Nonlinear quasilocalized excitations in glasses: True representatives of soft spots

Structural glasses formed by quenching a melt possess a population of soft quasilocalized excitations—often called “soft spots”—that are believed to play a key role in various thermodynamic, transport, and mechanical phenomena. Under a narrow set of circumstances, quasilocalized excitations assume the form of vibrational (normal) modes, that are readily obtained by a harmonic analysis of the multidimensional potential energy. In general, however, direct access to the population of quasilocalized modes via harmonic analysis is hindered by hybridizations with other low-energy excitations, e.g., phonons. In this series of papers we reintroduce and investigate the statistical-mechanical properties of a class of low-energy quasilocalized modes—coined here nonlinear quasilocalized excitations (NQEs)—that are defined via an anharmonic (nonlinear) analysis of the potential-energy landscape of a glass, and do not hybridize with other low-energy excitations. In this paper, we review the theoretical framework that embeds a micromechanical definition of NQEs. We demonstrate how harmonic quasilocalized modes hybridize with other soft excitations, whereas NQEs properly represent soft spots without hybridization. We show that NQEs' energies converge to the energies of the softest, nonhybridized harmonic quasilocalized modes, cementing their status as true representatives of soft spots in structural glasses. Finally, we perform a statistical analysis of the mechanical properties of NQEs, which results in a prediction for the distribution of potential-energy barriers that surround typical inherent states of structural glasses, as well as a prediction for the distribution of local strain thresholds to plastic instability.

Figure 1

NQEs are indifferent to hybridization with excitations of nearby frequencies. Panel (a) shows a normal mode that consists of two hybridized quasilocalized excitations of similar frequency. In panel (b), we show that these excitations are cleanly separated as two NQEs. Panel (c) shows a normal mode that is a mixture of a quasilocalized excitation and a nearby phononic excitation; panel (d) shows the dehybridized NQE. In panels (a) and (b), only the largest 0.1% of components of each mode is shown. For visual purposes, the modes are scaled so that the largest component of the NQE is the same size as the corresponding component of the harmonic mode.

Figure 2

The quartic expansion coefficients ${\chi }_4$ are primarily sensitive to the localization $N{e}_4$ of the associated NQEs, independent of preparation protocol or system size of the glass.

Figure 3

Harmonic QLM $\widehat{\mathit{\psi }}$ and quartic NQE ${\widehat{\mathit{\pi }}}_4$ converge energetically and structurally in the limit of low frequencies. Panels (a) and (c) show the softest harmonic QLM in the entire ensemble of solids of the IPL model in 3D and 2D. Panels (b) and (d) show their quartic nonlinear counterparts. The relative difference in energy between the harmonic and quartic excitations $({\omega }_4^2-{\omega }_2^2)/{\omega }_4^2$ is 0.15% in 3D and 2.1% in 2D; the structural difference $1-|\widehat{\mathit{\psi }}·{\widehat{\mathit{\pi }}}_4|$ is $3.6\times {10}^{-6}$ in 3D and $2.4\times {10}^{-4}$ in 2D.

Figure 4

Every harmonic QLM $\widehat{\mathit{\psi }}$ has a corresponding quartic NQE ${\widehat{\mathit{\pi }}}_4$ of similar frequency and participation in 3D (a) and 2D (b). Because the harmonic QLM represents the direction of lowest stiffness in the entire system, the frequency of the NQE is always slightly higher, whereas the NQE—which is free of hybridization—has a slightly lower participation ratio. When the QLM's frequency approaches the frequency of the first phonon band, hybridization becomes stronger. This is reflected in a higher participation $Ne$ [and see also Fig. 1].

Figure 5

Harmonic QLM $\widehat{\mathit{\psi }}$ and quartic NQE ${\widehat{\mathit{\pi }}}_4$ converge energetically and structurally in the limit of low frequencies. Panels (a) and (e) show the energetic convergence [Eq. (15)] in 3D and 2D. Panels (b) and (f) show the structural convergence [Eq. (16)] in 3D and 2D. Panels (c) and (d) show that the prefactors $f\left(L\right)$ and $g\left(L\right)$ of the energetic and structural convergence are weakly $L$ dependent in 3D, whereas panels (g) and (h) show they follow $f\left(L\right)\sim L^2$ and $g\left(L\right)\sim L^4$ in 2D. For panels (a), (b), (e), and (f), the data points represent the running median.

