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# Local thermal energy as a structural indicator in glasses

Edited by James S. Langer, University of California, Santa Barbara, CA, and approved May 25, 2017 (received for review March 16, 2017)

## Significance

When liquids are cooled sufficiently fast, they fail to crystalize and form disordered solids—glasses. Understanding the physical properties of glasses remains a major challenge, in large part because of the lack of tools to characterize the emerging disordered structures and well-established structure–properties relations. In this work, we propose that fluctuational thermal energy reveals highly localized and soft structures in glasses. We show that the degree of softness of these “soft spots” follows a universal fat-tailed statistical distribution and relate it to the density of noncrystalline vibrational states. The softest spots are shown to predict the loci of irreversible plastic rearrangements in sheared glasses, thus offering a generic structure–properties approach to glassy materials.

## Abstract

Identifying heterogeneous structures in glasses—such as localized soft spots—and understanding structure–dynamics relations in these systems remain major scientific challenges. Here, we derive an exact expression for the local thermal energy of interacting particles (the mean local potential energy change caused by thermal fluctuations) in glassy systems by a systematic low-temperature expansion. We show that the local thermal energy can attain anomalously large values, inversely related to the degree of softness of localized structures in a glass, determined by a coupling between internal stresses—an intrinsic signature of glassy frustration—anharmonicity and low-frequency vibrational modes. These anomalously large values follow a fat-tailed distribution, with a universal exponent related to the recently observed universal

Understanding the glassy state of matter remains one of the greatest challenges in condensed matter physics and materials science (1⇓⇓⇓–5). In large part, the elusive nature of the glassy state is due to the absence of well-established tools and concepts to quantify the disordered structures characterizing glassy materials—in sharp contrast to their ordered crystalline counterparts—and because of the lack of understanding of the relations between glassy structures and dynamics. Over the years, many attempts have been made to identify physical quantities that can indicate underlying local structures within glassy materials (6⇓⇓–9). These indicators include, among others, free volume (10⇓–12), internal stresses (13), local elastic moduli (14), local Debye–Waller factor (15), coarse-grained energy and density (16, 17), locally favored structures (18⇓–20), short- and medium-range order (21⇓–23), and various weighted sums over a system-dependent number of low-frequency normal modes (24⇓⇓⇓⇓⇓–30).

These quantities measure some properties of quiescent glasses evaluated at or in the near vicinity of a mechanically (meta-)stable state of a glass (an inherent structure). Some of these indicators are purely structural in nature (i.e., they are obtained from the knowledge of particle positions alone), whereas others require in addition the knowledge of interparticle interactions. Recently, the local yield stress—the minimal local stress needed to trigger an irreversible plastic rearrangement—has been proposed as a structural indicator (31). It is required, however, to externally drive each local region in a glass to its nonlinear rearrangement threshold and hence, belongs to a different class of structural indicators compared with those previously mentioned. The utility of each of the proposed indicators is usually assessed by looking for correlations between the revealed structures—typically localized soft spots—and glassy dynamics, either thermally activated relaxation in the absence of external driving forces or localized irreversible plastic rearrangements under the application of global driving forces. In fact, a recent study established such structure–dynamics correlations by machine-learning techniques, leaving the precise physical nature of the underlying structural indicator unspecified (32, 33). These machine learning-based structural indicators also belong to a different class of structural indicators, because the training stage of the machine-learning algorithm requires knowledge of the plastic rearrangements themselves.

Some of the previously proposed structural indicators have revealed a certain degree of correlation between identified soft spots and dynamics, providing important evidence that preexisting localized structures in a glass significantly affect its dynamics. However, oftentimes, the physical foundations of the structural indicators remain unclear, and they are sometimes defined algorithmically but are not derived from well-established physical observables. Moreover, their statistical properties are not commonly addressed, the relations between them and other basic physical quantities are not established, and the fundamental reasons for them being particularly sensitive to underlying heterogeneous structures in glasses remain elusive.

