Wave attenuation in glasses: Rayleigh and generalized-Rayleigh scattering scaling

Self-consistent theory for sound propagation in a simple model of a disordered, harmonic solid

We present a self-consistent theory for sound propagation in a simple model of a disordered solid. The solid is modeled as a collection of randomly distributed particles connected by harmonic springs with strengths that depend on the interparticle distances, i.e., the Euclidean random matrix model of Mézard et al. [Nucl. Phys. B 559, 689 (1999)0550-321310.1016/S0550-3213(99)00428-9]. The derivation of the theory combines two exact projection operator steps and a factorization approximation. Within our approach, the square of the speed of sound is non-negative. We expect that an unjamming transition would manifest itself through the vanishing of the speed of sound.

Vibrational lifetimes and viscoelastic properties of ultrastable glasses

Amorphous solids are viscoelastic. They dissipate energy when deformed at finite rate and finite temperature. We here use analytic theory and molecular simulations to demonstrate that linear viscoelastic dissipation can be directly related to the static and dynamic properties of the fundamental vibrational excitations of an amorphous system. We study ultrastable glasses that do not age, i.e., that remain in stable minima of the potential energy surface at finite temperature. Our simulations show four types of vibrational modes, which differ in spatial localization, similarity to plane waves and vibrational lifetimes. At frequencies below the Boson peak, the viscoelastic response can be split into contributions from plane-wave and quasilocalized modes. We derive a parameter-free expression for the viscoelastic storage and loss moduli for both of these modes. Our results show that the dynamics of microscopic dissipation, in particular the lifetimes of the modes, determine the viscoelastic response only at high frequency. Quasilocalized modes dominate the linear viscoelastic response at intermediate frequencies below the Boson peak.

Elasticity of self-organized frustrated disordered spring networks

There have been some interesting recent advances in understanding the notion of mechanical disorder in structural glasses and the statistical mechanics of these systems' low-energy excitations. Here we contribute to these advances by studying a minimal model for structural glasses' elasticity in which the degree of mechanical disorder-as characterized by recently introduced dimensionless quantifiers-is readily tunable over a very large range. We comprehensively investigate a number of scaling laws observed for various macro, meso and microscopic elastic properties, and rationalize them using scaling arguments. Interestingly, we demonstrate that the model features the universal quartic glassy vibrational density of states as seen in many atomistic and molecular models of structural glasses formed by cooling a melt. The emergence of this universal glassy spectrum highlights the role of self-organization (toward mechanical equilibrium) in its formation, and elucidates why models featuring structural frustration alone do not feature the same universal glassy spectrum. Finally, we discuss relations to existing work in the context of strain stiffening of elastic networks and of low-energy excitations in structural glasses, in addition to future research directions.

Properties of stable ensembles of Euclidean random matrices

We study the spectrum of a system of coupled disordered harmonic oscillators in the thermodynamic limit. This Euclidean random matrix ensemble has been suggested as a model for the low temperature vibrational properties of glass. Exact numerical diagonalization is performed in three and two spatial dimensions, which is accompanied by a detailed finite size analysis. It reveals a low-frequency regime of sound waves that are damped by Rayleigh scattering. At large frequencies localized modes exist. In between, the central peak in the vibrational density of states is well described by Wigner's semicircle law for not too large disorder, as is expected for simple random matrix systems. We compare our results with predictions from two recent self-consistent field theories.

Key factors impacting treatment efficiency in actual copper mineral processing wastewater by ozonation

Ozone advanced oxidation has been widely used in water treatment, but little research has been reported on the application of ozone to difficult-to-degrade mineral wastewater. In this paper, the effect of ozonation application in the treatment of copper mineral processing wastewater, which is difficult to be effectively treated by traditional processes due to its complex composition, was investigated. The effects of ozonation time, ozone concentration, temperature and pH on the degradation of organic compounds in the wastewater by ozonation were researched. It was found that the chemical oxygen demand (COD) of the wastewater could be reduced by 83.02% by ozonation under optimal treatment conditions. In addition, the mechanism of ozone degradation of the difficult-to-degrade wastewater was studied, and the reasons for the fluctuating variations of COD and ammonia nitrogen during ozonation treatment were explained.

