Two-Dimensional Melting of Colloidal Hard Spheres

Phase transitions and dimensional cross-over in layered confined solids

The nature of solid phases and cross-over of order-disorder phase transitions from two-dimensional (2D) layers to three-dimensional (3D) bulk in confined atomic systems remain largely unexplained. To this end, we consider noble gases and aluminum confined between graphene sheets at different pressures and temperatures. Using crystal structure search methods and molecular dynamics based on machine-learned potentials with quantum-mechanical accuracy, we identify structures of multilayer confined solids that deviate from simple close packing. Upon heating, we find that confined 2D monolayers melt according to the two-step continuous Kosterlitz-Thouless-Halperin-Nelson-Young theory. However, multilayer solids transition continuously into an intermediate layered-hexatic phase before melting discontinuously into an isotropic liquid. This intermediate phase persists at least up to 12 layers studied here. This change can be qualitatively understood based on the cross-over from 2D topological defects toward 3D ones during melting as the number of layers increases.

Phase diagram of the hard-sphere potential model in three and four dimensions using a pseudo-hard-sphere potential

The hard-sphere potential has become a cornerstone in the study of both molecular and complex fluids. Despite its mathematical simplicity, its implementation in fixed time step molecular simulations remains a formidable challenge due to the discontinuity at contact. To avoid the issues associated with the ill-defined force at contact, a continuous potential has recently been proposed-here referred to as the pseudo-hard-sphere potential (pHS) [Báez et al., J. Chem, Phys. 149, 164907 (2018)]. This potential is constructed to match the second virial coefficient of the hard-sphere potential and is expected to mimic its thermodynamic properties. However, this hypothesis has only been partially validated within the fluid region of the phase diagram for hard-sphere dispersions in two and three dimensions. In this contribution, we examine the ability of the continuous pHS potential to reproduce the equation of state of a hard-sphere fluid, not only in the fluid phase but also across the fluid-solid coexistence region. Our focus is primarily on the phase diagram of hard-sphere systems in three and four dimensions; however, we also report on the feasibility of the pHS to reproduce the long time dynamics of a three-dimensional colloidal dispersion. We compare the thermodynamic properties obtained from Brownian dynamics simulations of the pHS potential with those derived from refined event-driven simulations of the corresponding hard-sphere potential. Furthermore, we provide a comparative analysis with theoretical equations of state based on both mean-field and integral equation approximations.

Tetratic phase in 2D crystals of squares

Melting in two-dimensional (2D) systems is described by the celebrated Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory, which explains how the unbinding of two types of topological defects destroys translational and orientational order at distinct temperatures. The intermediate hexatic phase, a fluid with six-fold quasi-long-ranged orientational order, has been observed in 2D colloidal monolayers of isotropic particles. In this study, we investigate the melting of a quadratic crystal with four-fold symmetry, composed of square particles of approximately 4 × 4 μm in size. These anisotropic particles were fabricated from photoresist using 3D nanoprinting. In an aqueous solution, the particles sediment onto a cover slide, forming a monolayer. The adjustable curvature of the cover slide precisely controls the monolayer density. At low densities, the particles exhibit free diffusion, forming a 2D fluid, while at high densities, they assemble into a quadratic crystal. Using a four-fold bond-order correlation function, we identify the tetratic phase with quasi-long ranged orientational order in close analogy to the hexatic phase in systems with six-fold symmetry.

Simulation Studies of Dynamical Heterogeneity in a Dense Two-Dimensional Dimer-Solvent System with Obstacles

A coarse-grained model of a two-dimensional colloidal suspension was designed. The model was athermal and, in addition, a lattice approximation was introduced. It consisted of solvent (monomer) molecules, dimer molecules, and immobile impenetrable obstacles that introduced additional heterogeneity into the system. Dynamic properties were determined by a Monte Carlo simulation using the dynamic lattice liquid simulation algorithm. It is shown that there is a range of obstacle concentrations in which different diffusion characteristics were observed for dimers and solvents. In the system studied, it is possible to define the ranges of concentrations of individual components (solvent, dimers, and obstacles), in which the nature of the movement of dimers and solvents is different (normal diffusion vs. subdiffusion). The ratio of diffusion coefficients of solvent molecules and dimers for short times does not depend on the concentration of obstacles, while for long times, the ratio increases but remains independent of the concentration of the dimer.

