Chakraborty lab provides new understanding on the physics of granular materials

14 07 2017

By Kabir Ramola, Ph.D

In the late 1980’s Sir Sam Edwards proposed a framework for describing the large scale properties of granular materials, such as sand or salt. In this description, similar to the well-established framework of statistical mechanics, the global properties of a complex system are determined by an average over all possible microscopic configurations consistent with a given global property. This is usually attributable to the very fast dynamics of the constituent particles making up the system. The extension of such treatments to granular systems where particles are static or ‘jammed’ represents a fundamental challenge in this field. Even so, Edwards’ conjecture postulated that for given external parameters such as volume, all possible packings of a granular material are equally likely. Such a conjecture, like Boltzmann’s hypothesis in statistical mechanics, can then be used as a starting point to develop new physical theories for such materials based on statistical principles. Indeed, several frameworks have been developed assuming this conjecture to be true.

Figure 1 : Snapshot of the system studied and illustration of the associated energy landscape at different volume fractions.

A simple illustration of this conjecture would be, if one were to pour sand into a bowl, and not bias the preparation in any way, then all the trillion trillions of configurations allowed for the grains would be equally likely. Clearly such a conjecture is utterly infeasible to test experimentally.  In a recent paper that appeared in Nature Physics, we instead performed detailed numerical computations on a theoretical system of soft disks (in two dimensions) with hard internal cores. We focused on a system of 64 disks which already pushed the limits of current computational power. We found that if one fixes the density of a given system of disks, the probability of a packing occurring depends on the pressure, violating Edwards’ proposition. However, at a critical density, where particles just begin to touch or ‘jam’, this probability remarkably becomes independent of the pressure, and all configurations are indeed equally likely to occur. This jamming point is in fact very interesting in its own right since most granular materials are found at the threshold of being jammed and ‘unjammed’. To be fair to Edwards, the hypothesis was made for ‘hard’ grains in which particles are precisely at this threshold, and therefore our numerics seem to confirm the original statement. This is the first time that this statement has been out to a direct test and will no doubt lead to many interesting directions in the future.

Links to news sources describing this article:

doi: 10.1038/nphys4168
Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming.
Stefano Martiniani, K. Julian Schrenk, Kabir Ramola, Bulbul Chakraborty & Daan Frenkel.
Nature Physics


“Granular Materials” video by NSF highlights research by Chakraborty group

19 06 2012

From NSF Multimedia Gallery: ”Granular materials — like sand, rice, or powdered pharmaceuticals — are everywhere, yet their behavior is poorly understood.  In some ways behaving like liquids, in other ways behaving like solids, such materials have unique properties and pose unique questions to answer.  From clogged coal hoppers to powdered-snow avalanches, scientists and engineers are gaining new perspective on the fundamental nature of grains.  In this video, see some of the latest research into the behavior of granular materials.”

Prof. Bulbul Chakraborty’s group website

Nature article

Shear-induced jamming

19 12 2011

From breakfast cereals to sand on a beach, granular materials are all around us. Under different conditions, these materials can exhibit liquid-like behavior (flowing) as well as solid-like behavior. The transition between solid and liquid phases has been known as the jamming transition.

The basic concept of jamming is pretty intuitive. A simple example of what can induce jamming is the following: compacting loose sand inside a container increases its density. When the container is removed, the sand can form a self-supporting pile, hence becoming jammed. Jamming has been studied extensively in numerical simulations of systems composed of idealized grains without frictional forces.  These studies find a critical density at which jamming occurs. Since these idealized granular materials are non-cohesive (no attractive forces between them)  they can become solids only through externally imposed pressure, such as through compaction, and therefore a critical density makes sense.  Real granular materials, however, have friction, and how this affects jamming is not well understood.

An experimental image of typical Shear Jammed state in a 2-D frictional granular material. The shear strain is applied in the horizontal direction. Red colored grains form the backbone of the system, which provides rigidity with respect to external shear

Newly published in Nature, are results of a collaboration between Prof. Bulbul Chakraborty’s group at Brandeis and Prof. Behringer’s group at Duke University, which show a new class of jammed states in frictional granular materials. This new class of “Shear-Jammed” states exhibits a richer phenomenology than previously seen. An initially unjammed or loose granular material can become jammed not just by increasing its density, but by applying shear strain on it while holding the density fixed. Shear-Jammed states are inherently anisotropic in their stress and grain-to-grain contact network (see photo above). The transition from an unjammed to shear-jammed state is clearly marked by a percolation of the strong force chains in all directions (see video below). The phenomenon of shear-jamming does not currently have a fundamental theoretical description. Ongoing work in Prof. Chakraborty’s group attempts to construct a theoretical framework for this non-equilibrium phase transition using a generalization of equilibrium statistical ensembles.

