur where the particles possess different phononic and phasonic offsets. Our results are important to understand how soft matter quasicrystals can be grown as perfectly as possible and to obtain a deeper insight into the mechanisms of the growth of quasicrystals in general.
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How quasicrystal structures form has always been a mysterious since the discovery of these magic structures. In this work, the nucleation of decagonal, dodecagonal, heptagonal, and octagonal quasicrystal structures controlled by the coupling among multiple length scales is investigated using a dynamic phase-field crystal model. We observe that the nucleation of quasicrystals is realized through local rearrangement of length scales, i.e., the generation, merge and stack of 3-atom building blocks constructed by the length scales, and accordingly, propose a geometric model to describe the cooperation of length scales during structural transformation in quasicrystal nucleation. Essentially, such cooperation is crucial to quasicrystal formation, and controlled by the match and balance between length scales. These finds clarify the scenario and microscopic mechanism about the structural evolution during quasicrystal nucleation, and help us to understand the common rule for the formation of periodic crystal and quasicrystal structures with various symmetries.

This article presents a Hamiltonian architecture based on vertex types and empires for demonstrating the emergence of aperiodic order in one dimension by a suitable prescription for breaking translation symmetry. At the outset, the paper presents different algorithmic, geometrical, and algebraic methods of constructing empires of vertex configurations of a given lattice. These empires have non-local scope and form the building blocks of the proposed lattice model. This model is tested via Monte Carlo simulations beginning with randomly arranged N tiles. The simulations clearly establish the Fibonacci configuration, which is a one-dimensional quasicrystal of length N, as the final relaxed state of the system. The Hamiltonian is promoted to a matrix operator form by performing dyadic tensor products of pairs of interacting empire vectors followed by a summation over all permissible configurations. A spectral analysis of the Hamiltonian matrix is performed and a theoretical method is presented to find the exact solution of the attractor configuration that is given by the Fibonacci chain as predicted by the simulations. Finally, a precise theoretical explanation is provided which shows that the Fibonacci chain is the most probable ground state. The proposed Hamiltonian is a mathematical model of the one dimensional Fibonacci quasicrystal.

In this work, we report results of extensive computer simulations regarding the phase behavior of a core-softened system. By using structural and thermodynamic descriptors, as well as self-diffusion coefficients, we provide a comprehensive view of the rich phase behavior displayed by the particular instance of the model studied in here. Our calculations agree with previously published results focused on a smaller region in the temperature-density parameter space [Dudalov et al. Soft Matter \textbf{10}, 4966 (2014)]. In this work, we explore a broader region in this parameter space, and uncover interesting fluid phases with low-symmetry local order, that were not reported by previous works. Solid phases were also found, and have been previously characterized in detail by Kryuchkov et al. [Soft Matter \textbf{14}, 2152 (2018)]. Our results support previously reported findings, and provide new physical insights regarding the emergence of order as disordered phases transform into solids by providing radial distribution function maps and specific heat data. Our results are summarized in terms of a phase diagram.

Understanding the microscopic kinetics of quasicrystal formation via nucleation and growth is crucial. Here, we report unusual pathways to nucleation and growth of dodecagonal quasicrystals via a phase field crystal model incorporating a two-length-scale potential. Under certain thermodynamic parameters, both quasicrystal growths via heterogeneous and homogeneous nucleation may be associated with a multistep behavior and the transient appearance of triangular and intermediate phases, different from classical nucleation pathways. The metastable intermediate phase spontaneously occurs to bridge the triangular phase and quasicrystal nuclei of different orientations to reduce the total free energy of the system. Decomposition of an undercooled fluid phase into quasicrystal phase shows a multistep pathway wherein the triangular phase and the intermediate phase may occur faster than the quasicrystal phase, when the growth rate of one length-scale ordering is significantly different from the other and the subsequent competing and coupling of both length scales are involved. The calculated structure factor, radial distribution function, and the aperiodic tiling structure of the intermediate phase explain why it appears during the quasicrystal formation.

Solid structures with different types of ordering (crystals, random tilings and glasses) are considered. We show that the short-range order structure of random tilings characterised by radial distribution functions looks very similar to the ones of glasses. However, the dynamical properties of random tilings look qualitatively identical to the ones of crystals. Because of this basing on radial distribution functions, only the system can be erroneously identified as glass or as random tiling. However, combining it with the dynamical properties, such as mean square displacement of stress autocorrelation function one can unambiguously distinguish glass and random tiling structure.