Sound attenuation in two-dimensional glasses at finite temperatures

The thermal conductivity of glasses exhibits an unusual temperature dependence compared to their crystalline counterparts. Sound attenuation due to disorder in glasses was proposed to be important in rationalizing this special behavior. Simulation studies suggest that in the harmonic approximation, the sound attenuation follows Rayleigh scattering scaling at small wave vector in both two-dimensional (2D) and 3D glasses. The influence of the anharmonicity on sound attenuation has very recently been investigated numerically, but only in 3D glasses. Hence, it remains unknown in simulations how sound attenuation changes with the wave vector in 2D glasses when the anharmonicity comes into play. Here, we address this issue by performing computer simulations in low-temperature 2D glasses over a large range of glass stabilities. We find that the way the anharmonicity affects sound attenuation in 2D glasses is the same as that in 3D, thus revealing that numerically the influence of the anharmonicity on sound attenuation does not rely on the spatial dimension.

Density of states below the first sound mode in 3D glasses

Glasses feature universally low-frequency excess vibrational modes beyond Debye prediction, which could help rationalize, e.g., the glasses’ unusual temperature dependence of thermal properties compared to crystalline solids. The way the density of states of these low-frequency excess modes D( ω) depends on the frequency ω has been debated for decades. Recent simulation studies of 3D glasses suggest that D( ω) scales universally with ω4 in a low-frequency regime below the first sound mode. However, no simulation study has ever probed as low frequencies as possible to test directly whether this quartic law could work all the way to extremely low frequencies. Here, we calculated D( ω) below the first sound mode in 3D glasses over a wide range of frequencies. We find D( ω) scales with ω β with β < 4 at very low frequencies examined, while the ω4 law works only in a limited intermediate-frequency regime in some glasses. Moreover, our further analysis suggests our observation does not depend on glass models or glass stabilities examined. The ω4 law of D( ω) below the first sound mode is dominant in current simulation studies of 3D glasses, and our direct observation of the breakdown of the quartic law at very low frequencies thus leaves an open but important question that may attract more future numerical and theoretical studies.

Phonon transport properties of particulate physical gels

Particulate physical gels are sparse, low-density amorphous materials in which clusters of glasses are connected to form a heterogeneous network structure. This structure is characterized by two length scales, ξ s and ξ G: ξ s measures the length of heterogeneities in the network structure and ξ G is the size of glassy clusters. Accordingly, the vibrational states (eigenmodes) of such a material also exhibit a multiscale nature with two characteristic frequencies, [Formula: see text] and ω G, which are associated with ξ s and ξ G, respectively: (i) phonon-like vibrations in the homogeneous medium at [Formula: see text], (ii) phonon-like vibrations in the heterogeneous medium at [Formula: see text], and (iii) disordered vibrations in the glassy clusters at ω > ω G. Here, we demonstrate that the multiscale characteristics seen in the static structures and vibrational states also extend to the phonon transport properties. Phonon transport exhibits two distinct crossovers at frequencies ω* and ω G (or at wavenumbers of [Formula: see text] and [Formula: see text]). In particular, both transverse and longitudinal phonons cross over between Rayleigh scattering at [Formula: see text] and diffusive damping at [Formula: see text]. Remarkably, the Ioffe–Regel limit is located at the very low frequency of ω*. Thus, phonon transport is localized above ω*, even where phonon-like vibrational states persist. This markedly strong scattering behavior is caused by the sparse, porous structure of the gel.

