Dynamical criterion for two-dimensional freezing

Diffusion, density, and defects on spheres

We simulate and model diffusion of spherical colloids of radius, a, on spherical surfaces of radius, R, as a function of relative size and surface concentration. Using Brownian dynamics simulations, we quantify diffusion and microstructure at different concentrations ranging from single particles to dense crystalline states. Self-diffusion and structural metrics (pair distribution, local density, and topological charge) are indistinguishable between spheres and planes for all concentrations up to dense liquid states. For concentrations approaching and greater than the freezing transition, smaller spheres with higher curvature show increased diffusivities and nonuniform density/topological defect distributions, which differ qualitatively from planar surfaces. The total topological charge varies quadratically with sphere radius for dense liquid states and linearly with sphere radius for dense crystals with icosahedrally organized grain scars. Between the dense liquid and dense crystal states on spherical surfaces is a regime of fluctuating and interacting defect clusters. We show local density governs self-diffusion in dense liquids on flat and spherical surfaces via the pair distribution. In contrast, dynamic topological defects couple to finite diffusivities through freezing and in low density crystal states on spherical surfaces, where neither exist on flat surfaces.

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.

Melting, freezing, and dynamics of two-dimensional dipole systems in screening bulk media

This paper reports on the molecular dynamics simulations of classical two-dimensional (2D) electric dipole systems. The properties of 2D systems with bare (nonscreened) and screened dipole-dipole interactions have been investigated. Based on the polygon construction method, we present simulation results on the phase transition, and we locate the melting and freezing points of 2D dipole systems in terms of a polygon disorder parameter, with the polygon disorder parameter being the sum of nontriangular polygon order parameters. It was found that the phase transition of the system occurs when the polygon disorder parameter has a value 0.165. This result was cross-checked by using both local and overall orientational order parameters. We also identified that the value of the average local orientational order parameter at the phase transition point is 0.67. These results are valid for the ordinary (bare) dipole-dipole interaction as well as the screened dipole-dipole interaction, and they are expected to be general for other 2D systems with repulsive pair interaction. We observed that both melting and freezing points shift to lower values of temperature due to screening. In the liquid state, the radial distribution function and polygon construction method show the loss of order in a structure as screening becomes more severe. Furthermore, the impact of screening on the system's collective excitation spectra and diffusive characteristics at liquid and solid states has been studied. Results show the decrease in the values of both longitudinal and transverse sound speeds and the emergence of anomalous superdiffusive motion in the liquid state due to screening.

Thermodynamics and dynamics of two-dimensional systems with dipolelike repulsive interactions

Thermodynamics and dynamics of a classical two-dimensional system with dipolelike isotropic repulsive interactions are studied systematically using extensive molecular dynamics (MD) simulations supplemented by appropriate theoretical approximations. This interaction potential, which decays as an inverse cube of the interparticle distance, belongs to the class of very soft long-ranged interactions. As a result, the investigated system exhibits certain universal properties that are also shared by other related soft-interacting particle systems (like, for instance, the one-component plasma and weakly screened Coulomb systems). These universalities are explored in this article to construct a simple and reliable description of the system thermodynamics. In particular, Helmholtz free energies of the fluid and solid phases are derived, from which the location of the fluid-solid coexistence is determined. The quasicrystalline approximation is applied to the description of collective modes in dipole fluids. Its simplification, previously validated on strongly coupled plasma fluids, is used to derive explicit analytic dispersion relations for the longitudinal and transverse wave modes, which compare satisfactory with those obtained from direct MD simulations in the long-wavelength regime. Sound velocities of the dipole fluids and solids are derived and analyzed.

Crystallization and flow in active patch systems

Based upon recent experiments in which Janus particles are made into active swimmers by illuminating them with laser light, we explore the effect of applying a light pattern on the sample, thereby creating activity inducing zones or active patches. We simulate a system of interacting Brownian diffusers that become active swimmers when moving inside an active patch and analyze the structure and dynamics of the ensuing stationary state. We find that, in some respects, the effect of spatially inhomogeneous activity is qualitatively similar to a temperature gradient. For asymmetric patches, however, this analogy breaks down because the ensuing stationary state is specific to partial active motion.

