Research Projects
Granular systems are composed of discrete, macroscopic grains that interact via dissipative and short-range interactions. Our work has focused on dry assemblies, which have purely repulsive interactions. Examples range from pharmaceutical powders where the particle sizes are on the micrometer scale, to geological flows where the particles may be several meters in size. Since individual grains are large, thermal energy at room temperature is unable to displace them. Thus, granular materials are often termed `athermal'. A key feature of granular packings is that they are jammed and possess a nonzero yield stress. When stresses below the yield stress are applied these systems remain static, while particle rearrangements occur and these systems flow when stresses above the yield stress are applied. However, it is often difficult to predict the precise value of the yield stress because it can depend on the construction history, local structural properties, and how stresses are applied. In addition, the response of granular systems to applied forces is often spatially nonuniform and time-dependent. For example, shear banding and stick-slip behavior are common in slowly sheared granular systems. In our recent studies, we have employed computer simulations (for example molecular dynamics) to model granular materials and gain a deeper understanding of the complex phenomena that occur near jamming in these systems. We are also interested in using these simulations to identify interesting and new research directions that can be explored in future experimental and theoretical studies of these systems. We are pursuing several projects related to jamming in granular media and other athermal systems.
A. Random Close Packing
Many experimental and computational studies have found that dense amorphous packings of smooth, hard particles frequently possess packing fractions near what has been termed `random close packing', which is approximately 0.64 in 3d monodisperse and 0.84 in 2d polydisperse systems. However, recent studies have pointed out that different protocols can yield jammed configurations over a range of packing fractions and, thus the concept of random close packing is somewhat murky. We are interested in determining under what circumstances the random close packed state can be reinterpreted as the most likely random jammed state. This idea is motivated by our recent results that show that the probability to obtain a jammed (mechanically stable) configuration at a particular packing fraction becomes sharply peaked near the random close packed density in the large system limit as shown in Fig. 1.
Fig. 1: Probability to obtain a jammed configuration at each packing fraction as a function of system size N for 2D bidisperse systems.
Does the shape and location of the distribution of jammed mechanically stable states depend on the algorithm used to create them? To investigate this, we studied small 2d systems in which we can enumerate nearly all of the possible jammed states. We generate the jammed granular packings by combining successive growth/shrinkage of soft particles and energy minimization until the system is in a mechanically stable state with infinitesimal particle overlaps.
After carrying out the requisite exponentially large number of trials, we decomposed the probability to obtain a jammed state at a given packing fraction into the density of jammed packing fractions and the frequency with which each distinct jammed state occurs. The frequency distribution depends on the protocol, while the density of jammed packing fractions does not. In 2d bidisperse systems, we have shown that the shape and location of the probability distribution are controlled by the protocol-independent contribution. Thus, defining random close packing as the most likely random jammed state appears to be robust at least in the 2d bidisperse systems we studied. We are currently studying the distribution of jammed states in 2d and 3d systems composed of frictionless, monodisperse particles to determine whether there is a connection between the most likely jammed configuration and random close packing in these systems. These studies are more difficult because monodisperse systems are prone to crystallization, especially in 2d. We are also investigating systems composed of anisotropic particles such as rigid dimers to understand how rotational degrees of freedom affect the distribution and properties of jammed packings. Packings of monodisperse disks and rigid dimers are shown below in Fig. 2.
Fig. 2: (a) Polycrystalline domains in a jammed packing of 256 monodisperse particles in 2d. (b) Amorphous packing of 32 rigid dimers in 2d.
B. Frequency Distribution
Do different jammed mechanically stable configurations occur with equal probabilities? We have found that certain jammed states are extremely frequent, while others are orders of magnitude more rare. For example, as shown in the figure below, the frequency distribution of jammed configurations depends exponentially on packing fraction with the most frequent states occurring at the highest packing fractions. Even more striking, we have found that distinct jammed configurations at the same packing fraction can occur with drastically different frequencies. How can distinct jammed states at nearly the same density have such different frequencies? We are currently carrying out studies to understand what sets the frequency with which jammed states occur. We have evidence that suggests that features of the potential energy landscape in the case of soft particles or the free energy landscape in the case of hard particles control the frequency distribution. These features do not depend on the protocol used to generate the jammed configurations.
Fig. 3: (a) Frequency distribution as a function of packing fraction for a 10-particle bidisperse system in 2d. (b) Accumulated frequency distribution over a narrow range of packing fraction for the same system in (a).
