One-stage continuous melting transition in two dimensions

Anomalous flocking in nonpolar granular Brownian vibrators

Using Brownian vibrators, we investigated the structures and dynamics of quasi-2d granular materials, with packing fractions (ϕ) ranging from 0.111 to 0.832. Our observations revealed a remarkable large-scale flocking behavior in hard granular disk systems, encompassing four distinct phases: granular fluid, flocking fluid, poly-crystal, and crystal. Anomalous flocking emerges at ϕ = 0.317, coinciding with a peak in local density fluctuations, and ceased at ϕ = 0.713 as the system transitioned into a poly-crystal state. The poly-crystal and crystal phases resembled equilibrium hard disks, while the granular and flocking fluids differed significantly from equilibrium systems and previous experiments involving uniformly driven spheres. This disparity suggests that collective motion arises from a competition controlled by volume fraction, involving an active force and an effective attractive interaction resulting from inelastic particle collisions. Remarkably, these findings align with recent theoretical research on the flocking motion of spherical active particles without alignment mechanisms.

Bond-orientational order and Frank's constant in two-dimensional colloidal hard spheres

Recently, the full phase behaviour of 2D colloidal hard spheres was experimentally established, and found to involve a first order liquid to hexatic transition and a continuous hexatic to crystal transition (Thorneywork et al 2017 Phys. Rev. Lett. 118 158001). Here, we expand upon this work by considering the behaviour of the bond-orientational correlation time and Frank's constant in the region of these phase transitions. We also consider the excess entropy, as calculated from the radial distribution functions, for a wide range of area fractions covering the liquid, hexatic and crystal phases. In all cases, the behaviour of these quantities further corroborates the previously reported melting scenario of 2D colloidal hard spheres.

Two-Dimensional Melting of Colloidal Hard Spheres

We study the melting of quasi-two-dimensional colloidal hard spheres by considering a tilted monolayer of particles in sedimentation-diffusion equilibrium. In particular, we measure the equation of state from the density profiles and use time-dependent and height-resolved correlation functions to identify the liquid, hexatic, and crystal phases. We find that the liquid-hexatic transition is first order and that the hexatic-crystal transition is continuous. Furthermore, we directly measure the width of the liquid-hexatic coexistence gap from the fluctuations of the corresponding interface, and thereby experimentally establish the full phase behavior of hard disks.

Two-dimensional melting of colloids with long-range attractive interactions

The solid-liquid melting transition in a two-dimensional (2-D) attractive colloidal system is visualized using superparamagnetic colloids that interact through a long-range isotropic attractive interaction potential, which is induced using a high-frequency rotating magnetic field. Various experiments, supported by Monte Carlo simulations, are carried out over a range of interaction potentials and densities to determine structure factors, Lindermann parameters, and translational and orientational order parameters. The system shows a first-order solid-liquid melting transition. Simulations and experiments suggest that dislocations and disclinations simultaneously unbind during melting. This is in direct contrast with reports of 2-D melting of paramagnetic particles that interact with a repulsive interaction potential.

Kosterlitz-Thouless physics: a review of key issues

This article reviews, from a very personal point of view, the origins and the early work on transitions driven by topological defects such as vortices in the two dimensional planar rotor model and in (4)Helium films and dislocations and disclinations in 2D crystals. I cover the early papers with David Thouless and describe the important insights but also the errors and oversights since corrected by other workers. I then describe some of the experimental verifications of the theory and some numerical simulations. Finally applications to superconducting arrays of Josephson junctions and to recent cold atom experiments are described.

Destabilisation of the hexatic phase in systems of hard disks by quenched disorder due to pinning on a lattice

We investigate the effect of quenched disorder on the melting mechanism of two-dimensional hard disks using large-scale event-driven molecular dynamics simulations. The two-stage melting scenario of a continuous solid-hexatic and a first-order hexatic-liquid transition for a 2D system of hard disks does not persist in the case of quenched disorder, which arises by pinning less than one percent of the particles on a triangular lattice. Based on the Halperin-Nelson-Young (HNY) renormalization group equation, we observe that a first-order solid-liquid transition preempts the Kosterlitz-Thouless-type solid-hexatic transition in a 2D system of hard disks with quenched disorder as the stiffness of the crystal is increased by the presence of pinned particles.

Two-stage melting induced by dislocations and grain boundaries in monolayers of hard spheres

Melting in two-dimensional systems has remained controversial as theory, simulations, and experiments show contrasting results. One issue that obscures this discussion is whether or not theoretical predictions on strictly 2D systems describe those on quasi-2D experimental systems, where out-of-plane fluctuations may alter the melting mechanism. Using event-driven molecular dynamics simulations, we find that the peculiar two-stage melting scenario of a continuous solid-hexatic and a first-order hexatic-liquid transition as observed for a truly 2D system of hard disks [Bernard and Krauth, Phys. Rev. Lett., 2011, 107, 155704] persists for a quasi-2D system of hard spheres with out-of-plane particle motions as high as half the particle diameter. By calculating the renormalized Young's modulus, we show that the solid-hexatic transition is of the Kosterlitz-Thouless type and occurs via dissociation of bound dislocation pairs. In addition, we find a first-order hexatic-liquid transition that seems to be driven by spontaneous proliferation of grain boundaries.