Computation of shear viscosity of colloidal suspensions by SRD-MD

The behaviour of sheared colloidal suspensions with full hydrodynamic interactions (HIs) is numerically studied. To this end, we use the hybrid stochastic rotation dynamics-molecular dynamics (SRD-MD) method. The shear viscosity of colloidal suspensions is computed for different volume fractions, both for dilute and concentrated cases. We verify that HIs help in the collisions and the streaming of colloidal particles, thereby increasing the overall shear viscosity of the suspension. Our results show a good agreement with known experimental, theoretical, and numerical studies. This work demonstrates the ability of SRD-MD to successfully simulate transport coefficients that require correct modelling of HIs.

Specific heat in two-dimensional melting

We report the specific heat cN around the melting transition(s) of micrometer-sized superparamagnetic particles confined in two dimensions, calculated from fluctuations of positions and internal energy, and corresponding Monte Carlo simulations. Since colloidal systems provide single particle resolution, they offer the unique possibility to compare the experimental temperatures of the peak position of cN(T) and symmetry breaking, respectively. While order parameter correlation functions confirm the Kosterlitz-Thouless-Halperin-Nelson-Young melting scenario where translational and orientational order symmetries are broken at different temperatures with an intermediate so called hexatic phase, we observe a single peak of the specific heat within the hexatic phase, with excellent agreement between experiment and simulation. Thus, the peak is not associated with broken symmetries but can be explained with the total defect density, which correlates with the maximum increase of isolated dislocations. The absence of a latent heat strongly supports the continuous character of both transitions.

Hydrodynamic interactions slow down crystallization of soft colloids

Colloidal suspensions are often argued to be an ideal model for studying phase transitions such as crystallization, as they have the advantage of tunable interactions and experimentally tractable time and length scales. Because crystallization is assumed to be unaffected by details of particle transport other than the bulk diffusion coefficient, findings are frequently argued to be transferable to pure melts without solvent. In this article, we present molecular dynamics simulations of crystallization in a suspension of colloids with Yukawa interactions which challenge this assumption. In order to investigate the role of hydrodynamic interactions mediated by the solvent, we model the solvent both implicitly and explicitly, using Langevin dynamics and the fluctuating lattice Boltzmann method, respectively. Our simulations show a significant reduction of the crystal growth velocity due to hydrodynamic interactions even at moderate hydrodynamic coupling. This slowdown is accompanied by a reduction of the width of the layering region in front of the growing crystal. Thus the dynamics of a colloidal suspension differ strongly from that of a melt, making it less useful as a model for solvent-free melts than previously thought.

Freezing of a two-dimensional fluid into a crystalline phase: density functional approach

A free-energy functional for a crystal proposed by Singh and Singh [Europhys. Lett. 88, 16005 (2009)] which contains both the symmetry conserved and symmetry broken parts of the direct pair correlation function has been used to investigate the crystallization of a two-dimensional fluid. The results found for fluids interacting via the inverse power potential u(r)=ε(σ/r)(n) for n=3,6, and 12 are in good agreement with experimental and simulation results. The contribution made by the symmetry broken part to the grand thermodynamic potential at the freezing point is found to increase with the softness of the potential. Our results explain why the Ramakrishnan-Yussouff [Phys. Rev. B 19, 2775 (1979)] free-energy functional gave good account of freezing transitions of hard-core potentials but failed for potentials that have soft core and/or attractive tail.

Comparison of 2D melting criteria in a colloidal system

We use super-paramagnetic spherical particles which are arranged in a two-dimensional monolayer at a water/air interface to investigate the crystal to liquid phase transition. According to the KTHNY theory a crystal melts in thermal equilibrium by two continuous phase transitions into the isotropic liquid state with an intermediate phase, commonly known as the hexatic phase. We verify the significance of several criteria based on dynamical and structural properties to identify the crystal-hexatic and hexatic-isotropic liquid phase transitions for the same experimental data of the given setup. The criteria are the bond orientational correlation function, the Larson-Grier criterion, the 2D dynamic Lindemann parameter, the bond orientational susceptibility, the 2D Hansen-Verlet rule, the Löwen-Palberg-Simon criterion as well as a criterion based on the shape factor of Voronoi cells and Minkowski functionals. For our system with long-range repulsion, the bond order correlation function and bond order susceptibility work best to identify the hexatic-isotropic liquid transition and the 2D dynamic Lindemann parameter identifies unambiguously the hexatic-crystalline transition.