We are currently performing studies to probe the topographical features of the potential energy surface near jammed states. To do this, we have added thermal fluctuations to jammed configurations (as shown in Movie 2 below) and measured how long it takes these systems to break, i.e. switch from one jammed state to another. These measurements will allow us to obtain the heights of energy barriers that separate potential energy minima. We have evidence that the energy barriers near infrequent states are relatively low, while those near frequent states are relatively high. We have also measured the shape of configuration space near jammed states and have evidence that it is not isotropic. Most directions are highly constrained; large fluctuations only occur along a few directions in configuration space. We believe that these results for small systems are generic and may help explain slow dynamics and aging behavior in large jammed and glassy systems.
Movie 2: The breaking of a jammed configuration in the presence of thermal fluctuations.
C. Shear Banding
In contrast to simple Newtonian fluids, sheared granular materials frequently form shear bands, in which shear is localized to several layers near the shearing wall while the remainder of the system is nearly static. An improved understanding of why shear bands form is important to the myriad industries that process and transport granular materials. However, there are many factors that can contribute to the formation of highly nonlinear velocity profiles in sheared granular materials, such as dynamic and static friction, gravity, inelasticity, and the Couette geometry of the shearing cell. In an attempt to isolate the primary causes of shear banding, we have carried out a series of molecular dynamics simulations of model granular materials undergoing boundary-driven shear flow. We first focused on frictionless systems in a 2d planar geometry in the absence of gravity. We found that nonlinear velocity profiles are unstable in these systems. Nonlinear velocity profiles formed at short times, but evolved toward linear profiles on time scales much longer than the time it takes a shear wave to traverse the system. In addition, the granular temperature profiles became spatially uniform at long times. The time evolution of the velocity and granular temperature profiles are shown in Fig. 4 below.
Fig. 4: (a) Velocity and (b) granular temperature profiles as a function of time after the initiation of boundary-driven planar shear flow in 2d frictionless systems. The filled (dashed) lines indicate the profile at the longest (shortest) times.
These results imply that in order to obtain stable nonlinear velocity profiles, the system must be able to maintain a sufficiently large granular temperature difference across the system at long times. To demonstrate this, we have also studied granular systems undergoing both boundary-driven shear flow and vertical vibrations. We found that nonlinear velocity profiles were stable at long times in these systems because the vertical vibrations could maintain a sufficiently large granular temperature gradient across the system. These results emphasize that studying the transport of granular temperature (and other measures of temperature) is important to understanding velocity profiles in these systems. More recently, we have begun investigating sheared granular systems composed of particles with dynamic and static friction and in Couette shearing cells. Movie 3 below shows that stable shear bands form readily in these systems. These studies will address which of the features, inelasticity, friction, or the curved geometry, determine the shape of the velocity profiles in granular systems.
Movie 3: Frictionless granular materials undergoing Couette shear flow in 2d. The inner ring rotates at fixed angular velocity, while the outer ring is stationary. The tagged particles identify the location of the shear band.
D. Definitions of Yield Stress
A distinctive feature of dense granular systems is that they possess a nonzero yield stress. They are able to resist stresses below the yield stress, but flow when stresses above the yield stress are applied. Does the value of the yield shear stress depend on how it is measured? Is more force required to initiate shear flow rather than to maintain slow steady shear flow? To begin to address these questions, we performed two different measurements of the yield shear stress in frictionless granular systems. First, we measured the shear stress required to initiate sustained shear flow (for more than say a strain of 10). We also measured the average shear stress in steadily sheared systems in the limit of zero shearing velocity. We found that these two measurements give different results, but the difference approaches zero in the large system size limit. The rate of convergence of the two measurements depends on important parameters such as the density of the system and hardness of the interaction potential between grains. In future studies, we will investigate whether friction causes these two measures of the yield shear stress to differ even in the infinite system size limit.
E. Cohesive Interactions between Particles
Purely repulsive interactions provide an excellent starting point to study the fundamental properties of jamming in granular materials. This model is useful because it captures the effects of geometric constraints on particle rearrangement and mechanical response under confinement, while using very simple assumptions about the microscopic interactions between particles.
In many experimental settings, however, there is also cohesion between particles. Cohesion arises from many sources, including the formation of liquid bridges in humid conditions, van der Waals forces between small particles, and interactions between asperities on grains with sub-micron roughness. This attractive force has been neglected in many previous studies since it is assumed to produce only second-order effects. While many features of the jamming transition do remain unchanged with the addition of a small, short-ranged attractive force between grains, there are important effects that alter even the qualitative features of both the attainable packings and their mechanical response.
