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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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... When s l (i, j) b c , the bond between atoms i and j is considered crystalline or quasicrystalline, otherwise it is considered disordered. In this paper, we find l = 6 and b c = 0.7 can effectively detect the formation of crystal structures (fcc, hcp, and bcc) [31], while l = 12 and b c = 0.45 can detect the formation of quasicrystalline structures [32]. To detect quasicrystalline structures, we calculate s 12 (i, j) including neighbors with separations r i j < 2.31σ as in Ref. [32]. ...

... R c decreases modestly (by less than an order of magnitude) for f ICO 0.5, and decreases dramatically (by more than two orders of magnitude) for 0.5 f ICO 0.8. When f ICO 0.8, the system can form quasicrystals [31], which causes R c to increase as f ICO → 1. Note that R c for elements with ICO symmetry is much lower than that for elements with crystalline symmetry. ...

... We find that R c ( f ICO ) possesses a minimum near f ICO ∼ 0.8. The nonmonotonic behavior of R c ( f ICO ) can be rationalized by considering the interfacial free energy barrier for crystal nucleation [31,[40][41][42]. In the crystal-forming regime with f ICO 0.8, local ICO order is incompatible with crystalline symmetry, and thus increasing f ICO enhances the free energy barrier for crystal nucleation, leading to decreases in R c . ...

Prediction of the glass-forming ability (GFA) of alloys remains a major challenge. We are not yet able to predict the composition dependence of the GFA of even binary alloys. To investigate the effect of each element's propensity to form particular crystal structures on glass formation, we focus on binary alloys composed of elements with the same size but different atomic symmetries using the patchy-particle model. For mixtures with atomic symmetries that promote different crystal structures, the minimum critical cooling rate Rc is only a factor of 5 lower than that for the pure substances. For mixtures with different atomic symmetries that promote local crystalline and icosahedral order, the minimum Rc is more than three orders of magnitude lower than that for pure substances. Results for Rc for the patchy-particle model are consistent with those from embedded atom method simulations and sputtering experiments of NiCu, TiAl, and high entropy alloys.

... 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.

... Despite the intensive studies on the structures of quasicrystals, kinetics and dynamics of quasicrystal growth, however, are still incomplete [31][32][33][34]. It is not clear how the quasicrystals appear from a liquid or crystalline state, what is happening at the periphery of quasicrystals duing their growth, and how the structures fluctuate. ...

... In ref. [26], the growth of DDQC was studied by particle deposition on a prepared quasicrystalline substrate. The growth of a three-dimensional DDQC was studied in ref. [31], and it was suggested that characteristic structures (icosahedrons in that system) preferentially appear around the nucleus of DDQC. The growth of a two-dimensional DDQC was also studied in ref. [35]. ...

The self-assembly of two-dimensional dodecagonal quasicrystal (DDQC) from patchy particles are investigated by Brownian dynamics simulations. The patchy particle has a five-fold rotational symmetry pattern described by the spherical harmonics $Y_{55}$. From the formation of the DDQC obtained by an annealing process, we find the following mechanism. The early stage of the dynamics is dominated by hexagonal structures. Then, nucleation of dodecagonal motifs appears by particle rearrangement, and finally the motifs expand whole system. The transition from the hexagonal structure into the dodecagonal motif is made by the collective rational motion of the particles. The DDQC consists of clusters of dodecagonal motifs, which can be classified into several packing structures. By the analyses of the DDQC under fixed temperature, we find the fluctuations are characterised by changes in the network of the dodecagonal motifs. Finally we compare the DDQC assembled from the patchy particle system and isotropic particle system. The two systems both share a similar mechanism of the formation and fluctuation of DDQC.

... Recently, quasicrystals have also been created in soft material systems [6][7][8] such as colloidal systems 9 , micellar systems 10,11 , polymer melts [12][13][14] , and DNA motifs 15,16 . Quasicrystal formation have been numerically simulated, [17][18][19][20][21][22][23] using simple models. When the Lennard-Jones-Gauss (LJG) potential, which has two minima, is used as the interaction potential [20][21][22] , decagonal or dodecagonal quasicrystals are created by controlling parameters at high temperature. ...

... A periodic boundary condition was used in the x-direction. Quasicrystals do not have translational symmetry, but a similar pattern is approximately repeated with intervals such as 4,11,15,19,26,38, and 52 in dodecagonal quasicrystals. We set the size in the x-direction L x to 38. ...

