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Publication# Scaling for hard-sphere colloidal glasses near jamming

Abstract

Hard-sphere colloids are model systems in which to study the glass transition and universal properties of amorphous solids. Using covariance matrix analysis to determine the vibrational modes, we experimentally measure here the scaling behavior of the density of states, shear modulus, and mean-squared displacement (MSD) in a hard-sphere colloidal glass. Scaling the frequency with the boson-peak frequency, we find that the density of states at different volume fractions all collapse on a single master curve, which obeys a power law in terms of the scaled frequency. Below the boson peak, the exponent is consistent with theoretical results obtained by real-space and phase-space approaches to understanding amorphous solids. We find that the shear modulus and the MSD are nearly inversely proportional, and show a singular power-law dependence on the distance from random close packing. Our results are in very good agreement with the theoretical predictions. Copyright (C) EPLA, 2016

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Amorphous solid

In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid) is a solid that lacks the long-range order that is characteristic of a crystal. The terms "glass" and

Normal mode

A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by th

Shear modulus

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to

Silicon nitride based ceramics (SiAlONs); tetragonal polycrystalline zirconia (3Y‐TZP); alumina and their composites reinforced with different amount of multi‐ walled carbon‐nanotubes (CNTs) have been processed by Spark Plasma Sintering (SPS). High temperature mechanical spectroscopy measurements were performed in each material between room temperature and 1600 K. In each of these materials, anelastic and viscoplastic relaxation phenomena were investigated and responsible mechanisms were explained. In particular, grain boundary (GB) sliding, which is responsible for high temperature plasticity in fine grained ceramics, gives rise to a peak or an exponential increase in the high temperature mechanical loss. Peak and exponential background depend on two forces, which control the GB sliding: a friction force due to the GB viscosity and a restoring force due to the elasticity of the surrounding grains. In the present thesis, ceramics and composites have been processed, which allow one to study the role played by these forces on the thermo‐ mechanical behavior of these refractory materials. SiAlON ceramics were chosen for studying the viscous force due to the inter‐granular glassy phase. In zirconia and alumina, it is shown how the CNT reinforcements may improve the restoring force and consequently the creep resistance. Different grades of SiAlON ceramics, processed with different sintering aids (Ca2+, Y3+, Yb3+), with oxygen rich (CaO, Y2O3 and Yb2O3) and nitrogen rich (CaH2, YN and YbN) compounds, have been studied. Mechanical spectroscopy has been used to analyze the behavior of the residual glassy phase, present after sintering, either as grain‐boundary glassy (GB) films or glass pockets located in GB triple junctions. The mechanical loss spectra show a relaxation peak, which is due to a relaxation phenomenon (the so called “α‐relaxation”) associated with the glass transition in the amorphous phase. The peak position depends on the glass viscosity and the peak height is mainly affected by the glassy phase amount and the SiAlON bimodal microstructure, namely equiaxed versus elongated grains. Moreover the peak height depends on the restoring force provided by the neighboring grains, which limit the GB sliding process. As a matter of fact, a good correlation between the mechanical loss peak and plastic deformation in a compression test has been observed. It is concluded that the elongated grains in the bimodal microstructure provide a higher restoring force, which limits the GB sliding of smaller equiaxed grains. In the case of equiaxed fine grained oxide ceramics, such as zirconia and alumina, the restoring force has been increased by CNT additions, which can reinforce the GBs. 3Y‐TZPcomposites with a homogenous distributions of CNTs, ranging within 0.5 – 5 wt%, were processed by SPS. A significant improvement in room temperature fracture toughness and shear modulus as well as creep performance at high temperature were obtained, and interpreted as due to the GB reinforcing role of the CNTs. To support this interpretation, high‐resolution electron microscopy and Raman spectroscopy have been carried out. Moreover, a remarkable enhancement of the electrical conductivity up to ten orders of magnitude due to CNT additions has been obtained with respect to the pure ceramics. The isothermal spectrum of the 3Y‐TZP composites (measured at 1600 K) is composed of a mechanical loss peak at a frequency of about 0.1 Hz, which is superimposed on an exponential increase at lower frequency. This is interpreted as primarily due to GB sliding. The absence of a well‐marked peak in monolithic 3Y‐ TZP is justified by considering that the restoring force decreases at low frequencies or high temperatures. Therefore, GB sliding is no more restricted and the mechanical loss increases exponentially, which is correlated to macroscopic creep. With CNT additions the mechanical loss decreases and the relaxation peak is better resolved with respect to the background. This is interpreted by the pinning effect of CNTs on GBs, providing addition source of restoring force, which can hinder GB sliding at high temperature, resulting in a creep resistance improvement. Similarly to 3Y‐TZP based composites, alumina specimens (3 different types of alumina powders) reinforced with different amount of CNTs were sintered by SPS. It is shown how the initial particle size of the powders may affect the dispersion of CNTs. Measurement of different properties, such as hardness, fracture toughness and mechanical loss at high temperature, have evidenced the crucial role of the CNT dispersion in the obtained mechanical properties.

