Jamming III: Characterizing randomness via the entropy of jammed matter

The nature of randomness in disordered packings of frictional and frictionless spheres is investigated using theory and simulations of identical spherical grains. The entropy of the packings is defined through the force and volume ensemble of jammed matter and this is shown to be difficult to calculate analytically. A mesoscopic ensemble of isostatic states is then utilized in an effort to predict the entropy through the definition of a volume function that is dependent on the coordination number. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter, and elucidating the phase diagram for jammed granular matter. Analytical calculations are compared to numerical simulations using volume fluctuation analysis and graph theoretical methods, with reasonable agreement. The entropy of the jammed system reveals that random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits. Ensemble calculations show that the entropy vanishes at random close packing (RCP), while numerical simulations show that a finite entropy remains in the microscopic states at RCP. The notion of a negative compactivity, which explores states with volume fractions below those achievable by existing simulation protocols, is also explored, expanding the equations of state. The mesoscopic theory reproduces the simulations results in shape well, though a difference in magnitude implies that the entire entropy of the packing may not be captured by the methods presented herein. We discuss possible extensions to the present mesoscopic approach describing packings from random loose packing (RLP) to RCP to the ordered branch of the equation of state in an effort to understand the entropy of jammed matter in the full range of densities from RLP to face-centered cubic (FCC) packing.     

Micromechanical properties of carbon–silica aerogel composites

Materials’ endurance to mechanical stress is desirable from a technological point of view. In particular, in the case of silica aerogels, an improvement of the material elasticity is needed for some applications. Carbon–silica aerogel composites have been obtained and their mechanical properties, Young’s modulus, elastic parameter and hardness, have been evaluated with a dynamical, non-destructive microindentation technique. Large changes are found in Young’s modulus when only a small amount of carbon is added. This is clearly shown in the shape of the indentation curves as well as in the increase of the elastic parameter value, which evaluates the percentage of elasticity versus plasticity. Young’s modulus values obtained for carbon–silica aerogels show a similar variation with the carbon mass fraction to that predicted by a commonly used model for composite materials. The measured hardness values corresponding to the total elastoplastic deformation do not show such a prominent dependency on the carbon mass fraction as the elastic parameter and Young’s modulus do and they are similar to those measured for the pure-silica aerogel. Received: 18 May 2001 / Accepted: 30 July 2001 / Published online: 30 October 2001     

Entropy of jammed matter

We investigate the nature of randomness in disordered packings of frictional spheres. We calculate the entropy of 3D packings through the force and volume ensemble of jammed matter, a mesoscopic ensemble and numerical simulations using volume fluctuation analysis and graph theoretical methods. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter. At the mesoscopic level the entropy vanishes at random close packing, while the microscopic states contribute to a finite entropy. The entropy of the jammed system reveals that the random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits.     

The role of Frank elasticity in cholesteric elastomers

The effect of Frank elasticity on deformations of cholesteric elastomers by mechanical stress applied perpendicular to the helix axis is studied by numerical minimization of the free energy. Above a critical strain, a solution with an only oscillating director is found to be stable in comparison to a distorted helix. At the critical strain, the contractions perpendicular to stress change discontinuously. The critical strain is found to increase with increasing Frank elasticity contribution to the free energy density, and to diverge when the conformation anisotropy of the polymer backbone vanishes. The results are compared with recent experiments which indicated that, in case of weak conformation anisotropies, the Frank elasticity contribution to the free energy cannot be neglected.     

Elasticity from the force network ensemble in granular media

Transmission of forces in static granular materials is studied within the framework of the force network ensemble, by numerically evaluating the mechanical response of hexagonal packings of frictionless grains and rectangular packings of frictional grains. In both cases, close to the point of application of the overload, the response is nonlinear and displays two peaks, while at larger length scales, it is linear and elasticlike. The crossover between these two behaviors occurs at a depth that increases with the magnitude of the overload and decreases with increasing friction.     

The statistical mechanics of granular systems composed of spheres and elongated grains

In this paper we will apply a statistical mechanical theory of granular materials to a system composed of a mixture of elongated rods and spheres (both mono- and polydisperse). We will show for a system of rods and spheres that the system's constituents do not completely phase segregate even at minimum close packing. The generalisation of this to packings of rods and bi-disperse spheres produces a similar result except that the two species of sphere do phase segregate at some particular packing.     