This video shows the evolution of the strong force cluster and transition from unjammed to fragile and eventually to SJ. The video shows experimental states created under pure shear. Green colored grains form the strong force cluster defined in the paper. Initially, the system is unjammed. As the fraction of force bearing grains increases with increasing strain, the strong force cluster percolates in the compressive (vertical) direction and we call the state fragile.  Eventually the system becomes percolated in all directions with sufficient number of force bearing grains. We call these states Shear Jammed.

see also:

Brandeis in Aspen II: Physics of granular materials

1 07 2011

by Aparna Baskaran

This post is a companion to Brandeis in Aspen I, and describes a workshop attended by Bulbul Chakraborty and Aparna Baskaran at the Aspen Center for Physics. The format of Aspen workshops is different from the usual academic workshop.  Each day has just one or two talks, which are primarily self-organized on a volunteer basis among the participants.  The format is designed to encourage  physicists working in a particular area to share research findings and enable cross-pollination of ideas in an informal and loosely structured setting. The workshop attended by Chakraborty and Baskaran was entitled “Fluctuation and Response in granular materials”. Granular materials are ubiquitous in nature and industry. Examples range from sand and other geological materials, food and consumer products, and pebble beds in nuclear reactors. Understanding and controlling the properties of granular materials impacts such diverse processes as oil recovery, nuclear pebble bed reactors, printing and copying, and pharmaceutical processing. Granular media pose difficult and unique scientific challenges that distinguish them from atomic, nano-scale, and colloidal materials. Being intrinsically out of thermal equilibrium, assemblies of grains readily become trapped in metastable states, are extremely sensitive to preparation conditions, and can have strongly time-dependent properties.  Relaxing the constraints of thermal equilibrium, however, offers an advantage by opening up possibilities for creating novel static and dynamic phases that have distinctive functional properties.
At Aspen, the one-on-one and small sub group interactions among the participants covered a wide range of topics that are at the forefront of materials research, however, the program as a whole primarily focused on two questions. The first question was: What do we understand about jamming of granular materials? Jamming is what occurs in everyday life when we are trying to get coffee beans out of a hopper and they suddenly stop flowing. We fix this by tapping on the hopper. But this same phenomenon when it happens in giant grain silos causes them to collapse. So, one of the challenges is to be able to predict jamming events. The role of the physicist here is to design and carry out experiments in minimal model systems and develop theoretical frameworks that lead to predictive models of observed phenomena. Statistical Mechanics provides a powerful theoretical tool to address this question and our own Professor Chakraborty is one of the leading experts in the theory of jamming. The participants at the workshop had several robust discussions on the current understanding of this phenomenon and theoretical and experimental challenges that remain to be addressed. The second question that the workshop focused on was : How does a dense granular material behave when sheared? Granular materials are called rheological fluids in that they exhibit shear-thinning and shear thickening behavior. In everyday life, we are all familiar with shear thinning. When we squeeze a tube of toothpaste, we are shearing it and it flows onto our brush. But once on the brush it stays put. This behavior is called shear thinning. Understanding rheology of granular materials is important for diverse applications ranging from pharmaceutical processes to being able to print well. The participants discussed in detail the physics of sheared granular materials and shared insight obtained from theory, simulations and experiments. All participants departed the workshop invigorated by the robust exchange of ideas, ready to address the challenges presented by these complex materials.

Collective behaviors in active matter

13 06 2011

L. Giomi, L. Mahadevan, B. Chakraborty, M. Hagan

Active matter is describes systems whose constituent elements consume energy and are thus out-of-equilibrium. Examples include flocks or herds of animals, collections of cells, and components of the cellular cytoskeleton. When these objects interact with each other, collective behavior can emerge that is unlike anything possible with an equilibrium system. The types of behaviors and the factors that control them however, remain incompletely understood. In a recent paper in Physical Review Letters, “Excitable patterns in active nematics“, Giomi and coworkers develop a continuum theoretical description motivated by recent experiments from the Hagan Group and Chakraborty Group at Brandeis in which microtubules (filamentous cytoskeletal molecules) and clusters of kinesin (a molecular motor) exhibit dramatic spatiotemporal fluctuations in density and alignment. Specifically, they consider a hydrodynamic description for density, flow, and nematic alignment. In contrast to previous theories of this type, the degree of nematic alignment is allowed to vary in space and time.  Remarkably, the theory predicts that the interplay between non-uniform nematic order, activity and flow results in spatially modulated relaxation oscillations, similar to those seen in excitable media and biological examples such as the cardiac cycle. At even higher activity the dynamics is chaotic and leads to large-scale swirling patterns which resemble those seen in recent experiments. An example of the flow pattern is shown below left, and the nematic order parameter, which describes the degree of alignment of the filaments, as shown for the same configuration below right. These predictions can be tested in future experiments on systems of microtubules and motor proteins.