Universal density of low-frequency states in silica glass at finite temperatures

The theoretical understanding of the low-frequency modes in amorphous solids at finite temperature is still incomplete. The study of the relevant modes is obscured by the dressing of interparticle forces by collision-induced momentum transfer that is unavoidable at finite temperatures. Recently, it was proposed that low-frequency modes of vibrations around the thermally averaged configurations deserve special attention. In simple model glasses with bare binary interactions, these included quasilocalized modes whose density of states appears to be universal, depending on the frequencies as D(ω)∼ω^{4}, in agreement with the similar law that is obtained with bare forces at zero temperature. In this paper, we report investigations of a model of silica glass at finite temperature; here the bare forces include binary and ternary interactions. Nevertheless, we can establish the validity of the universal law of the density of quasilocalized modes also in this richer and more realistic model glass.

Microscopic analysis of sound attenuation in low-temperature amorphous solids reveals quantitative importance of non-affine effects

Sound attenuation in low-temperature amorphous solids originates from their disordered structure. However, its detailed mechanism is still being debated. Here, we analyze sound attenuation starting directly from the microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping coefficient. We verify that the sound damping coefficients calculated from our expression agree very well with results from independent simulations of sound attenuation. Small wavevector analysis of our expression shows that sound attenuation is primarily determined by the non-affine displacements’ contribution to the sound wave propagation coefficient coming from the frequency shell of the sound wave. Our expression involves only quantities that pertain to solids’ static configurations. It can be used to evaluate the low-temperature sound damping coefficients without directly simulating sound attenuation.

Structural, mechanical, and vibrational properties of particulate physical gels

Our lives are surrounded by a rich assortment of disordered materials. In particular, glasses are well known as dense, amorphous materials, whereas gels exist in low-density, disordered states. Recent progress has provided a significant step forward in understanding the material properties of glasses, such as mechanical, vibrational, and transport properties. In contrast, our understanding of particulate physical gels is still highly limited. Here, using molecular dynamics simulations, we study a simple model of particulate physical gels, the Lennard-Jones (LJ) gels, and provide a comprehensive understanding of their structural, mechanical, and vibrational properties, all of which are markedly different from those of LJ glasses. First, the LJ gels show sparse, heterogeneous structures, and the length scale ξ_s of the structures grows as the density is lowered. Second, the LJ gels are extremely soft, with both shear G and bulk K moduli being orders of magnitude smaller than those of LJ glasses. Third, many low-frequency vibrational modes are excited, which form a characteristic plateau with the onset frequency ω_* in the vibrational density of states. Structural, mechanical, and vibrational properties, characterized by ξ_s, G, K, and ω_*, respectively, show power-law scaling behaviors with the density, which establishes a close relationship between them. Throughout this work, we also reveal that LJ gels are multiscale, solid-state materials: (i) homogeneous elastic bodies at long lengths, (ii) heterogeneous elastic bodies with fractal structures at intermediate lengths, and (iii) amorphous structural bodies at short lengths.

Low-Frequency Excess Vibrational Modes in Two-Dimensional Glasses

Glasses possess more low-frequency vibrational modes than predicted by Debye theory. These excess modes are crucial for the understanding of the low temperature thermal and mechanical properties of glasses, which differ from those of crystalline solids. Recent simulational studies suggest that the density of the excess modes scales with their frequency ω as ω^{4} in two and higher dimensions. Here, we present extensive numerical studies of two-dimensional model glass formers over a large range of glass stabilities. We find that the density of the excess modes follows D_{exc}(ω)∼ω^{2} up to around the boson peak, regardless of the glass stability. The stability dependence of the overall scale of D_{exc}(ω) correlates with the stability dependence of low-frequency sound attenuation. However, we also find that, in small systems, where the first sound mode is pushed to higher frequencies, at frequencies below the first sound mode, there are excess modes with a system size independent density of states that scales as ω^{3}.