Contact forces in a granular packing

We present the results of a systematic numerical investigation of force distributions in granular packings. We find that all the main features of force transmission previously established for two-dimensional systems of hard particles hold in three-dimensional systems and for soft particles, too. In particular, the probability distribution of normal forces falls off exponentially for forces above the mean force. For forces below the mean, this distribution is either a decreasing power law when the system is far from static equilibrium, or nearly uniform at static equilibrium, in agreement with recent experiments. Moreover, we show that the forces below the mean do not contribute to the shear stress. The subnetwork of the contacts carrying a force below the mean thus plays a role similar to a fluid surrounding the solid backbone composed of the contacts carrying a force above the mean. We address the issue of the computation of contact forces in a packing at static equilibrium. We introduce a model with no local simplifying force rules, that allows for an exact computation of contact forces for given granular texture and boundary conditions. (c) 1999 American Institute of Physics.     

Quantum transport in honeycomb lattice ribbons with armchair and zigzag edges coupled to semi-infinite linear chain leads

We study quantum transport in honeycomb lattice ribbons with either armchair or zigzag edges. The ribbons are coupled to semi-infinite linear chains serving as the input and output leads and we use a tight-binding Hamiltonian with nearest-neighbor hops. The input and output leads are coupled to the ribbons through bar contacts. In narrow ribbons we find transmission gaps for both types of edges. The appearance of this gap is due to the enhanced quantum interference coming from the multiple channels in bar contacts. The center of the gap is at the middle of the band in ribbons with armchair edges. This particle-hole symmetry is because bar contacts do not mix the two sublattices of the underlying bipartite honeycomb lattice when the ribbon has armchair edges. In ribbons with zigzag edges the gap center is displaced to the right of the band center. This breakdown of particle-hole symmetry is the result of bar contacts now mixing the two sublattices. We also find transmission oscillations and resonances within the transmitting region of the band for both types of edges. Extending the length of a ribbon does not affect the width of the transmission gap, as long as the ribbon’s length is longer than a critical value when the gap can form. Increasing the width of the ribbon, however, changes the width of the gap. In ribbons with zigzag edges the gap width systematically shrinks as the width of the ribbon is increased. In ribbons with armchair edges the gap is not well-defined because of the appearance of transmission resonances. We also find only evanescent waves within the gap and both evanescent and propagating waves in the transmitting regions.     

Suppressed compressibility at large scale in jammed packings of size-disperse spheres

We analyze the large-scale structure and fluctuations of jammed packings of size-disperse spheres, produced in a granular experiment as well as numerically. While the structure factor of the packings reveals no unusual behavior for small wave vectors, the compressibility displays an anomalous linear dependence at low wave vectors and vanishes when q→0. We show that such behavior occurs because jammed packings of size-disperse spheres have no bulk fluctuations of the volume fraction and are thus hyperuniform, a property not observed experimentally before. Our results apply to arbitrary particle size distributions. For continuous distributions, we derive a perturbative expression for the compressibility that is accurate for polydispersity up to about 30%.     

Size-topology correlations in disk packings: terminal bidispersity in order–disorder transitions

Abstract

In random packings or tilings, the size distribution of individual elements (domains) and the statistics of numbers of neighbours of those domains are strongly correlated. In the case of circular disks forming a random packing in the plane, it has long been known empirically that a certain critical amount of bidispersity avoids crystallization of the packing. We demonstrate how the formalism of a simplified granocentric model allows for an analytical computation of the size-topology correlation as a function of both size ratio and frequency of small disks. The results, obtained without free parameters, are in excellent agreement with the empirical findings of packing simulations concerning critical (terminal) bidispersity. It is also shown that, at equal size variance, the discrete (bidisperse) disk size distributions induce stronger disorder than continuously polydisperse disks.     

Discrete Element Method studies of the collision of one rapid sphere on 2D and 3D packings

We performed numerical simulations of one-bead collision on the surface of a static granular medium. The simulations have been done for two- and three-dimensional packings of beads. The effect of the incident bead velocity, the shot angle, the mechanical parameters and the packing structure are analyzed for ordered and disordered 2D packings and only disordered 3D packings. The 2D results are in good agreement with experimental available data. The 3D simulations give good preliminaries results about the shock-wave propagation through the stacking and provides new insights in the ejection process (“splash function”).     

Strain stiffening in random packings of entangled granular chains

Random packings of granular chains are presented as a model system to investigate the contribution of entanglements to strain stiffening. The chain packings are sheared in uniaxial compression experiments. For short chain lengths, these packings yield when the shear stress exceeds the scale of the confining pressure, similar to granular packings of unconnected particles. In contrast, packings of chains which are long enough to form loops exhibit strain stiffening, in which the effective stiffness of the material increases with strain, similar to many polymer materials. The latter packings can sustain stresses orders-of-magnitude greater than the confining pressure, and do not yield until the chain links break. X-ray tomography measurements reveal that the strain-stiffening packings contain system-spanning clusters of entangled chains.     

Echo tracer dispersion in flows of polymer solutions through porous media: A tool for detecting weak permeability heterogeneities?

