Breakdown of the Debye approximation for the acoustic modes with nanometric wavelengths in glasses

The nature of non-phononic excitations in disordered systems

The frequency scaling exponent of low-frequency excitations in microscopically small glasses, which do not allow for the existence of waves (phonons), has been in the focus of the recent literature. The density of states g(ω) of these modes obeys an ωs scaling, where the exponent s, ranging between 2 and 5, depends on the quenching protocol. The orgin of these findings remains controversal. Here we show, using heterogeneous-elasticity theory, that in a marginally-stable glass sample g(ω) follows a Debye-like scaling (s = 2), and the associated excitations (type-I) are of random-matrix type. Further, using a generalisation of the theory, we demonstrate that in more stable samples, other, (type-II) excitations prevail, which are non-irrotational oscillations, associated with local frozen-in stresses. The corresponding frequency scaling exponent s is governed by the statistics of small values of the stresses and, therefore, depends on the details of the interaction potential.

Heterogeneous attenuation of sound waves in three-dimensional amorphous solids

Sound waves are attenuated as they propagate in amorphous materials. We investigate the mechanism driving sound attenuation in the Rayleigh scattering regime by resolving the dynamics of an excited phonon in time and space via numerical simulations. We find sound attenuation is spatiotemporally heterogeneous. It starts in localized regions, which identify soft regions within the material and correlate with low-frequency vibrational modes. As time progresses, the regions where sound is primarily attenuated invade the system via an apparent diffusive process.

Intra-protein interacting collective modes in the terahertz frequency region

Understanding how proteins work requires a thorough understanding of their internal dynamics. Proteins support a wide range of motions, from the femtoseconds to seconds time scale, relevant to crucial biological functions. In this context, the term "protein collective dynamics" refers to the complex patterns of coordinated motions of numerous atoms throughout the protein in the sub-picosecond time scale (terahertz frequency region). It is hypothesized that these dynamics have a substantial impact on the regulation of functional dynamical mechanisms, including ligand binding and allosteric signalling, charge transport direction, and the regulation of thermodynamic and thermal transport properties. Using the theoretical framework of hydrodynamics, the collective dynamics of proteins had previously been described in a manner akin to that of simple liquids, i.e. in terms of a single acoustic-like excitation, related to intra-protein vibrational motions. Here, we employ an interacting-mode model to analyse the results from molecular dynamics simulations and we unveil that the vibrational landscape of proteins is populated by multiple acoustic-like and low-frequency optic-like modes, with mixed symmetry and interfering with each other. We propose an interpretation at the molecular level of the observed scenario that we relate to the side-chains and the hydrogen-bonded networks dynamics. The present insights provide a perspective for understanding the molecular mechanisms underlying the energy redistribution processes in the interior of proteins.

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.

Phonon behavior in a random solid solution: a lattice dynamics study on the high-entropy alloy FeCoCrMnNi

High-Entropy Alloys (HEAs) are a new family of crystalline random alloys with four or more elements in a simple unit cell, at the forefront of materials research for their exceptional mechanical properties. Their strong chemical disorder leads to mass and force-constant fluctuations which are expected to strongly reduce phonon lifetime, responsible for thermal transport, similarly to glasses. Still, the long range order would associate HEAs to crystals with a complex disordered unit cell. These two families of materials, however, exhibit very different phonon dynamics, still leading to similar thermal properties. The question arises on the positioning of HEAs in this context. Here we present an exhaustive experimental investigation of the lattice dynamics in a HEA, Fe₂₀Co₂₀Cr₂₀Mn₂₀Ni₂₀, using inelastic neutron and X-ray scattering. We demonstrate that HEAs present unique phonon dynamics at the frontier between fully disordered and ordered materials, characterized by long-propagating acoustic phonons in the whole Brillouin zone.

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.

A universal dimensionless length scale in medium range order amorphous structures

Our theoretical analysis reveals the existence of a previously unknown universality, namely, a dimensionless length scale, related to a ratio of medium and short range order structures of amorphous systems that originates from a conspiracy between dispersion forces among molecules and their phonon mediated coupling. The scale in turn gives rise to the structure lying underneath the low temperature universalities of many other properties e.g. specific heat, internal friction, boson peak characteristics, Meissner-Berret ratio etc.