Low-energy quasilocalized excitations in structural glasses

Glassy solids exhibit a wide variety of generic thermomechanical properties, ranging from universal anomalous specific heat at cryogenic temperatures to nonlinear plastic yielding and failure under external driving forces, which qualitatively differ from their crystalline counterparts. For a long time, it has been believed that many of these properties are intimately related to nonphononic, low-energy quasilocalized excitations (QLEs) in glasses. Indeed, recent computer simulations have conclusively revealed that the self-organization of glasses during vitrification upon cooling from a melt leads to the emergence of such QLEs. In this Perspective, we review developments over the past three decades toward understanding the emergence of QLEs in structural glasses and the degree of universality in their statistical and structural properties. We discuss the challenges and difficulties that hindered progress in achieving these goals and review the frameworks put forward to overcome them. We conclude with an outlook on future research directions and open questions.

Unifying Description of the Vibrational Anomalies of Amorphous Materials

The vibrational density of states D(ω) of solids controls their thermal and transport properties. In crystals, the low-frequency modes are extended phonons distributed in frequency according to Debye's law, D(ω)∝ω^{2}. In amorphous solids, phonons are damped, and at low frequency D(ω) comprises extended modes in excess over Debye's prediction, leading to the so-called boson peak in D(ω)/ω^{2} at ω_{bp}, and quasilocalized ones. Here we show that boson peak and phonon attenuation in the Rayleigh scattering regime are related, as suggested by correlated fluctuating elasticity theory, and that amorphous materials can be described as homogeneous isotropic elastic media punctuated by quasilocalized modes acting as elastic heterogeneities. Our numerical results resolve the conflict between theoretical approaches attributing amorphous solids' vibrational anomalies to elastic disorder and localized defects.

Unified quantifier of mechanical disorder in solids

Mechanical disorder in solids, which is generated by a broad range of physical processes and controls various material properties, appears in a wide variety of forms. Defining unified and measurable dimensionless quantifiers, allowing quantitative comparison of mechanical disorder across widely different physical systems, is therefore an important goal. Two such coarse-grained dimensionless quantifiers (among others) appear in the literature: one is related to the spectral broadening of discrete phononic bands in finite-size systems (accessible through computer simulations) and the other is related to the spatial fluctuations of the shear modulus in macroscopically large systems. The latter has been recently shown to determine the amplitude of wave attenuation rates in the low-frequency limit (accessible through laboratory experiments). Here, using two alternative and complementary theoretical approaches linked to the vibrational spectra of solids, we derive a basic scaling relation between the two dimensionless quantifiers. This scaling relation, which is supported by simulational data, shows that the two apparently distinct quantifiers are in fact intrinsically related, giving rise to a unified quantifier of mechanical disorder in solids. We further discuss the obtained results in the context of the unjamming transition taking place in soft sphere packings at low confining pressures, in addition to their implications for our understanding of the low-frequency vibrational spectra of disordered solids in general, and in particular those of glassy systems.

Sound damping in soft particle packings: the interplay between configurational disorder and inelasticity

We numerically investigate sound damping in disordered two-dimensional soft particle packings. We simulate evolution of standing waves of particle displacements and analyze time correlation functions of particle velocities and power spectra. We control the strength of inelastic interactions between the particles in contact to show how the inelasticity affects anomalous sound characteristics of disordered systems: Increasing the strength of inelastic interactions, we find that (i) sound softening vanishes and (ii) attenuation coefficients exhibit a transition from the Rayleigh law to quadratic growth. We also report (iii) how the Ioffe-Regel limit frequencies depend on the strength of inelasticity as useful information for experiments and applications of the sound in disordered media. Our findings suggest that sound damping in soft particle packings is determined by the interplay between elastic heterogeneities and inelasticity.

