Theory Of Two-dimensional Melting Research Articles

Topological phase diagrams of the frustrated Ising ferromagnet

The emergence of complex modulated structures in the magnetization pattern of thin films is a well-established experimental phenomenology caused by the frustrating effects of competing interactions. Using a coarse-grained version of the Ising ferromagnet with dipolar interactions, we develop a method that uses the information from the microscopic Hamiltonian to predict the specific topological phases present in the temperature-external magnetic field phase diagram. This is done by the combination of mean-field variational calculations and the renormalization group equations from the classical theory of two-dimensional melting. In this framework, we are able to distinguish when the orientational and translational symmetries are broken, discriminating between the ordered and disordered states of the system for all temperatures and fields. We observe that the reentrance developed by the $H$-$T$ phase diagrams in the regime of weak dipolar interactions is directly related with the appearance of anomalous topological transitions. These results motivate the realization of new experiments on magnetic thin films in order to explore the topological properties of the magnetic textures, allowing to identify new exotic phases in these materials.

Hexatic smectic phase with algebraically decaying bond-orientational order

The hexatic phase predicted by the theories of two-dimensional melting is characterized by the power-law decay of the orientational correlations, whereas the in-layer bond orientational order in all the hexatic smectic phases observed so far was found to be long range. We report a hexatic smectic phase where the in-layer bond orientational correlations decay algebraically, in quantitative agreement with the hexatic ordering predicted by the theory for two dimensions. The phase was formed in a molecular dynamics simulation of a one-component system of particles interacting via a spherically symmetric potential. The present results thus demonstrate that the theoretically predicted two-dimensional hexatic order can exist in a three-dimensional system.

Note: Melting criterion for soft particle systems in two dimensions

A simple criterion for melting of two-dimensional crystals with soft long-ranged interactions is proposed. It states that the ratio of the transverse sound velocity of an ideal crystalline lattice to the thermal velocity is a quasi-universal number close to $4.3$ at melting. This criterion is arrived by reference to the Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young theory of two-dimensional melting, combined with the observation that the ratio of transverse-to-longitudinal sound velocities is small in the soft interaction limit. Application of this criteria allows estimating melting lines in a simple yet relatively accurate manner. Two-dimensional weakly screened Yukawa systems represent one relevant example considered.

Observation of the Chiral and Achiral Hexatic Phases of Self-assembled Micellar polymers.

We report the discovery of a thermodynamically stable line hexatic (N + 6) phase in a three-dimensional (3D) system made up of self-assembled polymer-like micelles of amphiphilic molecules. The experimentally observed phase transition sequence nematic (N) N + 6 two-dimensional hexagonal (2D-H) is in good agreement with the theoretical predictions. Further, the present study also brings to light the effect of chirality on the N + 6 phase. In the chiral N + 6 phase the bond-orientational order within each “polymer” bundle is found to be twisted about an axis parallel to the average polymer direction. This structure is consistent with the theoretically envisaged Moiré state, thereby providing the first experimental demonstration of the Moiré structure. In addition to confirming the predictions of fundamental theories of two-dimensional melting, these results are relevant in a variety of situations in chemistry, physics and biology, where parallel packing of polymer-like objects are encountered.

Modes of surface premelting in colloidal crystals composed of attractive particles.

