# Pinching a glass reveals key properties of its soft spots

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Edited by James S. Langer, University of California, Santa Barbara, CA, and approved January 27, 2020 (received for review November 13, 2019)

## Significance

Glasses form when liquids are quickly cooled. Many of the properties of glasses are universal—i.e., independent of their composition and the liquid-phase temperature

## Abstract

It is now well established that glasses feature quasilocalized nonphononic excitations—coined “soft spots”—, which follow a universal

Understanding the micromechanical, statistical, and thermodynamic properties of soft, nonphononic excitations in structural glasses remains one of the outstanding challenges in glass physics, despite decades of intensive research (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–19). Soft, nonphononic excitations are believed to give rise to a broad range of glassy phenomena, many of which are still poorly understood; some noteworthy examples include the universal thermodynamic and transport properties of glasses at temperatures of 10 K and lower (2, 4, 20⇓–22); the low-temperature yielding transition in which a mechanically loaded brittle glass fails via the formation of highly localized bands of plastic strain (23, 24); and anomalous, non-Rayleigh wave-attenuation rates (25⇓–27).

Computational studies have been invaluable in advancing our knowledge about the statistical and mechanical properties of soft, glassy excitations and in revealing the essential roles that these excitations play in various glassy phenomena. Schober and Laird (28, 29) were the first to reveal the existence of soft spots in the form of low-frequency, quasilocalized vibrational modes in a model computer glass. Soon later, Schober et al. (30) showed that relaxation events deep in the glassy state exhibit patterns that resemble quasilocalized modes (QLMs), suggesting a link between soft, glassy structures and dynamics. In an important subsequent work (31), this link was further strengthened by showing that relaxational dynamics in supercooled liquids strongly correlates with quasilocalized, low-frequency vibrational modes measured in underlying inherent states. Some years later, it was shown that plastic activity in model structural glasses and soft-sphere packings is intimately linked to nonphononic, low-frequency modes (32⇓–34).

It was, however, only recently that the universal statistical and structural properties of soft QLMs in glasses were revealed, first in a Heisenberg spin glass in a random field (35), and later in model structural glasses (13, 14, 36⇓–38). It is now well accepted that the density of nonphononic QLMs of frequency ω grows from zero (i.e., without a gap) as

The key challenge in revealing the statistical, structural, and energetic properties of soft QLMs in computer investigations lies in the abundance of spatially extended low-frequency phonons in structural glasses (36, 39). These phononic excitations hybridize with quasilocalized excitations, as pointed out decades ago by Schober and Oligschleger (40). These hybridization processes hinder the accessibility of crucial information regarding characteristic length and frequency scales of QLMs and regarding their prevalence.

While promising attempts to overcome the aforementioned hybridization issues have been put forward (36, 40⇓–42), a complete statistical–mechanical picture of QLMs is still lacking. In particular, recent work has revealed that annealing processes affect QLMs in three ways: Firstly, the number of QLMs appears to decrease upon deeper annealing—i.e., they are depleted—as first pointed out in refs. 15 and 43. Secondly, the core size of QLMs,

In this work, we investigate the effect of very deep supercooling/annealing on the statistical, structural, and energetic properties of QLMs in a model computer glass (see Materials and Methods for details). First, we explain why information regarding the number of QLMs cannot typically be obtained from the universal vibrational density of states (vDOS) of QLMs alone. Instead, we show that the vDOS grants access to a composite physical observable, which encodes information regarding both the characteristic frequency scale of QLMs,

Interestingly, this analysis reveals that N follows an equilibrium-like Boltzmann relation

Furthermore, we show that

## The QLMs Depletion vs. Stiffening Conundrum

It is now established that the vDOS of QLMs, **1** (denoted by

In Fig. 2, we plot the cumulative vDOS calculated for glassy samples rapidly quenched from parent equilibrium temperatures *Inset* shows that the prefactor

What physics is encapsulated in **1** is valid—i.e., in the range **1** should be rewritten as**2** and Debye’s vDOS of (acoustic) phononic excitations in crystalline solids (52). The latter takes the form **2**, and between

In order to disentangle the number of QLMs (N) and their characteristic frequency (

In general, though, the lowest phonon frequency is, in fact, smaller than

## Estimating QLMs’ Frequency Scale by Pinching a Glass

The previous discussion showed that the *SI Appendix*). By averaging

In Fig. 4, *Left*, we plot the characteristic frequency *Left*, *Inset*, also plateaus at the same

We conclude that, in the

The stiffening of QLMs by a factor of approximately two accounts for an approximate 30-fold variation of **2**. The remaining variation is attributed to the number of QLMs, *Right*. The result indicates that QLMs are depleted by slightly less than two orders of magnitude in the simulated *Right* appears to be consistent with a very recent study (61) of the depletion of tunneling two-level systems in stable computer glasses, possibly indicating that a subset of the QLMs is associated with tunneling two-level systems (1, 2, 36, 62).