Self-Assembly of Colloidal Dumbbell Isomers and Plasmonic Properties for Optical Metamaterials

In this study, we explore the self-assembly of various colloidal symmetric dumbbell (DB) isomers, including dipole Janus, cis-Janus, trans-Janus, apolar-inward and polar-inward perpendicular Janus, and alternating perpendicular Janus DBs. Using dissipative particle dynamics (DPD) simulations under conditions mimicking experimental setups, we investigate cluster formation driven by emulsion droplet evaporation. Our findings reveal a diverse set of cluster structures, which are in good agreement with experimental and simulation results reported in the literature while also predicting the formation of novel cluster configurations. These structures, characterized by well-defined and predictable patterns, are potentially applicable to creating colloidal molecules and crystals. Furthermore, we examine the dynamics of cluster formation to gain insight into the mechanisms guiding the self-assembly of these diverse colloidal DBs. The study highlights the impact of particle isomerism on the resulting assembly structures. We further select a set of typical nanoclusters obtained, including a tetrahedral cluster, which is the simplest, to study its plasmonic properties. Our findings indicate that increasing the nanoparticle (NP) radius or decreasing the gap between NPs leads to a red shift in the plasmonic resonance wavelength and enhances the resonance strength. We identify critical parameter regions where the electric-dipole and magnetic-dipole resonances can be engineered to achieve negative dielectric permittivity and magnetic permeability, which are essential for developing negative-index metamaterials.

Self-Assembly at Curved Biointerfaces

Most of the biological interfaces are curved. Understanding the organizational structures and interaction patterns at such curved biointerfaces is therefore crucial not only for deepening our comprehension of the principles that govern life processes but also for designing and developing targeted drugs aimed at diseased cells and tissues. Despite the considerable efforts dedicated to this area of research, our understanding of curved biological interfaces is still limited. Many aspects of these interfaces remain elusive, presenting both challenges and opportunities for further exploration. In this review, we summarize the structural characteristics of biological interfaces found in nature, the current research status of materials associated with curved biointerfaces, and the theoretical advancements achieved to date. Finally, we outline future trends and challenges in the theoretical and technological development of curved biointerfaces. By addressing these challenges, people could bridge the knowledge gap and unlock the full potential of curved biointerfaces for scientific and technological advancements, ultimately benefiting various fields and improving human health and well-being.

Observation of the hexatic phase in a two-dimensional complex plasma using machine learning

Complex plasmas consist of ionized gas and charged solid microparticles, representing the plasma state of soft matter. We apply machine learning methods to investigate a melting transition in a two-dimensional complex plasma. A convolutional neural network is constructed and trained with the numerical simulation. The hexatic phase is successfully identified and the evolution of topological defects is studied during melting transition in both simulations and experiments.

Random close packing of semi-flexible polymers in two dimensions: Emergence of local and global order

Through extensive Monte Carlo simulations, we systematically study the effect of chain stiffness on the packing ability of linear polymers composed of hard spheres in extremely confined monolayers, corresponding effectively to 2D films. First, we explore the limit of random close packing as a function of the equilibrium bending angle and then quantify the local and global order by the degree of crystallinity and the nematic or tetratic orientational order parameter, respectively. A multi-scale wealth of structural behavior is observed, which is inherently absent in the case of athermal individual monomers and is surprisingly richer than its 3D counterpart under bulk conditions. As a general trend, an isotropic to nematic transition is observed at sufficiently high surface coverages, which is followed by the establishment of the tetratic state, which in turn marks the onset of the random close packing. For chains with right-angle bonds, the incompatibility of the imposed bending angle with the neighbor geometry of the triangular crystal leads to a singular intra- and inter-polymer tiling pattern made of squares and triangles with optimal local filling at high surface concentrations. The present study could serve as a first step toward the design of hard colloidal polymers with a tunable structural behavior for 2D applications.

Deformation of confined liquid interfaces by inhomogeneous electric fields and localized particle forces

HYPOTHESIS

Oil-water interfaces that are created by confining a certain amount of oil in a square shaped pixel (∼200 x 200 μm2 with a height of ∼10 μm) topped by a layer of water, have a curvature that depends on the amount of oil that happens to be present in the confining area. Under the application of an electric field normal to the interface, the interface will deform due to inhomogeneities in the electric field. These inhomogeneities are expected to arise from the initial curvature of the meniscus, from fringe fields that emerge at the confining pixel walls and, if applicable, from interfacially adsorbed particles.

