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Resource Letter GP-1: Granular physics or nonlinear dynamics in a sandbox
James Kakalios
Citation: American Journal of Physics 73, 8 (2005); doi: 10.1119/1.1810154
View online: http://dx.doi.org/10.1119/1.1810154
View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/73/1?ver=pdfcov
Published by the American Association of Physics Teachers
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Submerged‐Flow Dynamics of Granular Media in a Rotating Drum
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Understanding the breakdown of Fourier’s law in granular fluids
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Transient spiral and target patterns in vertically oscillated thin granular layers
AIP Conf. Proc. 501, 43 (2000); 10.1063/1.59949
Contact forces in a granular packing
Chaos 9, 544 (1999); 10.1063/1.166428
Resource Letter: ND-1: Nonlinear Dynamics
Am. J. Phys. 65, 822 (1997); 10.1119/1.18544
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RESOURCE LETTER
Roger H. Stuewer, Editor
School of Physics and Astronomy, 116 Church Street SE,
University of Minnesota, Minneapolis, Minnesota 55455
This is one of a series of Resource Letters on different topics intended to guide college physicists,
astronomers, and other scientists to some of the literature and other teaching aids that may help
improve course content in specified fields. 关The letter E after an item indicates elementary level or
material of general interest to persons becoming informed in the field. The letter I, for intermediate
level, indicates material of somewhat more specialized nature; and the letter A indicates rather
specialized or advanced material.兴 No Resource Letter is meant to be exhaustive and complete; in time
there may be more than one letter on some of the main subjects of interest. Comments on these
materials as well as suggestions for future topics will be welcomed. Please send such communications
to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School of Physics and Astronomy,
University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455; e-mail:
rstuewer@physics.umn.edu.
Resource Letter GP-1: Granular physics or nonlinear dynamics
in a sandbox
James Kakaliosa)
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455
共Received 25 June 2003; accepted 3 September 2004兲
This Resource Letter provides a guide to the literature on the statics and dynamics of granular
media. Journal article and book references are provided for the following topics: Packing, Angle of
Repose, Avalanches and Granular Flow, Hoppers and Jamming, Vertically Vibrated Induced
Phenomena, Avalanche Stratification, and Axial Segregation. © 2005 American Association of Physics
Teachers.
关DOI: 10.1119/1.1810154兴
I. INTRODUCTION
Within every boulder, a grain of sand lies waiting. Large
rocks are lifted to the tops of distant mountains through plate
tectonics, where vegetation and erosion break them into
smaller and smaller pieces. Gravity pulls them down into
valleys, where water runoff from rain or melting snow carries the smaller particles 共the size distribution can range from
meters to fine silt兲 toward the ocean.1 A given grain of sand
takes approximately 10 000 years to be transported one mile
closer to the ocean. Smaller particles are easier to move, so
that the average size of the grains continually decreases in
the river currents, until at the ocean’s edge there is a fairly
monodisperse size distribution of sand grains. Granular material extends underwater approximately 100 miles from the
water–sand interface 共termed the beach兲 before oceanic pressure compresses the granular material back into solid rock.
At that point plate tectonics results in either subduction with
consequent melting of the material into its constituent elements, or uplifting during the formation of new mountain
ranges. In the latter case the entire cycle may then be repeated. If a complete cycle requires 100 million years, then
certain grains of sand on your favorite beach may have undergone several such round trips to bring them to their
present state.1,2
A typical grain of sand has a diameter ranging from 1 mm
to 400 ␮m. According to Brown and Richards,3 ‘‘granular
media’’ consist of discrete solids in direct physical contact
most of the time, as compared to slurries, fluidized beds and
a兲
Electronic mail: Kakalios@umn.edu
suspensions. Granular material of diameter less than 100 ␮m
are referred to as powders, while those of diameter less than
10 ␮m are superfine powders and those below 1 ␮m are
hyperfine powders. At the other extreme, any particulate with
a diameter greater than 3 mm is a pebble, an aggregate 共such
as employed in concrete兲 or a rock, depending on its size.
There is a lot of sand on the planet–approximately 10 million cubic miles, all told. Over 10% of the Earth’s surface
consists of deserts, and if all the sand in the world were
deposited on the United States it would cover this country to
a thickness three miles high.1
A select group of physicists have investigated the properties of granular media, dating back to the 17th century. Coulomb addressed the issue of the origin of the static angle of
repose,77 defined as the maximum angle that a sandpile can
be constructed while remaining stable against gravity-driven
avalanches. The nature of the mechanisms underlying a
sandpile’s angle of repose remains a topic of intense research
interest today. Once the sandpile begins to flow, the moving
sand expands under shear, so that the grains of sand move
out of the way of each other. This dilatancy was noted and
described by Osborne Reynolds,91 whose contributions to
fluid dynamics are honored, in part, by the dimensionless
number that bears his name. The patterns and structures that
wind-driven sand create were catalogued and analyzed by
Ralph A. Bagnold,4 whose pioneering investigations of Aeolian geomorphology were similarly recognized and commemorated through a dimensionless number 共the Bagnold
number, expressing the ratio of the forces due to friction and
collisions between solids to the forces arising from the surrounding viscous fluid兲. So closely is Bagnold associated
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Am. J. Phys. 73 共1兲, January 2005
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© 2005 American Association of Physics Teachers
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with the study of sand dunes that he appears as a minor
character in Michael Odjede’s 1992 novel The English Patient 共Alfred A. Knopf Inc., New York兲.
In addition to its intrinsic scientific interest and geological
significance, the study of the statics and dynamics of granular materials has profound industrial and commercial applications. Nearly 80% of everything manufactured or grown in
this country exists in at least one stage in its development as
a granular material. Expenses involved with powder processing in the pharmaceutical, agricultural, and construction industries in this country alone are estimated to be on the order
of $80 billion a year. Approximately 3% of all the electrical
power generated in this country is utilized in the single application of grinding metal ores into powders, which are then
separated using surfactants before alloying and recasting.33,34
Nearly 20 pounds of surfactants for every man, woman, and
child on the planet are manufactured every year, with their
largest application being the flotation of metal particles from
low-yield ores.12 No doubt owing to its broad and immediate
relevance for practical applications, the study of granular
materials has been effectively ignored by physicists 共aside
from the few noteworthy exceptions mentioned above兲 until
fairly recently, and investigations of the properties of granular systems have been primarily carried out by chemical,
civil, and mechanical engineers.
This situation changed in 1987, with the publication in
Physical Review Letters of ‘‘Self Organized Criticality: A
Universal Explanation for 1/f Noise’’ by Per Bak, Chao
Tang, and Kurt Wiesenfeld,92 who presented a mathematical
model for fluctuations in disordered systems, in particular for
their tendency to exhibit spectral densities that vary as the
inverse of the frequency f 共hence the descriptive term ‘‘1/f
noise’’兲. Bak, Tang, and Wiesenfeld used as their representative model the size distributions of avalanches for a sandpile
at the critical angle of repose to which additional sand is
added at a uniform rate. Their model predicted that this
simple system would, without any external tuning, organize
itself into a critical state 共as in a ‘‘critical’’ or second-order
phase transition兲 whereby the mass distribution of the avalanches would exhibit a 1/f power spectrum. Computer
simulations of one-dimensional cellular automata sandpiles
provided support for their model.92 The great excitement engendered in the physics community by the possibility that
‘‘self-organized criticality’’ was the start of a new theory for
complex systems was tempered two years later. By then experimental evidence indicated that fluctuations in the magnitude of avalanches in real sandpiles are described by power
spectra characterized by a Lorentzian frequency dependence
共that is, constant for low frequencies and a 1/f 2 dependence
above a characteristic frequency determined by the system’s
properties兲, rather than a power-law frequency dependence.93
Notwithstanding the controversy over the avalanche fluctuations, the publication of the Self-Organized Criticality 共SOC兲
model can be considered a seminal event in the study of the
physics of granular media. Regardless of whether or not the
SOC model provides any insight into 1/f noise, it was at
least partly responsible for the subsequent wealth of attention
paid by physicists to granular systems, inspiring experimental and theoretical investigations of the fascinating nonlinear
dynamics exhibited by these deceptively simple seeming systems. The SOC paper, referring to 1/f noise phenomena,
stimulated the interest of those physicists primarily interested
in the electronic properties of solids. However, at the same
time ‘‘soft’’ condensed matter physicists were also turning
their attention to granular media, motivated by their interest
in pattern formation in dynamically driven systems.122,132
These papers, also published in 1989, clearly demonstrated,
independent of the question of the applicability of SOC, that
there was interesting physics waiting to be uncovered in the
study of granular systems.
Despite extensive and thorough investigations, both theoretical and experimental, carried out by the engineering community over the past 50 years, there remain many important
unanswered questions in granular media that physicists have
begun to address. One of the more striking phenomena exhibited by granular materials is the size or mass segregation
of two or more different granular species when dynamically
driven. Rather than leading to further mixing, as commonsense might suggest, spontaneous segregation can be observed when mixtures are vertically shaken, rotated in a horizontal cylinder about its long axis, or simply poured into a
vertical Hele–Shaw cell with narrow plate separations. The
simple process of pouring a mixture from a discharge hopper
turns out to be not so simple, as convective rolls will develop
within the hopper, leading to size segregation. The occurrence of jamming of hopper discharges at the orifice has been
studied as an example of a kinetically driven transition, not
unlike the glass transition. The segregation of granular materials that differs by size, shape, mass, density, or surface
roughness is a major concern for the pharmaceutical industry, for example, where granular systems need to be well
mixed and homogeneous over length scales of a pill diameter
or less. The elucidation of the basic mechanisms underlying
this phenomenon, in which physicists have played an important role, will lead to reduced costs and improved efficiencies
in this and other important industries.
The following discussion is a selective overview of some
of the fascinating characteristics and phenomena found in
granular media. The breadth of this field, coupled with the
rapid pace of publications, has led to the regrettable yet inevitable exclusion of many significant topics. It would be all
too easy to expand the length of this Resource Letter into a
full monograph. That would be unnecessary, as many excellent books on this subject are already in print, as indicated in
Sec. II B. The reader is advised to begin with Jacques Duran’s Sands, Powder and Grains.5 At just over 200 pages, it
provides a highly readable and comprehensive introduction
to the physics of granular media. I will consider in this Resource Letter only granular systems for which the interstitial
fluid between the grains is dry air. In this case, the forces
acting upon any given grain include only gravity, electrostatics, and contact forces from neighboring grains. The presence of water between the granular contacts leads to cohesive forces between grains that can alter dramatically both
the statics and dynamics of a sandpile. A complete discussion
of the influence of humidity and its attendant cohesiveness
on granular dynamics would warrant a separate Resource
Letter. Moreover, I will focus on properties of granular media easily accessible through table-top experiments. A comprehensive review of geomorphological phenomena resulting
from Aeolian granular dynamics could readily occupy several Resource Letters.
This Resource Letter is organized as follows. A listing of
general resources available on granular media precedes separate sections devoted to Packing, Angle of Repose, Avalanches and Granular Flow, Hoppers and Jamming, Vertically Vibrated Induced Phenomena, Avalanche Stratification,
Axial Segregation, and Granular Media and Traffic. There is
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Am. J. Phys., Vol. 73, No. 1, January 2005
James Kakalios
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considerable overlap in the physical concepts in some of
these sections, and to some readers the delimitation may
seem arbitrary. For example, Secs. III and IV discuss issues
related to ‘‘Packing’’ and the ‘‘Angle of Repose’’ of sandpiles. These two topics have much in common, and the assignment of certain references to one section or the other
necessarily involved a subjective judgment call; as did certain references in Secs. V and VI, ‘‘Avalanches and Granular
Flows’’ and ‘‘Hoppers and Jamming.’’ It is hoped that breaking these topics into shorter sections results in the material
being more accessible than coarse-grained divisions of
‘‘Statics,’’ ‘‘Dynamics,’’ and ‘‘Segregation Phenomena.’’
II. RESOURCES
A. Journals
Annual Review of Fluid Mechanics
CHAOS
Chemical Engineering Science
Europhysics Letters
Granular Matter
Nature
Journal of Applied Physics A
Journal of Engineering Mechanics
Journal of Fluid Mechanics
Journal of Statistical Physics
Physica A and D
Physical Review Letters
Physical Review E (occasionally A and B as
well)
Physics of Fluids A
Powder Technology
Reviews of Modern Physics
Transactions of the Institute of Chemical Engineering Science
B. Books and major compilations
1. A Scientist at the Seashore, James S. Trefil 共Collier Books, New
York, 1984兲. A popular science book for the general public. Chapters
10–12 describe phenomena observed at the beach. 共E兲
2. Sand, Raymond Siever 共Scientific American Library, New York,
1988兲. Combines striking photographs with a scholarly discussion of
granular properties. 共E兲
3. Principles of Powder Mechanics, R. L. Brown and J. C. Richards
共Pergamon, Oxford, 1966兲. An early, classic text. 共I兲
4. The Physics of Blown Sand and Desert Dunes, R. A. Bagnold
共Methuen, London, 1941兲. A classic text discussing physical mechanisms underlying Aeolian geomorphology. 共I兲
5. Sands, Powders, and Grains: An Introduction to the Physics of
Granular Materials, Jacques Duran 共Springer, New York, 1997兲. An
excellent introduction to the field, clearly and entertainingly written.
