Session D53: Focus Session: Jamming -- Nonlinear Acoustics and Vibrational Response

Vibrational Modes in Colloidal Crystals

We investigate vibrational modes in quasi-two-dimensional colloidal crystals using video microscopy and displacement covariance matrix analysis. Debye scaling in the phonon density of states and the dispersion curve for two-dimensional hexagonal crystals are recovered for both mono-layer and double layer colloidal crystals. Using ``soft spots'' analysis, low-frequency quasi-localized phonon modes, which were found to coincide with fragile regions in glasses [1] appear to be spatially correlated with structural defects in colloidal crystals. Thus, ``soft spots'' may be a useful general identifier for defects in both crystalline and amorphous solids. This work is supported by NSF DMR 0804881, MRSEC DMR11-20901, and by NASA NNX08AO0G. \\[4pt] [1] K. Chen et al, Phys. Rev. Lett. 107, 108301 (2011)   

Acoustic measurement of a granular density of state

Measurements of the vibrational density of states $D(\omega)$ in glasses reveal that an excess number of low-frequency modes is associated with a loss of mechanical rigidity. An excess number of such modes has also been observed in both simulations of idealized granular materials near the jamming point, and in experiments on colloids. We experimentally investigate similar features in a jammed, quasi-two-dimensional granular material. We mimic thermal fluctuations using an electromagnetic driver to inject acoustic white noise, while piezoelectric sensors embedded inside a subset of the particles provide measurements of single-particle velocities. By analogy with conventional thermal techniques, we calculate a $D(\omega)$-like quantity via the spectrum of the velocity autocorrelation function. We measure $D(\omega)$ as a function of the confining pressure and find that the peak in the density of states shifts to higher frequency with system pressure. At low pressure, disordered systems have more low frequency modes than do hexagonally-packed systems.   

Elastic weakening of a dense granular medium by acoustic fluidization

Elastic waves propagating through a dense granular pack provide a unique probe of the elastic properties and internal dissipation of the medium [1], and also allow investigating the irreversible rearrangement of the contact network at large vibration amplitude. In this talk, we describe two distinct types of nonlinearity, i.e. hertzian and frictional, at the grain contact by sound amplitude and velocity measurements, respectively, under different confining pressure [2]. Beyond certain wave amplitude, the sound-matter interaction becomes irreversible, leaving the medium in a weakened and slightly compacted state. A slow recovery of the initial elastic modulus is observed after acoustic perturbation, revealing the plastic creep growth of microcontacts. The cross-correlation function of configuration-specific acoustic speckles highlights the relationship between the macroscopic elastic weakening and the local change of the contact networks, induced by strong sound vibration, in the absence of appreciable grain motion. We show that the softening of elastic modulus is much more pronounced with the shear wave (up to 20{\%}) than with the compressional wave (to 10{\%}). \\[4pt] [1] Th. Brunet, X. Jia and P. Mills, Phys. Rev. Lett \textbf{101}, 138001 (2008) \\[0pt] [2] Th. Brunet, X. Jia and P. Johnson, Geophys. Res. Lett \textbf{35}, L19308 (2008); X. Jia, Th. Brunet and J. Laurent, Phys. Rev. E \textbf{00}, 000300(R) (2011)   

Nonharmonicity in vibrated granular solids

We have shown that granular packings composed of frictionless particles with repulsive contact interactions are strongly nonharmonic. When infinitesimally perturbed along linear response eigenmodes of the static packing, energy leaks from the original mode of vibration to a continuum of frequencies due solely to contact breaking even when the system is under significant compression. Further, vibrated packings possess well-defined equilibrium positions that are different than those of the unperturbed packing. The vibrational density of states obtained using the displacement matrix and velocity autocorrelation function methods exhibit an increase in the number of low-frequency modes over that obtained from linear response of the static packing. The form of the density of states in vibrated granular packings is reminiscent of the low-frequency behavior of the vibrational density of states in fluid systems. We also investigate the effects of inter-particle friction, dissipation, particle shape, and degree of positional order on the density of states and thermal transport properties in driven granular packings.   

Irreversible Incremental Behavior in a Granular Material

We test the elasticity of dense, isotropic, compressed aggregates of frictional spheres using cyclic increments of shear and volume strain in a numerical simulation. For both types of increments, we measure irreversibility in relative displacements and contact forces that is stronger for the increments in shear. The strength of the irreversibility increases as the average number of contacts per particle (the coordination number) decreases. This irreversibility may be associated with the opening of contacts in an increment of loading, pointed out in a recent paper of Schreck et al. (PRL, 2011); such contact opening could lead to irreversible rearrangement of the contact network when the increment is relaxed.   

Experimental Measurements of Force Propagation in Vibrated Photoelastic Disks

We measure and analyze the propagation forces in vibrated disks under constant pressure with different amplitudes and frequencies. We use photoelastic particles to visualize the stress within each particle using a high-speed video camera. From the images we can extract the time dependent force at each contact to determine how force propagates through the contact network. Using mono-disperse particles we focus on force propagation during the phase transition from an ordered solid-like state to a disordered fluid-like state as we change the vibration amplitude. With bi-disperse particles we compare with the transition to a disordered solid-like state.   

