Measuring spatial distribution of the local elastic modulus in 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.

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.

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.

Thawed matrix method for computing local mechanical properties of amorphous solids

We present a method for computing locally varying nonlinear mechanical properties in particle simulations of amorphous solids. Plastic rearrangements outside a probed region are suppressed by introducing an external field that directly penalizes large nonaffine displacements. With increasing strength of the field, plastic deformation can be localized. We characterize the distribution of local plastic yield stresses (residual local stresses to instability) with our approach and assess the correlation of their spatial maps with plastic activity in a model two-dimensional amorphous solid. Our approach reduces artifacts inherent in a previous method known as the "frozen matrix" approach that enforces fully affine deformation and improves the prediction of plastic rearrangements from structural information.

Necking and failure of a particulate gel strand: signatures of yielding on different length scales

"Sticky" spheres with a short-ranged attraction are a basic model of a wide range of materials from the atomic to the granular length scale. Among the complex phenomena exhibited by sticky spheres is the formation of far-from-equilibrium dynamically arrested networks which comprise "strands" of densely packed particles. The aging and failure of such gels under load is a remarkably challenging problem, given the simplicity of the model, as it involves multiple length- and time-scales, making a single approach ineffective. Here we tackle this challenge by addressing the failure of a single strand with a combination of methods. We study the mechanical response of a single strand of a model gel-former to deformation, both numerically and analytically. Under elongation, the strand breaks by a necking instability. We analyse this behaviour at three different length scales: a rheological continuum model of the whole strand; a microscopic analysis of the particle structure and dynamics; and the local stress tensor. Combining these different approaches gives a coherent picture of the necking and failure. The strand has an amorphous local structure and has large residual stresses from its initialisation. We find that neck formation is associated with increased plastic flow, a reduction in the stability of the local structure, and a reduction in the residual stresses; this indicates that the system loses its solid character and starts to behave more like a viscous fluid. These results will inform the development of more detailed models that incorporate the heterogeneous network structure of particulate gels.

Effective elastic moduli of composites with a strongly disordered host material

The local elastic properties of strongly disordered material are investigated using the theory of correlated random matrices. A significant increase in stiffness is shown in the interfacial region, the thickness of which depends on the strength of disorder. It is shown that this effect plays a crucial role in nanocomposites, in which interfacial regions are formed around each nanoparticle. The studied interfacial effect can significantly increase the influence of nanoparticles on the macroscopic stiffness of nanocomposites. The obtained thickness of the interfacial region is determined by the heterogeneity lengthscale and is of the same order as the lengthscale of the boson peak.

Strain correlation functions in isotropic elastic bodies: large wavelength limit for two-dimensional systems

Strain correlation functions in two-dimensional isotropic elastic bodies are shown both theoretically (using the general structure of isotropic tensor fields) and numerically (using a glass-forming model system) to depend on the coordinates of the field variable (position vector r in real space or wavevector q in reciprocal space) and thus on the direction of the field vector and the orientation of the coordinate system. Since the fluctuations of the longitudinal and transverse components of the strain field in reciprocal space are known in the long-wavelength limit from the equipartition theorem, all components of the correlation function tensor field are imposed and no additional physical assumptions are needed. An observed dependence on the field vector direction thus cannot be used as an indication for anisotropy or for a plastic rearrangement. This dependence is different for the associated strain response field containing also information on the localized stress perturbation.