共E,I兲
6. Physics of Granular Media, edited by D. Bideau and J. Dodds 共Nova
Science, Commack, NY, 1991兲. Proceedings from one of the first conferences on granular media following the Self-Organized Criticality
paper. 共A兲
7. Flows of Granular Materials, S. B. Savage 共Udine, Italy, 1992兲. An
engaging perspective by one of the field’s top researchers. 共I兲
8. Statics and Kinematics of Granular Materials, R. M. Nedderman
共Cambridge U.P., Cambridge, UK, 1992兲. Primarily concerned with
packing and flows through hoppers. 共A兲
9. Granular Matter: An Interdisciplinary Approach, edited by Anita
Mehta 共Springer-Verlag, Berlin, 1993兲. Collection of review articles.
共I兲
10. Non-Linearity and Breakdown in Soft Condensed Matter, edited
by K. K. Bardhan, B. K. Chakrabarti, and A. Hansen 共Springer-Verlag,
Berlin, 1994兲. 共I兲
11. Mobile Particulate Systems, edited by E. Guazzelli and L. Oger 共Kluwer Academic, Dordrecht, 1995兲. 共A兲
12. Fragile Objects, P.-G. deGennes and J. Badoz 共Springer-Verlag, New
York, 1996兲. An excellent overview of soft-condensed matter systems.
共E兲
13. How Nature Works: The Science of Self Organized Criticality, Per
Bak 共Springer-Verlag, Berlin, 1996兲. A history of Self-Organized Criticality by one of its originators, for a general audience. 共E兲
14. Statics of Granular Media, V. V. Sokolovski 共Pergamon, New York,
1965兲. An early, geometric view of granular packing. 共I兲
15. Self-Organized Criticality, Emergent Complex Behavior in Physical and Biological Systems, H. J. Jensen 共Cambridge U.P., Cambridge, 1998兲. Overview of experimental and simulation studies of the
Self-Organized Criticality model, including but not restricted to granular media systems. 共E, I兲
16. The Pursuit of Perfect Packing, Tomasa Aste and Denis Weaire 共IOP,
New York, 2000兲. A readable account of the geometrical issues related
to granular packing, with many historical anecdotes. 共E, I兲
17. Jamming and Rheology: Constrained Dynamics on Microscopic
and Macroscopic Scales, edited by A. J. Liu and S. R. Nagel 共Taylor
& Francis, New York, 2001兲. A collection of articles providing an
overview of granular flow, edited by two of the field’s top researchers.
共I兲
C. Conference proceedings
18. Powders and Grains 93, edited by C. Thorton 共Balkena, Rotterdam,
1993兲. 共A兲
19. Traffic and Granular Flow, edited by D. Wolf 共World Scientific,
Singapore, 1995兲. 共A兲
20. Powders and Grains 97, edited by R. P. Behringer and J. T. Jenkins
共Balkena, Rotterdam, 1997兲. 共A兲
21. Statistical Mechanics in Physics and Biology, Mater. Res. Soc.
Symp. Proc. Vol. 463 共Materials Research Society, Pittsburgh, PA,
1997兲. 共A兲
22. Physics of Dry Granular Media, edited by H. J. Herrmann, S. Luding, and J. P. Hovi, NATO ASI Series E Vol. 350 共Kluwer, Amsterdam,
1998兲. 共A兲
23. The Granular State, Mater. Res. Soc. Symp. Proc. Vol. 627 共Materials Research Society, Pittsburgh, PA, 2001兲. 共A兲
24. Powders and Grains 01, edited by Yuji Kishino 共Balkena, Rotterdam,
2001兲. 共A兲
D. Review articles
25. ‘‘Powder Mixing–A Literature Survey,’’ M. H. Cooke, D. J. Stephens,
and J. Bridgwater, Powder Technol. 15, 1–20 共1976兲. 共I兲
26. ‘‘The Segregation of Particulate Materials. A Review,’’ J. C. Williams,
Powder Technol. 15, 245–254 共1976兲. 共I兲
27. ‘‘Computer Simulation of Granular Shear Flows,’’ C. S. Campbell and
C. E. Brennen, J. Fluid Mech. 151, 167–188 共1985兲. A good introduction of simulations of granular dynamics. 共I兲
28. ‘‘Physics of the Granular State,’’ H. M. Jaeger and S. R. Nagel, Science 255, 1523–1531 共1992兲. A highly cited and influential review. 共E兲
29. ‘‘Instabilites in a Sandpile,’’ S. R. Nagel, Rev. Mod. Phys. 64, 321–
325 共1992兲. 共E兲
30. ‘‘The Dynamics of Flowing Sand,’’ R. P. Behringer, Nonlinear Sci.
Today 3, 1–15 共1993兲. 共E兲
31. ‘‘The Dynamics of Sand,’’ Anita Mehta and G. C. Barker, Rep. Prog.
Phys. 57, 383– 416 共1994兲. 共I兲
32. ‘‘What is Shaking in the Sandbox,’’ H. M. Jaeger, J. B. Knight, C.-h.
Liu, and S. R. Nagel, MRS Bulletin, 19 共5兲, 25–31 共1994兲. 共E兲
33. ‘‘The Legacy of Neglect in the U. S.,’’ J. B. Ennis, J. Green, and R.
Davis, Chem. Eng. Prog. 90, 32– 43 共1994兲. A good overview of the
industrial applications and societal perspectives of granular media. 共E兲
34. ‘‘The Importance of Storage, Transfer and Collection,’’ T. M. Knowlton, J. W. Carson, G. E. Klinzing, and W. -C. Yang, Chem. Eng. Prog.
90, 44 –54 共1994兲. Another review of industrial applications of granular media. 共E兲
35. ‘‘The Physics of Granular Materials,’’ H. M. Jaeger, S. R. Nagel, and
R. P. Behringer, Phys. Today 49 共4兲, 32–38 共1996兲. 共E兲
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36. ‘‘Granular Solids, Liquids and Gases,’’ H. M. Jaeger, S. R. Nagel, and
R. P. Behringer, Rev. Mod. Phys. 68, 1259–1273 共1996兲. A more technical treatment of the topics considered in Ref. 35. 共I兲
37. A broad collection of review articles is contained in a special issue of
Chaos 9, 共3兲共1999兲. 共A兲
38. ‘‘Built Upon Sand: Theoretical Ideas Inspired by Granular Flows,’’
Leo P. Kadanoff, Rev. Mod. Phys. 71, 435– 444 共1999兲. 共I兲
39. ‘‘Granular Matter: A Tentative View,’’ P. G. de Gennes, Rev. Mod.
Phys. 71, S374 –S382 共1999兲. 共I兲
40. ‘‘Nonequilibrium Patterns in Granular Mixing and Segregation,’’ T.
Shinbrot and F. J. Muzzio, Phys. Today 53 共3兲, 25–30 共2000兲. 共E兲
E. Web sites
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
http://www.granular.com
http://chaos.ph.utexas.edu/research/granular/granular.html
http://mrsec.uchicago.edu/granular
http://sol.rutgers.edu/⬃shinbrot/Group – Index.html
http://summa.physik.hu-berlin.de/⬃kies/
http://www.haverford.edu/physics-astro/Gollub/lab.html
http://www.ica1.uni-stuttgart.de/
http://pg.chem-eng.nwu.edu/mixing/
http://widget.ecn.purdue.edu/⬃psl/home.html
http://physics.clarku.edu/⬃akudrolli/silo.html
http://super.phys.northwestern.edu/⬃pbu/
http://www.phy.duke.edu/⬃bob/
http://www.phys.ntnu.no/⬃fossumj/cpx/index.html
http://www.physics.umn.edu/groups/sand
III. PACKING
The volume occupied by granular particles in a container
is determined by their configuration, and depends on local
contact forces, the container’s geometry, friction between
grains and with the container walls, and the details of the
system’s history. In this sense granular media have more in
common with nonequilibrium melt-quenched glasses than
thermodynamic solids or liquids, whose properties are independent of thermal history. A grain of sand or powder is a
macroscopic object, with many internal degrees of freedom.
Consequently any kinetic energy introduced into the granular
system is rapidly dissipated through inelastic collisions between particles. Therefore, in the absence of an external driving force, the configuration of the pile will remain unchanged.
For a granular system with dry inert air 共that is, considering only the static case of no net air flow兲 as the interstitial
medium, the intergrain forces are entirely classical, consisting of contact and frictional forces, both being electrostatic
in nature. In the absence of cohesive forces, the geometry of
a granular system is determined solely by gravity and the
boundaries of the container. The gravitational potential energy of a particle in a sandpile is approximately 1012 times
greater than the thermal energy kT at room temperature.29,36
One can therefore consider granular media to be thermodynamically at zero degrees, and the properties of the sandpile
are determined by the specific details under which the pile
was constructed. An important consequence is that within a
granular system, stable arches and voids may develop, so
that a vertical load owing to a mass of grains can have a
significant horizontal component. A graphic illustration of
this phenomenon is reflected in the collapse of grain silos,
which frequently fail owing to large pressures on their sides
rather than at their base.52 These arches play a role in
dilatancy,91 in which a vertical force applied to the top of a
granular system leads to the creation of new voids, which
decreases the system’s density. That is, sand deforms in the
presence of shear stresses, and will only compact under isotropic pressure. This accounts for the dry footprints one
leaves behind when walking along a wet, sandy shoreline.
The water-saturated sand underfoot expands and develops
new voids and pores, into which water drains, leaving the top
surface drier than its surroundings. Capillary action will
eventually restore the top surface to its prior wet appearance.
The possibility that stable arches can form indicates that
the volume an ensemble of granular material occupies is a
highly sensitive function of the connections between grains.
The densest packing of granular material corresponds to a
crystalline hexagonal close-packed 共or face-centered-cubic兲
structure for spherical grains, which has a packing fraction of
0.74. The random close packing of spheres has a packing
fraction of 0.63, while a packing fraction of 0.55 is found for
a random loose packing in the limit of zero gravitational
force, determined by studying glass spheres in a liquid
whose density is chosen to approach neutral buoyancy
conditions.55,56 The difference between these two packing
extremes corresponds to a variation in the average intergrain
separation of only ⬃10%. 57,58 A granular system prepared in
a random configuration and then disturbed, such as by periodic vertical taps to the container, will explore some portion
of configurational space in a limited way as it settles to a
lower volume 共‘‘contents may have settled during shipping’’兲. The dynamics of a granular medium as it slowly
relaxes with a logarithmic time dependence to a denser configuration has been mapped to the ‘‘random-parking’’
problem.73 Even after 106 separate taps, a granular system in
a cylindrical container whose height is very much larger than
its diameter shows no evidence of reaching a timeindependent stable final state.59– 63,68 –76
The large number of internal degrees of freedom for any
given grain of sand results in collisions being highly inelastic. In this way kinetic energy between interacting particles is
rapidly lost, and clustering or clumping of particles moving
in a restricted geometry is observed. This effect, termed
‘‘inelastic-collapse,’’ leads to a divergence of the collision
frequency for some of the particles. This significant experimental phenomenon is a major complication in numerical
simulations.64 – 67
55. ‘‘Random Loose Packings of Uniform Spheres and the Dilatancy Onset,’’ George Y. Onoda and Eric G. Lininger, Phys. Rev. Lett. 64,
2727–2730 共1990兲. 共I兲
56. ‘‘Random Packings of Spheres and Fluidity Limits of Monodisperse
and Bidisperse Suspensions,’’ Andrew P. Shapiro and Ronald F. Probstein, Phys. Rev. Lett. 68, 1422–1425 共1992兲. 共A兲
57. ‘‘A Model for the Packing of Irregularly Shaped Grains,’’ C. C. Mounfield and S. F. Edwards, Physica A 210, 301–316 共1994兲. 共I兲
58. ‘‘Perturbative Theory of the Packing of Mixtures and of Non-Spherical
Particles,’’ R. B. S. Oakeshott and S. F. Edwards, Physica A 202,
482– 498 共1994兲; ‘‘The Statistical Mechanics of Granular Systems
Composed of Spheres and Elongated Grains,’’ S. F. Edwards and C. C.
Mounfield, ibid. 210, 290–300 共1994兲. These papers describe an innovative attempt to develop a ‘‘statistical mechanics’’ for powders. While
interactions in a thermodynamic system lead to a decrease of the system’s energy, the corresponding variable, these authors suggest, is the
volume occupied by a granular material. Theoretical discussions of the
variation in the packing fraction as the system is perturbed are discussed. 共I兲
59. ‘‘Density Relaxation in a Vibrated Granular Material,’’ James B.
Knight, Christopher G. Fandrich, Chen Ning Lau, Heinrich M. Jaeger,
and Sidney R. Nagel, Phys. Rev. E 51, 3957–3963 共1995兲. Experimental measurements of the time dependence of the volume occupied by
monodisperse granular media in a tall narrow cylinder as a function of
periodic vibrations. 共A兲
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60. ‘‘Intermittencies in the Compression Process of a Model Granular Medium,’’ A. Ngadi and J. Rajchenbach, Phys. Rev. Lett. 80, 273–276
共1998兲. 共A兲
61. ‘‘Fluctuations in Granular Media,’’ Daniel W. Howell, R. P. Behringer,
and C. T. Veje, Chaos 9, 559–572 共1999兲. Review of experimental
studies and numerical simulations of force chains in granular media
slowly sheared by a rotating annulus. These forces are quantified by
shearing polymer disks and viewing them through crossed polarizers.
