Two-dimensional melting: from liquid-hexatic coexistence to continuous transitions

Influence of anomalous agents on the dynamics of an active system

Swarming behavior in systems of self-propelled particles, whether biological or artificial, has received increased attention in recent years. Here, we show that even a small number of particles with anomalous behavior can change dramatically collective dynamics of the swarming system and can impose unusual behavior and transitions between dynamic states. Our results pave the way to practical approaches and concepts of multiagent dynamics in groups of flocking animals: birds, insects, and fish, i.e., active and living soft matter.

Necessary and sufficient symmetries in Event-Chain Monte Carlo with generalized flows and application to hard dimers

Event-Chain Monte Carlo (ECMC) methods generate continuous-time and non-reversible Markov processes, which often display significant accelerations compared to their reversible counterparts. However, their generalization to any system may appear less straightforward. In this work, our aim is to distinctly define the essential symmetries that such ECMC algorithms must adhere to, differentiating between necessary and sufficient conditions. This exploration intends to delineate the balance between requirements that could be overly limiting in broad applications and those that are fundamentally essential. To do so, we build on the recent analytical description of such methods as generating piecewise deterministic Markov processes. Therefore, starting with translational flows, we establish the necessary rotational invariance of the probability flows, along with determining the minimum event rate. This rate is identified with the corresponding infinitesimal Metropolis rejection rate. Obeying such conditions ensures the correct invariance for any ECMC scheme. Subsequently, we extend these findings to encompass schemes involving deterministic flows that are more general than mere translational ones. Specifically, we define two classes of interest of general flows: the ideal and uniform-ideal ones. They, respectively, suppress or reduce the event rates. From there, we implement a comprehensive non-reversible sampling of a system of hard dimers by introducing rotational flows, which are uniform-ideal. This implementation results in a speed-up of up to ∼3 compared to the state-of-the-art ECMC/Metropolis hybrid scheme.

Nernst Sign Reversal in the Hexatic Vortex Phase of Weakly Disordered a-MoGe Thin Films

The hexatic phase is an intermediate stage in the melting process of a 2D crystal due to topological defects. Recently, this exotic phase was experimentally identified in the vortex lattice of 2D weakly disordered superconducting MoGe by scanning tunneling microscopic measurements. Here, we study this vortex state by the Nernst effect, which is an effective and sensitive tool to detect vortex motion, especially in the superconducting fluctuation regime. We find a surprising Nernst sign reversal at the melting transition of the hexatic phase. We propose that they are a consequence of vortex dislocations in the hexatic state which diffuse preferably from the cold to hot.

Two-Step Melting in a Bulk Crystal via Intermediate Metastable Liquid

The mechanism of melting is significant, as it links the structure and dynamics between crystal and liquid. In two dimensions, the crystal could first melt into a hexatic liquid before finally reaching a disordered liquid. However, such a hexatic liquid phase is unstable in three dimensions, and melting is recognized as a one-step process. Here we report a two-step melting process in a three-dimensional system, (S)-(+)-ibuprofen. The crystal melts through an indirect pathway that first transforms into an intermediate liquid phase exhibiting an extremely long lifetime followed by the transition to the ordinary liquid phase at a spinodal point with the occurrence of long-range fluctuations. Such observations suggest that the complexity of liquid could affect the transition pathway of melting. These results could lead us to hypothesize the existence of continuous melting in three dimensions.

Liquid-hexatic transition for soft disks

We study the liquid-hexatic transition of soft disks with massively parallel simulations and determine the equation of state as a function of system size. For systems with interactions decaying as the inverse mth power of the separation, the liquid-hexatic phase transition is continuous for m=12 and m=8, while it is of first order for m=24. The critical power m for the transition between continuous and first-order behavior is larger than previously reported. The continuous transition for m=12 implies that the two-dimensional Lennard-Jones model has a continuous liquid-hexatic transition at high temperatures. We also study the Weeks-Chandler-Andersen model and find a continuous transition at high temperatures that is consistent with the soft-disk case for m=12. Pressure data as well as our implementation are available from an open-source repository.

Two-Dimensional Melting of Two- and Three-Component Mixtures

We elucidate the interplay between diverse two-dimensional melting pathways and establish solid-hexatic and hexatic-liquid transition criteria via the numerical simulations of the melting transition of two- and three-component mixtures of hard polygons and disks. We show that a mixture's melting pathway may differ from its components and demonstrate eutectic mixtures that crystallize at a higher density than their pure components. Comparing the melting scenario of many two- and three-component mixtures, we establish universal melting criteria: the solid and hexatic phases become unstable as the density of topological defects, respectively, overcomes ρ_{d,s}≃0.046 and ρ_{d,h}≃0.123.

