Robert P. Behringer, PhD
Dr. Behringer’s research interests include granular and soft condensed matter physics (click here
to see a video demonstrating granular friction). Below, he discusses work on various ongoing projects in the lab:
Experiments on Jamming: Of particular importance is our work on the nature of jamming in granular systems. Briefly, jamming occurs when collections of classical particles, such as sand grains, are in a state that is mechanically rigid and stable. Many systems that can jam occur in nature, including granular materials, foams, colloids, and glasses (e.g. a liquid that has been super-cooled below the temperature for which a transition to a crystalline solid normally occurs). Much work has been focused on systems consisting of frictionless particles that are easily modeled in Molecular Dynamics (MD) simulations. Such model systems are jammed for high enough densities, low enough temperatures, and small shear stresses. There is a special density where, for zero temperature and zero shear stress, idealized system undergo a transition between jammed—i.e. mechanically robust, and unjammed—i.e. deforming without resistance to applied stresses.
We have shown experimentally that physical granular systems exhibit many of the features found in simulations. For these experiments, we used systems of photoelastic particles that allow us to provide all relevant physical data at the particle scale. What is particularly important for these experiments is the development of inverse algorithms that allow us to determine contact forces acting between individual grains—in detail and with good resolution. An amusing use of such particles forms part of a recently unveiled, prize-winning exhibit at the Museum of Science and Industry in Chicago. We helped in the conceptual design of this project, and with the help of the Physics Instrument Shop, supplied the necessary photoelastic particles.
Jamming under shear: In recent work, we have shown that systems of frictional particles exhibit entirely unanticipated states when shear is applied. Specifically, for densities which are too low to be jammed under isotropic conditions, the application of shear leads to states that are jammed and highly anisotropic, both in terms of the geometric structure and in terms of the stresses. These states are not consistent with the jamming picture of frictionless particles and require a significant rethinking of jamming diagram. Ongoing analysis of this data with theorists Bulbul Chakraborty and student Max Bi at Brandeis, Corey O’Hern at Yale University, and Duke post-doc Jie Zhang (now Assistant Professor at Indiana University Purdue-Fort Wayne) is yielding truly important new insights about physical granular systems, and novel statistical approaches to granular matter.
Cyclic shear and jamming: The work described in the above paragraph was carried out in a specially designed apparatus that allows us to apply arbitrary strains, including shear, to a sample of our photoelastic particles. In a new apparatus, Ph.D. student Jie Ren, post-doc Joshua Dijksman and I are setting about a systematic study of the sensitivity of granular systems to small highly uniform strains. It is the ability to apply shear uniformly and precisely that is the special feature of this new apparatus. We are currently testing new ideas about reversibility and granular entropy. This experiment has come on-line in the past month or two, and represents a true tour de force: the system is automated in such a way that it successively takes three different image types for tracking contact forces, particle displacements and particle rotations, for a range of shear strains.
Glassy behavior and jamming: For spatially disordered packings of particles, jamming and glassy behavior are intertwined, and this is certainly the case for granular systems. In an ordinary glass forming liquid, that is super-cooled or compressed below the freezing point, molecules become trapped in ‘cages’ formed by surrounding neighbor molecules. Self diffusion of molecules provides a key measure of this structural limitation on molecular motion. In fact, caging occurs gradually as a fluid is super-cooled. As a consequence, there is not a sharp glass transition compared to the sharp transition between fluid and crystalline solid. This means that subtle measures are need for characterizing the glass transition. A new so-called fourth-order approach (χ4) has been used recently to characterize length and time scales in glassy systems. During my sabbatical leave starting in May and again in the Fall of 2010, I joined forces with Olivier Dauchot and Ph.D. student Corentin Coulais to apply ideas from glassy systems, to driven systems of photoelastic disks. An overarching concept is that structures such as force chains, which are characteristic of how stresses/forces are carried in granular materials, must also be intimately connected with caging—and escape from cages, which dominates long-time diffusive behavior. During my leave, we created an experiment using photoelastic disks made according to procedures that we have established, and an apparatus at the C.E.A (Commisariat à l’Energie Atomique) in Saclay, south of Paris. These state-of-the-art experiments are now yielding data that address the issue of the interconnectivity of force chains and glassy dynamics.
Does dimensionality matter?