The spectral density of the stress fluctuations is found to exhibit a
power-law frequency dependence 共A兲.
62. ‘‘A Tetris-Like Model for the Compaction of Dry Granular Media,’’ E.
Caglioti, V. Loreto, H. J. Herrmann, and M. Nicodemi, Phys. Rev. Lett.
79, 1575–1578 共1997兲. A theoretical investigation of compaction and
packing as grains are added from the top of a pile, using a Tetris-like
model to mimic the influence of varying particle shapes. 共A兲
63. ‘‘Studies of Columns of Beads Under External Vibrations,’’ S. Luding,
E. Clement, A. Blumen, J. Rajchenbach, and J. Duran, Phys. Rev. E
49, 1634 –1646 共1994兲. Experimental and numerical studies of a onedimensional column of steel beads experiencing external vibrations, as
the number of beads, the degree of agitation, and the restitution coefficient are systematically varied. Clustered states are observed for high
dissipation and/or large numbers of particles, as well as a transition
from a condensed phase to a fluidized phase as the acceleration of the
base plate is increased. 共I, A兲
64. ‘‘Clustering Instability in Dissipative Gases,’’ I. Goldhirsch and G.
Zanetti, Phys. Rev. Lett. 70, 1619–1622 共1993兲. Analytical calculations and numerical simulations of a gas of inelastically colliding particles, finding that the system is unstable against the formation of
high-density clusters. 共A兲
65. ‘‘Inelastic Collapse and Clumping in a One-Dimensional Granular Medium,’’ Sean McNamara and W. R. Young, Phys. Fluids A 4, 496
共1992兲; ‘‘Inelastic Collapse in Two-Dimensions,’’ Phys. Rev. E 50,
R28 –R31 共1994兲. Theoretical modeling and molecular dynamic simulations for one- and two-dimensional gases of inelastic disks are described. From an initially random configuration, a finite-time singularity can appear, leading to increasing numbers of collisions and the
resulting formation of clusters. 共A兲
66. ‘‘Cluster Formation Due to Collisions in Granular Material,’’ A.
Kudrolli, M. Wolpert, and J. P. Gollub, Phys. Rev. Lett. 78, 1383–1386
共1997兲. Experimental investigation of spherical particles rolling on a
smooth surface, driven by an oscillating sidewall. Inelastic collisions
lead to the formation of clusters. 共I兲
67. ‘‘Cluster-Growth in Freely Cooling Granular Media,’’ S. Luding and
H. J. Herrmann, Chaos 9, 673– 681 共1999兲. Numerical simulations of
the time dependence of inelastic collapse of hard spheres and subsequent aggregate formation are described. 共A兲
68. ‘‘Uniaxial Compression Effects on 2D Mixtures of ‘Hard’ and ‘Soft’
Cylinders,’’ T. Travers, D. Bideau, A. Gervois, J. P. Troadec, and J. C.
Messager, J. Appl. Phys. A 19, L1033 共1986兲. An experimental study
of the macroscopic stress–strain law for a mixture of ‘hard’ and ‘soft’
cylinders examines the role that geometric and compositional heterogeneities play. 共I兲
69. ‘‘Force Fluctuations in Bead Packs,’’ C.-H. Liu, S. R. Nagel, D. A.
Schecter, S. N. Coppersmith, S. Majumdar, O. Narayan, and T. A.
Witten, Science 269, 513 共1995兲. Experimental studies and numerical
simulations of force chains in bead packs are described. The fluctuations in the force distribution arise from variations in contact angles
between beads along with the constraint of the force balance due to
every other bead in the pile. 共I兲
70. ‘‘Contact Forces in a Granular Packing,’’ Farhang Radjai, Stephane
Roux, and Jean Jacques Moreau, Chaos 9, 544 –550 共1999兲. Numerical
simulations of two- and three-dimensional granular packings, confirming the existence of force chains and an exponential distribution of
contact forces. 共A兲
71. ‘‘Granule-by-Granule Reconstruction of a Sandpile from X-Ray Microtomography Data,’’ G. T. Seidler, G. Martinez, L. H. Seeley, K. H.
Kim, E. A. Behne, S. Zaranek, B. D. Chapman, S. M. Heald, and D. L.
Brewe, Phys. Rev. E 62, 8175– 8181 共2000兲. 共A兲
72. ‘‘Footprints in Sand: The Response of a Granular Material to Local
Perturbations,’’ Junfei Geng, D. Howell, E. Longhi, R. P. Behringer, G.
Reydellet, L. Vanel, E. Clement, and S. Luding, Phys. Rev. Lett. 87,
35506 –35510 共2001兲. Experimental report of the ensemble-averaged
response of granular packings to point forces applied at the top of the
pile of polymer discs, whereby the force chains are imaged using
73.
74.
75.
76.
crossed polarizers. Ordered packings have a long-range propagative
force component not found in disordered sandpiles. 共A兲
‘‘Granular Relaxation Under Tapping and the Traffic Problem,’’ D. C.
Hong, S. Yue, M. Y. Choi, and Y. W. Kim, Phys. Rev. E 50, 4123–
4135 共1994兲. The volume relaxation of a one-dimensional granular
column is investigated within the context of a diffusing void model.
The volume occupied by a granular system as a function of perturbations is related to the problem of partitioning automobiles in a disordered parking lot. Steady-state traveling wave solutions are found that
can then account for the discontinuous stick–slip reduction of the column’s height under continuous tapping. 共A兲
‘‘Packing of Compressible Granular Materials,’’ Hernan A. Makse,
David L. Johnson, and Lawrence M. Schwartz, Phys. Rev. Lett. 84,
4160– 4163 共2000兲. Computer simulations are compared to experimental measurements of the packing fraction of three-dimensional granular
media as a function of external confining stress, indicating the presence of localized force chains. 共A兲
‘‘Influence of Shape on Ordering of Granular Systems in Two Dimensions,’’ I. C. Rankenburg and R. J. Zieve, Phys. Rev. E 63, 61303–
61312 共2001兲. Experiments and simulations are reported for twodimensional granular materials consisting of various noncircular
shapes. 共A兲
‘‘Random Packings of Frictionless Particles,’’ C. S. O’Hern, S. A.
Langer, A. J. Liu, and S. R. Nagel, Phys. Rev. Lett. 88, 95703–95707
共2002兲. Numerical simulations of random ensembles of frictionless
granular media, intended to model observed exponential distribution of
forces within pile. 共A兲
IV. ANGLE OF REPOSE
One of the earliest scientific studies of granular media addressed the basic physical mechanisms underlying the angle
of repose of a dry, cohesiveless sandpile.77 When individual
grains of sand are dropped from a given vertical height onto
a horizontal surface, they come to rest through inelastic collisions with the horizontal base plate and with each other. If
a grain lands on top of another grain, it will most likely
tumble off, unless the surrounding area is already occupied
with other sand grains. In this way the dropped grains form a
sandpile. If the location of the source of falling grains does
not vary, the resulting pile will be approximately conical in
structure, with a triangular vertical cross section where the
free surface makes an angle ␪ m with the horizontal. This is
termed the maximum angle of stability. For a sandpile whose
free surface makes an angle shallower than ␪ m with the horizontal, the random configuration of the sand grains held together by contact normal forces, intergrain friction, and gravity will be stable 共excepting the possibility of frictional
creep101兲. The pile will remain in this configuration indefinitely unless the base plate is disturbed or further grains are
added to the pile. In this sense the sandpile can be considered
a ‘‘solid,’’ for no fluid, regardless of its viscosity, will retain
its shape against gravity for sufficiently long times. If the
base plate is now tilted, so that the free surface of the conical
pile makes an angle larger than ␪ m with the horizontal, gravitational forces overwhelm frictional drag and the sand grains
on the surface of the pile form an avalanche down the
surface.78,79 At this point the sandpile pours like a fluid,
though with a crucial distinction from Newtonian fluids discussed in Sec. V. This avalanche continues until the free
surface makes a smaller angle ␪ r , termed the angle of repose, with the horizontal surface. Experimental and numerical investigations have found that the angle of repose ␪ r for
a dry sandpile depends on the density of the grains, the coefficient of sliding and rolling friction, particle size, and surface roughness and shape.80– 86 The presence of a small
amount of interstitial liquid introduces cohesive forces owing
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Fig. 1. Image of a two-dimensional pile of polymer discs 共whose axis of
polarization rotates under pressure兲 when viewed between crossed polarizers. The bright regions indicate discs under compressive strain, indicating
the force chains in the pile 共Reprinted with permission from Ref. 83. Copyright 2001 by the A.P.S.兲
to the liquid bridges between neighboring particles. This cohesive force can overwhelm gravity and intergrain friction,
and the free surface of a moist sandpile can be nearly perpendicular to the horizontal, a beneficial property during the
construction of sandcastles at the beach.
The distribution of applied loads is highly nonuniform for
a random geometry of contacts in a dry, cohesionless sandpile. The inhomogeneous network of contacts leads to the
development of ‘‘force chains,’’ through which loads percolate within the sandpile. In this way regions within a sandpile
may be shielded from forces, whether an externally applied
load or the weight of the material above it. Detailed experimental investigations have explored the influence of intergrain friction and packing order on the length and distribution of these force chains. The ‘‘packing-order’’ term refers
to the detailed history of the grains of sand within the pile.
Two sandpiles, containing the same number of the same type
of particle but differing in their placement, either in a crystalline configuration or an amorphous pile, will differ greatly
in their force network distribution.80,83,87,88 These force
chains can be imaged in two-dimensional systems using
polymer discs whose polarization axis rotates under strain.
As shown in Fig. 1, when a two-dimensional sandpile is
viewed through crossed polarizers, the weight of the pile is
selectively carried by a subset of random contacts 共which
appear bright in this image兲 throughout the system.52,83
The sensitivity of the sandpile’s mechanical properties on
the intergrain connections is also reflected in its ability to
propagate sound.89,90 Acoustic vibrations depend critically on
the compression and expansion of contacts between neighboring grains. Consequently, in the long-wavelength limit
共long, that is, compared to both the diameter of the granular
species and any arches兲 the variation of contact pressure with
depth in the granular system leads to a nonlinear dispersion
of acoustic waves, the net effect being to deflect horizontal
sound waves orthogonally to the direction of propagation.
Owing to this acoustical dispersion, granular media function
as highly effective vibration-isolation systems. Time-of-flight
experiments find that the leading edge of an acoustical pulse
in granular material corresponds to a ‘‘speed of sound’’ of
⬃280 m/s. In addition, at lower wavelengths, the acoustic
response displays a ‘‘noisy’’ frequency dependence that is
reproducible and stable in time. However, if the granular
system is disturbed 共such as by running one’s finger through
the sandbox兲 the ‘‘noisiness’’ of the frequency response
changes to another, stable, reproducible pattern. Clearly, the
detailed nature of the force chains throughout the granular
system plays a central role in determining the properties of
the fundamental excitations of the system.
77. ‘‘Essay on the Rules of Maximis and Minimis Applied to Some Problems of Equilibrium Related to Architecture,’’ C. A. Coulomb, Acad.
R. Mem. Phys. Divers Savants 7, 343 共1773兲. Frequently cited and
seldom read, historically significant as one of the first scientific papers
on the physical mechanisms underlying granular packing. 共A兲
78. ‘‘Stochastic Model for the Motion of a Particle on an Inclined Rough
Plane and the Onset of Viscous Friction,’’ G. G. Batrouni, S. Dippel,
and L. Samson, Phys. Rev. E 53, 6496 – 6503 共1996兲. A stochastic
model is developed of particles moving down a roughened inclined
plane that finds that the frictional force is proportional to the velocity
rather than the expected square of the velocity. These results are in
agreement with experimental measurements. 共A兲
79. ‘‘Experiments on a Gravity-Free Dispersion of Large Solid Spheres in
a Newtonian Fluid Under Shear,’’ R. A. Bagnold, Proc. R. Soc. London, Ser. A 225, 49– 63 共1954兲. Of early, historical significance. 共E兲
80. ‘‘Force Distributions in Three-Dimensional Granular Assemblies: Effects of Packing Order and Interparticle Friction,’’ D. L. Blair, N. W.
Mueggenburg, A. H. Marshall, H. M. Jaeger, and S. R. Nagel, Phys.
Rev. E 63, 41304 – 41311 共2001兲. Experimental measurements of the
normal force distribution for a vertical container with a triangular cross
section are reported. The force-distribution function is essentially identical for random and hexagonal close-packed crystalline arrangements.
共I兲
81. ‘‘Interfacial Friction of Powders on Concave Counterfaces,’’ B. J.
Briscoe, L. Pope, and M. J. Adams, Powder Technol. 37, 169–181
共1984兲. An experimental study of frictional forces, dynamic angle of
repose, and normal forces at the container walls of granular media in a
rotating cylinder are reported as a function of particle mass and rotational velocity. 共I兲
82. ‘‘Friction in Granular Flows,’’ H. M. Jaeger, Chu-Heng Liu, S. R.
