Granular Crystals Research Articles

Tunable Wave Propagation in Granular Crystals by Altering Lattice Network Topology

We develop a framework for wave tailoring by altering the lattice network topology of a granular crystal consisting of spherical granules in contact. The lattice topology can alternate between two stable configurations, with the spherical granules of the lattice held in stable equilibrium in each configuration by gravity. Under impact, the first configuration results in a wave with rapidly decaying amplitude as it propagates along a primary chain, while the second configuration results in a solitary wave propagating along the primary chain with no decay. The mechanism to achieve such tunability is by having energy diverted to the granules adjacent to the primary chain in the first case but not the second. The tunable design of the proposed network is validated using both numerical simulations and experiments. In terms of potential applications, the proposed bistable lattice network can be viewed either as a wave attenuator or as a device that allows higher amplitude wave propagation in one direction than in the opposite direction. The lattice is analogous to a crystal phase transformation due to the change in atomic configurations, leading to the change in properties at the macroscale.

Nanocrystal growth and morphology of PbTeSe-ZnSe composite thin films prepared by one-step synthesis method

The microstructure of polycrystalline PbTe1−xSex-ZnSe composite thin films has been studied by scanning transmission electron microscopy and electron diffraction. The films were prepared by the one-step synthesis method using simultaneous evaporation of PbTe and ZnSe. The nanocrystals of PbTe1−xSex are formed in a ZnSe matrix. Tellurium concentration can be tuned by controlling the PbTe evaporation source temperatures between 753 K and 793 K. Binary PbSe nanocrystals were formed at 753 K, while ternary PbTe1−xSex nanocrystals were formed at 793 K. The nanocrystals grow in a granular shape at the initial stage of film growth, and the morphology changes to nanowire-shape as the film grows, irrespective of the Te concentration. The ternary PbTe1−xSex nanocrystals were composed of two phases with different Te concentration; Te-rich (Se-poor) granular crystals were formed near the bottom half parts of the film and Te-poor (Se-rich) nanowires were formed at the upper half parts of the film. Columnar ZnSe crystals contain high-density {111} stacking faults due to the low stacking fault energy of ZnSe. A balance of deposition and re-evaporation on the substrate during the film growth will be responsible for the resultant nanocrystal morphology.

Nonlinear wave propagation and rotational dynamics In microscale granular media

Granular media have been shown to exhibit complex dynamics stemming from dispersion and nonlinear particulate interactions. Studies of wave propagation in granular materials composed of ordered arrays of spherical particles interacting via Hertzian contact, known as “granular crystals,” have demonstrated rich nonlinear phenomena, including amplitude-dependent and tunable sound speeds, solitary waves, and discrete breathers. However, most of the past work on granular crystals has considered macroscale spheres, with diameters on the order of centimeters or millimeters. More recent works have explored wave phenomena in microscale granular crystals, having particle diameters of a few microns, which can be fabricated using self-assembly methods, and characterized experimentally using laser ultrasonic techniques. The physics of microscale granular media have been shown to vary drastically from their macroscale counterparts; for example, interparticle adhesion, which is negligible at macroscale, creates an intrinsic static load, which linearizes the contact forces at low amplitude, and can increase the importance of shear interactions and particle rotations. In this work, we present theoretical and computational analyses of nonlinear wave propagation in microscale granular crystals, examining the interplay between nonlinearity and rotational dynamics. In particular, we focus on amplitude-dependent distribution of energy into the translational and rotational modes of a hexagonally close-packed granular crystal.

Quantitative tuning nanoscale solitary waves

One-dimensionally arrayed C60 system, a nanoscale counterpart of traditional macroscale granular crystals, supports solitary wave propagation. In this paper, it is shown that inserting a series of short carbon nanotubes to C60 system may achieve quantitative tuning of C60-based solitary wave based on molecular dynamics simulations. A concept of energy tunnel is suggested which explains the change of transmitted energy in carbon nanotubes. Based on the energy tunnel theory, the amplitude and wave speed can be quantitatively tuned, which, to the authors' knowledge, has not been achieved at macroscale. This study may be instructive to the design of nanoscale wave-tailoring devices as well as stress wave monitoring measurements.