We are studying the effects of cohesion on granular solids using a minimal model that introduces a small attractive force between contacting particles. Packings are obtained numerically using a conjugate gradient algorithm to minimize the energy at constant density. We then conduct mechanical tests to determine the elastic moduli, and to observe how elasticity gives way to plasticity when the material breaks. The results from these tests are then compared to the behavior of materials with purely repulsive interactions.
One of the most striking observations is that packings in cohesive systems can have a very wide range of packing fractions. While the packing fraction of purely repulsive granular solids tends to random-close-packing in the thermodynamic limit, the packing fraction in cohesive systems can take on almost any value. This arises from the formation of gel-like networks stabilized by the attractive bonds between particles. In our preliminary studies on two-dimensional systems we have observed mechanically stable packings of cohesive disks with packing fractions as low as 0.51 and as high as 0.85.
Fig. 1E: Mechanically stable packings of cohesive grains with (a) large packing fraction 0.85 and (b) small packing fraction 0.51. Note the single bond in the center that holds the packing together.
In Fig. 1E we display two pictures of cohesive packings: one at a high packing fraction of 0.84 and another at a low packing fraction of 0.51. As the packing fraction is reduced the mechanically stable states tend to become more heterogeneous, containing void spaces of various sizes and long quasi-one-dimensional chains. The relative number of attractive bonds also increases.
Fig. 2E: Pressure as a function of packing fraction during isotropic expansion of a mechanically stable packing for systems with (a) purely repulsive interactions and (b) both repulsive and attractive interactions.
In addition to characterizing the distribution of packing fractions at which mechanically stable cohesive grain packings occur, we are interested in the mechanical response of the material to uniform compression/expansion and shear strain. In Fig. 2E we plot the response of both cohesive and non-cohesive granular materials to uniform, quasi-static expansion. We peform these mechanical tests by reducing the packing fraction at a slow rate, minimizing the potential energy at each step. In both plots we see that the pressure has a series of continuous (elastic) branches, separated by sudden plastic rearrangements. It is immediately evident that the elastic and plastic properties of cohesive systems are drastically different from those in purely repulsive systems. First, plasticity plays a more important role in cohesive granular packings. Second, in the cohesive case, the continuous segments have significantly different slopes, which shows that the network geometry affects the elastic moduli. Finally, cohesion presents the possibility of jamming under tension, and therefore the pressure can (and does) become negative. In the purely repulsive case, the network is always destroyed when the pressure tends to zero.
II. Jamming in quasi-one dimensional and other confined systems
In the previous section, we discussed our ongoing research of jamming phenomena in dense granular materials in 2d and 3d. Because these systems are in 2d and 3d and have dissipative interactions, analytical calculations in these systems are often extremely difficult. Thus, we are also performing investigations of jamming and glassy behavior in simpler quasi-1d systems, in which particles are not allowed to switch order. In these systems, we are able to compare our numerical simulations with analytical results.
Movie 4: Brownian motion in figure-8 system with 10 particles. The figures from left to right shows the dilute, near dynamical arrest and close packing density of figure-8 system
Click here for high res version
As an example, we show a simulation of a 1d system with an intersection in Movie 4. The geometry is in the shape of a figure-8. The dynamical rules for this system are simple: each particle performs a random walk, but the particles are hard and not allowed to overlap one another. The presence of the intersection causes the dynamics to become dramatically slower than Brownian motion in a 1d system without an intersection. The mean-square displacement versus time for the figure-8 system is shown in Fig. 5 for several different packing fractions. We are currently performing calculations of the probability for gaps of specific sizes to occur and the length of time that such gaps remain open. Measuring these quantities will allow us to better understand the slow dynamics displayed by these systems. Future projects in this area include studying the dynamics of Brownian particles in quasi-1d dimensional systems with multiple intersections and in the presence of a biasing field and in narrow channels with varying widths.
Fig. 5: Mean-square displacement as a function of time for several packing fractions in a figure-8 system with 10 particles.
Trajectory of particles in figure-8 system with 10 particles. The left figure shows the trajectories in the short time scale whereas the right figure shows the trajectories in the diffusive time scale. In the diffusive time scale particles tend to make collective jump.
The Manhattan Model
Manhattan Model consists of hard particles moving in narrow
channels with multiple intersections.
click for here The Manhattan Model movie
III. Effective temperatures in driven, dissipative systems
In athermal systems such as granular materials, thermal energy at room temperature is unable to displace individual grains. Therefore, externally applied forces such as shear stress, vibration, gravity, or air drag are required to induce motion in these systems. When sufficiently large external forces are applied, athermal systems flow, causing grain rearrangements and fluctuations in macroscopic quantities like shear stress and pressure. For example, in Movie 5 below, we show a molecular dynamics simulation of a model granular system undergoing linear shear flow. Are the shear-induced fluctuations that occur in this system similar to thermal fluctuations in equilibrium systems? Can a thermodynamically consistent effective temperature be defined for such driven, dissipative systems?