Microrheological consequences of attractive colloid-colloid potentials in a two-dimensional Brownian fluid

By using microrheological methods commonly employed in videomicroscopy experiments, we study the rheology of a two-dimensional computational fluid formed by Brownian disks with the aim of exploring the influence of some effective colloid-colloid attractive interactions. The model of fluid is developed by Brownian dynamics simulations without hydrodynamical interactions, and it is characterized by calculating its equation of state from the pair distribution function. Micromechanical properties, relative and intrinsic viscosity and freezing are discussed. Then, we include attractive forces such a Asakura-Oosawa depletion force or an empiric expression proposed by Grier and Hal (GH) for an anomalous electrostatic potential observed in confined and charged colloids. By using both potentials, viscosity is clearly increased, but when the GH potential is included, viscoelastic gel state is reached for intermediate values of surface concentration. Finally, we analyse the influence of the attractive potentials in the breaking-up by thermal fluctuations of linear chains formed by 2D particles, finding that the GH potential reduces the characteristical time at which the disks can be considered as disaggregated. In this work, we employ an experimental-like methodology for the study of a Brownian hard-disk fluid, providing a very useful link with experimental procedures.

Crystallization in a dense suspension of self-propelled particles

Using Brownian dynamics computer simulations, we show that a two-dimensional suspension of self-propelled ("active") colloidal particles crystallizes at sufficiently high densities. Compared to the equilibrium freezing of passive particles, the freezing density is both significantly shifted and depends on the structural or dynamical criterion employed. In nonequilibrium the transition is accompanied by pronounced structural heterogeneities. This leads to a transition region between liquid and solid in which the suspension is globally ordered but unordered liquidlike "bubbles" still persist.

Freezing transition and interaction potential in monolayers of microparticles at fluid interfaces

The structure and the interaction potential of monolayers of charged polystyrene microparticles at fluid interfaces have been studied by optical microscopy. Microparticles of different sizes have been studied over a broad range of surface particle densities. The structural characterization is based on the analysis of images obtained by digital optical microscopy. From the experimental images, radial distribution functions, hexagonal bond order correlation functions, and temporal orientational correlation functions have been calculated for different monolayer states at both the air/water and oil/water interfaces. The interaction potential has been calculated from the structure factor using integral equations within the hypernetted chain closure relationship. For particles trapped at the oil-water interface, it was found that, upon increasing the surface coverage, a freezing transition occurs, that leads to the formation of a 2D crystalline structure. We have studied the freezing densities of particle monolayers at the oil/water interface and compared them with Monte Carlo simulation results reported by H. Löwen. In contrast, at the air-water interface, freezing is inhibited due to the formation of particle aggregates.

Eighth-order phase-field-crystal model for two-dimensional crystallization

We present a derivation of the recently proposed eighth-order phase-field crystal model [A. Jaatinen, Phys. Rev. E 80, 031602 (2009)] for the crystallization of a solid from an undercooled melt. The model is used to study the planar growth of a two-dimensional hexagonal crystal, and the results are compared against similar results from dynamical density functional theory of Marconi and Tarazona, as well as other phase-field crystal models. We find that among the phase-field crystal models studied, the eighth-order fitting scheme gives results in good agreement with the density functional theory for both static and dynamic properties, suggesting it is an accurate and computationally efficient approximation to the density functional theory.

Identifying anomalous diffusion and melting in dusty plasmas

Anomalous diffusion in liquids and the solid-liquid phase transition (melting) are studied in two-dimensional Yukawa systems. The self-intermediate scattering function (self-ISF), calculated from simulation data, exhibits a temporal decay, or relaxation, with a characteristic relaxation time. This decay is found to be useful for distinguishing normal and anomalous diffusion in a liquid, and for identifying the solid-liquid phase transition. For liquids, a scaling of the relaxation time with length scale is found. For the solid-liquid phase transition, the shape of the self-ISF curve is found to be a sensitive indicator of phase. Friction has a significant effect on the timing of relaxation, but not the melting point.