Nagel, and T. A. Witten, Europhys. Lett. 11, 619– 624 共1990兲. Early
attempt to develop analytical model for angle of repose. 共E兲
83. ‘‘Memory in Two-Dimensional Heap Experiments,’’ Junfei Geng,
Emily Longhi, R. P. Behringer, and D. W. Howell, Phys. Rev. E 64,
60301– 60304 共2001兲. Measurements of the force-chain distribution in
a two-dimensional sandpile using photoelastic discs viewed through
crossed polarizers 共see also Ref. 52兲, as a function of the sandpileconstruction history. 共A兲
84. ‘‘Grain Non-Sphericity Effects on the Angle of Repose of Granular
Material,’’ Jason A. C. Gallas and Stefan Sokolowski, Int. J. Mod.
Phys. B 7, 2037–2046 共1993兲. Simulations of two-dimensional sandpiles, taking into account that real sand grains are not perfect spheres.
共A兲
85. ‘‘Numercial Investigation of the Angle of Repose of Monosized
Spheres,’’ Y. C. Zhou, B. H. Xu, and A. B. Yu, Phys. Rev. E 64,
21301–21309 共2001兲. Simulations, using discrete element method, of
the angle of repose, investigating sensitivity to various parameters including coefficient of sliding and rolling friction, particle size, and
container thickness 共A兲.
86. ‘‘Effect of Grain Anisotropy on Ordering, Stability and Dynamics in
Granular Systems,’’ C. J. Olson, C. Reichhardt, M. McCloskey, and R.
J. Zieve, Europhys. Lett. 57, 904 共2002兲. Simulations and experiments
investigating the role of grain-aspect ratio on the angle of repose in
two-dimensional sandpiles. 共A兲
87. ‘‘Signatures of Granular Microstructure in Dense Shear Flows,’’ D. M.
Mueth, G. F. Debregeas, G. S. Karczmar, P. J. Eng, S. R. Nagel, and H.
M. Jaeger, Nature 共London兲 406, 385–389 共2000兲. Noninvasive measurements are employed to characterize the steady-state shear flow of
granular media in a three-dimensional Couette geometry. The shape of
the velocity profile is characterized by two length scales independent
of the height and shear rate, but is sensitive to the grain’s morphology.
共A兲
88. ‘‘Stresses Developed by Dry Cohesionless Granular Materials Sheared
in an Annular Shear Cell,’’ S. B. Savage and M. Sayed, J. Fluid Mech.
142, 391– 430 共1982兲. Experimental studies of granular material in
Couette-geometry shear cells. 共I兲
89. ‘‘Sound in Sand,’’ Chu–heng Liu and Sidney R. Nagel, Phys. Rev.
Lett. 68, 2301–2304 共1992兲. Measurements of the spectral density of
vibrations recorded through a three-dimensional granular system are
reported. The power spectra decay with a power-law frequency dependence over five decades in frequency. The stability of the frequency
response was an early indication of the influence of force chains on the
granular system’s dynamical behavior. 共A兲
90. ‘‘Spatial Patterns of Sound Propagation in Sand,’’ Chu-heng Liu, Phys.
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Fig. 2. Time-lapse image of a pile of mustard seeds, tilted to an angle
greater than the angle of maximum stability. The sharp boundary between
avalanching seeds and those maintaining a rigid pile is evident 共Reprinted
with permission from Ref. 36. Copyright 1996 by the A.P.S.兲
Rev. B 50, 782–794 共1994兲. Experimental investigation of the fundamental acoustic excitations of granular media. Time-of-flight measurements are described characterizing the speed of sound in granular media. 共I, A兲
V. AVALANCHES AND GRANULAR FLOW
As mentioned in Sec. I, physical sandpiles do not display
the 1/f distribution in avalanche size that the Self-Organized
Criticality model92 was intended to account for, using cellular automata simulations of sandpiles as the archetypal example. Initially, there was some controversy over the Chicago results, which found that the distribution of avalanches
had a Lorentzian frequency dependence rather than a powerlaw frequency dependence.93 A Lorentzian power spectrum
for the distribution of avalanches indicates that there is a
characteristic size to the granular avalanches. This agrees
with the observation that only grains within a wedge between
the maximum angle of stability ␪ m and the angle of repose ␪ r
participate in an avalanche in a continuously driven system.
However, Held and co-workers at I.B.M. in 1990 observed
avalanche distributions consistent with SOC predictions.94
After a dispute over whether the findings depended on the
use of ‘‘Chicago’’ sand or ‘‘New York’’ sand, a consensus
developed that Held’s results were due to finite-size effects,
and when the size of the experimental system is increased,
the agreement with SOC disappears.94 –97 A recent development is the report of avalanche distributions as predicted by
Self-Organized Criticality when the granular system consists
of long-grain uncooked rice, but not when spherical beads or
sand grains are employed. The role that the highly asymmetric geometry of the granular media may play in the avalanche dynamics remains under investigation.98,99
One of the complications in developing a hydrodynamic
model for granular materials is that the flow of powders or
grains is strongly non-Newtonian.100–109 For example, when
a static sandpile at the angle of marginal stability is further
tilted, the resulting movement, driven by gravity, is restricted
to a narrow region typically ten grains down from the free
surface, as shown in Fig. 2.36 This is in marked contrast to
the manner in which a Newtionian fluid, such as water,
would flow. In that case only the fluid in direct contact with
the rigid base plate would be stationary 共forming the ‘‘no-
slip zone’’兲 and the velocity of the water increases with distance from the base plate. For the granular system, the entire
pile, save the top surface, is immobile, and there is a fairly
sharp boundary, roughly one or two grains wide, separating
the flowing region from the underlying static, rigid pile.
Magnetic-resonance-imaging studies of continuously avalanching grains in a rotating cylinder find that the particle’s
velocity increases quadratically with distance from the
boundary separating the flowing and static grains.110 This
velocity discontinuity yields a large shear force at this interface. As in the case of uniaxial compression, sand expands
owing to shear, and this flow-driven dilatancy can lead to
sieving and segregation of avalanching granular material owing to differing size or density. Recent studies have found
that particles below this interface are not perfectly static, but
rather exhibit creep, that is, slow motion observed on long
time scales, which can extend to arbitrary depth within the
sandpile.101
The sensitivity to external driving, coupled with a nonlinear dependence on external constraints and sample history,
has complicated efforts to develop a continuum model for
granular dynamics.111–114 One of the more successful attempts to model granular flow analytically is by Bouchaud
and co-authors,113 who represent a sandpile as consisting of
two constituents, sticking or rolling grains. The latter are
described by a mean velocity with a dispersion factor. The
two hydrodynamical variables in the model are the height of
the sandpile 共that is, the density of immobile grains兲 and the
density of rolling grains. By allowing for the interconversion
between sticking and rolling states, they obtain a hysteresis
whereby an angle of maximum stability, greater than the
angle of repose, must be exceeded to induce an avalanche of
flowing sand.
91. ‘‘On the Dilatancy of Media Composed of Rigid Particles in Contact,’’
O. Reynolds, Philos Mag. 20, 469 共1885兲. Of historical significance.
共A兲
92. ‘‘Self-Organized Criticality: An Explanation of the 1/f Noise,’’ P. Bak,
C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381–384 共1987兲.
Describes cellular-automata simulations of distributions of avalanches
from a continually driven sandpile at the angle of maximum stability.
共I兲
93. ‘‘Relaxation at the Angle of Repose,’’ H. M. Jaeger, C.-h. Liu, and S.
R. Nagel, Phys. Rev. Lett. 62, 40– 43 共1989兲. Experimental results
which contradict the predictions of Self-Organized Criticality model in
Ref. 92. 共I兲
94. ‘‘Experimental Study of Critical-Mass Fluctuations in an Evolving
Sandpile,’’ G. A. Held, D. H. Solina II, D. T. Keane, W. J. Haag, P. M.
Horn, and G. Grinstein, Phys. Rev. Lett. 65, 1120–1123 共1990兲. Experimental test of the Self-Organized Criticality model, which agrees
with the predictions of Ref. 92. 共I兲
95. ‘‘Finite-Size Effects in a Sandpile,’’ C.-h. Liu, H. M. Jaeger, and S. R.
Nagel, Phys. Rev. A 43, 7091–7092 共1991兲. Experimental investigation of the discrepancy between Refs. 93 and 94, attributing the discrepancy to finite-size effects in Ref. 94. 共I兲
96. ‘‘Tracer Dispersion in a Self-Organized Critical System,’’ Kim Christensen, A̧lvaro Corral, Vidar Frette, Jens Feder, and Torstein Jøssang,
Phys. Rev. Lett. 77, 107–110 共1996兲. Experimental report that avalanches of long-grained rice agree with Self-Organized Criticality
model, while similar studies with spherical grains do not. 共A兲
97. ‘‘Avalanches in One-Dimensional Piles with Different Types of
Bases,’’ E. Altshuler, O. Ramos, C. Martinez, L. E. Flores, and C.
Noda, Phys. Rev. Lett. 86, 5490–5493 共2001兲. Experimental investigation examining whether the discrepancy between granular avalanches that agree with the SOC model and those that do not can be
attributed to a sensitivity to the properties of the baseplate on which
the pile resides. The authors conclude that the proper choice of a
baseplate can indeed lead to scaling properties of granular avalanches
that agree with SOC predictions. 共A兲
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98. ‘‘Dynamics of a Grain on a Sandpile Model,’’ L. Quartier, B. Andreotti, S. Douady, and A. Daer, Phys. Rev. E 62, 8299– 8307 共2000兲.
Experimental and theoretical investigations of the motion of a grain
moving down an inclined plane comprised of identical grains, wherein
the transient and steady-state motions are examined. 共A兲
99. ‘‘Avalanche Dynamics in a Pile of Rice,’’ V. Frette, K. Christensen, A.
Malthe-Sørenssen, J. Feder, T. Jøssang, and P. Meakin, Nature 共London兲 379, 49–52 共1996兲. Experimental studies of the size distribution
of avalanches for a pile at the angle of maximum stability, when the
material consists of long-grained rice. The observed avalanche distribution agrees with predictions of the Self-Organized Criticality model.
共I兲
100. ‘‘Grain Flows as a Fluid Mechanical Phenomenon,’’ P. K. Haff, J.
Fluid Mech. 134, 401– 430 共1983兲. An important, early attempt to
describe granular flow using the phenomenology of hydrodynamics.
共E, I兲
101. ‘‘Creep Motion in a Granular Pile Exhibiting Steady Surface Flow,’’
Teruhisa S. Komatsu, Shio Inagaki, Naoko Nakagawa, and Satoru
Nasuno, Phys. Rev. Lett. 86, 1757–1760 共2001兲. Experimental studies that clearly demonstrate that even below the angle of maximum
stability, slow relaxation and granular flow occur. 共I兲
102. ‘‘Role of Surface Diffusion as a Mixing Mechanism in a BarrelMixer: Part One,’’ B. H. Kaye and D. B. Sparrow, Ind. Chem. 40,
200–205 共1964兲. Experimental study of the curvature of the profile of
flowing granular particles in a horizontal rotating cylinder as a function of rotation speed. Surface diffusion is found to extend a distance
of eight or nine particles beneath the free surface. 共I兲
103. ‘‘Flow Regimes in Fine Cohesive Powders,’’ A. Castellanos, J. M.
Valverde, A. T. Perez, A. Ramos, and P. Keith Watson, Phys. Rev.
Lett. 82, 1156 –1159 共1999兲. Experimental studies of fine, cohesive
grains in a horizontal rotating drum are employed to map out the
dynamical phase diagram for the transition from solid-like to fluidized behavior as a function of velocity and particle size. 共A兲
104. ‘‘Sensitivity of Granular Surface Flows to Preparation,’’ A. Daerr and
S. Douady, Europhys. Lett. 47, 324 –330 共1999兲. Experimental studies of transient avalanches arising when a cylindrical sandpile collapses under its own weight to form a conical structure. 共A兲
105. ‘‘Scales and Kinetics of Granular Flows,’’ I. Goldhirsch, Chaos 9,
659– 672 共1999兲. A review of theoretical attempts to understand clustering and inelastic collapse through kinetic theory for the flow of
dilute granular media. 共A兲
106. ‘‘Rapid Gravity Flow of Cohesionless Granular Materials Down Inclined Chutes,’’ M. Sayed and S. B. Savage, J. Appl. Math. Phys. 34,
84 –100 共1983兲. 共A兲
107. ‘‘Dynamics of Grain Avalanches,’’ J. Rajchenbach, Phys. Rev. Lett.