Local and global measures of the shear moduli of jammed disk packings

Strain-controlled isotropic compression gives rise to jammed packings of repulsive, frictionless disks with either positive or negative global shear moduli. We carry out computational studies to understand the contributions of the negative shear moduli to the mechanical response of jammed disk packings. We first decompose the ensemble-averaged, global shear modulus as 〈G〉=(1-F_{-})〈G_{+}〉+F_{-}〈G_{-}〉, where F_{-} is the fraction of jammed packings with negative shear moduli and 〈G_{+}〉 and 〈G_{-}〉 are the average values from packings with positive and negative moduli, respectively. We show that 〈G_{+}〉 and 〈|G_{-}|〉 obey different power-law scaling relations above and below pN^{2}∼1. For pN^{2}>1, both 〈G_{+}〉N and 〈|G_{-}|〉N∼(pN^{2})^{β}, where β∼0.5 for repulsive linear spring interactions. Despite this, 〈G〉N∼(pN^{2})^{β^{'}} with β^{'}≳0.5 due to the contributions from packings with negative shear moduli. We show further that the probability distribution of global shear moduli P(G) collapses at fixed pN^{2} and different values of p and N. We calculate analytically that P(G) is a Γ distribution in the pN^{2}≪1 limit. As pN^{2} increases, the skewness of P(G) decreases and P(G) becomes a skew-normal distribution with negative skewness in the pN^{2}≫1 limit. We also partition jammed disk packings into subsystems using Delaunay triangulation of the disk centers to calculate local shear moduli. We show that the local shear moduli defined from groups of adjacent triangles can be negative even when G>0. The spatial correlation function of local shear moduli C(r[over ⃗]) displays weak correlations for pn_{sub}^{2}<10^{-2}, where n_{sub} is the number of particles within each subsystem. However, C(r[over ⃗]) begins to develop long-ranged spatial correlations with fourfold angular symmetry for pn_{sub}^{2}≳10^{-2}.

Residual stress in athermal soft disordered solids: insights from microscopic and mesoscale models

In soft amorphous materials, shear cessation after large shear deformation leads to configurations having residual shear stress. The origin of these states and the distribution of the local shear stresses within the material is not well understood, despite its importance for the change in material properties and consequent applications. In this work, we use molecular dynamics simulations of a model dense non-Brownian soft amorphous material to probe the non-trivial relaxation process towards a residual stress state. We find that, similar to thermal glasses, an increase in shear rate prior to the shear cessation leads to lower residual stress states. We rationalise our findings using a mesoscopic elasto-plastic description that explicitly includes a long range elastic response to local shear transformations. We find that after flow cessation the initial stress relaxation indeed depends on the pre-sheared stress state, but the final residual stress is majorly determined by newly activated plastic events occurring during the relaxation process, a scenario consistent with the phenomenology of avalanche dynamics in the low shear rate limit of steadily sheared amorphous solids. Our simplified coarse grained description not only allows capturing the phenomenology of residual stress states but also rationalising the altered material properties that are probed using small and large deformation protocols applied to the relaxed material.

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.

Static self-induced heterogeneity in glass-forming liquids: Overlap as a microscope

We propose and numerically implement a local probe of the static self-induced heterogeneity characterizing glass-forming liquids. This method relies on the equilibrium statistics of the overlap between pairs of configurations measured in mesoscopic cavities with unconstrained boundaries. By systematically changing the location of the probed cavity, we directly detect spatial variations of the overlap fluctuations. We provide a detailed analysis of the statistics of a local estimate of the configurational entropy, and we infer an estimate of the surface tension between amorphous states, ingredients that are both at the basis of the random first-order transition theory of glass formation. Our results represent the first direct attempt to visualize and quantify the self-induced heterogeneity underpinning the thermodynamics of glass formation. They pave the way for the development of coarse-grained effective theories and for a direct assessment of the role of thermodynamics in the activated dynamics of deeply supercooled liquids.

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.

Local elastic properties of polystyrene nanocomposites increase significantly due to nonaffine deformations

We investigate the local elastic properties of polystyrene doped with SiO_{2} nanoparticles by analyzing the local density fluctuations. The density fluctuations were established from coarse-grained molecular dynamics simulations performed with the MARTINI force field. A significant increase in polystyrene stiffness was revealed within a characteristic range of 1.4 nm from the nanoparticle, while polystyrene density saturates to the bulk value at significantly shorter distances. The enhancement of the local elastic properties of the polymer was attributed to the effect of nonaffine deformations at the length scale below 1 nm, which was further confirmed through the random matrix model with variable strength of disorder.

