Quasicrystal of Binary Hard Spheres on a Plane Stabilized by Configurational Entropy.

Because of their aperiodic nature, quasicrystals are one of the least understood phases in statistical physics. One significant complication they present in comparison to their periodic counterparts is the fact that any quasicrystal can be realized as an exponentially large number of different tilings, resulting in a significant contribution to the quasicrystal entropy. Here, we use free-energy calculations to demonstrate that it is this configurational entropy which stabilizes a dodecagonal quasicrystal in a binary mixture of hard spheres on a plane. Our calculations also allow us to quantitatively confirm that in this system all tiling realizations are essentially equally likely, with free-energy differences less than 0.0001k_{B}T per particle-an observation that could be related to the observation of only random tilings in soft-matter quasicrystals. Owing to the simplicity of the model and its available counterparts in colloidal experiments, we believe that this system is an excellent candidate to achieve the long-awaited quasicrystal self-assembly on the micron scale.

In search of a precursor for crystal nucleation of hard and charged colloids.

The interplay between crystal nucleation and the structure of the metastable fluid has been a topic of significant debate over recent years. In particular, it has been suggested that even in simple model systems such as hard or charged colloids, crystal nucleation might be foreshadowed by significant fluctuations in local structure around the location where the nucleus first arises. We investigate this using computer simulations of spontaneous nucleation events in both hard and charged colloidal systems. To detect local structural variations, we use both standard and unsupervised machine learning methods capable of finding hidden structures in the metastable fluid phase. We track numerous nucleation events for the face-centered cubic and body-centered cubic crystals on a local level and demonstrate that all signs of crystallinity emerge simultaneously from the very start of the nucleation process. We thus conclude that we observe no precursor for the crystal nucleation of hard and charged colloids.

Improving the prediction of glassy dynamics by pinpointing the local cage.

The relationship between structure and dynamics in glassy fluids remains an intriguing open question. Recent work has shown impressive advances in our ability to predict local dynamics using structural features, most notably due to the use of advanced machine learning techniques. Here, we explore whether a simple linear regression algorithm combined with intelligently chosen structural order parameters can reach the accuracy of the current, most advanced machine learning approaches for predicting dynamic propensity. To achieve this, we introduce a method to pinpoint the cage state of the initial configuration-i.e., the configuration consisting of the average particle positions when particle rearrangement is forbidden. We find that, in comparison to both the initial state and the inherent state, the structure of the cage state is highly predictive of the long-time dynamics of the system. Moreover, by combining the cage state information with the initial state, we are able to predict dynamic propensities with unprecedentedly high accuracy over a broad regime of time scales, including the caging regime.

Self-assembly of dodecagonal and octagonal quasicrystals in hard spheres on a plane.

Hard spheres are one of the most fundamental model systems in soft matter physics, and have been instrumental in shedding light on nearly every aspect of classical condensed matter. Here, we add one more important phase to the list that hard spheres form: quasicrystals. Specifically, we use simulations to show that an extremely simple, purely entropic model system, consisting of two sizes of hard spheres resting on a flat plane, can spontaneously self-assemble into two distinct random-tiling quasicrystal phases. The first quasicrystal is a dodecagonal square-triangle tiling, commonly observed in a large variety of colloidal systems. The second quasicrystal has, to our knowledge, never been observed in either experiments or simulations. It exhibits octagonal symmetry, and consists of three types of tiles: triangles, small squares, and large squares, whose relative concentration can be continuously varied by tuning the number of smaller spheres present in the system. The observed tile composition of the self-assembled quasicrystals agrees very well with the theoretical prediction we obtain by considering the four-dimensional (lifted) representation of the quasicrystal. Both quasicrystal phases form reliably and rapidly over a significant part of parameter space. Our results demonstrate that entropy combined with a set of geometrically compatible, densely packed tiles can be sufficient ingredients for the self-assembly of colloidal quasicrystals.

Microtubule nucleation complex behavior is critical for cortical array homogeneity and xylem wall patterning.

Plant cell walls are versatile materials that can adopt a wide range of mechanical properties through controlled deposition of cellulose fibrils. Wall integrity requires a sufficiently homogeneous fibril distribution to cope effectively with wall stresses. Additionally, specific conditions, such as the negative pressure in water transporting xylem vessels, may require more complex wall patterns, e.g., bands in protoxylem. The orientation and patterning of cellulose fibrils are guided by dynamic cortical microtubules. New microtubules are predominantly nucleated from parent microtubules causing positive feedback on local microtubule density with the potential to yield highly inhomogeneous patterns. Inhomogeneity indeed appears in all current cortical array simulations that include microtubule-based nucleation, suggesting that plant cells must possess an as-yet unknown balancing mechanism to prevent it. Here, in a combined simulation and experimental approach, we show that a limited local recruitment of nucleation complexes to microtubules can counter the positive feedback, whereas local tubulin depletion cannot. We observe that nucleation complexes preferentially appear at the plasma membrane near microtubules. By incorporating our experimental findings in stochastic simulations, we find that the spatial behavior of nucleation complexes delicately balances the positive feedback, such that differences in local microtubule dynamics-as in developing protoxylem-can quickly turn a homogeneous array into a banded one. Our results provide insight into how the plant cytoskeleton has evolved to meet diverse mechanical requirements and greatly increase the predictive power of computational cell biology studies.