MODELING AND EXPERIMENTS

We model the shape of the confined oil-water interface invoking capillarity and electrostatics. Furthermore, we measure the initial curvature by tracking the position of interfacially adsorbed particles depending on sample tilt.

FINDINGS

We found that the pixels exhibited meniscus curvature radii ranging from 0.6-7 mm. The corresponding model based minimum oil film thicknesses range between 0.7 and 9 μm. Furthermore, the model shows that the initial meniscus curvature can increase up to 76 percent relative to the initial curvature by the electric field before the oil film becomes unstable. The pixel wall and particles are shown to have minimal impact on the interface deformation.

Rotational diffusion of colloidal microspheres near flat walls

Recently, colloids with an off-center fluorescent core and homogeneous composition have been developed to measure the rotational diffusivity of microparticles using 3D confocal microscopy in refractive index-matched suspensions. Here, we show that the same particles may be imaged using a standard fluorescence microscope to yield their rotational diffusion coefficients. Trajectories of the off-center core may be combined with known expressions for the correlation decay of particle orientations to determine an effective rotational diffusivity. For sedimented particles, we also find the rotational diffusivity about axes perpendicular and parallel to the interface by adding some bright field illumination and simultaneously tracking both the core and the particle. Trajectories for particles of different sizes yield excellent agreement with hydrodynamic models of rotational diffusion near flat walls, taking the sedimentation-diffusion equilibrium into account. Finally, we explore the rotational diffusivity of particles in crowded two-dimensional monolayers, finding a different reduction of the rotational motion about the two axes depending on the colloidal microstructure.

Interfacial Fluid Rheology of Soft Particles

In situ interfacial rheology and numerical simulations are used to investigate microgel monolayers in a wide range of packing fractions, ζ_{2D}. The heterogeneous particle compressibility determines two flow regimes characterized by distinct master curves. To mimic the microgel architecture and reproduce experiments, an interaction potential combining a soft shoulder with the Hertzian model is introduced. In contrast to bulk conditions, the elastic moduli vary nonmonotonically with ζ_{2D} at the interface, confirming long-sought predictions of reentrant behavior for Hertzian-like systems.

Melting Is Well-Known, but Is It Also Well-Understood?

Contrary to continuous phase transitions, where renormalization group theory provides a general framework, for discontinuous phase transitions such a framework seems to be absent. Although the thermodynamics of the latter type of transitions is well-known and requires input from two phases, for melting a variety of one-phase theories and models based on solids has been proposed, as a generally accepted theory for liquids is (yet) missing. Each theory or model deals with a specific mechanism using typically one of the various defects (vacancies, interstitials, dislocations, interstitialcies) present in solids. Furthermore, recognizing that surfaces are often present, one distinguishes between mechanical or bulk melting and thermodynamic or surface-mediated melting. After providing the necessary preliminaries, we discuss both types of melting in relation to the various defects. Thereafter we deal with the effect of pressure on the melting process, followed by a discussion along the line of type of materials. Subsequently, some other aspects and approaches are dealt with. An attempt to put melting in perspective concludes this review.

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.

Optothermal crystallization of hard spheres in an effective bidimensional geometry

Using colloids effectively confined in two dimensions by a cell with a thickness comparable to the particle size, we investigate the nucleation and growth of crystallites induced by locally heating the solvent with a near-infrared laser beam. The particles, which are "thermophilic," move towards the laser spot solely because of thermophoresis with no convection effects, forming dense clusters whose structure is monitored using two order parameters that gauge the local density and the orientational ordering. We find that ordering takes place when the cluster reaches an average surface density that is still below the upper equilibrium limit for the fluid phase of hard disks, meaning that we do not detect any sign of a proper "two-stage" nucleation from a glass or a polymorphic crystal structure. The crystal obtained at late growth stage displays a remarkable uniformity with a negligible amount of defects, arguably because the incoming particles diffuse, bounce, and displace other particles before settling at the crystal interface. This "fluidization" of the outer crystal edge may resemble the surface enhanced mobility giving rise to ultra-stable glasses by physical vapor deposition.

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.

