Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

Local Structural Features Elucidate Crystallization of Complex Structures

Complex crystal structures are composed of multiple local environments, and how this type of order emerges spontaneously during crystal growth has yet to be fully understood. We study crystal growth across various structures and along different crystallization pathways, using self-assembly simulations of identical particles that interact via multiwell isotropic pair potentials. We apply an unsupervised machine learning method to features from bond-orientational order metrics to identify different local motifs present during a given structure's crystallization process. In this manner, we distinguish different crystallographic sites in highly complex structures. Tailoring this order parameter to structures of varying complexity and coordination number, we study the emergence of local order along a multistep crystal growth pathway─from a low-density fluid to a high-density, supercooled amorphous liquid droplet and to a bulk crystal. We find a consistent under-coordination of the liquid relative to the average coordination number in the bulk crystal. We use our order parameter to analyze the geometrically frustrated growth of a Frank-Kasper phase and discover how structural defects compete with the formation of crystallographic sites that are more high-coordinated than the liquid environments. The method presented here for classifying order on a particle-by-particle level has broad applicability to future studies of structural self-assembly and crystal growth, and they can aid in the design of building blocks and for targeting pathways of formation of soft-matter structures.

Classification of complex local environments in systems of particle shapes through shape symmetry-encoded data augmentation

Detecting and analyzing the local environment is crucial for investigating the dynamical processes of crystal nucleation and shape colloidal particle self-assembly. Recent developments in machine learning provide a promising avenue for better order parameters in complex systems that are challenging to study using traditional approaches. However, the application of machine learning to self-assembly on systems of particle shapes is still underexplored. To address this gap, we propose a simple, physics-agnostic, yet powerful approach that involves training a multilayer perceptron (MLP) as a local environment classifier for systems of particle shapes, using input features such as particle distances and orientations. Our MLP classifier is trained in a supervised manner with a shape symmetry-encoded data augmentation technique without the need for any conventional roto-translations invariant symmetry functions. We evaluate the performance of our classifiers on four different scenarios involving self-assembly of cubic structures, two-dimensional and three-dimensional patchy particle shape systems, hexagonal bipyramids with varying aspect ratios, and truncated shapes with different degrees of truncation. The proposed training process and data augmentation technique are both straightforward and flexible, enabling easy application of the classifier to other processes involving particle orientations. Our work thus presents a valuable tool for investigating self-assembly processes on systems of particle shapes, with potential applications in structure identification of any particle-based or molecular system where orientations can be defined.

Anoplophora graafi longhorn beetle coloration is due to disordered diamond-like packed spheres

While artificial photonic materials are typically highly ordered, photonic structures in many species of birds and insects do not possess a long-range order. Studying their order-disorder interplay sheds light on the origin of the photonic band gap. Here, we investigated the scale morphology of the Anoplophora graafi longhorn beetle. Combining small-angle X-ray scattering and slice-and-view FIB-SEM tomography with molecular dynamics and optical simulations, we characterised the chitin sphere assemblies within blue and green A. graafi scales. The low volume fraction of spheres and the number of their nearest neighbours are incompatible with any known close-packed sphere morphology. A short-range diamond lattice with long-range disorder best describes the sphere assembly, which will inspire the development of new colloid-based photonic materials.

Fast crystal growth in deeply undercooled ZrTi melts

We investigate the growth of crystals in Zr50Ti50 melts by classical molecular-dynamics simulations with an embedded atom method and a Stillinger-Weber potential model. Both models display fast solidification rates that can be captured by the transition state theory or the Ginzburg-Landau theory at small undercoolings. Fast crystal-growth rates are found to be affected by the pre-existing ordering in liquids, such as the body-centered cubic-like and icosahedral-like structures. The interface-induced ordering unveiled by the crystal-freezing method can explain the rate difference between these two models. However, these orderings fail to rationalize the temperature evolution of the growth rate at deep undercoolings. We correlate the growth kinetics with the detailed dynamical processes in liquids, finding the decoupling of hierarchic relaxation processes when collective motion emerges in supercooled liquids. We find that the growth kinetics is nondiffusive, but with a lower activation barrier corresponding to the structural relaxation or the cage-relative motion in ZrTi melts. These results explore a new relaxation mechanism for the fast growth rate in deeply undercooled liquids.

Phase Reentrances and Solid Deformations in Confined Colloidal Crystals

A simple geometric constraint often leads to novel, complex crystalline phases distinct from the bulk. Using thin-film charge colloidal crystals, a model system with tunable interactions, we study the effects of geometric constraints. Through a combination of experiments and simulations, we systematically explore phase reentrances and solid deformation modes concerning geometrical confinement strength, identifying two distinct categories of phase reentrances below a characteristic layer number, N_{c}: one for bcc bulk-stable and another for fcc bulk-stable systems. We further verify that the dominant thermodynamic origin is the nonmonotonic dependence of solids' free energy on the degree of spatial confinement. Moreover, we discover transitions in solid deformation modes between interface-energy and bulk-energy dominance: below a specific layer number, N_{k}, geometric constraints generate unique soft deformation modes adaptive to confinement. These findings on the N-dependent thermodynamic and kinetic behaviors offer fresh insights into understanding and manipulating thin-film crystal structures.

Rationalizing Euclidean Assemblies of Hard Polyhedra from Tessellations in Curved Space

Entropic self-assembly is governed by the shape of the constituent particles, yet a priori prediction of crystal structures from particle shape alone is nontrivial for anything but the simplest of space-filling shapes. At the same time, most polyhedra are not space filling due to geometric constraints, but these constraints can be relaxed or even eliminated by sufficiently curving space. We show using Monte Carlo simulations that the majority of hard Platonic solids self-assemble entropically into space-filling crystals when constrained to the surface volume of a 3-sphere. As we gradually decrease curvature to "flatten" space and compare the local morphologies of crystals assembling in curved and flat space, we show that the Euclidean assemblies can be categorized as either remnants of tessellations in curved space (tetrahedra and dodecahedra) or nontessellation-based assemblies caused by large-scale geometric frustration (octahedra and icosahedra).