Study on preparation of thermal storage ceramic by using clay shale

Shale was used as main raw material for developing thermal storage ceramics. The samples were fabricated via semi-dry pressing followed by pressureless sintering. The result showed that the sample (75% shale, 10% kaolin, 10% potash feldspar and 5% soda feldspar) fired at 1080°C exhibited the best comprehensive performance. Ocular examination reveals that no cracks were observed after 30 cycle times thermal shock tests (wind cooling from 600°C to room temperature). The results presented that the high bending strength remained after 20 cycle times thermal shock tests but plummeted at the thirtieth time. Other properties were given as follows: bulk density: 2.60g/cm3; thermal conductivity: 2.33W/(m°C); and heat storage density: 578.50mJ/m3. XRD analysis indicated that the quartz and hematite were the main solid phases in the sample. Some isolated pores, quartz crystals, granular hematite crystals and needle-like mullite crystals were observed in the matrix according to the SEM (Scanning Electron Microscope) analysis. More pores were found with temperature rizing according to SEM analysis. The relatively high content of Fe2O3 contributed to the formation of the vitreous phase and favored the densification. Overall, the introduction of shale effectively reduced the firing temperature and performed the better thermal storage properties.

Shear to longitudinal mode conversion via second harmonic generation in a two-dimensional microscale granular crystal

Shear to longitudinal mode conversion via second harmonic generation is studied theoretically and computationally for plane waves in a two-dimensional, adhesive, hexagonally close-packed microscale granular medium. The model includes translational and rotational degrees of freedom, as well as normal and shear contact interactions. We consider fundamental frequency plane waves in all three linear modes, which have infinite spatial extent and travel in one of the high-symmetry crystal directions. The generated second harmonic waves are longitudinal for all cases. For the lower transverse–rotational mode, an analytical expression for the second harmonic amplitude, which is derived using a successive approximations approach, reveals the presence of particular resonant and antiresonant wave numbers, the latter of which is prohibited if rotations are not included in the model. By simulating a lattice with adhesive contact force laws, we study the effectiveness of the theoretical analysis for non-resonant, resonant, and antiresonant cases. This work is suitable for the analysis of microscale and statically compressed macroscale granular media, and should inspire future studies on nonlinear two- and three-dimensional granular systems in which interparticle shear coupling and particle rotations play a significant role.

Crystal structure of TiBi2.

Black granular single crystals of monotitanium dibismuth, TiBi2, were synthesized by slow cooling of a mixture of Bi and Ti from 693 K. The title compound is isostructural with CuMg2 (ortho-rhom-bic Fddd symmetry). Ti atoms are located in square anti-prisms of Bi atoms. The network of one type of Bi atom spirals along the a-axis direction while honeycomb layers of the other type of Bi atom spreading in the ab plane inter-lace one another.

Exceptional elastic anisotropy of hybrid organic–inorganic perovskite CH3NH3PbBr3 measured by laser ultrasonic technique

The results of the direct experimental evaluation of all elastic constants of single crystal hybrid organic–inorganic perovskite (HOIP) methylammonium lead bromide, a material known due to its possible solar‐energy, optoelectronics, X‐ray detector and thermoelectricity applications, are reported. The measurements of anisotropic elasticity of CH3NH3PbBr3by the technique of laser ultrasonics demonstrate that properties of HOIPs can be even more remarkable than the theoretical expectations: the extracted shear modulus is more than twice smaller, the universal anisotropy is more than twice higher, while the Debye temperature is more than 50° lower. Thus HOIPs can exhibit extremely low shear rigidity and extremely high anisotropy, both strongly overrunning the parameters which have been expected based on earlier first principles theoretical predictions. A simple theoretical model of granular crystal indicates that these observations could be related to the contributions of rotations/tilts of PbBr6octahedra to elastic response of cubic CH3NH3PbBr3. Another experimental observation is strong stiffening of shear rigidity with temperature increase from its room value up to 120 °C. Discovered elastic properties of HOIPs characterize them as exceptionally ductile/flexible/adaptive materials which could be deposited on corrugated/structured surfaces. (© 2016 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)