Vibrational disorder and densification-induced homogenization of local elasticity in silicate glasses

We report the effect of structural compaction on the statistics of elastic disorder in a silicate glass, using heterogeneous elasticity theory with the coherent potential approximation (HET-CPA) and a log-normal distribution of the spatial fluctuations of the shear modulus. The object of our study, a soda lime magnesia silicate glass, is compacted by hot-compression up to 2 GPa (corresponding to a permanent densification of ~ 5%). Using THz vibrational spectroscopic data and bulk mechanical properties as inputs, HET-CPA evaluates the degree of disorder in terms of the length-scale of elastic fluctuations and the non-affine part of the shear modulus. Permanent densification decreases the extent of non-affine elasticity, resulting in a more homogeneous distribution of strain energy, while also decreasing the correlation length of elastic heterogeneity. Complementary 29Si magic angle spinning NMR spectroscopic data provide a short-range rationale for the effect of compression on glass structure in terms of a narrowing of the Si-O-Si bond-angle and the Si-Si distance.

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.

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.

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.

Emergence of linear isotropic elasticity in amorphous and polycrystalline materials

We investigate the emergence of isotropic linear elasticity in amorphous and polycrystalline solids via extensive numerical simulations. We show that the elastic properties are correlated over a finite length scale ξ_{E}, so that the central limit theorem dictates the emergence of continuum linear isotropic elasticity on increasing the specimen size. The stiffness matrix of systems of finite size L>ξ_{E} is obtained, adding to that predicted by linear isotropic elasticity a random one of spectral norm (L/ξ_{E})^{-3/2} in three spatial dimensions. We further demonstrate that the elastic length scale corresponds to that of structural correlations, which in polycrystals reflect the typical size of the grain boundaries and length scales characterizing correlations in the stress field. We finally demonstrate that the elastic length scale affects the decay of the anisotropic long-range correlations of locally defined shear modulus and shear stress.

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.

Mechanical disorder of sticky-sphere glasses. I. Effect of attractive interactions

Recent literature indicates that attractive interactions between particles of a dense liquid play a secondary role in determining its bulk mechanical properties. Here we show that, in contrast with their apparent unimportance to the bulk mechanics of dense liquids, attractive interactions can have a major effect on macro- and microscopic elastic properties of glassy solids. We study several broadly applicable dimensionless measures of stability and mechanical disorder in simple computer glasses, in which the relative strength of attractive interactions-referred to as "glass stickiness"-can be readily tuned. We show that increasing glass stickiness can result in the decrease of various quantifiers of mechanical disorder, on both macro- and microscopic scales, with a pair of intriguing exceptions to this rule. Interestingly, in some cases strong attractions can lead to a reduction of the number density of soft, quasilocalized modes, by up to an order of magnitude, and to a substantial decrease in their core size, similar to the effects of thermal annealing on elasticity observed in recent works. Contrary to the behavior of canonical glass models, we provide compelling evidence indicating that the stabilization mechanism in our sticky-sphere glasses stems predominantly from the self-organized depletion of interactions featuring large, negative stiffnesses. Finally, we establish a fundamental link between macroscopic and microscopic quantifiers of mechanical disorder, which we motivate via scaling arguments. Future research directions are discussed.

Novel elastic instability of amorphous solids in finite spatial dimensions

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

Intermittent rearrangements accompanying thermal fluctuations distinguish glasses from crystals

Glasses exhibit vibrational and thermal properties that are markedly different from those of crystals. While recent works have advanced our understanding of vibrational excitations in glasses in the harmonic approximation limit, efforts in understanding finite-temperature anharmonic processes have been limited. In crystals, phonon-phonon coupling provides an extremely efficient mechanism for anharmonic decay that is also important in glasses. By using extensive molecular dynamics simulation of model atomic systems, here we first describe, both numerically and analytically, the anharmonic couplings in the crystal and the glass by focusing on the temperature dependence of the associated decay rates. Next, we show that an additional anharmonic channel of different origin emerges in the amorphous case, which induces unconventional intermittent rearrangements of particles. We have found that thermal vibrations in glasses trigger transitions among numerous different local minima of the energy landscape, which, however, are located within the same wide (meta)basin. These processes generate motions that are different from both diffusive and out-of-equilibrium aging dynamics. We suggest that (i) the observed intermittent rearrangements accompanying thermal fluctuations are crucial features distinguishing glasses from crystals and (ii) they can be considered as relics of the liquid state that survive the complete dynamic arrest taking place at the glass transition temperature.