Ordered ground state configurations of the asymmetric Wigner bilayer system-Revisited with unsupervised learning

We have reanalyzed the rich plethora of ground state configurations of the asymmetric Wigner bilayer system that we had recently published in a related diagram of states [Antlanger et al., Phys. Rev. Lett. 117, 118002 (2016)], comprising roughly 60 000 state points in the phase space spanned by the distance between the plates and the charge asymmetry parameter of the system. In contrast to this preceding contribution where the classification of the emerging structures was carried out "by hand," we have used for the present contribution machine learning concepts, notably based on a principal component analysis and a k-means clustering approach: using a 30-dimensional feature vector for each emerging structure (containing relevant information, such as the composition of the configuration as well as the most relevant order parameters), we were able to reanalyze these ground state configurations in a considerably more systematic and comprehensive manner than we could possibly do in the previously published classification scheme. Indeed, we were now able to identify new structures in previously unclassified regions of the parameter space and could considerably refine the previous classification scheme, thereby identifying a rich wealth of new emerging ground state configurations. Thorough consistency checks confirm the validity of the newly defined diagram of states.

Lightweight Extendable Stacking Framework for Structure Classification in Atomistic Simulations

Identifying an atom's local crystal structure is one crucial step in many atomistic simulation analyses. However, many traditional methods are available to only a few limited types of structures, and their performance often relies on manually determined parameters, which may lead to poor classification results in complex material systems. Machine learning models can enhance accuracy and generalizability, but they typically require large amounts of data and computation. This issue could be more severe for deep-learning-based frameworks, especially when confronted with unfamiliar crystal structures. To address this challenge, we propose a lightweight and extendable stacked structure (LESS) classifier, which adopts bond orientational order parameters as features and assembles several efficient machine learning methods as based models. The LESS classifier can recognize a variety of crystal structures, e.g., amorphous, mono, and binary structures, with over 98.8% accuracy on our validation data set, outperforming many current methods even including some deep-learning methods. Our model can also conduct probabilistic classification that aids in the interpretation of atomic structures in complicated environments such as heterogeneous interfaces. Furthermore, when exposed to a completely unknown crystal structure, the LESS framework can efficiently incorporate this new knowledge with generative sampled data from the current model. Overall, our model exhibits great potential as an accurate and flexible atomic structure identification tool featuring high efficiency in both learning and retraining.

Template-induced crystallization of charged colloids: a molecular dynamics study

By using a large enough number of particles and implementing a parallel algorithm on the CUDA platform, we have performed brute-force molecular dynamics simulations to study the template-induced heterogeneous crystallization in charged colloids. Six kinds of templates, whose patterns include the planes of fcc(100), fcc(110), fcc(111), bcc(100), bcc(110) and bcc(111), have been implanted into the middle of the simulation box. Except the fcc(111) template, whose structure benefits not only fcc but also hcp crystals resulting in a similar behavior to homogeneous crystallization, bcc-type templates favor the formation of bcc crystals and bcc-like precursors while fcc-type templates favor the formation of fcc crystals and fcc-like precursors. Therefore, for fcc(100) and fcc(110) templates, heterogeneous crystallization will definitely result in a fcc crystallite. However, the results of heterogeneous crystallization that are induced by bcc-type templates are subtly different at different state points. At the state points where the interaction strength of charged colloids is weak and the fcc phase is thermodynamically stable, the bcc crystals formed with the promotion of bcc-type templates are not stable so as to tend to transform into fcc or hcp crystals. When the interaction strength of charged colloids is high, the predominant bcc crystals formed with the promotion of bcc-type templates can always persist within the time scale of simulation although not bcc but fcc crystals are thermodynamically stable.

The thermodynamic-pathway-determined microstructure evolution of copper under shock compression

Shock-induced structural transformations in copper exhibit notable directional dependence and anisotropy, but the mechanisms that govern the responses of materials with different orientations are not yet well understood. In this study, we employ large-scale non-equilibrium molecular dynamics simulations to investigate the propagation of a shock wave through monocrystal copper and analyse the structural transformation dynamics in detail. Our results indicate that anisotropic structural evolution is determined by the thermodynamic pathway. A shock along the [Formula: see text] orientation causes a rapid and instantaneous temperature spike, resulting in a solid-solid phase transition. Conversely, a liquid metastable state is observed along the [Formula: see text] orientation due to thermodynamic supercooling. Notably, melting still occurs during the [Formula: see text]-oriented shock, even if it falls below the supercooling line in the thermodynamic pathway. These results highlight the importance of considering anisotropy, the thermodynamic pathway and solid-state disordering when interpreting phase transitions induced by shock. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Multiple Scenarios of Low-Temperature Nucleation in Simple Liquids

Usually, sufficient supercooling of a liquid is employed to bypass the free energy barrier and speed up crystallization. However, lowering the temperature T induces new issues competing with the crystallization, e.g., slow particle motion, geometric frustration, and the glass formation, which complicates our understanding of crystal growth. Here we systematically study the low-temperature nucleation kinetics discriminated by the maximum nucleation rate temperature T_{d} and the glass transition temperature T_{g}. At T_{d}, the ratio of the precursor and geometrically frustrated particles reaches the maximum. When T_{g}<T<T_{d}, nucleation kinetics is characterized by the subdiffusive slow particle motion, the high degrees of geometric frustration, and the saturation of precursors. In this regime, nucleation can proceed through the diffusionless-like ordering of precursors. Near T_{g}, there is a crossover regime, where geometrically frustrated particles percolate and the glass formation strongly slows down the nucleation. When T<T_{g}, diffusionless nucleation is obstructed due to the weak vibrational motion and the mechanical stability of the glassy state.

