Measuring spatial distribution of the local elastic modulus in glasses

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.

Cutoff nonlinearities in the low-temperature vibrations of glasses and crystals

We present a computer simulation study of glassy and crystalline states using the standard Lennard-Jones interaction potential that is truncated at a finite cutoff distance, as is typical of many computer simulations. We demonstrate that the discontinuity at the cutoff distance in the first derivative of the potential (corresponding to the interparticle force) leads to the appearance of cutoff nonlinearities. These cutoff nonlinearities persist into the very-low-temperature regime thereby affecting low-temperature thermal vibrations, which leads to a breakdown of the harmonic approximation for many eigenmodes, particularly for low-frequency vibrational modes. Furthermore, while expansion nonlinearities which are due to higher order terms in the Taylor expansion of the interaction potential are usually ignored at low temperatures and show up as the temperature increases, cutoff nonlinearities can become most significant at the lowest temperatures. Anharmonic effects readily show up in the elastic moduli which not only depend on the eigenfrequencies, but are crucially sensitive to the eigenvectors of the normal modes. In contrast, those observables that rely mainly on static structural information or just the eigenfrequencies, such as the vibrational density of states, total potential energy, and specific heat, show negligible dependence on the presence of the cutoff. Similar aspects of nonlinear behavior have recently been reported in model granular materials, where the constituent particles interact through finite-range, purely repulsive potentials. These nonlinearities have been ascribed to the nature of the sudden cutoff at contact in the force law. As a consequence, we demonstrate that cutoff nonlinearities emerge as a general feature of ordered and disordered solid state systems interacting through truncated potentials.

Spatial Distributions of Local Elastic Moduli Near the Jamming Transition

Recent progress on studies of the nanoscale mechanical responses in disordered systems has highlighted a strong degree of heterogeneity in the elastic moduli. In this contribution, using computer simulations, we study the elastic heterogeneities in athermal amorphous solids--composed of isotropic static sphere packings--near the jamming transition. We employ techniques based on linear response methods that are amenable to experimentation. We find that the local elastic moduli are randomly distributed in space and are described by Gaussian probability distributions, thereby lacking any significant spatial correlations, that persist all the way down to the transition point. However, the shear modulus fluctuations grow as the jamming threshold is approached, which is characterized by a new power-law scaling. Through this diverging behavior we are able to identify a characteristic length scale, associated with shear modulus heterogeneities, that distinguishes between bulk and local elastic responses.

Correlation between rearrangements and soft modes in polymer glasses during deformation and recovery

We explore the link between soft vibrational modes and local relaxation events in polymer glasses during physical aging, active deformation at constant strain rate, and subsequent recovery. A softness field is constructed out of the superposition of the amplitudes of the lowest energy normal modes, and found to predict up to 70% of the rearrangements. Overlap between softness and rearrangements increases logarithmically during aging and recovery phases as energy barriers rise due to physical aging, while yielding rapidly rejuvenates the overlap to that of a freshly prepared glass. In the strain hardening regime, correlations rise for uniaxial tensile deformation but not for simple shear. These trends can be explained by considering the differing degrees of localization of the soft modes in the two deformation protocols.

Probing relevant ingredients in mean-field approaches for the athermal rheology of yield stress materials

Although the notion of mechanical noise is expected to play a key role in the non-linear rheology of athermally sheared amorphous systems, its characterization has so far remained elusive. Here, we show using molecular dynamic simulations that in spite of the presence of strong spatio-temporal correlations in the system, the local stress exhibits normal diffusion under the effect of the mechanical noise in the finite driving regime. The diffusion constant appears to be proportional to the mean plastic activity. Our data suggests that the corresponding proportionality constant is density independent, and can be directly related to the specific form of the rheological flow curve, pointing the way to a generic way of modeling mechanical noise in mean-field equations.