88, 14301–14304 共2002兲. Experimental study of the nucleation and
growth of avalanches of noncohesive grains in a two-dimensional
rotating cylinder. The downward front of the avalanche is found to
propagate with a velocity approximately twice the average avalanche
velocity, while the upper front flows with a speed roughly equal to
the average avalanche speed. 共A兲
108. ‘‘Flow in Powders: From Discrete Avalanches to Continuous Regime,’’ J. Rajchenbach, Phys. Rev. Lett. 65, 2221–2224 共1990兲. Excellent paper investigating the conditions at which discrete, intermittent avalanches transform into a continuous flow. The transformation
is found to be hysteretic. 共I兲
109. ‘‘Evolution of a Sandpile in a Thick-Flow Regime,’’ S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E 61, 2909–2919 共2000兲. Analytic calculations of the evolution of a one-dimensional sandpile,
when the thickness of the avalanching layer is large compared to the
grain size, for a uniform input flow. 共A兲
110. ‘‘Flow Measurements by NMR,’’ A. Caprihan and E. Fukushima,
Phys. Rep. 198, 195–235 共1990兲; ‘‘Nuclear Magnetic Resonance as a
Tool to Study Flow,’’ E. Fukushima, Annu. Rev. Fluid Mech. 31,
95–123 共1999兲. Comprehensive reviews of the theory and applications of pulsed nuclear-magnetic resonance as a noninvasive probe of
the velocity distributions of liquid flow. Extension of this technique to
granular particles that contain a liquid center 共where the protons in
the fluid are able to follow the rapidly changing magnetic-field gradients兲 enables determinations of the position and velocity of flowing
granular media. 共E, I兲
111. ‘‘A Theory of Rapid Flow of Identical, Smooth, Nearly Elastic Particles,’’ J. T. Jenkins and S. B. Savage, J. Fluid Mech. 130, 187–202
共1983兲. 共A兲
112. ‘‘Grain Dynamics in a Two-Dimensional Granular Flow,’’ S. Horluck
and P. Dimon, Phys. Rev. E 63, 31301–31317 共2001兲. Experimental
studies of individual spheres in a granular flow in a small-angle twodimensional funnel, with particular attention to the mechanisms underlying the creation and interaction of shock waves. 共I兲
113. ‘‘A Model for the Dynamics of Sandpile Surfaces,’’ J.-P. Bouchaud,
M. E. Cates, J. Ravi Prakash, and S. F. Edwards, J. Phys. I. 4 共10兲
1383–1410 共1994兲; ‘‘Hysteresis and Metastability in a Continuum
Sandpile Model,’’ Phys. Rev. Lett. 74, 1982–1985 共1995兲. A continuum model is described for sandpile surfaces. By including two
distinct phases of grains; trapped and mobile; and allowing for their
interconversion, a realistic description of the observed angles of
maximum stability and repose are obtained. 共A兲
114. ‘‘Surface Flows of Granular Mixtures. I. General Principles and Minimal Model,’’ T. Boutreux and P. G. de Gennes, J. Phys. I 6, 1295–
1307 共1996兲. The model of Ref. 105 is extended to include mixtures
of differing granular media, leading to a continuum description of
segregation of the avalanching material. 共A兲
VI. HOPPERS AND JAMMING
The combination of inhomogeneous force chains in a
granular system and non-Newtonian fluid dynamics lead to
significant properties when granular media drain gravitationally through an orifice in the bottom of a hopper. While the
grains flow through the orifice as would a fluid, the detailed
nature of the behavior of the granular material is quite different from that of a Newtonian liquid. For example, a container filled with water, open to atmospheric pressure at the
top, will initially discharge rapidly through an opening in the
baseplate owing to the large pressure of the volume of water
in the tank. As this pressure head drops, the rate of discharge
of the water decreases. In contrast, to first order, the flow of
grains of sand through an equivalent open orifice occurs at
the same rate regardless of the quantity of sand above the
opening.8,115–121 This is because of the arching and force
chains mentioned in Sec. III. The irregular contacts between
grains leads to a fraction of the weight of the volume of sand
above the orifice being transferred horizontally to the sidewalls of the container, rather than hydrodynamically to the
material at the opening. The pressure on the grains of sand
near the orifice arises only from a restricted, hemispherical
volume approximately ten grains in radius from the opening,
and is roughly independent of the amount of additional material in the container. This region is referred to as the ‘‘freefall arch,’’ which forms the boundary separating particles that
are not in direct contact and accelerate freely owing to gravity, and the packed bed of compressed particles
above.30,122–124 The mass-flow rate for granular material in a
hopper with an orifice of diameter D o is observed to vary as
1/2 2 1/2
g 1/2D 5/2
for
o as compared to a discharge rate of g D o H
liquids 共where H is the height of the fluid in the container兲.8
Because the discharge rate of granular systems is independent of the amount of material in the container, we have the
practical application of the hour glass. When half of the sand
has flowed through an ‘‘egg timer,’’ one equates this with
half of the time necessary to fully drain the timer, which
would only be true if the top container discharged at a constant rate.
Another indication that one is not dealing with a Newtonian fluid when considering the draining of granular material
from a hopper is the evolution of the top surface of granular
media. For example, the top surface of grains in a cylindrical
hopper with a circular orifice in its baseplate initially will
maintain a horizontal profile. As the amount of granular material in the cylinder decreases, a conical indentation forms
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as the material directly above the orifice feels its influence,
while those grains off to the side remain stationary. The exact
shape of this indentation along with the timing of its development is sensitive to both the container and orifice geometries. At the shear interface between the rigid granular material within the hopper and the flowing material draining
through the bottom opening there can be significant convection rolls, which can lead to size segregation of a binary
mixture of different size or mass granular materials.8,125,126
An initially homogeneous mixture in a hopper will remain
mixed as it empties from a hopper when the orifice is first
opened, but will alternate between large and small particles
共for example兲 as the hopper continues to drain. An understanding of this segregation phenomenon as a mixture empties from a hopper is obviously of importance for many industrial applications, from agricultural to pharmaceutical.
The suggestion that granular systems might be considered
as analogous to melt-quenched glasses 共whose properties are
highly sensitive to the material’s prior thermal history兲 has
been extended by Liu and Nagel.127 These authors suggest
that the jammed state of a granular material is akin to a
frozen glass at low temperatures. Just as raising the temperature of a glass leads to unjamming of the thermally arrested
material, lowering the density of a jammed granular system
similarly leads to fluid-like behavior. While both glasses and
sand jam at high densities, the role of temperature in a glass
is filled by an externally applied load for a sandpile. Further
work is needed to determine whether this conceptual framework can serve as the basis of a more complete theory of
granular media.128 –130
115. ‘‘Flow of Granular Material Through Horizontal Apertures,’’A. Harmens, Chem. Eng. Sci. 18, 297–306 共1963兲. Analytical calculation of
the mass flow rate of granular matter driven by gravity through a
horizontal orifice. 共I兲
116. ‘‘The Flow of Granular Materials. I. Discharge Rates from Hoppers,’’
R. M. Nedderman, U. Tuzun, S. B. Savage, and G. T. Houlsby, Chem.
Eng. Sci. 37 共11兲, 1597–1609 共1982兲. The first of three review papers
on the flow of granular media through hoppers, with emphasis on the
influence of the container and orifice geometry. 共I兲
117. ‘‘The Flow of Granular Solids Through Orifices,’’ W. A. Beverloo, H.
A. Leniger, and J. van de Velde, Chem. Eng. Sci. 15, 260–269
共1961兲. Experimental investigation of the discharging from a hopper
by differing seeds, with the development of a phenomenological expression for the discharge rate. 共A兲
118. ‘‘The Hour-Glass Theory of Hopper Flow,’’ J. F. Davidson and R. M.
Nedderman, Trans. Inst. Chem. Eng. 51, 29–35 共1973兲. Expressions
for the mass-flow rate and stress distributions are presented; based
upon experimental investigations of cohesionless granular media
flowing from a smooth-walled conical hopper. 共A兲
119. ‘‘Funnel Flows in Hoppers,’’ T. V. Nguyen, C. E. Brennen, and R. H.
Sabersky, J. Appl. Mech. 47, 729 共1980兲. 共A兲
120. ‘‘Shocks in Sand Flowing in a Silo,’’ A. Samandi, L. Mahadevan, and
A. Kudrolli, J. Fluid Mech. 452, 293–301 共2002兲. An experimental
study of the granular dynamics on the top surface of material emptying from a quasi-two-dimensional silo. 共I兲
121. ‘‘Rate of Discharge of Granular Materials from Bins and Hoppers,’’
H. E. Rose and T. Tanaka, Engineer 208, 465– 469 共1959兲. Early,
careful study of discharge rates as the particle diameter and coefficient of friction are systematically varied. 共A兲
122. ‘‘Pattern Formation in Flowing Sand,’’ G. William Baxter, R. P. Behringer, Timothy Fagert, and G. Allan Johnson, Phys. Rev. Lett. 62,
2825–2828 共1989兲. X-ray transmission studies of granular material
draining owing to gravity from a conical hopper, demonstrating that
the density waves that propagate during discharge are sensitive to the
granular materials’ surface roughness. 共I兲
123. ‘‘Experimental Test of Time Scales in Flowing Sand,’’ G. W. Baxter,
R. Leone, and R. P. Behringer, Europhys. Lett. 21, 569–574 共1993兲.
Power spectrum analysis of granular discharge. 共I兲
124. ‘‘Granular Flow: Friction and the Dilatancy Transition,’’ Peter A. Thompson and Gary S. Grest, Phys. Rev. Lett. 67, 1751–1754 共1991兲.
Molecular dynamics simulations of two-dimensional granular systems under shear are described. 共A兲
125. ‘‘The Discharge of Fine Sands from Conical Hoppers,’’ T. M. Verghese and R. M. Nedderman, Chem. Eng. Sci. 50, 3143–3153 共1995兲.
共A兲
126. ‘‘Effects of Horizontal Vibration on Hopper Flows of Granular Materials,’’ M. L. Hunt, R. C. Weathers, A. T. Lee, and C. E. Brennen,
Phys. Fluids 11, 68 –75 共1999兲. Modifications to the mass-flow
discharge-rate expression for quasi-two-dimensional hoppers are investigated as periodic horizontal vibrations are applied. 共I兲
127. ‘‘Jamming is Not Just Cool Any More,’’ A. J. Liu and S. R. Nagel,
Nature 共London兲 396, 21–22 共1998兲. Presentation of a potentially
significant model likening a jamming transition in granular media to
a glass transition in a thermally arrested liquid. 共E兲
128. ‘‘Force Distributions Near the Jamming and Glass Transitions,’’ C. S.
O’Hern, S. A. Langer, A. J. Liu, and S. R. Nagel, Phys. Rev. Lett. 86,
111–114 共2001兲. Experimental and theoretical determinations of interparticle normal-force distributions near the jamming transition. 共A兲
129. ‘‘Jamming of Granular Flow in a Two-Dimensional Hopper,’’ Kiwing
To, Pik-Yin Lai, and H. K. Pak, Phys. Rev. Lett. 86, 71–74 共2001兲.
The jamming probability in a quasi-two-dimensional hopper is measured, and the arch at the orifice is described using the formalism of
a self-avoiding random walker. 共A兲
130. ‘‘Jamming Phase Diagram for Attractive Particles,’’ V. Trappe, V.
Prasad, L. Cipelletti, P. N. Segre, and D. A. Weitz, Nature 共London兲
411, 772–775 共2001兲. An experimental examination of the solid–
liquid transition for weakly attractive colloidal particles that indicates
support for the model of Liu and Nagel 共Ref. 127兲. 共I兲
VII. VERTICALLY VIBRATED INDUCED
PHENOMENA
One consequence of the highly inelastic collisions between grains in a granular system is that any fluid-like behavior is only observed when the system is continuously
dynamically driven. As mentioned in Sec. II, the configuration of a granular system is determined in large part by the
boundary conditions of the container that constrains the particles. One convenient technique for adding external energy
to a granular system is through oscillations of the baseplate
of the container in either the vertical 共orthogonal to the plane
of the baseplate兲 or horizontal 共within the plane of the baseplate兲 directions. Michael Faraday reported in 1831 that
when a granular system is sinusoidally, vertically vibrated
through its base plate, convective rolls and heaping are
observed.131 For large amplitude A and high-frequency oscillations ␻, where the acceleration of the container ␻ 2 A is
larger than the acceleration owing to gravity g, the granular
material achieves lift-off from the bottom of the container
during certain phases of the vertical shaking cycle. The
granular medium then dilates and generates a large-scale
convective roll that transports granular material upwards in
the center of the system and downwards at the sidewalls in
rectangular or cylindrical containers. The net effect is the
formation of a stable heap in the center.132–137 A change of
the container’s boundary conditions can reverse the direction
of the convective rolls, so that an indentation occurs in the
center.134 Grains are continuously flowing along the free surface of this heap. While there is no question that this heaping
phenomenon results from convective rolls, as confirmed by
Magnetic-Resonance-Imaging 共MRI兲 studies,138 the mechanism by which these rolls are created remains an open question. Friction at the sidewalls and intergrain pressure from
the interstitial gas play a role in heap formation as well.153
When an identical system is subjected to horizontal periodic
vibrations of the baseplate, convective patterns also have
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Fig. 4. Image of a localized vibration-induced excitation, termed an ‘‘oscillon’’ 共Reprinted with permission from Ref. 141兲.
Fig. 3. Patterns observed for a cylindrical cell filled with ⬃100 glass particles, whose baseplate is sinusoidally vibrated, as the vibration frequency
and amplitude are varied 共Reprinted with permission from Ref. 140. Copyright 1994 by the A.P.S.兲
been observed. Though not as extensively investigated as
vertical vibrations, the smoothness of the boundaries is crucial for determining the details of the observed patterns in
horizontally shaken systems.139
For vertical vibrations in a large, shallow cylinder containing granular material, the free surface can exhibit a rich collection of standing-wave patterns as the oscillation amplitude
and frequency are varied. The similarities to Faraday patterns
in vibrated fluids is striking in certain circumstances, as
shown in Fig. 3, though the physical mechanisms underlying
these surface structures is very different in the granular
system.140
Umbanhowar, Melo, and Swinney reported141 that for a
narrow but definite range of vibration amplitude and frequency, stable, two-dimensional localized excitations in a vibrating layer of sand could be observed. These excitations,
which they termed ‘‘oscillons’’ 共see Fig. 4兲, have a propensity to assemble into ‘‘molecular’’ and ‘‘crystalline’’ structures. That is, a pair of oscillons beating 180 degrees out of
phase with each other, can form a bound dimer pair, similar
to a vortex–anti-vortex bound pair. Trimers and more complicated structures have also been reported. The fascinating
zoology of patterns obtained from vibrating granular materials is not close to being explored exhaustively.141–144
A striking segregation effect, whereby large particles in a
mixture of granular media of differing sizes rise to the top of
a cylinder when continuously, periodically shaken in the vertical direction, is known as ‘‘The Brazil Nut Problem.’’ 145,146
This whimsical title reflects the observation that in a large
can of mixed nuts, the bigger, heavier Brazil nuts will be
found at the top of the mixture, rather than having settled to
the bottom of the container. This phenomenon was first reported when studying the separation of pharmaceuticals as a
function of vertical shaking in an effort to mix a disperse
system.145 The rising of larger particles to the top of a granular mixture has been attributed to a ratcheting mechanism.
