Intermediate Hexatic Phase Research Articles

Melting and freezing of a skyrmion lattice

We report comprehensive Monte-Carlo studies of the melting of skyrmion lattices (SkL) in systems of small, medium, and large sizes with the number of skyrmions ranging from 103to over 105. Large systems exhibit hysteresis similar to that observed in real experiments on the melting of SkLs. For sufficiently small systems which achieve thermal equilibrium, a fully reversible sharp solid-liquid transition on temperature with no intermediate hexatic phase is observed. A similar behavior is found on changing the magnetic field that provides the control of pressure in the SkL. We find that on heating the melting transition occurs via a formation of grains with different orientations of hexagonal axes. On cooling, the fluctuating grains coalesce into larger clusters until a uniform orientation of hexagonal axes is slowly established. The observed scenario is caused by collective effects involving defects and is more complex than a simple picture of a transition driven by the unbinding and annihilation of dislocation and disclination pairs.

The influence of Gaussian pinning on the melting scenario of a two-dimensional soft-disk system: First-order versus continuous transition

Two-dimensional systems are realized experimentally as thin layers on a substrate. The substrate can have some imperfections (defects of the crystalline structure, chemical impurities, etc.) with strong interactions with particles within and near the defects. Such randomly distributed centers of strong interactions are called ”pinning centers”. The presence of random pinning can substantially change the behavior of the system. It not only shifts the melting point of the system, but can also change the melting scenario itself. In the present paper the influence of Gaussian pinning on the melting scenario of a two-dimensional system of soft disks is studied by means of molecular dynamics simulation. We randomly introduce into the system of soft disks a set of “pinning centers” which attract the particles via the Gauss potential. We observe that increasing the depth of a Gaussian well leads to a change in the melting scenario of the system: from melting occurring via a first-order liquid-hexatic transition followed by a continuous hexatic-crystal transition to two continuous transitions through an intermediate hexatic phase. We show that there are still two transition stages for strong disorder, but the melting scenario is qualitatively different from that for weak disorder. The results demonstrate that the simple kind of quenched disorder can significantly affect the melting scenario of two-dimensional systems and offer the possibility of their application for the interpretation of complex experiments.

Electronic topology with bound defect charges promotes intermediate hexatic phase in two-dimensional melting

Electronic topology with bound defect charges promotes intermediate hexatic phase in two-dimensional melting

Observation of two-step melting on a sphere

Melting in two-dimensional flat space is typically two-step and via the hexatic phase. How melting proceeds on a curved surface, however, is not known. Topology mandates that crystalline particle assemblies on these surfaces harbor a finite density of defects, which itself can be ordered, like the icosahedral ordering of 5-coordinated disclination defects on a sphere. Thus, melting even on a sphere, the simplest closed surface, involves the loss of both crystalline and defect order. Probing the interplay of these two forms of order, however, requires a system in which melting can be performed in situ, and this has not been achieved hitherto. Here, by tuning interparticle interactions in situ, we report an observation of an intermediate hexatic phase during the melting of colloidal crystals on a sphere. Remarkably, we observed a precipitous drop in icosahedral defect order in the hexatic phase where the shear modulus is expected to vanish. Furthermore, unlike in flat space, where disorder can fundamentally alter the nature of the melting process, on the sphere, we observed the signature characteristics of ideal melting. Our findings have profound implications for understanding, for instance, the self-assembly and maturation dynamics of viral capsids and also phase transitions on curved surfaces.

Scanning-probe and information-concealing machine learning intermediate hexatic phase and critical scaling of solid-hexatic phase transition in deformable particles

We investigate the two-dimensional melting of deformable polymeric particles with multi-body interactions described by the Voronoi model. We report machine learning evidence for the existence of the intermediate hexatic phase in this system, and extract the critical exponent for the divergence of the correlation length of the associated solid-hexatic phase transition. Moreover, we clarify the discontinuous nature of the hexatic-liquid phase transition in this system. These findings are achieved by directly analyzing system's spatial configurations with two generic machine learning approaches developed in this work, dubbed “scanning-probe” via which the possible existence of intermediate phases can be efficiently detected, and “information-concealing” via which the critical scaling of the correlation length in the vicinity of generic continuous phase transition can be extracted. Our work provides new physical insights into the fundamental nature of the two-dimensional melting of deformable particles, and establishes a new type of generic toolbox to investigate fundamental properties of phase transitions in various complex systems.