Beating the amorphous limit in thermal conductivity by superlattices design

The value measured in the amorphous structure with the same chemical composition is often considered as a lower bound for the thermal conductivity of any material: the heat carriers are strongly scattered by disorder, and their lifetimes reach the minimum time scale of thermal vibrations. An appropriate design at the nano-scale, however, may allow one to reduce the thermal conductivity even below the amorphous limit. In the present contribution, using molecular-dynamics simulation and the Green-Kubo formulation, we study systematically the thermal conductivity of layered phononic materials (superlattices), by tuning different parameters that can characterize such structures. We have discovered that the key to reach a lower-than-amorphous thermal conductivity is to block almost completely the propagation of the heat carriers, the superlattice phonons. We demonstrate that a large mass difference in the two intercalated layers, or weakened interactions across the interface between layers result in materials with very low thermal conductivity, below the values of the corresponding amorphous counterparts.

Elasticity of frictionless particles near jamming

We study the linear elastic response of harmonic disk packings near jamming via three types of probes: (i) point forcing, (ii) constrained homogeneous deformation of subregions of large systems, and (iii) unconstrained deformation of the full system subject to periodic boundary conditions. For the point forcing, our results indicate that the transverse component of the response is governed by a lengthscale ξT, which scales with the confining pressure, p, as ξT∼p-0.25, while the longitudinal component is governed by ξL, which scales as ξL∼p-0.4. The former scaling is precisely the transverse lengthscale, which has been invoked to explain the structure of normal modes near the density of states anomaly in sphere packings, while the latter is much closer to the rigidity length, l*∼p-0.5, which has been invoked to describe the jamming scenario. For the case of constrained homogeneous deformation, we find that μ(R), the value of the shear modulus measured in boxes of size R, gives a value much higher than the continuum result for small boxes and recedes to its continuum limit only for boxes bigger than a characteristic length, which scales like p-0.5, precisely the same way as l*. Finally, for the case of unconstrained homogeneous deformation, we find displacement fields with power spectra, which are consistent with independent, uncorrelated Eshelby transformations. The transverse sector is amazingly invariant with respect to p and very similar to what is seen in Lennard-Jones glasses. The longitudinal piece, however, is sensitive to p. It develops a plateau at long wavelength, the start of which occurs at a length that grows in the p→0 limit. Strikingly, the same behavior is observed both for applied shear and dilation.

Heterogeneous Viscoelasticity: A Combined Theory of Dynamic and Elastic Heterogeneity

We present a heterogeneous version of Maxwell's theory of viscoelasticity based on the assumption of spatially fluctuating local viscoelastic coefficients. The model is solved in coherent-potential approximation. The theory predicts an Arrhenius-type temperature dependence of the viscosity in the vanishing-frequency limit, independent of the distribution of the activation energies. It is shown that this activation energy is generally different from that of a diffusing particle with the same barrier-height distribution, which explains the violation of the Stokes-Einstein relation observed frequently in glasses. At finite but low frequencies, the theory describes low-temperature asymmetric alpha relaxation. As examples, we report the good agreement obtained for selected inorganic, metallic, and organic glasses. At high frequencies, the theory reduces to heterogeneous elasticity theory, which explains the occurrence of the boson peak and related vibrational anomalies.

Shear-stress relaxation and ensemble transformation of shear-stress autocorrelation functions

We revisit the relation between the shear-stress relaxation modulus G(t), computed at finite shear strain 0<γ≪1, and the shear-stress autocorrelation functions C(t)|(γ) and C(t)|(τ) computed, respectively, at imposed strain γ and mean stress τ. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general G(t)=C(t)|(τ)=C(t)|(γ)+G(eq) for t>0 with G(eq) being the static equilibrium shear modulus. G(t) and C(t)|(γ) thus must become different for solids and it is impossible to obtain G(eq) alone from C(t)|(γ) as often assumed. We comment briefly on self-assembled transient networks where G(eq)(f) must vanish for a finite scission-recombination frequency f. We argue that G(t)=C(t)|(τ)=C(t)|(γ) should reveal an intermediate plateau set by the shear modulus G(eq)(f=0) of the quenched network.