Langevin dynamics simulations are performed to examine how impurities affect two-dimensional dodecagonal quasicrystals. We assumed that the interaction potential between two particles is the Lennard-Jones-Gauss potential if at least one of these particles is a matrix particle and that the interaction potential between two impurities is the Lennard-Jones potential. Matrix particles and impurities impinge with constant rates on the substrate created by a part of a dodecagonal quasicrystal consisting of square and triangular tiles. The dependences of the twelve-fold rotational order and the number of shield-like tiles on the impurity density are examined after sufficient solid layers are grown. While the change in the twelve-fold rotational symmetry is small, the number of shield-like tiles in the solid increases greatly with increasing impurity density.

... These changes in tiling configurations are representative of phason excitations and relaxations, and do not introduce defects into the crystal 6 . Previous studies suggest that entropic contributions can have a significant influence on QC stability 1,2,9 . This suggests there is degeneracy in tile configurations that preserves quasiperiodicity and that QCs can grow into low or zero phason strain structures without the need for complex phason flip sequences 9 . ...

... Although past studies on QC growth mechanisms extend our understanding of phason contributions to stability in a bulk QC 1,2,9 , studies on phason contributions to the formation and motion of grain boundaries (GBs) remain limited 10 . Yet the latter is critically important from a practical standpoint since the formation of polycrystals is unavoidable due to finite nucleation rates below the melting point. ...

Quasicrystals exhibit long-range order but lack translational symmetry. When grown as
single crystals, they possess distinctive and unusual properties owing to the absence of grain boundaries. Unfortunately, conventional methods such as bulk crystal growth or thin film deposition only allow us to synthesize either polycrystalline quasicrystals or quasicrystals that are at most a few centimeters in size. Here, we reveal through real-time and 3D imaging the formation of a single decagonal quasicrystal arising from a hard collision between multiple growing quasicrystals in an Al-Co-Ni liquid. Through corresponding molecular dynamics simulations, we examine the underlying kinetics of quasicrystal coalescence and investigate the effects of initial misorientation between the growing quasicrystalline grains on the formation of grain boundaries. At small misorientation, coalescence occurs following rigid rotation that is facilitated by phasons. Our joint experimental-computational discovery paves the way toward fabrication of single, large-scale quasicrystals for novel applications.

... The emergence of a DDQC prior to σ may bear some similarities to simulations of colloidal quasicrystals. 101,102 Because the DDQC is created from the same square and triangular tiles as σ (see the Supporting Information), it is plausible that the DDQC represents a way for the system to first form the requisite tiles, which then reorganize themselves into the equilibrium, periodic σ phase. 102 Thermal processing also provides a route for accessing the C14 and C15 Laves phases as metastable states. ...

... 101,102 Because the DDQC is created from the same square and triangular tiles as σ (see the Supporting Information), it is plausible that the DDQC represents a way for the system to first form the requisite tiles, which then reorganize themselves into the equilibrium, periodic σ phase. 102 Thermal processing also provides a route for accessing the C14 and C15 Laves phases as metastable states. 60 As illustrated in Figure 8, cooling a PI−PLA diblock copolymer melt with f A = 0.15 from disorder produces bcc and then σ, behavior that would be anticipated from the SCFT phase diagram. ...

The discovery of the Frank–Kasper σ phase in diblock copolymer and tetrablock terpolymer melts catalyzed a renewed interest over the past decade in understanding particle-forming phases in block polymer systems. This Perspective provides a concise overview of the Frank–Kasper phases seen to date in block polymers (A15, σ, and the C14 and C15 Laves phases) and mechanisms known to produce them: conformational asymmetry in neat diblock copolymer melts, interfacial segregation effects in diblock copolymer blends, particle swelling in diblock copolymer/homopolymer blends, and matrix segregation effects in neat tetrablock terpolymer melts. While a qualitative understanding of the emergence of Frank–Kasper phases in block polymer systems has been achieved, a number of outstanding questions remain, in particular those arising from the low degree of polymerization used in experiments, nonequilibrium effects during thermal processing, and the large design space available in blends and multiblock systems. This Perspective discusses potential avenues for future research related to these areas as well as overarching issues underlying the connections between Frank–Kasper phase formation in block polymers to other soft matter and metals.

... It has been discovered that in dense regions, quite interesting phases can be found even for simple potentials such as Lennard-Jones (LJ), Square-Well (SW), Square-Shoulder (SS), Square-Shoulder + Square-Well (S + SW), Triangle-Well (TW), Yukawa and Mie potentials [27][28][29][30][31][32][33][34][35][36][37][38][39][40]. There are still some routes that have not been sufficiently studied even for the simple spherical potentials. ...