Universality of ultrasonic attenuation coefficient of amorphous systems at low temperatures

The competition between unretarded dispersion interactions between molecules prevailing at medium range order length scales and their phonon induced coupling at larger scales leads to appearance of nano-scale sub structures in amorphous systems. The complexity of intermolecular interactions gives rise to randomization of their operators. Based on a random matrix modelling of the Hamiltonian and its linear response to an external strain field, we show that the ultrasonic attenuation coefficient can be expressed as a ratio of two crucial length-scales related to molecular dynamics. A nearly constant value of the ratio for a wide range of materials then provides a theoretical explanation of the experimentally observed qualitative universality of the ultrasonic attenuation coefficient at low temperatures.

Average density of states of amorphous Hamiltonians: role of phonon mediated coupling of nano-clusters

Based on a description of an amorphous solid as a collection of coupled nanosize molecular clusters referred as basic blocks, we analyse the statistical properties of its Hamiltonian. The information is then used to derive the ensemble averaged density of the vibrational states (non-phonon) which turns out to be a Gaussian in the bulk of the spectrum and an Airy function in the low frequency regime. A comparison with experimental data for six glasses confirms validity of our theoretical predictions.

Probing nonaffine expansion with light scattering

In disordered materials under mechanical stress, the induced deformation can deviate from the affine one, even in the elastic regime. The nonaffine contribution was observed and characterized in numerical simulations for various systems and reported experimentally in colloidal gels. However, low amplitude of nonaffinity and its local character makes the experimental study challenging. We present a method based on the phase compensation of the wave scattered from a thermally dilated amorphous material using fine wavelength tuning of the optical probe beam. Using a glass frit as a sample, we ensure complete reversibility of the material deformation, while experimental observations enable us to confirm the occurrence of nonaffinity in the elastic regime. We develop a model for the coupled effect of the thermal expansion or contraction of the material and the dilatation of the incident wavelength, which allows us to estimate the magnitude of the nonaffine displacement and the spatial extent of its correlation domain.

Transverse Viscous Transport in Classical Solid States

The transverse velocity time correlation function C[over ˜]_{T}(k,ω) with k and ω being the wave number and the frequency, respectively, is a fundamental quantity in determining the transverse mechanical and transport properties of materials. In ordinary liquids, a nonzero value of C[over ˜]_{T}(k,0) is inevitably linked to viscous material flows. Even in solids where significant material flows are precluded due to almost frozen positional degrees of freedom, our molecular dynamics simulations reveal that C[over ˜]_{T}(k,0) takes a nonzero value, whereby the time integration of the velocity field shows definite diffusive behavior with diffusivity C[over ˜]_{T}(k,0)/3. This behavior is attributed to viscous transport accompanying a small random convection of the velocity field (the inertia effect), and the resultant viscosity is measurable in the Eulerian description: the constituent particles that substantially carry momenta fluctuate slightly around their reference positions. In the Eulerian description, the velocity field is explicitly associated with such fluctuating instantaneous particle positions, whereas in the Lagrangian description, this is not the case. The present study poses a fundamental problem for continuum mechanics: reconciling liquid and solid descriptions in the limit of the infinite structural relaxation time.

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.

Low temperature heat capacity of nanosize amorphous solids

Contrary to previous studies based on bulk materials, we analyze the behavior of the vibrational density of states (VDOS) and specific heat of an amorphous solid of nano-scales (∼3 nm) at low temperature. With Hamiltonian formulation based on the intermolecular dispersion forces, our analysis indicates a universal semi-circle form of the average VDOS in the bulk of the spectrum along with a super-exponentially increasing behavior in its edge. The latter in turn leads to a specific heat with a superlinear temperature (T) dependence belowT< 1Keven at nano-scales, and, surprisingly agreeing with the experiments although the latter are carried out at macroscopic scales. The omnipresence of dispersion forces at microscopic scales indicates the application of our results to other disordered materials too.