Topological Defect Guided Order Evolution across the Nematic-Smectic Phase Transition

Topological defects usually emerge and vary during the phase transition of ordered systems. Their roles in thermodynamic order evolution keep being the frontier of modern condensed matter physics. Here, we study the generations of topological defects and their guidance on the order evolution during the phase transition of liquid crystals (LCs). With a given preset photopatterned alignment, two different types of topological defects are achieved depending on the thermodynamic process. Because of the memory effect of LC director field across the Nematic-Smectic (N-S) phase transition, a stable array of toric focal conic domains (TFCDs) and a frustrated one are generated in S phase, respectively. The frustrated one transfers to a metastable TFCD array with a smaller lattice constant, and further changes to a crossed-walls type N state due to the inheritance of orientational order. A free energy on temperature diagram and corresponding textures vividly describe the phase transition process and the roles of topological defects in the order evolution across the N-S phase transition. This Letter reveals the behaviors and mechanisms of topological defects on order evolution during phase transitions. It paves a way for investigating topological defect guided order evolution which is ubiquitous in soft matter and other ordered systems.

Topological phase transition in a quasi-two-dimensional Coulomb gas

A system with an equal number of positive and negative charges confined in a box with a small but finite thickness is modeled as a function of temperature using mesoscale numerical simulations for various values of the charges. The Coulomb interaction is used in its three-dimensional form, U(r)∼1/r. A topological phase transition is found in this quasi-two-dimensional (2D) system. The translational order parameter, spatial correlation function, specific heat, and electric current show qualitatively different trends below and above a critical temperature. We find that a 2D logarithmic Coulomb interaction is not essential for the appearance of this transition. This work suggests new experimental tests of our predictions, as well as novel theoretical approaches to probe quasi-2D topological phase transitions.

Packing and emergence of the ordering of rods in a spherical monolayer

Spatially ordered systems confined to surfaces such as spheres exhibit interesting topological structures because of curvature induced frustration in orientational and translational order. The study of these structures is important for investigating the interplay between the geometry, topology, and elasticity, and for their potential applications in materials science, such as engineering directionally binding particles. In this work, we numerically simulate a spherical monolayer of soft repulsive spherocylinders (SRSs) and study the packing of rods and their ordering transition as a function of the packing fraction. In the model that we study, the centers of mass of the spherocylinders (situated at their geometric centers) are constrained to move on a spherical surface. The spherocylinders are free to rotate about any axis that passes through their respective centers of mass. We show that, up to moderate packing fractions, a two dimensional liquid crystalline phase is formed whose orientational ordering increases continuously with increasing density. This monolayer of orientationally ordered SRS particles at medium densities resembles a hedgehog-long axes of the SRS particles are aligned along the local normal to the sphere. At higher packing fractions, the system undergoes a transition to the solid phase, which is riddled with topological point defects (disclinations) and grain boundaries that divide the whole surface into several domains.

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.

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.

Freezing phase transition in hard-core lattice gases on the triangular lattice with exclusion up to seventh next-nearest neighbor

Hard-core lattice-gas models are minimal models to study entropy-driven phase transitions. In the k-nearest-neighbor lattice gas, a particle excludes all sites up to the kth next-nearest neighbors from being occupied by another particle. As k increases from one, it extrapolates from nearest-neighbor exclusion to the hard-sphere gas. In this paper we study the model on the triangular lattice for k≤7 using a flat histogram algorithm that includes cluster moves. Earlier studies focused on k≤3. We show that for 4≤k≤7, the system undergoes a single phase transition from a low-density fluid phase to a high-density sublattice-ordered phase. Using partition function zeros and nonconvexity properties of the entropy, we show that the transitions are discontinuous. The critical chemical potential, coexistence densities, and critical pressure are determined accurately.

