Observation of the onset of strong scattering on high frequency acoustic phonons in densified silica glass

Long-range correlations in elastic moduli and local stresses at the unjamming transition

We explore the behaviour of spatially heterogeneous elastic moduli as well as the correlations between local moduli in model solids with short-range repulsive potentials. We show through numerical simulations that local elastic moduli exhibit long-range correlations, similar to correlations in the local stresses. Specifically, the correlations in local shear moduli exhibit anisotropic behavior at large lengthscales characterized by pinch-point singularities in Fourier space, displaying a structural pattern akin to shear stress correlations. Focussing on two-dimensional jammed solids approaching the unjamming transition, we show that stress correlations exhibit universal properties, characterized by a quadratic p2 dependence of the correlations as the pressure p approaches zero, independent of the details of the model. In contrast, the modulus correlations exhibit a power-law dependence with different exponents depending on the specific interaction potential. Furthermore, we illustrate that while affine responses lack long-range correlations, the total modulus, which encompasses non-affine behavior, exhibits long-range correlations.

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.

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.

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.

Probing the non-Debye low-frequency excitations in glasses through random pinning

We investigate the properties of the low-frequency spectrum in the density of states [Formula: see text] of a 3D model glass former. To magnify the non-Debye sector of the spectrum, we introduce a random pinning field that freezes a finite particle fraction to break the translational invariance and shifts all of the vibrational frequencies of the extended modes toward higher frequencies. We show that non-Debye soft localized modes progressively emerge as the fraction p of pinned particles increases. Moreover, the low-frequency tail of [Formula: see text] goes to zero as a power law [Formula: see text], with [Formula: see text] and [Formula: see text] above a threshold fraction [Formula: see text].

Elastic moduli and vibrational modes in jammed particulate packings

When we elastically impose a homogeneous, affine deformation on amorphous solids, they also undergo an inhomogeneous, nonaffine deformation, which can have a crucial impact on the overall elastic response. To correctly understand the elastic modulus M, it is therefore necessary to take into account not only the affine modulus M_{A}, but also the nonaffine modulus M_{N} that arises from the nonaffine deformation. In the present work, we study the bulk (M=K) and shear (M=G) moduli in static jammed particulate packings over a range of packing fractions φ. The affine M_{A} is determined essentially by the static structural arrangement of particles, whereas the nonaffine M_{N} is related to the vibrational eigenmodes. We elucidate the contribution of each vibrational mode to the nonaffine M_{N} through a modal decomposition of the displacement and force fields. In the vicinity of the (un)jamming transition φ_{c}, the vibrational density of states g(ω) shows a plateau in the intermediate-frequency regime above a characteristic frequency ω^{*}. We illustrate that this unusual feature apparent in g(ω) is reflected in the behavior of M_{N}: As φ→φ_{c}, where ω^{*}→0, those modes for ω<ω^{*} contribute less and less, while contributions from those for ω>ω^{*} approach a constant value which results in M_{N} to approach a critical value M_{Nc}, as M_{N}-M_{Nc}∼ω^{*}. At φ_{c} itself, the bulk modulus attains a finite value K_{c}=K_{Ac}-K_{Nc}>0, such that K_{Nc} has a value that remains below K_{Ac}. In contrast, for the critical shear modulus G_{c}, G_{Nc} and G_{Ac} approach the same value so that the total value becomes exactly zero, G_{c}=G_{Ac}-G_{Nc}=0. We explore what features of the configurational and vibrational properties cause such a distinction between K and G, allowing us to validate analytical expressions for their critical values.