Here, we propose a structural indicator of glassy “softness”—the local thermal energy (LTE)—which is a transparent physical observable derived by a systematic low-temperature expansion. We use the exact expression for the LTE of interacting particles to elucidate the underlying physical factors—most notably internal stresses, anharmonicity, and nonlinear coupling to low-frequency vibrational modes—that give rise to significant spatial heterogeneities of softness. We show that the LTE can attain anomalously large values directly related to particularly soft regions in a glass, which follow a fat-tailed distribution. The power law exponent characterizing this distribution is shown to be universal and directly related to the recently observed universal

## Physical Observables in the Low-Temperature Limit

Our starting point is the idea that the thermal average of local physical observables in a system equilibrated at a low temperature ** x** represent the deviations of the system’s degrees of freedom from a (possibly local) minimum of its energy

*SI Appendix*)

**1**, higher-order terms in

**1**represents an intrinsic property of an inherent structure, independent of temperature.

To gain some understanding of the physics encapsulated in Eq. **1**, let us briefly consider a few physical observables. Consider first the total energy **1** equals the number of degrees of freedom **1** vanishes, and we obtain **1** is fully consistent with well-established results (equipartition and thermal expansion) and highlight the anharmonic nature of the second term on the RHS of Eq. **1**.

The examples presented above focused on macroscopic (global) scalar observables. Because our main interest is in spatial heterogeneity, we consider now microscopic (local) observables defined at the particles’ level. We thus focus on the microscopic generalization of **1**, the normalized thermal average of ** x** in the

**2**—which features a quadratic (nonlinear) coupling between the anharmonicity tensor

**2**should be distinguished from the local Debye–Waller factor

**1**that is given by

**1**, which involves a single contraction of the inverse of the dynamical matrix

## Local Thermal Energy

The normalized thermal displacement vector **2** and shown to exhibit strong spatial heterogeneity in Fig. 1, contributes to the thermal average of any physical observable

Our goal now is to identify a physical observable **1** and **2**, we then define

Mechanical equilibrium at particle **3** vanishes. Such internal stress-free disordered systems were studied in ref. 38, where it was shown that, under these conditions,

An intrinsic signature of glassy systems is the existence of internal frustration (40) that leads to the emergence of internal forces/stresses, **3** to be generically nonzero for glasses. Because *Left*, we expect *Left*—are expected to feature much larger values.

To test these ideas, we plot in Fig. 1, *Right* the normalized LTE *Left*. The result is striking: *Inset*, showing perfect agreement with the exact expression in Eq. **3**.

## Universal Anomalous Statistics

To quantify the degree of softness of soft spots revealed by *Right*—and its probability of occurrence, we focus next on the statistical properties of **2** and **3** suggests that soft vibrational modes (i.e., modes with small frequencies

To proceed, note that **3** has one contribution that involves a single contraction with

We argue that low-frequency plane waves and quasilocalized soft glassy modes make qualitatively different contributions to the double sum in Eq. **4**. In order to support this claim, we note that, similarly to the discussion about the dipolar nature of **5**, which is verified below, establishes an important relation between the LTE

Using Eq. **5** and the universal relation *SI Appendix*. The prediction in Eq. **6** has far-reaching implications. First, it suggests that the physical observable **6** assumed that **6** rationalizes the existence of anomalously soft localized spots in glassy materials and predicts its probability.

To test the prediction in Eq. **6** and its degree of universality, we performed extensive numerical simulations of different computer glass-forming models [(*i*) a binary system of point-like particles interacting via inverse power law purely repulsive pairwise potentials in 2D (2DIPL) and 3D (3DIPL) (34) and (*ii*) the canonical Kob–Andersen binary Lennard–Jones (3DKABLJ) system (44) in 3D (*SI Appendix* has details about models and methods)] to extract the statistics of **3**. The results are summarized in Fig. 2. All of the glasses considered exhibit a power law tail with a universal exponent fully consistent with the theoretically predicted **6** and therefore, also implicitly, its underlying assumptions. The results presented in this section explain the physical origin of the sensitivity of

