An accurate and simple equation of state for hard disks

Santos, Andrés; Haro, M. López de; Yuste, S. B.

doi:10.1063/1.470649

Cited by 105 publications

(96 citation statements)

References 33 publications

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“…However, its magnitude (amplitude) does depend on the area fraction both directly and indirectly via the dimensionless isothermal compressibility, S(0). One can gain intuition concerning this variation from the 2D hard disk equation of state [36]. The amplitude is plotted in the inset of Fig.…”

Section: B Analytic Limitsmentioning

confidence: 99%

Correlated two-particle diffusion in dense colloidal suspensions at early times: Theory and comparison to experiment

et al. 2015

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The spatially resolved diffusive dynamic cross correlations of a pair of colloids in dense quasi-two-dimensional monolayers of identical particles are studied experimentally and theoretically at early times where motion is Fickian. In very dense systems where strong oscillatory equilibrium packing correlations are present, we find an exponential decay of the dynamic cross correlations on small and intermediate length scales. At large separations where structure becomes random, an apparent power law decay with an exponent of approximately −2.2 is observed. For a moderately dense suspension where local structural correlations are essentially absent, this same apparent power law decay is observed over all probed interparticle separations. A microscopic nonhydrodynamic theory is constructed for the dynamic cross correlations which is based on interparticle frictional effects and effective structural forces. Hydrodynamics enters only via setting the very short-time single-particle self-diffusion constant. No-adjustable-parameter quantitative predictions of the theory for the dynamic cross correlations are in good agreement with experiment over all length scales. The origin of the long-range apparent power law is the influence of the constraint of fixed interparticle separation on the amplitude of the mean square force exerted on the two tagged particles by the surrounding fluid. The theory is extended to study high-packing-fraction 3D hard sphere fluids. The same pattern of an oscillatory exponential form of the dynamic cross correlation function is predicted in the structural regime, but the long-range tail decays faster than in monolayers with an exponent of −3.

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Section: B Analytic Limitsmentioning

confidence: 99%

Correlated two-particle diffusion in dense colloidal suspensions at early times: Theory and comparison to experiment

et al. 2015

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“…al. [23] for ρ ≤ 0.85. From this equation of state we can obtain the Helmholtz and hence the Gibbs free energy by integrating starting from the value given by Eq.…”

Section: A Equation Of State Free Energy and First Order Meltingmentioning

confidence: 99%

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

2000

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Elastic moduli and dislocation core energy of the triangular solid of hard disks of diameter σ are obtained in the limit of vanishing dislocation-antidislocation pair density, from Monte Carlo simulations which incorporates a constraint, namely that all moves altering the local connectivity away from that of the ideal triangular lattice are rejected. In this limit, we show that the solid is stable against all other fluctuations at least upto densities as low as ρσ 2 = 0.88. Our system does not show any phase transition so diverging correlation lengths leading to finite size effects and slow relaxations do not exist. The dislocation pair formation probability is estimated from the fraction of moves rejected due to the constraint which yields, in turn, the core energy E c and the (bare) dislocation fugacity y. Using these quantities, we check the relative validity of first order and Kosterlitz -Thouless -HalperinNelson -Young (KTHNY) melting scenarios and obtain numerical estimates of the typical expected transition densities and pressures. We conclude that a KTHNY transition from the solid to a hexatic phase preempts the solid to * on leave from: Material Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India 1 liquid first order transition in this system albeit by a very small margin, easily masked by crossover effects in unconstrained "brute -force" simulations with small number of particles.

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“…It is then straightforward to calculate the average overlap energy for a given aggregate morphology. Moreover, for each morphology, we consider either fluid or crystalline order in the clusters; the entropy is estimated using an empirical equation of state for the hard-disk fluid [17] or the cellular theory of the hard-disk crystal [18]. The result is a closed, albeit cumbersome, analytic form as a function of temperature, density, and the two structural parameters, cluster size and lattice spacing (the latter are fixed via Lagrange multipliers).…”

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confidence: 99%

Soft spheres make more mesophases

et al. 2007

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We use both mean-field methods and numerical simulation to study the phase diagram of classical particles interacting with a hard-core and repulsive, soft shoulder. Despite the purely repulsive interaction, this system displays a remarkable array of aggregate phases arising from the competition between the hard-core and shoulder length scales. In the limit of large shoulder width to core size, we argue that this phase diagram has a number of universal features, and classify the set of repulsive shoulders that lead to aggregation at high density. Surprisingly, the phase sequence and aggregate size adjusts so as to keep almost constant inter-aggregate separation.PACS numbers: 61.30. Dk, 61.30.St,61.46.Bc Entropy is a potent force in the theory of selfassembly. It can be argued, through entropic considerations alone, that hard spheres will self-assemble into the face-centered-cubic (fcc) lattice or any of its many variants related through stacking faults. As a result, when a material exhibits an fcc phase, it is often attributed to the optimality of the close-packed lattice. When less common or less dense lattices are formed, all sundry of explanations are invoked, ranging from quantum mechanics [1], lattice effects [2], partially filled Landau levels [3], and even soft interactions [4,5]. While there has been concerted effort to tailor the pair interaction to achieve a desired periodic arrangement [6] this must be done in the context of those packing motifs that arise from generic interactions. For instance, it would be no trick to tailor a potential to make an fcc lattice.With this in mind, here we consider a seemingly simple extension of the hard sphere model, namely a hard core interaction of radius σ with a soft shoulder of radius σ s and height ǫ (HCSS):When σ s /σ 1 this potential models hard spheres with a soft pair repulsion and was used to study isostructural transitions in Cs and Ce [7]. In generic repulsive potentials, it has been shown that as the range of the soft repulsion grows (corresponding to σ s /σ ≈ 2) a rich variety of density-modulated ground states appear [8, 9, 10] which can be characterized as periodic arrangements of regular sized clusters of the original spheres. In this Letter we establish a sufficient condition on the pair potential for clustering and the subsequent ordering of the clusters which generalizes results on soft potentials without hard cores [11]. We develop a self-consistent field theory for soft repulsion and use it to study the formation of striped phases. We corroborate our analytic treatment with numerical solutions that also predict the existence of hexagonal and inverted hexagonal phases with both fluid and crystalline order in the clusters, as shown in Fig. 1. We also present results from Monte Carlo simulations of the HCSS potential which both stimulate and support the more general results. In all cases, we find that over the range of stable aggregate structures the lattice constant remains fixed while the clusters change their size and morphology so as to maint...

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An accurate and simple equation of state for hard disks

Cited by 105 publications

References 33 publications

Correlated two-particle diffusion in dense colloidal suspensions at early times: Theory and comparison to experiment

Correlated two-particle diffusion in dense colloidal suspensions at early times: Theory and comparison to experiment

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

Soft spheres make more mesophases

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