Crystal surfaces typically melt into a thin liquid layer at temperatures slightly below the melting point of the crystal. Such surface premelting is prevalent in all classes of solids and is important in a variety of metallurgical, geological and meteorological phenomena. Premelting has been studied using X-ray diffraction and differential scanning calorimetry, but the lack of single-particle resolution makes it hard to elucidate the underlying mechanisms. Colloids are good model systems for studying phase transitions because the thermal motions of individual micrometre-sized particles can be tracked directly using optical microscopy. Here we use colloidal spheres with tunable attractions to form equilibrium crystal-vapour interfaces, and study their surface premelting behaviour at the single-particle level. We find that monolayer colloidal crystals exhibit incomplete premelting at their perimeter, with a constant liquid-layer thickness. In contrast, two- and three-layer crystals exhibit conventional complete melting, with the thickness of the surface liquid diverging as the melting point is approached. The microstructures of the surface liquids differ in certain aspects from what would be predicted by conventional premelting theories. Incomplete premelting in the monolayer crystals is triggered by a bulk isostructural solid-solid transition and truncated by a mechanical instability that separately induces homogeneous melting within the bulk. This finding is in contrast to the conventional assumption that two-dimensional crystals melt heterogeneously from their free surfaces (that is, at the solid-vapour interface). The unexpected bulk melting that we observe for the monolayer crystals is accompanied by the formation of grain boundaries, which supports a previously proposed grain-boundary-mediated two-dimensional melting theory. The observed interplay between surface premelting, bulk melting and solid-solid transitions challenges existing theories of surface premelting and two-dimensional melting.

Membrane Fluctuations Destabilize Clathrin Protein Lattice Order

We develop a theoretical model of a clathrin protein lattice on a flexible cell membrane. The clathrin subunit is modeled as a three-legged pinwheel with elastic deformation modes and intersubunit binding interactions. The pinwheels are constrained to lie on the surface of an elastic sheet that opposes bending deformation and is subjected to tension. Through Monte Carlo simulations, we predict the equilibrium phase behavior of clathrin lattices at various levels of tension. High membrane tensions, which correspond to suppressed membrane fluctuations, tend to stabilize large, flat crystalline structures similar to plaques that have been observed in vivo on cell membranes that are adhered to rigid surfaces. Low tensions, on the other hand, give rise to disordered, defect-ridden lattices that behave in a fluidlike manner. The principles of two-dimensional melting theory are applied to our model system to further clarify how high tensions can stabilize crystalline order on flexible membranes. These results demonstrate the importance of environmental physical cues in dictating the collective behavior of self-assembled protein structures.

Quantitative Characterization of Hexagonal Packings in Nanoporous Alumina Arrays: A Case Study

Herein is presented a methodology to quantify the degree of hexagonal order in nanoporous alumina arrays (NAA). The approach is inspired by the theory of two-dimensional melting, developed to describe phase transitions in two-dimensional systems that present liquid-crystal-like structures. A local order parameter (LOP) is defined to quantify the degree of hexagonal order of each pore without any arbitrary parameters. Using this LOP, three main qualitative and quantitative analytical tools were developed: (i) a color code to create a map of the LOP, which is a visual tool to identify the degree of order; (ii) quantitative measurements of the average hexagonal order of the sample by measuring the distribution of the LOP and the distribution of the number of neighbors of each pore, and (iii) a quantification of the spatial correlation of the LOP, which indicates how far the hexagonal order is spread in a sample. Because this approach has a strong support on tools developed in statistical mechanics, one can go beyond a simple characterization and interpret the results in terms of phases, as in other physical systems. This may help to unveil the mechanisms behind the self-organization process and long-range order observed in NAA. Moreover, this approach can be trivially extended to characterize other physical systems that form hexagonal packings.

10.1007/s11447-008-2018-8

Highly ordered three-dimensional dust structures are created in a striated glow discharge, and their horizontal cross-sectional images are analyzed. Calculated correlation functions, local correlation parameters, and corresponding approximations are used to classify the state of a structure according to the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) two-dimensional melting theory and a phenomenological approach. An orientational map based on an orientational parameter is proposed to expose domains in a cross section of a structure. It is shown that a plasma crystal is a polycrystal consisting of hexagonal domains (crystallites). Thermophoretic forces are used to create corners of various angles in the perimeter of the structure. Transition between hexagonal and square cell shapes is observed.

