Granular Materials

 

Introduction    Major results    Collaborations     Publications  

Introduction

Dry granular materials are very simple: they are large conglomerations of discrete macroscopic particles. If they are non-cohesive, then the forces between them are essentially only repulsive so that the shape of the material is determined by external boundaries and gravity. 

Yet despite this seeming simplicity, granular materials behave differently from any of the other standard and familiar forms of matter : solids, liquids or gases, and should therefore be considered as an additional state of matter in its own right. At the root of this unique status are three important aspects: the existence of static friction, the fact that temperature is effectively zero and, for moving grains, the inelastic nature of their collisions. In some cases, such as a sandpile at rest with a slope less than the angle of repose, static friction produces solid-like behavior: the material remains at rest even though gravitational forces create macroscopic stresses on its surface. If the pile is tilted several degrees above the angle of repose grains start to flow, like in a fluid.

However, this flow is clearly not that of an ordinary fluid because it only exists in a boundary layer at the pile's surface. Also, unlike in an ordinary fluid, kT plays no role in a granular material and entropy considerations can easily be outweighted by dynamical effects that now become of paramount importance. Unless perturbed by external disturbances, each metastable configuration of the material will last indefinitely, and no thermal averaging over nearby configurations will take place. Because each configuration has its unique properties, the reproducibility of granular behavior, even on large scales and certainly near the static limit, can only be defined in terms of ensemble averages.

In order to better understand the dynamical and statistical properties of granular materials, we have investigated surface flows in a rotating drum, then studied the response of a granular pile to a localised surface disturbance. We have also considered the relaxation of a granular pile, following the occurence of an avalanche. Then we concentrated on the so-called jamming transition and the glassy behaviours of granular media. This has lead us to investigate the statistics of the free volumes inside a static pile. Finally, we have looked at some applications to dunes dynamics and industrial powders flows.

 

Major results
Surface flows

Surface flow on a granular pile inside a rotating drum
We investigate steady granular surface flows in a rotating drum, measuring the velocity profile in the flowing layer at the center of the drum, the flowing layer thickness and the static/flowing boundary profiles. The velocity varies linearly with depth, with a gradient independent of both the flowing layer thickness and the static/flowing boundary local slope. Then, writing the depth averaged conservation equations for granular surface flows, it was possible to find experimentally the constitutive relations needed to close these equations. Altogether it has brought the evidence that the relation between stress and strain is non local: 
(i) the velocity gradient is found to be constant in the flowing layer whereas momentum balance predicts a linear variation of the shear stress with depth;

(ii) the velocity gradient presents a different scaling with the depth for dense granular flows down a rough inclined plane indicating the non local influence of boundary conditions on internal rheology inside the flowing layer;

(iii) the velocity gradient does not vanish at the free surface at variance with typical fluids.

Measuring correlations among moving grains in the flowing layer, we could observe the first experimental evidence of rigid clusters of grains embedded in the flow and characterize their geometrical and statistical properties.. Although clustering instabilities driven by the inelasticity of grain collision are well known in granular gases, they have never been observed in dense surface flows. We find that these clusters are fractal and their size is power-law distributed from the microscopic scale - the diameter of a grain - up to the macroscopic scale - the flowing layer thickness. Therefore, no characteristic correlation length can be defined in the flowing layer. These results clearly call for further work in modelizing non-local effects in granular flows. A collective paper on the subject of granular flows has been written by the participants to the GDR Milieux Divisés.

Figure:  Presence of rigid clusters of beads in the flowing layer of 2D flows
Relaxation and response to a localized perturbation

A number of experimental works have studied surface instability of a granular heap, namely avalanches, and have led to the conclusion that such avalanches do not behave in a critical manner at all. In recent years, it has also been suggested that the jamming transition of granular materials could be analog to a glass transition. Further works on mean field glass models have developed this analogy and tried to unify concepts that had emerged in both fields, such as the dynamical temperature defined from fluctuation dissipation relations, and that related to Edwards' statistical ensemble.

First, the micro-displacements generated by a small localized overload at the free surface are visualized experimentally inside a packing of steel beads. For a triangular packing, beads rearrangements remain confined in two inverted triangles on both sides of the applied overload. This pattern disappears for stronger disorder. A simple model allows us to account for these observations and to relate them to the stress function response measured via photoelastic visualizations.

