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Penrose tilings as jammed solids.

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This summary is machine-generated.

Penrose tilings, similar to jammed systems, exhibit vanishing elastic moduli despite having many zero modes. Randomizing their structure reveals properties akin to particulate jamming, with a bulk modulus but no shear modulus.

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Area of Science:

  • Physics
  • Materials Science
  • Mathematical Physics

Background:

  • Penrose tilings possess unique properties like fivefold symmetry and isotropic elasticity.
  • Their inhomogeneous coordination resembles force networks in jammed systems.
  • Under periodic boundary conditions, these lattices exhibit an average coordination of four.

Purpose of the Study:

  • Investigate the elastic and vibrational properties of rational approximants to Penrose tilings.
  • Analyze how these properties scale with unit-cell size.
  • Compare the behavior of idealized Penrose lattices with randomized, jammed particulate systems.

Main Methods:

  • Studied rational approximants of Penrose tilings.
  • Analyzed elastic and vibrational properties as a function of unit-cell size N(S).
  • Introduced randomization to site positions to model generic forms.

Main Results:

  • Rational approximants exhibit of order sqrt[N(S)] zero modes and states of self-stress.
  • All elastic moduli vanish for these approximants.
  • Randomized Penrose lattices show a non-zero bulk modulus and a vanishing shear modulus.
  • A flat density of states was observed in the randomized systems.

Conclusions:

  • Penrose tiling approximants share characteristics with jammed systems, including numerous zero modes and vanishing elastic moduli.
  • Randomized Penrose lattices mimic the mechanical behavior of particulate systems at jamming.
  • The findings bridge concepts from quasicrystals, network mechanics, and jamming physics.