Impact of pressure on plastic yield in amorphous solids with open structure

Plasticity in amorphous silica is unusual: The yield stress decreases with hydrostatic pressure, in contrast to the Mohr-Coulomb response commonly found in more compact materials such as bulk metallic glasses. To better understand this response, we have carried out molecular dynamics simulations of plastic response in a model glass with open structure. The simulations reproduce the anomalous dependence of yield stress with pressure and also correctly predict that the plastic response turns to normal once the material has been fully compacted. We also show that the overall shape of the yield surface is consistent with a quadratic behavior predicted assuming local buckling of the structure, a point of view that fits well into the present understanding of the deformation mechanisms of amorphous silica. The results also confirm that free volume is an adequate internal variable for a continuum scale description of the plastic response of amorphous silica. Finally, we also investigate the long-range correlations between rearrangement events. We find that strong intermittency is observed when the structure remains open, while compaction results in more homogeneous rearrangements. These findings are in agreement with recent results on the effect of compression on the middle range order in silicate glasses and also suggest that the well-known volume recovery of densified silica at relatively low temperatures is in fact a form of aging.

Evaluation of x-ray Brillouin scattering data

Making use of the classical second-moment sum rule, it is possible to convert a series of constant-Q x-ray Brillouin scattering scans (Q momentum transfer) into a series of constant frequency scans over the measured Q range. The method is applied to literature results for the longitudinal phonon dispersion in several glass formers. The constant frequency scans are well fitted in terms of a Q-independent phonon damping depending exclusively on the frequency, in agreement with two recent theories of the boson peak. The method allows us to link the x-ray Brillouin scattering to the diffuse Umklapp scattering from the boson peak vibrations at higher momentum transfer on an absolute intensity scale.

Anharmonic damping of terahertz acoustic waves in a network glass and its effect on the density of vibrational states

We report the observation, by means of high-resolution inelastic x-ray scattering, of an unusually large temperature dependence of the sound attenuation of a network glass at terahertz frequency, an unprecedentedly observed phenomenon. The anharmonicity can be ascribed to the interaction between the propagating acoustic wave and the bath of thermal vibrations. At low temperatures the sound attenuation follows a Rayleigh-Gans scattering law. As the temperature is increased the anharmonic process sets in, resulting in an almost quadratic frequency dependence of the damping in the entire frequency range. We show that the temperature variation of the sound damping accounts quantitatively for the temperature dependence of the density of vibrational states.

Acoustic excitations in glassy sorbitol and their relation with the fragility and the boson peak

We report a detailed analysis of the dynamic structure factor of glassy sorbitol by using inelastic X-ray scattering and previously measured light scattering data [B. Ruta, G. Monaco, F. Scarponi, and D. Fioretto, Philos. Mag. 88, 3939 (2008)]. The thus obtained knowledge on the density-density fluctuations at both the mesoscopic and macroscopic length scale has been used to address two debated topics concerning the vibrational properties of glasses. The relation between the acoustic modes and the universal boson peak (BP) appearing in the vibrational density of states of glasses has been investigated, also in relation with some recent theoretical models. Moreover, the connection between the elastic properties of glasses and the slowing down of the structural relaxation process in supercooled liquids has been scrutinized. For what concerns the first issue, it is here shown that the wave vector dependence of the acoustic excitations can be used, in sorbitol, to quantitatively reproduce the shape of the boson peak, supporting the relation between BP and acoustic modes. For what concerns the second issue, a proper study of elasticity over a wide spatial range is shown to be fundamental in order to investigate the relation between elastic properties and the slowing down of the dynamics in the corresponding supercooled liquid phase.

Emergence of crystal-like atomic dynamics in glasses at the nanometer scale

The vibrational dynamics of a permanently densified silica glass is compared to the one of an α-quartz polycrystal, the silica polymorph of the same density and local structure. The combined use of inelastic x-ray scattering experiments and ab initio numerical calculations provides compelling evidence of a transition, in the glass, from the isotropic elastic response at long wavelengths to a microscopic regime as the wavelength decreases below a characteristic length ξ of a few nanometers, corresponding to about 20 interatomic distances. In the microscopic regime the glass vibrations closely resemble those of the polycrystal, with excitations related to the acoustic and optic modes of the crystal. A coherent description of the experimental results is obtained assuming that the elastic modulus of the glass presents spatial heterogeneities of an average size a ~ ξ/2 π.