We have then reconsidered the avalanching pile problem but, instead of focusing on avalanches statistics, we have studied the relaxation of the pile towards mechanical equilibrium following an avalanche. We first have shown that the dynamics exhibits non-trivial relaxations and is much slower than expected given the  "microscopic" time-scale. We observed intermittent bursts which reactivate the pile activity and identified them with correlated movements in space and time.
Mobile volume fraction evolution for two different realizations at similar pile slope : (o) q=15°; (+) q=16.5°. Inset is the log-lin plot of the same data. Displacements in the pile at time step (a) and (b). The dark pixels corresponds to positions where a displacement has occurred. The red overlay indicates temporally correlated displacements. 

Perturbation experiments have allowed us to confirm the emergence of strong spatial correlations when increasing the pile slope, which are reminiscent of the clusters that we observed in flows. Finally we have extracted in the unperturbed case a two-time relaxation function, which can be shown to rescale in an aging like manner with a `reparametrization' of time suggested by a simple model that we introduce in order to describe our experimental data. Moreover, this model allows to relate the above relaxation function to a two-time correlation function which shares strong similarities with auto-correlation functions appearing in aging systems.

 
Glassy behaviour of a bidimensional granular media


Granular media, commonly refered to as a-thermal systems, obey a dissipative dynamics a priori very different from an Hamiltonian evolution. However everyday life and recent experiments suggest that a thermodynamical description of granular media might be feasible. Especially in the context of gentle compaction of grains, strong similarities with the behaviour of thermal glassy systems have been underlined.

We study the microscopic behaviour of granular materials under cyclic shear. The goal of these experiments is to find a microscopic ground for these similarities evidenced in previous studies.

Following tracers, we show that the particle motion is subdiffusive and that the trajectories exhibit strong cage effect of the kind observed in colloidal glasses. By direct visual observations, we investigate the cage properties (size, elapsed time in a cage...) and relate them to the global properties of the material. Further statistical characterization of the motion are studied in an attempt to identify relevant lengthscales and ultimately the physical nature of a ”cage”. More specifically, it was possible following all particles to compute four-points correlation functions, the behaviour of which allows to identify a dynamical lengthscale. Also the experiment allows both temporal and spatial resolution so that we could observe the spatial heterogeneities of the time relaxation.

On the left : prototype of the experimental set up. On the right (top): Cage trajectories at different amplitude of shear cycles. On the right (bottom): Diffusion properties.
wpeD.jpg (34853 octets) wpeC.jpg (31362 octets)
On the left : dynamical structure factor for different wave vectors; On the right : dynamical susceptibility

 

 Field of the temporal decorrelation and evidence of the dynamical heterogeneities
Statistics of free volumes inside a static pile


Given the above similarities between granular media and glass formers, and given that granular media consist in a large number of grains, there is a strong motivation for providing a statistical ground to a  thermodynamical description. It has been argued by Edwards and collaborators that the dynamics is controlled by the mechanically stable — the so-called blocked — configurations and that all such configurations of a given volume are statistically equivalent This immediately leads to the definition of a configurational entropy and the associated state variable, the compactivity, the formal analogy of a temperature. First attempts to test this flat measure assumption have been conducted. However, clear evidence in real granular media is still lacking.

We study the full probability distribution of the volume, not only at the scale of the packing but for subsystems of increasing sizes. Apart from the expected exponential weight, most of the information about the system lies in the entropic pre-factor dependance. We have shown that the specific free volume per particule inside a cluster of size N follows a Gamma-law distribution, which allow the proper definition of two intensive parameters. However there is strong evidence of non extensivity suggesting long range correlations.
wpe12.jpg (39660 octets)

Free volume statistics. (a): distributions of the free volume per grain inside clusters of N grains; the larger N, the narrower the distribution. (b): dependence on N of the first (top) and second (bottom) moments of the free volume distributions.
 