Collective Variables for Crystallization Simulations-from Early Developments to Recent Advances

Crystallization is an important physicochemical process which has relevance in material science, biology, and the environment. Decades of experimental and theoretical efforts have been made to understand this fundamental symmetry-breaking transition. While experiments provide equilibrium structures and shapes of crystals, they are limited to unraveling how molecules aggregate to form crystal nuclei that subsequently transform into bulk crystals. Computer simulations, mainly molecular dynamics (MD), can provide such microscopic details during the early stage of a crystallization event. Crystallization is a rare event that takes place in time scales much longer than a typical equilibrium MD simulation can sample. This inadequate sampling of the MD method can be easily circumvented by the use of enhanced sampling (ES) simulations. In most of the ES methods, the fluctuations of a system's slow degrees of freedom, called collective variables (CVs), are enhanced by applying a bias potential. This transforms the system from one state to the other within a short time scale. The most crucial part of such CV-based ES methods is to find suitable CVs, which often needs intuition and several trial-and-error optimization steps. Over the years, a plethora of CVs has been developed and applied in the study of crystallization. In this review, we provide a brief overview of CVs that have been developed and used in ES simulations to study crystallization from melt or solution. These CVs can be categorized mainly into four types: (i) spherical particle-based, (ii) molecular template-based, (iii) physical property-based, and (iv) CVs obtained from dimensionality reduction techniques. We present the context-based evolution of CVs, discuss the current challenges, and propose future directions to further develop effective CVs for the study of crystallization of complex systems.

Plasma Oxidation of Copper: Molecular Dynamics Study with Neural Network Potentials

The formation of thin oxide films is of significant scientific and practical interest. In particular, the semiconductor industry is interested in developing a plasma atomic layer etching process to pattern copper, replacing the dual Damascene process. Using a nonthermal oxygen plasma to convert the metallic copper into copper oxide, followed by a formic acid organometallic reaction to etch the copper oxide, this process has shown great promise. However, the current process is not optimal because the plasma oxidation step is not self-limiting, hampering the degree of thickness control. In the present study, a neural network potential for the binary interaction between copper and oxygen is developed and validated against first-principles calculations. This potential covers the entire range of potential energy surfaces of metallic copper, copper oxides, atomic oxygen, and molecular oxygen. The usable kinetic energy ranges from 0 to 20 eV. Using this potential, the plasma oxidation of copper surfaces was studied with large-scale molecular dynamics at atomic resolution, with an accuracy approaching that of the first principle calculations. An amorphous layer of CuO is formed on Cu, with thicknesses reaching 2.5 nm. Plasma is found to create an intense local heating effect that rapidly dissipates across the thickness of the film. The range of this heating effect depends on the kinetic energy of the ions. A higher ion energy leads to a longer range, which sustains faster-than-thermal rates for longer periods of time for the oxide growth. Beyond the range of this agitation, the growth is expected to be limited to the thermally activated rate. High-frequency, repeated ion impacts result in a microannealing effect that leads to a quasicrystalline oxide beneath the amorphized layer. The crystalline layer slows down oxide growth. Growth rate is fitted to the temperature gradient due to ion-induced thermal agitations, to obtain an apparent activation energy of 1.0 eV. A strategy of lowering the substrate temperature and increasing plasma power is proposed as being favorable for more self-limited oxidation.

Molecular cluster analysis using local order parameters selected by machine learning

Accurately extracting local molecular structures is essential for understanding the mechanisms of phase and structural transitions. A promising method to characterize the local molecular structure is defining the value of the local order parameter (LOP) for each particle. This work develops the Molecular Assembly structure Learning package for Identifying Order parameters (MALIO), a machine learning package that can propose an optimal (set of) LOP(s) quickly and automatically for a huge number of LOP species and various methods of selecting neighboring particles for the calculation. We applied this package to distinguish between the nematic and smectic phases of uniaxial liquid crystal molecules, and selected candidate LOPs that could be used to precisely observe the nematic-smectic phase transition. The LOP candidates were used to observe the nucleation and subsequent percolation transition, and the effect of the choice of LOP species and neighboring particles on the statistics of local molecular structures (clusters) was examined. The procedure revealed the time evolution of the number of clusters and the dependence of the percolation curve on the number of neighboring particles for each LOP species. The LOP species with the lowest dependence on the number of neighboring particles was the best-performing LOP species in the MALIO screening strategy. These results not only show that machine learning can powerfully screen a huge number of LOP species and suggest only a few promising candidates, but also indicate that MALIO can select the best LOP species.

Dimensionality reduction of local structure in glassy binary mixtures

We consider unsupervised learning methods for characterizing the disordered microscopic structure of supercooled liquids and glasses. Specifically, we perform dimensionality reduction of smooth structural descriptors that describe radial and bond-orientational correlations and assess the ability of the method to grasp the essential structural features of glassy binary mixtures. In several cases, a few collective variables account for the bulk of the structural fluctuations within the first coordination shell and also display a clear connection with the fluctuations of particle mobility. Fine-grained descriptors that characterize the radial dependence of bond-orientational order better capture the structural fluctuations relevant for particle mobility but are also more difficult to parameterize and to interpret. We also find that principal component analysis of bond-orientational order parameters provides identical results to neural network autoencoders while having the advantage of being easily interpretable. Overall, our results indicate that glassy binary mixtures have a broad spectrum of structural features. In the temperature range we investigate, some mixtures display well-defined locally favored structures, which are reflected in bimodal distributions of the structural variables identified by dimensionality reduction.