Measuring spatial distribution of the local elastic modulus in glasses

Glasses exhibit spatially inhomogeneous elastic properties, which can be investigated by measuring their elastic moduli at a local scale. Various methods to evaluate the local elastic modulus have been proposed in the literature. A first possibility is to measure the local stress-local strain curve and to obtain the local elastic modulus from the slope of the curve or, equivalently, to use a local fluctuation formula. Another possible route is to assume an affine strain and to use the applied global strain instead of the local strain for the calculation of the local modulus. Most recently, a third technique has been introduced, which is easy to be implemented and has the advantage of low computational cost. In this contribution, we compare these three approaches by using the same model glass and reveal the differences among them caused by the nonaffine deformations.

Sound attenuation at terahertz frequencies and the boson peak of vitreous silica

The propagation and damping of the acoustic excitations in vitreous silica is measured at terahertz frequencies using inelastic x-ray scattering. The apparent sound velocity shows a marked dispersion with frequency while the sound attenuation undergoes a crossover from a fourth to a second power law frequency dependence. This finding solves a recent controversy concerning the location of this crossover in vitreous silica, clarifying that it occurs at the position of the glass-characteristic excess of vibrational modes known as boson peak, and thus establishing a direct connection between boson peak and acoustic dispersion curves.

Anomalous properties of the acoustic excitations in glasses on the mesoscopic length scale

The low-temperature thermal properties of dielectric crystals are governed by acoustic excitations with large wavelengths that are well described by plane waves. This is the Debye model, which rests on the assumption that the medium is an elastic continuum, holds true for acoustic wavelengths large on the microscopic scale fixed by the interatomic spacing, and gradually breaks down on approaching it. Glasses are characterized as well by universal low-temperature thermal properties that are, however, anomalous with respect to those of the corresponding crystalline phases. Related universal anomalies also appear in the low-frequency vibrational density of states and, despite a longstanding debate, remain poorly understood. By using molecular dynamics simulations of a model monatomic glass of extremely large size, we show that in glasses the structural disorder undermines the Debye model in a subtle way: The elastic continuum approximation for the acoustic excitations breaks down abruptly on the mesoscopic, medium-range-order length scale of approximately 10 interatomic spacings, where it still works well for the corresponding crystalline systems. On this scale, the sound velocity shows a marked reduction with respect to the macroscopic value. This reduction turns out to be closely related to the universal excess over the Debye model prediction found in glasses at frequencies of approximately 1 THz in the vibrational density of states or at temperatures of approximately 10 K in the specific heat.

Boson peak and its relation to acoustic attenuation in glasses

Experimental results on the density of states and on the acoustic modes of glasses in the THz region are compared to the predictions of two categories of models. A recent one, solely based on an elastic instability, does not account for most observations. Good agreement without adjustable parameters is obtained with models including the existence of nonacoustic vibrational modes at THz frequency, providing in many cases a comprehensive picture for a range of glass anomalies.

Acoustic properties of a porous glass (vycor) at hypersonic frequencies

Brillouin scattering experiments have been performed from 5 to 1600 K in vycor, a porous silica glass. The acoustic velocity and attenuation at hypersonic frequencies are compared to those of bulk silica and others porous silica samples. The experimental evidence for the influence of porosity on the scattering by acoustic waves is compared to calculations. The correlation between internal friction and thermal conductivity at low temperature is discussed.

Acoustic attenuation in glasses and its relation with the boson peak

A theory for the vibrational dynamics in disordered solids [W. Schirmacher, Europhys. Lett. 73, 892 (2006), based on the random spatial variation of the shear modulus, has been applied to determine the wave vector (k) dependence of the Brillouin peak position (Omega(k)) and width (Gamma(k)), as well as the density of vibrational states [g(omega)], in disordered systems. As a result, we give a firm theoretical ground to the ubiquitous k2 dependence of Gamma(k) observed in glasses. Moreover, we derive a quantitative relation between the excess of the density of states (the boson peak) and Gamma(k), two quantities that were not considered related before. The successful comparison of this relation with the outcome of experiments and numerical simulations gives further support to the theory.