Understanding the scaling of boson peak through insensitivity of elastic heterogeneity to bending rigidity in polymer glasses

Amorphous materials exhibit peculiar mechanical and vibrational properties, including non-affine elastic responses and excess vibrational states, i.e., the so-called boson peak (BP). For polymer glasses, these properties are considered to be affected by the bending rigidity of the constituent polymer chains. In our recent work [Tomoshige,et al2019,Sci. Rep.919514], we have revealed simple relationships between the variations of vibrational properties and the global elastic properties: the response of the BP scales only with that of the global shear modulus. This observation suggests that the spatial heterogeneity of the local shear modulus distribution is insensitive to changes in the bending rigidity. Here, we demonstrate the insensitivity of elastic heterogeneity by directly measuring the local shear modulus distribution. We also study transverse sound wave propagation, which is also shown to scale only with the global shear modulus. Through these analyses, we conclude that the bending rigidity does not alter the spatial heterogeneity of the local shear modulus distribution, which yields vibrational and acoustic properties that are controlled solely by the global shear modulus of a polymer glass.

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.

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.

Onset of interfacial waves in the terahertz spectrum of a nanoparticle suspension

We used inelastic x-ray scattering to gain insight into the complex terahertz dynamics of a diluted Au-nanoparticle suspension in glycerol. We observe that, albeit sparse, Au nanoparticles leave clear signatures on the dynamic response of the system, the main one being an additional mode propagating at the nanoparticle-glycerol interface. A Bayesian inferential analysis of the line shape reveals that such a mode, at variance with conventional acoustic modes, keeps a hydrodynamiclike behavior well beyond the continuous limit and down to subnanometer distances.

Boson peak, elasticity, and glass transition temperature in polymer glasses: Effects of the rigidity of chain bending

The excess low-frequency vibrational spectrum, called boson peak, and non-affine elastic response are the most important particularities of glasses. Herein, the vibrational and mechanical properties of polymeric glasses are examined by using coarse-grained molecular dynamics simulations, with particular attention to the effects of the bending rigidity of the polymer chains. As the rigidity increases, the system undergoes a glass transition at a higher temperature (under a constant pressure), which decreases the density of the glass phase. The elastic moduli, which are controlled by the decrease of the density and the increase of the rigidity, show a non-monotonic dependence on the rigidity of the polymer chain that arises from the non-affine component. Moreover, a clear boson peak is observed in the vibrational density of states, which depends on the macroscopic shear modulus G. In particular, the boson peak frequency ω_BP is proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{G}$$\end{document}G. These results provide a positive correlation between the boson peak, shear elasticity, and the glass transition temperature.

Sound attenuation in stable glasses

Understanding the difference between the universal low-temperature properties of amorphous and crystalline solids requires an explanation for the stronger damping of long-wavelength phonons in amorphous solids. A longstanding sound attenuation scenario, resulting from a combination of experiments, theories, and simulations, leads to a quartic scaling of sound attenuation with the wavevector, which is commonly attributed to the Rayleigh scattering of sound. Modern computer simulations offer conflicting conclusions regarding the validity of this picture. We simulate glasses with an unprecedentedly broad range of stabilities to perform the first microscopic analysis of sound damping in model glass formers across a range of experimentally relevant preparation protocols. We present convincing evidence that quartic scaling is recovered for small wavevectors irrespective of the glass's stability. With increasing stability, the wavevector where the quartic scaling begins increases by approximately a factor of three and the sound attenuation decreases by over an order of magnitude. Our results uncover an intimate connection between glass stability and sound damping.

Fluctuating Elasticity Fails to Capture Anomalous Sound Scattering in Amorphous Solids

The fluctuating elasticity (FE) model, introduced phenomenologically and developed by Schirmacher [J. Non-Cryst. Solids 357, 518 (2011)JNCSBJ0022-309310.1016/j.jnoncrysol.2010.07.052], is today the only theoretical framework available to analyze low-temperature elastic acoustic scattering in glasses. Its existing formulations, which neglect the tensorial nature of elasticity and exclude long-range disorder correlations, predict that the acoustic damping coefficients obey the standard Rayleigh scaling law: Γ∼k^{d+1}, with k the acoustic wave vector, in dimension d. However, recent numerical data, supported by the analysis of existing experimental results, show that Γ does not obey this scaling law but Γ∼-k^{d+1}lnk. Here we analyze in detail how a fully tensorial FE model can be constructed as a long wavelength approximation of the elastic response of the discrete, atomistic, problem. We show that, although it incorporates all long-range correlations, it fails to capture the observed damping in two respects: (i) it misses the anomalous scaling, and predicts the standard Rayleigh law; (ii) it grossly underestimates the amplitude of scattering by about 2 orders of magnitude. This brings clear evidence that the small scale nonaffine displacement fields, although not simply reducible to local defects, play a crucial role in acoustic wave scattering and hence cannot be ignored.