Local orderings in the melting process of a square crystal and in the resulting liquid

Using Brownian dynamics simulations we investigate the melting processes of a square crystalline lattice of colloidal particles interacting via an isotropic potential, which comprises both a hard-core repulsion and an additional softened square-well potential. For temperatures slightly lower than the transition one, we found a proliferation of small liquid clusters surrounded by the square lattice. These clusters are not static, quite the opposite, they have an intense dynamics and are continuously formed and destroyed over time. However, no unbound topological defects are observed. At the transition temperature, one of these liquid clusters starts to grow, until the entire system becomes in the liquid phase, then, characterizing a first-order phase transition. The tetratic intermediate phase, as given by the KTHNY theory, was not observed. Moreover, the liquid phase exhibits a considerable number of crystalline clusters having square and triangular orderings, which remain present even when increasing temperature by an order of magnitude. As the temperature increases, structural changes within the liquid phase are analyzed by evaluating the number and sizes of the square and triangular clusters. A transition of the dominant clusters is observed.

Sequence of phase transitions in a model of interacting rods

In a system of interacting thin rigid rods of equal length 2ℓ on a two-dimensional grid of lattice spacing a, we show that there are multiple phase transitions as the coupling strength κ=ℓ/a and the temperature are varied. There are essentially two classes of transitions. One corresponds to the Ising-type spontaneous symmetry-breaking transition and the second belongs to less-studied phase transitions of geometrical origin. The latter class of transitions appears at fixed values of κ irrespective of the temperature, whereas the critical coupling for the spontaneous symmetry-breaking transition depends on it. By varying the temperature, the phase boundaries may cross each other, leading to a rich phase behavior with infinitely many phases. Our results are based on Monte Carlo simulations on the square lattice and a fixed-point analysis of a functional flow equation on a Bethe lattice.

Thermal Hysteresis and Ordering Behavior of Magnetic Skyrmion Lattices

The physics of phase transitions in two-dimensional (2D) systems underpins research in diverse fields including statistical mechanics, nanomagnetism, and soft condensed matter. However, many aspects of 2D phase transitions are still not well understood, including the effects of interparticle potential, polydispersity, and particle shape. Magnetic skyrmions are chiral spin-structure quasi-particles that form two-dimensional lattices. Here, we show, by real-space imaging using in situ cryo-Lorentz transmission electron microscopy coupled with machine learning image analysis, the ordering behavior of Néel skyrmion lattices in van der Waals Fe₃GeTe₂. We demonstrate a distinct change in the skyrmion size distribution during field-cooling, which leads to a loss of lattice order and an evolution of the skyrmion liquid phase. Remarkably, the lattice order is restored during field heating and demonstrates a thermal hysteresis. This behavior is explained by the skyrmion energy landscape and demonstrates the potential to control the lattice order in 2D phase transitions.

Kinetics of inherent processes counteracting crystallization in supercooled monatomic liquid

Crystallization of supercooled liquids is mainly determined by two competing processes associated with the transition of particles (atoms) from liquid phase to crystalline one and, vice versa, with the return of particles from crystalline phase to liquid one. The quantitative characteristics of these processes are the so-called attachment rateg+and the detachment rateg-, which determine how particles change their belonging from one phase to another. In the present study, acorrespondence rulebetween the ratesg+andg-as functions of the sizeNof growing crystalline nuclei is defined for the first time. In contrast to the well-known detailed balance condition, which relatesg+(N)andg-(N)atN=nc(wheren_cis the critical nucleus size) and is satisfied only at the beginning of the nucleation regime, the foundcorrespondence ruleis fulfilled at all the main stages of crystallization kinetics (crystal nucleation, growth and coalescence). On the example of crystallizing supercooled Lennard-Jones liquid, the rateg-was calculated for the first time at different supercooling levels and for the wide range of nucleus sizesN∈[nc;40nc]. It was found that for the whole range of nucleus sizes, the detachment rateg-is only≈2% less than the attachment rateg+. This is direct evidence that the role of the processes that counteract crystallization remains significant at all the stages of crystallization. Based on the obtained results, a kinetic equation was formulated for the time-dependent distribution function of the nucleus sizes, that is an alternative to the well-known kinetic Becker-Döring-Zeldovich-Frenkel equation.

Revealing the role of liquid preordering in crystallisation of supercooled liquids

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.

Mesoscale simulation approach for assembly of small deformable objects

We adapt Vertex models to understand the physical origin of the formation of long-range ordered structures in repulsive soft particles. The model incorporates contributions from the volume and surface area of each particle. Sampling using Monte Carlo simulations allows the system to naturally select preferred structures. We observe transitions between a body-centered cubic ordered state and a disordered state. Constraints to the simulation domain can suppress or allow the system to follow a path similar to Martensitic transformations from one ordered state to another ordered state. Finally, we show that rapid quenches from a disordered state into the ordered region lead to metastable local particle arrangements instead of a large-scale single crystal.

Effect of the Interface on the Compressibility of Substances with Spherical Nano-Inhomogeneities on the Example of Al/C60

The paper examines the compressibility of media with nano-inhomogeneities using the example of an aluminum melt and C₆₀ fullerenes immersed in it. The results of molecular dynamics simulations indicate a significant effect of the interface on the effective compressibility of a heterogeneous medium. It is found that the application of the rule of mixture for the Al/C₆₀ system results in an incorrect qualitative picture of the dependence of compressibility on the concentration of fullerenes. To explain this effect, an analytical model is proposed that takes into account the reduction in distances between atoms of different components during compression. The model makes it possible to estimate the effective mechanical characteristics of a liquid with nano-inhomogeneities within the framework of the mechanical approach, and correctly predicts the nature of the change in the dependence of compressibility on concentration.

The role of the diffusion in the predictions of the classical nucleation theory for quasi-real systems differ in dipole moment value

In this paper, we examine the crystallization tendency for two quasi-real systems, which differ exclusively in the dipole moment's value. The main advantage of the studied system is the fact that despite that their structures are entirely identical, they exhibit different physical properties. Hence, the results obtained for one of the proposed model systems cannot be scaled to reproduce the results for another corresponding system, as it can be done for simple model systems, where structural differences are modeled by the different parameters of the intermolecular interactions. Our results show that both examined systems exhibit similar stability behavior below the melting temperature. This finding is contrary to the predictions of the classical nucleation theory, which suggests a significantly higher crystallization tendency for a more polar system. Our studies indicate that the noted discrepancies are caused by the kinetic aspect of the classical nucleation theory, which overestimates the role of diffusion in the nucleation process.