Downsizing granular crystals and their kin

In their interesting article on granular crystals (Physics Today, November 2015, page 44), Mason Porter, Panayotis Kevrekidis, and Chiara Daraio presented a rich set of recent developments in the field. They noted that switches and Boolean logic elements are among the possible applications of the propagation of nonlinear waves in granular media. Readers might be interested to know that Boolean logic operations have also been performed in related phononic crystals, although in that setting the functions do not rely on nonlinear phenomena.11. J. O. Vasseur, B. Morvan, A. Tinel, N. Swinteck, A. C. Hladky-Hennion, P. A. Deymier, Phys. Rev. B 86, 134305 (2012). https://doi.org/10.1103/PhysRevB.86.134305 ,22. S. Bringuier, N. Swinteck, J. O. Vasseur, J.-F. Robillard, K. Runge, K. Muralidharan, P. A. Deymier, J. Acoust. Soc. Am. 130, 1919 (2011). https://doi.org/10.1121/1.3631627 We strongly agree with the authors that the emerging possibility of downsizing granular crystals to the micron and perhaps submicron scale is full of potential. The richer physics of granular interactions at the smaller scales could serve as a bridge between macroscopic crystals and atomistic crystals. For instance, the rotational degrees of freedom inherent in granular media33. A. Merkel, V. Tournat, V. Gusev, Phys. Rev. Lett. 107, 225502 (2011). https://doi.org/10.1103/PhysRevLett.107.225502 offer unique opportunities to break the time-reversal symmetry of the dynamical equations. That could lead to topologically constrained wave propagation and the possibility of information-processing components that exploit the phase of waves instead of their amplitude.44. P. A. Deymier, K. Runge, N. Swinteck, K. Muralidharan, C. R. Méc. 343, 700 (2015). https://doi.org/10.1016/j.crme.2015.07.003 We believe, and we hope the authors will agree, that small-scale granular crystals and their phononic-crystal and acoustic-metamaterial cousins constitute a new frontier in the manipulation of acoustic waves and their utilization in technological progress.REFERENCESSection:ChooseTop of pageREFERENCES <<1. J. O. Vasseur, B. Morvan, A. Tinel, N. Swinteck, A. C. Hladky-Hennion, P. A. Deymier, Phys. Rev. B 86, 134305 (2012). https://doi.org/10.1103/PhysRevB.86.134305 , Google ScholarCrossref, ISI2. S. Bringuier, N. Swinteck, J. O. Vasseur, J.-F. Robillard, K. Runge, K. Muralidharan, P. A. Deymier, J. Acoust. Soc. Am. 130, 1919 (2011). https://doi.org/10.1121/1.3631627 , Google ScholarCrossref, ISI3. A. Merkel, V. Tournat, V. Gusev, Phys. Rev. Lett. 107, 225502 (2011). https://doi.org/10.1103/PhysRevLett.107.225502 , Google ScholarCrossref, ISI4. P. A. Deymier, K. Runge, N. Swinteck, K. Muralidharan, C. R. Méc. 343, 700 (2015). https://doi.org/10.1016/j.crme.2015.07.003 , Google ScholarCrossref, ISI© 2016 American Institute of Physics.

Complex Contact-Based Dynamics of Microsphere Monolayers Revealed by Resonant Attenuation of Surface Acoustic Waves.