Granular materials are large sets of macroscopic particles that interact solely via contact forces. The static behavior depends on the contact network and on the surface friction forces between grains; when they are set in motion (typically by vibrations) their dynamics is dominated by inelastic collisions. For these reasons granular media show an extremely rich phenomenology, ranging from fluid-like properties (if strongly vibrated), to "jamming", glassy, behavior (if weakly vibrated), to aging and hysteretical phenomena observed when they become trapped in frozen, amorphous states. The objective of this work is to study these states and transitions, and to characterize the analogies found between the dynamic behavior of vibrated granular media and the glass transition observed in thermal glass-formers. These analogies justify the interest in granular materials, because granular media can be seen as simplified model systems useful in the study of out of equilibrium thermodynamics, and, in general, to the larger framework known as "complexity". The granular materials considered here are composed of spheric, polished glass spheres. Since the surface state plays an important role in the grain-grain interaction, some measurements were also performed with acid etched beads, having different surface roughness. The samples are vertically vibrated to achieve vibrofluidization. Different kinds of vibration are used, to highlight different properties of the system. We first consider the transition between the fluid and the subcooled glassy phase, using different experimental techniques. The most important one is a torsion oscillator, that interacts with the granular media via immersed probes. The torsion oscillator can be used in forced mode. A torque is applied on the probe, and we measure the mechanical response function (complex susceptibility). In general, a relaxation is found and it is interpreted as the signature of the irreversible energy loss (damping) in granular collisions. This relaxation has an intrinsic time scale, and systematic analysis of it shows that a clear parallel can be traced to the behavior of "strong" glasses. In particular, it is found that (i) the relaxation time is a function of a unified control parameter, proportional to the square root of the average vibration, and phenomenologically equivalent to an effective temperature; (ii) the functional form with which the relaxation times approach the final "frozen" state has an Arrhenius, or Vögel-Fulcher-Tamman (VFT) behavior. The same torsion oscillator is employed in free mode. In this case, no external torque is applied, and the probe moves adapting its position under the effect of the continuous rearrangements in the sample. The system is studied by computing the power spectral density of the (angular position) time series. The resulting spectra represent a "configurational noise" as the system randomly hops from one configuration to the following. This allows to define, using a completely different approach, the same intrinsic time scale observed in forced mode measurements. The comparison of the two techniques allows to obtain a more complete and detailed picture of the dynamics in the jamming region. From this comparison, it was inferred that the system is also influenced by an effective vibration frequency, and that the relaxation time has indeed a non-Arrhenius behavior as a function of a control parameter defined as as = √ Γ/ωs. A model was developed combining rheological observations to a statistic approach describing extremal phenomena. This model justifies the appearance of both the control parameter and the VFT evolution of the relaxation. Furthermore, the model is predictive and exposes the effect of a few other rheologic properties of granular system. The effect of surface roughness are considered, showing that the static and dynamic surface friction coefficients are well described by the model. A second relevant part of this work is devoted to an explicit verification that macroscopic probes act as Brownian objects. This fact is often used to interpret experimental data (also in the present work) and to propose theoretical model. However, no explicit evidence has ever been discussed. This is hard to do, using a constrained system such as the torsion oscillator, because the restoring coefficient influences the dynamics of diffusion. To overcome the problem we built a different apparatus, called "Brownian motor", where the probes are mounted on ball bearings, so that they are free to turn without constraint. The properties of the time series of the position of the free turning probe and of the torsionally constrained oscillator can finally be analyzed and compared with simple simulations. The data show an overall diffusion-like behavior, that is influenced by the presence of constraints. Using fractal analysis we estimate the diffusion, or Hurst exponent. This allows to verify that a "macroscopic" object (the probe) immersed in the "microscopic" granular medium indeed behaves as a Brownian object, and that its dynamics can be studied in detail, showing that it undergoes anomalous diffusion. This work is concluded with a discussion on a few possible developments. The most promising idea is a novel approach to the study of the geometrical properties of the contact network of granular assemblies, that is responsible for many of the properties of the granular sample. By using Magnetic Resonance Imaging, the static 3-D structure of granular media can be reconstructed with unprecedented accuracy, resolution and ease of reproducibility. From the spatial information we can extract all the properties of static granular media: the compaction factor, the grain-grain correlation function, the free volume and other observables. Systematic studies could allow experimental confirmations of the many theoretical models that have been proposed in the last years and that still lack a thorough comparison with experiments. This idea does not conclude the perspectives of this work, that are vast and intriguing. A few promising subjects are reviewed more into detail in the corresponding Perspective section. To name a few we cite: measurements of induced aging in non-vibrated samples, the Brownian motor, stick and slip phenomena and their comparison with earthquakes.

Gérard Gremaud, Daniele Mari, Alessandro Luigi Sellerio

We investigate a vibrated granular system composed of millimeter-size glass beads. When the system is submitted to a perturbation with decreasing intensity, below the fluidization limit, it evolves in a way similar to glass-forming liquids until it reaches an amorphous jammed state. This jamming transition is observed by the means of an immersed oscillator, either in the free or forced mode, while the granular medium is submitted to an external perturbation at a fixed frequency or within a frequency band. The complex susceptibility of the oscillator is measured as a function of the probe forcing frequency or as a function of the perturbation intensity. Data show that the jamming dynamics is "activated," similarly to thermal systems. The empirical control parameter is found proportional to the square root of the vibration intensity and inversely proportional to the vibration frequency. In the case of broadband external vibration, the average frequency of the power spectrum has to be considered.

2011