Ordered dust structures in a glow discharge

Highly ordered three-dimensional dust structures are created in a striated glow discharge, and their horizontal cross-sectional images are analyzed. Calculated correlation functions, local correlation parameters, and corresponding approximations are used to classify the state of a structure according to the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) two-dimensional melting theory and a phenomenological approach. An orientational map based on an orientational parameter is proposed to expose domains in a cross section of a structure. It is shown that a plasma crystal is a polycrystal consisting of hexagonal domains (crystallites). Thermophoretic forces are used to create corners of various angles in the perimeter of the structure. Transition between hexagonal and square cell shapes is observed.

Thickness-Dependent Thermal Behavior due to Molecular Tilt Coupling Strength in Free-Standing 70.7 Thin Films

ABSTRACT We report for the first time the thermal study near the Sm-C–Sm-I critical point in free-standing thin films of 70.7 compound. The profiles of the Sm-C–Sm-I heat-capacity anomalies are found to exhibit an unexpected crossover behavior with film thickness in reduced dimensions. The data from two-layer films are not consistent with the predictions of two-dimensional melting theory. Our results also provide evidence in support of the recent theory proposed by Defontaines and Prost. Moreover, we confirm the existence of a surface hexatic Sm-L phase in 70.7 films of less than eight layers whose phase sequence is however dramatically different from that observed in the thicker films.

Liquid crystalline states for two-dimensional electrons in strong magnetic fields

Based on the Kosterlitz-Thouless-Halperin-Nelson-Young theory of two-dimensional melting and the analogy between Laughlin states and the two-dimensional one-component plasma, we investigate the possibility of liquid crystalline states in a single Landau level (LL). We introduce many-body trial wave functions that are translationally invariant but possess twofold (i.e., nematic), fourfold (tetratic), or sixfold (hexatic) broken rotational symmetry at respective filling factors $\ensuremath{\nu}=1/3,$ 1/5, and 1/7 of the valence LL. We find that the above liquid crystalline states exhibit a soft charge-density wave (CDW) which underlies the translationally invariant state but which is destroyed by quantum fluctuations. By means of Monte Carlo simulations, we determine that, for a considerable variety of interaction potentials, the anisotropic states are energetically unfavorable for the lowest and first excited LL's (with index $L=0,1),$ whereas the nematic is favorable at the second excited LL $(L=2).$

Melting of two-dimensional systems: Dependence of the type of transition on the radius of the potential

In the framework of the two-dimensional melting theory based on the density functional approach, we use a Monte Carlo computer simulation to study melting of a two-dimensional hard disk system with a rectangular-well attraction potential. We show that depending on the attraction radius, the melting can occur via a single first-order transition as well as continuously in accordance with the Kosterlitz–Thouless–Halperin–Nelson–Young theory.

Calorimetric and Optical-Reflectivity Study of Thin Free-Standing 64COOBC Liquid-Crystal Films

High-resolution heat-capacity and optical-reflectivity measurements have been conducted on thin free-standing films of n-hexyl-4′-n-pentanoyloxy-biphenyl-4-carboxylate (64COOBC). The data suggest incomplete layer-by-layer liquid-hexatic transitions. The transition in extremely thin films shows a diverging heat-capacity anomaly. The results are compared with those reported in other compounds, and discussed in the context of the theory of two-dimensional melting.

Heat Capacity Study of Monolayer Propane on Graphite

Heat capacity measurements have been conducted for monolayer propane films adsorbed on graphite in the vicinity ofthe melting transition. Films at ten coverages between 0.4 and 1.07 layers were studied. For coverages below n = 1.0 a very weak heat capacity peak of height ∼2 C/Nk B , is found at melting; the transition is centered about 67 K, in good agreement with the melting temperature found in structural studies. Above n = 1.0 the melting peak moves to higher temperatures with increasing coverage. Data for propane are compared to those found for other hydrocarbon systems on graphite and to current theories of two-dimensional melting.

