Crystallization in a One-Dimensional Periodic Landscape

Friedrich, Manuel; Stefanelli, Ulisse

doi:10.1007/s10955-020-02537-9

Cited by 9 publications

(8 citation statements)

References 21 publications

(35 reference statements)

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“…Brascamp and Lieb [11] as well as Aizenman and Martin [1] gave important results on the optimality of the equidistant configuration concerning the Jellium model, see also [31]. The emergence of crystallized state for one-dimensional system embedded in a periodic energy landscape is furthermore studied in [18] by Friedrich and Stefanelli where it is shown that equidistant ground states are generally not expected. Furthermore, Blanc and Le Bris [9] proved the periodicity of the ground state for the one-dimensional Thomas-Fermi-von-Weizsäcker energy.…”

Section: Introductionmentioning

confidence: 99%

Note on Crystallization for Alternating Particle Chains

2020

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We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

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Section: Introductionmentioning

confidence: 99%

Note on Crystallization for Alternating Particle Chains

2020

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“…Recent results for positive temperature including an analysis of boundary layers are obtained in [34,35]. For results on dimers we refer to [6,29].) The first rigorous results for a two-dimensional system were achieved in [32,33,43]; see also the recent paper [18].…”

Section: Introductionmentioning

confidence: 99%

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Friedrich

Kreutz

Schmidt

2021

Arch Rational Mech Anal

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We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of $$\Gamma $$ Γ -convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, that is, on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum–solid and solid–solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that, due to the extremely brittle setup, interpolating boundary layers near cracks are energetically not favorable.

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“…The mechanical behaviour of one-dimensional systems has been of interest for decades. Such systems serve as toy models for higher-dimensional theoretical investigations and are of interest with respect to one-dimensional structures, see, e.g., [7,8,10,11,19]. In order to understand the effective behaviour of materials, the systems are studied as the number of particles tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

Onset of fracture in random heterogeneous particle chains

Lauerbach¹,

Neukamm²,

Schäffner³

et al. 2021

Preprint

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In mechanical systems it is of interest to know the onset of fracture in dependence of the boundary conditions. Here we study a one-dimensional model which allows for an underlying heterogeneous structure in the discrete setting. Such models have recently been studied in the passage to the continuum by means of variational convergence (Γ-convergence). The Γ-limit results determine thresholds of the boundary condition, which mark a transition from purely elastic behaviour to the occurrence of a crack. In this article we provide a notion of fracture in the discrete setting and show that its continuum limit yields the same threshold as that obtained from the Γ-limit. Since the calculation of the fracture threshold is much easier with the new method, we see a good chance that this new approach will turn out useful in applications.

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Crystallization in a One-Dimensional Periodic Landscape

Cited by 9 publications

References 21 publications

Note on Crystallization for Alternating Particle Chains

Note on Crystallization for Alternating Particle Chains

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Onset of fracture in random heterogeneous particle chains

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