The system behavior for an active nematic at high activity. (left) The velocity field (arrows) is superimposed on a plot of the concentration of active nematogens (green=large concentration, red=small concentration). (right) A plot of the nematic order parameter, S,  (blue=large S, brown=small S) is superimposed on a plot of the nematic director (arrows). The flow under high activity is characterized by large vortices that span lengths of the order of the system size and the director field is organized in grains.


Prolonging assembly through dissociation

10 06 2011

by division of science

Microtubules are semiflexible polymers that serve as structural components inside the eukaryotic cell and are involved in many cellular processes such as mitosis, cytokinesis, and vesicular transport. In order to perform these functions, microtubules continually rearrange through a process known as dynamic instability, in which they switch from a phase of slow elongation to rapid shortening (catastrophe), and from rapid shortening to growth (rescue). The basic self-assembly mechanism underlying this process, assembly mediated by nucleotide phosphate activity, is omnipresent in biological systems.  A recent paper, Prolonging assembly through dissociation: A self-assembly paradigm in microtubules ,  published in the May 3 issue of Physical Review E,  presents a new paradigm for such self-assembly in which increasing depolymerization rate can enhance assembly.  Such a scenario can occur only out of equilibrium. Brandeis Physics postdoc Sumedha, working with Chakraborty and Hagan, carried out theoretical analysis of a stochastic hydrolysis model to demonstrate the effect and predict features of growth fluctuations, which should be measurable in experiments that probe microtubule dynamics at the nanoscale.

Model for microtuble dynamics. All activity is assumed to occur at the right end of the microtubule (denoted as ">")

The essential features of the model that leads to the counterintuitive result of depolymerization helping assembly are (a) stochastic hydrolysis that allows GTP to transform into GDP  in any part of the microtubule, and (b) a much higher rate of GTP attachment if the end of the microtubule has a GTP-bound tubulin dimer, compared to a GDP-bound tubulin dimer.    Process (a) leads to islands of GTP-bound tubulins to be buried deep in the microtubule.   Depolymerization from the end reveals these islands and enhances assembly because of the biased attachment rate (b).  The simplicity of the model lent itself to analytical results for various aspects of the growth statistics in particular parameter regimes.   Simulations of the model supported these analytical results, and extended them to regimes where it was not possible to solve the model analytically.  The statistics of the growth fluctuations in this stochastic hydrolysis model are very different from “cap models” which do not have GTP remnants buried inside a growing microtubule.   Testing the predictions in experiments could, therefore, lead to a better understanding of the processes underlying dynamical instability in-vivo and in-vitro.   An interesting question to explore is whether the bias in the attachment rates is different under different conditions of microtubule growth.

A lattice of interacting chemical oscillators

24 05 2011

by Zahera Jabeen

At Brandeis, there is a long tradition of interesting experiments on the Belousov-Zhabostinsky reaction system, with the legendary Zhabotinsky himself having been a part of the fraternity. This reaction system shows interesting oscillatory and stable patterns (see videos on Youtube). In the Fraden lab, an oil emulsion of micron-sized water droplets containing the BZ reactions, was shown to show interesting synchronization properties and complex spatial patterns [Toiya et al, J. Phys. Chem. Lett. 1, 1241 (2010)]. A coupling between the droplets due to preferential diffusion of an inhibitory reactant (bromine) in the oil medium was seen to be responsible for these collective phenomena. In a new paper titled “Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions” in Phys. Rev. E, Physics Ph. D student Michael Giver, postdoc Zahera Jabeen and Prof. Bulbul Chakraborty show that neighboring oscillators can be modeled as Kuramoto phase oscillators, coupled nonlinearly to its nearest neighbors. The form of the coupling chosen is repulsive, which favors out of phase synchronization. They show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. A linear chain of these repulsively coupled oscillators shows anti-phase synchronization, in which neighboring oscillators show a phase difference of π The phase difference between the neighboring oscillators when placed on a ring however depends on the number of oscillators. In such a case, the locally preferred phase difference of π is ruled out for an odd number of oscillators, as this may lead to frustration. When these oscillators are placed on a triangular lattice in two dimensions, the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2 π /3. Interestingly, domains with different helicities form in the lattice. In each domain, the phases of any three neighboring oscillators can vary continuously in either clockwise or an anti-clockwise direction. Hence, phase difference between the nearest neighbors are seen to be ±2π /3 in the two domains (See figure). A phase difference of π is seen at the interfaces of these domains. These domains can grow in time, resembling domain coarsening in other statistical studies. At large coupling strengths, the domains freeze in size due to frequency synchronization of all the oscillators. Hence, an interplay between frequency synchronization and phase synchronization was seen in this system. Ongoing studies in the BZ experimental setup at the Fraden Lab, find correlations with the above results. Hence, insights into a complex system like the BZ oscillators could be gained using the phase oscillator formalism. The research was supported by the ACS Petroleum Research Fund and the Brandeis MRSEC. Michael Giver is a trainee in the Brandeis NSF-sponsored IGERT program Time, Space & Structure: Physics and Chemistry of BIological Systems

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