Aging evolution of the low-frequency non-Debye vibrational density of states in glasses

Glasses obtained by a rapid quench are out of equilibrium and age: their properties keep drifting with the waiting time $t_w$ as the system explores progressively deeper minima of its potential energy landscape. Experiments show that this aging reaches the vibrational spectrum: physical aging modifies the low-frequency vibrational density of states $D(\omega)$ of glassy polymers, and geologically hyperaged amber (110 million years of natural aging) shows a depletion of the terahertz density of states. Numerical work has meanwhile established, at equilibrium, that the low-frequency spectrum of a model glass below the boson peak obeys a non-Debye power law $D(\omega)\sim\omega^{s}$ whose exponent grows from the Debye value $s=2$ to $s\approx4$ as the parental temperature of the equilibrated supercooled liquid approaches the dynamical crossover temperature, a crossover that can be mapped onto the size of dynamically heterogeneous frozen regions through a random-pinning construction. What is missing — and what Paoluzzi, Angelani, Parisi, and Ruocco explicitly proposed as future work in Phys. Rev. Lett. 123, 155502 (2019) — is the numerical investigation of $D(\omega)$ in the aging regime: how the non-Debye spectrum, its exponent and prefactor, the population and localization of soft quasilocalized modes, and the associated heterogeneity length evolve as a glass ages after a quench. This problem asks for that age-resolved numerical characterization and for the connection it establishes between the equilibrium spectroscopic phenomenology and the hyperaging observations.

Background

At small frequencies, the vibrational density of states $D(\omega)$ — the distribution of normal-mode frequencies of a solid — of a three-dimensional glass is expected to follow Debye's law $D(\omega)\sim\omega^{2}$, because on long length scales any solid behaves as an elastic continuum whose only low-energy excitations are sound waves (phonons, the Goldstone modes associated with broken translational symmetry). Glasses, however, also display a universal excess of vibrational states over this law at terahertz frequencies, the boson peak, and numerical work has revealed a further non-Debye (non-phononic) sector below it: a population of soft modes that are spatially localized rather than extended, whose density vanishes as a higher power law $D(\omega)\sim\omega^{s}$, with $s$ reported near 4 for well-annealed glasses and debated in the range from 3 to 5 depending on preparation. These quasilocalized soft modes are believed to originate in locally unstable, elastically heterogeneous regions of the glass.

A standard tool for studying them is the inherent-structure protocol: equilibrate a model glass former — here a three-dimensional binary mixture simulated with hybrid Brownian/Swap Monte Carlo dynamics, swap moves (particle-exchange moves) being a device that accelerates the equilibration of deeply supercooled liquids — at a parental temperature $T^{*}$, then minimize the potential energy to a nearby local minimum (the inherent structure) and diagonalize the Hessian matrix of second derivatives of the potential (the dynamical matrix) to obtain the harmonic spectrum. In this framework Paoluzzi, Angelani, Parisi, and Ruocco (2019) showed that the exponent of the low-frequency spectrum below the boson peak grows continuously from the Debye value $s=2$ to $s\approx4$ as the parental temperature $T^{*}$ is lowered toward the dynamical crossover temperature $T_d$ (the mode-coupling-like temperature at which the liquid's relaxation time rises steeply). Comparing this with an earlier random-pinning study — in which a fraction $p$ of particles is frozen in place during the minimization, producing the same crossover $s(p)$ from 2 to 4 above a threshold $p_{th}\sim0.5$ (Angelani, Paoluzzi, Parisi, and Ruocco 2018) — they matched $s(T^{*})=s(p)$ and interpreted the resulting fraction $p(T^{*})$ as the fraction of particles belonging to dynamically heterogeneous frozen regions. This yields a length scale $\xi_{pin}=(pN/\rho)^{1/3}$ (for $N$ particles at number density $\rho$) that grows mildly as $T^{*}$ approaches $T_d$, connecting the non-Debye spectrum to the spatial heterogeneity of the supercooled liquid. Related work has since mapped how the non-phononic spectrum responds to the quench rate (Lerner and Bouchbinder 2017), to glass stability from poorly annealed to ultrastable (Wang et al. 2019; Xu et al. 2024) — an ultrastable glass being one prepared by extremely slow annealing or vapor deposition, thermodynamically equivalent to a glass aged for astronomically long times — and to elastic heterogeneity through generalized heterogeneous-elasticity theory (Schirmacher et al. 2024).

All of these studies share one structural limitation: each spectrum is that of a final inherent structure, characterized by its preparation endpoint, with no time resolution. Real glasses, however, age: after a quench below the glass transition, structural relaxation continues and observables drift with the waiting time $t_w$ (the time elapsed since preparation), the phenomenon of physical aging. Experiments show that this aging reaches the vibrational spectrum: physical aging modifies the low-frequency $D(\omega)$ of glassy polymers (Duval, Saviot, David, Etienne, and Jal 2003), and geologically hyperaged amber displays a depletion of the terahertz density of states together with a less disordered, longer-range-correlated elastic matrix (Pogna, Chumakov, Ferrante, Ramos, and Scopigno 2019). Paoluzzi et al. (2019) therefore closed their paper by proposing, as an explicit future direction, the numerical investigation of $D(\omega)$ in the aging regime, in order to understand the relation between the non-Debye spectrum and aging in glasses. That age-resolved numerical investigation — connecting the equilibrium length-scale phenomenology to the hyperaging observations — is the open problem stated here.

Problem Statement

Perform the numerical investigation of the low-frequency vibrational density of states $D(\omega)$ of a structural glass in the aging regime, proposed as future work in Paoluzzi et al. (Phys. Rev. Lett. 123, 155502 (2019)), and establish how aging modifies the non-Debye spectrum. Concretely, all four of the following parts are required:

(1) Age-resolved spectra. Prepare a three-dimensional model glass former by a quench from an equilibrated configuration, let it age isothermally in the glassy state, and at a controlled set of waiting times $t_w$ minimize to inherent structures and compute $D(\omega)$ from the dynamical matrix, together with a localization measure of the normal modes. The measurement must be resolved by age, not only by preparation endpoint.

(2) Non-Debye exponent and soft modes. Determine how the non-Debye sector below the boson peak evolves with $t_w$: the effective exponent $s$ in $D(\omega)\sim\omega^{s}$ (or equivalently the full shape and weight of the non-phononic spectrum), its prefactor, and the abundance and degree of localization of the soft quasilocalized modes.

(3) Length scale. Establish whether the pinning-based inference of the heterogeneous-region length, defined at equilibrium through the mapping $s(T^{*})=s(p)$, extends to aged out-of-equilibrium configurations, and whether the inferred length grows, saturates, or shrinks with age.

(4) Interpretation and connection. Relate the spectral evolution to aging-induced structural relaxation (the drift of the glass toward deeper inherent structures), and compare it with the equilibrium parental-temperature phenomenology and with the experimental observation that physical aging and geological hyperaging deplete the terahertz density of states.

Parts (1)-(3) are the required numerical results and part (4) is the required interpretive discussion, grounded in them; no proper subset of the four parts resolves the task.

The common objective is the relation between the non-Debye spectrum and aging: whether the frozen-heterogeneity physics that governs the equilibrium crossover of $s(T^{*})$ also governs the time-dependent spectral evolution of an aging glass, so that the natural hyperaging experiments can be connected to the spectroscopic and length-scale findings of the equilibrium theory.

References

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