... Specially, the high density regions have been less explored. A very successful tool used to determine phase diagrams are molecular simulation methods [27][28][29][30][31][32][33][34][35][36][37][38][39][40]. A very important finding obtained by using simulations is that for some two dimensional (2D) systems the phase diagram is richer than their three dimensional (3D) version, particularly, for the high density region. ...

The ability to tailor effective interactions at the molecular level to provide a platform to create advanced functional materials is a challenge for the scientific community. The main goal is to develop a good interaction potential model driving the formation of a given set of target structures. In this work, we propose a simple method to design interaction potentials able to generate a desired geometrical pattern. The methodology combines the Ground State Energy (GSE) method with simulation techniques. To illustrate the method we explored a subfamily of square-shoulder + square potentials, our target are square and triangular particle arrays. We found that the GSE method predictions were qualitatively correct at non-zero temperatures. The proposed methodology allowed us to build the phase diagram of one member of this family of potentials that exhibits: vapour, liquid, and different crystalline lattices (triangular, square, rhomboid). We show that the GSE method is a powerful and inexpensive computational tool to test the viability of the desired particle array at zero-temperature and also as an initial step to design phase diagrams of model potentials.

... This problem has been of interest since the discovery of QCs and it has been discussed theoretically [4][5][6][7][8][9][10][11][12] and studied numerically. 13) Onoda et al. 4) reported an algorithm for the growth of a perfect Penrose tiling, a typical 2D QC, using local rules. This algorithm has since been extended for the growth of a 3D decagonal QC, 5) which consists of a periodic stacking of 2D Penrose tilings, and a 3D icosahedral QC of Ammann tiling. ...

... Each structure in the growth process was analyzed based on a high-dimensional description of the i-QC structure. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] In general, a set of particle positions r k 2 E k (where E k is physical space) in an i-QC structure corresponds to projections of part of a six-dimensional hypercubic lattice with points R ðn 1 ;...;n 6 Þ given by ...

... 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. ...

A series of noncrystalline ABn dendron-like giant molecules DPOSS-MPOSSn (n = 2-6, DPOSS: hydrophilic polyhedral oligomeric silsesquioxane (POSS) cage; MPOSS: hydrophobic POSS cage) were synthesized. These samples present a thermodynamically stable phase formation sequence from the hexagonal cylinder phase (plane group of P6mm), to the Frank-Kasper (F-K) A15 phase (space group of Pm3̅n), and further to the F-K σ phase (space group of P42/mnm), with increasing the number of MPOSS in a single molecule (n, from 2 to 6). Moreover, for DPOSS-MPOSS5 and DPOSS-MPOSS6, an intriguing dodecagonal quasicrystal (DQC) structure has been identified and revealed as a kinetic favorable metastable phase at lower temperatures, while the thermodynamically stable phase is the σ phase. The detailed investigation of the transition kinetics between the DQC and σ phase in these samples makes it possible to identify how the self-assembly directs the phase transition in terms of molecular and supramolecular aspects.

... This picture is consistent with the diffuse interface theory and has been confirmed by the studies on liquid metals [40][41][42] . The role of icosahedra in quasicrystal formation is similar to that of crystal-like preordering in ordinary crystallisation 28,29,37,43 . Such examples have been experimentally observed for ZrCu-based and Al-based bulk metallic glasses 44,45 . ...

The recent discovery of non-classical crystal nucleation pathways has revealed the role of fluctuations in the liquid structural order, not considered in classical nucleation theory. On the other hand, classical crystal growth theory states that crystal growth is independent of interfacial energy, but this is questionable. Here we elucidate the role of liquid structural ordering in crystal nucleation and growth using computer simulations of supercooled liquids. We find that suppressing the crystal-like structural order in the supercooled liquid through a new order-killing strategy can reduce the crystallisation rate by several orders of magnitude. This indicates that crystal-like liquid preordering and the associated interfacial energy reduction play an essential role in nucleation and growth processes, forcing critical modifications of the classical crystal growth theory. Furthermore, we evaluate the importance of this additional factor for different types of liquids. These findings shed new light on the fundamental understanding of crystal growth kinetics. In classical nucleation theory, structural order in the liquid phase is not considered. But simulations of supercooled liquids now show that crystal-like liquid preordering play an essential role in nucleation and growth processes - calling for extensions of the classical theory.