Reentrant phase separation of a sparse collection of nonreciprocally aligning self-propelled disks

We study a model of aligning self-propelled disks that nonreciprocally reorient the self-propulsion directions along the interparticle separation and towards the other disks. In the limit of small inertia and large softness, where conventional motility-induced phase separation is absent, we demonstrate that the homogeneous system at a small area fraction phase-separates into clusters and a low-density phase that, eventually, reenters the homogeneous phase with a monotonic increase in alignment strength. The disks inside the clusters move with a finite space-dependent speed, constantly shuttling between clusters through the surrounding low-density homogeneous phase while maintaining the hexatic structure properties within the clusters. The area fraction gradually increases from the periphery towards the center of the clusters with a negligible correlation of the velocity and propulsion direction inside the clusters. The novel collective behavior of reentrant phase separation is found to follow from both the limits of hard disks and extremely small inertia, tending towards the overdamped limit. However, important differences in the structural and dynamical properties are shown in the limit of hard disks and extremely small inertia, as compared to that for soft disks at finite inertia. We show that the cluster phase is associated with an effective temperature for a wide range of values of alignment strength, whereas an effective temperature is associated with the specific range of alignment in the low-density phase. We believe that the reentrant phase behavior in the limit of small area fraction and the remarkable properties of the clusters should be useful in understanding a wide range of physics issues, ranging from clogging and unclogging to information exchange and transport, in biological and synthetic self-propelled systems.

Hydrodynamic effects on the liquid-hexatic transition of active colloids

We study numerically the role of hydrodynamics in the liquid-hexatic transition of active colloids at intermediate activity, where motility induced phase separation (MIPS) does not occur. We show that in the case of active Brownian particles (ABP), the critical density of the transition decreases upon increasing the particle's mass, enhancing ordering, while self-propulsion has the opposite effect in the activity regime considered. Active hydrodynamic particles (AHP), instead, undergo the liquid-hexatic transition at higher values of packing fraction [Formula: see text] than the corresponding ABP, suggesting that hydrodynamics have the net effect of disordering the system. At increasing densities, close to the hexatic-liquid transition, we found in the case of AHP the appearance of self-sustained organized motion with clusters of particles moving coherently.

Geometry-controlled phase transition in vibrated granular media

We report experiments on the dynamics of vibrated particles constrained in a two-dimensional vertical container, motivated by the following question: how to get the most out of a given external vibration to maximize internal disorder (e.g. to blend particles) and agitation (e.g. to absorb vibrations)? Granular media are analogs to classical thermodynamic systems, where the injection of energy can be achieved by shaking them: fluidization arises by tuning either the amplitude or the frequency of the oscillations. Alternatively, we explore what happens when another feature, the container geometry, is modified while keeping constant the energy injection. Our method consists in modifying the container base into a V-shape to break the symmetries of the inner particulate arrangement. The lattice contains a compact hexagonal solid-like crystalline phase coexisting with a loose amorphous fluid-like phase, at any thermal agitation. We show that both the solid-to-fluid volume fraction and the granular temperature depend not only on the external vibration but also on the number of topological defects triggered by the asymmetry of the container. The former relies on the statistics of the energy fluctuations and the latter is consistent with a two-dimensional melting transition described by the KTHNY theory.

Anomalous behavior of a two-dimensional Hertzian disk system

The anomalous behavior of a two-dimensional system of Hertzian disks with exponent α=7/2 has been studied using the method of molecular dynamics. The phase diagram of this system is the melting line of a triangular crystal with several maxima and minima. Waterlike density and diffusion anomalies have been found in the reentrant melting regions. Noteworthy, a density anomaly has been observed not only in the liquid and hexatic but also in the solid phase. The calculations of the phonon spectra of longitudinal and transverse modes have yielded negative dependence of the frequency of transverse modes on density along all directions in the regions with a density anomaly. This indicates an association of the density anomaly with transverse oscillations of the crystal lattice. The regions of density and diffusion anomalies have been drawn on the phase diagram. It has been found that the stability regions of anomalous diffusion extend to temperatures well above the maximum melting point T=0.0058 of the triangular crystal.

Emergent elastic fields induced by topological phase transitions: Impact of molecular chirality and steric anisotropy

Chirality, the property of an object that cannot be superimposed on its mirror image, plays an essential role in condensed matter, such as magnetic, electronic, and liquid crystal systems. Topological phases emerge in such chiral materials, wherein helical and vortex-like structures—called skyrmions—are observed. However, the role of elastic fields in these topological phases remains unexplored. Here, we construct a molecular model of two-dimensional crystals incorporating steric anisotropy and chiral interactions to elucidate this problem. The coupling between the elastic fields and phase transitions between uniform, helical, and half-skyrmion phases can be utilized to switch these topological phases by external forces. Our results provide a fundamental physical principle for designing topological materials using chiral molecular and colloidal crystals.