Internally Stressed and Positionally Disordered Minimal Complexes Yield Glasslike Nonphononic Excitations

Glasses, unlike their crystalline counterparts, exhibit low-frequency nonphononic excitations whose frequencies ω follow a universal D(ω)∼ω^{4} density of states. The process of glass formation generates positional disorder intertwined with mechanical frustration, posing fundamental challenges in understanding the origins of glassy nonphononic excitations. Here we suggest that minimal complexes-mechanically frustrated and positionally disordered local structures-embody the minimal physical ingredients needed to generate glasslike excitations. We investigate the individual effects of mechanical frustration and positional disorder on the vibrational spectrum of isolated minimal complexes, and demonstrate that ensembles of marginally stable minimal complexes yield D(ω)∼ω^{4}. Furthermore, glasslike excitations emerge by embedding a single minimal complex within a perfect lattice. Consequently, minimal complexes offer a conceptual framework to understand glasslike excitations from first principles, as well as a practical computational method for introducing them into solids.

Mechanical disorder of sticky-sphere glasses. II. Thermomechanical inannealability

Many structural glasses feature static and dynamic mechanical properties that can depend strongly on glass formation history. The degree of universality of this history dependence and what it is possibly affected by are largely unexplored. Here we show that the variability of elastic properties of simple computer glasses under thermal annealing depends strongly on the strength of attractive interactions between the glasses' constituent particles-referred to here as glass "stickiness." We find that in stickier glasses the stiffening of the shear modulus with thermal annealing is strongly suppressed, while the thermal-annealing-induced softening of the bulk modulus is enhanced. Our key finding is that the characteristic frequency and density per frequency of soft quasilocalized modes becomes effectively invariant to annealing in very sticky glasses; the latter are therefore deemed "thermomechanically inannealable." The implications of our findings and future research directions are discussed.

Novel elastic instability of amorphous solids in finite spatial dimensions

Recently, progress has been made in the understanding of anomalous vibrational excitations in amorphous solids. In the lowest-frequency region, the vibrational spectrum follows a non-Debye quartic law, which persists up to zero frequency without any frequency gap. This gapless vibrational density of states (vDOS) suggests that glasses are on the verge of instability. This feature of marginal stability is now highlighted as a key concept in the theories of glasses. In particular, the elasticity theory based on marginal stability predicts the gapless vDOS. However, this theory yields a quadratic law and not the quartic law. To address this inconsistency, we presented a new type of instability, which is different from the conventional one, and proposed that amorphous solids are marginally stable considering the new instability in the preceding study [M. Shimada, H. Mizuno and A. Ikeda, Soft Matter, 2020, 16, 7279]. In this study, we further extend and detail the results for these instabilities. By analyzing various examples of disorder, we demonstrate that real glasses in finite spatial dimensions can be marginally stable by the proposed novel instability.

Simple and Broadly Applicable Definition of Shear Transformation Zones

Plastic deformation in amorphous solids is known to be carried by stress-induced localized rearrangements of a few tens of particles, accompanied by the conversion of elastic energy to heat. Despite their central role in determining how glasses yield and break, the search for a simple and generally applicable definition of the precursors of those plastic rearrangements-the so-called shear transformation zones (STZs)-is still ongoing. Here we present a simple definition of STZs-based solely on the harmonic approximation of a glass's energy. We explain why and demonstrate directly that our proposed definition of plasticity carriers in amorphous solids is more broadly applicable compared to anharmonic definitions put forward previously. Finally, we offer an open-source library that analyzes low-lying STZs in computer glasses and in laboratory materials such as dense colloidal suspensions for which the harmonic approximation is accessible. Our results constitute a physically motivated methodological advancement towards characterizing mechanical disorder in glasses, and understanding how they yield.