... Monte Carlo methods have had a lasting success in studying statistical physics. To name a few, Monte Carlo simulation has found applications in binary alloys, quasicrystals, diffusion limited aggregation, liquidgas transitions, micelle formation, simple and complex fluids, magnetic materials, adsorbed surface layers, polymers, and protein folding [2,38,51,53,56,87]. Monte Carlo simulation has also become a pivotal tool for studying phase transitions [30,56]. ...

Monte Carlo sampling of the canonical distribution presents a formidable challenge when the potential energy landscape is characterized by a large number of local minima separated by high barriers. The principal observation of this work is that the multiple local minima and energy barriers in a landscape can often occur as a result of discrete symmetries in the potential energy function. A new Monte Carlo method is proposed, group action Markov chain Monte Carlo (GA-MCMC), which augments more conventional trial moves (e.g. random jumps, hybrid Monte Carlo, etc.) with the application of a group action from a well-chosen generating set of the discrete symmetry group; the result is a framework for symmetry-adapted MCMC. It is shown that conventional trial moves are generally optimal for "local mixing" rates, i.e. sampling a single energy well; whereas the group action portion of the GA-MCMC trial move allows the Markov chain to propagate between energy wells and can vastly improve the rate of "global mixing". The proposed method is compared with standard jumps and umbrella sampling (a popular alternative for energy landscapes with barriers) for potential energies with translational, reflection, and rotational symmetries. GA-MCMC is shown to consistently outperform the considered alternatives, even when the symmetry of the potential energy function is broken. The work culminates by extending GA-MCMC to a clustering-type algorithm for interacting dielectric polymer chains. Not only does GA-MCMC again outperform the considered alternatives, but it is the only method which consistently converges for all of the cases considered. Some new and interesting phenomena regarding the electro-elasticity of dielectric polymer chains, unveiled via GA-MCMC, is briefly discussed.

... Furthermore, according to the valuable insights from other soft matter systems, the interplays of local packings on both submesoatomic and mesoatomic level are critical for these intriguing assembling behaviors in our case (Fig. 4F): 1) Starting from the amorphous binary blends, the local configurations of the mixed giant molecules will transit to either single-shell or double-shell mesoatoms upon annealing. 2) Though the double-shell mesoatoms are in a nonequilibrium state, for certain local configurations, they are kinetically more favored within the reach of minimum rearrangement (40). Thus, these double-shell mesoatoms form once annealed at low temperature and persist because of the limited diffusion of the giant molecules (41,42). ...

Significance
As the formation of quasicrystalline ordering in both metal alloys and soft matters reinforces the scale-invariance principle, however, it remains unknown that why metallic quasicrystals prefer decagonal quasicrystalline (DQC, 10-fold) ordering, and the condensed soft maters only exhibit dodecagonal quasicrystalline (DDQC, 12-fold) ordering. By tuning the self-sorting ability in a pair of giant molecules, the well-mixed binary blends generate a metastable DQC phase and corresponding unprecedent phase sequence (DQC → DDQC → Frank–Kasper σ) during thermal annealing. These peculiar assembling behaviors result from interplays of submesoatomic and mesoatomic packings. Reducing the self-sorting strength has induced an extra complexity in the sub-mesoatomic packings of giant molecules, which further influences the clustering on the mesoatomic level.

... Quasicrystals have been described as a "path of least resistance" to formation of a solid phase, requiring markedly less structural rearrangement than forming a pure FK phase. 70 Accordingly, DQCs that transform into a σ phase over time have been observed experimentally. 34 In our simulations, we do not see any evolution of the DQC towards a σ phase; however, in experiment the DQC can persist for days to months depending on temperature, 34 so such a transition may not be accessible in silico. ...

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.

... Generally, the F-K phases can be considered as ordered approximates of quasi-periodic crystals because of the common construction rules of atomic clustering and stacking 17 . For instance, icosahedral atomic arrangements similar to those of F-K phases can also be observed in the three-dimensional (3D) five-fold quasicrystals [18][19][20] ; Self-assembly of the F-K A15 fundamental structures and Z fundamental structures can directly form the dodecagonal quasicrystals showing twelve-fold diffraction symmetry 21,22 . ...

... Generally, the Frank-Kasper phases can be considered as ordered approximates of quasi-periodic crystals due to some shared construction rules from atomic clustering and stacking [11]. Since the Al-Mn icosahedral quasicrystal discovered in 1984 by Shechtman et al [12], the complex architectures of Frank-Kasper phases are considered as a link between the traditional simple periodic structures (such as the face-centered cubic, hexagonal close-packed, and body-centered cubic structures) and quasicrystals [10,13]. ...