That is, an upward fluctuation of a large particle induced by
the vertical shaking results in a void directly beneath this
particle.133,136,146 Other larger particles cannot fit into this
narrow, constrained space, but smaller particles in the system
can. In this way the large particle is prevented from moving
downward with subsequent vertical shakes, and the large particle eventually creeps up to the top of the container. An
alternative explanation proposes that convective rolls, whose
presence is confirmed by MRI measurements, are continuously present as the system is vertically shaken.134,138,147
These convective rolls entrain the larger particles, bringing
them to the top surface. Once at the top of the container, the
larger particles are unable to follow the convective rolls of
the smaller grains that move downward at the sides of the
container, and upward in its center. Strong support for the
influence of convective rolls is the observation that alteration
of the boundary conditions of the container can cause the
rolls to reverse direction. Once the rolls move upward at the
sidewalls and downward in the center, larger particles move
to the bottom of the container with vertical shaking, rather
than to the top. However, the subject is not closed, and experimental and theoretical studies on this question
continue.137,148 –153
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131. ‘‘On a Peculiar Class of Acoustical Figures; and on Certain Forms
Assumed by Groups of Particles upon Vibrating Elastic Surfaces,’’
M. Faraday, Philos. Trans. R. Soc. London, 52, 299–340 共1831兲.
Historically significant. 共A兲
132. ‘‘Instability in a Sand Heap,’’ P. Evesque and J. Rajchenbach, Phys.
Rev. Lett. 62, 44 – 46 共1989兲. Early, important experimental study of
instabilities in free-surface profile that arise in granular system vertically vibrated. 共E兲
133. ‘‘Vibrated Powders: A Microscopic Approach,’’ Anita Mehta and C.
C. Barker, Phys. Rev. Lett. 67, 394 –397 共1991兲. A three-dimensional
microscopic model is described for a granular system subjected to
vertical vibration, with attention paid to the competition between collective and single-particle excitations. 共A兲
134. ‘‘Effects of Container Geometry on Granular Convection,’’ E. L.
Grossman, Phys. Rev. E 56, 3290–3300 共1997兲. 共A兲
135. ‘‘Bubbling in Vertically Vibrated Granular Materials,’’ H. K. Pak and
R. P. Behringer, Nature 共London兲 371, 231–233 共1994兲; ‘‘Effects of
Ambient Gases on Granular Materials Under Vertical Vibrations,’’ H.
K. Pak, E. v. Doorn, and R. P. Behringer, Phys. Rev. Lett. 74, 4643–
4646 共1995兲. Experimental studies that find that convection and heaping are sensitive to bubbling of the interstitial gas trapped within the
granular media. 共I兲
136. ‘‘Computer Simulation of the Mechanical Sorting of Grains,’’ P. K.
Haff and B. T. Werner, Powder Technol. 48, 239–245 共1986兲. 共I兲
137. ‘‘Experimental Study of Heaping in a Two-Dimensional ‘Sandpile,’ ’’
E. Clement, J. Duran, and J. Rajchenbach, Phys. Rev. Lett. 69, 1189–
1192 共1992兲. Experimental studies of heaping in two-dimensional
vertically vibrated systems indicate that a block-slip mechanism at
the container walls is responsible for heaping under certain conditions. 共I兲.
138. ‘‘Granular Convection Observed by Magnetic Resonance Imaging,’’
E. E. Ehrichs, H. M. Jaeger, Greg S. Karczmar, James B. Knight,
Vadim Yu. Kuperman, and Sidney R. Nagel, Science 267, 1632–1634
共1995兲. The first direct observation that convective rolls are present in
vertically vibrated granular media, imaged using MRI. 共E兲
139. ‘‘Convection in Horizontally Vibrated Granular Material,’’ Milica
Medved, Damien Dawson, Heinrich M. Jaeger, and Sidney R. Nagel,
Chaos 9, 691– 696 共1999兲. 共I兲
140. ‘‘Transition to Parametric Wave Patterns in a Vertically Oscillated
Granular Layer,’’ Francisco Melo, Paul Unbanhowar, and Harry L.
Swinney, Phys. Rev. Lett. 72, 172–175 共1994兲. Experimental study of
surface patterns formed by granular media in wide, shallow containers as a function of vertical oscillation amplitude and frequency. 共I兲
141. ‘‘Localized Excitations in a Vertically Vibrated Granular Layer,’’ P.
B. Umbanhowar, F. Melo, and H. L. Swinney, Nature 共London兲 382,
793–796 共1996兲. The first experimental report of the observation of
spatially localized excitations 共oscillons兲 in a vertically vibrated shallow granular system. 共I兲
142. ‘‘Spiral Patterns in Oscillated Granular Layers,’’ John R. de Bruyn, B.
C. Lewis, M. D. Shattuck, and Harry L. Swinney, Phys. Rev. E 63,
41305– 41316 共2001兲. 共A兲
143. ‘‘Electrostatically Driven Granular Media: Phase Transitions and
Coarsening,’’ I. S. Aranson, D. Blair, V. A. Kalatsky, G. W. Crabtree,
W.-K. Kwok, V. M. Vinokur, and U. Welp, Phys. Rev. Lett. 84,
3306 –3309 共2000兲; ‘‘Phase Separation and Coarsening in Electrostatically Driven Granular Media,’’ I. S. Aranson, B. Meerson, P. V.
Sasorov, and V. M. Vinokur, Phys. Rev. Lett. 88, 204301–204304
共2002兲. Instead of mechanically vibrating a baseplate, an applied
electric field is employed to oscillate a charged granular medium. A
dynamical phase diagram for nucleation and coarsening of dense
clusters as a function of frequency and maximum applied electric
field is presented. 共A兲
144. ‘‘Surface Waves in Vertically Vibrated Granular Materials,’’ H. K.
Pak and R. P. Behringer, Phys. Rev. Lett. 71, 1832–1835 共1993兲.
Experimental observation of surface waves that propagate up the
slope of a sandpile for a granular system in which an annular layer is
vertically vibrated. 共E兲
145. ‘‘Segregation Kinetics of Particulate Solids Systems. I. Influence of
Particle Size and Particle Size Distribution,’’ James L. Olsen and
Edward G. Rippie, J. Pharm. Sci. 53 共2兲, 147–150 共1964兲; ‘‘Segregation Kinetics of Particulate Solids Systems. II. Particle DensitySize Interactions and Wall Effects,’’ Edward G. Rippie, James L.
Olsen, and Morris D. Faiman, ibid. 53 共11兲, 1360–1363 共1964兲;
‘‘Segregation Kinetics of Particulate Solids Systems. III. Dependence
146.
147.
148.
149.
150.
151.
152.
153.
on Agitation Intensity,’’ Morris D. Faiman and Edward G. Rippie,
ibid. 54 共5兲, 719–722 共1965兲. The first experimental reports of vertical segregation of granular materials of differing sizes under vertical
vibrations, spread out over several publications. Noteworthy is the
journal chosen for publication, indicating that this phenomenon is not
simply of academic interest but has significant ramifications for industrial powder processing. 共I兲
‘‘Why the Brazil Nuts Are on Top: Size Segregation of Particulate
Matter by Shaking,’’ A. Rosato, K. J. Strandburg, F. Prinz, and R. H.
Swendsen, Phys. Rev. Lett. 58, 1038 –1040 共1987兲. Simulation study
of vertical segregation that supported the ratcheting mechanism. As
only periodic boundary conditions were investigated, convective rolls
would not have been observed. 共A兲
‘‘An Experimental Study of Granular Convection,’’ J. B. Knight, E.
E. Ehrichs, V. Yu. Kuperman, J. K. Flint, H. M. Jaeger, and S. R.
Nagel, Phys. Rev. E 54, 5726 –5738 共1996兲. Thorough overview, using MRI, of the dynamics of convective rolls in vertically vibrated
systems, characterizing both the radial and depth dependence of the
vertical flow velocity, as the container aspect ratio is systematically
varied. 共A兲
‘‘Size Segregation in a Two-Dimensional Sandpile: Convection and
Arching Effects,’’ J. Duran, T. Mazozi, E. Clement, and J. Rajchenbach, Phys. Rev. E 50, 5138 –5141 共1994兲. Experimental studies of
two-dimensional granular media vertically shaken demonstrate that
convection and heaping occur for differing regimes of excitation amplitude and frequency. 共I兲
‘‘Density-Noise Power Fluctuations in Vibrated Granular Media,’’ E.
R. Nowak, A. Grushin, A. C. B. Barnum, and M. B. Weissman, Phys.
Rev. E 63, 20301–20304 共2001兲. Experimental studies that find that
the spectral density of density fluctuations in vertically vibrated
granular systems are characterized by non-Gaussian statistics, indicating the presence of strongly cooperative interactions between fluctuators. 共A兲
‘‘Size Segregation and Convection,’’ T. Poschel and H. J. Herrmann,
Europhys. Lett. 29, 123–128 共1995兲. Molecular-dynamics simulations confirm that convective cells are associated with the rise of
larger particles in vertically vibrated systems. 共A兲
‘‘Rise-Time Regimes of a Large Sphere in Vibrated Bulk Solids,’’ L.
Vanel, A. D. Rosato, and R. N. Dave, Phys. Rev. Lett. 78, 1255–1258
共1997兲. Experimental studies of the rise of larger particles embedded
in a granular system when vertically vibrated. These studies find that
segregation occurs both for large-amplitude high-frequency oscillations, which induce convective rolls at the sidewalls, as well as for
low-amplitude low-frequency excitations that induce a ‘‘nonconvective regime.’’ 共A兲
‘‘Hydrodynamic Description of Granular Convection,’’ H. Hayakawa,
S. Yue, and D. C. Hong, Phys. Rev. Lett. 75, 2328 –2331 共1995兲;
‘‘Traffic Equations and Granular Convection,’’ D. C. Hong and S.
Yue, Phys. Rev. E 58, 4763– 4775 共1998兲. A hydrodynamic model,
along with numerical simulations for granular convection in a vertically vibrated bed is presented. 共A兲
‘‘Size Segregation of Granular Particles,’’ Matthias E. Mobius, Benjamin E. Lauderdale, Sidney R. Nagel, and Heinrich M. Jaeger, Nature 共London兲 414, 270 共2001兲. Brief report of experimental study of
vertical segregation effects as a function of ambient air pressure in
container. The timing and magnitude of the vertical segregation effect
are found to decrease at lower background pressures, indicating a role
played by the interstitial air. 共A兲
VIII. AVALANCHE STRATIFICATION
When a mixture of large and small granular media is
poured into a vertical Hele–Shaw cell, consisting of two vertical transparent plates held apart with a narrow separation,
mounted on a horizontal baseplate, another striking segregation effect may be observed.154 Typically, these two vertical
plates are closed at one end, but this is not crucial to observe
the segregation phenomenon. Initially, the granular mixture
forms a pile at the bottom of the cell. As the pile grows, two
segregation phenomena may be observed, even when care is
taken to ensure that the granular mixture is not segregating
as it leaves the hopper above the Hele–Shaw cell. The larger
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particles will tend to accumulate at the bottom of the growing pile, leaving the smaller particles separated near the top.
Since to first order the velocity of the falling grains is independent of their size, both large and small grains arrive at the
top of the pile with the same velocity. However, their mass
difference leads to different kinetic energies. As the particles
move in an avalanche down the sandpile’s free surface, they
suffer inelastic collisions with the other particles in the pile.
The larger particles require more collisions to exhaust their
initial kinetic energy and are more likely to continue moving
down to the bottom of the pile, where they come to rest and
accumulate, leaving the smaller particles on top.155
In addition to this segregation, certain granular mixtures
can display a stratification effect, whereby the avalanching
material separates into alternating layers of large and small
grains.154,156 –161 There are essentially two mechanisms that
have been proposed to account for this stratification effect.
One model suggests that stratification does not begin until
the pile has reached a size such that even the most energetic
particles cannot make it down to the bottom of the sandpile.