Vibrational density of states of amorphous solids with long-ranged power-law-correlated disorder in elasticity

A theory of vibrational excitations based on power-law spatial correlations in the elastic constants (or equivalently in the internal stress) is derived, in order to determine the vibrational density of states D([Formula: see text]) of disordered solids. The results provide the first prediction of a boson peak in amorphous materials where spatial correlations in the internal stresses (or elastic constants) are of power-law form, as is often the case in experimental systems, leading to a logarithmic enhancement of (Rayleigh) phonon attenuation. A logarithmic correction of the form [Formula: see text] is predicted to occur in the plot of the reduced excess DOS for frequencies around the boson peak in 3D. Moreover, the theory provides scaling laws of the density of states in the low-frequency region, including a [Formula: see text] regime in 3D, and provides information about how the boson peak intensity depends on the strength of power-law decay of fluctuations in elastic constants or internal stress. Analytical expressions are also derived for the dynamic structure factor for longitudinal excitations, which include a logarithmic correction factor, and numerical calculations are presented supporting the assumptions used in the theory.

Acoustic resonance in periodically sheared glass: damping due to plastic events

Using molecular dynamics simulation, we study acoustic resonance in a low-temperature model glass by applying a small periodic shear at a boundary wall. Shear wave resonance occurs as the frequency ω approaches ωl = πc⊥l/L (l = 1, 2…). Here, c⊥ is the transverse sound speed and L is the cell width. At resonance, large-amplitude sound waves appear after many cycles even if the applied strain γ0 is very small. They then induce plastic events, which are heterogeneous on the mesoscopic scale and intermittent on timescales longer than the oscillation period tp = 2π/ω. We visualize them together with the extended elastic strains around them. These plastic events serve to damp sounds. We obtain the nonlinear damping Q-1 = tan δ due to the plastic events near the first resonance at ω ≅ ω1, which is linear in γ0 and independent of ω. After many resonant cycles, we observe an increase in the shear modulus (measured after switching-off the oscillation). We also observe catastrophic plastic events after a very long time (∼103tp), which induce system-size elastic strains and cause a transition from resonant to off-resonant states. At resonance, stroboscopic diffusion becomes detectable.

Key role of retardation and non-locality in sound propagation in amorphous solids as evidenced by a projection formalism

We investigate acoustic propagation in amorphous solids by constructing a projection formalism based on separating atomic vibrations into two, "phonon" (P) and "non-phonon" (NP), subspaces corresponding to large and small wavelengths. For a pairwise interaction model, we show the existence of a "natural" separation lengthscale, determined by structural disorder, for which the isolated P subspace presents the acoustic properties of a nearly homogenous (Debye-like) elastic continuum, while the NP one encapsulates all small scale non-affinity effects. The NP eigenstates then play the role of dynamical scatterers for the phonons. However, at variance with a conjecture of defect theories, their spectra present a finite low frequency gap, which turns out to lie around the Boson peak frequency, and only a small fraction of them are highly localized. We then show that small scale disorder effects can be rigorously reduced to the existence, in the Navier-like wave equation of the continuum, of a generalized elasticity tensor, which is not only retarded, since scatterers are dynamical, but also non-local. The full neglect of both retardation and non-locality suffices to account for most of the corrections to Born macroscopic moduli. However, these two features are responsible for sound speed dispersion and have quite a significant effect on the magnitude of sound attenuation. Although it remains open how they impact the asymptotic, large wavelength scaling of sound damping, our findings rule out the possibility of representing an amorphous solid by an inhomogeneous elastic continuum with the standard (i.e., local and static) elastic moduli.

Extracting the properties of quasilocalized modes in computer glasses: Long-range continuum fields, contour integrals, and boundary effects

Low-frequency nonphononic modes and plastic rearrangements in glasses are spatially quasilocalized, i.e., they feature a disorder-induced short-range core and known long-range decaying elastic fields. Extracting the unknown short-range core properties, potentially accessible in computer glasses, is of prime importance. Here we consider a class of contour integrals, performed over the known long-range fields, which are especially designed for extracting the core properties. We first show that, in computer glasses of typical sizes used in current studies, the long-range fields of quasilocalized modes experience boundary effects related to the simulation box shape and the widely employed periodic boundary conditions. In particular, image interactions mediated by the box shape and the periodic boundary conditions induce the fields' rotation and orientation-dependent suppression of their long-range decay. We then develop a continuum theory that quantitatively predicts these finite-size boundary effects and support it by extensive computer simulations. The theory accounts for the finite-size boundary effects and at the same time allows the extraction of the short-range core properties, such as their typical strain ratios and orientation. The theory is extensively validated in both two and three dimensions. Overall, our results offer a useful tool for extracting the intrinsic core properties of nonphononic modes and plastic rearrangements in computer glasses.