Surface tension of bulky colloids, capillarity under gravity, and the microscopic origin of the Kardar-Parisi-Zhang equation

Experimental measurements of the surface tension of colloidal interfaces have long been in conflict with computer simulations. In this Letter we show that the surface tension of colloids as measured by surface fluctuations picks up a gravity-dependent contribution which removes the discrepancy. The presence of this term puts a strong constraint on the structure of the interface which allows one to identify corrections to the fundamental equation of equilibrium capillarity and deduce bottom up the microscopic origin of a growth model with close relation to the Kardar-Parisi-Zhang equation.

Two-Dimensional Crystals far from Equilibrium

When driven by nonequilibrium fluctuations, particle systems may display phase transitions and physical behavior with no equilibrium counterpart. We study a two-dimensional particle model initially proposed to describe driven non-Brownian suspensions undergoing nonequilibrium absorbing phase transitions. We show that when the transition occurs at large density, the dynamics produces long-range crystalline order. In the ordered phase, long-range translational order is observed because equipartition of energy is lacking, phonons are suppressed, and density fluctuations are hyperuniform. Our study offers an explicit microscopic model where nonequilibrium violations of the Mermin-Wagner theorem stabilize crystalline order in two dimensions.

Hydrodynamic coupling melts acoustically levitated crystalline rafts

Going beyond the manipulation of individual particles, first steps have recently been undertaken with acoustic levitation in air to investigate the collective dynamical properties of many-body systems self-assembled within the levitation plane. However, these assemblies have been limited to two-dimensional, close-packed rafts where forces due to scattered sound pull particles into direct frictional contact. Here, we overcome this restriction using particles small enough that the viscosity of air establishes a repulsive streaming flow at close range. By tuning the particle size relative to the characteristic length scale for viscous streaming, we control the interplay between attractive and repulsive forces and show how particles can be assembled into monolayer lattices with tunable spacing. While the strength of the levitating sound field does not affect the particles' steady-state separation, it controls the emergence of spontaneous excitations that can drive particle rearrangements in an effectively dissipationless, underdamped environment. Under the action of these excitations, a quiescent particle lattice transitions from a predominantly crystalline structure to a two-dimensional liquid-like state. We find that this transition is characterized by dynamic heterogeneity and intermittency, involving cooperative particle movements that remove the timescale associated with caging for the crystalline lattice. These results shed light on the nature of athermal excitations and instabilities that can arise from strong hydrodynamic coupling among interacting particles.

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.

Geometric conjecture about phase transitions

As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach the steady state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and ε-mixing time of the configuration spaces of hard-disk and hard-sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems and numerical evidence suggests that a discontinuity in the ε-mixing time coincides with the solid-fluid phase transition in the thermodynamic limit.

Theory and Simulations of Ionic Liquids in Nanoconfinement

Room-temperature ionic liquids (RTILs) have exciting properties such as nonvolatility, large electrochemical windows, and remarkable variety, drawing much interest in energy storage, gating, electrocatalysis, tunable lubrication, and other applications. Confined RTILs appear in various situations, for instance, in pores of nanostructured electrodes of supercapacitors and batteries, as such electrodes increase the contact area with RTILs and enhance the total capacitance and stored energy, between crossed cylinders in surface force balance experiments, between a tip and a sample in atomic force microscopy, and between sliding surfaces in tribology experiments, where RTILs act as lubricants. The properties and functioning of RTILs in confinement, especially nanoconfinement, result in fascinating structural and dynamic phenomena, including layering, overscreening and crowding, nanoscale capillary freezing, quantized and electrotunable friction, and superionic state. This review offers a comprehensive analysis of the fundamental physical phenomena controlling the properties of such systems and the current state-of-the-art theoretical and simulation approaches developed for their description. We discuss these approaches sequentially by increasing atomistic complexity, paying particular attention to new physical phenomena emerging in nanoscale confinement. This review covers theoretical models, most of which are based on mapping the problems on pertinent statistical mechanics models with exact analytical solutions, allowing systematic analysis and new physical insights to develop more easily. We also describe a classical density functional theory, which offers a reliable and computationally inexpensive tool to account for some microscopic details and correlations that simplified models often fail to consider. Molecular simulations play a vital role in studying confined ionic liquids, enabling deep microscopic insights otherwise unavailable to researchers. We describe the basics of various simulation approaches and discuss their challenges and applicability to specific problems, focusing on RTIL structure in cylindrical and slit confinement and how it relates to friction and capacitive and dynamic properties of confined ions.