Contact-based vibrations play an essential role in the dynamics of granular materials. Significant insights into vibrational granular dynamics have previously been obtained with reduced-dimensional systems containing macroscale particles. We study contact-based vibrations of a two-dimensional monolayer of micron-sized spheres on a solid substrate that forms a microscale granular crystal. Measurements of the resonant attenuation of laser-generated surface acoustic waves reveal three collective vibrational modes that involve displacements and rotations of the microspheres, as well as interparticle and particle-substrate interactions. To identify the modes, we tune the interparticle stiffness, which shifts the frequency of the horizontal-rotational resonances while leaving the vertical resonance unaffected. From the measured contact resonance frequencies we determine both particle-substrate and interparticle contact stiffnesses and find that the former is an order of magnitude larger than the latter. This study paves the way for investigating complex contact-based dynamics of microscale granular crystals and yields a new approach to studying micro- to nanoscale contact mechanics in multiparticle networks.

Universal relations for solitary waves in granular crystals under shocks with finite rise and decay times.

We focus on solitary waves generated in arrays of lightly contacting spherical elastic granules by shock forces of steep rise and slow decay durations and establish a priori: (i) whether the peak value of the resulting solitary wave would be greater than, equal to, or less than the peak value of the input shock force; (ii) the magnitude of the peak value of the solitary waves; (iii) the magnitude of the linear momentum in each solitary wave; (iv) the magnitude of the linear momentum added to the remaining granules if the first granule is ejected; and (v) a quantitative estimate of the effect of the granules' radius, density, and stiffness on force amplification or mitigation. We have supported the analytical results by direct numerical simulations.

Stresses and orientational order in shearing flows of granular liquid crystals.

We perform discrete element simulations of homogeneous shearing of frictionless cylinders and show that the particles are characterized by orientational order and form a granular liquid crystal. For elongated and flat cylinders, the alignment is in the plane of shearing, while cylinders having an aspect ratio equal to 1 and 0.8 show no orientational order. We show that the particle pressure is insensitive to the cylinder aspect ratio and well predicted by the kinetic theory of granular gases, with a singularity in the radial distribution function at contact different from that for frictionless spheres. The numerical results quantitatively agree with physical experiments on different geometries. The particle shear stress is affected by orientational anisotropy. We postulate that, for frictionless cylinders, the viscosity is roughly due to the motion of the orientationally disordered fraction of the particles, and show that it is proportional, through the order parameter, to the expression of kinetic theory. Finally, we suggest that the orientational order is the result of the competing effects of the shear rate, which induces alignment, and the granular temperature, which ramdomizes.

Surface waves in granular phononic crystals.

The existence of surface elastic waves at a mechanically free surface of granular phononic crystals is studied. The granular phononic crystals are made of spherical particles distributed periodically on a simple cubic lattice. It is assumed that the particles are interacting by means of normal, shear, and bending contact rigidities. First, Rayleigh-type surface acoustic waves, where the displacement of the particles takes place in the sagittal plane while the particles possess one rotational and two translational degrees of freedom, are analyzed. Second, shear-horizontal-type waves, where the displacement of the particles is normal to the sagittal plane while the particles possess one translational and two rotational degrees of freedom are studied. The existence of zero-group-velocity surface acoustic waves of Rayleigh type is theoretically predicted and interpreted. A comparison with surface waves predicted by the reduced Cosserat theory is performed, and some limitations of the latter are established.

Superdiffusive transport and energy localization in disordered granular crystals.

We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder-an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)-and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment m̃(2) of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m̃(2)∼t(γ) (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m̃(2) and the inverse participation ratio P(-1) satisfy scaling relations m̃(2)∼t(γ) and P(-1)∼t(-η), and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η<0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ≈1.7 and η≈1) for all three types of disorder and for both types of initial conditions.