Nonlinear acoustics of glass bead packings at vanishing static pressures

We present here a set of recent results obtained in three experimental configurations: linear and nonlinear acoustic probing of a granular slab at different compacities, surface acoustic waves in granular layers submitted to gravity, resonances of a granular layer with an in-depth elasticity gradient. We succeeded to overcome the experimental issues associated to the dramatic increase of acoustic wave attenuation when the confining pressure diminishes. Interpretations reveal that the manifestations of nonlinear effects (self-demodulation, nonlinear resonance, second harmonic or subharmonic generation...) allow to isolate the different types of nonlinearities involved (Hertz, clapping, stick-slip, hysteresis...). Also, some discrepancies are observed for the extracted linear elastic parameters scaling laws as a function of vanishing pressure (lower than 100 Pa typically) between the different developed experimental configurations and the theoretical predictions. Explanations for these discrepancies are given. We show than under some conditions, it is necessary to take into account the coupling of grain motion with that of the saturating air. Application of these results to the probing of granular layers under destabilization will be presented.   

Acoustic Echoes in Model Glasses

At low temperatures, glasses and crystals behave in qualitatively different ways. In particular, glasses have a great many more low-energy excitations that have traditionally been explained in terms of a distribution of dilute, two-level quantum states that are created by clusters of particles tunneling between two nearly degenerate ground states. Strong evidence for this model has come from the saturation effects and acoustic echoes [1] observed in these excitations. We show that, in contrast to conventional wisdom, the quasi-localized, strongly anharmonic, normal modes of jammed systems [2] can produce acoustic echoes due to the shift in the mode frequency with increasing amplitude. We observe this both in jammed packings of spherical particles with finite-range, Hertzian repulsions, and in model glasses interacting with a Lennard-Jones potential. In contrast to pulse echoes in two-level systems, a distinguishing feature of these ``anharmonic echoes'' is the appearance of multiple echoes after two excitation pulses, a feature also observed in experiments [1].\\[4pt] [1] B. Golding and J. E. Graebner. Phys. Rev. Lett. \textbf{37}, 852 (1976).\\[0pt] [2] N. Xu, V. Vitelli, A. J. Liu, and S. R. Nagel. Europhys. Lett. \textbf{90}, 56001 (2010).   

Extreme Physics and Rearrangements near Jamming

Near jamming, linear response becomes irrelevant. I briefly discuss how this gives rise to a range of intrinsically nonlinear, even extreme, phenomena. Moreover, reversibility also breaks down, and I will discuss the nature of rearrangements near jamming.   

Formation, Propagation, and Attenuation of Shocks Waves in Jammed Matter

We study the formation and propagation of fully non-linear waves in jammed granular media. Close to the jamming point, an arbitrary initial distortion of the media will induce the formation of non-linear finite amplitude waves. There are two regimes in the evolution of these waves. At early times non-linear interactions dominate the propagation, leading to a temporal evolution strongly dependent on the initial distortion. At long times the propagation is characterized by a new universal regime, dominated by hydrodynamical attenuation. Here the non-linear waves evolve in a self-similar fashion, characterized by a power law attenuation whose exponent is weakly dependent on the initial pressure of the system.   

Jamming of soft spheres at finite temperature : a granular experiment

At large packing fraction, disordered packings of particles with repulsive contact interactions jam into a rigid state where they withstand finite shear stresses before yielding. For frictionless particles and at zero temperature, the jamming transition coincides with the onset of iso-staticity and many geometrical and mechanical properties scale with the distance to the jamming point. What are the vestige of jamming at finite temperature and how jamming impacts the thermodynamics of glasses remain open issues. We address these questions experimentally by investigating the dynamics of both the density field and the force network of an horizontally shaken bi-disperse packing of photo-elastic disks. The average number of contact clearly displays an abrupt transition which we interpret as the jamming transition. Besides, dynamical heterogeneities are observed and their amplitude exhibits a maximum, which, in turn, signs a dynamical transition. We discuss in detail the interplay between these two transitions and how they depend on the particle softness and amplitude of the horizontal vibration.   

Density of vibrational modes in partially crystalline granular packings

Numerous numerical results have shown that systems of monodisperse frictionless disks crystallize readily and that disordered mechanically stable packings are rarely obtained. We numerically investigate the dependence of the cluster size distribution on system size and quench rate. We also investigate the effect of crystallization on the vibrational response outside the linear response regime. We study changes in the density of vibrational modes due to changes in the average crystallite size and perturbation amplitude in partially crystalline granular packings. In particular we determine how the number of contacts (above the isostatic value) affects anharmonic response in granular packings.   

Studying the low-frequency quasilocalized modes in disordered colloidal systems

In disordered colloidal systems, we experimentally measure the normal modes with covariance matrix method, and clarify the origin of low-frequency quasilocalization at single-particle level. We observe important features from both jamming and glass simulations: there is a plateau in the density of states which is suppressed upon ompression, as predicted by jamming; within the same systems, we also find that the low-frequency quasilocalization originates from the coupling between large vibrations of defective structures and transverse excitations, consistent with recent glass simulation. The coexistence of these features demonstrates an experimental link between jamming and glass. Extensive simulations further show that such structural origin of quasilocalization is universally valid for various temperatures and volume fractions.