Analytical prediction of logarithmic Rayleigh scattering in amorphous solids from tensorial heterogeneous elasticity with power-law disorder

The damping or attenuation coefficient of sound waves in solids due to impurities scales with the wavevector to the fourth power, also known as Rayleigh scattering. In amorphous solids, Rayleigh scattering may be enhanced by a logarithmic factor although computer simulations offer conflicting conclusions regarding this enhancement and its microscopic origin. We present a tensorial replica field-theoretic derivation based on heterogeneous or fluctuating elasticity (HE), which shows that long-range (power-law) spatial correlations of the elastic constants, is the origin of the logarithmic enhancement to Rayleigh scattering of phonons in amorphous solids. We also consider the case of zero spatial fluctuations in the elastic constants, and of power-law decaying fluctuations in the internal stresses. Also in this case the logarithmic enhancement to the Rayleigh scattering law can be derived from the proposed tensorial HE framework.

Universal Low-Frequency Vibrational Modes in Silica Glasses

It was recently shown that different simple models of glass formers with binary interactions define a universality class in terms of the density of states of their quasilocalized low-frequency modes. Explicitly, once the hybridization with standard Debye (extended) modes is avoided, a number of such models exhibit a universal density of states, depending on the mode frequencies as D(ω)∼ω^{4}. It is unknown, however, how wide this universality class is, and whether it also pertains to more realistic models of glass formers. To address this issue we present analysis of the quasilocalized modes in silica, a network glass that has both binary and ternary interactions. We conclude that in three dimensions silica exhibits the very same frequency dependence at low frequencies, suggesting that this universal form is a generic consequence of amorphous glassiness.

Sound attenuation in finite-temperature stable glasses

The temperature dependence of the thermal conductivity of amorphous solids is markedly different from that of their crystalline counterparts, but exhibits universal behaviour. Sound attenuation is believed to be related to this universal behaviour. Recent computer simulations demonstrated that in the harmonic approximation sound attenuation Γ obeys quartic, Rayleigh scattering scaling for small wavevectors k and quadratic scaling for wavevectors above the Ioffe-Regel limit. However, simulations and experiments do not provide a clear picture of what to expect at finite temperatures where anharmonic effects become relevant. Here we study sound attenuation at finite temperatures for model glasses of various stability, from unstable glasses that exhibit rapid aging to glasses whose stability is equal to those created in laboratory experiments. We find several scaling laws depending on the temperature and stability of the glass. First, we find the large wavevector quadratic scaling to be unchanged at all temperatures. Second, we find that at small wavevectors Γ∼k^1.5 for an aging glass, but Γ∼k² when the glass does not age on the timescale of the calculation. For our most stable glass, we find that Γ∼k² at small wavevectors, then a crossover to Rayleigh scattering scaling Γ∼k⁴, followed by another crossover to the quadratic scaling at large wavevectors. Our computational observation of this quadratic behavior reconciles simulation, theory and experiment, and will advance the understanding of the temperature dependence of thermal conductivity of glasses.

