Granular Matter

"To see a world in a grain of sand, 
And a heaven in a wild flower, 
Hold infinity in the palm of your hand, 
     And eternity in an hour. "

- William Blake, Auguries of Innocence (1863)

Granular materials are collections of discrete macroscopic solid particles. They are everywhere in our everyday life and in nature. Sand, rice and nuts are just a few common examples, and even the rings of Saturn are composed of granular materials. Although granular material is ubiquitous, it shows unique behaviors different from ordinary phases of matter. First, the temperature T plays no role in granular materials. The typical energy scale determined by the potential energy of one granule is at least 1012 times kBT at room desert.jpgtemperature, therefore granular systems are always out-of-equilibrium. Furthermore, the interaction between the granules are highly dissipative and has a non-linear form.

Research on the granular materials has a long history. Coulomb, Faraday and Reynolds are the pioneers in this field among others. P.G. de Gennes first brought physicists attention into this originally engineering field. Since then, the granular physics has developed into a rich field. The industrial applications are also motivation for studying granular materials. Many of our industries rely on transporting and storing granular materials, so a comprehensive understanding of the properties of these materials can make us handle them more effectively. 

We study several projects relating to the glassy dynamics and flow behaviors of granular materials:

(1) Jamming transition of granular materials
(2) Packing of granular polymer chains
(3) Singularity and fractals in sand
(4) Impact of a granular jet: Emergence of a liquid with zero surface tension
(5) Probe 3D granular flow with magnetic resonance imaging (MRI)
(6) Bubbles in sand
(7) Brazil nut effect (Granular size separation)

Jamming of tapioca pearls

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Left: Jamming transition of automatically swelling granular particles. Right: Voronoi construction of a granular pack (Yellow cells for hexagons, blue cells for pentagons, red cells for heptagons. There is also one octagon (green) and one square (magenta).

Glasses return to the liquid state upon heating - they become soft and can flow. Sand flowing through a pipe or out of an orifice can easily jam and become rigid. Actually, the analogy between the glass transition of supercooled liquids and the jamming transition of granular materials is much deeper than what we thought intuitively based on their superficial similarities. The recently proposed jamming phase diagram provides an approach to unify studies in these two diverse fields (A. J. Liu and S. R. Nagel, Nature 396, 21 (1998)). Here, we experimentally investigate jamming in a quasi-two-dimensional granular system composed of automatically swelling particles — tapioca pearls. First, by analyzing the Voronoi structure of packs going through the jamming transition, we show that the local configuration of jamming packs is strikingly similar to that of glass-forming liquids, both in terms of their universal area distribution and process of defect annealing. Furthermore, we demonstrate that an unambiguous structural signature of the jamming transition can be obtained from the firt peak of the pair correlation functions. The same signature is also found in the second peak of the pair correlation function, but not in the third peak, reflecting the underlying singularity of the jamming transition. We also study the development of clusters in this system. A static length scale extracted from the cluster structure reaches the size of the system when the system approaches the jamming point. Finally, we show that in a highly inhomogeneous system, friction causes the system to jam in series of steps. In this case, jamming may be obtained through successive buckling of force chains. Our study provides insights into the structural properties of general jamming system.

  • Experimental study of the jamming transition at zero temperature, X. Cheng, Phys. Rev. E 81, 031301 (2010).
  • Packing structure of a two-dimensional granular system through the jamming transition, X. Cheng, Soft Matter 6, 2931 (2010).

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Packing of key chains

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X-ray tomography of packing of granular polymer chains

Rigid particles pack into structures, such as sand dunes on the beach, whose overall stability is determined by the average number of contacts between particles. However, when packing spatially extended objects with flexible shapes, additional concepts must be invoked to understand the stability of the resulting structure.  We examine the disordered packing of chains (ball-chains commonly used as key chains or window shade pulls) constructed out of flexibly connected hard spheres. Using x-ray tomography, we find that long chains pack into a low-density structure whose mechanical rigidity is mainly provided by the backbone. On compaction, randomly oriented, semi-rigid loops form along the chain, and the packing of chains can be understood as the jamming of these elements. Building on this result and also on the fact that polymer molecules are often modeled as flexible chains, we uncover close similarities between the packing of chains and the glass transition in polymers, which provides a beautiful illustration of how molecular geometry and symmetry, independent of the specific microscopic interactions, can influence the structure of condensed matter. Read more in the perspectives of Science and also in the APS Highlight.  

  • The Packing of Granular Polymer Chains, L.-N. Zou, X. Cheng, M. L. Rivers, H. M. Jaeger and S. R. Nagel, Science 326, 408 (2009).

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Singularity and fractals in sand

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Fractal formed when air (black) displacing
sand (grey) confined in a Hele-Shaw cell

The finger-like branching pattern that occurs when a less viscous fluid displaces a more viscous one confined between two parallel plates has been studied widely and with various normal fluids. We have investigated the granular analogue in such Hele–Shaw geometry, where air (the low-viscosity fluid) displaces glass beads (the high viscosity granular "fluid"). Because granular fluids composed of dry non-cohesive grains exhibit negligible surface tension, this allows us to explore a regime not accessible with ordinary fluids. We demonstrate that the grain–gas interface exhibits a fractal structure and sharp cusps, which are associated with the hitherto-unrealizable singular hydrodynamics predicted for the zero-surface-tension limit of normal fluid fingering. The scaling for the finger width is distinct from that for ordinary fluids, reflecting unique granular properties such as friction-induced dissipation as opposed to viscous damping. However, the fractal dimension of the fingering pattern and the shape of the singular cusps on the interface agree with the theories based on simple Laplacian growth of conventional fluid fingering in the zero-surface-tension limit. Read more in the press: Nature News.