The results presented in Fig. 4 demonstrate that pinching a glass may offer a procedure to separate the depletion and stiffening processes that take place with progressive supercooling. Next, we aim at exploring the physical implications of disentangling N and

## A Thermodynamic Signature of the QLMs

QLMs correspond to compact zones (though they also have long-range elastic manifestations), which are embedded inside a glass, and characterized by particularly soft structures. It is tempting, then, to think of them as quasiparticles that feature well-defined properties (e.g., formation energy). If true, one may hypothesize that QLMs can be created and annihilated by thermodynamic fluctuations and follow an equilibrium distribution at the parent equilibrium temperatures

As we have now at hand an estimate of the number N as a function of *Right*), we can start testing these ideas. To this aim, we plot in Fig. 5 N vs. *parent temperature* **3** is that QLMs seem to feature a well-defined formation energy,

The results in Eq. **3** and Fig. 5 indicate that QLMs might indeed correspond to a subset of configurational degrees of freedom that equilibrate at the parent temperature

The Boltzmann-like relation in Eq. **3**, when interpreted in terms of STZs, is a cornerstone of the nonequilibrium thermodynamic STZ theory of the glassy deformation (46⇓–48), where

It is natural to define a length scale corresponding to the typical distance between QLMs as

## A Glassy Length Scale Revealed by Pinching a Glass

What additional physics can pinching a glass reveal? Up to now, we explored the physics of the QLMs number N; we now turn to the other contribution to *Left*); is this stiffening related to other properties of QLMs that vary with

To that aim, we construct a length scale

In order to shed light on the physical meaning of *SI Appendix* for details). These three length scales feature very similar variations with **4**)—seems to provide a measure of the core size of QLMs, and in light of the suggested relation between the latter and STZs, also of the size of STZs.

Additional insight may be gained by invoking the relation—established in ref. 45—between

## Summary and Outlook

In this work, we have employed a computer-glass model, which can be deeply annealed (81), to quantitatively study the variation of the properties of QLMs (soft spots) with the depth of annealing. Most notably, we calculated the variation of the number, characteristic frequency, and core size of QLMs with the parent temperature from which the glass is formed. This has been achieved by assuming that the characteristic frequency scale of QLMs can be estimated through the bulk-average response of a glass to a local pinch. This frequency scale, in turn, allowed us to disentangle the apparently inseparable effects of the depletion and stiffening of QLMs, which are both encoded in the prefactor of the universal

We found that the number of QLMs follows a Boltzmann-like factor, with the parent temperature—from which equilibrium configurations were vitrified—playing the role of the equilibrium temperature. Consequently, the parent temperature may be regarded as a nonequilibrium temperature that characterizes QLMs deep inside the glassy state. Furthermore, our analysis reveals that both the core size of QLMS and the mesoscopic length scale that marks the cross-over between atomistic-disorder-dominated responses near local perturbations, and continuum-like responses far away from local perturbations, can be estimated by using the characteristic frequency of QLMs—obtained by pinching the glass—and the speed of shear waves.

Our results may have important implications for various basic problems in glass physics. We mention a few of them here; first, the Boltzmann-like law of the number of QLMs may play a major role in theories of the relaxation, flow, and deformation of glasses and may support some existing approaches. Second, together with other available observations (38, 45, 80), our results may suggest that the boson-peak frequency could be robustly probed by pinching glassy samples, instead of the more involved analysis required otherwise (9, 38). Finally, the variation of the energy scale proportional to *Left* with figure 8 of ref. 82). If valid, our results appear to support elasticity-based theories of the glass transition (83⇓–85) and indicate that QLMs play important roles in relaxation processes in deeply supercooled liquids (31). We hope that these interesting investigation directions will be pursued in the near future.

## Materials and Methods

We employed a computer-glass-forming model in three dimensions, simulated by using the swap Monte Carlo method, explained, e.g., in ref. 81. The model consists of soft repulsive spheres interacting via a *N* = 2,000; 8,000; and 16,000 particles, respectively, by instantaneously quenching (to zero temperature) independent configurations equilibrated at various parent temperatures *Left*, *Inset* and in ref. 86. In our model, we find

## Acknowledgments

We thank David Richard for his help with our graphics. Fruitful discussions with David Richard and Geert Kapteijns are warmly acknowledged. E.B. was supported by the Minerva Foundation with funding from the Federal German Ministry for Education and Research, the Ben May Center for Chemical Theory and Computation, and the Harold Perlman Family. E.L. was supported by the Netherlands Organisation for Scientific Research (Vidi Grant 680-47-554/3259).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: e.lerner{at}uva.nl.

Author contributions: C.R., E.B., and E.L. designed research; C.R. and E.L. performed research; C.R., E.B., and E.L. discussed the results; and C.R., E.B., and E.L. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1919958117/-/DCSupplemental.

Published under the PNAS license.

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