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.

Stable glassy configurations of the Kob-Andersen model using swap Monte Carlo

The swap Monte Carlo algorithm allows the preparation of highly stable glassy configurations for a number of glass-formers but is inefficient for some models, such as the much studied binary Kob-Andersen (KA) mixture. We have recently developed generalizations to the KA model where swap can be very effective. Here, we show that these models can, in turn, be used to considerably enhance the stability of glassy configurations in the original KA model at no computational cost. We successfully develop several numerical strategies both in and out of equilibrium to achieve this goal and show how to optimize them. We provide several physical measurements indicating that the proposed algorithms considerably enhance mechanical and thermodynamic stability in the KA model, including a transition toward brittle yielding behavior. Our results thus pave the way for future studies of stable glasses using the KA model.

Residual stress distributions in amorphous solids from atomistic simulations

The distribution of local residual stresses (threshold to instability) that controls the statistical properties of plastic flow in athermal amorphous solids is examined with an atomistic simulation technique. For quiescent configurations, the distribution has a pseudogap (power-law) form with an exponent that agrees well with global yielding statistics. As soon as deformation sets in, the pseudogap region gives way to a system size dependent plateau at small residual stresses that can be understood from the statistics of local residual stress differences between plastic events. Results further suggest that the local yield stress in amorphous solids changes even if the given region does not participate in plastic activity.

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.

Statistical mechanics of local force dipole responses in computer glasses

Soft quasilocalized modes (QLMs) are universally featured by structural glasses quenched from a melt, and are involved in several glassy anomalies such as the low-temperature scaling of their thermal conductivity and specific heat, and sound attenuation at intermediate frequencies. In computer glasses, QLMs may assume the form of harmonic vibrational modes under a narrow set of circumstances; however, direct access to their full distribution over frequency is hindered by hybridizations of QLMs with other low-frequency modes (e.g., phonons). Previous studies to overcome this issue have demonstrated that the response of a glass to local force dipoles serves as a good proxy for its QLMs; we, therefore, study here the statistical-mechanical properties of these responses in computer glasses, over a large range of glass stabilities and in various spatial dimensions, with the goal of revealing properties of the yet-inaccessible full distribution of QLMs' frequencies. We find that as opposed to the spatial-dimension-independent universal distribution of QLMs' frequencies ω (and, consequently, also of their stiffness κ = ω²), the distribution of stiffnesses associated with responses to local force dipoles features a (weak) dependence on spatial dimension. We rationalize this dependence by introducing a lattice model that incorporates both the real-space profiles of QLMs-associated with dimension-dependent long-range elastic fields-and the universal statistical properties of their frequencies. Based on our findings, we propose a conjecture about the form of the full distribution of QLMs' frequencies and its protocol-dependence. Finally, we discuss possible connections of our findings to basic aspects of glass formation and deformation.

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.

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.

Revealing Inherent Structural Characteristics of Jammed Particulate Packings

We look for inherent structural characteristics hidden behind amorphous solid formation by using zero-temperature jammed packings of frictionless particles as models. Differently from previous geometrical approaches, we introduce a microscopic mechanical or vibrational order parameter Ψ, which characterizes the susceptibility of particle motion to infinitesimal thermal excitation. We show that (i) the distribution of Ψ has a power-law tail toward high Ψ and (ii) the spatial organization of Ψ is characterized by a nontrivial scale-free correlation. Both findings (i) and (ii) are regarded as a real-space manifestation of marginal stability due to critical self-organization of jammed packings toward mechanical equilibrium. Furthermore, we find that the power-law exponent of the Ψ distribution tail shows a critical-like scaling behavior toward the unjamming transition, which unveils an intriguing interplay between jamming criticality and marginal stability. Our microscopic order parameter provides new structural insights into the marginal stability and instability of jammed packings and may shed light on the important common structural feature of amorphous solids.