This question was posed as the focus of a conference organized by Patrick Charbonneau (Duke Chemistry) Karen Daniels (NC State and former Duke post-doc) and Matthias Schroeter, University of Gottingen. There is little question that our experiments using photoelastic disks and other quasi-two-dimensional particles have yielded data that have illuminated the nature of granular behavior. But what differences occur between 2D to 3D systems? This is a very important question, since the geometric structures that appear in jammed 2D systems, force chains, may have different spatial properties when there is an added dimension. To address the issue of 3D granular force structures, we are developing a novel apparatus that will allow particle imaging in 3D (with some photoelastic information) and photoelastic information at the walls, where individual particles make contact. The particles must be confined so that we can apply arbitrary strains, something that is difficult to achieve in 2D, and even more difficult in 3D. Post-doc Joshua Dijksman, who is already an expert on 3D imaging, is developing this new experiment.
Network theory and granular mechanics: In other work this year, I have collaborated with mathematician Antoinette Tordesillas and her group at the University of Melbourne, Australia, and Jie Zhang in novel applications of network theory to granular packings and their mechanical stability. Network theory is a natural concept for granular ‘solids’ because a stressed granular system exhibits force chains—sets of grains which carry above average forces along filamentary networks. Our experimental data is unique in providing grain-scale information on inter-grain forces, which is essential to this type of analysis. Ongoing work in this direction has characterized key geometric configurations of grains that sustain stability of stressed packings, and also identified novel evolution of the associated networks as the system is deformed.
Granular materials as a model for avalanches: Many systems exhibit avalanche phenomena. For instance, when a magnetic sample is placed in a changing magnetic field, oriented domains re-align with the new field direction in a complex collective manner that is responsible for Barkhausen noise. Real avalanches, bubble rafts, sheared granular materials, and many other locally constrained systems exhibit similar behavior, for a given class of system (universality). Data from shear experiments carried out several years ago by Duke Ph.D. Robert Hartley and me is providing an excellent test bed for avalanches models developed by theorist Karin Dahmen and her student Tyler Earnest at the University of Illinois at Urbana-Champagne. In particular, these experiments show good scaling behavior as a function of the density of the system, a natural control parameter.
Jamming of ‘non-spherical’ particles: The majority of work on jamming has concentrated on particles with spherical symmetry—disks in 2D, or spheres in 3D. However, recent work by O’Hern et al. and Torquato et al. has shown that the nature of jamming is significantly altered if the shape of the particles is no longer spherical. Likewise, it is known that something as simple as the packing fraction at jamming is a sensitive function of the aspect ratio for ellipses. Graduate student Somayeh Farhadi and I carried out experiments to test predictions and more generally to explore jamming stress states for elliptical particles made of a photoelastic material. In order to obtain the same quality of data for ellipses that we can obtain for disks, there are important mathematical and computational challenges. We have made good progress on these, particularly on the mathematical problem of finding a computationally efficient solution for the stresses inside an ellipse that is subject to point forces. We are currently collecting a large body of data characterizing jamming and sheared states near jamming.
Characterizing granular networks with Computational Homology:
With mathematicians Konstantin Mischaikov at Rugers, Lou Kondic at NJIT, and theorist Corey O’Hern, we are applying novel ideas from computational homology to characterize granular systems. Both the stress chains seen in jammed granular materials, and the structure of the network formed by contacting grains are of
interest. The goal of this work is to relate key features in the granular system to Betti numbers, which can be computed very efficiently. At the least, we expect to obtain more efficient methods for computing information such as g(r), and more deeply we will seek measures that can help define a new statistical approach to granular materials, including generalized entropies. In our collaboration, data from our experiments and MD simulations by Lou Kondic become the input to algorithms that yield the Betti numbers. Then, the challenge is to relate these measures to important physical characteristics of the granular systems.
High speed impacts on granular materials: Duke is the lead institution in a multi-university project to understand the response of a granular material to a high speed impact. This work involves experiments at Duke and the University of Maryland that will characterize impacts in 2D and 3D respective, and theory/simulations generated at Yale and NJIT. The collaboration involves my lab at Duke, Wolfgang Losert at the Maryland, Corey O’Hern at Yale, and Lou Kondic at NJIT. The dynamics of impact on a granular material is highly complex, since sound speeds in granular materials are slow, and the impactors can easily arrive at supersonic speeds. Moreover, the complex propagation of sound through chains in granular system can lead to unusual acoustic propagation. At Duke, we have constructed a 2D impacting system using photoelastic particles.