To address these questions, we have investigated effective temperatures defined from equilibrium fluctuation-dissipation (FD) relations and measured them in sheared and vibrated granular systems as a function of the driving intensity, density, and material properties such as inelasticity and friction. Our preliminary results indicate a number of surprising and noteworthy results:
- The effective temperature depends on the time scale over which it is measured, i.e. the effective temperature measured at short times can be very different from that measured at long times.
- The long-time effective temperature is controlled by the pressure of the system in the quasi-static regime, while the short-time granular temperature is not.
- The shape of the response and correlation functions used to measure the long-time effective temperature depend on the observable studied and whether the measurements are performed at constant volume or constant pressure.
These results point out that a compelling thermodynamic description of these systems is yet to emerge, and we are actively pursuing this line of research.
Movie 5: Linear shear flow in an athermal system with periodic boundary conditions. The shear induces fluctuations in the pressure, shear stress, and other physical quantities.
IV. Phase transitions in charged colloids
If a charged colloidal system is prepared so that particle surface charge fluctuations are small, we expect that an equilibrium phase transition from a disordered liquid to an ordered crystalline phase will occur as temperature is lowered. When the system is quenched below the freezing temperature, this transition happens rapidly and there are clear static structural signatures of the phase transition, for example in the pair correlation function or scattering intensity. Freezing can also be detected in dynamic measurements, for example, in a liquid the mean-square discplacement behaves diffusively at long times whereas it is constant at long times in a crystalline state.
Now consider increasing the charge polydispersity in colloidal systems while holding other variables fixed. As long as the magnitude of the charge polydispersity is small, the freezing transition will not be altered significantly. As the polydispersity is increased further, the system slows down enormously. With strong charge polydispersity, the system can become trapped in a metastable disordered glassy state characterized by anomalously large viscosities and relaxation times that exceed experimental time scales. In these highly polydisperse systems, the lowest energy equilibrium phase is often not known.
Several short-term goals of this project include
- Determining numerically the phase diagram of charged colloids as a function of temperature and charge polydispersity.
- Measuring the mean-square displacement and intermediate scattering intensity to study the slow dynamics near the glass transition in these systems.
- Studying the aging behavior in long-range repulsive glasses and comparing this behavior to aging phenomena found in glassy systems with short-range repulsions.
- Measuring dynamical heterogeneities and lengthscales that characterize them in repulsive glasses.
- Comparing these findings to experiments on charged colloidal systems in collaboration with Prof. Eric Dufresne from the Departments of Mechanical Engineering and Physics at Yale.
In Fig. 4a and Fig. 4b, we show movies from our molecular dynamics simulations for colloidal systems interacting via Yukawa potentials for small and large charge polydispersity. For small polydispersity a crystal structure, with small defects, is clearly visible. For large polydispersity and the same temperature, the crystal structure no longer exists, but there are mesoscopic regions that seem to be denser than their surroundings. This suggests that dynamic correlation lengths may arise in the glass state.
Fig. 4a: Movie from molecular dynamics simulation of a charged colloidal system with small charge polydispersity.
Fig. 4b: Movie from molecular dynamics simulation of a charged colloidal system with large charge polydispersity.
V. Constrictions in narrow channels
A general feature of particulate systems is that the dynamics becomes sluggish as the density increases. In cases where there is no significant particle ordering, theoretical treatments of the slowdown are difficult to formulate. Therefore, simple models can serve to elucidate the necessary features of dynamic slowdown and provide new insights to theory.
Here we study the flow of a collection of particles that are constrained to the geometry of Fig. 5. The particle-particle and particle-wall interactions are infinitely stiff and elastic, and the particles exhibit Brownian motion. In addition there is a small mean force on each particle towards the left.
Fig. 5a: Movie of particulate flow in the presence of a constriction.
For low densities and/or small mean force the system flows and reaches a steady state that is determined by the mean flux through the constriction. For intermediate values of the density or mean force, the flow becomes non-stationary and intermittent. The mean flux fluctuates between finite values (flowing states) and zero (jammed states). For large values of the density or mean force, the flow becomes completely jammed.
We are interested in understanding the origin of the slowdown and jamming, as a function of density, mean force, and the size of constriction. In addition to motivating theoretical advances in more general cases, this geometry is quite relevant in biological and engineering contexts. Moreover, it can be accurately studied in the context of colloidal experiments.
login to post comments