Z phase is one of the three basic units by which the Frank-Kasper phases are generally assembled. Compared to the other two basic units, i.e., A15 and C15 structures, the Z phase structure is rarely experimentally observed because of a relatively large volume ratio among the constituents to inhibit its formation. Moreover, the discovered Z structures are generally the three-dimensional (3D) ordered Gibbs bulk phases to conform to their thermodynamic stability. Herein, we confirmed the existence of a metastable two-dimensional (2D) Frank-Kasper Z phase with one unit-cell height in the crystallography in a model Mg-Sm-Zn system, by using aberration-corrected high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) combined with density functional theory (DFT) calculations. This finding is important for understanding the relationship between the traditional crystal structures and the quasicrystals, and it is also expected to provide a new insight to understand the clustering and stacking behavior of atoms in condensed matters.

... Thus, by choosing the simplest models, it would be possible to provide predictions about the macroscopic behavior of many different systems. Indeed, by using models with simple interactions (purely attractive, repulsive, or mixed potentials) computer simulations have found a variety of different mesophases [41][42][43][44][45][46][47][48], that are also found by using more elaborated models [49][50][51][52][53]. Also, it was assumed that an important ingredient to obtain complex lattices is the introduction of anisotropic interactions between constituent particles [54][55][56][57]. However, experimental evidence showed the formation of complex lattices in spherical colloidal particles (including colloidal nanocrystals) grafted with polymers and/or DNA strands [5,[58][59][60]. ...

The self-assembly of colloidal nanocrystals at interfaces provides a bottom-up ap-proach to create functional materials for developing next-generation flexible optoelec-tronic devices and sensors. In this work, we report phase diagrams of simple models ofcolloidal nanocrystals confined at a flat interface. By performing extensive computersimulations we elucidate the mesoscale organization that takes place as different pa-rameters are varied. Our simulation results uncover rich phase diagrams where hexag-onal, rhomboid, honeycomb and stripe phases as well as hierarchical self-assembly arefound. Our results could serve as a guideline for experimentalists to design colloidalnanocrystal arrangements to target specific applications.

... 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.

... 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.

... 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.

... 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 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.

This review summarizes the self-assembly of block molecules forming unconventional two-dimensional (2D) periodic nanopatterns. Especially, we emphasize the structural evolution from simple columnar phases to complex 2D tiling morphologies in soft materials including block copolymers, liquid crystals, giant molecules, etc. Then, the state-of-the-art nanofabrication technologies for making sophisticated nanostructures with specific functions via combining both bottom-up assembly and top-down lithography-based methods are discussed, highlighting the use of directed self-assembly processes. Finally, we provide our perspective on this area. By further increasing the complexity of block molecules and the designability of lithography, low-dimensional ordered morphologies will be particularly promising for further application in nanotechnology.

Mg-Zn alloys form the basis of a wide variety of commercial light-weight Mg alloys due to their precipitation hardenability, biocompatibility, and low cost. Despite significant progress, there exist controversies over the crystal structures and stabilities of various complex precipitates in this important binary system. In this work, the information about crystal structures and stabilities of phases in Mg-Zn system is critically reviewed and three key open questions are identified: (1) What are crystal structures of Guinier-Preston (GP) zones? (2) What are relative stabilities of a myriad of phases observed for β1′ precipitates? (3) Why does the β2′ phase have two distinct orientation relationships (ORs) with α-Mg? To shed light on these questions, comprehensive first-principles calculations based on density functional theory, cluster expansion, and Monte Carlo simulations are performed. The atomic structures of GP zones are predicted, and the effect of coherency strain on their stabilities are analyzed. The structures of β1′precipitates composed of the rhombic MgZn2 and the elongated hexagonal Mg6Zn7 units are provided. It is shown that the β1′precipitate can be stabilized with increased fraction of rhombic MgZn2 units, which leads to local regions of the C14 MgZn2 Laves phase. The origin of the two distinct ORs between β2′ phase and the matrix is traced back to two formation paths, i.e., relaxation of the coherent Zn ordering on HCP matrix and coarsening of C14 MgZn2 region in β1′precipitates. Finally, a feasible precipitation sequence in Mg-Zn alloys is suggested.