Theory of applying shear strains from boundary walls: Linear response in glasses

We construct a linear response theory of applying shear deformations from boundary walls in the film geometry in Kubo's theoretical scheme. Our method is applicable to any solids and fluids. For glasses, we assume quasi-equilibrium around a fixed inherent state. Then, we obtain linear-response expressions for any variables including the stress and the particle displacements, even though the glass interior is elastically inhomogeneous. In particular, the shear modulus can be expressed in terms of the correlations between the interior stress and the forces from the walls. It can also be expressed in terms of the inter-particle correlations, as has been shown in the previous literature. Our stress relaxation function includes the effect of the boundary walls and can be used for inhomogeneous flow response. We show the presence of long-ranged, long-lived correlations among the fluctuations of the forces from the walls and the displacements of all the particles in the cell. We confirm these theoretical results numerically in a two-dimensional model glass. As an application, we describe emission and propagation of transverse sounds after boundary wall motions using these time-correlation functions. We also find resonant sound amplification when the frequency of an oscillatory shear approaches that of the first transverse sound mode.

Local versus Global Stretched Mechanical Response in a Supercooled Liquid near the Glass Transition

Amorphous materials have a rich relaxation spectrum, which is usually described in terms of a hierarchy of relaxation mechanisms. In this work, we investigate the local dynamic modulus spectra in a model glass just above the glass transition temperature by performing a mechanical spectroscopy analysis with molecular dynamics simulations. We find that the spectra, at the local as well as on the global scale, can be well described by the Cole-Davidson formula in the frequency range explored with simulations. Surprisingly, the Cole-Davidson stretching exponent does not change with the size of the local region that is probed. The local relaxation time displays a broad distribution, as expected based on dynamic heterogeneity concepts, but the stretching is obtained independently of this distribution. We find that the size dependence of the local relaxation time and moduli can be well explained by the elastic shoving model.

Boson Peak Decouples from Elasticity in Glasses with Low Connectivity

We perform molecular-dynamics simulations of the vibrational and elastoplastic properties of polymeric glasses and crystals and the corresponding atomic systems. We evidence that the elastic scaling of the density of states in the low-frequency boson peak (BP) region is different in crystals and glasses. Also, we see that the BP of the polymeric glass is nearly coincident with the one of the atomic glasses, thus revealing that the former-unlike the elasticity-is controlled by nonbonding interactions only. Our results suggest that the interpretation of the BP in terms of the macroscopic elasticity, discussed in highly connected systems, does not hold for systems with low connectivity.

Mechanical heterogeneity in ionic liquids

Molecular dynamics (MD) simulations of five ionic liquids based on 1-alkyl-3-methylimidazolium cations, [C _n C₁im]⁺, have been performed in order to calculate high-frequency elastic moduli and to evaluate heterogeneity of local elastic moduli. The MD simulations of [C _n C₁im][NO₃], n = 2, 4, 6, and 8, assessed the effect of domain segregation when the alkyl chain length increases, and [C₈C₁im][PF₆] assessed the effect of strength of anion-cation interaction. Dispersion curves of excitation energies of longitudinal and transverse acoustic, LA and TA, modes were obtained from time correlation functions of mass currents at different wavevectors. High-frequency sound velocity of LA modes depends on the alkyl chain length, but sound velocity for TA modes does not. High-frequency bulk and shear moduli, K _∞ and G _∞ , depend on the alkyl chain length because of a density effect. Both K _∞ and G _∞ are strongly dependent on the anion. The calculation of local bulk and shear moduli was accomplished by performing bulk and shear deformations of the systems cooled to 0 K. The simulations showed a clear connection between structural and elastic modulus heterogeneities. The development of nano-heterogeneous structure with increasing length of the alkyl chain in [C _n C₁im][NO₃] implies lower values for local bulk and shear moduli in the non-polar domains. The mean value and the standard deviations of distributions of local elastic moduli decrease when [NO₃]^- is replaced by the less coordinating [PF₆]^- anion.

