Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

Widom, Michael; Destainville, Nicolas; Mosseri, Rémy; Bailly, Francis

doi:10.1007/s10955-005-6998-x

J Stat Phys

2005

DOI: 10.1007/s10955-005-6998-x

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Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

Michael Widom

Nicolas Destainville

Rémy Mosseri

et al.

Abstract: We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in th… Show more

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Cited by 5 publications

(16 citation statements)

References 30 publications

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“…-The computational power provided by the Transition matrix Monte Carlo technique allowed us to investigate random tilings with 2D-fold rotational symmetry in their large size, large D limit. The numerical entropies obtained strongly support the analytic predictions of previous publications [10,11] and go into the direction of a large D "universal" entropy independent of size, shape, tile fractions and boundary conditions. This work shows that the knowledge of a few adjustable parameters that can be estimated by finite D and finite p fits is sufficient to get a good estimate of finite D entropies of physical interest (see Eq.…”

supporting

confidence: 84%

“…To verify them further, we also calculate the large D limit for finite p. The case p = 1 was solely considered in Ref. [11]. We were able to perform more intensive numerical calculations here.…”

mentioning

confidence: 99%

“…In Refs. [10,11], an original analytic mean-field theory for plane rhombus tilings with 2D-fold symmetry, in the large D limit, was proposed. Its ultimate goal is to derive valuable results on finite D tilings by estimating finite D corrections to the c EDP Sciences infinite D limit.…”

mentioning

confidence: 99%

“…Therefore we anticipate that the relative errors for larger values of D are also smaller than 1%. To test the predictions stated above, we also extrapolate the limit σ ∞ = lim D→∞ σ D = 0.563±0.006 by fitting the finite D values at order 2 in 1/D [11]. The error bar also corresponds to a relative error of 1%.…”

mentioning

confidence: 99%

“…We find consistently σ ∞ (p = 2) = σ ∞ (p = 3) = 0.568 ± 0.006. Then we consider boundary strains [11], with high frequency, (p α ) = (1, p, . .…”

mentioning

confidence: 99%

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Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry

Destainville

2006

Eur. Phys. J. B

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Abstract. -We perform Transition matrix Monte Carlo simulations to evaluate the entropy of rhombus tilings with fixed polygonal boundaries and 2D-fold rotational symmetry. We estimate the large-size limit of this entropy for D = 4 to 10. We confirm analytic predictions of N. Destainville et al., J. Stat. Phys. 120, 799 (2005) and M. Widom et al., J. Stat. Phys. 120, 837 (2005), in particular that the large size and large D limits commute, and that entropy becomes insensible to size, phason strain and boundary conditions at large D. We are able to infer finite D and finite size scalings of entropy. We also show that phason elastic constants can be estimated for any D by measuring the relevant perpendicular space fluctuations.Random tiling models [1, 2] have been intensively studied since the discovery of quasicrystals in 1984, because they are good paradigmatic models of quasicrystals. These metallic compounds exhibit exotic symmetries (e.g. icosahedral) which are classically forbidden by crystallographic rules. This is accounted for by the existence of underlying quasiperiodic or random tilings. When tiles are decorated in some manner by atoms, these tilings become excellent candidates for modeling real quasycristalline compounds [3]. As compared to perfectly quasiperiodic tilings, some specific degrees of freedom, the so-called phason flips, are activated in random tilings, giving access to a large number of microscopic configurations. The number of configurations grows exponentially with the system size in contrast to perfectly quasiperiodic tilings where it only grows polynomially. The resulting configurational entropy lowers the free energy as compared to competing crystalline phases. Despite their random character, random tilings still display the required macroscopic point symmetries in their Fourier spectra. They are as good candidates as perfectly quasiperiodic tilings for modeling quasicrystals [1,2]. The statistical mechanics of random tilings is of central interest for quasi-crystallography. But the calculation of the thermodynamical observables of interest in random tiling models has turned out to be a formidable task. Even the calculation of configurational entropy in the case of maximally random tilings where all tilings have the same energy is a notoriously difficult problem. Very few models are exactly solvable [4][5][6][7][8][9], and a large majority of calculations rely on numerical simulations. In Refs. [10,11], an original analytic mean-field theory for plane rhombus tilings with 2D-fold symmetry, in the large D limit, was proposed. Its ultimate goal is to derive valuable results on finite D tilings by estimating finite D corrections to the c EDP Sciences