The self-assembly of two-dimensional dodecagonal quasicrystals (DDQCs) from patchy particles is investigated by Brownian dynamics simulations. The patchy particle has a five-fold rotational symmetry pattern described by the spherical harmonics Y55. From the formation of the DDQC obtained by an annealing process, we find the following mechanism. The early stage of the dynamics is dominated by hexagonal structures. Then, nucleation of dodecagonal motifs appears by particle rearrangement, and finally the motifs span the whole system. The transition from the hexagonal structure into the dodecagonal motif is coincident with the collective motion of the particles. The DDQC consists of clusters of dodecagonal motifs, which can be classified into several packing structures. By the analyses of the DDQC under fixed temperature, we find that the fluctuations are characterised by changes in the network of the dodecagonal motifs. Finally we compare the DDQCs assembled from the patchy particle system and isotropic particle system. The two systems share a similar mechanism of the formation and fluctuation of DDQCs.

Langevin dynamics simulations are performed to examine how impurities affect two-dimensional dodecagonal quasicrystals. We assumed that the interaction potential between two particles is the Lennard-Jones-Gauss potential if at least one of these particles is a matrix particle and that the interaction potential between two impurities is the Lennard-Jones potential. Matrix particles and impurities impinge with constant rates on the substrate created by a part of a dodecagonal quasicrystal consisting of square and triangular tiles. The dependences of the twelve-fold rotational order and the number of shield-like tiles on the impurity density are examined after sufficient solid layers are grown. While the change in the twelve-fold rotational symmetry is small, the number of shield-like tiles in the solid increases greatly with increasing impurity density.

Main observation and conclusion
We report herein the precision synthesis and phase behaviors of multi-tailed, B2AB2-type regio-isomeric giant surfactants consisting of a hydrophilic polyhedral oligomeric silsesquioxane (POSS) head tethered with four hydrophobic polystyrene (PS) tails. The synthesis was accomplished through two sequential “click” reactions to give a series of regio-isomeric giant surfactants S2DS2 (where S is short for PS tails and D for hydroxyl-functionalized POSS) in para-, meta-, and ortho-configurations. Their phase structures and phase behaviors at the columnar-spherical boundary were investigated with a single PS tail molecular weight of 1.4 kDa. Specifically, the para-and meta-isomers show hexagonally packed cylinders phases with slightly different order-disorder transition temperature (~120 °C and ~130 °C) and the ortho-isomer exhibits an order-order transition from a kinetically favored, metastable dodecagonal quasi-crystal phase to a thermodynamically stable sigma phase at ~120 °C, as well as a further transition into the disordered state at ~140 °C. The phase diagram was constructed and their differences were rationalized based on the calculated interfacial area per molecule. This work demonstrates that tiny structural disparity could not only lead to unconventional phase formation in single-component macromolecules, but also render dynamic and rich phase behaviors in these macromolecular assemblies.
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The self-assembly of lanthanide ions with ditopic organic spacers results in the formation of complex tiling patterns that mimic the structural motifs of quasi-periodic 2D materials. The linking of trans-{LnI2}+ nodes (Ln = Gd, Dy) by both closed-shell and anion radicals of 4,4'-bipyridine affords rare examples of Archimedean tessellations in a metal-organic framework. We furthermore demonstrate the occurrence of sizable magnetic exchange interactions and slow relaxation of magnetization behavior in a complex tessellation pattern. The implementation of Archimedean tessellations in lanthanide(III) coordination solids couriers a strategy to design elusive quasi-periodic metal-organic frameworks with inimitable magnetic properties.

Significance
Quasicrystals, which exhibit local rotational symmetries and infinite unit cells, are intermediate states of matter between glasses and periodic three-dimensional (3D) crystals. Despite their ubiquity in metal alloys, self-assembled micellar quasicrystals have only serendipitously emerged from intricate molecular building blocks or complex materials-processing protocols. We demonstrate that hydration of a simple oil and surfactant (“soap”) mixture stimulates dodecagonal quasicrystalline ordering of oil-swollen spherical micelles, while oil addition to surfactant/water mixtures instead yields only periodic 3D crystals. Aperiodic order thus emerges as a scale-invariant feature of materials spanning metal alloys to self-assembled soft particles. The straightforward and path-dependent preparation of nonequilibrium quasicrystalline states from simple building blocks suggests that these complex sphere packings may often lurk in plain sight.

