Table of contents

Granular materials are of great importance to many industrial and natural processes, and thus attract considerable attention over a broad range of disciplines, from engineering to chemical and physical sciences. As their physical properties are different from known thermal systems, they also raise deep basic scientific issues.

This special issue of Journal of Physics: Condensed Matter collects together a group of papers from leading researchers in the physics community which discuss recent new findings and ideas introduced in this rapidly evolving field. Although many topics are still not fully developed, the last decade has witnessed an increasing understanding of a variety of theoretical and experimental issues, ranging from the properties of granular packing, stress distribution and force networks to the dynamics and rheology of granular flow, instabilities and jamming; from the statistical mechanics of granular packs, fluids and gases to applications to problems of geophysical and industrial relevance.

The papers in this issue provide an introduction to these topics, summarizing the state-of-the-art and addressing some of the many, still open, questions by featuring an up-to-date presentation of new developments. A deeper test of the theories discussed here and an understanding of the new experimental results reported are certainly among the relevant open research directions ahead in this field.

I wish to thank all the contributing authors to this special issue of Journal of Physics: Condensed Matter devoted to granular media.

The most suitable paradigms and tools for investigating the structure of granular packings are reviewed and discussed in the light of some recent empirical results. I examine the 'typical' local configurations, their relative occurrences, their correlations, their organization and the resulting overall hierarchical structure. The analysis of very large samples of monosized spheres packed in a disorderly fashion, with packing fractions ρ ranging from 0.58 to 0.64, demonstrates the existence of clear relations between the geometrical structure and the packing density ρ. It is observed that the local Delaunay and Voronoï volumes have distributions which decay exponentially at large volumes, and have characteristic coefficients proportional to ρ. The average number of contacts per sphere grows linearly with ρ. The topological density increases with ρ and it is larger than the corresponding one for lattice sphere packing. The heights and the shapes of the peaks in the radial distribution function also depend on ρ. Moreover, the study of several other quantities, such as the dihedral angle distribution, the fractions of four-, five-and six-membered rings and the shape of the Voronoï cells, shows that the local organization has characteristic patterns which depend on the packing density. All the empirical evidence excludes the possibility of the presence of any crystalline order, even at the minimal correlation length (one bead diameter). Moreover, icosahedral order and other close packed configurations have no statistical significance, excluding therefore the possibility that geometrical frustration could play any significant role in the formation of such amorphous structures. On the other hand, the analysis of the local geometrical 'caging' discloses that, at densities above 0.601, the system can no longer explore the phase space by means of local moves only and it becomes trapped either in the basin of attractions of inherent states, with limiting densities in the range 0.61–0.64, or in the crystalline branch.

A granular assembly composed of a collection of identical grains may pack under different microscopic configurations with microscopic features that are sensitive to the preparation history. A given configuration may also change in response to external actions such as compression and shearing. We show, using a mechanical response function method developed experimentally and numerically, that the macroscopic stress profiles are strongly dependent on these preparation procedures. These results were obtained for both two and three dimensions. The method reveals that, under a given preparation history, the macroscopic symmetries of the granular material is affected, and in most cases significant departures from isotropy should be observed. This suggests a new path towards a non-intrusive test of granular material constitutive properties.

In this paper we discuss whether thermodynamical concepts and in particular the notion of temperature could be relevant for the dynamics of granular systems. We briefly review how a temperature-like quantity can be defined and measured in granular media in very different regimes, namely the glassy-like, the liquid-like and the granular gas. The common denominator will be given by the fluctuation–dissipation theorem, whose validity is explored by means of both numerical and experimental techniques. It turns out that, although a definition of a temperature is possible in all cases, its interpretation is far from being obvious. We discuss the possible perspectives both from the theoretical and, more importantly, from the experimental point of view.

We present a biased review of some of the most 'spectacular' effects appearing in the dynamics of granular gases where the dissipative nature of the collisions leads to a rich phenomenology, exhibiting striking differences with equilibrium gases. Among these differences, the focus here is on the illustrative examples of the 'Maxwell demon'-like experiment, modification of Fourier's law, non-equipartition of energy and non-Gaussianity of the velocity distributions. The presentation remains as non-technical as possible.