Inhomogeneous elastic response of silica glass

Using large scale molecular dynamics simulations we investigate the properties of the nonaffine displacement field induced by macroscopic uniaxial deformation of amorphous silica, a strong glass according to Angell's classification. We demonstrate the existence of a length scale xi characterizing the correlations of this field (corresponding to a volume of about 1000 atoms), and compare its structure to the one observed in a standard fragile model glass. The "boson-peak" anomaly of the density of states can be traced back in both cases to elastic inhomogeneities on wavelengths smaller than xi where classical continuum elasticity becomes simply unapplicable.

Evidence for a crossover in the frequency dependence of the acoustic attenuation in vitreous silica

We report measurements of the sound attenuation coefficient in vitreous silica, for sound waves of wavelength between 50 and 80 nm, performed with the new inelastic UV light scattering technique. These data indicate that in silica glass a crossover between a temperature-dependent (at low frequency) and a temperature-independent (at high frequency) acoustic attenuation mechanism occurs at Q approximately equal to 0.15 nm(-1). The absence of any signature in the static structure factor at this Q value suggests that the observed crossover should be associated with local elastic constant fluctuations.

Glass-specific behavior in the damping of acousticlike vibrations

High frequency sound is observed in lithium diborate glass, Li2O-2B2O3, using Brillouin scattering of light and x rays. The sound attenuation exhibits a nontrivial dependence on the wave vector, with a remarkably rapid increase towards a Ioffe-Regel crossover as the frequency approaches the boson peak from below. An analysis of literature results reveals that the boson-peak frequency is closely related with a Ioffe-Regel limit for sound in many glasses. We conjecture that this relation, specific to glassy materials, might be rather common among them.

Universal sound absorption in low-temperature amorphous solids

A general elastohydrodynamic theory is developed based on the phenomenological assumption of a sharp decrease of shear relaxation time at large wave vectors k>k(xi), where k(xi) is of order of inverse of several interatomic distances a. This theory describes the low-energy excitations of glassy and amorphous solids, which contribute to anomalous linear-in-temperature specific heat and limit phonon thermal conductivity. The ratio of the wavelength of the phonon, lambda, to its mean free path, l, which is the universal property of sound absorption in glasses, is derived in this theory to be lambda/l=(2/3)(c(t)/c(l))(2)(k(xi)a)(3), where c(t) and c(l) are transverse and longitudinal sound velocities correspondingly.

Inelastic ultraviolet scattering from high frequency acoustic modes in glasses

The dynamic structure factor of vitreous silica and glycerol has been measured as a function of temperature and of the momentum transfer up to Q=0.105 nm(-1) using a novel experimental technique, the inelastic ultraviolet scattering. As in the case of Brillouin light scattering and ultrasonic measurements, the temperature dependence of the acoustic attenuation shows a plateau below the glass transition whose amplitude scales as Q2. Moreover, a slight temperature dependence of attenuation has been found in vitreous silica at about 130 K, which seems to be reminiscent of the peak measured at lower Qs. These two findings strongly support the idea that anharmonicity is responsible for sound attenuation at ultrasonic and hypersonic frequencies. Finally, we demonstrate that the attenuation mechanism should show a change of regime between 0.105 and 1 nm(-1).

High frequency dynamics in a monatomic glass

The high frequency dynamics of glassy selenium has been studied by inelastic x-ray scattering at beam line BL35XU (SPring-8). The high quality of the data allows one to pinpoint the existence of a dispersing acoustic mode for wave vectors (Q) of 1.5<Q<12.5 nm(-1), helping to clarify a previ-ous contradiction between experimental and numerical results. The sound velocity shows a positive dispersion, exceeding the hydrodynamic value by approximately 10% at Q<3.5 nm(-1). The Q2 dependence of the sound attenuation Gamma(Q), reported for other glasses, is found to be the low-Q limit of a more general Gamma(Q) proportional, variant Omega(Q)(2) law, which applies also to the higher Q region, where Omega(Q) proportional, variant Q no longer holds.