Topological phase transitions into skyrmion and half-skyrmion (meron) phases are widely observed in condensed matter, such as chiral magnets and liquid crystals. They are utilized to design magnetoelectric, optical, and mechanoresponsive materials by controlling such topological phases. However, the role of the elastic field in nonuniform topological phases is elusive, though the essential role of crystal elasticity in uniform ordered crystal phase has been recognized. To elucidate this problem, we construct a model describing chiral molecules and colloids in quasi-two-dimensional molecular crystals, which incorporates intermolecular chiral twisting and spheroidal steric interactions. We reveal that emergence of the elastic fields from the competition between steric anisotropy and intermolecular twisting is a key to control uniform, helical, and half-skyrmion structures. By utilizing the coupling between the spheroidal orientations and the elastic fields, these topological phases are switched by temperature, external electromagnetic fields, and anisotropic stresses, where a re-entrant phase transition between the helical and the half-skyrmion phases is discovered. Our results imply that controlling the emergent elastic fields is crucial for obtaining a fundamental physical principle for controlling topological phases using chiral molecular and colloidal crystals.

The role of attraction in the phase diagrams and melting scenarios of generalized 2D Lennard-Jones systems

Monolayer and two-dimensional (2D) systems exhibit rich phase behavior, compared with 3D systems, in particular, due to the hexatic phase playing a central role in melting scenarios. The attraction range is known to affect critical gas–liquid behavior (liquid–liquid in protein and colloidal systems), but the effect of attraction on melting in 2D systems remains unstudied systematically. Here, we have revealed how the attraction range affects the phase diagrams and melting scenarios in a 2D system. Using molecular dynamics simulations, we have considered the generalized Lennard-Jones system with a fixed repulsion branch and different power indices of attraction from long-range dipolar to short-range sticky-sphere-like. A drop in the attraction range has been found to reduce the temperature of the gas–liquid critical point, bringing it closer to the gas–liquid–solid triple point. At high temperatures, attraction does not affect the melting scenario that proceeds through the cascade of solid–hexatic (Berezinskii–Kosterlitz–Thouless) and hexatic–liquid (first-order) phase transitions. In the case of dipolar attraction, we have observed two triple points inherent in a 2D system: hexatic–liquid–gas and crystal–hexatic–gas, the temperature of the crystal–hexatic–gas triple point is below the hexatic–liquid–gas triple point. This observation may have far-reaching consequences for future studies, since phase diagrams determine possible routes of self-assembly in molecular, protein, and colloidal systems, whereas the attraction range can be adjusted with complex solvents and external electric or magnetic fields. The results obtained may be widely used in condensed matter, chemical physics, materials science, and soft matter.

Unified analysis of topological defects in 2D systems of active and passive disks

We provide a comprehensive quantitative analysis of localized and extended topological defects in the steady state of 2D passive and active repulsive Brownian disk systems. We show that, both in and out-of-equilibrium, the passage from the solid to the hexatic is driven by the unbinding of dislocations, in quantitative agreement with the KTHNY singularity. Instead, extended clusters of defects largely dominate below the solid-hexatic critical line. The latter percolate in the liquid phase very close to the hexatic-liquid transition, both for continuous and discontinuous transitions, in the homogeneous liquid regime. At critical percolation the clusters of defects are fractal with statistical and geometric properties that are independent of the activity and compatible with the universality class of uncorrelated critical percolation. We also characterize the spatial organization of point-like defects and we show that the disclinations are not free, but rather always very near more complex defect structures. At high activity, the bulk of the dense phase generated by Motility-Induced Phase Separation is characterized by a density of point-like defects, and statistics and morphology of defect clusters, set by the amount of activity and not the packing fraction. Hexatic domains within the dense phase are separated by grain-boundaries along which a finite network of topological defects resides, interrupted by gas bubbles in cavitation. This structure is dynamic in the sense that the defect network allows for an unzipping mechanism that leaves free space for gas bubbles to appear, close, and even be released into the dilute phase.

Structural defects of monolayer wet particles during melting under vertical vibration

The evolution of structural defects during phase transition is of great significance to the understanding of the mechanism of solid-liquid transition. However, the current research on topological defects still uses the pair-correlation function and the orientational order correlation function, so it is difficult to quantify the detailed changes of structural defects locally. In this paper, the local volume fraction is proposed as a key parameter to accurately quantify the variation of structural defects. The experimental results indicate that the evolution of structural defects in the particle system is caused by the decrease of local volume fraction, so the critical value of phase transition could be determined by the minimum local volume fraction ϕ_{min}. Furthermore, according to the evolution law of structural defects, it can be deduced that the phase transition is continuous, which is consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young theory. Therefore, the quantitative analysis of structural defects by using local volume fraction can help make the mechanism of solid-liquid phase transformation clearer.