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supporting

confidence: 84%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…We find consistently σ ∞ (p = 2) = σ ∞ (p = 3) = 0.568 ± 0.006. Then we consider boundary strains [11], with high frequency, (p α ) = (1, p, . .…”

mentioning

confidence: 99%

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Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry

Destainville

2006

Eur. Phys. J. B

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Random Tilings of High Symmetry: I. Mean-Field Theory

et al. 2005

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We study random tiling models in the limit of high rotational symmetry. In this limit a mean-field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions. We base our mean-field theory on an iterative tiling construction inspired by the work of de Bruijn. In addition to the entropy, we consider correlation functions, phason elasticity and the thermodynamic limit. Tilings of dimension other than two are considered briefly. FIG. 1: Examples of free boundary d = 2 tilings with D = 3, 5, 7 and 9.

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Entropically stabilized growth of a two-dimensional random tiling

et al. 2010

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The assembly of molecular networks into structures such as random tilings and glasses has recently been demonstrated for a number of two-dimensional systems. These structures are dynamicallyarrested on experimental timescales so the critical regime in their formation is that of initial growth. Here we identify a transition from energetic to entropic stabilisation in the nucleation and growth of a molecular rhombus tiling. Calculations based on a lattice gas model show that clustering of topological defects and the formation of faceted boundaries followed by a slow relaxation to equilibrium occurs under conditions of energetic stabilisation. We also identify an entropicallystabilised regime in which the system grows directly into an equilibrium configuration without the need for further relaxation. Our results provide a methodology for identifying equilibrium and nonequilibrium randomness in the growth of molecular tilings, and we demonstrate that equilibrium spatial statistics are compatible with exponentially slow dynamical behaviour.The properties of two-dimensional supramolecular networks have been the focus of growing interest in recent years with most efforts directed towards the controlled introduction of translational order into such systems [1,2]. However there have been several recent observations of surface-bound supramolecular arrays which assemble into dynamically-arrested structures akin to glasses [3][4][5] which lack translational order. Such arrangements raise many interesting questions related to the growth of random systems [6,7]. In particular it is important to distinguish randomising effects which arise from kinetic effects, such as nucleation [8,9], from equilibrium disorder due to entropic terms in the free energy. Entropically-stabilised disorder may be regarded as intrinsic randomness, whereas kinetically-driven disorder is often determined by sample history and preparative conditions. In one recent study [3] a random molecular rhombus tiling was shown to have equilibrium (maximum entropy) spatial correlations, despite being frozen on an experimental timescale. In such a system the maximum entropy configuration must form, and be frozen in, during the initial growth, since the spatio-temporal fluctuations which normally facilitate the evolution of kinetically-trapped configurations to equilibrium are absent. However, it is not clear, a priori, that there is a set of local rules for molecular attachment which can lead to the direct growth of a 'perfect', i.e. maximum entropy, configuration.In this paper we address this question and show that equilibrium and non-equilibrium effects in the growth of a rhombus tiling [10-17] may be distinguished using tiletile correlations of arrays simulated using a lattice gas model [18]. Direct growth to a configuration with equilibrium statistics occurs when entropic terms dominate the free energy, while non-equilibrium effects result in faceted islands and clustering of topological defects.The parameters which control growth are the tile-tile interaction...

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Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

Cited by 5 publications

References 30 publications

Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry

Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry

Random Tilings of High Symmetry: I. Mean-Field Theory

Entropically stabilized growth of a two-dimensional random tiling

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