The QLMs Depletion vs. Stiffening Conundrum

It is now established that the vDOS of QLMs, D(ω), follows a universal gapless law (13, 14, 36⇓–38)D(ω)=Ag ω4  for  0≤ω≤ωg,[1]where ωg is the upper cutoff of this scaling regime, and the prefactor Ag is extensively discussed below. The ω4 law has been rationalized by various models (1⇓⇓–4, 16⇓⇓–19) and is known to be intimately related to the existence of frustration-induced internal stresses in glasses (44), but its theoretical foundations are not yet fully developed. The prefactor Ag in Eq. 1 (denoted by A4 in refs. 18, 38, and 51) is a nonuniversal quantity that encodes information about a particular glassy state, most notably its composition (constituent elements, interaction potential, etc.) and its preparation protocol (15, 38, 43). The ultimate goal of this work is to explore the physical information encapsulated in Ag and its dependence on the glass-preparation protocol.

In Fig. 2, we plot the cumulative vDOS calculated for glassy samples rapidly quenched from parent equilibrium temperatures Tp (as appears in the figure legend) to zero temperature. The system size was chosen so as to avoid hybridization with phonons at the lowest frequencies, as explained in ref. 13. The figure shows, in agreement with ref. 38, that the ω4 scaling persists all the way down to the deepest supercooled states accessible to us, Tp=1/3 (the units used to report Tp are defined below). Fig. 2, Inset shows that the prefactor Ag varies by nearly three orders of magnitude in the simulated Tp range. The huge variability of Ag with the preparation protocol, here quantified by the parent equilibrium temperature Tp, indicates dramatic changes in the resulting glassy states, despite the fact that all of them follow the universal ω4 law.

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Fig. 2.

Cumulative density of states CDF≡∫0ωD(ω′)dω′ for various parent temperatures Tp (see values in the legend). Here, we present data measured in 10,000 glassy samples of N = 2,000 particles for Tp≤11/18 and 2,000 samples of N = 16,000 particles for Tp>11/18. (Inset) The prefactors Ag vs. Tp; see text for discussion.

What physics is encapsulated in Ag? To start addressing this question, let us first consider the dimensions of Ag. When D(ω) is integrated over the frequency range in which Eq. 1 is valid—i.e., in the range 0≤ω≤ωg—one obtains an estimate for the total number of QLMs, N. Consequently, Ag has the dimensions of an inverse frequency to the fifth power, where the dimensionless prefactor is proportional to N. Since D(ω) follows a power law—i.e., it is scale-free in the range 0≤ω≤ωg—the only possible frequency scale that can appear in it is the upper cutoff ωg. Hence, we expect to have Ag∼Nωg−5, which implies that Eq. 1 should be rewritten asD(ω)∼N ωg−5ω4  for  0≤ω≤ωg.[2]We would like to note the analogy, and the fundamental difference, between Eq. 2 and Debye’s vDOS of (acoustic) phononic excitations in crystalline solids (52). The latter takes the form D(ω)=AD ω2 (in three dimensions), with AD=9N/ωD3, where ωD is Debye’s frequency and N is the number of particles. The integral over D(ω) in the range 0≤ω≤ωD equals the number of degrees of freedom in the system, 3N. The analogy between Debye’s vDOS and the glassy vDOS in Eq. 2, and between ωg and Debye’s frequency, is evident. Yet, there is a crucial difference between the two cases; in Debye’s theory, the number of phononic excitations is a priori known to equal the number of degrees of freedom 3N (in fact, ωD is precisely defined so as to ensure the latter). In the glassy case, however, there is neither an a priori constraint on the number of QLMs N, nor on the upper-frequency cutoff ωg (the total number of vibrational modes, both glassy and phononic, is, of course, still determined by the total number of degrees of freedom, but there is no a priori constraint on the fraction of QLMs out of the total number of vibrational modes). Hence, N and ωg should be treated as independent quantities that can feature different dependencies on the glass history (preparation protocol).

In order to disentangle the number of QLMs (N) and their characteristic frequency (ωg) contributions to Ag∼Nωg−5, one needs to estimate one of them—i.e., either N or ωg—independently of Ag. In principle, as the characteristic frequency ωg represents the upper cutoff on the ω4 scaling regime (as explained above), one can try to estimate it through the deviation from the universal ω4 law. This has been, in fact, demonstrated in ref. 14 for rapidly quenched glassy samples in a narrow range of system sizes, in three and four dimensions. Some of the data appearing in figure 2 b and c of ref. 14 are reproduced here in Fig. 3, where the lowest phononic band is shown in orange in each graph. It is observed that, in these examples, the vDOS deviates from the ω4 scaling at a frequency smaller than the lowest phonon frequency, which can be identified with ωg (marked by the vertical dashed lines).

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Fig. 3.

The vDOS D(ω) of small glassy samples (the number of particles N is specified in each graph) in three dimensions (3D; Left) and four dimensions (4D; Right), obtained by a rapid quench in ref. 14. The data are adapted from figure 2 b and c of ref. 14, where frequencies are normalized as detailed in SI Appendix. The vertical continuous lines indicate the position of the first phonon band, whereas the dashed lines mark the breakdown of the ω4 scaling regime.