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

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

Field-driven cluster formation in two-dimensional colloidal binary mixtures

We study size- and charge-asymmetric oppositely charged colloids driven by an external electric field. The large particles are connected by harmonic springs, forming a hexagonal-lattice network, while the small particles are free of bonds and exhibit fluidlike motion. We show that this model exhibits a cluster formation pattern when the external driving force exceeds a critical value. The clustering is accompanied with stable wave packets in vibrational motions of the large particles.

Hydrodynamic Enhancement of p-atic Defect Dynamics

We investigate numerically and analytically the effects of hydrodynamics on the dynamics of topological defects in p-atic liquid crystals, i.e., two-dimensional liquid crystals with p-fold rotational symmetry. Importantly, we find that hydrodynamics fuels a generic passive self-propulsion mechanism for defects of winding number s=(p-1)/p and arbitrary p. Strikingly, we discover that hydrodynamics always accelerates the annihilation dynamics of pairs of ±1/p defects and that, contrary to expectations, this effect increases with p. Our Letter paves the way toward understanding cell intercalation and other remodeling events in epithelial layers.

Probing Temperature Responsivity of Microgels and Its Interplay with a Solid Surface by Super-Resolution Microscopy and Numerical Simulations

Super-resolution microscopy has become a powerful tool to investigate the internal structure of complex colloidal and polymeric systems, such as microgels, at the nanometer scale. An interesting feature of this method is the possibility of monitoring microgel response to temperature changes in situ. However, when performing advanced microscopy experiments, interactions between the particle and the environment can be important. Often microgels are deposited on a substrate, since they have to remain still for several minutes during the experiment. This study uses direct stochastic optical reconstruction microscopy (dSTORM) and advanced coarse-grained molecular dynamics simulations to investigate how individual microgels anchored on hydrophilic and hydrophobic surfaces undergo their volume phase transition with temperature. We find that, in the presence of a hydrophilic substrate, the structure of the microgel is unperturbed and the resulting density profiles quantitatively agree with simulations performed under bulk conditions. Instead, when a hydrophobic surface is used, the microgel spreads at the interface and an interesting competition between the two hydrophobic strengths,monomer-monomer vs monomer-surface,comes into play at high temperatures. The robust agreement between experiments and simulations makes the present study a fundamental step to establish this high-resolution monitoring technique as a platform for investigating more complex systems, these being either macromolecules with peculiar internal structure or nanocomplexes where molecules of interest can be encapsulated in the microgel network and controllably released with temperature.

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.

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.

Long-Ranged Order and Flow Alignment in Sheared p-atic Liquid Crystals

We formulate a hydrodynamic theory of p-atic liquid crystals, namely, two-dimensional anisotropic fluids endowed with generic p-fold rotational symmetry. Our approach, based on an order parameter tensor that directly embodies the discrete rotational symmetry of p-atic phases, allows us to unveil several unknown aspects of flowing p-atics, that previous theories, characterized by O(2) rotational symmetry, could not account for. This includes the onset of long-ranged orientational order in the presence of a simple shear flow of arbitrary shear rate, as opposed to the standard quasi-long-ranged order of two-dimensional liquid crystals, and the possibility of flow alignment at large shear rates.

Defect absorption and emission for p-atic liquid crystals on cones

We investigate the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle β, the apex develops an effective topological charge of -χ, where 2πχ=2π(1-sinβ) is the deficit angle of the cone, and a topological defect of charge σ behaves as if it had an effective topological charge Q_{eff}=(σ-σ^{2}/2) when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g., imposed by tangential boundary conditions at the edge, we find that for a disk the ground-state configuration consists of p defects each of charge +1/p lying equally spaced on a concentric ring of radius d=(p-1/3p-1)^{1/2p}R, where R is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground-state configurations as a function of cone angle: (i) for sharp cones, all of the +1/p defects are absorbed by the apex; (ii) at intermediate cone angles, some of the +1/p defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (iii) for nearly flat cones, all of the +1/p defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on p and the deficit angle, which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement.