Double-well potentials are used for molecular dynamics simulation in monatomic systems. The potentials change as their parameters are adjusted, resulting in different structures. Of particular interest, we obtain decagonal and dodecagonal quasicrystals by simulations. We also verify the results and explain the formation of quasicrystals from the perspective of potential energy.

Quasicrystals and their approximants have triggered widespread interest due to the challenge of solving their complex crystal structures as well as their possibly exceptional properties. The structural motifs of approximants are similar to those of the corresponding quasicrystals, but to what extent are their crystallization pathways the same? Unfortunately, there have been very few in situ experimental investigations to answer this question. Here, by leveraging the high penetrating power of hard X-rays, synchrotron-based X-ray tomography was conducted in order to capture the nucleation and growth of a decagonal quasicrystal and its related approximant. The combination of data-driven computational analysis with new thermodynamic databases allowed the characterization, with high precision, of the constitutional and kinetic driving forces for crystallization. The experimental results prove that the growth of both crystals from a liquid is dominated by first-order kinetics. Nevertheless, and somewhat surprisingly, significant differences were observed in their rates of nucleation and growth. The reasons for such divergent behaviours are discussed in light of contemporary theories of intermetallic crystallization.

The growth of quasicrystals, i.e., structures with long-range positional order but no periodic translational symmetry, is more complex than the growth of periodic crystals. By employing Brownian dynamics simulations in two dimensions for colloidal particles that interact according to an isotropic pair potential with two incommensurate lengths, we study the growth of quasicrystalline structures by sequentially depositing particles at their surface. We quantify the occurrence of quasicrystalline order as a function of the temperature and the rate of added particles. In addition, we explore the defects like local triangular order or gaps within the quasicrystalline structure. Furthermore, we analyze the shapes of the surfaces in grown structures which tend to build straight lines along the symmetry axes of the quasicrystal. Finally, we identify phasonic flips which are rearrangements of the particles due to additional degrees of freedom. The number of phasonic flips decreases with the distance to the surface.

We study the relation of crystal-liquid interfacial free energy and medium range order in the quasicrystal-forming Ti37Zr42Ni21 liquid from undercooling experiment and ab initio molecular dynamics (MD) simulation. Adding a small amount of Ag to the liquid significantly reduces the degree of undercooling, which is suggestive of small interfacial free energy, and thus very similar atomic configuration between the liquid and the icosahedral quasicrystal phases. Using ab initio MD study, we find that Ag atoms predominantly form a bond with Zr atoms in the short range and, further, Ag-Zr pairs are extended in the liquid, as a medium range order which is identical to the global structural feature reported recently [Liu et al., Phys. Rev. Lett. 105, 155501 (2010)]. This result may expect extremely small undercooling if the icosahedral medium range order exists in a liquid forming an icosahedral quasicrystal, which implies the ambiguity of clear distinction of heterogeneous and homogeneous nucleation.

We discuss how a machine learning approach based on relative entropy optimization can be used as an inverse design strategy to discover isotropic pair interactions that self-assemble single- or multi-component particle systems into Frank-Kasper phases. In doing so, we also gain insights into self-assembly of quasicrystals.

How does a quasicrystal grow? Despite the decades of research that have been dedicated to this area of study, it remains one of the fundamental puzzles in the field of crystal growth. Although there has been no lack of theoretical studies on quasicrystal growth, there have been very few experimental investigations with which to test their various hypotheses. In particular, evidence of the in situ and three-dimensional (3D) growth of a quasicrystal from a parent liquid phase is lacking. To fill-in-the-gaps in our understanding of the solidification and melting pathways of quasicrystals, we performed synchrotron-based X-ray imaging experiments on a decagonal phase with composition of Al-15at%Ni-15at%Co. High-flux X-ray tomography enabled us to observe both growth and melting morphologies of the 3D quasicrystal at temperature. We determined that there is no time-reversal symmetry upon growth and melting of the decagonal quasicrystal. While quasicrystal growth is predominantly dominated by the attachment kinetics of atomic clusters in the liquid phase, melting is instead barrier-less and limited by buoyancy-driven convection. These experimental results provide the much-needed benchmark data that can be used to validate simulations of phase transformations involving this unique phase of matter.