Topographic Control of Order in Quasi-2D Granular Phase Transitions

We experimentally investigate the nature of 2D phase transitions in a quasi-2D granular fluid. Using a surface decorated with periodically spaced dimples we observe interfacial tension between coexisting granular liquid and crystal phases. Measurements of the orientational and translational order parameters and associated susceptibilities indicate that the surface topography alters the order of the phase transition from a two-step continuous one to a first-order liquid-solid one. The interplay of boundary inelasticity and geometry, either order promoting or inhibiting, controls whether it is the granular crystal or the granular fluid which makes contact with the edge. This order induced wetting has important consequences, determining how coexisting phases separate spatially.

Hard core lattice gas with third next-nearest neighbor exclusion on triangular lattice: One or two phase transitions?

We obtain the phase diagram of the hard core lattice gas with third nearest neighbor exclusion on the triangular lattice using Monte Carlo simulations that are based on a rejection-free flat histogram algorithm. In a recent paper [Darjani et al., J. Chem. Phys. 151, 104702 (2019)], it was claimed that the lattice gas with third nearest neighbor exclusion undergoes two phase transitions with increasing density with the phase at intermediate densities exhibiting hexatic order with continuously varying exponents. Although a hexatic phase is expected when the exclusion range is large, it has not been seen earlier in hard core lattice gases with short range exclusion. In this paper, by numerically determining the entropies for all densities, we show that there is only a single phase transition in the system between a low-density fluid phase and a high density ordered sublattice phase and that a hexatic phase is absent. The transition is shown to be first order in nature, and the critical parameters are determined accurately.

Data-driven criterion for the solid-liquid transition of two-dimensional self-propelled colloidal particles far from equilibrium

We establish an explicit data-driven criterion for identifying the solid-liquid transition of two-dimensional self-propelled colloidal particles in the far from equilibrium parameter regime, where the transition points predicted by different conventional empirical criteria for melting and freezing diverge. This is achieved by applying a hybrid machine learning approach that combines unsupervised learning with supervised learning to analyze a huge amount of the system's configurations in the nonequilibrium parameter regime on an equal footing. Furthermore, we establish a generic data-driven evaluation function, according to which the performance of different empirical criteria can be systematically evaluated and improved. In particular, by applying this evaluation function, we identify a new nonequilibrium threshold value for the long-time diffusion coefficient, based on which the predictions of the corresponding empirical criterion are greatly improved in the far from equilibrium parameter regime. These data-driven approaches provide a generic tool for investigating phase transitions in complex systems where conventional empirical ones face difficulties.

Emergence and melting of active vortex crystals

Melting of two-dimensional (2D) equilibrium crystals is a complex phenomenon characterized by the sequential loss of positional and orientational order. In contrast to passive systems, active crystals can self-assemble and melt into an active fluid by virtue of their intrinsic motility and inherent non-equilibrium stresses. Currently, the non-equilibrium physics of active crystallization and melting processes is not well understood. Here, we establish the emergence and investigate the melting of self-organized vortex crystals in 2D active fluids using a generalized Toner-Tu theory. Performing extensive hydrodynamic simulations, we find rich transition scenarios. On small domains, we identify a hysteretic transition as well as a transition featuring temporal coexistence of active vortex lattices and active turbulence, both of which can be controlled by self-propulsion and active stresses. On large domains, an active vortex crystal with solid order forms within the parameter range corresponding to active vortex lattices. The melting of this crystal proceeds through an intermediate hexatic phase. Generally, these results highlight the differences and similarities between crystalline phases in active fluids and their equilibrium counterparts.

Structural transition in two-dimensional Hertzian spheres in the presence of random pinning

Using molecular dynamics simulation we have investigated the influence of random pinning on the phase diagram and melting scenarios of a two-dimensional system with the Hertz potential for α=5/2. It has been shown that random pinning can cardinally change the mechanism of first-order transition between the different crystalline phases (triangular and square) by virtue of generating hexatic and tetratic phases: a triangular crystal to hexatic transition is of the continuous Berezinskii-Kosterlitz-Thouless (BKT) type, a hexatic to tetratic transition is of first order, and finally, there is a continuous BKT-type transition from tetratic to the square crystal.