Figure 6

Representative examples of fields $\mathit{\Delta }\equiv {\widehat{\mathit{\pi }}}_4-\widehat{\mathit{\psi }}$ [see Eq. (17) and discussion in text] in (a) 3D and (b) 2D are shown in black. Shown in green are the corresponding NQEs ${\widehat{\mathit{\pi }}}_4$. The magnitude of $\mathit{\Delta }$ is much smaller than 1, so for visual purposes it has been scaled so that the size of the largest component of the core of $\mathit{\Delta }$ matches the corresponding component of ${\widehat{\mathit{\pi }}}_4$. For visualization purposes, we show only the largest 2% of components of $\mathit{\Delta }$ and ${\widehat{\mathit{\pi }}}_4$ in 3D, and the largest 1.4% of ${\widehat{\mathit{\pi }}}_4$ in 2D.

Figure 7

Difference between cubic and quartic NQEs. Panel (a) shows that the relative energy difference between cubic and quartic modes is approximately 30%, independent of frequency, corresponding to a relative frequency difference of about 14%. Panel (b) shows that the third-order coefficients ${\tau }_3\text{and}{\tau }_4$, which are a measure of the asymmetry of the energy landscape along the mode, are highly sensitive to the modes' structure, as quantified by the difference from unity of their overlap, $1-|{\widehat{\mathit{\pi }}}_3·{\widehat{\mathit{\pi }}}_4|$. Panel (c) shows that the fourth-order coefficients ${\chi }_3\text{and}{\chi }_4$ are insensitive to small changes in the modes' structure, consistent with the fact that $\chi$ scales inversely with the participation ratio (see Fig. 2), which is a global geometric measure. We emphasize that in panel (b) the data span almost five orders of magnitude, whereas in panel (c) the data span less than one. All data are reported as 2D histograms, with the bin counts given by the color of the bin.

Figure 8

The ensemble of cubic modes follows ${\tau }_3^2\sim {\kappa }_3{\chi }_3$, independent of glass stability. The continuous line marks the bound given by Eq. (32), at which the mode—in the fourth-order expansion of the potential energy—is a symmetric double-well potential. Modes that fall below this bound either are single-well potentials or have a second minimum that has a higher energy than the inherent state. Modes that exceed this bound have a second minimum with a lower energy than the inherent state (see illustrations). In our ensemble of solids, only 6.6% ($T_{\text{p}}=1.30$), 4.6% ($T_{\text{p}}=0.50$), and 0.5% ($T_{\text{p}}=0.35$) of cubic modes exceed this bound. In Fig. 9 (in Appendix pp5), we demonstrate that ${\chi }_3$ is independent of ${\kappa }_3$, which together with the data in this figure demonstrates that ${\tau }_3\sim \sqrt{{\kappa }_3}\sim {\omega }_3$, which is central to the predictions that follow in the next section.

Figure 9

(a) The magnitude of the field ${\mathit{U}}^{\left(4\right)}\begin{array}{c}\\ .\\ .\\ .\end{array}{\widehat{\mathit{\pi }}}_4{\widehat{\mathit{\pi }}}_4{\widehat{\mathit{\pi }}}_4/{\chi }_4$ is of order $O\left(1\right)$, independent of ${\omega }_4$. (b) The shear-strain-coupling parameter ${\nu }_3\equiv \frac{{\partial }^{2}U}{\partial \gamma \partial \mathit{x}}·{\widehat{\mathit{\pi }}}_3$ associated with cubic modes ${\widehat{\mathit{\pi }}}_3$ is independent of ${\omega }_3$. (c) The fourth-order coefficient of cubic modes ${\chi }_3$ is independent of ${\omega }_3$. (d) The entire ensemble of cubic modes follows the relation ${\tau }_3^2\sim {\kappa }_3{\chi }_3$ in 2D, 3D, and 4D. (e) The ensemble of quartic modes does not follow this relation. (f) The fourth-order coefficient of quartic modes ${\chi }_4$ is independent of ${\omega }_4$. In panels (a), (e), and (f), the data are presented as a 2D histogram, where each bin is colored according to the number of hits in that bin (the color coding is the same as in Fig. 7).