The break-up of a two-dimensional circular disc by normal and oblique impact on a hard frictionless plate is investigated by molecular dynamics simulations. The disc is composed of numerous unbreakable randomly shaped convex polygons connected together by simple elastic beams that break when bent or stretched beyond a certain limit. It is found that for both normal and oblique impacts the crack patterns are the same and depend solely on the normal component of the impact velocity. Analysing the pattern of breakage, amount of damage, fragment masses and velocities, we show the existence of a critical velocity which separates two regimes of the impact process: below the critical point only a damage cone is formed at the impact site (damage), cleaving of the particle occurs at the critical point, while above the critical velocity the disc breaks into several pieces (fragmentation). In the limit of very high impact velocities the disc suffers complete disintegration (shattering) into many small fragments. In agreement with experimental results, fragment masses are found to follow the Gates–Gaudin–Schuhmann distribution (power law) with an exponent independent of the velocity and angle of impact. The velocity distribution of fragments exhibits an interesting anomalous scaling behaviour when changing the impact velocity and the size of the disc.

We present experimental and numerical results on 2D and 3D confined granular chute flows. We address the issue of the role of the lateral boundaries. In particular, we find that the presence of flat frictional lateral walls greatly alters the flow features as soon as the width of the flowing layer is of the order of the spacing between the walls or greater. First, steady and fully developed (SFD) flows are observed up to very large inclination angles where accelerated flows would have been expected. Second, at given inclination angle, there exists an upper bound on the flow rate for SFD flows to occur. When one approaches this critical flow rate, a static heap forms along the chute base, on which is the flowing layer. The heap is stabilized by the flow atop it and was named a sidewall-stabilized heap (SSH) since its angle is much greater than those usually exhibited by granular heaps. Both kinds of flow have been studied in 2D and 3D confined configurations. In particular, it is found that these flows exhibit either a Bagnold velocity profile or an exponential one. Moreover, we identify a dimensionless parameter, depending crucially on the sidewall friction, that is expected to drive the transition between these two regimes. We also point out the differences between purely 2D flows and 3D confined flows.

Model experiments are reported on the build-up of granular piles in two dimensions. These show that, as the initial density of falling grains is increased, the resulting pile has decreasing final density and its coordination number approaches the low value predicted for the theoretical marginal rigidity state. This provides the first direct experimental evidence for this state of granular matter. We trace the decrease in the coordination number to the dynamics within an advancing yield front between the consolidated pile and the falling grains. We show that the front's size diverges as the marginal rigidity state is approached, suggesting a critical phenomenon.

It is shown that the hydrodynamic modes of a dilute granular gas of inelastic hard spheres can be identified, and calculated in the long wavelength limit. Assuming they dominate at long times, formal expressions for the Navier–Stokes transport coefficients are derived. They can be expressed in a form that generalizes the Green–Kubo relations for molecular systems, and it is shown that they can also be evaluated by means of N-particle simulation methods. The form of the hydrodynamic modes to zeroth order in the gradients is used to detect the presence of inherent velocity correlations in the homogeneous cooling state, even in the low density limit. They manifest themselves in the fluctuations of the total energy of the system. The theoretical predictions are shown to be in agreement with molecular dynamics simulations. Relevant related questions deserving further attention are pointed out.

We present different experiments on dense granular assemblies with the aim of clarifying the notion of 'jamming transition' for these assemblies of non-Brownian particles. The experimental set-ups differ in the way in which external perturbations are applied in order to unjam the systems. The first experiment monitors the response to a locally applied deformation of a model packing at rest. The two other experiments study local and collective dynamics in a granular assembly weakly excited by vibration.

Granulation is a process whereby a dense colloidal suspension is converted into pasty granules (surrounded by air) by application of shear. Central to the stability of the granules is the capillary force arising from the interfacial tension between solvent and air. This force appears capable of maintaining a granule in a jammed solid state, under conditions where the same amount of solvent and colloid could also exist as a flowable droplet. We argue that in the early stages of granulation the physics of dilatancy, which requires that a powder expand on shearing, is converted by capillary forces into the physics of arrest. Using a schematic model of colloidal arrest under stress, we speculate upon various jamming and granulation scenarios. Some preliminary experimental results on aspects of granulation in hard-sphere colloidal suspensions are also presented.

