Comment on "Penrose Tilings as Jammed Solids"

TL;DR

This paper refutes a claim that Penrose tilings have a nonzero bulk modulus when disordered, demonstrating through numerical analysis that their bulk modulus actually vanishes asymptotically, aligning them with other disordered isostatic networks.

Contribution

The authors provide a detailed numerical analysis showing that Penrose tilings do not have a nonzero bulk modulus when disordered, countering previous claims and clarifying their suitability as models of jammed packings.

Findings

Bulk modulus of Penrose tilings vanishes asymptotically.

Previous analysis only considered one disorder level, leading to incorrect conclusions.

Penrose tilings are not superior models of jammed packings compared to other isostatic networks.

Abstract

In a recent letter, Stenull and Lubensky claim that periodic approximants of Penrose tilings, which are generically isostatic, have a nonzero bulk modulus B when disordered, and, therefore, Penrose tilings are good models of jammed packings. The claim of a nonzero B, which is made on the basis of a normal mode analysis of periodic Penrose approximants for a single value of the disorder epsilon, is the central point of their letter: other properties of Penrose tilings, such as the vanishing of the shear modulus, and a flat density of vibrational states, are already shared by most geometrically disordered isostatic networks studied so far. In this comment, Conjugate Gradient is used to solve the elastic equations on approximants with up to 8x10^4 sites for several values of epsilon, to show beyond reasonable doubt that Stenull and Lubensky's claim is incorrect. The bulk modulus of generic Penrose tilings is zero asymptotically. According to our results, B grows as (epsilon^2 L^3) when (epsilon^2 L^3) << 10^2, then saturates, and finally decays as (epsilon^2 L^3)^{-2/3} ~ 1/L^2 for epsilon^2 L^3 >> 10^2. Stenull and Lubensky seem to have only analyzed one value of epsilon for which saturation is reached at the largest size studied. This led them to a wrong conclusion. We support our results by also considering generic Penrose approximants with fixed boundaries, whose bulk modulus constitutes a strict upper bound for that of periodic systems, finding that these have a vanishing B as well for large L. We conclude that the main point in Stenull and Lubensky letter is unjustified. Penrose tilings are no better models of jammed packings than any of the previously studied isostatic networks with geometric disorder.