The role of pressure and defects in the wurtzite to rock salt transition in cadmium selenide

Using molecular dynamics and path sampling techniques we investigated the effect of pressure and defects in the wurtzite to rock salt transition in cadmium selenide (CdSe). In the pressure range 2-10 GPa, rate constants of transition are in the order of 10^-23 to 10⁵ s^-1 for the transformation of a relatively small wurtzite crystal consisting of 1024 atoms with periodic boundary conditions. The transition paths predominantly evolve through an intermediate 5-coordinated structure, as reported before, though its typical lifetime within the transition paths is particularly long in the intermediate pressure range (4-6 GPa). The defects were created by removing Cd-Se pairs from an otherwise perfect crystal. The removals were either selected fully randomized or grouped in clusters (cavity creation). We find that the rate of transition due to the defects increases by several orders of magnitude even for a single pair removal. This is caused by a change in the transition mechanism that no longer proceeds via the intermediate 5-coordinated structure, when defects are present. Further, the cavity creation yields a lower rate than the fully randomized removal.

Controlling disorder in self-assembled colloidal monolayers via evaporative processes

Monolayers of assembled nano-objects with a controlled degree of disorder hold interest in many optical applications, including photovoltaics, light emission, sensing, and structural coloration. Controlled disorder can be achieved through either top-down or bottom-up approaches, but the latter is more suited to large-scale, low-cost fabrication. Disordered colloidal monolayers can be assembled through evaporatively driven convective assembly, a bottom-up process with a wide range of parameters impacting particle placement. Motivated by the photonic applications of such monolayers, in this review we discuss the quantification of monolayer disorder, and the assembly methods that have been used to produce them. We review the impact of particle and solvent properties, as well as the use of substrate patterning, to create the desired spatial distributions of particles.

Hard superellipse phases: particle shape anisotropy & curvature

We report computer simulations of two-dimensional convex hard superellipse particle phases vs. particle shape parameters including aspect ratio, corner curvature, and sidewall curvature. Shapes investigated include disks, ellipses, squares, rectangles, and rhombuses, as well as shapes with non-uniform curvature including rounded squares, rounded rectangles, and rounded rhombuses. Using measures of orientational order, order parameters, and a novel stretched bond orientational order parameter, we systematically identify particle shape properties that determine liquid crystal and crystalline phases including their coarse boundaries and symmetry. We observe phases including isotropic, nematic, tetratic, plastic crystals, square crystals, and hexagonal crystals (including stretched variants). Our results catalog known benchmark shapes, but include new shapes that also interpolate between known shapes. Our results indicate design rules for particle shapes that determine two-dimensional liquid, liquid crystalline, and crystalline microstructures that can be realized via particle assembly.

The L-G phase transition in binary Cu-Zr metallic liquids

The authors recently reported that undercooled liquid Ag and Ag-Cu alloys both exhibit a first order phase transition from the homogeneous liquid (L-phase) to a heterogeneous solid-like G-phase under isothermal evolution. Here, we report a similar L-G transition and heterogenous G-phase in simulations of liquid Cu-Zr bulk glass. The thermodynamic description and kinetic features (viscosity) of the L-G-phase transition in Cu-Zr simulations suggest it corresponds to experimentally reported liquid-liquid phase transitions in Vitreloy 1 (Vit1) and other Cu-Zr-bearing bulk glass forming alloys. The Cu-Zr G-phase has icosahedrally ordered cores versus fcc/hcp core structures in Ag and Ag-Cu with a notably smaller heterogeneity length scale Λ. We propose the L-G transition is a phenomenon in metallic liquids associated with the emergence of elastic rigidity. The heterogeneous core-shell nano-composite structure likely results from accommodating strain mismatch of stiff core regions by more compliant intervening liquid-like medium.

Self-propulsion and self-navigation: Activity is a precursor to jamming

Traffic jams are an everyday hindrance to transport and typically arise when many vehicles have the same or a similar destination. We show, however, that even when uniformly distributed in space and uncorrelated, targets have a crucial effect on transport. At modest densities an instability arises leading to jams with emergent correlations between the targets. By considering limiting cases of the dynamics which map onto active Brownian particles, we argue that motility induced phase separation is the precursor to jams. That is, jams are MIPS seeds that undergo an extra instability due to target accumulation. This provides a quantitative prediction of the onset density for jamming, and suggests how jamming might be delayed or prevented. We study the transition between jammed and flowing phase, and find that transport is most efficient on the cusp of jamming.

Pressure Dependence of the Crystallization Rate for the S-Enantiomer and a Racemic Mixture of Ibuprofen

This paper examines the pressure effect on the crystallization rate of the pharmaceutically active enantiomerically pure S-enantiomer and the racemic mixture of the well-known drug ibuprofen. Performed experimental studies revealed that at ambient pressure S-ibuprofen crystallizes faster than the racemic mixture. When the pressure increases, the crystallization rate slows down for both systems, but interestingly it is more apparent in the case of the S-enantiomer. It is found that this experimentally observed trend can be understood based on the predictions of the classical nucleation theory. We suggest that the solid-liquid interfacial free energy is the main reason for the observed variations in S- and RS-ibuprofen's stability behaviors. Employing a special method of computational studies, i.e., the capillary fluctuation method, we show that the increase in pressure affects the solid-liquid interfacial free energy for S- and RS-ibuprofen in an entirely different way. Importantly, the detected differences correspond to the experimentally observed variations in the overall crystallization rates.