Crystal nucleation of highly screened charged colloids.

We study the nucleation of nearly hard charged colloidal particles. We use Monte Carlo simulations in combination with free-energy calculations to accurately predict the phase diagrams of these particles and map them via the freezing density to hard spheres, then we use umbrella sampling to explore the nucleation process. Surprisingly, we find that even very small amounts of charge repulsion can have a significant effect on the phase behavior. Specifically, we find that phase boundaries and nucleation barriers are mostly dependent on the Debye screening length and that even screening lengths as small as 2% of the particle diameter are sufficient to show marked differences in both. This work demonstrates clearly that even mildly charged colloids are not effectively hard spheres.

Comparing machine learning techniques for predicting glassy dynamics.

In the quest to understand how structure and dynamics are connected in glasses, a number of machine learning based methods have been developed that predict dynamics in supercooled liquids. These methods include both increasingly complex machine learning techniques and increasingly sophisticated descriptors used to describe the environment around particles. In many cases, both the chosen machine learning technique and choice of structural descriptors are varied simultaneously, making it hard to quantitatively compare the performance of different machine learning approaches. Here, we use three different machine learning algorithms-linear regression, neural networks, and graph neural networks-to predict the dynamic propensity of a glassy binary hard-sphere mixture using as structural input a recursive set of order parameters recently introduced by Boattini et al. [Phys. Rev. Lett. 127, 088007 (2021)]. As we show, when these advanced descriptors are used, all three methods predict the dynamics with nearly equal accuracy. However, the linear regression is orders of magnitude faster to train, making it by far the method of choice.

Inverse design of soft materials via a deep learning-based evolutionary strategy.

Colloidal self-assembly­the spontaneous organization of colloids into ordered structures­has been considered key to produce next-generation materials. However, the present-day staggering variety of colloidal building blocks and the limitless number of thermodynamic conditions make a systematic exploration intractable. The true challenge in this field is to turn this logic around and to develop a robust, versatile algorithm to inverse design colloids that self-assemble into a target structure. Here, we introduce a generic inverse design method to efficiently reverse-engineer crystals, quasicrystals, and liquid crystals by targeting their diffraction patterns. Our algorithm relies on the synergetic use of an evolutionary strategy for parameter optimization, and a convolutional neural network as an order parameter, and provides a way forward for the inverse design of experimentally feasible colloidal interactions, specifically optimized to stabilize the desired structure.

Machine learning many-body potentials for colloidal systems.

Simulations of colloidal suspensions consisting of mesoscopic particles and smaller species such as ions or depletants are computationally challenging as different length and time scales are involved. Here, we introduce a machine learning (ML) approach in which the degrees of freedom of the microscopic species are integrated out and the mesoscopic particles interact with effective many-body potentials, which we fit as a function of all colloid coordinates with a set of symmetry functions. We apply this approach to a colloid-polymer mixture. Remarkably, the ML potentials can be assumed to be effectively state-independent and can be used in direct-coexistence simulations. We show that our ML method reduces the computational cost by several orders of magnitude compared to a numerical evaluation and accurately describes the phase behavior and structure, even for state points where the effective potential is largely determined by many-body contributions.

Averaging Local Structure to Predict the Dynamic Propensity in Supercooled Liquids.

Predicting the local dynamics of supercooled liquids based purely on local structure is a key challenge in our quest for understanding glassy materials. Recent years have seen an explosion of methods for making such a prediction, often via the application of increasingly complex machine learning techniques. The best predictions so far have involved so-called Graph Neural Networks (GNNs) whose accuracy comes at a cost of models that involve on the order of 10^{5} fit parameters. In this Letter, we propose that the key structural ingredient to the GNN method is its ability to consider not only the local structure around a central particle, but also averaged structural features centered around nearby particles. We demonstrate that this insight can be exploited to design a significantly more efficient model that provides essentially the same predictive power at a fraction of the computational complexity (approximately 1000 fit parameters), and demonstrate its success by fitting the dynamic propensity of Kob-Andersen and binary hard-sphere mixtures. We then use this to make predictions regarding the importance of radial and angular descriptors in the dynamics of both models.