Reverse order-disorder transition of Janus particles confined in two dimensions

Janus particles with different patch sizes, confined to two dimensions, generate a series of patterns of interest to the field of nanoscience. Here we observe reverse melting, where for some densities the system melts under cooling. For a broad range of hydrophobic patch sizes (60^{∘}<θ_{0}<90^{∘}), a reentrant transition from solid to liquid and then to an ordered phase emerges as temperature (T) decreases due to the formation of rhombus chains at low T. This reentrant phase has pseudo long-range orientational order but short-range translational order, similar to a hexatic phase. Our work provides guidelines to study the melting and assembly of Janus particles in two dimensions, as well as mechanisms to generate phases with specific symmetry.

Collective excitations in active fluids: Microflows and breakdown in spectral equipartition of kinetic energy

The effect of particle activity on collective excitations in active fluids of microflyers is studied. With an in silico study, we observed an oscillating breakdown of equipartition (uniform spectral distribution) of kinetic energy in reciprocal space. The phenomenon is related to short-range velocity-velocity correlations that were realized without forming of long-lived mesoscale vortices in the system. This stands in contrast to well-known mesoscale turbulence operating in active nematic systems (bacterial or artificial) and reveals the features of collective dynamics in active fluids, which should be important for structural transitions and glassy dynamics in active matter.

Influence of interaction softness on phase separation of active particles

Using a minimal model of active Brownian particles, we study the effect of a crucial parameter, namely the softness of the interparticle repulsion, on motility-induced phase separation. We show that an increase in particle softness reduces the ability of the system to phase separate and the system exhibits a delayed transition. After phase separation, the system state properties can be explained by a single relevant length scale, the effective interparticle distance. We estimate this length scale analytically and use it to rescale the state properties at dense phase for systems with different interaction softness. Using this length scale, we provide a scaling relation for the time taken to phase separate which shows a high sensitivity to the interaction softness.

Elongation and percolation of defect motifs in anisotropic packing problems

We examine the regime between crystalline and amorphous packings of anisotropic objects on surfaces of different genus by continuously varying their size distribution or shape from monodispersed spheres to bidispersed mixtures or monodispersed ellipsoidal particles; we also consider an anisotropic variant of the Thomson problem with a mixture of charges. With increasing anisotropy, we first observe the disruption of translational order with an intermediate orientationally ordered hexatic phase as proposed by Nelson, Rubinstein and Spaepen, and then a transition to amorphous state. By analyzing the structure of the disclination motifs induced, we show that the hexatic-amorphous transition is caused by the growth and connection of disclination grain boundaries, suggesting this transition lies in the percolation universality class in the scenarios considered.

Signatures of continuous hexatic-liquid transition in two-dimensional melting

Recent studies have shown that the melting of two-dimensional crystals can be either continuous or discontinuous, relying on multiple parameters such as particle stiffness, density, and particle size dispersity. However, what determines the continuity or discontinuity of the two-dimensional melting remains elusive. Here we study the two-dimensional melting of binary mixtures of soft-core particles. The two particle species are different in either particle size or particle stiffness. Starting with the mono-component systems which exhibit discontinuous hexatic-liquid transition, we gradually increase the particle size or stiffness dispersity and find that the hexatic-liquid coexistent region shrinks and eventually vanishes above a critical dispersity. Therefore, the growth of disorder caused by the particle size or stiffness dispersity leads to the discontinuous-continuous transition of the two-dimensional melting. We further find that as long as the melting is continuous the defect concentrations on the boundary between hexatic and liquid phases remain almost constant, accompanied by an almost constant correlation length. These characteristic defect concentrations and correlation length are universal and independent of particle interactions, temperature, and type of particle dispersity, which act as signatures of the continuous two-dimensional melting.

Two-step melting of the Weeks-Chandler-Anderson system in two dimensions

We present a detailed numerical simulation study of a two-dimensional system of particles interacting via the Weeks-Chandler-Anderson potential, the repulsive part of the Lennard-Jones potential. With the reduction of density, the system shows a two-step melting: a continuous melting of solid to hexatic phase, followed by a first-order melting of hexatic to liquid. The solid-hexatic melting is consistent with the Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young (BKTHNY) scenario and shows dislocation unbinding. The first-order melting of the hexatic to the liquid phase, on the other hand, displays defect-strings formed at the hexatic-liquid interfaces. We present a detailed phase diagram in the density-temperature plane.