We perform ab initio molecular dynamics simulations in stable and undercooled Al1-xCux liquids, with copper composition x = 0.3 and 0.4, to examine the relationship between structure and dynamics as a function of temperature. The temperature evolution of the diffusivity, relaxation times and viscosity display a change from an Arrhenius behavior at high temperature to a non-Arrhenius one at low temperature, which occurs at a crossover temperature TX yet in the vicinity of the liquidus temperature. We find that TX corresponds also to an onset of dynamic heterogeneities (DHs) that develop in the undercooled states. The structure factors and pair-correlation functions display characteristic features compatible with the occurrence of the icosahedral short-range order (ISRO) as well as the development of a medium range order (MRO) upon cooling. A common neighbor analysis further identifies that ISRO undergoes a strong increase below TX. Finally, we demonstrate that the medium-range order is formed by interconnected fivefold bipyramids and is correlated with the emergence of DHs.

For many years, quasicrystals were only observed as solid-state metallic alloys, yet current research is actively exploring their formation in a variety of soft materials, including systems of macromolecules, nanoparticles, and colloids. Much effort is being invested in understanding the thermodynamic properties of these soft-matter quasicrystals in order to predict and possibly control the structures that form, and to hopefully shed light on the broader, yet unresolved, general questions of quasicrystal formation and stability. Moreover, the ability to control the self-assembly of soft quasicrystals may contribute to the development of novel photonic or other applications based on self-assembled metamaterials. Here we follow a path, leading to quantitative stability predictions, that starts with a model developed two decades ago to treat the formation of multiple-scale quasiperiodic Faraday waves---standing wave patterns in vibrated fluid surfaces---that was later mapped onto systems of soft particles, interacting via multiple-scale pair-potentials. We review, and substantially expand, the quantitative predictions of these models, while correcting a few discrepancies in earlier calculations, and presenting new analytical methods for treating the models. In so doing, we find a number of new stable quasicrystalline structures with octagonal, octadecagonal and higher-order symmetries, some of which may hopefully be observed in future experiments.

Using ab initio molecular dynamics, we present a systematic study of structural and transport properties of liquidAl90Zn10 and Al83Zn10Cr7 alloys. In the liquid phase, we find that Cr additions promote the formation of a heterogeneous local ordering characterized by a strong five-fold symmetry (icosahedral short-range order (ISRO)) around Cr atoms. In the undercooled phase, we observe the extension of ISRO to icosahedral medium-range order (IMRO) length scale referring to Cr atoms. In examining dynamic properties, we show that this Cr induced structural heterogeneity leads to a substantial decoupling of Cr diffusion from the diffusion of Al and Zn components by a factor of 3 at 1000 K, the liquidus temperature. Below this temperature, the formation of IMRO gives rise to a non-Arrhenian temperature dependence of diffusivity and viscosity, a breakdown of the Stokes-Einstein relation, as well as the onset of dynamic heterogeneities. Using the isoconfigurational ensemble method, we evidence that the structural origin of dynamics heterogeneities is clearly related to IMRO. Finally we discuss the role of IMRO in a quasicrystal-enhanced nucleation mechanism discovered recently in Al–Zn–Cr alloys.

Although the use of mathematical models is ubiquitous in modern science, the involvement of mathematical modeling in the sciences is rarely seen as cases of interdisciplinary research. Often, mathematics is “applied” in the sciences, but mathematics also features in open-ended, truly interdisciplinary collaborations. The present paper addresses the role of mathematical models in the open-ended process of conceptualizing new phenomena. It does so by suggesting a notion of cultures of mathematization, stressing the potential role of the mathematical model as a boundary object around which negotiations of different desiderata can take place. This framework is then illustrated by a case study of the early efforts to produce a mathematical model for quasi-crystals in the first two decades after Dan Shechtman’s discovery of this new phenomenon in 1984.

Complex metallic alloys (CMAs) are important engineering materials with great scientific and technological significance. Accurate description and better understanding of the atomic structure and structure-related properties is essential for CMAs’ future design and practical application. Atomic-level structure of CMAs remains as a mystery to be uncovered due to their complicated atomic configuration. Since atomic clusters are advocated as basic building blocks of materials, various cluster-based models have been developed during the past decades, among which the “cluster-plus-glue-atom” model has been proven as an effective one to describe the atomic structural features of CMAs. Meanwhile, many intriguing rules correlating the atomic cluster structure with the composition and physicochemical property of CMAs have been revealed in light of the cluster-plus-glue-atom model. Besides, the cluster-plus-glue-atom model has exerted profound influence on the theoretical design of CMAs with desired properties. In this review article, the atomic cluster structural features of CMAs described by the cluster-plus-glue-atom model, and the cluster structure related composition rules, physicochemical properties, together with their correlations have been outlined.