q-Independent Slow Dynamics in Atomic and Molecular Systems

Investigating million-atom systems for very long simulation times, we demonstrate that the collective density-density correlation time (τ_{α}) in simulated supercooled water and silica becomes wave-vector independent (q^{0}) when the probing wavelength is several times larger than the interparticle distance. The q independence of the collective density-density correlation functions, a feature clearly observed in light-scattering studies of some soft-matter systems, is thus a genuine feature of many (but not all) slow-dynamics systems, either atomic, molecular, or colloidal. Indeed, we show that when the dynamics of the density fluctuations includes particle-type diffusion, as in the case of the Lennard-Jones binary-mixture model, the q^{0} regime does not set in and the relaxation time continues to scale as τ_{α}∼q^{-2} even at small q.

Universal Origin of Boson Peak Vibrational Anomalies in Ordered Crystals and in Amorphous Materials

The vibrational spectra of solids, both ordered and amorphous, in the low-energy regime, control the thermal and transport properties of materials, from heat capacity to heat conduction, electron-phonon couplings, conventional superconductivity, etc. The old Debye model of vibrational spectra at low energy gives the vibrational density of states (VDOS) as proportional to the frequency squared, but in many materials the spectrum departs from this law which results in a peak upon normalizing the VDOS by frequency squared, which is known as the "boson peak." A description of the VDOS of solids (both crystals and glasses) is presented starting from first principles. Without using any assumptions whatsoever of disorder in the material, it is shown that the boson peak in the VDOS of both ordered crystals and glasses arises naturally from the competition between elastic mode propagation and diffusive damping. The theory explains the recent experimental observations of boson peak in perfectly ordered crystals, which cannot be explained based on previous theoretical frameworks. The theory also explains, for the first time, how the vibrational spectrum changes with the atomic density of the solid, and explains recent experimental observations of this effect.

Rotational sound in disordered granular materials

We numerically study the evolution of elastic standing waves in disordered disk systems with a focus on the dispersion relations of rotational sound. As on a lattice, the rotational mode exhibits an optical-like dispersion relation in the high frequency regime, representing a shoulder in the vibrational density of states and fast oscillations of the autocorrelations of rotational velocities. If tangential stiffness between the disks is large enough, a lattice-based model perfectly describes the dispersion relation of the rotational mode. If it is comparable to or smaller than the normal stiffness, the model fails for short wavelengths. However, the dispersion relation then follows the model prediction for the transverse mode, implying that the fast oscillations of disks' rotations switch to acousticlike behavior. We evidence such a transition from rotational to transverse modes by analyzing their respective participation of different degrees of freedom to the eigenvectors.

Propagative and diffusive regimes of acoustic damping in bulk amorphous material

In amorphous solids, a non-negligible part of thermal conductivity results from phonon scattering on the structural disorder. The conversion of acoustic energy into thermal energy is often measured by the dynamical dtructure factor (DSF) thanks to inelastic neutron or x-ray scattering. The DSF is used to quantify the dispersion relation of phonons, together with their damping. However, the connection of the dynamical structure factor with dynamical attenuation of wave packets in glasses is still a matter of debate. We focus here on the analysis of wave-packet propagation in numerical models of amorphous silicon. We show that the damped harmonic oscillator model fits of the dynamical structure factors give a good estimate of the wave packets mean free path, only below the Ioffe-Regel frequency. Above the Ioffe-Regel frequency and below the mobility edge, a pure diffusive regime without a definite mean free path is observed. The high-frequency mobility edge is characteristic of a transition to localized vibrations. Below the Ioffe-Regel frequency, a mixed regime is evidenced at intermediate frequencies, with a coexistence of propagative and diffusive wave fronts. The transition between these different regimes is analyzed in detail and reveals a complex dynamics for energy transport, thus raising the question of the correct modeling of thermal transport in amorphous materials.