2D melting of plasma crystals: equilibrium and nonequilibrium regimes

Comprehensive experimental investigations of melting in two-dimensional complex plasmas were carried out. Different experiments were performed in steady and unsteady heating regimes. We demonstrate an Arrhenius dependence of the defect concentration on the kinetic temperature in steady-state experiments, and show the evidence of metastable quenching in unsteady experiments, where the defect concentration follows a power-law temperature scaling. In all experiments, independent indicators suggest a grain-boundary-induced melting scenario.

Diffusion in two-dimensional colloidal systems on periodic substrates

We study the diffusive behavior of two-dimensional charged colloidal suspensions subjected to a sinusoidal substrate by means of Brownian dynamics simulations. We mainly focus on the dependence of the mean-square displacement on the substrate strength. Our findings show a variation in the particle diffusion due to a substrate-induced distortion of the dynamic cage of nearest-neighbor colloids. This mechanism leads to a transition from normal diffusion at short times to subdiffusion on intermediate time scales. However, at long times normal diffusion is recovered. We also show that the variation in the long-time self-diffusion coefficient may be associated with the freezing and re-entrant melting transitions.

Experimental study of structure and dynamics in a monolayer of paramagnetic colloids confined by parallel hard walls

We study the structural and dynamical properties of paramagnetic colloidal spheres interacting as repulsive dipoles in two dimensions and confined between parallel hard walls. We observed that the structure and dynamics of the self-assembled colloids are strongly dependent upon the width of the confining channel. The system exhibits re-entrant behavior as a function of the channel width, transitioning from solid-like to liquid-like repeatedly in excellent agreement with simulation results. For large channels, an ordered layered structure self-assembles near the walls, but this local structure is not commensurate with the bulk structure, leading to localized stable defects.

Structure and dynamics of repulsive magnetorheological colloids in two-dimensional channels

We study a system of colloidal spheres with induced magnetic dipoles confined in two-dimensional (2D) hard-wall channels using Brownian dynamics simulations. The external magnetic field is directed normal to the 2D plane and therefore the colloids interact with a purely repulsive r(-3) potential. The effects of confinement between parallel walls are determined by analyzing the structure and dynamics of these confined systems and comparing to the unbounded (infinite) 2D plane limit. The bond-order correlation function is analyzed as a function of time and exhibits unique characteristics associated with the channel-like confinement. The existence of a plateau in this correlation function is observed over an intermediate time scale and the fate of the plateau (decay or persistence) depends upon the channel width, the strength of the external magnetic field, and the number density. The plateau is analyzed in further detail and an explanation is put forth for its existence and subsequent long time behavior. Additionally, re-entrant behavior with respect to dimensionless channel width is observed in the structural properties and an associated state-diagram is presented for these systems.

Structural analysis of a dipole system in two-dimensional channels

A system of magnetic dipoles in two-dimensional (2D) channels was studied using Brownian dynamics simulations. The dipoles interact with a purely repulsive r(-3) potential and are confined by two hard walls in one of the dimensions. Solid crystals were annealed in the 2D channels and the structural properties of the crystals were investigated. The long-ranged nature of the purely repulsive dipoles combined with the presence of hard walls led to structural deviations from the unbounded (infinite) 2D dipolar crystal. The structures in the channels were characterized by a high density of particles along the walls. The particles along the wall became increasingly localized as the channel width was increased. The spacing of the walls was important in determining the properties of the structures formed in the channel. Small changes in the width of the channel induced significant structural changes in the crystal. These structural changes were manifested in the density profiles, defect concentrations, and local bond-orientation order of the system. Oscillations in the structural properties were observed as the channel width was increased, indicating the existence of magic-number channel widths for this system. As the channel width was increased the properties of the confined system approached those of the unbounded system surprisingly slowly.

Locked-floating-solid to locked-smectic transition in colloidal systems

We investigate two-dimensional melting of a colloidal system in the presence of a one-dimensional periodic substrate potential created by two interfering laser beams. We study the commensurability ratio p=sqrt[3]a/2d=2 with a the mean particle distance and d the period of the periodic potential. In contrast with the previously investigated case p=1, here we observe that melting of the locked-floating solid occurs via a novel locked-smectic phase, predicted by recent theoretical studies.