Melting of two-dimensional tunable-diameter colloidal crystals

Melting of two-dimensional colloidal crystals is studied by video microscopy. The samples were composed of microgel spheres whose diameters could be temperature tuned, and whose pair potentials were measured to be short ranged and repulsive. We observed two-step melting from the crystal to a hexatic phase and from the hexatic to the liquid phase as a function of the temperature-tunable volume fraction. The translational and orientational susceptibilities enabled us to definitively determine the phase transition points, avoiding ambiguities inherent in other analyses and resolving a "dislocation precursor stage" in the solid phase that some of the traditional analyses may incorrectly associate with the hexatic phase. A prefreezing stage of the liquid with ordered patches was also found.

Crystallization and melting of bacteria colonies and Brownian bugs

Motivated by the existence of remarkably ordered cluster arrays of bacteria colonies growing in Petri dishes and related problems, we study the spontaneous emergence of clustering and patterns in a simple nonequilibrium system: the individual-based interacting Brownian bug model. We map this discrete model into a continuous Langevin equation which is the starting point for our extensive numerical analyses. For the two-dimensional case we report on the spontaneous generation of localized clusters of activity as well as a melting-freezing transition from a disordered or isotropic phase to an ordered one characterized by hexagonal patterns. We study in detail the analogies and differences with the well-established Kosterlitz-Thouless-Halperin-Nelson-Young theory of equilibrium melting, as well as with another competing theory. For that, we study translational and orientational correlations and perform a careful defect analysis. We find a nonstandard one-stage, defect-mediated transition whose nature is only partially elucidated.

Positional order and diffusion processes in particle systems

The relaxation of a nonequilibrium solid to a fluid is determined by observing the positional order parameter in Monte Carlo simulations, and discussed based on diffusion processes in the hard-particle systems. From the cumulant expansion up to the second order, the relation between the positional order parameter Psi and the mean square displacement <mui2> is obtained to be Psi approximately exp(-K2<mui2>/2d) with a reciprocal vector K and the dimension of the system d. On the basis of this relation, the positional order should decay exponentially as Psi is approximately exp(-K2Dt) when the system involves normal diffusion with a diffusion constant D. A diffusion process with swapping positions of particles is also discussed. The swapping of particles contributes to the higher orders of the cumulants, and swapping positions allows particles to diffuse without destroying the positional order while the normal diffusion destroys it.

Computer simulations of two-dimensional melting with dipole-dipole interactions

We perform molecular dynamics and Monte Carlo simulations of two-dimensional melting with dipole-dipole interactions. Both static and dynamic behaviors are examined. In the isotropic liquid phase, the bond orientational correlation length xi 6 and susceptibility chi 6 are measured, and the data are fitted to the theoretical ansatz. An algebraic decay is detected for both spatial and temporal bond orientational correlation functions in an intermediate temperature regime, and it provides an explicit evidence for the existence of the hexatic phase. From the finite-size scaling analysis of the global bond orientational order parameter, the disclination unbinding temperature Ti is estimated. In addition, from dynamic Monte Carlo simulations of the positional order parameter, we extract the critical exponents at the dislocation unbinding temperature Tm. All the results are in agreement with those from experiments and support the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory.

Large-scale simulations of the two-dimensional melting of hard disks

Large-scale computer simulations with more than four million particles have been performed to study the melting transition in a two-dimensional hard disk fluid. The van der Waals loop previously observed in the pressure-density relationship of smaller simulations is shown to disappear systematically with increase in sample size, but even with these large system sizes, the freezing transition still exhibits what appears to be weakly first-order behavior, though the scaling of the bond orientation order is consistent with the Halperin-Nelson-Young picture. Above this freezing transition region, scaling analysis of the translational order yields a lower bound for the melting density that is much higher than previously thought and provides compelling evidence that the solid phase first melts into a hexatic phase via a continuous transition, before it goes into the isotropic phase.

Lateral organization of cholesterol molecules in lipid-cholesterol assemblies

We present results of an off-lattice simulation of a two-component planar system, as a model for lateral organization of cholesterol molecules in lipid-cholesterol assemblies. We explore the existence of "superlattice" structures even in fluid systems, in the absence of an underlying translational long-range order, and study their coupling to hexatic or bond-orientational order. We discuss our results in context of geometric superlattice theories and "condensation complexes" in understanding a variety of experiments in artificial lipid-cholesterol assemblies.