Quasicrystals exhibit long-range order but lack translational symmetry. When grown as single crystals, they possess distinctive and unusual properties owing to the absence of grain boundaries. Unfortunately, conventional methods such as bulk crystal growth or thin film deposition only allow us to synthesize either polycrystalline quasicrystals or quasicrystals that are at most a few centimeters in size. Here, we reveal through real-time and 3D imaging the formation of a single decagonal quasicrystal arising from a hard collision between multiple growing quasicrystals in an Al-Co-Ni liquid. Through corresponding molecular dynamics simulations, we explore the preconditions required for quasicrystal coalescence and investigate the effects of initial misorientation between the growing quasicrystalline grains on the formation of grain boundaries. At small misorientation, coalescence occurs through a rigid rotation mechanism that is facilitated by phason defects. Our joint experimental-computational discovery paves the way toward fabrication of single, large-scale quasicrystals for novel applications.

Understanding the 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 water molecules incompletely. In this paper, we develop GCIceNet, which automatically generates machine-based order parameters for classifying the phases of water molecules via supervised and unsupervised learning. The multiple graph convolutional layers in GCIceNet can learn topological information on the complex hydrogen bond networks. It shows a substantial improvement in accuracy for predicting the phase of water molecules in a bulk system and an ice/vapor interface system. A relative importance analysis shows that 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, GCIceNet is expected to serve as a powerful tool for the fields of glassy liquids and hydration layers around biomolecules.

The unconventional spherical assembly of discotic perylene bisimides (PBIs), as opposed to common columnar assembly, was successfully achieved by the incorporation of bulky units at the periphery. The peripheric steric hindrance can be tuned by varying the linkage structures, resulting in diverse spherical supramolecular structures in PBIs.
Abstract
Like other discotic molecules, self‐assembled supramolecular structures of perylene bisimides (PBIs) are commonly limited to columnar or lamellar structures due to their distinct π‐conjugated scaffolds and unique rectangular shape of perylene cores. The discovery of PBIs with supramolecular structures beyond layers and columns may expand the scope of PBI‐based materials. A series of unconventional spherical packing phases in PBIs, including A15 phase, σ phase, dodecagonal quasicrystalline (DQC) phase, and body‐centered cubic (BCC) phase, is reported. A strategy involving functionalization of perylene core with several polyhedral oligomeric silsesquioxane (POSS) cages achieved spherical assemblies of PBIs, instead of columnar assemblies, due to the significantly increased steric hindrance at the periphery. This strategy may also be employed for the discovery of unconventional spherical assemblies in other related discotic molecules by the introduction of similar bulky functional groups at their periphery. An unusual inverse phase transition sequence from a BCC phase to a σ phase was observed by increasing annealing temperature.

Like other discotic molecules, self‐assembled supramolecular structures of perylene bisimides (PBIs) are commonly limited to columnar or lamellar structures due to their distinct π‐conjugated scaffolds and unique rectangular shape of perylene cores. The discovery of PBIs with supramolecular structures beyond layers and columns may expand the scope of PBI‐based materials. Herein, we report a series of unconventional spherical packing phases in PBIs including A15 phase, σ phase, dodecagonal quasicrystalline (DQC) phase, and body‐centered cubic (BCC) phase. By the functionalization of perylene core with several polyhedral oligomeric silsesquioxane (POSS) cages, our strategy successfully achieves spherical assemblies of PBIs, instead of columnar assemblies, due to the significantly increased steric hindrance at the periphery. This strategy may also be employed for the discovery of unconventional spherical assemblies in other related discotic molecules by the introduction of similar bulky functional groups at their periphery. In addition, an unusual inverse phase transition sequence from BCC phase to σ phase by increasing annealing temperature is observed.

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.

Quasicrystals are solid structures with symmetry forbidden by crystallographic rules. Because of this some structural characteristics of quasicrystals, for instance, radial distribution function, can look similar to the ones of amorphous phases. This is of principal importance since radial distribution function is the main property to characterize the structure in molecular simulation. In the present paper we compare the radial distribution functions and dynamical properties of three systems in the vicinity of glass transition, quasicrystal formation and crystallization. We show that in spite of similarity of radial distribution functions the dynamical properties of a system in the vicinity of quasicrystal are qualitatively equivalent to the ones of crystal. Because of this combination the radial distribution functions with investigation of dynamics of the liquid allows unambiguously distinguish glass and quasicrystal.