Apparently universal changes in the elastic properties of monomolecular films

Surface light scattering has been used to investigate various spread monolayers at the air-water interface. In particular, the variation of the dilatational elastic modulus has been studied as the films undergo a phase transition from the vapor to a condensed phase. In all cases studied this modulus increases discontinuously on condensation of the film. The increases are, within the uncertainties, apparently universal, being of magnitude 4\ensuremath{\pi} when expressed in units of ${\mathrm{k}}_{\mathrm{B}}$T/A, where A is the molecular area in the monolayer. This value accords with the predictions of certain theories of two-dimensional melting.

Nature of the smectic-A-hexatic-B-crystal-B transitions of a liquid-crystal compound.

High-resolution heat-capacity, optical reflectivity, and electron-diffraction measurements have been conducted on thin free-standing films of a liquid-crystal compound exhibiting the smectic-$A$---hexatic-$B$---crystal-$B$ transitions. Despite the absence of herringbone order, the liquid-hexatic transition in extremely thin films has been found to show a pronounced pretransitional heat-capacity anomaly, in disagreement with the predictions of two-dimensional melting theory. In addition, both the surface-enhanced layer-by-layer transition near the smectic-$A$---hexatic-$B$ and an unexpected layer thinning transition above the bulk isotropic-smectic-$A$ transition temperature have been observed.

Melting transition in two dimensions: A finite-size scaling analysis of bond-orientational order in hard disks.

We describe a general and efficient method, based on computer simulations and applicable to a general class of fluids, that allows us to determine (i) bounds on the transition densities of the melting transition that are valid in the thermodynamic limit and (ii) the order of the phase transition. The bond-orientational order parameter, its susceptibility, and the compressibility are measured simulataneously on many length scales, and the latter two quantities are extrapolated to the thermodynamic limit by application of the subblock analysis method of finite-size scaling. We include a detailed analysis, related to the subblock method, of the cross correlations of the fluctuations of the density and the order parameter. The behavior of the extrapolated order parameter susceptibility yields precise upper and lower bounds for the melting and freezing densities, respectively. We apply these techniques to the two-dimensional melting transition in large systems of 16 384 hard disks using canonical Monte Carlo computer simulations. The measured bond-orientational susceptibilities are found to be incompatible with predictions of the Halperin-Nelson-Young theory of two-dimensional melting. This and the behavior of the bond-orientational cumulant are two strong pieces of evidence of a first-order phase transition.

Thermal properties of ‘stacked hexatic phases’ in liquid crystals

Abstract Ever since the hexatic phase was proposed more than 14 years ago in the context of two-dimensional melting theory, the concept of hexatic order has offered unique insight into the nature of several liquid-crystal phases, namely the hexatic-B, smectic-I and smectic-F phases. Moreover, these liquid-crystal phases have offered numerous opportunities to explore and investigate the nature of hexatic order and the related phase transitions. In this paper we present an extensive review of the thermal properties associated with the phase transitions related to these liquid-crystal hexatic phases. Although both electron-beam and X-ray diffraction studies have indicated the importance of hexatic ordering in understanding these phases, the existing experimental data from numerous thermal measurements fail to demonstrate that hexatic order, which can be represented by a complex variable and therefore belongs to the XY universality class, is the only symmetry-breaking field associated with the hexatic-B—smectic-A transition in liquid crystals.

Measurement of the dynamics of a two-dimensional dislocation-mediated melting.

The Kosterlitz-Thouless theory of two-dimensional melting predicts the proliferation of free dislocations in the hexatic phase. We have studied a number of these free dislocations in second-stage nitric-acid-intercalated graphite by two different techniques. We have measured their distance by determining the correlation length with neutron diffraction and independently the number of dislocation cores by measuring the diffusion of the nitric-acid molecules using incoherent quasielastic neutron scattering. These two measurements lead to the same values, which shows the validity of the microscopic assumptions of the theory in terms of dislocations.

Hexatic phase: microscopic approach to the Frank constant

The Frank constant of the hexatic phase appearing in the phenomenological theories of two-dimensional melting is derived in terms of microscopic characteristics of the system such as the potential and the radial distribution function.