Hydrodynamic theory of p-atic liquid crystals

We formulate a comprehensive hydrodynamic theory of two-dimensional liquid crystals with generic p-fold rotational symmetry, also known as p-atics, of which nematics (p=2) and hexatics (p=6) are the two best known examples. Previous hydrodynamic theories of p-atics are characterized by continuous O(2) rotational symmetry, which is higher than the discrete rotational symmetry of p-atic phases. By contrast, here we demonstrate that the discrete rotational symmetry allows the inclusion of additional terms in the hydrodynamic equations, which, in turn, lead to novel phenomena, such as the possibility of flow alignment at high shear rates, even for p>2. Furthermore, we show that any finite imposed shear will induce long-ranged orientational order in any p-atic liquid crystal, in contrast to the quasi-long-ranged order that occurs in the absence of shear. The induced order parameter scales like a nonuniversal power of the applied shear rate at small shear rates.

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.

Observing capture with a colloidal model membrane channel

We use video microscopy to study the full capture process for colloidal particles transported through microfluidic channels by a pressure-driven flow. In particular, we obtain trajectories for particles as they move from the bulk into confinement, using these to map in detail the spatial velocity and concentration fields for a range of different flow velocities. Importantly, by changing the height profiles of our microfluidic devices, we consider systems for which flow profiles in the channel are the same, but flow fields in the reservoir differ with respect to the quasi-2D monolayer of particles. We find that velocity fields and profiles show qualitative agreement with numerical computations of pressure-driven fluid flow through the systems in the absence of particles, implying that in the regimes studied here particle-particle interactions do not strongly perturb the flow. Analysis of the particle flux through the channel indicates that changing the reservoir geometry leads to a change between long-range attraction of the particles to the pore and diffusion-to-capture-like behaviour, with concentration fields that show qualitative changes based on device geometry. Our results not only provide insight into design considerations for microfluidic devices, but also a foundation for experimental elucidation of the concept of a capture radius. This long standing problem plays a key role in transport models for biological channels and nanopore sensors.

Fractional defect charges in liquid crystals with p-fold rotational symmetry on cones

Conical surfaces, with a δ function of Gaussian curvature at the apex, are perhaps the simplest example of geometric frustration. We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones. For free boundary conditions at the base, we find both the ground state(s) and a discrete ladder of metastable states as a function of both the cone angle and the liquid crystal symmetry p. We find that these states are characterized by a set of fractional defect charges at the apex and that the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction of the cone. We check our predictions for the ground-state energies numerically for a set of commensurate cone angles (corresponding to a set of commensurate Gaussian curvatures concentrated at the cone apex), whose surfaces can be polygonized as a perfect triangular or square mesh, and find excellent agreement with our theoretical predictions.

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.

Self-Assembly Dynamics of Reconfigurable Colloidal Molecules

Colloidal molecules are designed to mimic their molecular analogues through their anisotropic shape and interactions. However, current experimental realizations are missing the structural flexibility present in real molecules thereby restricting their use as model systems. We overcome this limitation by assembling reconfigurable colloidal molecules from silica particles functionalized with mobile DNA linkers in high yields. We achieve this by steering the self-assembly pathway toward the formation of finite-sized clusters by employing high number ratios of particles functionalized with complementary DNA strands. The size ratio of the two species of particles provides control over the overall cluster size, i.e., the number of bound particles N, as well as the degree of reconfigurability. The bond flexibility provided by the mobile linkers allows the successful assembly of colloidal clusters with the geometrically expected maximum number of bound particles and shape. We quantitatively examine the self-assembly dynamics of these flexible colloidal molecules by a combination of experiments, agent-based simulations, and an analytical model. Our "flexible colloidal molecules" are exciting building blocks for investigating and exploiting the self-assembly of complex hierarchical structures, photonic crystals, and colloidal metamaterials.

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.

Large-scale dynamics of event-chain Monte Carlo

Event-chain Monte Carlo (ECMC) accelerates the sampling of hard-sphere systems, and has been generalized to the potentials used in classical molecular simulations. Rather than imposing detailed balance on the transition probabilities, the method enforces a weaker global-balance condition in order to guarantee convergence to equilibrium. In this paper, we generalize the factor-field variant of ECMC to higher space dimensions. In the two-dimensional fluid phase, factor-field ECMC saturates the lower bound z=0 for the dynamical scaling exponent for local dynamics, whereas molecular dynamics is characterized by z=1 and local Metropolis Monte Carlo by z=2. In the presence of hexatic order, factor fields are not found to speed up the convergence. We note that generalizations of factor fields could couple to orientational order.