Defects in crystals of soft colloidal particles.

In this paper we use computer simulations to examine point defects in systems of "soft" colloidal particles including Hertzian spheres, and star polymers. We use Monte Carlo simulations to determine the deformation of the different crystals associated with vacancies and interstitials and use thermodynamic integration to predict the equilibrium concentrations of such defects. We find that the nature of the lattice distortion is mainly determined by the crystal structure and not by the specifics of the interaction potential. We can distinguish one-, two-, and three-dimensional lattice distortions and find that the range of the distortion generally depends on the dimensionality. We find that in both model systems the deformation of the body-centered cubic (BCC) crystal caused by an interstitial is one-dimensional and we show that its structure is well described as a crowdion. Similarly, we show that the one-dimensional deformation of the hexagonal (H) crystal of Hertzian spheres caused by a vacancy can be characterized as a voidion. Interestingly, with the exception of the FCC crystal in the Hertzian sphere model, in all cases we find that the interstitial concentration is higher than the vacancy concentration. Most noteworthy, the concentration of interstitials in the BCC crystals can reach up to 1%.

Point defects in crystals of charged colloids.

Charged colloidal particles-on both the nano and micron scales-have been instrumental in enhancing our understanding of both atomic and colloidal crystals. These systems can be straightforwardly realized in the lab and tuned to self-assemble into body-centered-cubic (BCC) and face-centered-cubic (FCC) crystals. While these crystals will always exhibit a finite number of point defects, including vacancies and interstitials-which can dramatically impact their material properties-their existence is usually ignored in scientific studies. Here, we use computer simulations and free-energy calculations to characterize vacancies and interstitials in FCC and BCC crystals of point-Yukawa particles. We show that, in the BCC phase, defects are surprisingly more common than in the FCC phase, and the interstitials manifest as so-called crowdions: an exotic one-dimensional defect proposed to exist in atomic BCC crystals. Our results open the door to directly observe these elusive defects in the lab.

Modeling of effective interactions between ligand coated nanoparticles through symmetry functions.

Ligand coated nanoparticles are complex objects consisting of a metallic or semiconductor core with organic ligands grafted on their surface. These organic ligands provide stability to a nanoparticle suspension. In solutions, the effective interactions between such nanoparticles are mediated through a complex interplay of interactions between the nanoparticle cores, the surrounding ligands, and the solvent molecules. While it is possible to compute these interactions using fully atomistic molecular simulations, such computations are too expensive for studying self-assembly of a large number of nanoparticles. The problem can be made tractable by removing the degrees of freedom associated with the ligand chains and solvent molecules and using the potentials of mean force (PMF) between nanoparticles. In general, the functional dependence of the PMF on the inter-particle distance is unknown and can be quite complex. In this article, we present a method to model the two-body and three-body PMF between ligand coated nanoparticles through a linear combination of symmetry functions. The method is quite general and can be extended to model interactions between different types of macromolecules.

Autonomously revealing hidden local structures in supercooled liquids.

Few questions in condensed matter science have proven as difficult to unravel as the interplay between structure and dynamics in supercooled liquids. To explore this link, much research has been devoted to pinpointing local structures and order parameters that correlate strongly with dynamics. Here we use an unsupervised machine learning algorithm to identify structural heterogeneities in three archetypical glass formers-without using any dynamical information. In each system, the unsupervised machine learning approach autonomously designs a purely structural order parameter within a single snapshot. Comparing the structural order parameter with the dynamics, we find strong correlations with the dynamical heterogeneities. Moreover, the structural characteristics linked to slow particles disappear further away from the glass transition. Our results demonstrate the power of machine learning techniques to detect structural patterns even in disordered systems, and provide a new way forward for unraveling the structural origins of the slow dynamics of glassy materials.

Predicting the phase behavior of mixtures of active spherical particles.

An important question in the field of active matter is whether or not it is possible to predict the phase behavior of these systems. Here, we study the phase coexistence of binary mixtures of torque-free active Brownian particles for both systems with purely repulsive interactions and systems with attractions. Using Brownian dynamics simulations, we show that phase coexistences can be predicted quantitatively for these systems by measuring the pressure and "reservoir densities." Specifically, in agreement with the previous literature, we find that the coexisting phases are in mechanical equilibrium, i.e., the two phases have the same pressure. Importantly, we also demonstrate that the coexisting phases are in chemical equilibrium by bringing each phase into contact with particle reservoirs and show that for each species, these reservoirs are characterized by the same density for both phases. Using this requirement of mechanical and chemical equilibrium, we accurately construct the phase boundaries from properties that can be measured purely from the individual coexisting phases. This result highlights that torque-free active Brownian systems follow simple coexistence rules, thus shedding new light on their thermodynamics.