Isolating long-wavelength fluctuation from structural relaxation in two-dimensional glass: cage-relative displacement

It has recently been revealed that long-wavelength fluctuation exists in two-dimensional (2D) glassy systems, having the same origin as that given by the Mermin-Wagner theorem for 2D crystalline solids. In this paper, we discuss how to characterise quantitatively the long-wavelength fluctuation in a molecular dynamics simulation of a lightly supercooled liquid. We employ the cage-relative mean-square displacement (MSD), defined on relative displacement to its cage, to quantitatively separate the long-wavelength fluctuation from the original MSD. For increasing system size the amplitude of acoustic long wavelength fluctuations not only increases but shifts to later times causing a crossover with structural relaxation of caging particles. We further analyse the dynamic correlation length using the cage-relative quantities. It grows as the structural relaxation becomes slower with decreasing temperature, uncovering an overestimation by the four-point correlation function due to the long-wavelength fluctuation. These findings motivate the usage of cage-relative MSD as a starting point for analysis of 2D glassy dynamics.

Inherent stress correlations in a quiescent two-dimensional liquid: Static analysis including finite-size effects

After constructing a formalism to analyze spatial stress correlations in two-dimensional equilibrated liquids, we show that the sole conjunction of mechanical balance and material isotropy demands all anisotropic components of the inherent state (IS) stress autocorrelation matrix to decay at long range as 1/r^{2} in the large system size limit. Furthermore, analyzing numerical simulation data for an equilibrated supercooled liquid, we bring evidence that, in finite-sized periodic systems, the autocorrelations of pressure and shear stresses present uniform backgrounds of amplitudes proportional to the inverse cell area. These backgrounds bring relevant contributions to macroscopic IS stress fluctuations, with the consequence that the latter scale as inverse area, yet in an anomalous way, inconsistent with viewing an IS as equivalent, in the thermodynamic limit, to an ensemble of independent finite-sized subsystems. In that sense, ISs are not spatially ergodic.

Vibrational coupling in plasmonic molecules

Plasmon hybridization theory, inspired by molecular orbital theory, has been extremely successful in describing the near-field coupling in clusters of plasmonic nanoparticles, also known as plasmonic molecules. However, the vibrational modes of plasmonic molecules have been virtually unexplored. By designing precisely configured plasmonic molecules of varying complexity and probing them at the individual plasmonic molecule level, intramolecular coupling of acoustic modes, mediated by the underlying substrate, is observed. The strength of this coupling can be manipulated through the configuration of the plasmonic molecules. Surprisingly, classical continuum elastic theory fails to account for the experimental trends, which are well described by a simple coupled oscillator picture that assumes the vibrational coupling is mediated by coherent phonons with low energies. These findings provide a route to the systematic optical control of the gigahertz response of metallic nanostructures, opening the door to new optomechanical device strategies.

Pressure Dependence of the Boson Peak of Glassy Glycerol

The pressure dependence of the boson peak (BP) of glycerol, including its behavior across the liquid-glass transition, has been studied using Raman scattering. A significant increase of the BP frequency was observed with pressure up to 11 GPa at room temperature. The pressure dependence of BP frequency ν_BP is proportional to (1+P/P₀)^1/3, where P and P₀ are the pressure and a constant, respectively, consistent with a soft potential model. The characteristic length of medium range order is close in size to a cyclic trimer of glycerol molecules, as predicted by the medium range order of a BP excitation using molecular dynamics simulations, and the pressure dependence of a characteristic medium range order is nearly constant. The pressure induced structural changes in glycerol can be understood in terms of the shrinkage of voids with cyclic trimers persisting to at least 11 GPa. Pressure dependence of the intermolecular O-H stretching mode indicates that the intermolecular hydrogen bond distances gradually decrease up to the glass transition pressure of ∼5 GPa and become nearly constant in the glassy state, indicating the disappearance of free volume in the dense glass.

Unveiling Dimensionality Dependence of Glassy Dynamics: 2D Infinite Fluctuation Eclipses Inherent Structural Relaxation

By using large-scale molecular dynamics simulations, the dynamics of two-dimensional (2D) supercooled liquids turns out to be dependent on the system size, while the size dependence is not pronounced in three-dimensional (3D) systems. It is demonstrated that the strong system-size effect in 2D amorphous systems originates from the enhanced fluctuations at long wavelengths which are similar to those of 2D crystal phonons. This observation is further supported by the frequency dependence of the vibrational density of states, consisting of the Debye approximation in the low-wave-number limit. However, the system-size effect in the intermediate scattering function becomes negligible when the length scale is larger than the vibrational amplitude. This suggests that the finite-size effect in a 2D system is transient and also that the structural relaxation itself is not fundamentally different from that in a 3D system. In fact, the dynamic correlation lengths estimated from the bond-breakage function, which do not suffer from those enhanced fluctuations, are not size dependent in either 2D or 3D systems.