At this point a metastable wedge forms. When the top surface of this wedge exceeds the maximum angle of stability,
the material within the wedge moves in an avalanche down
the top surface. As the granular mixture flows, it dilates owing to the shear force arising from the sharp velocity discontinuity between the flowing layer and the rigid pile. In this
expanded, flowing layer, the smaller particles may drop
down to the rigid surface beneath the flowing layer. The
metastable wedge thereby separates into two layers of small
particles underneath a layer of large particles.157,160 The alternative model posits that stratification occurs continuously
as the material flows down the top surface, with the smaller
particles, which are more susceptible to inelastic
collisions,162 being caught in small gaps, and stopping
first.156,158,161 The ‘‘metastable-wedge’’ argument accounts
for the observation that stratification only occurs once a critical height of the pile is reached, and that its onset may be
varied by changing the kinetic energy of the incoming granular mixture added to the top of the growing pile.160 However,
the observation of an upward-moving density wave during
the layering process is well described by the ‘‘continuousflow’’ model. Which mechanism dominates turns out to be a
sensitive function of the cell geometry and granular-flow
rates.160 For certain configurations both mechanisms compete, leading to ‘‘pairing’’ of small particle bands.157,160
Shear flows as granular material moves in an avalanche
down an inclined slope, either in chute flow or along the
surface of a sandpile tilted above the angle of maximum
stability, can exhibit cluster formation, roll waves, or a fingering instability as the granular front propagates.163
154. ‘‘The Segregation of Particulate Materials,’’ J. C. Williams, Powder
Technol. 15, 245–256 共1976兲. The first experimental report of stratification of binary mixture of granular media when poured into a
narrow vertical Hele-Shaw cell. 共I兲
155. ‘‘Interparticle Percolation and Segregation in Granular Materials: A
Review,’’ S. B. Savage in Developments in Engineering Mechanics,
edited by A. P. S. Selvadurai 共Elsevier, Amsterdam, 1987兲, pp. 347–
363; ‘‘Particle Size Segregation in Inclined Chute Flow of Cohesionless Granular Solids,’’ S. B. Savage and C. K. K. Lun, J. Fluid Mech.
189, 311–335 共1988兲. Experimental investigation of percolation of
finer particles to the bottom of a flowing granular layer during an
avalanche. 共I兲
156. ‘‘Spontaneous Stratification in Granular Mixtures,’’ Hernan A.
Makse, Shlomo Havlin, Peter R. King, and H. Eugene Stanley, Nature 共London兲 386, 379–382 共1997兲. Experimental report of avalanche stratification and segregation, supported by cellular-automata
157.
158.
159.
160.
161.
162.
163.
simulations. Proposed mechanism for stratification involves continuous trapping of finer particles during avalanche, with larger grains
residing on top. 共I兲
‘‘Phase Diagram for Avalanche Stratification of Granular Media,’’ J.
P. Koeppe, M. Enz, and J. Kakalios, Phys. Rev. E 58, R4104 –R4107
共1998兲. Experimental investigation and numerical simulations of avalanche stratification as the plate separation of the Hele–Shaw cell and
flow rate of addition of granular mixture are systematically varied.
First report of ‘‘pairing’’ of small particle layers for certain plate
separations and flow rates. Model proposed involves the development
of metastable wedge between angles of maximum stability and repose. During flow of material in wedge, sieving percolation segregation occurs as described by Savage 共Ref. 155兲. 共I兲
‘‘Dynamics of Granular Stratification,’’ H. A. Makse, R. C. Ball, H.
Eugene Stanley, and S. Warr, Phys. Rev. E 58, 3357–3367 共1998兲;
‘‘Mechanisms of Granular Spontaneous Stratification and Segregation in Two-Dimensional Silos,’’ Pierre Cizeau, Hernan A. Makse,
and H. Eugene Stanley, ibid. 59, 4408 – 4421 共1999兲. Elucidation of
cellular-automata model, supported by analytical calculations and experimental observations, in support of ‘‘continuous-flow’’ model of
avalanche stratification. 共A兲
‘‘Stripes Ordering in Self-Stratification Experiments of Binary and
Ternary Granular Mixtures,’’ N. Lecocq and N. Vandewalle, Phys.
Rev. E 62, 8241– 8244 共2000兲. Experimental study of avalanche
stratification when ternary granular mixtures are poured into a vertical Hele–Shaw cell. Depending on the characteristics of the granular
material employed, differing layering schema are observed. 共I兲
‘‘Avalanche Stratification-Experimental Tests of the ‘Metastable
Wedge’ and ‘Continuous Flow’ Models,’’ M. E. Swanson, M. Landreman, J. Michel, and J. Kakalios, Mater. Res. Soc. Symp. Proc.
627, BB2.6 共2001兲. Experimental attempts to reconcile two differing
models for avalanche stratification. Which mechanism dominates is
sensitive function of plate separations and flow rates. 共I兲
‘‘Microscopic Model for Granular Stratification and Segregation,’’ H.
A. Makse and H. J. Herrmann, Europhys. Lett. 43, 1– 4 共1998兲. Numerical simulations supporting the ‘‘continuous-flow’’ model for avalanche stratification. 共A兲
‘‘Different Characteristics of the Motion of a Single Particle on a
Bumpy Inclined Line,’’ G. H. Ristow, F.-X. Riguidel, and D. Bideau,
J. Phys. I 4, 1161–1172 共1994兲. Experimental and theoretical investigation of a single ball rolling down a rough inclined plane, to elucidate the energy-dissipation mechanisms during real granular avalanches. 共A兲
‘‘Fingering in Granular Flows,’’ O. Pouliquen, J. Delour, and S. B.
Savage, Nature 共London兲 386, 816 共1997兲; ‘‘Segregation Induced Instabilities of Granular Fronts,’’ O. Pouliquen and J. W. Vallance,
Chaos 9, 621– 630 共1999兲; ‘‘Longitudinal Vortices in Granular
Flows,’’ Y. Forterre and O. Pouliquen, Phys. Rev. Lett. 86, 5886 –
5889 共2001兲. Experimental studies elucidating mechanisms underlying fingering instability of advancing front of granular material flowing down an inclined plane, driven by convection transverse to the
direction of downward flow. 共A兲
IX. AXIAL SEGREGATION
Consider a cylinder partially filled with a binary mixture
of granular materials differing in size or density and positioned so that its long axis is parallel to the horizontal plane.
The cylinder is now rotated about its long axis, like a drum
mixer. If the cylinder is more than 50% filled by volume,
then either segregation of the mixture of granular materials
about the axis of rotation or if the system is initially prepared
in a segregated state, patterning may occur.164 –168 This is
because only a narrow wedge near the top surface is free to
flow as the system is rotated. Geometric shadowing will determine the amount of material affected by the rotation. In
this way a central segregated core near the axis of rotation
will either become mixed or patterned depending on the extent to which the material within an avalanching wedge intersects the central region. Depending on the configuration of
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Fig. 5. Images of a horizontal cylinder, 2 ft long and 5 in. in diameter,
one-third filled with a 50/50 mixture by volume of uncooked rice and split
peas. The top image is in an initial mixed state and the lower image is of the
same system following rotation about the long horizontal axis at 15 rpm for
2 h. See K. M. Hill and J. Kakalios, http://www.physics.umn.edu/groups/
sand 共Ref. 54兲.
the container and the amount of material present, this shadowing can lead to a variety of patterns of the binary granular
mixtures as the cylinder is rotated.
If the system is less than 50% filled with a granular mixture by volume, then two other distinct types of segregation
may occur. Initially, within the first few rotations of the cylinder, a fraction of the smaller particles will concentrate near
the axis of rotation.169–173 MRI studies have confirmed that
this ‘‘radial segregation’’ occurs along the entire length of the
cylinder,174 though initially it had been studied in quasi-twodimensional cylinders that are only a few particle diameters
long. This radial segregation results from the sieving of the
smaller grains through the dilated, top avalanching surface.
In addition, depending on the rotation speed of the cylinder,
a second segregation effect may be observed.175–186 In this
case the granular material forms nearly homogeneous alternating bands of large and small particles, which occur along
the length of the cylinder like rings on a finger, as illustrated
in Fig. 5. This phenomenon is termed ‘‘axial segregation.’’
The location and width of the bands varies randomly from
trial to trial. MRI studies find that the radially segregated
state persists even in the presence of the axial segregation,174
so that a band of large particles is actually an outer ring of
large beads, with a core of small particles near the axis of
rotation. Certain systems display a reversible axialsegregation effect, in that a homogeneous mixed state will
form alternating segregated bands when the cylinder is rotated at a relatively high rotation speed 共typically ⬃15 rpm
or higher兲, while if rotation is then continued but at a slower
speed 共such as ⬃5 rpm) the segregated bands disappear and
the mixed state is restored.184,185 However, MRI studies find
that even in this ‘‘re-mixed’’ state the radial segregated core
remains.174
The standard model for axial segregation involves the
angle the avalanching material makes with the back wall of
the cylinder. As the cylinder rotates, the granular material
maintains contact with the back wall of the cylinder until the
angle of maximum stability is reached, at which point the
grains flow down the top surface, and are then brought back
up again as the cylinder turns. Even at 5 rpm, the granular
material is in constant motion down the free surface. Viewed
from the side, the granular material forms a ‘‘dynamic angle
of repose,’’ which depends on the properties of the granular
material, the rotation speed of the cylinder, the frictional
characteristics of the cylinder wall, and the concentration
共ratio of large to small beads, for example兲 of the mixture. It
is found that, for a given rotation speed, a mixture of large
and small beads that have the same density 共mass/bead volume兲 maintains contact with the back wall to a greater
height, than for an identical cylinder filled either with all
large or all small beads. A collection of all large particles, for
example, will have several large voids and arches, while a
50/50 mixture can have these open spaces filled with smaller
beads. In this way the granular mixture is more compact and
stiffer than the same volume of monodisperse large beads,
and hence can be tilted to a larger angle before an avalanche
occurs. This concentration dependence of the dynamic angle
of repose accounts for the axial-segregation effect. At a relatively high rotation speed, such as 15 rpm, the granular mixture forms a larger dynamic angle of repose than if only large
particles are present. However, random collisions between
the avalanching beads can result in concentration fluctuations, where the number of large beads in one section of the
cylinder is greater than the average. In this case the dynamic
angle of repose of the section enriched with large particles
will be lower than the adjoining regions, which consist of the
average mixed state. A large bead at the interface of this
fluctuation-induced large-bead-enhanced region can therefore lower its gravitational potential energy by falling from
the mixed state into this region. In this way the initially
narrow large-bead-rich region acquires additional large beads
and grows in width. Moreover, as more and more large beads
drift from the mixed regions to the large-bead band, the
mixed state becomes richer in small beads. In this way, fluctuations of more ordered regions 共that is, concentrations differing from the average兲 will be stabilized and grow, owing
to interactions at its interface. Of course, random collisions
at the interface also will lead to standard Fickian diffusion,
which will tend to re-mix the large-bead band back into the
mixed state. When these two effects of gravitational drift and
Fickian diffusion are combined, the resulting time dependence for the spatial concentration of large beads is described by a ‘‘diffusion equation’’ with an effective diffusion
coefficient that is the difference of two terms reflecting drift
and diffusion. At 15 rpm, drift dominates over diffusion and
the effective diffusion coefficient is negative, describing a
situation where an ordered fluctuation is stable and grows in
time. For certain granular mixtures at slower rotation speeds
共such as 5 rpm兲 there is no difference in the dynamic angle of
repose between the mixed and all-large-bead states. In this
case there is no drift term. If the state is mixed, it will stay
mixed, and if axially segregated, random collisions return the
alternating bands to a homogeneous mixed state. Systems for
which the mixed-state dynamic angle of repose is larger than
the large-bead case at all rotation speeds demonstrate a nonreversible axial-segregation effect, and those for which there
is no angle difference at any speed never segregate 共aside
from radial segregation兲.185
While this explanation is consistent with observations of
axial segregation and measurements of the concentration dependence of the dynamic angle of repose, MRI studies have
found that concentration modulations in the bulk of the
granular system may be present that do not extend to the top
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of the free-flowing rotating surface.174 That is, there can be
more axially segregated structure within the bulk than
present on the top surface, which casts doubt on a ‘‘dynamicangle-of-repose’’ model that posits that the segregation
mechanism is purely a surface effect, driven by height variations of the top flowing layer. Rather, MRI studies indicate
that the depth of the avalanching material is greater for binary mixtures than for single-phase regions. This suggests
that churning of the radially segregated core by the avalanching wedge induces concentration fluctuations, which in turn
are reflected in the surface dynamic angle of repose.189,190
These height variations may then stabilize and reinforce the
concentration modulations, leading to a quasi-stable axially
segregated banding pattern.
In addition to the radial segregation of smaller particles
within the interior of the rotating granular mixture, there is
another phenomenon that is observed prior to the formation
of stable axially segregated bands. This is transient bands of
segregated materials that appear as bidirectional traveling
waves when viewed from the top flowing surface.191,192
These initial traveling-wave patterns eventually stabilize to
form the pattern of axially segregated bands discussed above.
Continued rotation for longer times leads to coarsening and
evolution of this banding pattern.193,194
164. ‘‘Avalanche Mixing of Granular Solids,’’ G. Metcalfe, T. Shinbrot, J.
J. McCarthy, and J. M. Ottino, Nature 共London兲 374, 39– 42 共1995兲.
Experimental study of geometric shadowing for granular media in a
slowly rotating cylinder as a function of filling height. 共I兲
165. ‘‘Size Segregation and Convection of Granular Mixtures Almost
Completely Packed in a Thin Rotating Box,’’ Akinori Awazu, Phys.
