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

Weber, H.; Marx, Dominik; Binder, Kurt

doi:10.1103/physrevb.51.14636

Cited by 174 publications

(201 citation statements)

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“…At higher densities there is another secondorder phase transition into a solid that has long-range orientational order and quasi-long range translational order, mediated by dislocation pair binding. The validity of this theory is still contested even for hard disks, 23 and its applicability to systems where there is strong coupling between orientational and translational molecular degrees of freedom is questionable. Additionally, the basic theory needs to be modified to include three independent elastic moduli as opposed to only two in the case of six-fold rotational symmetry.…”

Section: B Orientational Ordermentioning

confidence: 99%

Tetratic order in the phase behavior of a hard-rectangle system

et al. 2006

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Previous Monte Carlo investigations by Wojciechowski et al. have found two unusual phases in two-dimensional systems of anisotropic hard particles: a tetratic phase of four-fold symmetry for hard squares [Comp. Methods in Science and Tech., 10: 235-255, 2004 ], and a nonperiodic degenerate solid phase for hard-disk dimers [Phys. Rev. Lett., 66: 3168-3171, 1991 ]. In this work, we study a system of hard rectangles of aspect ratio two, i.e., hard-square dimers (or dominos), and demonstrate that it exhibits a solid phase with both of these unusual properties. The solid shows tetratic, but not nematic, order, and it is nonperiodic having the structure of a random tiling of the square lattice with dominos. We obtain similar results with both a classical Monte Carlo method using true rectangles and a novel molecular dynamics algorithm employing rectangles with rounded corners. It is remarkable that such simple convex two-dimensional shapes can produce such rich phase behavior. Although we have not performed exact free-energy calculations, we expect that the random domino tiling is thermodynamically stabilized by its degeneracy entropy, well-known to be 1.79k B per particle from previous studies of the dimer problem on the square lattice. Our observations are consistent with a KTHNY two-stage phase transition scenario with two continuous phase transitions, the first from isotropic to tetratic liquid, and the second from tetratic liquid to solid.

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Section: B Orientational Ordermentioning

confidence: 99%

Tetratic order in the phase behavior of a hard-rectangle system

et al. 2006

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“…The hexatic phase is characterized by exponential positional but quasi-long range orientational correlations. It has long been discussed whether the melting transition follows a one-step first-order scenario between the liquid and the solid (without the hexatic) as in three spatial dimensions [6]), or whether it agrees with the celebrated Kosterlitz, Thouless [7], Halperin, Nelson [8] and Young [9] (KTHNY) two-step scenario with a hexatic phase separated by continuous transitions from the liquid and the solid [10][11][12][13][14][15][16][17][18]. Two-dimensional melting was discovered [4] in the simplest particle system, the hard-disk model.…”

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Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition

2011

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Melting in two spatial dimensions, as realized in thin films or at interfaces, represents one of the most fascinating phase transitions in nature, but it remains poorly understood. Even for the fundamental hard-disk model, the melting mechanism has not been agreed on after fifty years of studies. A recent Monte Carlo algorithm allows us to thermalize systems large enough to access the thermodynamic regime. We show that melting in hard disks proceeds in two steps with a liquid phase, a hexatic phase, and a solid. The hexatic-solid transition is continuous while, surprisingly, the liquid-hexatic transition is of first-order. This melting scenario solves one of the fundamental statistical-physics models, which is at the root of a large body of theoretical, computational and experimental research. Generic two-dimensional particle systems cannot crystallize at finite temperature [1][2][3] because of the importance of fluctuations, yet they may form solids [4]. This paradox has provided the motivation for elucidating the fundamental melting transition in two spatial dimensions. A crystal is characterized by particle positions which fluctuate about the sites of an infinite regular lattice. It has long-range positional order. Bond orientations are also the same throughout the lattice. A crystal thus possesses long-range orientational order. The positional correlations of a two-dimensional solid decay to zero as a power law at large distances. Because of the absence of a scale, one speaks of "quasi-long range" order. In a two-dimensional solid, the lattice distortions preserve long-range orientational order [5], while in a liquid, both the positional and the orientational correlations decay exponentially.Besides the solid and the liquid, a third phase, called "hexatic", has been discussed but never clearly identified in particle systems. The hexatic phase is characterized by exponential positional but quasi-long range orientational correlations. It has long been discussed whether the melting transition follows a one-step first-order scenario between the liquid and the solid (without the hexatic) as in three spatial dimensions Two-dimensional melting was discovered [4] in the simplest particle system, the hard-disk model. Hard disks (of radius σ) are structureless and all configurations of nonoverlapping disks have zero potential energy. Two isolated disks only feel the hard-core repulsion, but the other disks mediate an entropic "depletion" interaction (see, e.g., [19]). Phase transitions result from an "order from disorder" phenomenon: At high density, ordered configurations can allow for larger local fluctuations, thus higher entropy, than the disordered liquid. For hard disks, no difference exists between the liquid and the gas. At fixed density η, the phase diagram is independent of temperature T = 1/k B β, and the pressure is proportional to T , as discovered by D. Bernoulli in 1738. Even for this basic model, the nature of the melting transition has not been agreed on.The hard-disk model has been simulated with the...

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“…However, the data are not in the scaling region, since their largest system contained only 400 particles. MC investigations of the bond-orientational order parameter via finite-size scaling (FSS) with the block analysis technique of 16384 particle systems were done by Weber, Marx and Binder [13,14]. They also favoured a first-order phase transition.…”

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confidence: 99%

“…But the results of such small systems are affected by large finite-size effects. Simulations performed in the last years used Monte Carlo (MC) techniques either with constant volume (NV T ensemble) [11,12,13,14,15,16,17,18,19,20,21] or constant pressure (NpT ensemble) [22,23,24]. Zollweg, Chester and Leung [11] made detailed investigations of large systems up to 16384 particles, but draw no conclusives about the order of the phase transition.…”

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The hexatic phase of the two-dimensional hard disk system

Jaster¹

2004

Physics Letters A

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We report Monte Carlo results for the two-dimensional hard disk system in the transition region. Simulations were performed in the N V T ensemble with up to 1024 2 disks. The scaling behaviour of the positional and bond-orientational order parameter as well as the positional correlation length prove the existence of a hexatic phase as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young theory.The analysis of the pressure shows that this phase is outside a possible first-order transition.Key words: Hard disk model, Two-dimensional melting, KTHNY theory PACS: 64.70.Dv, 64.60.CnThe nature of the two-dimensional melting transition has been an unsolved problem for many years [1,2]. The Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory [3,4,5,6] predicts two continuous transitions. The first transition occurs when the solid (quasi-long-range positional order, long-range orientational order) undergoes a dislocation unbinding transition to the hexatic phase (short-range positional order, quasi-long-range orientational order). The second transition is the disclination unbinding transition which transforms this hexatic phase into an isotropic phase (short-range positional and orientational order). An alternative scenario has been proposed by Chui [7,8]. He presented a theory via spontaneous generation of grain boundaries, i.e. collective excitations of dislocations. He found that grain boundaries may be generated before the dislocations unbind if the core energy of dislocations is sufficiently small, and predicted a first-order transition. This is characterized by a coexistence region of the solid and isotropic phase, while no hexatic phase exists. Another proposal was given by Glaser and Clark [2]. They considered a detailed theory where the transition is handled as a condensation of localized, thermally generated geometrical defects and found also a first-order transition. Calculations based on the density-functional approach were done by Ryzhov and

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

Cited by 174 publications

References 55 publications

Tetratic order in the phase behavior of a hard-rectangle system

Tetratic order in the phase behavior of a hard-rectangle system

Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition

The hexatic phase of the two-dimensional hard disk system

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