Crossover of collective modes and positive sound dispersion in supercritical state

Supercritical state has been viewed as an intermediate state between gases and liquids with largely unknown physical properties. Here, we address the important ability of supercritical fluids to sustain collective excitations. We directly study propagating modes on the basis of correlation functions calculated in molecular dynamics simulations and find that the supercritical system sustains propagating solid-like transverse modes below the Frenkel line but not above where there is one longitudinal mode only. Important thermodynamic implications of this finding are discussed. We directly detect positive sound dispersion (PSD) below the Frenkel line where transverse modes are operative and quantitatively explain its magnitude on the basis of transverse and longitudinal velocities. PSD disappears above the Frenkel line which therefore demarcates the supercritical phase diagram into two areas where PSD does and does not operate.

Anomalous phonon scattering and elastic correlations in amorphous solids

A major issue in materials science is why glasses present low-temperature thermal and vibrational properties that sharply differ from those of crystals. In particular, long-wavelength phonons are considerably more damped in glasses, yet it remains unclear how structural disorder at atomic scales affects such a macroscopic phenomenon. A plausible explanation is that phonons are scattered by local elastic heterogeneities that are essentially uncorrelated in space, a scenario known as Rayleigh scattering, which predicts that the damping of acoustic phonons scales with wavenumber k as k^d+1 (in dimension d). Here we demonstrate that phonon damping scales instead as - k^d+1 ln k, with this logarithmic enhancement originating from long-range spatial correlations of elastic disorder caused by similar stress correlations. Our work suggests that the presence of long-range spatial correlations of local stress and elasticity may well be the crucial feature that distinguishes amorphous solids from crystals.

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.

Probing the Role of Mobility in the Collective Motion of Nonequilibrium Systems

By systematically varying the mobility of self-propelled particles in a 2D lattice, we experimentally study the influence of particle mobility on system's collective motion. Our system is intrinsically nonequilibrium due to the lack of energy equipartition. By constructing the covariance matrix of spatial fluctuations and solving for its eigenmodes, we obtain the collective motions of the system with various magnitudes. Interestingly, our structurally ordered nonequilibrium system exhibits properties almost identical to disordered glassy systems under thermal equilibrium: the modes with large overall motions are spatially correlated and quasilocalized while the modes with small collective motions are highly localized, resembling the low- and high-frequency modes in glass. More surprisingly, a peak similar to the boson peak forms in our nonequilibrium system as the number of mobile particles increases, revealing the possible origin of the boson peak from a dynamic aspect. We further illustrate that the spatially correlated large-movement modes can be produced by the cooperation of highly active particles above a threshold fraction, while the localized small-movement modes can be created by adding individual inactive particles. Our study clarifies the role of mobility in collective motions, and further suggests a promising possibility of extending the powerful mode analysis approach to nonequilibrium systems.

Collective modes and thermodynamics of the liquid state

Strongly interacting, dynamically disordered and with no small parameter, liquids took a theoretical status between gases and solids with the historical tradition of hydrodynamic description as the starting point. We review different approaches to liquids as well as recent experimental and theoretical work, and propose that liquids do not need classifying in terms of their proximity to gases and solids or any categorizing for that matter. Instead, they are a unique system in their own class with a notably mixed dynamical state in contrast to pure dynamical states of solids and gases. We start with explaining how the first-principles approach to liquids is an intractable, exponentially complex problem of coupled non-linear oscillators with bifurcations. This is followed by a reduction of the problem based on liquid relaxation time τ representing non-perturbative treatment of strong interactions. On the basis of τ, solid-like high-frequency modes are predicted and we review related recent experiments. We demonstrate how the propagation of these modes can be derived by generalizing either hydrodynamic or elasticity equations. We comment on the historical trend to approach liquids using hydrodynamics and compare it to an alternative solid-like approach. We subsequently discuss how collective modes evolve with temperature and how this evolution affects liquid energy and heat capacity as well as other properties such as fast sound. Here, our emphasis is on understanding experimental data in real, rather than model, liquids. Highlighting the dominant role of solid-like high-frequency modes for liquid energy and heat capacity, we review a wide range of liquids: subcritical low-viscous liquids, supercritical state with two different dynamical and thermodynamic regimes separated by the Frenkel line, highly-viscous liquids in the glass transformation range and liquid-glass transition. We subsequently discuss the fairly recent area of liquid-liquid phase transitions, the area where the solid-like properties of liquids have become further apparent. We then discuss gas-like and solid-like approaches to quantum liquids and theoretical issues that are similar to the classical case. Finally, we summarize the emergent view of liquids as a unique system with a mixed dynamical state, and list several areas where interesting insights may appear and continue the extraordinary liquid story.