Since the discovery of quasicrystals began to percolate through the scientific community, thirty years ago, Delaunay sets have been the tool of choice for describing their structures geometrically. These descriptions have gradually evolved from tiling vertex models to random cluster models, as the structures of real and simulated quasicrystals have been clarified experimentally. In this paper I outline these developments and explain why this productive dialogue between mathematicians and materials scientists will continue.

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- M Dzugutov
- S I Simdyankin

F. H. M. Zetterling, M. Dzugutov, and S. I. Simdyankin,
Journal of Non-Crystalline Solids 293, 39 (2001).

- P R Ten Wolde
- M J Ruiz-Montero
- D Frenkel

P. R. ten Wolde, M. J. Ruiz-Montero, and D. Frenkel,
The Journal of Chemical Physics 104, 9932 (1996), ISSN
0021-9606.

A metallic solid (Al-14-at. pct.-Mn) with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations, has been observed. Its diffraction spots are as sharp as those of crystals but cannot be indexed to any Bravais lattice. The solid is metastable and forms from the melt by a first-order transition.

A structural model of a one-component $\sigma$-phase crystal has been
constructed by means of molecular dynamics simulation. The phonon dispersion
curves and the vibrational density of states were computed for this model. The
dependence of the vibrational properties on the thermodynamical parameters was
investigated. The vibrational density of states of the $\sigma$-phase structure
is found to be similar to that of a one-component glass with icosahedral local
order. On the basis of this comparison it is concluded that the $\sigma$ phase
can be considered to be a good crystalline reference structure for this glass.

Bond-orientational order in molecular-dynamics simulations of supercooled liquids and in models of metallic glasses is studied. Quadratic and third-order invariants formed from bond spherical harmonics allow quantitative measures of cluster symmetries in these systems. A state with short-range translational order, but extended correlations in the orientations of particle clusters, starts to develop about 10% below the equilibrium melting temperature in a supercooled Lennard-Jones liquid. The order is predominantly icosahedral, although there is also a cubic component which we attribute to the periodic boundary conditions. Results are obtained for liquids cooled in an icosahedral pair potential as well. Only a modest amount of orientational order appears in a relaxed Finney dense-random-packing model. In contrast, we find essentially perfect icosahedral bond correlations in alternative "amorphon" cluster models of glass structure.

We examine the favored cluster structures for two new interatomic potentials, which both behave as monatomic model glass formers in bulk. We find that the oscillations in the potential lead to global minima that are noncompact arrangements of linked 13-atom icosahedra. The structural properties of the clusters correlate with the glass forming propensities of the potentials, and with the fragilities of the corresponding supercooled liquids. © 2003 American Institute of Physics.

We report a computer-simulation study of the rate of homogeneous crystal nucleation and the structure of crystal nuclei in a Lennard-Jones system at moderate undercooling. The height of the nucleation barrier has been determined using umbrella sampling, whereas the barrier crossing rate is calculated using molecular dynamics simulation. The simulations clearly show that the barrier crossing is a diffusive process. Nevertheless, the kinetic prefactor in the nucleation rate is found to be some two orders of magnitude larger than predicted by classical nucleation theory. The height of the barrier is in good agreement with the theoretical prediction. Although the Lennard-Jones system has a stable face-centered cubic (fcc) phase below the melting line, the precritical nuclei are found to be mainly body-centered cubic (bcc) ordered. As they grow to their critical size, they become more fcc ordered in the core. However, the critical and postcritical nuclei retain a high degree of bcc ordering in the interface. Furthermore it is found that in the interface the density falls off faster than the structural order parameter, which is in agreement with the predictions of density functional calculations.

Random tilings that comprise squares and equilateral triangles can model quasicrystals with twelvefold symmetry. A (phason) elastic theory for such tilings is constructed, whose order parameter is the phason field, and whose entropy density includes terms up to third order in the phason strain. Due to an unusual constraint, the phason field of any square-triangle tiling is irrotational and, as a result, the form of the entropy density is simpler than the general form that is required by twelvefold symmetry alone. Using an update move, which rearranges a closed, nonlocal, one-dimensional chain of squares and triangles, the unknown parameters of the elastic theory are estimated via Monte Carlo simulations: (i) One of the two second-order elastic constants and the third-order elastic constant are found by measuring phason fluctuations; athermal systems (maximally random ensembles) with the same background phason strain but different sizes of unit cell are simulated to distinguish the effects of a finite background phason strain from the effects of finite unit-cell size. (ii) The entropy per unit area at zero phason strain and the other second-order elastic constant are found from the entropies that thermal systems (canonical ensembles) gain between zero and infinite temperature, which are estimated using Ferrenberg and Swendsen's histogram method. A way to set up transfer-matrix calculations for random square-triangle tilings is also presented.