We measure the flow of granular materials inside a quasi-two-dimensional silo as it drains and compare the data with some existing models. The particles inside the silo are imaged and tracked with unprecedented resolution in both space and time to obtain their velocity and diffusion properties. The data obtained by varying the orifice width and the hopper angle allow us to thoroughly test models of gravity driven flows inside these geometries. All of our measured velocity profiles are smooth and free of the shock-like discontinuities ('rupture zones') predicted by critical state soil mechanics. On the other hand, we find that the simple kinematic model accurately captures the mean velocity profile near the orifice, although it fails to describe the rapid transition to plug flow far away from the orifice. The measured diffusion length b, the only free parameter in the model, is not constant as usually assumed, but increases with both the height above the orifice and the angle of the hopper. We discuss improvements to the model to account for the differences. From our data, we also directly measure the diffusion of the particles and find it to be significantly less than predicted by the void model, which provides the classical microscopic derivation of the kinematic model in terms of diffusing voids in the packing. However, the experimental data are consistent with the recently proposed spot model, based on a simple mechanism for cooperative diffusion. Finally, we discuss the flow rate as a function of the orifice width and hopper angles. We find that the flow rate scales with the orifice size to the power of 1.5, consistent with dimensional analysis. Interestingly, the flow rate increases when the funnel angle is increased.

We study the segregation process of a granular mixture of discs on an horizontally oscillating tray by realistic molecular dynamics simulations. As found in the experiments, we observe the initially disordered mixture to segregate via the formation of stripes perpendicular to the driving direction. The segregation process turns out to be the result of a dynamical instability very similar to the Kelvin–Helmholtz instability observed in fluid mechanics, and it is not due to the depletion potential, i.e., to the effective potential existing between two big particles in a bath of smaller ones, but rather to a dynamical process whose nature is discussed.

We discuss some recent results on the statistical mechanics approach to dense granular media. In particular, by analytical mean field investigation we derive the phase diagram of monodisperse and bidisperse granular assemblies. We show that 'jamming' corresponds to a phase transition from a 'fluid' to a 'glassy' phase, observed when crystallization is avoided. The nature of such a 'glassy' phase turns out to be the same as found in mean field models for glass formers. This gives quantitative evidence for the idea of a unified description of the 'jamming' transition in granular media and thermal systems, such as glasses. We also discuss mixing/segregation transitions in binary mixtures and their connections to phase separation and 'geometric' effects.

The zero-temperature dynamics of simple models such as Ising ferromagnets provides, as an alternative to the mean-field situation, interesting examples of dynamical systems with many attractors (absorbing configurations, blocked configurations, zero-temperature metastable states). After a brief review of metastability in the mean-field ferromagnet and of the droplet picture, we focus our attention onto zero-temperature single-spin-flip dynamics of ferromagnetic Ising models. The situations leading to metastability are characterized. The statistics and the spatial structure of the attractors thus obtained are investigated, and put in perspective with uniform a priori ensembles. We review the vast amount of exact results available in one dimension, and present original results on the square and honeycomb lattices.

One of the goals of the present paper is to present a brief, and admittedly somewhat biased, review of some recent theoretical advances in the field of granular gases. Another goal is to highlight some challenges facing this field. A third goal is to present some new results concerning the Chapman–Enskog expansion. These include an extension to weakly frictional granular gases, and a study of a stationary granular gas subject to a heat flux (an earlier study describing a stationary sheared granular gas is presented as well). In addition, a computer-aided method for inverting the linearized Boltzmann operator, which should be useful for kinetic theorists, is presented in an appendix. A further goal is to briefly discuss possible extensions beyond Boltzmann kinetics, as well as beyond hydrodynamics, i.e., the moderately dense regime on one hand, and the Knudsen regime on the other hand. An attempt is made throughout this paper to put granular gases in a general context, by distinguishing between those features that are common to granular and molecular gases and those that are not. Also, stress is put on the physics (and methods) rather than phenomena. While this is not a review article, parts of it are intended for the uninitiated in the field.