Structural relaxation in glassy polymers predicted by soft modes: a quantitative analysis

We present a quantitative analysis of the correlation between quasi-localized, low energy vibrational modes and structural relaxation events in computer simulations of a quiescent, thermal polymer glass. Our results extend previous studies on glass forming binary mixtures in 2D, and show that the soft modes identify regions that undergo irreversible rearrangements with up to 7 times the average probability. We study systems in the supercooled- and aging-regimes and discuss temperature- as well as age-dependence of the correlation. In addition to the location of rearrangements, we find that soft modes also predict their direction on the molecular level. The soft regions are long lived structural features, and the observed correlations vanish only after >50% of the system has undergone rearrangements.

Acoustic excitations and elastic heterogeneities in disordered solids

In the recent years, much attention has been devoted to the inhomogeneous nature of the mechanical response at the nanoscale in disordered solids. Clearly, the elastic heterogeneities that have been characterized in this context are expected to strongly affect the nature of the sound waves which, in contrast to the case of perfect crystals, cannot be completely rationalized in terms of phonons. Building on previous work on a toy model showing an amorphization transition, we investigate the relationship between sound waves and elastic heterogeneities in a unified framework by continuously interpolating from the perfect crystal, through increasingly defective phases, to fully developed glasses. We provide strong evidence of a direct correlation between sound wave features and the extent of the heterogeneous mechanical response at the nanoscale.

Evolution of elastic heterogeneity during aging in metallic glasses

The properties of glasses vary widely depending on the way they are prepared, even though their structures appear similar. We show that the local potential energy landscape (PEL) sensitively reflects the stability differences through simulation of local structural excitation in a model metallic glass. It is observed that the spectrum of local structural excitation develops a pseudogap at low energies as the glass becomes more stable. We also demonstrate that the spatial variation of the atomic level shear modulus, rather than the distribution of the magnitude of the single atom shear modulus, is more closely related to the nature of the PEL and the stabilities of glasses. In particular, local aggregation of atoms with low shear modulus greatly contributes to instability of the system.

Time-dependent elastic response to a local shear transformation in amorphous solids

The elastic response of a two-dimensional amorphous solid to induced local shear transformations, which mimic the elementary plastic events occurring in deformed glasses, is investigated via molecular-dynamics simulations. We show that for different spatial realizations of the transformation, despite relative fluctuations of order one, the long-time equilibrium response averages out to the prediction of the Eshelby inclusion problem for a continuum elastic medium. We characterize the effects of the underlying dynamics on the propagation of the elastic signal. A crossover from a propagative transmission in the case of weakly damped dynamics to a diffusive transmission for strong damping is evidenced. In the latter case, the full time-dependent elastic response is in agreement with the theoretical prediction, obtained by solving the diffusion equation for the displacement field in an elastic medium.

Thermomechanical properties and deformation of coarse-grained models of hard-soft block copolymers

In this paper, we investigate the enhancement mechanism of the mechanical properties for hard-soft block copolymers by using molecular dynamics simulations at various temperatures. A coarse-grained approach is adopted to study sufficiently generic models. Our numerical experiments demonstrate that the nonbond potential plays a more significant role in mechanical properties compared to the bond potential. This finding serves as a cornerstone to understand the hard-soft materials. To explore the effect of hard segments, four copolymers with different concentrations and energy factors that describe the interaction between hard beads are conducted. Simulation results show that the mechanical performances of the system with large attractive force and small concentration of hard segments could be improved dramatically in conjunction with a moderate increment of the glass transition temperature. In particular, the energy factor shows a substantial influence in determining the microphase separation as well as the morphology of hard domains. These observations are believed to provide design guidelines for polymeric materials in engineering practice.