In general, though, the lowest phonon frequency is, in fact, smaller than ωg, which obscures the identification of the latter due to hybridizations (39). Indeed, in Fig. 2, it is observed that, as Tp decreases, the lowest phonon band pushes the vDOS upwards in the middle of the scaling regime, disallowing to extract ωg. Hence, we conclude that the vDOS alone does not allow one to distinguish between changes in the number of QLMs (e.g., a decrease, i.e., depletion) and in their characteristic frequency (e.g., an increase, i.e., stiffening). How to disentangle the N and ωg dependence of Ag, and the possible depletion and stiffening of QLMs associated with them, is the question we address next.

Estimating QLMs’ Frequency Scale by Pinching a Glass

The previous discussion showed that the Tp dependence of Ag∼Nωg−5 cannot be readily used to extract the Tp dependence of N and ωg separately. Consequently, one needs additional physical input in order to disentangle the two quantities. Here, we follow the suggestion put forward in ref. 15 that the characteristic frequency ωg of QLMs can be probed through pinching a glass. Formally, by pinching, we mean applying a force dipole d(ij) to a pair of interacting particles i,j in a glassy sample. The displacement response to d(ij), which was shown to closely resemble the spatial pattern of QLMs (15), can be associated with a frequency ωg(ij) (see additional details in SI Appendix). By averaging ωg(ij) over many interacting pairs i,j in a glassy sample, one obtains a characteristic frequency scale, which was proposed to represent ωg. This suggestion was discussed at length and tested, under various circumstances, in ref. 15; here, we follow it—i.e., assume that the Tp dependence of the dipole response is proportional to ωg(Tp). The remainder of the paper is devoted to exploring the implications of this assumption.

In Fig. 4, Left, we plot the characteristic frequency ωg vs. the parent temperature Tp, where ωg(Tp) is estimated by the pinching procedure just described. It is observed that ωg varies by nearly a factor of two at low parent temperatures Tp and reaches a plateau at higher Tp. We further find that the sample-to-sample mean athermal shear modulus, G, shown in Fig. 4, Left, Inset, also plateaus at the same Tp as ωg does. Consequently, in what follows, we conveniently express temperatures in terms of the onset temperature Tonset of the high-Tp plateaus of G and ωg.

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Fig. 4.

(Left) The characteristic frequency ωg of QLMs, estimated by the pinching procedure discussed in the text, plotted vs. the parent temperature Tp. (Left, Inset) The sample-to-sample mean athermal shear modulus, G, plotted against Tp. (Right) N is proportional to the number of QLMs and is plotted here against Tp.

We conclude that, in the Tp range considered here, QLMs appear to stiffen by a factor of approximately two with decreasing Tp. Interestingly, in ref. 38, it was reported that the boson peak frequency ωBP varies by approximately a factor of two over a similar range of Tp, suggesting that ωBP and ωg might be related. In ref. 45, a similar proposition was put forward in the context of the unjamming transition (53⇓–55), where it was argued that the renowned “unjamming” frequency scale ω* (5, 53) can be extracted by considering the frequencies associated with the responses to a local pinch. However, since ω* and ωBP may differ (10), it is not currently clear which of these frequencies is better represented by ωg.

The stiffening of QLMs by a factor of approximately two accounts for an approximate 30-fold variation of Ag, due to the ωg−5 dependence in Eq. 2. The remaining variation is attributed to the number of QLMs, N=Ag ωg5 (note that here, we use an equality, as the Tp-independent prefactor is of no interest), plotted in Fig. 4, Right. The result indicates that QLMs are depleted by slightly less than two orders of magnitude in the simulated Tp range. The strong depletion of QLMs upon deeper supercooling has dramatic consequences for the properties of the resulting glassy states. For example, brittle failure (56, 57) and reduced fracture toughness (58⇓–60) are claimed to be a consequence of this depletion. It is interesting to note that the range of variability observed in Fig. 4, 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 ωg.

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 Tp. Moreover, their equilibrium thermodynamic nature might be manifested in nonequilibrium glassy states as they become frozen in during the rapid quench upon glass formation.

As we have now at hand an estimate of the number N as a function of Tp (cf. Fig. 4, Right), we can start testing these ideas. To this aim, we plot in Fig. 5 N vs. Tp−1 on a semilogarithmic scale; the outcome reveals a key result: the number of QLMs follows a Boltzmann-like law, with the parent temperature Tp playing the role of the equilibrium temperature, namely,N∝exp −EQLMkB Tp.[3]A possibly related Boltzmann-like law, albeit for Ag(Tp) itself, was observed in ref. 51 for reheated stable glasses (63). A corollary of Eq. 3 is that QLMs seem to feature a well-defined formation energy, EQLM≈3.3 (in units of kBTonset). It is surprising that EQLM appears to be independent of Tp, while the characteristic energy scale associated with ωg does appear to depend on it. Future research should shed additional light on this nontrivial observation.