Simulating the effects of carbon nanotube continuity and interfacial bonding on composite strength and stiffness

Molecular dynamics simulations of carbon nanotube (CNT) composites, in which the CNTs are continuous across the periodic boundary, overestimate the experimentally measured mechanical properties of CNT composites along the fiber direction. Since the CNTs in these composites are much shorter than the composite dimensions, load must be transferred either directly between CNTs or through the matrix, a mechanism that is absent in simulations of effectively continuous CNTs. In this study, the elastic and fracture properties of high volume fraction discontinuous carbon nanotube/amorphous carbon composite systems were compared to those of otherwise equivalent continuous CNT composites using ReaxFF reactive molecular dynamics simulations. These simulations were used to show how the number of nanotube-matrix interfacial covalent bonds affect composite mechanical properties. Furthermore, the mechanical impact of interfacial bonding was decomposed to reveal its effect on the properties of the CNTs, the interfacial layer of matrix, and the bulk matrix. For the composites with continuous reinforcement, it was found that any degree of interfacial bonding has a negative impact on axial tensile strength and stiffness. This is due to disruption of the structure of the CNTs and interfacial matrix layer by the interfacial bonds. For the discontinuous composites, the modulus was maximized between 4%-7% interfacial bonding and the strength continues to increase up to the highest levels of interfacial bonding studied. Areas of low stress and voids were observed in the simulated discontinuous composites at the ends of the tubes, from which fracture was observed to initiate. Experimental carbon nanotube yarn composites were fabricated and tested. The results were used to illustrate knockdown factors relative to the mechanical performance of the tubes themselves.

Stress-stress fluctuation formula for elastic constants in the NPT ensemble

Several fluctuation formulas are available for calculating elastic constants from equilibrium correlation functions in computer simulations, but the ones available for simulations at constant pressure exhibit slow convergence properties and cannot be used for the determination of local elastic constants. To overcome these drawbacks, we derive a stress-stress fluctuation formula in the NPT ensemble based on known expressions in the NVT ensemble. We validate the formula in the NPT ensemble by calculating elastic constants for the simple nearest-neighbor Lennard-Jones crystal and by comparing the results with those obtained in the NVT ensemble. For both local and bulk elastic constants we find an excellent agreement between the simulated data in the two ensembles. To demonstrate the usefulness of the formula, we apply it to determine the elastic constants of a simulated lipid bilayer.

Linking microscopic and macroscopic response in disordered solids

The modulus of a rigid network of harmonic springs depends on the sum of the energies in each of the bonds due to an applied distortion such as compression in the case of the bulk modulus or shear in the case of the shear modulus. However, the distortion need not be global. Here we introduce a local modulus, L_{i}, associated with changing the equilibrium length of a single bond, i, in the network. We show that L_{i} is useful for understanding many aspects of the mechanical response of the entire system. It allows an efficient computation of how the removal of any bond changes the global properties such as the bulk and shear moduli. Furthermore, it allows a prediction of the distribution of these changes and clarifies why the changes of these two moduli due to removal of a bond are uncorrelated; these are the essential ingredients necessary for the efficient manipulation of network properties by bond removal.