Rev. Lett. 84, 4585– 4588 共2000兲. Simulations of a quasi-twodimensional rotating box, where the axis of rotation passes through
center of square face of box, nearly full of granular media. A global
convection is observed following the formation of the radially segregated state, which in turn induces an axially segregated banding pattern. 共A兲
166. ‘‘Comparing Extremes: Mixing of Fluids, Mixing of Solids,’’ Julio
M. Ottino and Troy Shinbrot, in Mixing: Chaos and Turbulence, edited by Chante et al. 共Kluwer Academic, New York, 1999兲, pp. 163–
186. Comparison of fluid–fluid mixing and granular solid–solid segregation and mixing in rotating horizontal cylinders. Radial and axial
segregation are discussed, as the influence of geometric shadowing of
top flowing surface intersecting the central stationary core. 共E兲
167. ‘‘Pattern Formation during Mixing and Segregation of Flowing
Granular Media,’’ G. Metcalfe and M. Shattuck, Physica A 233, 709–
717 共1996兲. MRI study of axial segregation for binary granular mixtures as the particle size and density are systematically varied. 共A兲
168. ‘‘Transverse Flow and Mixing of Granular Materials in a Rotating
Cylinder,’’ D. V. Khakhar, J. J. McCarthy, T. Shinbrot, and J. M.
Ottino, Phys. Fluids 9, 31– 43 共1997兲. Surface flow for monodisperse
granular media in a horizontal rotating cylinder is investigated using
digital-image analysis. The variation of the average velocity of the
flowing material as a function of rotation speed, particle properties,
and depth toward the axis of rotation is described. 共I兲
169. ‘‘An Analysis of Radial Segregation for Different Sized Spherical
Solids in Rotary Cylinders,’’ N. Nityanand, B. Manley, and H. Henein, Metall. Trans. B 17B, 247–257 共1986兲. Experimental observations of radial segregation for two-dimensional cylinders. 共I兲
170. ‘‘Particle Mass Segregation in a Two-Dimensional Rotating Drum,’’
G. H. Ristow, Europhys. Lett. 28, 97–101 共1994兲. Experimental study
of radial segregation in two-dimensional cylinders for binary granular
mixtures as the ratio of particle mass is systematically varied. 共A兲
171. ‘‘Radial Segregation in a Two-Dimensional Rotating Drum,’’ C. M.
Dury and G. H. Ristow, J. Phys. I 7, 737–745 共1997兲. Simulations of
radial segregation using the discrete element method. 共A兲
172. ‘‘Radial Segregation in a 2D Drum: Experimental Analysis,’’ F. Cantelaube and D. Bideau, Europhys. Lett. 30, 133–138 共1995兲. Experimental study of large and small discs in a two-dimensional 共i.e.,
short兲 rotating cylinder. 共I兲
173. ‘‘Mixing of a Granular Material in a Bi-Dimensional Rotating
Drum,’’ E. Clement, J. Rajchenbach, and J. Duran, Europhys. Lett.
30, 7–12 共1995兲. Using motion capture by a CCD camera, trajectories of tracer particles are measured for rotation in a two-dimensional
cylinder. When the tracer is the same size as the other beads, two
competing attractive regions, near the cylinder center and the outer
wall, are identified. 共I兲
174. ‘‘Bulk Segregation in Rotated Granular Material Measured by Magnetic Resonance Imaging,’’ K. M. Hill, A. Caprihan, and J. Kakalios,
Phys. Rev. Lett. 78, 50–53 共1997兲. MRI study of concentration variations beneath the top surface for radial and axial segregation in a
horizontal cylinder. Study found evidence of segregated banding
within the bulk that did not extend to the top surface, raising questions concerning models for axial segregation that suggest the phenomenon is surface-flow-driven. 共A兲
175. ‘‘Horizontal Rotating Cylinder,’’ Y. Oyama, Bull. Inst. Phys. Chem.
Res. Jpn. Rep. 18, 600 共1939兲共in Japanese兲. First experimental report
of axial segregation of binary mixture of granular media in a horizontal rotating drum. 共A兲
176. ‘‘Mixing of Solids,’’ S. S. Weidenbaum, Adv. Chem. Eng. 2, 211
共1958兲. 共I兲.
177. ‘‘Mixing and De-Mixing of Solid Particles: Part I. Mechanisms in a
Horizontal Drum Mixer,’’ M. B. Donald and B. Roseman, B. Chem.
Eng. 7 共10兲 749–753 共1962兲; ‘‘Mixing and De-Mixing of Solid Particles: Part II. Effects of Varying the Operating Conditions of a Horizontal Drum Mixer,’’ 7 共11兲, 823– 827 共1962兲. Early experimental
study of axial-segregation effect. A model is proposed that is based
upon velocity gradients created owing to drag on the granular material by the cylinder walls. 共I兲
178. ‘‘Particle Mixing by Percolation,’’ J. Bridgwater, N. W. Sharpe, and
D. C. Stocker, Trans. Inst. Chem. Eng. 47, T114 –T119 共1969兲. Experimental studies of mechanisms underlying radial segregation. 共A兲
179. ‘‘Fundamental Powder Mixing Mechanisms,’’ J. Bridgwater, Powder
Technol. 15, 215–231 共1976兲. 共I兲
180. ‘‘Axial Transport of Granular Solids in Horizontal Rotating Cylinders. Part 1: Theory,’’ S. Das Gupta, D. V. Khakhar, and S. K. Bhatia,
Powder Technol. 67, 145–151 共1991兲; ‘‘Axial Segregation of Particles in a Horizontal Rotating Cylinder,’’ Chem. Eng. Sci. 46, 1513–
1517 共1991兲; ‘‘Axial Transport of Granular Solids in Rotating Cylinders. Part 2: Experiments in a Non-Flow System,’’ S. J. Rao, S. K.
Bhatia, and D. V. Khakhar, Powder Technol. 67, 153–162 共1991兲.
Axial segregation effect is described for binary granular mixtures
with varying particle diameters. Results are described in terms of a
‘‘dynamical-angle-of-repose’’ model. 共I兲
181. ‘‘Dynamics of Avalanches in a Rotating Cylinder,’’ S. Fauve, C.
Laroche, and S. Douady, in Physics of Granular Media, edited by
Daniel Bideau and John Dodds 共Nova Science, Commack, NY,
1991兲, p. 277. 共I兲
182. ‘‘Disorder, Diffusion and Structure Formation in Granular Flows,’’
Stuart B. Savage, in Disorder and Granular Media, edited by D.
Bideau and A. Hansen 共North-Holland, Amsterdam, 1993兲, pp. 255–
285. Clear discussion of ‘‘dynamical-angle-of-repose’’ model for
axial segregation, supported by cellular-automata simulations, as well
as an exposition of granular flow from a hopper. 共E兲
183. ‘‘Rotationally Induced Segregation of Granular Materials,’’ O. Zik,
Dov Levine, S. G. Lipson, S. Shtrikman, and J. Stavans, Phys. Rev.
Lett. 73, 644 – 647 共1994兲. Experimental observation and analytical
theory for axial segregation of binary mixtures of granular media in
rotating cylinders. 共A兲
184. ‘‘Reversible Axial Segregation of Binary Mixtures of Granular Materials,’’ K. M. Hill and J. Kakalios, Phys. Rev. E 49, R3610–R3613
共1994兲. First report of an axial segregation effect that is removed
upon continued rotation at slower rotation speeds. Measurements of
continuous flowing profiles support ‘‘dynamical-angle-of-repose’’
model for segregation effect. 共E兲
185. ‘‘Reversible Axial Segregation of Rotating Granular Media,’’ K. M.
Hill and J. Kakalios, Phys. Rev. E 52, 4393– 4400 共1995兲. Study of
reversible axial segregation effect as the relative diameters of binary
mixtures of granular materials are systematically varied. Further experimental support of ‘‘dynamical-angle-of-repose’’ model for segregation effect is presented. 共I兲
186. ‘‘Axial Segregation of Granular Materials,’’ Dov Levine, Chaos 9,
573–580 共1999兲. Review article of advances in understanding of
axial segregation. 共I兲
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187. ‘‘Non-Invasive Measurements of Granular Flows by Magnetic Resonance Imaging,’’ M. Nakagawa, S. A. Altobelli, A. Caprihan, E.
Fukushima, and E. K. Jeong, Exp. Fluids 16, 54 – 60 共1993兲. Early
study of sub-surface concentration and velocity variations of granular
media in a horizontal rotating cylinder using Magnetic Resonance
Imaging. 共A兲
188. ‘‘Axial Segregation of Granular Flows in a Horizontal Rotating Cylinder,’’ Masami Nakagawa, Chem. Eng. Sci. 49, 2540–2544 共1994兲.
MRI study of axial segregation for granular mixture of a continuous
size distribution. 共A兲
189. ‘‘Structure and Kinematics in Dense Free-Surface Granular Flow,’’ K.
M. Hill, G. Gioia, and V. V. Tota, Phys. Rev. Lett. 91, 64302– 64305
共2003兲. Experimental study of the velocity profile of flowing beads in
a rotating cylinder, demonstrating that the self-diffusion coefficient
scales with the bead’s mean velocity. 共I兲
190. ‘‘Solid-Fluid Transition in a Granular Shear Flow,’’ Ashish V. Orpe
and D. V. Khakhar, Phys. Rev. Lett. 93, 68001– 68004 共2004兲. Experimental study of velocity profile as a function of depth from the
free-surface for granular media in a horizontal rotating cylinder, finding a sharp transition from fluid-like behavior to a fluid⫹solid state
with decreasing velocity beneath the freely flowing surface. 共I兲
191. ‘‘Traveling Granular Segregation Patterns in a Long Drum Mixer,’’
Kiam Choo, T. C. A. Molteno, and Stephen W. Morris, Phys. Rev.
Lett. 79, 2975–2978 共1997兲. Experimental observation of travelingwave patterns during initial transients, prior to formation of stable
axial segregated band pattern, for long narrow horizontal cylinders.
共I兲
192. ‘‘Axial Segregation of Powders in a Horizontal Rotating Tube,’’ Joel
Stavens, J. Stat. Phys. 93, 467– 475 共1998兲. A brief review of experimental and theoretical studies of axial segregation. 共E兲
193. ‘‘Axial Segregation of Granular Media in a Drum Mixer: Pattern
Evolution,’’ K. M. Hill, A. Caprihan, and J. Kakalios, Phys. Rev. E
56, 4386 – 4393 共1997兲. Surface observations and MRI studies of evolution of axially segregated banding pattern for extended rotation
times. 共A兲
194. ‘‘Avalanche-Mediated Transport in a Rotated Granular Mixture,’’ Vidar Frette and Joel Stavans, Phys. Rev. E 56, 6981– 6990 共1997兲.
Experimental study of axial-segregation band merging for extended
rotation of the horizontal cylinder. The authors propose that axially
propagating avalanches are responsible for the band coarsening. 共I兲
X. GRANULAR MEDIA AND TRAFFIC
The connection between granular media and other complex, nonequilibrium systems, such as glasses or automobile
traffic, have been made by many authors.19,73,185 Indeed, the
nonlinear diffusion equation that describes the conditions for
which axial segregation is observed has also been employed
to describe the spontaneous formation of traffic jams in highway flow.195–199 At large auto densities, drivers pack their
cars to a point of marginal stability, limited by the drivers’
awareness of traffic conditions ahead of them and their finite
response time. This densely packed state is unstable against
fluctuations, so that one driver suddenly slowing down or
even just tapping his or her brakes can lead to a backward
propagating avalanche of stopped cars 共that the lead car
which instigated the jam will not be part of兲. However, one
should approach cautiously this or any other metaphor for
granular systems. To say that a sandpile is similar to traffic
共or that protein dynamics is similar to glassy relaxation兲 is
simply transferring ignorance from one field to another 关S. R.
Nagel, private communication, 1987兴. Unless the simile
leads to new insights into either granular media or highway
flow, the mere demonstration that two distinct, complex systems may be described by the same differential equation is
not overly instructive. Better to study a sandpile and learn its
ways. If nothing else, scientific justifications for a trip to the
beach are far too rare to be passed up.
195. ‘‘Traffic Dynamics: Studies in Car Following,’’ R. E. Chandler, R.
Herman, and E. W. Montroll, Oper. Res. 6, 165–184 共1958兲. Nonlinear diffusion-equation analysis of instabilities in highway flow. 共I兲
196. ‘‘On Kinematic Waves. II. A Theory of Traffic Flow on Long
Crowded Roads,’’ M. J. Lighthill and G. B. Whitham, Proc. R. Soc.
London, San. A 229, 317–345 共1955兲. 共I兲
197. ‘‘Shock Waves on the Highway,’’ Paul I. Richards, Oper. Res. 4,
42–51 共1956兲. Analytical modeling of highway flow as a
‘‘continuous-fluid’’ with emphasis on the development of waves in
space and time resulting from perturbations such as traffic signals. 共I兲
198. ‘‘Nonlinear Effects in the Dynamics of Car Following,’’ G. F. Newell,
Oper. Res. 9, 209–229 共1961兲. Analysis of the propagation of smallamplitude disturbances through traffic, when modeled by a nonlinear
diffusion equation. 共I兲
199. ‘‘Traffic and Related Self-Driven Many-Particle Systems,’’ D. Helbing, Rev. Mod. Phys. 73, 1067–1141 共2001兲. A review, for the expert, of the theoretical advancements in understanding traffic dynamics since the pioneering studies in Refs. 195–198. 共A兲
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