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.

Nearly hyperuniform, nonhyperuniform, and antihyperuniform density fluctuations in two-dimensional transition metal dichalcogenides with defects

Hyperuniform many-body systems in $d$-dimensional Euclidean space are characterized by completely suppressed (normalized) infinite-wavelength density fluctuations, and appear to be endowed with novel exotic physical properties. In this work, we analyze the effects of localized defects on the density fluctuations across length scales and on the hyperuniformity property of experimental samples of two-dimensional transition metal dichalcogenides. In particular, we extract atomic coordinates from time series annular dark field-scanning transmission electron microscopy imaging data of 2D tungsten chalcogenides with the 2H structure showing continuous development and evolution of electron-beam induced defects, and construct the corresponding chemical-bonding informed coordination networks between the atoms. We then compute a variety of pair statistics and bond-orientational statistics to characterize the samples. At low defect concentrations, the corresponding materials are nearly hyperuniform. As more defects are introduced, the (approximate) hyperuniformity of the materials is gradually destroyed, and the system becomes non-hyperuniform even when the material still contains a significant amount of crystalline regions. At high defect concentrations, the structures become antihyperuniform with diverging (normalized) large-scale density fluctuations. Overall, the defected materials possess varying degrees of orientation order, and there is apparently no intermediate hexatic phase emerging. We also construct a minimalist structural model and demonstrate that the experimental samples can be essentially viewed as perturbed honeycomb crystals with small correlated displacements and double chalcogen 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.

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.

Hexatic phase in a model of active biological tissues.

In many biological processes, such as wound healing, cell tissues undergo an epithelial-to-mesenchymal transition, which is a transition from a more rigid to a more fluid state. Here, we investigate the solid/fluid transition of cell tissues within the framework of the self-propelled Voronoi model, which accounts for the deformability of the cells, for their many-body interactions, and for their polarized motility. The transition is controlled by two parameters, respectively accounting for the strength of the self-propelling force of the cells, and for the mechanical rigidity of the cells. We find the melting transition to occur via a continuous solid-hexatic transition followed by a continuous hexatic-liquid transition, as in the Kosterlitz, Thouless, Halperin, Nelson, and Young scenario. This finding indicates that the hexatic phase may have an unexpected biological relevance.

Thermally Driven Order-Disorder Transition in Two-Dimensional Soft Cellular Systems.

Many systems, including biological tissues and foams, are made of highly packed units having high deformability but low compressibility. At two dimensions, these systems offer natural tesselations of a plane with fixed density, in which transitions from ordered to disordered patterns are often observed, in both directions. Using a modified cellular Potts model algorithm that allows rapid thermalization of extensive systems, we numerically explore the order-disorder transition of monodisperse, two-dimensional cellular systems driven by thermal agitation. We show that the transition follows most of the predictions of Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory developed for melting of 2D solids, extending the validity of this theory to systems with many-body interactions. In particular, we show the existence of an intermediate hexatic phase, which preserves the orientational order of the regular hexagonal tiling but loses its positional order. In addition to shedding light on the structural changes observed in experimental systems, our study shows that soft cellular systems offer macroscopic systems in which the KTHNY melting scenario can be explored, in the continuation of Bragg's experiments on bubble rafts.

A kinetic-Monte Carlo perspective on active matter.

We study non-equilibrium phases for interacting two-dimensional self-propelled particles with isotropic pair-wise interactions using a persistent kinetic Monte Carlo approach. We establish the quantitative phase diagram, including the motility-induced phase separation (MIPS) that is a commonly observed collective phenomenon in active matter. In addition, we demonstrate for several different potential forms the presence of two-step melting, with an intermediate hexatic phase, in regions far from equilibrium. Increased activity can melt a two-dimensional solid and the melting lines remain disjoint from MIPS. We establish this phase diagram for a range of the inter-particle potential stiffnesses and identify the MIPS phase even in the hard-disk limit. We establish that the full description of the phase behavior requires three independent control parameters.