Thermodynamic nature of vitrification in a 1D model of a structural glass former

We propose a new spin-glass model with no positional quenched disorder which is regarded as a coarse-grained model of a structural glass-former. The model is analyzed in the 1D case when the number N of states of a primary cell is large. For N → ∞, the model exhibits a sharp freezing transition of the thermodynamic origin. It is shown both analytically and numerically that the glass transition is accompanied by a significant growth of a static length scale ξ pointing to the structural (equilibrium) nature of dynamical slowdown effects in supercooled liquids.

Collective ion dynamics in liquid zinc: evidence for complex dynamics in a non-free-electron liquid metal

A detailed inelastic neutron scattering investigation of the THz dynamics of liquid zinc is presented. The observed Q dependence clearly reveals the existence of a complex dynamics made up of two distinct excitations. The highest energy mode is the prolongation of the longitudinal acoustic density fluctuations whereas the comparison with the phonon dynamics of crystalline hcp zinc suggests a transverse acousticlike nature for the second one. This mode seems related to peculiar anisotropic interactions, possibly connected to the behavior of the crystalline phase.

Structural heterogeneities at the origin of acoustic and transport anomalies in glycerol glass-former

By means of large scale molecular dynamics simulations, we explore mesoscopic properties of prototypical glycerol glass-former above and below the glass transition. The model used, in excellent agreement with various experimental techniques, permits to carefully study the structure and the vibrational dynamics. We find that a medium range order is present in glycerol glass-former and arises from hydrogen bond network extension. The characteristic size of the structural heterogeneities is related to the anomalous properties of acoustic vibrations (Rayleigh scattering, "mode softening," and Boson Peak) in the glassy state. Finally the characteristic size of these heterogeneities, nearly constant in temperature, is also connected to the cross-over between structural relaxation and diffusion in liquid glycerol.

Unusual specific heat of almost dry L-cysteine and L-cystine amino acids

A detailed quantitative analysis of the specific heat in the 0.5- to 200-K temperature range for almost dry L-cysteine and its dimer, L-cystine, amino acids is presented. We report the occurrence of a sharp first-order transition at ∼76 K for L-cysteine associated with the thiol group ordering which was successfully modeled with the two-dimensional Ising model. We demonstrated that quantum rotors, two-level systems (TLS), Einstein oscillators, and acoustic phonons (the Debye model) are essential ingredients to correctly describe the overall experimental data. Our analysis pointed out the absence of the TLS contribution to the low temperature specific heat of L-cysteine. This result was similar to that found in other noncrystalline amorphous materials, e.g., amorphous silicon, low density amorphous water, and ultrastable glasses. L-cystine presented an unusual nonlinear acoustic dispersion relation ω(q)=vq0.95 and a Maxwell-Boltzmann-type distribution of tunneling barriers. The presence of Einstein oscillators with ΘE∼70 K was common in both systems and adequately modeled the boson peak contributions.

Rigidity, secondary structure, and the universality of the boson peak in proteins

Complementary neutron- and light-scattering results on nine proteins and amino acids reveal the role of rigidity and secondary structure in determining the time- and lengthscales of low-frequency collective vibrational dynamics in proteins. These dynamics manifest in a spectral feature, known as the boson peak (BP), which is common to all disordered materials. We demonstrate that BP position scales systematically with structural motifs, reflecting local rigidity: disordered proteins appear softer than α-helical proteins; which are softer than β-sheet proteins. Our analysis also reveals a universal spectral shape of the BP in proteins and amino acid mixtures; superimposable on the shape observed in typical glasses. Uniformity in the underlying physical mechanism, independent of the specific chemical composition, connects the BP vibrations to nanometer-scale heterogeneities, providing an experimental benchmark for coarse-grained simulations, structure/rigidity relationships, and engineering of proteins for novel applications.