How a meteorite or missile transfers impact energy to sand and dirt grains:
Using high-speed video of projectiles slamming into a bed of disks has given Behringer, graduate student Abe Clark, and Lou Kondic of New Jersey Institute of Technology a new microscopic picture of the way a meteorite or missile transfers the energy of its impact to sand and dirt grains. It was previously assumed that the slowing down would be smooth and that any sound wave would travel through a granular material in a regular, uniform pattern. But what this research supported by the Defense Threat Reduction Agency shows is that the bronze disk loses most of its energy in intense, sporadic acoustic pulses along networks of grains, or force chains, in the bed of beads. For a video demonstrating this, click here.
Sound propagation near jamming: We are seeking to understand the nature of sound propagation in the opposite limit of high speed impact, namely in systems of grains that are just above jamming. This project is an international collaboration with Eric Clément at the ESPCI in Paris, Matthias Sperl at the DLR (NASA equivalent) in Cologne Germany, Stefan Luding at the University of Twente in the Netherlands, and Matthias Schroeter at the University of Gottingen. Theory predicts the presence of a high number of low frequency (soft) vibrational modes in systems just above jamming. However, experimentally identifying these modes is difficult because gravity compacts ordinary materials (in 3D) to densities well above jamming. The European collaborators on this project are developing experiments that will be carried on parabolic flights that can attain very low gravity for 10’s of seconds. And, it is possible that some of these experiments will fly on the International Space Station. Work at Duke will take a different tack: because we can carry out very sensitive 2D experiments, it is possible to study the same phenomena without the use of expensive and complex flights.
Earthquakes and friction: We have developed experiments at the interface between microscopic friction seen in atomic systems, and macroscopic friction that is characteristic of earthquakes. In these experiments, we confine photoelastic disks in a vertical channel, and we pull a ‘slider’ across the surface, typically through a spring. This is a classic stick-slip friction apparatus, with the novel twist that one surface is a granular medium whose properties we can monitor using photoelasticity. With Jackie Krim of NC State University, we have explored the analogies to atomic friction, the transition from stick-slip to steady sliding, and the effects of external vibration. This project, which will continue to yield interesting data with graduate student Eric Sia working on it, formed the basis for a one-year sabbatical leave that Jackie spent with my group. With Chris Marone (Penn State, Geology) and Paul Johnson and Bob Ecke (Los Alamos National Lab) we are using this apparatus to pursue mechanisms that trigger earthquakes, and in particular, the sensitivity of faults (here modeled by our slider system) to external perturbations.
Homogenization for granular materials:
One of the standard approaches used in statistical physics is to seek the large-scale behavior of many-particle systems by an appropriate averaging process. The result of this approach is known as hydrodynamics when applied to fluids, elasticity theory for solids, and so forth. The idea is average over microscopic degrees of freedom of a many-body system to arrive at the ‘long wavelength’ continuum behavior that will be observed on the macroscopic scale. The application of this approach, called homogenization, to granular systems is of great theoretical interest. The engineering literature is replete with ad hoc
models for the continuum behavior of granular materials. However, these models are often of questionable validity, and the connection between the micro- and macro-scales is usually not possible. We have worked extensively with theorist Isaac Goldhirsh of the University of Tel Aviv to apply physically motivated homogenization procedures to our experimental data for granular systems. From this analysis, we obtained true continuum stress/strain and rotation data for a sheared granular system. Of particular note are the special and
unexpected connection between particle rotations and the macroscopic rotation of the strain field. Tragically, Isaac died in April of 2010. However, we are continuing to apply the approaches that he has developed to both the shear experiments and to other studies in our lab.
Active matter: Many systems consist of ‘objects’ that can communicate with each other, possibly through an external medium, to produce novel collective states. Examples include bacteria colonies, flocks of birds, and schools of fish. Granular particles can also exhibit behavior that resembles some of the phenomena observed for more biological systems. It is this analogy that I am pursuing with Eric Clément at the ESPCI-Paris. We have constructed systems of bacteria-like particles that consist of linear chains of grains (think lamp chain). We then place these ‘bacteria’ on a vibrated surface. Because these bacteria have a ‘head’ and a ‘tail’ they move in response to vibration. We are seeking the answers to questions such as: what are the ‘flocking’ modes of these objects? What is the nature of their motion through a constriction? What are the connections between the collective motion of these physical particles and their biological analogues? The initial stages of this work formed one half of my sabbatical effort during the Fall of 2010.
Granular projects with applications: In addition to the work above, that is directed at developing a basic statistical understanding of granular materials, we have several smaller projects that seek to apply basic concepts to important practical problems. There are three such projects, which I describe briefly below.