Influence of hydrodynamics on many-particle diffusion in 2D colloidal suspensions

We study many-particle diffusion in 2D colloidal suspensions with full hydrodynamic interactions through a novel mesoscopic simulation technique. We focus on the behaviour of the effective scaled tracer and collective-diffusion coefficients DT(rho)/D0 and DC(rho)/D0 respectively, where D0 is the single-particle diffusion coefficient, as a function of the density of the colloids rho. At low Schmidt numbers Sc - 1, we find that hydrodynamics has essentially no effect on the behaviour of DT (rho)/D0. At larger Sc, DT (rho)/D0 seems to be enhanced at all densities, although the differences compared to the case without hydrodynamics are rather minor. The collective-diffusion coefficient, on the other hand, is much more strongly coupled to hydrodynamical conservation laws and is distinctly different from the purely dissipative case without hydrodynamic interactions.

Universal scaling in complex (dusty) plasmas

The structural and dynamical properties of three-dimensional isotropic complex plasmas are investigated kinetically within the framework of a dissipative Yukawa model. A modified Coulomb coupling parameter is proposed whose value alone determines the location of the complex plasma melting line. This implies that the phase transition has a universal scaling at the kinetic level. In detail, our molecular dynamics investigations show that the system dynamics is universal (but different) in the limits of high as well as low-frictional dissipation, while in the intermediate case it depends considerably on the dissipation rate. Issues such as the influence of the interaction strength on the single particle diffusion constant and the applicability of dynamical criteria for freezing are discussed.

Dynamic scaling and freezing criteria in quasi-two-dimensional dispersions

We report on a Brownian dynamics simulation study of quasi-two-dimensional dispersions of colloidal spheres interacting by long-range electrostatic and dipolar magnetic forces. The calculated dynamic correlation functions are shown to obey dynamic scaling in terms of a characteristic relaxation time related to the mean particle distance and, due to hydrodynamic interactions, to the particle size. The dynamical freezing criterion of Löwen [Phys. Rev. E 53, R29 (1996)] is shown to be equivalent to a two-dimensional static freezing criterion.

Diffusion of hard disks and rodlike molecules on surfaces

We study the submonolayer diffusion of hard disks and rodlike molecules on smooth surfaces through numerical simulations and theoretical arguments. We concentrate on the behavior of the various diffusion coefficients as a function of the two-dimensional (2D) number density rho in the case where there are no explicit surface-particle interactions. For the hard disk case, we find that while the tracer diffusion coefficient D(T)(rho) decreases monotonically up to the freezing transition, the collective diffusion coefficient D(C)(rho) is wholly determined by the inverse compressibility which increases rapidly on approaching freezing. We also study memory effects associated with tracer diffusion, and present theoretical estimates of D(T)(rho) from the mode-mode coupling approximation. In the case of rigid rods with short-range repulsion and no orientational ordering, we find behavior very similar to the case of disks with the same repulsive interaction. Both D(T)(rho) and the angular diffusion coefficient D(R)(rho) decrease with rho. Also in this case D(C)(rho) is determined by inverse compressibility and increases rapidly close to freezing. This is in contrast to the case of flexible chainlike molecules in the lattice-gas limit, where D(C)(rho) first increases and then decreases as a function of the density due to the interplay between compressibility and mobility.

Long-time tracer diffusion of nonspherical brownian particles

The long-time tracer-diffusion properties of a nonspherical Brownian particle that interacts with a suspension of spherical particles are studied in terms of an idealized but nontrivial two-dimensional model system. For this system, the predictions of the generalized Langevin equation approach to tracer diffusion can be calculated, and compared with the (extrapolated) results of a computer simulation experiment. In the model, the nonspherical particle is represented by a rigid linear array of N(T) (=2 or 3) spherical particles with nearest-neighbor separation DeltaL. We calculate the long-time rotational and (transverse and longitudinal) translational diffusion coefficients. The theory is found to reproduce qualitatively and quantitatively the main features of the extrapolated results. Finally, we also present theoretical results that derive from still simpler approximate theoretical schemes.