Surface roughening, premelting and melting of monolayer and bilayer crystals

Dimensionality often strongly affects material properties and phase transition behaviors, but its effects on crystal surfaces, such as roughening and premelting, have been poorly studied. Our simulation revealed that these surface behaviors are distinct in monolayer and multilayer Lennard-Jones (LJ) crystals. Solid surfaces fluctuate as capillary waves during the roughening process, but complete roughening is preempted by premelting. As the melting temperature is approached, the thickness of the premelted liquid layer approaches a constant (i.e., blocked premelting) for monolayer crystals, but diverges as a power law (i.e., complete premelting) for bilayer and trilayer crystals. The surface liquids of monolayer crystals contain crystalline patches and exhibits rough liquid-vapour and liquid-crystal interfaces, in contrast to the normal surface liquids of bilayer and trilayer crystals. Monolayer crystals melt heterogeneously from the surface without forming a hexatic phase and produce many vacancies.

Laser-induced melting of two-dimensional dusty plasma system in RF discharge

We present a detailed analysis of experimental study, which shows clear evidence of a two-stage melting process of a quasi-two-dimensional dusty plasma system in a high-frequency gas discharge. We accurately calculated global parameters of the orientational and translational order, as well as their susceptibilities to determine two critical points, related to “solid-to-hexatic” and “hexatic-to-liquid” phase transitions. The nature of the emerging defects and changes in their mutual concentration, in addition to the estimate of core energy of free dislocations also counts in favor of the formation of an intermediate hexatic phase. These results are fully consistent with the Berezinsky–Kosterlitz–Thouless theory.

Phase behavior of Lennard-Jones particles in two dimensions

The phase diagram of the prototypical two-dimensional Lennard-Jones (LJ) system, while extensively investigated, is still debated. In particular, there are controversial results in the literature with regard to the existence of the hexatic phase and the melting scenario. Here we study the phase behavior of two-dimensional range-limited LJ particles via large-scale numerical simulations. We demonstrate that at a high temperature, when the attraction in the potential plays a minor role, melting occurs via a continuous solid-hexatic transition followed by a first-order hexatic-fluid transition. The hexatic phase occurs in a density range that vanishes as the temperature decreases so that at low-temperature melting occurs via a first-order liquid-solid transition. The temperature where the hexatic phase disappears is well above the liquid-gas critical temperature. The evolution of the density of topological defects confirms this scenario.

Three-step melting of hard superdisks in two dimensions

We explore the link between the melting scenarios of two-dimensional systems of hard disks and squares through replica-exchange Monte Carlo simulations of hard superdisks. The well-known melting scenarios are observed in the disk and square limits, while we observe an unusual three-step scenario for dual shapes. We find that two mesophases mediate the melting: a hexatic phase and another fluid phase with a D_{2} local symmetry, we call it rhombatic, where both bond and particle orientational orders are quasi-long-range. Our results show that not only can the melting process of liquid-crystal forming molecules be complicated, where elongated shapes stabilize several mesophases, but also that of anisotropic quasispherical molecules.

Effect of particle size distribution on polydisperse hard disks

Using Monte Carlo simulations, we systematically investigate the effect of particle size distribution on the phase behavior of polydisperse hard disks. Compared with the commonly used Gaussian-like polydisperse hard disks [P. Sampedro Ruiz, Q.-l. Lei, and R. Ni, Commun. Phys. 2, 70 (2019)], we find that the phase behavior of polydisperse hard-disk systems with lognormal and triangle distributions is significantly different. In polydisperse hard-disk systems of lognormal distributions, although the phase diagram appears similar to that of Gaussian-like polydisperse hard disks, the re-entrant melting of the hexatic or solid phase cannot be observed in sedimentation experiments. For polydisperse hard-disk systems of triangle distributions, the phase behavior is qualitatively different from the Gaussian-like and lognormal distributions, and we cannot reach any system of true polydispersity larger than 0.06, which is due to the special shape of the triangle distribution. Our results suggest that the exact particle size distribution is of primary importance in determining the phase behavior of polydisperse hard disks, and we do not have a universal phase diagram for different polydisperse hard-disk systems.

Colloidal assembly in droplets: structures and optical properties

Colloidal assembly in emulsion drops provides fundamental tools for studying optimum particle arrangement under spherical confinement and practical means for producing photonic microparticles. Recent progress has revealed that energetically favored cluster configurations are different from conventional supraballs, which could enhance optical performance. This paper reviews state-of-the-art emulsion-templated colloidal clusters, and particularly focuses on recently reported novel structures such as icosahedral, decahedral, and single-crystalline face-centered cubic (fcc) clusters. We classify the clusters according to the number of component particles as small (N < O(10²)), medium (O(10²) ≤N≤O(10⁴)), and large (N≥O(10⁵)). For each size of clusters, we discuss the detailed structures, mechanisms of cluster formation, and optical properties and potential applications. Finally, we outline current challenges and questions that require further investigation.

Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions

The creation of disordered hyperuniform materials with extraordinary optical properties (e.g., large complete photonic band gaps) requires a capacity to synthesize large samples that are effectively hyperuniform down to the nanoscale. Motivated by this challenge, we propose a feasible equilibrium fabrication protocol using binary paramagnetic colloidal particles confined in a 2D plane. The strong and long-ranged dipolar interaction induced by a tunable magnetic field is free from screening effects that attenuate long-ranged electrostatic interactions in charged colloidal systems. Specifically, we numerically find a family of optimal size ratios that makes the two-phase system effectively hyperuniform. We show that hyperuniformity is a general consequence of low isothermal compressibilities, which makes our protocol suitable to treat more general systems with other long-ranged interactions, dimensionalities, and/or polydispersity. Our methodology paves the way to synthesize large photonic hyperuniform materials that function in the visible to infrared range and hence may accelerate the discovery of novel photonic materials.

Otherwise identical particles with differing, fixed speeds demix under time-reversible dynamics

In recent years situations where elsewise identical particles demix when different degrees of freedom do not thermalize have become a research focus in nonequilibrium statistical mechanics. The majority of these models are formulated in the context of active particles, but the phenomenon also occurs for particles without driving. All the models studied so far share the property that they do not obey microscopic reversibility, and it may be thought that this is a necessary condition for such demixing to occur. We show here that such a demixing transition also occurs in a mixture of otherwise identical particles moving at two fixed but different speeds according to a time-reversible quasi-Newtonian dynamics. The mechanical instability underlying this behavior is generated by the lack of thermalization between the two subsystems, which is shared by all systems showing this behavior.

Solid-Liquid Transition of Deformable and Overlapping Active Particles

Experiments and theory have shown that cell monolayers and epithelial tissues exhibit solid-liquid and glass-liquid transitions. These transitions are biologically relevant to our understanding of embryonic development, wound healing, and cancer. Current models of confluent epithelia have focused on the role of cell shape, with less attention paid to cell extrusion, which is key for maintaining homeostasis in biological tissue. Here, we use a multiphase field model to study the solid-liquid transition in a confluent monolayer of deformable cells. Cell overlap is allowed and provides a way for modeling the precursor for extrusion. When cells overlap rather than deform, we find that the melting transition changes from continuous to first order like, and that there is an intermittent regime close to the transition, where solid and liquid states alternate over time. By studying the dynamics of five- and sevenfold disclinations in the hexagonal lattice formed by the cell centers, we observe that these correlate with spatial fluctuations in the cellular overlap, and that cell extrusion tends to initiate near fivefold disclinations.

Dense hard disk ordering: influence of bidispersity and quenched disorder

Using Monte Carlo simulations, the impact on structural ordering in two-dimensional systems via the interplay of size bidispersity and quenched disorder in the form of an externally applied spatially random potential, is studied for a system of hard disks. By scanning across a wide range of dense packing fractions, size ratios and roughness of the applied potential, the phase diagram is constructed, which demonstrates that both quenched and size disorders shift the onset of translational order to higher packings, while maintaining the presence of the intermediate hexatic phase. At larger disorder strengths, the signatures of structural order are absent within the range of investigated packing fractions. Further, the dynamics with increasing potential strength is analysed for the mono-component system to obtain a spatio-temporal description of the melting process. Finally, the influence of the externally rough field on the Mermin-Wagner fluctuations, characteristic to two-dimensional systems, is investigated.

Phase behavior of a two-dimensional core-softened system: new physical insights

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 2014 Soft Matter 10 4966). 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 2018 Soft Matter 14 2152). 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.

Attraction Tames Two-Dimensional Melting: From Continuous to Discontinuous Transitions

Two-dimensional systems may admit a hexatic phase and hexatic-liquid transitions of different natures. The determination of their phase diagrams proved challenging, and indeed, those of hard disks, hard regular polygons, and inverse power-law potentials have only recently been clarified. In this context, the role of attractive forces is currently speculative, despite their prevalence at both the molecular and colloidal scale. Here, we demonstrate, via numerical simulations, that attraction promotes a discontinuous melting scenario with no hexatic phase. At high-temperature, Lennard-Jones particles and attractive polygons follow the shape-dominated melting scenario observed in hard disks and hard polygons, respectively. Conversely, all systems melt via a first-order transition with no hexatic phase at low temperature, where attractive forces dominate. The intermediate temperature melting scenario is shape dependent. Our results suggest that, in colloidal experiments, the tunability of the strength of the attractive forces allows for the observation of different melting scenarios in the same system.