Sound attenuation in stable glasses

The origin of sound damping in amorphous solids: Defects and beyond

Comprehending sound damping is integral to understanding the anomalous low temperature properties of glasses. After decades of studies, Rayleigh scaling of the sound attenuation coefficient with frequency, [Formula: see text], became generally accepted. Rayleigh scaling invokes a picture of scattering from defects. It is unclear how to define glass defects. Here, we use a particle level contribution to sound damping to determine areas in the glass that contribute more to sound damping than other areas, which allows us to define defects. Over a range of stability, sound damping scales linearly with the fraction of particles in the defects. However, sound is still attenuated in ultrastable glasses where no defects are identified. We show that sound damping in these glasses is due to nearly uniformly distributed non-affine forces that arise after macroscopic deformation. To fully understand sound attenuation in glasses, one has to consider contributions from defects and a defect-free background, which represents a different paradigm of sound damping in glasses.

Defects, Sound Damping, and the Boson Peak in Amorphous Solids

Two nearly universal and anomalous properties of glasses, the peak in the specific heat and plateau of the thermal conductivity, occur around the same temperature. This coincidence suggests that the two phenomena are related. Both effects can be rationalized by assuming Rayleigh scaling of sound attenuation and this scaling leads one to consider scattering from defects. Identifying defects in glasses, which are inherently disordered, is a long-standing problem that was approached in several ways. We examine candidates for defects in glasses that represent areas of strong sound damping. We show that some defects are associated with quasi-localized excitations, which may be associated with modes in excess of the Debye theory. We also examine generalized Debye relations, which relate sound damping and the speed of sound to excess modes. We derive a generalized Debye relation that does not resort to an approximation used by previous authors. We find that our relation and the relation given by previous authors are almost identical at small frequencies and also reproduce the independently determined density of states. However, the different generalized Debye relations do not agree around the boson peak. While generalized Debye relations accurately predict the boson peak in two-dimensional glasses, they underestimate the boson peak in three-dimensional glasses.

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.

Scaling regimes and fluctuations of observables in computer glasses approaching the unjamming transition

Under decompression, disordered solids undergo an unjamming transition where they become under-coordinated and lose their structural rigidity. The mechanical and vibrational properties of these materials have been an object of theoretical, numerical, and experimental research for decades. In the study of low-coordination solids, understanding the behavior and physical interpretation of observables that diverge near the transition is of particular importance. Several such quantities are length scales (ξ or l) that characterize the size of excitations, the decay of spatial correlations, the response to perturbations, or the effect of physical constraints in the boundary or bulk of the material. Additionally, the spatial and sample-to-sample fluctuations of macroscopic observables such as contact statistics or elastic moduli diverge approaching unjamming. Here, we discuss important connections between all of these quantities and present numerical results that characterize the scaling properties of sample-to-sample contact and shear modulus fluctuations in ensembles of low-coordination disordered sphere packings and spring networks. Overall, we highlight three distinct scaling regimes and two crossovers in the disorder quantifiers χz and χμ as functions of system size N and proximity to unjamming δz. As we discuss, χX relates to the standard deviation σX of the sample-to-sample distribution of the quantity X (e.g., excess coordination δz or shear modulus μ) for an ensemble of systems. Importantly, χμ has been linked to experimentally accessible quantities that pertain to sound attenuation and the density of vibrational states in glasses. We investigate similarities and differences in the behaviors of χz and χμ near the transition and discuss the implications of our findings on current literature, unifying findings in previous studies.

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.

Detecting low-energy quasilocalized excitations in computer glasses

Soft, quasilocalized excitations (QLEs) are known to generically emerge in a broad class of disordered solids and to govern many facets of the physics of glasses, from wave attenuation to plastic instabilities. In view of this key role of QLEs, shedding light upon several open questions in glass physics depends on the availability of computational tools that allow one to study QLEs' statistical mechanics. The latter is a formidable task since harmonic analyses are typically contaminated by hybridizations of QLEs with phononic excitations at low frequencies, obscuring a clear picture of QLEs' abundance, typical frequencies, and other important micromechanical properties. Here we present an efficient algorithm to detect the field of quasilocalized excitations in structural computer glasses. The algorithm introduced takes a computer-glass sample as input and outputs a library of QLEs embedded in that sample. We demonstrate the power of the algorithm by reporting the spectrum of glassy excitations in two-dimensional computer glasses featuring a huge range of mechanical stability, which is inaccessible using conventional harmonic analyses due to phonon hybridizations. Future applications are discussed.