Applications: dunes dynamics and metallic powders


Dunes dynamics has strong impact on the ecology and the economy of sandy areas, but is far from being understood. Since the work on sand dune formation by R.A. Bagnold, a world wide inventory of dunes has been developed. The most common dune, the so-called barchan, has a typical crescent shape normal to the wind, with arms downwind and forms under mono-directional winds. Various models call for more experimental data. On one hand, field measurements are difficult to perform and often incomplete. On the other hand, it is believed that dunes have a minimal size of the order of one meter, not reducible to smaller laboratory scales. We have set up a wind channel where to conduct experiments with sand and we have shown experimentally that an initial sand pile, under a wind flow charged in sand, flattens and exhibits a typical barchan shape before disappearing. An evolution law has been proposed for the profile and the summit of the dune. The dune dynamics is shape invariant. The invariant shape, the "dune function", was isolated. Our results, which clearly demonstrates the feasibility of dunes in the lab, open a new way of investigations in desert studies.

Figure :Evolution of a sand pile towards a "barchan" dune when submitted to a wind saturated in sand
In collaboration with CEA/Grenoble (DRT/DTEN), we have studied problems related to metallic powder flows in industrial processes. In particular, the filling of containers must be reproducible, homogenous and as fast as possible while the powders consist in complex mixing of different components. We have started characterizing the behaviour of model powders using a rotating drum and flows in cylindrical tubes. Though the powders are very similar (composition, granulometry), very different behaviors can be observed.

Figure : Two very different behaviors after avalanche for almost identical powders. (a): static slopes; (b): relaxation front

            

Collaborations

Publications and Contributions to Scientific Meetings

You will find here a selection of our papers in the field,

  1. D. Bonamy, F. Daviaud, L. Laurent, M. Bonetti, J.P. Bouchaud, "Multi-scale clustering in granular surface flows", Phys. Rev. Lett. 89, 034301 2002. (pdf)
  2. D. Bonamy, F. Daviaud, L. Laurent, "Experimental study of granular surface flows via a fast camera: a continuous description", Phys. Fluids 14, 1666-1673, 2002  (pdf).
  3. D. Bonamy, S. Bernard-Bernardet, F. Daviaud, L. Laurent, "Micro-displacements induced by a local perturbation inside a granular packing", Phys. Rev E, (pdf)
  4. S. Deboeuf, E. Bertin, E. Lajeunesse, O. Dauchot, "Jamming transition of a granular pile below the angle of repose", accepted in Eur. Phys. J. B (pdf)
  5. O. Dauchot, F. Léchenault, C. Gasquet, F. Daviaud, "Barchan dunes in the lab", C. R. Mecanique 330, 185-191, 2002 (pdf).
  6. GDR Midi, a collective work, "On dense granular flows" EPJ E 14, 341-365, 2004 (pdf)
  7. E. Bertin, O. Dauchot & M. Droz, "Temperature in Nonequilibrium systems with conserved energy", PRL 93, 230601, 2004 (pdf)
  8. E. Bertin, O. Dauchot & M. Droz, "Non-equilibrium temperatures in steady-state systems with conserved energy", PRE, condmat/0412071 (pdf)
  9. G. Marty & O. Dauchot, "Subdiffusion and Cage Effect in a Sheared Granular Material", PRL 94, 015701, 2005 (pdf)
  10. O. Dauchot, G. Marty and G. Biroli, "Dynamical Heterogeneity close tot he jamming transition in a sheared granular material", accepted in PRL, 2005 (pdf)
  11. S. Deboeuf, O. Dauchot, L. Staron, A. Mangeney & J.P. Vilotte. "Memory of the Unjamming transition during cyclic tilting of a granular pile", accepted in Phys Rev E., 2005 condmat/00005480 (pdf)
  12. F. da Cruz, F. Lechenault, O. Dauchot and E. Bertin "Free volume distributions inside a bidimensional granular medium", in Proceedings of Powders and Grains, Stuttgart, 2005. (pdf)
  13. For a review see the Lecture Notes : O. Dauchot, "Glassy behaviours in a-thermal systems, the case of granular media", Luxembourg University, 2005, (pdf).

as well as pdf files of a selection of oral presentations

  1. Jamming transition of a granular pile. "A first step towards athermal glasses", UCGMG, 2003, ITALY (pdf).
  2. Clusters in granular flows. "Elements of a non-local rheology", University of Geneva, 2003, SUISSE (pdf)
  3. Subdiffusion and Cage effect, Oxford Meeting, 2004 (pdf)

  4. For a Review : Lecture in Luxembourg "Glassy behaviours in a-thermal systems, the case of granular media", 2005 Luxembourg (pdf)