Direct Liquid to Crystal Transition in a Quasi-Two-Dimensional Colloidal Membrane

Using synchrotron-based small-angle X-ray scattering, we study rigid fd viruses assembled into isolated monolayers from mixtures with a nonabsorbing polymer, which acts as an osmotic agent. As the polymer concentration increases, we observe a direct liquid to crystal transition, without an intermediate hexatic phase, in contrast with many other similar systems, such as concentrated DNA phases or packings of surfactant micelles. We tentatively attribute this effect to the difference in stiffness. The liquid phase can be well described by a hard-disk fluid, while we model the crystalline one as a hexagonal harmonic lattice and we evaluate its elastic constants.

The phase diagram and melting scenarios of two-dimensional Hertzian spheres

ABSTRACTWe present computer simulations of a system of purely repulsive soft colloidal particles interacting via the Hertz potential and constrained to a two-dimensional plane. This potential describes the elastic interaction of weakly deformable bodies and can be a reliable model for a qualitative description of behaviour of soft macromolecules, such as globular micelles and star polymers. We have found a large number of ordered phases, including dodecagonal quasicrystal, and analysed the melting scenarios of low-density triangle and square phases. It is interesting that depending on the position on the phase diagram, the system can melt both through the first-order transition and in accordance with the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young scenario (two continuous transitions with an intermediate hexatic phase) as well as according to recently proposed two-stage melting with the first-order hexatic–isotropic liquid transition and continuous solid–hexatic transition. We have also demonstrated the possibility of a tricritical point on the melting line and have found the line of a water-like density anomaly on the phase diagram.

Berezinskii–Kosterlitz–Thouless transition and two-dimensional melting**This review is an extended version of the report “Old and new in the physics of phase transitions” presented at the scientific session of the Physical Sciences Division of the Russian Academy of Sciences on 21 December 2016 (see Phys. Usp. 60 948–957 (2017); Usp. Fiz. Nauk 187 1021 (2017)). (Editor’s note.)

The main aspects of the theory of phase transitions in two-dimensional degenerate systems (Berezinskii–Kosterlitz–Thouless, or BKT, transitions) are reviewed in detail, including the transition mechanism, the renormalization group as a tool for describing the transition, and how the transition scenario can possibly depend on the core energy of topological defects (in particular, in thin superconducting films). Various melting scenarios in two-dimensional systems are analyzed, and the current status of actual experiments and computer simulations in the field is examined. Whereas in three dimensions melting always occurs as a single first-order transition, in two dimensions, as shown by Halperin, Nelson, and Young, melting via two continuous BKT transitions with an intermediate hexatic phase characterized by quasi-long-range orientational order is possible. But there is also a possibility for a first-order phase transition to occur. Recently, one further melting scenario, different from that occurring in the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young theory, has been proposed, according to which a solid can melt in two stages: a continuous BKT-type solid–hexatic transition and then a first-order hexatic-phase–isotropic-liquid phase transition. Particular attention is given to the melting scenario as a function of the potential shape and to the random pinning effect on two-dimensional melting. In particular, it is shown that random pinning can alter the melting scenario fundamentally in the case of a first-order transition. Also considered is the melting of systems with potentials having a negative curvature in the repulsion region–potentials that are successfully used in describing the anomalous properties of water in two dimensions.

Random pinning elucidates the nature of melting transition in two-dimensional core-softened potential system

Despite about forty years of investigations, the nature of the melting transition in two dimensions is not completely clear. In the framework of the most popular Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young (BKTHNY) theory, 2D systems melt through two continuous Berezinskii–Kosterlitz–Thouless (BKT) transitions with intermediate hexatic phase. The conventional first-order transition is also possible. On the other hand, recently on the basis of computer simulations the new melting scenario was proposed with continuous BKT type solid–hexatic transition and first order hexatic–liquid transition. However, in the simulations the hexatic phase is extremely narrow that makes its study difficult. In the present paper, we propose to apply the random pinning to investigate the hexatic phase in more detail. The results of molecular dynamics simulations of two dimensional system having core-softened potentials with narrow repulsive step which is similar to the soft disk system are outlined. The system has a small fraction of pinned particles giving quenched disorder. Random pinning widens the hexatic phase without changing the melting scenario and gives the possibility to study the behavior of the diffusivity and order parameters in the vicinity of the melting transition and inside the hexatic phase.