Vibrational Phenomena in Glasses at Low Temperatures Captured by Field Theory of Disordered Harmonic Oscillators

We investigate the vibrational properties of topologically disordered materials by analytically studying particles that harmonically oscillate around random positions. Exploiting classical field theory in the thermodynamic limit at T=0, we build up a self-consistent model by analyzing the Hessian utilizing Euclidean random matrix theory. In accordance with earlier findings [T. S. Grigera et al.J. Stat. Mech. (2011) P02015.JSMTC61742-546810.1088/1742-5468/2011/02/P02015], we take nonplanar diagrams into account to correctly address multiple local scattering events. By doing so, we end up with a first principles theory that can predict the main anomalies of athermal disordered materials, including the boson peak, sound softening, and Rayleigh damping of sound. In the vibrational density of states, the sound modes lead to Debye's law for small frequencies. Additionally, an excess appears in the density of states starting as ω^{4} in the low frequency limit, which is attributed to (quasi-) localized modes.

Non-phononic density of states of two-dimensional glasses revealed by random pinning

The vibrational density of states of glasses is considerably different from that of crystals. In particular, there exist spatially localized vibrational modes in glasses. The density of states of these non-phononic modes has been observed to follow g(ω) ∝ ω4, where ω is the frequency. However, in two-dimensional systems, the abundance of phonons makes it difficult to accurately determine this non-phononic density of states because they are strongly coupled to non-phononic modes and yield strong system-size and preparation-protocol dependencies. In this article, we utilize the random pinning method to suppress phonons and disentangle their coupling with non-phononic modes and successfully calculate their density of states as g(ω) ∝ ω4. We also study their localization properties and confirm that low-frequency non-phononic modes in pinned systems are truly localized without far-field contributions. We finally discuss the excess density of states over the Debye value that results from the hybridization of phonons and non-phononic modes.

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.

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}.

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.

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.

Physics of phonon-polaritons in amorphous materials

The nature of bosonic excitations in disordered materials has remained elusive due to the difficulties in defining key concepts such as quasi-particles in the presence of disorder. We report on an experimental observation of phonon-polaritons in glasses, including a prominent boson peak (BP), i.e., excess of THz modes over the Debye law. A theoretical framework based on the concept of diffusons is developed to describe the broadening linewidth of the polariton due to disorder-induced scattering. It is shown here for the first time that the BP frequency and the Ioffe-Regel (IR) crossover frequency of the polariton collapse onto the same power-law decay with the diffusivity of the bosonic excitation. This analysis dismisses the hypothesis of the BP being caused by a relic of the van Hove singularity. The presented framework establishes a new methodology to analyze bosonic excitations in amorphous media, well beyond the traditional case of acoustic phonons, and establishes the IR crossover as the fundamental physical mechanism behind the BP.

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.

Continuum constitutive laws to describe acoustic attenuation in glasses

Nowadays metamaterials are at the focus of an intense research as promising for thermal and acoustic engineering. However, the computational cost associated to the large system size required for correctly simulating them imposes the use of finite-elements simulations, developing continuum models, able to grasp the physics at play without entering in the atomistic details. Still, a correct description should be able to reproduce not only the extrinsic scattering sources on waves propagation, as introduced by the metamaterial microstructure, but also the intrinsic wave attenuation of the material itself. This becomes dramatically important when the metamaterial is made out of a glass, which is intrinsically highly dissipative and with a wave attenuation strongly dependent on frequency. Here we propose a continuum mechanical model for a viscoelastic medium, able to bridge atomic and macroscopic scale in amorphous materials and describe phonon attenuation due to atomistic mechanisms, characterized by a defined frequency dependence. This represents a first decisive step for investigating the effect of a complex nano- or microstructure on acoustic attenuation, while including the atomistic contribution as well.

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.

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.

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.

Stability dependence of local structural heterogeneities of stable amorphous solids

The universal anomalous vibrational and thermal properties of amorphous solids are believed to be related to the local variations of the elasticity. Recently it has been shown that the vibrational properties are sensitive to the glass's stability. Here we study the stability dependence of the local elastic constants of a simulated glass former over a broad range of stabilities, from a poorly annealed glass to a glass whose stability is comparable to laboratory exceptionally stable vapor deposited glasses. We show that with increasing stability the glass becomes more uniform as evidenced by a smaller variance of local elastic constants. We find that, according to the definition of local elastic moduli used in this work, the local elastic moduli are not spatially correlated.

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.