High antisite defect concentrations in hard-sphere colloidal Laves phases.

Binary mixtures of hard spheres can spontaneously self-assemble into binary crystals. Computer simulations have been especially useful in mapping out the phase behaviour of these mixtures, under the assumption that the stoichiometry of the binary crystal is ideal. Here we show that for a size ratio of q = 0.82 this assumption is not valid near the coexistence region between the fluid and the stable binary crystal, the MgZn2 Laves phase. Instead we find a surprisingly high number of antisite defects: up to 2% of the large spheres are replaced by small spheres in equilibrium. We demonstrate that the defect concentration can be estimated using simple approximations, providing an easy way to identify systems where antisite defects play an important role. Our results shed new light on the self-assembly of colloidal Laves phases, and demonstrate the importance of antisite defects in binary crystals.

Modeling of many-body interactions between elastic spheres through symmetry functions.

Simple models for spherical particles with a soft shell have been shown to self-assemble into numerous crystal phases and even quasicrystals. However, most of these models rely on a simple pairwise interaction, which is usually a valid approximation only in the limit of small deformations, i.e., low densities. In this work, we consider a many-body yet simple model for the evaluation of the elastic energy associated with the deformation of a spherical shell. The resulting energy evaluation, however, is relatively expensive for direct use in simulations. We significantly reduce the associated numerical cost by fitting the potential using a set of symmetry functions. We propose a method for selecting a suitable set of symmetry functions that capture the most relevant features of the particle's environment in a systematic manner. The fitted interaction potential is then used in Monte Carlo simulations to draw the phase diagram of the system in two dimensions. The system is found to form both a fluid and a hexagonal crystal phase.

Unsupervised learning for local structure detection in colloidal systems.

We introduce a simple, fast, and easy to implement unsupervised learning algorithm for detecting different local environments on a single-particle level in colloidal systems. In this algorithm, we use a vector of standard bond-orientational order parameters to describe the local environment of each particle. We then use a neural-network-based autoencoder combined with Gaussian mixture models in order to autonomously group together similar environments. We test the performance of the method on snapshots of a wide variety of colloidal systems obtained via computer simulations, ranging from simple isotropically interacting systems to binary mixtures, and even anisotropic hard cubes. Additionally, we look at a variety of common self-assembled situations such as fluid-crystal and crystal-crystal coexistences, grain boundaries, and nucleation. In all cases, we are able to identify the relevant local environments to a similar precision as "standard," manually tuned, and system-specific, order parameters. In addition to classifying such environments, we also use the trained autoencoder in order to determine the most relevant bond orientational order parameters in the systems analyzed.

Cuboidal Supraparticles Self-Assembled from Cubic CsPbBr3 Perovskite Nanocrystals.

Colloidal CsPbBr3 nanocrystals (NCs) have emerged as promising candidates for various opto-electronic applications, such as light-emitting diodes, photodetectors, and solar cells. Here, we report on the self-assembly of cubic NCs from an organic suspension into ordered cuboidal supraparticles (SPs) and their structural and optical properties. Upon increasing the NC concentration or by addition of a nonsolvent, the formation of the SPs occurs homogeneously in the suspension, as monitored by in situ X-ray scattering measurements. The three-dimensional structure of the SPs was resolved through high-angle annular dark-field scanning transmission electron microscopy and electron tomography. The NCs are atomically aligned but not connected. We characterize NC vacancies on superlattice positions both in the bulk and on the surface of the SPs. The occurrence of localized atomic-type NC vacancies-instead of delocalized ones-indicates that NC-NC attractions are important in the assembly, as we verify with Monte Carlo simulations. Even when assembled in SPs, the NCs show bright emission, with a red shift of about 30 meV compared to NCs in suspension.

Diffusion and interactions of interstitials in hard-sphere interstitial solid solutions.

Using computer simulations, we study the dynamics and interactions of interstitial particles in hard-sphere interstitial solid solutions. We calculate the free-energy barriers associated with their diffusion for a range of size ratios and densities. By applying classical transition state theory to these free-energy barriers, we predict the diffusion coefficients, which we find to be in good agreement with diffusion coefficients as measured using event-driven molecular dynamics simulations. These results highlight that transition state theory can capture the interstitial dynamics in the hard-sphere model system. Additionally, we quantify the interactions between the interstitials. We find that, apart from excluded volume interactions, the interstitial-interstitial interactions are almost ideal in our system. Lastly, we show that the interstitial diffusivity can be inferred from the large-particle fluctuations alone, thus providing an empirical relationship between the large-particle fluctuations and the interstitial diffusivity.