- Crater formation due to a gas jet: A problem of considerable consequence for any manned (and possibly unmanned) exploration of the moon or planets involves the formation of a crater on a lunar/planetary surface due to a retro-rocket on a lander. Graduate student Abe Clark and I have carried out experiments and analysis to understand this process using a simple quasi-2D experiment consisting of granular samples contained in a long narrow box that are subject to a plume of high-speed nitrogen gas. In fact, these studies, in collaboration with John Brandenburg of ORBITEC, have shown interesting dynamics which include a logarithmic evolution of the crater profile and more importantly, a symmetry breaking instability. The latter is of considerable practical importance, since this instability could drastically change the conditions beneath a real lunar or planetary lander.
- Jamming in hopper flow: The flow out of a hopper is of theoretical interest because it is a dynamical system that exhibits jamming, and of practical interest because jamming in an industrial setting can cause major difficulties. Graduate student Junyao Tang and I have carried out extensive experiments in 2D hoppers and associated theoretical analysis that now give a good picture of both the flow properties, including fluctuations and the nature of jamming.
- Shear flow and mixing of a granular material: In work with Paul Mort of Proctor & Gamble, graduate student Abe Clark and I are characterizing shear flow and mixing of dissimilar sizes of granular particles using our photoelastic approach. This work is intended to provide fundamental insights into mixing, segregation, force propagation and homogenization. At the same time, it will help to provide useful insight into flows that are of considerable industrial relevance.
Thin fluid films:
A smaller, but still important aspect of my research involves nonlinear dynamics and pattern formation for fluid flows, and in particular, the flow of thin films. I am part of a recently awarded NSF-DMS project focusing on thin films, and I am also continuing work in this area with former Ph.D. student Shomeek Mukhopadhyah, Lou Kondic at NJIT, Alex Oron at the Technion in Israel, and Nebo Murisic at UCLA. Senior collaborators in the new NSF-DMS project include Karen Daniels and Michael Shearer at NC State, Tom Witelski at Duke, and Rachel Levy at Harvey Mudd College. Work in the past year involved studies of spreading/retracting drops, rotating films, and convection in thin films. Post-doc Joshua Dijksman and I are setting up new experiments that will probe the properties of thin films under the combined driving forces due to the Marangoni effect, and centrifugal forcing
Joshua Dijksman, PhD:
on Marangoni driven contact lines in thin films, and imaging, in both two and three dimensions, of forces and deformations in slowly sheared granular packings. Click here
to see a You-Tube video of some of his recent research, here
to see a scanning video, here
to see a poster from the Dynamics Days 2011 Conference, and here
to see some of his recent work. His research interests extend to the field of disordered materials in general. In the image to the left you can see photoelastic particles marked by a fluorescent dye. By tracking a collection of these particles as they are sheared, one can obtain their position, orientation, and contact forces. This allows for the study of their diffusion, force network properties, non-affine rotations, jamming, reversibility, and chaos. This work is done together with Jie Ren, also in the Behringer group.
Nicolas Brodu, PhD: works on how to measure forces in full 3D inside granular materials and how to track the evolution of their structure over time during shear or compression cycles.
: has been studying the dynamics of crater formation by a jet of gas impinging on a two- dimensional granular bed, with applications to a retrograde rocket landing on a planetary surface. In addition to confirming some theoretical predictions made by a collaborator, he observed a horizontal symmetry breaking in the process which, as far as he knows, had not previously been observed. He is also studying a granular mixing project, where a central paddle-wheel drives a granular system consisting of the photo-elastic disks. Click here
for a poster from the Dynamics Days 2011 Conference and here
for a poster from the IFPRI Annual General Meeting in collaboration with NSF. Abe has also done experiments using a high-speed camera to learn exactly how grains absorb the blow of a falling rock hurtling into a bed of sand
is interested in the response to shear of a 2D photo-elastic granular system. She has built a novel experimental apparatus which can create quasi-static, nearly uniform simple shear to the system. By using photo-elastic particles and fluorescent labeling technique, she can obtain information about displacement, rotation and contact forces for each particle following each step of shear strain. From this information she is able to do various kinds of analysis to better understand the nature of jamming in granular systems. Click here
to see You-Tube videos of her research, here
to see a poster from the Dynamics Days 2011 Conference, and here
to see a poster from the IFPRI-NSF Conference 2011 (where Jie won the poster contest). Bottom left is a picture of her sample image; the image won first prize in a visualization contest sponsored by the Duke Engineering Department and was named "Picture of the Day" by the National Science Foundation.