In recent computer simulations of a simple monatomic system interacting via the Dzugutov pair potential, freezing of the fluid into an equilibrium dodecagonal quasicrystal has been reported [M. Dzugutov, Phys. Rev. Lett. 70, 2924 (1993)]. Here, using a combination of molecular dynamics simulation and thermodynamic perturbation theory, we conduct a detailed analysis of the relative stabilities of solid-phase structures of the Dzugutov-potential system. At low pressures, the most stable structure is found to be a bcc crystal, which gives way at higher pressures to a fcc crystal. Although a dodecagonal quasicrystal and a sigma-phase crystal compete with the bcc crystal for stability, they always remain metastable.

We report a molecular dynamics simulation of a simple monatomic glass-forming liquid. It is shown that transition to deeper minima in the energy landscape under supercooling results in the formation of icosahedrally structured domains with distinctly slow diffusion which grow with cooling in a low-dimensional manner and percolate around T(c), the critical temperature of the mode-coupling theory. Simultaneously, a sharp slowing down of the structural relaxation relative to diffusion is observed. It is concluded that this effect cannot be accounted for by the spatial variation in atomic mobility. The low-dimensional clustering is discussed as a possible mechanism of fragility.

Special computational techniques are required to compute absolute crystal nucleation rates of colloidal suspensions. Using crystal nucleation of hard-sphere colloids as an example, we describe in some detail the novel computational tools that are needed to perform such calculations. In particular, we focus on the definition of appropriate order parameters that distinguish liquid from crystal, and on techniques to compute the kinetic prefactor that enters in the expression for the nucleation rate. In addition, we discuss the relation between simulation results and theoretical predictions based on classical nucleation theory.

We study the kinetics of crystal nucleation of an undercooled Lennard-Jones liquid using various path-sampling methods. We obtain the rate constant and elucidate the pathways for crystal nucleation. Analysis of the path ensemble reveals that crystal nucleation occurs along many different pathways, in which critical solid nuclei can be small, compact, and face centered cubic, but also large, less ordered, and more body centered cubic. The reaction coordinate thus includes, besides the cluster size, also the quality of the crystal structure.

Using molecular dynamics, we investigate the crystal nucleation in a Lennard-Jones fluid as a function of the degree of supercooling. At moderate supercooling, a nucleation picture applies, while for deeper quenches, the phenomenon progressively acquires a spinodal character. We show that in the nucleation regime, the freezing is a two-step process. The formation of the critical nucleus is indeed preceded by the abrupt formation of a precritical crystallite from a density fluctuation in the fluid. In contrast, as the degree of supercooling is increased, crystallization proceeds in a more continuous and collective fashion and becomes more spatially diffuse, indicating that the liquid is unstable and crystallizes by a spinodal mechanism.

Complex alloy structures, particularly those of transition metals, are ; considered as determined by the geometricnl requirements for sphere packing. A ; characteristic of the class of structures discussed is that tetrahedral groupings ; of atoms occur everywhere in the structure--alternatively stated, coordination ; polyhedra have only triangular faces. The topological and geometrical properties ; of such polyhedra are examined and rules and theorems regarding them are deduced. ; Justification is given for the prominence of four such polyhedra (for ; coordination numbers of 12, 14, 15, and 16) in actual structures. General ; principles regarding the combination of these polyhedra into full structures are ; deduced and necessary definitions are given for terms that facilitate the ; detailed discussion of this class of structures. (auth);

A quasi-crystal is the natural extension of the notion of a crystal to structures with quasi-periodic, rather than periodic, translational order. Two and three-dimensional quasi-crystals are here classified by their symmetry under rotation, and it is shown that many disallowed crystals symmetries are allowed quasi-crystal symmetries. The diffraction pattern of an ideal quasi-crystal is analytically computed, and it is shown that the recently observed electron-diffraction pattern of an Al-Mn alloy is closely related to that of an icosahedral quasi-crystal.