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Fig. 5.

The density of QLMs, plotted against 1/Tp, revealing that it is controlled by a Boltzmann-like factor e−EQLM/kBTp, with the parent temperature playing the role of the equilibrium temperature. We find EQLM≈3.3, expressed in terms of kBTonset.

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 Tp and that carry memory of their equilibrium distribution when the glass goes out of equilibrium during a quench to lower temperatures. This physical picture strongly resembles the idea of a fictive/effective/configurational temperature, which was quite extensively used in models of the relaxation, flow, and deformation of glasses (46⇓–48, 64⇓⇓–67). This connection is further strengthened in light of available evidence indicating that the cores of deformation-coupled QLMs are the loci of irreversible plastic events that occur once a glass is driven by external forces (68⇓–70).

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 Tp is treated as a thermodynamic temperature that characterizes configurational degrees of freedom and that differs from the bath temperature. The strong depletion of STZs with decreasing Tp, as predicted by the Boltzmann-like relation, was shown to give rise to a ductile-to-brittle transition in the fracture toughness of glasses (58, 59). This prediction was recently supported by experiments on the toughness of bulk metallic glasses, where Tp was carefully controlled and varied (60).

It is natural to define a length scale corresponding to the typical distance between QLMs as ξs∼N−1/d-, once an estimate of their number N is at hand. Such a “site length” ξs was introduced in ref. 13, where it was related to the sample-to-sample average minimal QLM frequency ωmin, according to ωmin∼ωg(L/ξs)−d-/5. The latter implies that the lowest QLM frequency is selected among (L/ξs)d-∝N N possible candidates, which is directly related to the extreme value statistics of ωmin (13). The site length ξs is expected to control finite-size effects in studies of athermal plasticity in stable glasses, as discussed in detail in ref. 71. Similar definitions of a site length were proposed in refs. 71 and 72; an important message here is that the disentangling of the stiffening effect from the prefactor Ag is imperative for the purpose of obtaining a consistent definition of a length scale in such a setting.

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 Ag, i.e., to the frequency scale ωg that characterizes the typical stiffness of QLMs. ωg was shown to undergo stiffening with decreasing Tp (cf. Fig. 4, Left); is this stiffening related to other properties of QLMs that vary with Tp? An interesting possibility we explore here is whether it might be related to a glassy length scale that is associated with QLMs.

To that aim, we construct a length scale ξg asξg≡2π cs/ωg,[4]which corresponds to the wavelength of transverse phonons propagating at the shear wave-speed cs with an angular frequency ωg. This length is similar in spirit to the “boson peak” length ξBP∼cs/ωBP (73). The physical rationale behind our constructed length ξg is that the emerging length scale is expected to mark a cross-over in the elastic response of a glass to a local pinch, as discussed below. In Fig. 6, we plot ξg vs. the parent temperature Tp; we find that ξg decreases upon deeper annealing by ∼40%, a manifestation of the modest stiffening of the macroscopic shear modulus compared to that of QLMs (recall that cs is proportional to the square root of the shear modulus). This decreasing length is of unique character among the plethora of glassy length scales previously put forward in the context of the glass transition, most of which are increasing functions of decreasing temperature or parent temperature (74⇓⇓⇓⇓–79).

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Fig. 6.

The glassy length ξg, the cross-over length ξco, and the QLM core size ξQLM (see SI Appendix for details), plotted against the parent temperature Tp. These lengths vary together with parent temperature Tp, supporting their equivalence.

In order to shed light on the physical meaning of ξg, we consider also (i) the cross-over length ξco, as observed in the displacement response to local pinches, between an atomistic-disorder-dominated response at distances r≲ξco from the perturbation, to the expected continuum behavior seen at r>ξco, and (ii) the core size of QLMs, ξQLM, which is known to decrease upon annealing (38, 43, 44), as is also illustrated graphically in Fig. 1. In Fig. 6, we directly compare between ξg and our measurements of ξco and ξQLM (see SI Appendix for details). These three length scales feature very similar variations with Tp, strongly supporting their equivalence. Consequently, ξg—which was defined through the dipole response frequency ωg (cf. Eq. 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 ωg and the characteristic frequency ω* that emerges near the unjamming transition (53⇓–55). Indeed, in the unjamming scenario, the length ξg (often denoted ℓc) was shown to diverge upon approaching the unjamming point (50) and to mark the cross-over between disorder-dominated responses near a local perturbation and the continuum-like response observed in the far field, away from the perturbation. The same length was shown in ref. 80 to characterize the core size of QLMs near the unjamming point of harmonic-sphere packings. In light of the results shown in Fig. 6, we hypothesize that the fundamental cross-over length—below which responses to local perturbations are microstructural/disorder-dominated, and above which responses to local perturbations follow the expected continuum-like behavior—is, in fact, ξg, which, in turn, we show to agree well with the size of QLMs.

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 ω4 vDOS of QLMs.

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 ωg2 with annealing temperature appears to match very well the variation of activation barriers required to rationalize fragility measurements in laboratory glasses (compare Fig. 4, 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.