This paper reports on the behaviour of uniform granular particles of diameter d undergoing a gravity-free shear flow induced through parallel bumpy boundaries that move in opposite directions at constant velocity U. Non-equilibrium, discrete simulations are performed, in which particles are modelled as inelastic, frictional spheres. The flow is described using steady state profiles of mean velocity, granular temperature, solids fraction and normal pressure. A non-uniform local shear rate, characterized by an S-shaped mean velocity profile, produces an imbalance in the contact distribution of particles in the vicinity of the walls so that they drift toward the geometric centre of the flow. A spectral analysis of the transverse velocity provides evidence of convective cell structures in the secondary velocity field whose wavelength decreases with the effective shear rate. A typical tracer particle, whose trajectory has a power spectrum suggestive of persistent fractional Brownian motion, continually samples the entire shear region. In contrast, a large intruder , which also migrates away from energetic regions adjacent to the walls, eventually becomes trapped near the mid-plane of the flow with a speed that increases with ϕ and the effective shear rate . Fluctuations in the evolution of its transverse velocity component obey a power law of the form V_y^rms = Cϕ^−a, which is a consequence of its greater inertia, while fluctuations in its net force vary directly with ϕ.

The modelling of cohesive, frictional granular materials with a discrete particle molecular dynamics is reviewed. From the structure of the quasi-static granular solid, the fabric, stress, and stiffness tensors are determined, including both normal and tangential forces. The influence of the material properties on the flow behaviour is also reported, including relations between the microscopic attractive force and the macroscopic cohesion as well as the dependence of the macroscopic friction on the microscopic contact friction coefficient. Related to the dynamics, the anisotropy of both structure and stress are exponentially approaching the maximum.

This contribution considers recent developments in understanding the behaviour of vibro-fluidized beads in containers partitioned into several connected chambers. The system is studied theoretically by means of a phenomenological mean-field approach. The equations governing the evolution of the average occupancy and the average kinetic energy of each compartment are derived by means of a simplified treatment of the Boltzmann equation.

Some applications of the method are presented. These include the study of a simple granular gas in a many-compartment container and a binary granular mixture in a two-compartment container.

In this paper, we review some of our approaches to granular dynamics, now well known (Mehta 1994 Granular Matter: an Interdisciplinary Approach ed A Mehta (New York: Springer)) to consist of both fast and slow relaxational processes. In the first case, grains typically compete with each other, while in the second, they cooperate. A typical result of cooperation is the formation of stable bridges, signatures of spatiotemporal inhomogeneities; we review their geometrical characteristics and compare theoretical results with those of independent simulations. Cooperative excitations due to local density fluctuations are also responsible for relaxation at the angle of repose; the competition between these fluctuations and external driving forces can, on the other hand, result in a (rare) collapse of the sandpile to the horizontal. Both these features are present in a theory reviewed here. An arena where the effects of cooperation versus competition are felt most keenly is granular compaction; we review here a random graph model, where three-spin interactions are used to model compaction under tapping. The compaction curve shows distinct regions where 'fast' and 'slow' dynamics apply, separated by what we have called the single-particle relaxation threshold. In the final section of this paper, we explore the effect of shape—jagged versus regular—on the compaction of packings near their jamming limit. One of our major results is an entropic landscape that, while microscopically rough, manifests Edwards' flatness at a macroscopic level. Another major result is that of surface intermittency under low-intensity shaking.