Mining of Effective Local Order Parameters to Classify Ice Polymorphs

Order parameters make it possible to quantify the degree of structural ordering in a material and thus to apply as the reaction coordinates during the free-energy analysis of phase or structure transitions. Furthermore, order parameters are useful in determining the local structures of molecular groups during transition stages. However, identifying or developing local order parameters (LOPs) that are sensitive for specific materials and phases is a non-trivial task. In this study, the ability of LOPs to classify the solid and liquid structures of water at coexistence or triple points is investigated with the aid of supervised machine learning. The classification accuracy of a total of 179,738,433 combinations of 493 LOPs is automatically and systematically compared for water structures at the ice Ih-Ic-liquid coexistence point and the ice III-V-liquid and ice V-VI-liquid triple points. The optimal sets of two LOPs are found for each point, and sets of three LOPs are suggested for better accuracy.

Topological signatures and stability of hexagonal close packing and Barlow stackings

Two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals. These motifs, however, occupy an identical portion of space and bear identical first-order topological signatures as measured by persistent homology. Here we present a novel method based on k-fold covers that unambiguously distinguishes between these patterns. Moreover, our approach provides topological evidence that the FCC motif is the more stable of the two in the context of evolving experimental sphere packings during the transition from disordered to an ordered state. We conclude that our approach can be generalised to distinguish between various Barlow stackings manifested in minerals and metals.

Structural analysis of disordered dimer packings

Jammed disordered packings of non-spherical particles show significant variation in the packing density as a function of particle shape for a given packing protocol. Rotationally symmetric elongated shapes such as ellipsoids, spherocylinders, and dimers, e.g., pack significantly denser than spheres over a narrow range of aspect ratios, exhibiting a characteristic peak at aspect ratios of α_max ≈ 1.4-1.5. However, the structural features that underlie this non-monotonic behaviour in the packing density are unknown. Here, we study disordered packings of frictionless dimers in three dimensions generated by a gravitational pouring protocol in LAMMPS. Focusing on the characteristics of contacts as well as orientational and translational order metrics, we identify a number of structural features that accompany the formation of maximally dense packings as the dimer aspect ratio α is varied from the spherical limit. Our results highlight that dimer packings undergo significant structural changes as α increases up to α_max manifest in the reorganisation of the contact configurations between neighbouring dimers, increasing nematic order, and decreasing local translational order. Remarkably, for α > α_max our metrics remain largely unchanged, indicating that the peak in the packing density is related to the interplay of structural rearrangements for α < α_max and subsequent excluded volume effects with unchanged structure for α > α_max.

Fast crystal growth at ultra-low temperatures

It is believed that the slow liquid diffusion and geometric frustration brought by a rapid, deep quench inhibit fast crystallization and promote vitrification. Here we report fast crystal growth in charged colloidal systems under deep supercooling, where liquid diffusion is extremely low. By combining experiments and simulations, we show that this process occurs via wall-induced barrierless ordering consisting of two coupled steps: the step-like advancement of the rough interface that disintegrates frustration, followed by defect repairing inside the newly formed solid phase. The former is a diffusionless collective process, whereas the latter controls crystal quality. We further show that the intrinsic mechanical instability of a disordered glassy state subject to the crystal growth front allows for domino-like fast crystal growth even at ultra-low temperatures. These findings contribute to a deeper understanding of fast crystal growth and may be useful for applications related to vitrification prevention and crystal-quality control.

Atomic vibration as an indicator of the propensity for configurational rearrangements in metallic glasses

In a metallic glass (MG), the propensity for atomic rearrangements varies spatially from location to location in the amorphous solid, making the prediction of their likelihood a major challenge. One can attack this problem from the "structure controls properties" standpoint. But all the current structure-centric parameters are mostly based on local atomic packing information limited to short-range order, hence falling short in reliably forecasting how the local region would respond to external stimuli (e.g., temperature and/or stress). Alternatively, one can use indicators informed by physical properties to bridge the static structure on the one hand, and the response of the local configuration on the other. A sub-group of such physics-informed quantities consists of atomic vibration parameters, which will be singled out as the focus of this article. Here we use the Cu₆₄Zr₃₆ alloy to systematically demonstrate the following two points, all using a single model MG. First, we show in a comprehensive manner the interrelation among common vibrational parameters characterizing the atomic vibrational amplitude and frequency, including the atomic mean square displacement, flexibility volume, participation fraction in the low-frequency vibrational modes and boson peak intensity. Second, we demonstrate that these vibrational parameters fare much better than purely static structural parameters based on local geometrical packing in providing correlation with the propensity for local configurational transitions. These vibrational parameters also share a correlation length similar to that in structural rearrangements induced by external stimuli. This success, however, also poses a challenge, as it remains to be elucidated as to why short-time dynamical (vibrational) behavior at the bottom of the energy basin can be exploited to project the height of the energy barrier for cross-basin activities and in turn the propensity for locally collective atomic rearrangements.

The local and global geometry of trabecular bone

The organization and shape of the microstructural elements of trabecular bone govern its physical properties, are implicated in bone disease, and serve as blueprints for biomaterial design. To devise fundamental structure-property relationships and design truly bone-mimicking biomaterials, it is essential to characterize trabecular bone structure from the perspective of geometry, the mathematical study of shape. Using micro-CT images from 70 donors at five different sites, we analyze the local and global geometry of human trabecular bone in detail, respectively by quantifying surface curvatures and Minkowski functionals. We find that curvature density maps provide distinct and sensitive shape fingerprints for bone from different sites. Contrary to a common assumption, these curvature maps also show that bone morphology does not approximate a minimal surface but exhibits a much more intricate curvature landscape. At the global (or integral) perspective, our Minkowski analysis illustrates that trabecular bone exhibits other types of anisotropy/ellipticity beyond interfacial orientation, and that anisotropy varies substantially within the trabecular structure. Moreover, we show that the Minkowski functionals unify several traditional morphometric indices. Our geometric approach to trabecular morphometry provides a fundamental language of shape that could be useful for bone failure prediction, understanding geometry-driven tissue growth, and the design of bone-mimicking tissue scaffolds. STATEMENT OF SIGNIFICANCE: The architecture of trabecular bone is key in determining bone properties, and is often a starting point for the design of bone-substitutes. Despite the substantial history of bone morphometry, a fundamental characterization of trabecular bone geometry is still lacking. Therefore, we introduce a robust framework to quantify local and global trabecular bone geometry, which we apply to hundreds of micro-CT scans. Our approach relies on quantifying surface curvatures and Minkowski functionals, which are the most fundamental local and global shape quantifiers. Our results show that these shape metrics are sensitive to differences between bone types and unify traditional metrics within a single mathematical framework. This geometrical framework could also be useful to design bone-mimicking scaffolds and understand geometry-driven tissue growth.