## Softness Field and Predicting Plastic Rearrangements

The normalized LTE *Left*, we present yet another example of the spatial map of *Right*. A continuous field can be naturally constructed by coarse-graining *SI Appendix*). Applying this procedure to Fig. 3, *Left* yields Fig. 3, *Right*, which we treat as a softness field. Our goal now is to test the predictive powers of this softness field in relation to glassy dynamics. The latter, either thermally activated relaxation in nondriven conditions or plastic rearrangements under external driving forces, entails crossing some activation barriers. Activation barriers revealed by soft localized vibrational modes

To test the susceptibility of regions with large LTE to plastic rearrangements, we applied global quasistatic shear deformation in a certain direction under athermal conditions to each glass realization—such as the one shown in Fig. 3, *Right*—and measured the locations of the first few discrete irreversible plastic rearrangements as described in *SI Appendix*. The advantage of this *Right*. The first four plastic events overlap soft spots identified by the softness field, indicating a high degree of predictiveness of

To quantify the degree of predictiveness of the LTE *SI Appendix*. In addition to its location, each soft spot is characterized by its degree of softness, representing the average value of *SI Appendix*). Because the fat-tailed distribution in Eq. **6** predicts very large variability in the degree of softness of different soft spots within a single glass realization and among different realizations, we define *SI Appendix*). We stress that the soft spots are extracted for the nonsheared system and are not updated between plastic events.

The cumulative distribution function *Left*, closed symbols. As expected, the smaller the

Among the many structural indicators studied over the years (compare with the Introduction above), the normal modes-based approach (24⇓⇓–27) stands out according to the relatively high correlations between the structure and dynamics that it exhibits. The basic idea behind this approach is that, although a single low-lying normal mode *Left* and Fig. 1, *Right* by summing the norm squared of the components of low-lying normal modes

After the normal modes maps are constructed (*SI Appendix* has more details), we apply to them the same procedure described above and calculate the cumulative distribution function *Left*, open symbols. The comparison reveals that the thermal energy-based approach significantly outperforms the normal modes-based approach. The performance of both approaches is quantified in Fig. 4, *Right*, where we plot the ratio of

We thus conclude that the LTE has predictive powers that surpass those of the normal modes-based approach. Can we also assess its predictive powers in absolute terms? To address this question, one should note that soft spots are expected to be anisotropic objects (41, 46) characterized by orientation and polarity and hence, feature variable coupling to shearing in various directions. That is, they are expected to be spin-like objects. Consequently, a spot that is very soft in a given direction may not undergo a rearrangement if the projection of the driving force on its soft direction is small. Hence, the optimal predictive power based on the degree of softness alone—a scalar measure—may be significantly smaller than unity. In particular, assuming a uniform/isotropic orientational distribution of equally soft spots, a naive estimation indicates that only *Left*, closed symbols (

## Conclusion

We have shown that the low-temperature LTE

Although the problems of coexistence and hybridization of long-wavelength plane waves and soft vibrational modes, which have hampered a direct observation of soft quasilocalized glassy modes and their statistical distribution for a long time, will be addressed elsewhere, we stress that our results have potentially important implications in this context. The universal fat-tailed distribution

The universal anomalous distribution of

Our approach offers a general system/model-independent, physical/observable-based framework to identify structural properties of quiescent glasses and relate them to glassy dynamics. In particular, the identified field of soft spots and its time evolution under external driving forces should play a major role in theories of plasticity of amorphous materials, serving to define a population of STZ (37, 47⇓⇓–50). The predictive powers of our approach have been shown here for plastic rearrangements in athermal quasistatically driven systems. An important future challenge would be to test whether and to what extent these predictive powers persist at finite temperatures—possibly up to the glass transition region—and finite strain rates. It should also be tested against thermally activated relaxation in the absence of external driving forces. Finally, as mentioned above, an interesting direction would be to go beyond the scalar degree of softness measure by incorporating orientational information into a generalized structural indicator.

## Footnotes

↵

^{1}J.Z. and E.L. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: eran.bouchbinder{at}weizmann.ac.il.

Author contributions: E.L., Y.B.-S., and E.B. designed research; J.Z. and E.L. performed research; and J.Z., E.L., Y.B.-S., and E.B. 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.1704403114/-/DCSupplemental.

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