Melting of polydisperse hard disks

The melting of a polydisperse hard-disk system is investigated by Monte Carlo simulations in the semigrand canonical ensemble. This is done in the context of possible continuous melting by a dislocation-unbinding mechanism, as an extension of the two-dimensional hard-disk melting problem. We find that while there is pronounced fractionation in polydispersity, the apparent density-polydispersity gap does not increase in width, contrary to 3D polydisperse hard spheres. The point where the Young's modulus is low enough for the dislocation unbinding to occur moves with the apparent melting point, but stays within the density gap, just like for the monodisperse hard-disk system. Additionally, we find that throughout the accessible polydispersity range, the bound dislocation-pair concentration is high enough to affect the dislocation-unbinding melting as predicted by Kosterlitz, Thouless, Halperin, Nelson, and Young.

Critical exponents of isotropic-hexatic phase transition in the hard-disk system

The hard-disk system is studied by observing the nonequilibrium relaxation behavior of a bond-orientational order parameter. The density dependence of characteristic relaxation time tau is estimated from the finite-time scaling analysis. The critical point between the fluid and the hexatic phase is refined to be 0.899 (1) by assuming the divergence behavior of the Kosterlitz-Thouless transition. The value of the critical exponent eta is also studied by analyzing the fluctuation of the order parameter at the criticality and estimated as eta=0.25 (2). These results are consistent with the prediction by the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Nonequilibrium relaxation analysis of two-dimensional melting

The phase diagram of a hard-disk system is studied by observing nonequilibrium relaxation functions of a bond-orientational order parameter using particle dynamics simulations. From a finite-time scaling analysis, two Kosterlitz-Thouless transitions can be observed when the density is increased from the isotropic fluid phase to closest packing. The transition densities are estimated to be 0.901(2) and 0.910(2), where the density denotes the fraction of area occupied by the particles, the density is normalized to one for the quadratic packing configuration. These observations are consistent with the predictions of the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Melting of a quasi-two-dimensional metallic system

We analyze the melting of a quasi-two-dimensional metallic system using the results of a series of Monte Carlo simulations of an array of Pb atoms. The system was chosen to model the melting behavior observed for the monolayer of Pb that segregates in the liquid-vapor interface of a dilute Pb in Ga alloy [B. Yang et al., Proc. Natl. Acad. Sci. USA 96, 13 009 (1999)]. Our calculations employed a realistic pair interaction potential between lead pseudoatoms, one that is known to describe accurately the properties of the three-dimensional metal near the melting point. Our results reveal that in the quasi-two-dimensional Pb system melting is a two-stage process which proceeds through formation of a stable intermediate hexatic phase, in agreement with the prediction of the Kosterlitz-Thouless-Halperin-Nelson-Young theory. Both the solid-to-hexatic and the hexatic-to-liquid transitions are found to be first order in our simulations.

Global equation of state of two-dimensional hard sphere systems

Hard sphere systems in two dimensions are examined for arbitrary density. Simulation results are compared to the theoretical predictions for both the low- and the high-density limit, where the system is either disordered or ordered, respectively. The pressure in the system increases with the density, except for an intermediate range of volume fractions 0.65< or =nu< or =0.75, where a disorder-order phase transition occurs. The proposed global equation of state (which describes the pressure for all densities) is applied to the situation of an extremely dense hard sphere gas in a gravitational field and shows reasonable agreement with both experimental and numerical data.

Elastic moduli, dislocation core energy, and melting of hard disks in two dimensions

Elastic moduli and dislocation core energy of the triangular solid of hard disks of diameter sigma are obtained in the limit of vanishing dislocation-antidislocation pair density, from Monte Carlo simulations that incorporate a constraint, namely that all moves altering the local connectivity away from that of the ideal triangular lattice are rejected. In this limit we show that the solid is stable against all other fluctuations at least up to densities as low as rhosigma(2)=0.88. Our system does not show any phase transition so diverging correlation lengths leading to finite size effects and slow relaxations do not exist. The dislocation pair formation probability is estimated from the fraction of moves rejected due to the constraint which yields, in turn, the core energy E(c) and the (bare) dislocation fugacity y. Using these quantities, we check the relative validity of first order and Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) melting scenarios and obtain numerical estimates of the typical expected transition densities and pressures. We conclude that a KTHNY transition from the solid to a hexatic phase preempts the solid to liquid first order transition in this system albeit by a very small margin, easily masked by crossover effects in unconstrained "brute- force" simulations with a small number of particles.

Influence of vacancies on the melting transition of hard disks in two dimensions

We present the results of molecular dynamics simulations of two-dimensional (2D) hard disk systems in the vicinity of melting. The simulations are used to calculate the elastic constants, which can be used to estimate the location of the Kosterlitz-Thouless dislocation unbinding transition. Simulations on defect-free lattices indicate that this transition is expected to occur at essentially the same density as a first-order solid-isotropic transition and so it is not possible to rule out either a one step weak first-order transition between the solid and the isotropic fluid or a two step transition via a hexatic phase. Simulations performed on systems with vacancies indicate that the elastic constants are essentially unchanged at constant density. This result implies that vacancies have little influence on the melting of 2D hard disk solids.