We present two sets of rules for constructing quasiperiodic tilings that suggest a simpler structural model of quasicrystals and a more plausible explanation of why quasicrystals form. First, we show that quasiperiodic tilings can be constructed from a single prototile with matching rules which constrain the way that neighbors can overlap. Second, we show that maximizing the density of a certain cluster of fat and thin tiles can force a Penrose tiling without imposing the usual Penrose matching rules.

Over the past seven years, many examples of periodic crystals closely
related to quasicrystalline alloys have been discovered. These crystals
have been termed approximants, since the arrangements of atoms within
their unit cells closely approximate the local atomic structures in
quasicrystals. This colloquium focuses on these approximant structures,
their description, and their relationship to quasicrystals.

The free energy difference between a model system and some reference system can easily be written as an ensemble average, but the conventional Monte Carlo methods of obtaining such averages are inadequate for the free-energy case. That is because the Boltzmann-weighted sampling distribution ordinarily used is extremely inefficient for the purpose. This paper describes the use of arbitrary sampling distributions chosen to facilitate such estimates. The methods have been tested successfully on the Lennard-Jones system over a wide range of temperature and density, including the gas-liquid coexistence region, and are found to be extremely powerful and economical.

A number of singularities observed in super-cooled liquids approaching the glass transition point are commonly interpreted as indicating spatial heterogeneity. This conjecture assumes that the whole volume is decomposed into structurally distinct domains. In simple super-cooled liquids, the assumed domain structure is usually associated with icosahedral clustering; however, no evidence for growing length scale of icosahedral ordering in a super-cooled liquid has been found so far. We present a molecular dynamics simulation demonstrating formation of extended icosahedral configurations in a simple monatomic liquid approaching the glass transition temperature. An important observation is that these configurations show a tendency for low-dimensional growth.

In this Rapid Communication, a novel simple monatomic liquid, possessing the distinctive icosahedral inherent local order, is reported. It has been generated by a special form of pair potential employed in a molecular dynamics system of 16 384 particles, and remained stable within a wide range of temperatures explored. Pronounced stability of this model with respect to crystalline nucleation has been tested in a very long run under supercooling which was found to enhance its icosahedral inherent structure. The inherent structure factor exhibits an anomalous long-wavelength maximum which is interpreted as being indicative of the tendency for icosahedral clustering.

In this paper we examine the detailed relationship between the density-wave and unit-cell descriptions of quasicrystals. We show that phonons, phasons, and dislocations correspond to translations, distortions, and rearrangements of unit cells. The associated density-wave images closely resemble experimental electron micrographs of the icosahedral phase of aluminum-manganese and related alloys. Partial dislocations are also discussed and a natural classification scheme for partials is proposed.

A Comment on the Letter by D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).

An equilibrium uniaxial dodecagonal quasicrystal is reported to be formed by a freezing simple monatomic liquid in a molecular dynamics simulation. Its diffraction pattern closely resembles that of the dodecagonal quasicrystalline phase formed in V3Ni2 and V15Ni10Si alloys, and the high-resolution electron micrograph obtained for the latter is found to be consistent with the corresponding pattern of atomic layer in the simulated quasicrystal. The simulated structure is discussed in the context of existing models of quasicrystalline order.

We present results of simulations that predict the phases formed by the self-assembly of model nanospheres functionalized with a single polymer "tether," including double gyroid, perforated lamella, and crystalline bilayer phases. We show that microphase separation of the immiscible tethers and nanospheres causes confinement of the nanoparticles, which promotes local icosahedral packing that in turn stabilizes the gyroid. We present a new metric for determining the local arrangement of particles based on spherical harmonic "fingerprints," which we use to quantify the extent of icosahedral ordering.

By molecular dynamics (MD) simulation of the one-component Dzugutov liquid in a metastable equilibrium supercooled state approaching the glass transition, we investigate the structural properties of highly mobile particles moving in strings at low temperature T where string-like particle motion (SLM) is well developed. We find that SLM occurs most frequently in the boundary regions between clusters of icosahedrally-ordered particles and disordered, liquid-like, domains. Further, we find that the onset T for significant SLM coincides with the T at which clusters of icosahedrally-ordered particles begin to appear in considerable amounts, which in turn coincides with the onset T for non-Arrhenius dynamics. We find a unique structural environment for strings that is different from the structure of the bulk liquid at any T. This unique string environment persists from the melting T upon cooling to the lowest T studied in the vicinity of the mode-coupling temperature, and is explained by the existence of rigid elongated cages. We also form a criterion based solely on structural features of the local environment that allow the identification of particles with an increased propensity for mobility.