Continuous melting through a hexatic phase in confined bilayer water.

Liquid water is not only of obvious importance but also extremely intriguing, displaying many anomalies that still challenge our understanding of such an a priori simple system. The same is true when looking at nanoconfined water: The liquid between constituents in a cell is confined to such dimensions, and there is already evidence that such water can behave very differently from its bulk counterpart. A striking finding has been reported from computer simulations for two-dimensionally confined water: The liquid displays continuous or discontinuous melting depending on its density. In order to understand this behavior, we have analyzed the melting exhibited by a bilayer of nanoconfined water by means of molecular dynamics simulations. At high density we observe the continuous melting to be related to the phase change of the oxygens only, with the hydrogens remaining liquidlike throughout. Moreover, we find an intermediate hexatic phase for the oxygens between the liquid and a triangular solid ice phase, following the Kosterlitz-Thouless-Halperin-Nelson-Young theory for two-dimensional melting. The liquid itself tends to maintain the local structure of the triangular ice, with its two layers being strongly correlated yet with very slow exchange of matter. The decoupling in the behavior of the oxygens and hydrogens gives rise to a regime in which the complexity of water seems to disappear, resulting in what resembles a simple monoatomic liquid. This intrinsic tendency of our simulated water may be useful for understanding novel behaviors in other confined and interfacial water systems.

Direct observation of melting in a two-dimensional driven granular system.

Melting is considered to be one of the most fundamental problems in physical science. Generally, dimensionality plays an important role in melting. In three-dimension, it’s well known that a crystal melts directly into a liquid via a first-order transition. In two-dimension (2D), however, the melting process has been widely debated whether it is a first-order transition or a two-step transition with an intermediate hexatic phase. Experimentally 2D melting has been intensively studied in equilibrium systems such as molecular and colloidal crystals, but rarely been explored in non-equilibrium system such as granular materials. In this paper, we experimentally studied the 2D melting in a driven granular model system at single particle level using video recording and particle tracking techniques. Measurements of orientational/translational correlation functions show evidences that the melting is a two-step transition. A novel concept of orientational/translational susceptibilities enable us to clearly resolve the intermediate hexatic phase. Our results are in excellent agreement with the two-step melting scenario predicted by KTHNY theory, and demonstrate that the KTHNY melting scenario can be extended to non-equilibrium systems.

Random pinning changes the melting scenario of a two-dimensional core-softened potential system.

In experiments two-dimensional systems are realized mainly on solid substrates, which introduce quenched disorder due to some inherent defects. The defects of substrates influence the melting scenario of the systems and have to be taken into account in the interpretation of experimental results. We present the results of molecular dynamics simulations of a two-dimensional system with a core-softened potential in which a small fraction of the particles is pinned, inducing quenched disorder. Ppotentials of this type are widely used for the qualitative description of systems with waterlike anomalies. In our previous publications it was shown that the system demonstrates an anomalous melting scenario: at low densities the system melts through two continuous transitions in accordance with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory with an intermediate hexatic phase, while at high densities the conventional first-order melting transition takes place. We find that the well-known disorder-induced widening of the hexatic phase occurs at low densities, while in the high-density part of the phase diagram random pinning transforms the first-order melting into two transitions: a continuous KTHNY-like solid-hexatic transition and a first-order hexatic-isotropic liquid transition.

Hexatic phase and cluster crystals of two-dimensional GEM4 spheres

Two-dimensional crystals of classical particles are very peculiar in that melting may occur in two steps, in a continuous fashion, via an intermediate hexatic fluid phase exhibiting quasi-long-range orientational order. On the other hand, three-dimensional spheres repelling each other through a fast-decaying bounded potential of generalized-exponential shape (GEM4 potential) can undergo freezing into cluster crystals, allowing for more that one particle per lattice site. We hereby study the combined effect of low spatial dimensionality and extreme potential softness, by investigating the phase behavior of the two-dimensional (2D) GEM4 system. Using a combination of density-functional theory and numerical free-energy calculations, we show that the 2D GEM4 system displays one ordinary and several cluster triangular-crystal phases, and that only the ordinary crystal first melts into a hexatic phase. Upon heating, the difference between the various cluster crystals fades away, eventually leaving a single undifferentiated cluster phase with a pressure-modulated site occupancy.