Pinching a glass reveals key properties of its soft spots

It is now well established that glasses feature quasilocalized nonphononic excitations-coined "soft spots"-, which follow a universal [Formula: see text] density of states in the limit of low frequencies ω. All glass-specific properties, such as the dependence on the preparation protocol or composition, are encapsulated in the nonuniversal prefactor of the universal [Formula: see text] law. The prefactor, however, is a composite quantity that incorporates information both about the number of quasilocalized nonphononic excitations and their characteristic stiffness, in an apparently inseparable manner. We show that by pinching a glass-i.e., by probing its response to force dipoles-one can disentangle and independently extract these two fundamental pieces of physical information. This analysis reveals that the number of quasilocalized nonphononic excitations follows a Boltzmann-like law in terms of the parent temperature from which the glass is quenched. The latter, sometimes termed the fictive (or effective) temperature, plays important roles in nonequilibrium thermodynamic approaches to the relaxation, flow, and deformation of glasses. The analysis also shows that the characteristic stiffness of quasilocalized nonphononic excitations can be related to their characteristic size, a long sought-for length scale. These results show that important physical information, which is relevant for various key questions in glass physics, can be obtained through pinching a glass.

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.

Finite-size effects in the nonphononic density of states in computer glasses

The universal form of the density of nonphononic, quasilocalized vibrational modes of frequency ω in structural glasses, D(ω), was predicted theoretically decades ago, but only recently revealed in numerical simulations. In particular, it has been recently established that, in generic computer glasses, D(ω) increases from zero frequency as ω^{4}, independent of spatial dimension and of microscopic details. However, it has been shown [Lerner and Bouchbinder, Phys. Rev. E 96, 020104(R) (2017)2470-004510.1103/PhysRevE.96.020104] that the preparation protocol employed to create glassy samples may affect the form of their resulting D(ω): glassy samples rapidly quenched from high-temperature liquid states were shown to feature D(ω)∼ω^{β} with β<4, presumably limiting the degree of universality of the ω^{4} law. Here we show that exponents β<4 are seen only in small glassy samples quenched from high-temperature liquid states-whose sizes are comparable to or smaller than the size of the disordered core of soft quasilocalized vibrations-while larger glassy samples made with the same protocol feature the universal ω^{4} law. Our results demonstrate that observations of β<4 in the nonphononic density of states stem from finite-size effects, and we thus conclude that the ω^{4} law should be featured by any sufficiently large glass quenched from a melt.

Aging effects on thermal conductivity of glass-forming liquids

Thermal conductivity of a model glass-forming system in the liquid and glass states is studied using extensive numerical simulations. We show that near the glass transition temperature, where the structural relaxation time becomes very long, the measured thermal conductivity decreases with increasing age. Second, the thermal conductivity of the disordered solid obtained at low temperatures is found to depend on the cooling rate with which it was prepared. For the cooling rates accessible in simulations, lower cooling rates lead to lower thermal conductivity. Our analysis links this decrease of the thermal conductivity with increased exploration of lower-energy inherent structures of the underlying potential energy landscape. Further, we show that the lowering of conductivity for lower-energy inherent structures is related to the high-frequency harmonic modes associated with the inherent structure being less extended. Possible effects of considering relatively small systems and fast cooling rates in the simulations are discussed.

Energy transport in glasses

The temperature dependence of the thermal conductivity is linked to the nature of the energy transport at a frequency ω, which is quantified by thermal diffusivity d(ω). Here we study d(ω) for a poorly annealed glass and a highly stable glass prepared using the swap Monte Carlo algorithm. To calculate d(ω), we excite wave packets and find that the energy moves diffusively for high frequencies up to a maximum frequency, beyond which the energy stays localized. At intermediate frequencies, we find a linear increase of the square of the width of the wave packet with time, which allows for a robust calculation of d(ω), but the wave packet is no longer well described by a Gaussian as for high frequencies. In this intermediate regime, there is a transition from a nearly frequency independent thermal diffusivity at high frequencies to d(ω) ∼ ω-4 at low frequencies. For low frequencies the sound waves are responsible for energy transport and the energy moves ballistically. The low frequency behavior can be predicted using sound attenuation coefficients.