Physics-Inspired Structural Representations for Molecules and Materials

The first step in the construction of a regression model or a data-driven analysis, aiming to predict or elucidate the relationship between the atomic-scale structure of matter and its properties, involves transforming the Cartesian coordinates of the atoms into a suitable representation. The development of atomic-scale representations has played, and continues to play, a central role in the success of machine-learning methods for chemistry and materials science. This review summarizes the current understanding of the nature and characteristics of the most commonly used structural and chemical descriptions of atomistic structures, highlighting the deep underlying connections between different frameworks and the ideas that lead to computationally efficient and universally applicable models. It emphasizes the link between properties, structures, their physical chemistry, and their mathematical description, provides examples of recent applications to a diverse set of chemical and materials science problems, and outlines the open questions and the most promising research directions in the field.

Revealing thermally-activated nucleation pathways of diffusionless solid-to-solid transition

Solid-to-solid transitions usually occur via athermal nucleation pathways on pre-existing defects due to immense strain energy. However, the extent to which athermal nucleation persists under low strain energy comparable to the interface energy, and whether thermally-activated nucleation is still possible are mostly unknown. To address these questions, the microscopic observation of the transformation dynamics is a prerequisite. Using a charged colloidal system that allows the triggering of an fcc-to-bcc transition while enabling in-situ single-particle-level observation, we experimentally find both athermal and thermally-activated pathways controlled by the softness of the parent crystal. In particular, we reveal three new transition pathways: ingrain homogeneous nucleation driven by spontaneous dislocation generation, heterogeneous nucleation assisted by premelting grain boundaries, and wall-assisted growth. Our findings reveal the physical principles behind the system-dependent pathway selection and shed light on the control of solid-to-solid transitions through the parent phase's softness and defect landscape.

Searching for local order parameters to classify water structures at triple points

The diversity of ice polymorphs is of interest in condensed-matter physics, engineering, astronomy, and biosphere and climate studies. In particular, their triple points are critical to elucidate the formation of each phase and transitions among phases. However, an approach to distinguish their molecular structures is lacking. When precise molecular geometries are given, order parameters are often computed to quantify the degree of structural ordering and to classify the structures. Many order parameters have been developed for specific or multiple purposes, but their capabilities have not been exhaustively investigated for distinguishing ice polymorphs. Here, 493 order parameters and their combinations are considered for two triple points involving the ice polymorphs ice III-V-liquid and ice V-VI-liquid. Supervised machine learning helps automatic and systematic searching of the parameters. For each triple point, the best set of two order parameters was found that distinguishes three structures with high accuracy. A set of three order parameters is also suggested for better accuracy.

Semi-supervised learning for the study of structural formation in colloidal systems via image recognition

The analysis of the structural formation of colloidal systems using machine learning techniques has recently attracted much attention. In many of these studies, local bond-order parameters (LBOPs) were employed as descriptors, where such LBOPs are suitable mainly for the detection of crystal structures. On the other hand, image-based convolutional neural networks (CNNs) are quite effective in detecting not only crystals but also random structures, and the author demonstrated their efficiency in a previous paper. However, in supervised learning, it is difficult to obtain a correct result when there is an unexpected new phase that was unknown when training the CNN. In this paper, we propose a hybrid scheme that consists of supervised and unsupervised learning techniques, employing two different approaches: image-based CNN and generalized LBOP. The proposed method was applied to two-dimensional colloidal systems, and its efficiency was demonstrated.

From predictive modelling to machine learning and reverse engineering of colloidal self-assembly

An overwhelming diversity of colloidal building blocks with distinct sizes, materials and tunable interaction potentials are now available for colloidal self-assembly. The application space for materials composed of these building blocks is vast. To make progress in the rational design of new self-assembled materials, it is desirable to guide the experimental synthesis efforts by computational modelling. Here, we discuss computer simulation methods and strategies used for the design of soft materials created through bottom-up self-assembly of colloids and nanoparticles. We describe simulation techniques for investigating the self-assembly behaviour of colloidal suspensions, including crystal structure prediction methods, phase diagram calculations and enhanced sampling techniques, as well as their limitations. We also discuss the recent surge of interest in machine learning and reverse-engineering methods. Although their implementation in the colloidal realm is still in its infancy, we anticipate that these data-science tools offer new paradigms in understanding, predicting and (inverse) design of novel colloidal materials.

How real are liquid groundstates? Ultra-fast crystal growth and the susceptibility of energy minima in liquids

We calculate the degree to which the final structure of the local groundstate in a liquid is a function of the strength of a perturbing potential applied during energy minimization. This structural susceptibility is shown to correlate well with the observed tendency of the liquid adjacent to a crystal interface to exhibit a crystalline groundstate, a feature that has been strongly linked to the observation of ultra-fast crystal growth in pure metals and ionic melts. It is shown that the structural susceptibility increases dramatically as the interaction potential between atoms is softened.