An attempt is carried out to design metallic glasses by modifying the local structure of icosahedral quasicrystals. A series of metallic glasses are prepared by introducing beryllium to the quasicrystal former, Zr40Ti40Ni20, and the maximum diameter of the glasses reaches 20 mm. A phase transformation diagram is constructed for the alloys of (Zr40Ti40Ni20)100-xBex, revealing the transition from stable crystal to quasicrystal and then to metallic glasses with the increased Be content. This study shows a connection between the formation of metallic glasses and the specific thermodynamics of the initially precipitated quasicrystals. Interestingly, both improvements in the forming ability for glasses and quasicrystals are observed over a wide composition range. Ab initio molecular dynamic simulations show the evolution of the structural features of the metallic melts for the optimised glass forming region, which guarantee the primary precipitation of quasicrystal.

The self-assembly of two sizes of spherical nanocrystals has revealed a surprisingly diverse library of structures. To date at least fifteen distinct binary nanocrystal superlattice (BNSL) structures have been identified. The stability of these binary phases cannot be fully explained using the traditional conceptual framework treating the assembly process as entropy-driven crystallization of rigid spherical particles. Such deviation from hard sphere behavior may be explained by the soft and deformable layer of ligands that envelops the nanocrystals, which contributes significantly to the overall size and shape of assembling particles. In this work we describe a set of experiments designed to elucidate the role of the ligand corona in shaping the thermodynamics and kinetics of BNSL assembly. Using hydrocarbon-capped Au and PbS nanocrystals as a model binary system, we systematically tuned the core radius (R) and ligand chain length (L) of particles and subsequently assembled them into binary superlattices. The resulting database of binary structures enabled a detailed analysis of the role of effective nanocrystal size ratio, as well as softness expressed as L/R, in directing the assembly of binary structures. This catalog of superlattices allowed us to not only study the frequency of different phases but to also systematically measure the geometric parameters of the BNSLs. This analysis allowed us to evaluate new theoretical models treating the co-crystallization of deformable spheres and to formulate new hypotheses about the factors affecting the nucleation and growth of the binary superlattices. Among other insights, our results suggest that the relative abundance of the binary phases observed may be explained not only by considerations of thermodynamic stability, but also by a postulated pre-ordering of the binary fluid into local structures with icosahedral or polytetrahedral symmetry prior to nucleation.

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.

We present a detailed model to study the nucleation of triblock Janus particles from solution. The Janus particles are modeled as crosslinked polystyrene spheres whose poles are patched with sticky alkyl groups and their middle band is covered with negative charges. To mimic the experimental conditions, solvent, counterions and a substrate, on which the crystallization takes place, are included in the model. Many-body dissipative particle dynamics simulation technique is employed to include hydrodynamic and many-body interactions. Metadynamics simulations are done to explore the pathways for nucleation of Kagome and hexagonal lattices. In agreement with experiment, we found that nucleation of the Kagome lattice from solution follows a two-step mechanism. The connection of colloidal particles through their patches initially generates a disordered liquid network. Subsequently, orientational rearrangements in the liquid precursors lead to the formation of ordered nuclei. Biasing the potential energy of the largest crystal, a critical nucleus appears in the simulation box, whose further growth crystallizes the whole solution. The location of the phase transition point and its shift with patch width are in a very good agreement with experiment. The structure of the crystallized phase depends on the patch width; in the limit of very narrow patches strings are stable aggregates, intermediate patches stabilize the Kagome lattice, and wide patches nucleate the hexagonal phase. The scaling behavior of the calculated barrier heights confirms a first-order liquid-Kagome phase transition.

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.
Graphical abstract
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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.

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.

- M Dzugutov

M. Dzugutov, Physical Review Letters 70, 2924 (1993).

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F. Trudu, D. Donadio, and M. Parrinello, Physical Review Letters 97, 105701 (2006).

- J Roth
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J. Roth and A. R. Denton, Physical Review E 61, 6845
(2000).

- D Levine
- P J Steinhardt

D. Levine and P. J. Steinhardt, Physical Review Letters
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- D Moroni
- P R Ten Wolde
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D. Moroni, P. R. ten Wolde, and P. G. Bolhuis, Physical
Review Letters 94, 235703 (2005).

- A I Goldman
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A. I. Goldman and R. F. Kelton, Reviews of Modern
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M. Oxborrow and C. L. Henley, Physical Review B 48,
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- M Dzugutov

M. Dzugutov, Physical Review A 46, R2984 (1992).

- H C Jeong
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H. C. Jeong and P. J. Steinhardt, Physical Review B 55,
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- F H M Zetterling
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F. H. M. Zetterling, M. Dzugutov, and S. I. Simdyankin,
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- P R Ten Wolde
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- D Frenkel

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