  • Towards the zero-surface-tension limit in granular fingering instability, X. Cheng, L. Xu, A. Patterson, H. M. Jaeger and S. R. Nagel, Nature Phys. 4, 234 (2008).

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Impact of a granular jet: Emergence of a liquid with 
zero surface tension

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Images of granular jets impact on a solid target from high-speed photography 
showing typical particle behavior (left) and striking liquid-like behavior (right)

When one or two particles strike a smooth wall at normal incidence, they rebound in the direction whence they came. Yet, as we show here, a dense stream of non-cohesive particles hitting a target retains its integrity and deforms into a thin sheet with a shape resembling the structures created by an impinging liquid jet. However, with the granular materials, this “liquid” has special property of zero surface-tension. Furthermore, by decreasing the number of particles inside a jet, we can turn the behavior of the granular jet from the liquid-like behavior back to the normal particle-like behavior. Even more striking, this experiment can been seen as the classical analog of the much more microscopic high energy experiments done at RHIC (Relativistic Heavy Ion Collider) in the Brookhaven National Laboratory, where the quark-gluon plasma produced by the high-energy collisions of gold ions also shows the liquid-like scattering pattern, similar to what we observed with granular materials. Read more and watch movies in the press: UofC Chronicle and RHIC News.

  • Granular Jets as a Classical Analog of RHIC Collisions, X. Cheng, H. M. Jaeger and S. R. Nagel, RHIC News, January 15 (2008). http://www.bnl.gov/rhic/news/011508/story3.asp
  • Collective Behavior in a Granular Jet: Emergence of a Liquid with Zero Surface Tension, X. Cheng, G. Varas, D. Citron, H. M. Jaeger and S. R. Nagel, Phys. Rev. Lett. 99, 188001 (2007).

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Probe 3D granular flow with magnetic resonance imaging (MRI)

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MRI images of stationary pack (left) and granular flow under rotation (right)
Black lines in the images are created by spin tagging

Recent study of dense granular flow in a modified (split-bottomed) Couette Geometry brought a new insight into the concept of shear band in granular systems. The wide shear band away from any confining wall is crucial in understanding the behavior of the dense granular flow. However, till now the experimental results mainly focus on the flow on the surface due to the limit of traditional optical measurement. By employing the magnetic resonance imaging (MRI), we investigated the three-dimensional structure of the shear flow formed in this bottom-driven Couette cell. Combining with large-scale simulations, we showed that in spite of the presence of the vertical secondary flow in the steady state, the angular velocity profiles in horizontal plane still follow an error function as observed at the surface. The velocity scales of these two flows are well separated. The behavior of the shear band was investigated as a function of total filling height of the bulk (H). For each H, we mapped out the center and the width of the shear band at different heights in the bulk. We found that the shape of the shear band changes dramatically when its top detaches from the surface of the bulk. Our study provides the insights into the three-dimensional structure of complex granular flow.

  • Three-dimensional shear in granular flow, X. Cheng, J. B. Lechman, A. F. Barbero et al., Phys. Rev. Lett. 96, 038001 (2006).

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Bubbles in sand

Compaction of granular materials is of great importance in industries and attracts general interests in fundamental studies. We investigated the granular compaction by tapping a tilted tube filled with loosely compacted fine glass beads. We found that instead of uniformly compacted, the granular medium reduces its volume by forming a train of upward moving voids—the bed bubbles after taps! We found that the bubbling behavior of the granular bed is robust, i.e. the size and the velocity of bubbles are independent of the way of tapping. We investigated the properties of the bubbles as a function of the titled angle of the tube. A threshold angle related to the repose angle of the granular medium was found. Below this angle the bubbling behavior disappears. By changing the ambient pressure of the system, we found that the interstitial gas plays an essential role in generating bubbles. We proposed a simple mechanism for the phenomena by considering two curial factors—the avalanche of the glass beads and the presence of the interstitial gas. Movies of bubbling sand can be seen by following the links in the paper below.

  • Formation of air bubbles during compaction of a granular pack, X. Cheng, R. Smith, H. M. Jaeger and S. R. Nagel, Phys. Fluids 20, 123305 (2008).

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Brazil nut effect (Granular size separation)

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Rising of a larger intruder disk in a bed of smaller particles in a 2D cell

A big particle buried inside a bed of small particles tends to surface under vertical vibration. This phenomena is called Brazil Nut Effect (BNE), since Brazil nuts as big particles are always found at the surface when you open a can of mixed nuts. Although this effect was reported as early as 1930s, its mechanism is still not very clear. Recently, a group of scientists found that at certain special circumstances a big particle can also sink to the bottom of a bed (reverse BNE). Moreover, the rising speed of big particles is also non-monotonic: both light and heavy particles rise faster than particles of medium density. Using different techniques such as magnetic resonance imaging (MRI) and high speed photography, we studied this effect systematically. We mapped out a phase diagram which separates rising and sinking regime of big particles as a function of several control parameters. We clearly show the important effect of interstitial gas on BNE and reverse BNE. A simple model was proposed to quantitatively explain our finding.

  • The Effect of Air on Granular Size Separation in a Vibrated Granular Bed, M. E. Möbius, X. Cheng, P. Eshuis, G. S. Karczmar, S. R. Nagel and H. M. Jaeger, Phys. Rev. E 72, 011304 (2005).
  • Intruders in the Dust: Air-Driven Granular Size Separation, M.E. Möbius, X. Cheng, G. S. Karczmar, S. R. Nagel and H. M. Jaeger, Phys. Rev. Lett. 93, 198001 (2004). 

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