A jamming plane of sphere packings

The concept of jamming has attracted great research interest due to its broad relevance in soft-matter, such as liquids, glasses, colloids, foams, and granular materials, and its deep connection to sphere packing and optimization problems. Here, we show that the domain of amorphous jammed states of frictionless spheres can be significantly extended, from the well-known jamming-point at a fixed density, to a jamming-plane that spans the density and shear strain axes. We explore the jamming-plane, via athermal and thermal simulations of compression and shear jamming, with initial equilibrium configurations prepared by an efficient swap algorithm. The jamming-plane can be divided into reversible-jamming and irreversible-jamming regimes, based on the reversibility of the route from the initial configuration to jamming. Our results suggest that the irreversible-jamming behavior reflects an escape from the metastable glass basin to which the initial configuration belongs to or the absence of such basins. All jammed states, either compression- or shear-jammed, are isostatic and exhibit jamming criticality of the same universality class. However, the anisotropy of contact networks nontrivially depends on the jamming density and strain. Among all state points on the jamming-plane, the jamming-point is a unique one with the minimum jamming density and the maximum randomness. For crystalline packings, the jamming-plane shrinks into a single shear jamming-line that is independent of initial configurations. Our study paves the way for solving the long-standing random close-packing problem and provides a more complete framework to understand jamming.

Surfactants and rotelles in active chiral fluids

Surfactant molecules migrate to interfaces, reduce interfacial tension, and form micelles. All of these behaviors occur at or near equilibrium. Here, we describe active analogs of surfactants that operate far from equilibrium in active chiral fluids. Unlike molecular surfactants, the amphiphilic character of surfactants in active chiral fluids is a consequence of their activity. Our fluid of choice is a mixture of spinners that demixes into left-handed and right-handed chiral fluid domains. We realize spinners in experiment with three-dimensionally printed vibrots. Vibrot surfactants are chains of vibrots containing both types of handedness. Experiments demonstrate the affinity of double-stranded chains to interfaces, where they glide along and act as mixing agents. Simulations access larger systems in which single-stranded chains form spinning vesicles, termed rotelles. Rotelles are the chiral analogs of micelles. Rotelle formation is a ratchet mechanism catalyzed by the vorticity of the chiral fluid and only exist far from equilibrium.

Structure of the simple harmonic-repulsive system in liquid and glassy states studied by the triple correlation function

An efficient description of the structures of liquids and, in particular, the structural changes that happen with liquids on supercooling remains to be a challenge. The systems composed of soft particles are especially interesting in this context because they often demonstrate non-trivial local orders that do not allow to introduce the concept of the nearest-neighbor shell. For this reason, the use of some methods, developed for the structure analysis of atomic liquids, is questionable for the soft-particle systems. Here we report about our investigations of the structure of the simple harmonic-repulsive liquid in 3D using the triple correlation function (TCF), i.e., the method that does not rely on the nearest neighbor concept. The liquid is considered at reduced pressure (P = 1.8) at which it exhibits remarkable stability against crystallization on cooling. It is demonstrated that the TCF allows addressing the development of the orientational correlations in the structures that do not allow drawing definite conclusions from the studies of the bond-orientational order parameters. Our results demonstrate that the orientational correlations, if measured by the heights of the peaks in the TCF, significantly increase on cooling. This rise in the orientational ordering is not captured properly by the Kirkwood's superposition approximation. Detailed considerations of the peaks' shapes in the TCF suggest the existence of a link between the orientational ordering and the slowdown of the system's dynamics. Our findings support the view that the development of the orientational correlations in liquids may play a significant role in the liquids' dynamics and that the considerations of the pair distribution function may not be sufficient to understand intuitively all the structural changes that happen with liquids on supercooling. In general, our results demonstrate that the considerations of the TCF are useful in the discussions of the liquid's structures beyond the pair density function and interpreting the results obtained with the bond-orientational order parameters.

Physical origin of glass formation from multicomponent systems

The origin of glass formation is one of the most fundamental issues in glass science. The glass-forming ability (GFA) of multicomponent systems, such as metallic glasses and phase-change materials, can be enormously changed by slight modifications of the constituted elements and compositions. However, its physical origin remains mostly unknown. Here, by molecular dynamics simulations, we study three model metallic systems with distinct GFA. We find that they have a similar driving force of crystallization, but a different liquid-crystal interface tension, indicating that the latter dominates the GFA. Furthermore, we show that the interface tension is determined by nontrivial coupling between structural and compositional orderings and affects crystal growth. These facts indicate that the classical theories of crystallization need critical modifications by considering local ordering effects. Our findings provide fresh insight into the physical control of GFA of metallic alloys and the switching speed of phase-change materials without relying on experience.

Classifying Crystals of Rounded Tetrahedra and Determining Their Order Parameters Using Dimensionality Reduction

Using simulations we study the phase behavior of a family of hard spherotetrahedra, a shape that interpolates between tetrahedra and spheres. We identify 13 close-packed structures, some with packings that are significantly denser than previously reported. Twelve of these are crystals with unit cells of N = 2 or N = 4 particles, but in the shape regime of slightly rounded tetrahedra we find that the densest structure is a quasicrystal approximant with a unit cell of N = 82 particles. All 13 structures are also stable below close packing, together with an additional 14th plastic crystal phase at the sphere side of the phase diagram, and upon sufficient dilution to packing fractions below 50-60% all structures melt. Interestingly, however, upon compressing the fluid phase, self-assembly takes place spontaneously only at the tetrahedron and the sphere side of the family but not in an intermediate regime of tetrahedra with rounded edges. We describe the local environment of each particle by a set of l-fold bond orientational order parameters q̅l, which we use in an extensive principal component analysis. We find that the total packing fraction as well as several particular linear combinations of q̅l rather than individual q̅l's are optimally distinctive, specifically the differences q̅4 - q̅6 for separating tetragonal from hexagonal structures and q̅4-q̅8 for distinguishing tetragonal structures. We argue that these characteristic combinations are also useful as reliable order parameters in nucleation studies, enhanced sampling techniques, or inverse-design methods involving odd-shaped particles in general.