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# Force distribution affects vibrational properties in hard-sphere glasses

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 21, 2014 (received for review August 8, 2014)

## Significance

How a liquid becomes rigid at the glass transition is a central problem in condensed matter physics. In many scenarios of the glass transition, liquids go through a critical temperature below which minima of free energy appear. However, even in the simplest glass, hard spheres, what confers mechanical stability at large density is highly debated. In this work we show that to quantitatively understand stability at a microscopic level, the presence of weakly interacting pairs of particles must be included. This approach allows us to predict various nontrivial scaling behavior of the elasticity and vibrational properties of colloidal glasses that can be tested experimentally. It also gives a spatial interpretation to recent, exact calculations in infinite dimensions.

## Abstract

We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force *f* soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting *i*) the density of states has a low-frequency peak at a scale *ω* is the frequency, (*ii*) shear modulus and mean-squared displacement are inversely proportional with *iii*) continuum elasticity breaks down on a scale *z* is the coordination and *d* the spatial dimension. We numerically test (*i*) and provide data supporting that

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ed87{at}nyu.edu.

Author contributions: E.D., E.L., C.B., and M.W. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1415298111/-/DCSupplemental.

^{a}Center for Soft Matter Research, New York University, New York, NY 10003;^{b}Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, 94485 Amsterdam, The Netherlands; and^{c}Instituto de Fisica da Universidade Federal do Rio Grande do Sul, C.P. 15051 Porto Alegre, RS, Brazil

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 21, 2014 (received for review August 8, 2014)

## Significance

How a liquid becomes rigid at the glass transition is a central problem in condensed matter physics. In many scenarios of the glass transition, liquids go through a critical temperature below which minima of free energy appear. However, even in the simplest glass, hard spheres, what confers mechanical stability at large density is highly debated. In this work we show that to quantitatively understand stability at a microscopic level, the presence of weakly interacting pairs of particles must be included. This approach allows us to predict various nontrivial scaling behavior of the elasticity and vibrational properties of colloidal glasses that can be tested experimentally. It also gives a spatial interpretation to recent, exact calculations in infinite dimensions.

## Abstract

We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force *f* soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting $\mathbf{\mathbb{P}}\left(f\right)\sim {f}^{{\theta}_{e}}$, the force distribution of such pairs and ${\varphi}_{c}$ the packing fraction at which pressure diverges, we predict that (*i*) the density of states has a low-frequency peak at a scale $\omega *$, rising up to it as $D\left(\omega \right)\sim {\omega}^{2+a}$, and decaying above $\omega *$ as $D\left(\omega \right)\sim {\omega}^{-a}$ where $a=\left(1-{\theta}_{e}\right)/\left(3+{\theta}_{e}\right)$ and *ω* is the frequency, (*ii*) shear modulus and mean-squared displacement are inversely proportional with $\u3008\delta {R}^{2}\u3009\sim 1/\mu \sim {\left({\varphi}_{c}-\varphi \right)}^{\kappa}$, where $\kappa =2-2/\left(3+{\theta}_{e}\right)$, and (*iii*) continuum elasticity breaks down on a scale ${\ell}_{c}\sim 1/\sqrt{\delta z}\sim {\left({\varphi}_{c}-\varphi \right)}^{-b}$, where $b=\left(1+{\theta}_{e}\right)/\left(6+2{\theta}_{e}\right)$ and $\delta z=z-2d$, where *z* is the coordination and *d* the spatial dimension. We numerically test (*i*) and provide data supporting that ${\theta}_{e}\approx 0.41$ in our bidisperse system, independently of system preparation in two and three dimensions, leading to $\kappa \approx 1.41$, $a\approx 0.17$, and $b\approx 0.21$. Our results for the mean-square displacement are consistent with a recent exact replica computation for $d=\infty $, whereas some observations differ, as rationalized by the present approach.

- colloids
- glass transition
- marginal stability
- boson peak
- jamming

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ed87{at}nyu.edu.

Author contributions: E.D., E.L., C.B., and M.W. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1415298111/-/DCSupplemental.

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