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.

Curved colloidal crystals of discoids at near-critical liquid-liquid interface

The assembly of disc-shaped particles at curved liquid-liquid interfaces was studied by using confocal microscopy. The interface is formed by a phase-separating critical liquid mixture of 2,6-lutidine and heavy water, where the colloids spontaneously assembled forming a dome. The novelty of this system is three-fold. First, the domes can be constructed and annihilated remotely and reversibly, which allows dynamic control of the colloidal assembly. Second, the effect of curvature can be investigated by analyzing domes of different radii ranging from 5 μm to 125 μm. Third, the slow dynamics due to hydrodynamic interaction among the particles can be utilized to investigate the time-evolution of defect morphology. Unlike the widely studied repulsive colloids, the interparticle potential near the critical point has an attractive component. I contrasted the packing and defects morphology of a solid-like and liquid-like dome differing in particle number density. In the solid-like dome, a chain of 5- and 7-fold coordinated particles was observed. The analysis of trajectories showed that particles were bound in a potential well of a depth of about ten times the thermal energy, which matched well with the calculation of the pair-potential by considering the attractive critical Casimir force among the particles. In the liquid-like dome, 6-fold particles separated by clusters of 5- and 7-coordinated particles were observed, which is suggestive of liquid-solid coexistence. The uniqueness of this system will open up a new research avenue to investigate the effect of varying curvature on the crystallization, defects, and phase diagram of colloidal assemblies.

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.

Spatial Crossover Between Far-From-Equilibrium and Near-Equilibrium Dynamics in Locally Driven Suspensions

We examine the response of a quasi-two-dimensional colloidal suspension to a localized circular driving induced by optical tweezers. This approach allows us to resolve over 3 orders of magnitude in the Péclet number (Pe) and provide a direct observation of a sharp spatial crossover from far- to near-thermal-equilibrium regions of the suspension. In particular, particles migrate from high to low Pe regions and form strongly inhomogeneous steady-state density profiles with an emerging length scale that does not depend on the particle density and is set by Pe≈1. We show that the phenomenological two phase fluid constitutive model is in line with our results.

Lateral depletion effect on two-dimensional ordering of bacteriorhodopsins in a lipid bilayer: A theoretical study based on a binary hard-disk model

The 2D ordering of bacteriorhodopsins in a lipid bilayer was studied using a binary hard-disk model. The phase diagrams were calculated taking into account the lateral depletion effects. The critical concentrations of the protein ordering for monomers and trimers were obtained from the phase diagrams. The critical concentration ratio agreed well with the experiment when the repulsive core interaction between the depletants, namely, lipids, was taken into account. The results suggest that the depletion effect plays an important role in the association behaviors of transmembrane proteins.

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.

Ring structure of selected two-dimensional procrystalline lattices

Recent work has introduced the term "procrystalline" to define systems which lack translational symmetry but have an underlying high-symmetry lattice. The properties of five such two-dimensional (2D) lattices are considered in terms of the topologies of rings which may be formed from three-coordinate sites only. Parent lattices with full coordination numbers of four, five, and six are considered, with configurations generated using a Monte Carlo algorithm. The different lattices are shown to generate configurations with varied ring distributions. The different constraints imposed by the underlying lattices are discussed. Ring size distributions are obtained analytically for two of the simpler lattices considered (the square and trihexagonal nets). In all cases, the ring size distributions are compared to those obtained via a maximum entropy method. The configurations are analyzed with respect to the near-universal Lemaître curve (which connects the fraction of six-membered rings with the width of the ring size distribution) and three lattices are highlighted as rare examples of systems which generate configurations which do not map onto this curve. The assortativities are considered, which contain information on the degree of ordering of different sized rings within a given distribution. All of the systems studied show systematically greater assortativities when compared to those generated using a standard bond-switching method. Comparison is also made to two series of crystalline motifs which shown distinctive behavior in terms of both the ring size distributions and the assortativities. Procrystalline lattices are therefore shown to have fundamentally different behavior to traditional disordered and crystalline systems, indicative of the partial ordering of the underlying lattices.

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.