Tracer dispersion in Newtonian and shear-thinning fluids (scleroglucan-water polymer solutions) flowing through single and double porosity grain packings has been studied experimentally using both classical transmission dispersion and echo dispersion (in the latter, the concentration variation front is pumped back through a detector at the inlet after penetrating for a chosen distance into the sample). Transmission dispersion increases markedly in both types of samples with the shear thinning index of the fluid at all Péclet numbers (except when molecular diffusion is dominant). Echo and transmission experiments give nearly identical dispersivity values for Newtonian fluids while echo dispersivity is lower than transmission for shear thinning ones. The normalized dispersivity difference has same order of magnitude for single and double porosity samples and increases with the shear thinning exponent α (by a factor of 2 between α = 0.35 and α = 0.60). This difference may be due to heterogeneities inducing permeability variations of small amplitude over distances of the order of the sample section : their influence on tracer dispersion is partly reversible with respect to a change of the flow direction and is only detectable if it is amplified by the shear-thinning properties of the fluid. Received 19 September 2001     

Restructuring of force networks

The compaction of granular packings or soils is a collective process which for higher densities becomes increasingly slower reaching glassy behaviour. We present a study of this problem from various points of view, in particular we will represent the evolving force network that percolates through the system by an inverse fiber rupture model. Received 15 March 2002 and Received in final form 29 July 2002     

Influence of wall friction on granular column

Granular packings under gravity in frictional and frictionless silos were simulated and the influence of the wall friction on the normal force distribution was investigated. Although there is an obvious Janssen effect in frictional silos, only a slight influence on the geometry of packing was found. The law of normal force distribution is different for frictional and frictionless walls, which is related to the pressure profile. A modified formula with consideration of the pressure profile was well fitted to the simulation results.     

Non-Amontons behavior of friction in single contacts

We report on the frictional properties of a single contact between a glassy polymer lens and a flat silica substrate covered either by a disordered or by a self-assembled alkylsilane monolayer. We find that, in contrast to a widely spread belief, the Amontons proportionality between frictional and normal stresses does not hold. Besides, we observe that the velocity dependence of the sliding stress is strongly sensitive to the structure of the silane layer. Analysis of the frictional rheology observed on both disordered and self-assembled monolayers suggests that dissipation is controlled by the plasticity of a glass-like interfacial layer in the former case, and by pinning of polymer chains on the substrate in the latter one.     

Force chain splitting in granular materials: A mechanism for large-scale pseudo-elastic behaviour

We investigate both numerically and analytically the effect of strong disorder on the large-scale properties of the hyperbolic equations for stresses proposed in J.-P. Bouchaud, M.E. Cates, P. Claudin, J. Phys. I 5, 639 (1995), and J.P. Wittmer, P. Claudin, M.E. Cates, J.-P. Bouchaud, Nature 382, 336 (1996); J.P. Wittmer, P. Claudin, M.E. Cates, J. Phys. I 7, 39 (1997). The physical mechanism that we model is the local splitting of the force chains (the characteristics of the hyperbolic equation) by packing defects. In analogy with the theory of light diffusion in a turbid medium, we propose a Boltzmann-like equation to describe these processes. We show that, for isotropic packings, the resulting large-scale effective equations for the stresses have exactly the same structure as those of an elastic body, despite the fact that no displacement field needs to be introduced at all. Correspondingly, the response function evolves from a two-peak structure at short scales to a broad hump at large scales. We find, however, that the Poisson ratio is anomalously large and incompatible with classical elasticity theory that requires the reference state to be thermodynamically stable. Received 13 November 2000 and Received in final form 3 January 2001     

Statistical analysis of the transmission based on the DMPK equation: an application to Pb nano-contacts

The density of the transmission eigenvalues of Pb nano-contacts has been estimated recently in mechanically controllable break-junction experiments. Motivated by these experimental analyses, here we study the evolution of the density of the transmission eigenvalues with the disorder strength and the number of channels supported by the ballistic constriction of a quantum point contact in the framework of the Dorokhov-Mello-Pereyra-Kumar equation. We find that the transmission density evolves rapidly into the density in the diffusive metallic regime as the number of channels N_c of the constriction increase. Therefore, the transmission density distribution for a few N_c channels comes close to the known bimodal density distribution in the metallic limit. This is in agreement with the experimental statistical-studies in Pb nano-contacts. For the two analyzed cases, we show that the experimental densities are seen to be well described by the corresponding theoretical results.     

Violation of finite-size scaling in three dimensions

We reexamine the range of validity of finite-size scaling in the lattice model and the field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the lattice model and the field theory in the region . The non-scaling effects in the field theory and in the lattice model differ significantly from each other. Received 5 February 1999