We describe a series of experiments and computer simulations on vibrated granular media in a geometry chosen to eliminate gravitationally induced settling. The system consists of a collection of identical spherical particles on a horizontal plate vibrating vertically, with or without a confining lid. Previously reported results are reviewed, including the observation of homogeneous, disordered liquid-like states, an instability to a 'collapse' of motionless spheres on a perfect hexagonal lattice, and a fluctuating, hexagonally ordered state. In the presence of a confining lid we see a variety of solid phases at high densities and relatively high vibration amplitudes, several of which are reported for the first time in this article. The phase behaviour of the system is closely related to that observed in confined hard-sphere colloidal suspensions in equilibrium, but with modifications due to the effects of the forcing and dissipation. We also review measurements of velocity distributions, which range from Maxwellian to strongly non-Maxwellian depending on the experimental parameter values. We describe measurements of spatial velocity correlations that show a clear dependence on the mechanism of energy injection. We also report new measurements of the velocity autocorrelation function in the granular layer and show that increased inelasticity leads to enhanced particle self-diffusion.

The most striking phenomenon in the dynamics of granular gases is the formation of clusters and other structures. We investigate a gas of dissipatively colliding particles with a velocity dependent coefficient of restitution where cluster formation occurs as a transient phenomenon. Although for small impact velocity the particles collide elastically, surprisingly the temperature converges to zero.

We review some models of granular materials fluidized by means of external forces, such as random homogeneous forcing with damping, vibrating plates, flow in an inclined channel and flow in a double well potential. All these systems show the presence of density correlations and non-Gaussian velocity distributions. These models are useful in understanding the role of a kinetically defined 'temperature' (in this case the so-called granular temperature) in a non-equilibrium stationary state. In the homogeneously randomly driven gas the granular temperature is different from that of the driving bath. Moreover, two different granular materials mixed together may stay in a stationary state with different temperatures. At the same time, the granular temperature determines (as in equilibrium systems) the escape time in a double well potential.

In the first part we present experimental results concerning the flow of a densely packed grain collection down a two-dimensional inclined channel. For the range of inclinations corresponding to a steady, uniform regime and to nonsliding conditions at the bottom, we obtain quasi-linear profiles of velocity, that are in contradiction with the predictions of the kinetic theory. We attribute this discrepancy to the inadequacy of the binary collision picture in the case of dense packings. We also show that the various velocity profiles obtained for different flow rates and slopes merge onto a single master curve, according to the following law: (d being the grain diameter, θ the channel inclination angle and θ_c the maximal angle of repose), provided that the regime is steady and uniform. Arguing that continuous paths of transient contacts are effective for transporting momentum and energy through the bulk, and that the associated dissipation time is very short compared to the time associated with shearing, we succeed in explaining this scaling behaviour and the paradoxical nonzero shear rate in the vicinity of the free surface. We also show that for dense particulate flows, the dissipation is mainly due to frictional sliding.

In the second part, we emphasize some remarkable features exhibited by dry grain avalanches in laboratory experiments. According to the slope angle, the rear front propagates either upwards or downwards, with velocity approximately equal to the depth averaged velocity of the avalanche. As a counterpart, in both regimes, the velocity magnitude of the head front remains of the order of twice the depth averaged avalanche velocity. We suggest simple elementary mechanisms capable of accounting for these observations. We propose then an analytical modelling aimed at describing the combined processes governing the avalanche expansion. The two solutions that we obtain for the growth regimes and for the avalanche shapes resemble very closely the observations made in the laboratory and in the field.

The dynamics of granular compaction under vertical tapping is experimentally and numerically studied for isotropic and anisotropic grains. It is shown that if convection takes place in the medium, it has a strong influence on the compaction dynamics. The local behaviour of the grains during compaction and the consequences for the global properties of the medium are investigated.

Jamming is a common feature of out-of-equilibrium systems showing slow relaxation dynamics. Here we review our efforts in understanding jamming in granular materials using experiments and computer simulations. We first obtain an estimation of an effective temperature for a slowly sheared granular material very close to jamming. The measurement of the effective temperature is realized in the laboratory by slowly shearing a closely packed ensemble of spherical beads confined by an external pressure in a Couette geometry. All the probe particles, independent of their characteristic features, equilibrate at the same temperature, given by the packing density of the system. This suggests that the effective temperature is a state variable for the nearly jammed system. Then we investigate numerically whether the effective temperature can be obtained from a flat average over the jammed configuration at a given energy in the granular packing, as postulated by the thermodynamic approach to grains.