Traveling waves in trimer granular lattice I: Bifurcation structure of traveling waves in the unit-cell model

Present paper is the first one in the series devoted to the dynamics of traveling waves emerging in the uncompressed, tri-atomic granular crystals. This work is primarily concerned with the dynamics of one-dimensional periodic granular trimer (tri-atomic) chains in the state of acoustic vacuum. Each unit cell consists of three spherical particles of different masses subject to periodic boundary conditions. Hertzian interaction law governs the mutual interaction of these particles. Under the assumption of zero pre-compression, this interaction is modeled as purely nonlinear, which means the absence of linear force component. The dynamics of such chains is governed by the two system parameters that scale the mass ratios between the particles of the unit cell. Such a system supports two different classes of periodic solutions namely the traveling and standing waves. The primary objective of the present study is the numerical analysis of the bifurcation structure of these solutions with emphasis on the dynamics of traveling waves. In fact, understanding of the bifurcation structure of the traveling wave solutions emerging in the unit-cell granular trimer is rather important and can shed light on the more complex nonlinear wave phenomena emerging in semi-infinite trimer chains.

Unusual crystalline morphology of Poly Aryl Ether Ketones (PAEKs)

SEM image showing the granular crystal blocks in the spherulite of laser sintered PEEK.

Nonlinear resonances and antiresonances of a forced sonic vacuum.

We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude- and frequency-dependent boundary drive. Despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period and observe that this can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force ("antiresonances") between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings and Neimark-Sacker bifurcations as well as multiple isolas (e.g., of period-3, -4, or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports a notion of a (purely) "nonlinear spectrum" in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). We rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.

Nonlinear low-to-high-frequency energy cascades in diatomic granular crystals.

We study wave propagation in strongly nonlinear one-dimensional diatomic granular crystals under an impact load. Depending on the mass ratio of the "light" to "heavy" beads, this system exhibits rich wave dynamics from highly localized traveling waves to highly dispersive waves featuring strong attenuation. We demonstrate experimentally the nonlinear resonant and antiresonant interactions of particles, and we verify that the nonlinear resonance results in strong wave attenuation, leading to highly efficient nonlinear energy cascading without relying on material damping. In this process, mechanical energy is transferred from low to high frequencies, while propagating waves emerge in both ordered and chaotic waveforms via a distinctive spatial cascading. This energy transfer mechanism from lower to higher frequencies and wave numbers is of particular significance toward the design of novel nonlinear acoustic metamaterials with inherently passive energy redistribution properties.

Building a Better Battery: Ionic Conductivity of Granular Hydrate Crystals

Although solid state batteries are an alternative to conventional batteries, they generally produce low power due to their poor ionic conductivity. This study tested the ionic conductivities of powdered potassium sodium tartrate and potassium aluminum sulfate, two double salts that have garnered attention as potential solid state electrolytes. The substances were hypothesized on having abnormally high conductivities. Conductivity was measured using electrical impedance spectroscopy on an Arduino UNO. The results indicated that the double salts had indeed high conductivities for an ionic crystal at 5.6e-5±6e-6 S/m for PST and 7.8e-6±9e-7 S/m for PAS. However, the other 3 materials tested all produced similar values. Furthermore, one of the materials tested, sodium chloride, has a documented conductivity of 10-12 S/m. The high conductivities as well as the discrepancy between the measured and documented conductivities of sodium chloride were attributed to the powder form of the materials. Caking may have occurred and increased conductivity by increasing defect concentration as well as by creating moisture channels through which free ions could flow. These results have vast implications if a powdered electrolyte material may be successfully applied to existing fast ion conductors instead of insulators such as sodium chloride.

Zero-frequency and slow elastic modes in phononic monolayer granular membranes

We theoretically study the dispersion properties of elastic waves in hexagonal and honeycomb monolayer granular membranes with either out-of-plane or in-plane particle motion. The particles interact predominantly via normal and transverse contact rigidities. When rotational degrees of freedom are taken into account, the bending and torsional rigidities of the intergrain contacts can control some of the phononic modes. The existence of zero-frequency modes, zero-group-velocity modes and their transformation into slow propagating phononic modes due to weak bending and torsional intergrain interactions are investigated. We also study the formation and manipulation of Dirac cones and multiple degenerated modes. This could motivate variety of potential applications in elastic waves control by manipulating the contact rigidities in granular phononic crystals.