Local yield stress statistics in model amorphous solids

We develop and extend a method presented by Patinet, Vandembroucq, and Falk [Phys. Rev. Lett. 117, 045501 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.045501] to compute the local yield stresses at the atomic scale in model two-dimensional Lennard-Jones glasses produced via differing quench protocols. This technique allows us to sample the plastic rearrangements in a nonperturbative manner for different loading directions on a well-controlled length scale. Plastic activity upon shearing correlates strongly with the locations of low yield stresses in the quenched states. This correlation is higher in more structurally relaxed systems. The distribution of local yield stresses is also shown to strongly depend on the quench protocol: the more relaxed the glass, the higher the local plastic thresholds. Analysis of the magnitude of local plastic relaxations reveals that stress drops follow exponential distributions, justifying the hypothesis of an average characteristic amplitude often conjectured in mesoscopic or continuum models. The amplitude of the local plastic rearrangements increases on average with the yield stress, regardless of the system preparation. The local yield stress varies with the shear orientation tested and strongly correlates with the plastic rearrangement locations when the system is sheared correspondingly. It is thus argued that plastic rearrangements are the consequence of shear transformation zones encoded in the glass structure that possess weak slip planes along different orientations. Finally, we justify the length scale employed in this work and extract the yield threshold statistics as a function of the size of the probing zones. This method makes it possible to derive physically grounded models of plasticity for amorphous materials by directly revealing the relevant details of the shear transformation zones that mediate this process.

Anomalous vibrational properties in the continuum limit of glasses

The low-temperature thermal properties of glasses are anomalous with respect to those of crystals. These thermal anomalies indicate that the low-frequency vibrational properties of glasses differ from those of crystals. Recent studies revealed that, in the simplest model of glasses, i.e., the harmonic potential system, phonon modes coexist with soft localized modes in the low-frequency (continuum) limit. However, the nature of low-frequency vibrational modes of more realistic models is still controversial. In the present work, we study the Lennard-Jones (LJ) system using large-scale molecular-dynamics (MD) simulation and establish that the vibrational property of the LJ glass converges to coexistence of the phonon modes and the soft localized modes in the continuum limit as in the case of the harmonic potential system. Importantly, we find that the low-frequency vibrations are rather sensitive to the numerical scheme of potential truncation, which is usually implemented in the MD simulation, and this is the reason why contradictory arguments have been reported by previous works. We also discuss the physical origin of this sensitiveness by means of a linear stability analysis.

Continuum limit of the vibrational properties of amorphous solids

The low-frequency vibrational and low-temperature thermal properties of amorphous solids are markedly different from those of crystalline solids. This situation is counterintuitive because all solid materials are expected to behave as a homogeneous elastic body in the continuum limit, in which vibrational modes are phonons that follow the Debye law. A number of phenomenological explanations for this situation have been proposed, which assume elastic heterogeneities, soft localized vibrations, and so on. Microscopic mean-field theories have recently been developed to predict the universal non-Debye scaling law. Considering these theoretical arguments, it is absolutely necessary to directly observe the nature of the low-frequency vibrations of amorphous solids and determine the laws that such vibrations obey. Herein, we perform an extremely large-scale vibrational mode analysis of a model amorphous solid. We find that the scaling law predicted by the mean-field theory is violated at low frequency, and in the continuum limit, the vibrational modes converge to a mixture of phonon modes that follow the Debye law and soft localized modes that follow another universal non-Debye scaling law.

Shear-stress fluctuations in self-assembled transient elastic networks

Focusing on shear-stress fluctuations, we investigate numerically a simple generic model for self-assembled transient networks formed by repulsive beads reversibly bridged by ideal springs. With Δt being the sampling time and t_{☆}(f)∼1/f the Maxwell relaxation time (set by the spring recombination frequency f), the dimensionless parameter Δx=Δt/t_{☆}(f) is systematically scanned from the liquid limit (Δx≫1) to the solid limit (Δx≪1) where the network topology is quenched and an ensemble average over m-independent configurations is required. Generalizing previous work on permanent networks, it is shown that the shear-stress relaxation modulus G(t) may be efficiently determined for all Δx using the simple-average expression G(t)=μ_{A}-h(t) with μ_{A}=G(0) characterizing the canonical-affine shear transformation of the system at t=0 and h(t) the (rescaled) mean-square displacement of the instantaneous shear stress as a function of time t. This relation is compared to the standard expression G(t)=c[over ̃](t) using the (rescaled) shear-stress autocorrelation function c[over ̃](t). Lower bounds for the m configurations required by both relations are given.