Session H13: Focus Session: Jamming Theory and Experiment I

Jamming in Disordered and Ordered States: From RLP to FCC

The concept of jamming was originally introduced in the context of zero-temperature, frictionless sphere packings through which the jamming transition was identified with the more familiar idea of random close packing. More recently, the jamming behaviour for particles with friction has led to a practical definition of the less well-defined random loose packed limit. However, there are a number of subtleties associated with jamming that extend these concepts further. Here we implement a range of protocols to generate jammed packings both with and without friction, and find that the jamming transition actually consists of a finite region in packing fraction depending on the protocol used to create the jammed state. Furthermore, we examine how it is possible to tune the structural properties of jammed packings from the disordered regime through to the ordered face centred cubic lattice, and the subsequent changes in the jamming properties as the structure is manipulated.   

Long wavelength behavior of the static structure factor in jammed packings

There are several features associated with the jamming transition in monodisperse sphere packings. One recently reported property is the anomalous long wavelength behavior of the static structure factor, $S(k)$. An unusual linear dependence with the wavenumber $k$, becomes increasingly pronounced on approach to the jamming transition. However, it remains unclear how polydispersity and force model affect this behavior. Here, we study the structure factor of jammed disordered bidisperse sphere packings using computer simulations, especially its behavior in the long wavelength regime ($k \to 0$). We evaluated the structure factor using an appropiate formalism for polydisperse systems and extract information on the susceptibility in the low-$k$ limit.   

Topology of the force field in a jammed granular systems exposed to an intruder

It is well known that the structure of forces and stresses in granular systems goes through significant changes close to jamming. In this talk, we will present precise and objective measures of these changes based on topological properties of the force field in granular systems exposed to compression and shear. Then, we will discuss how these measures evolve in granular systems during an impact of a large intruder. We will particularly concentrate on the role of packing, polydispersity, and friction on structure of the force field.   

Critical Scaling of Shearing Rheology at the Jamming Transition of Soft Core Frictionless Disks

We perform numerical simulations to determine the shear stress and pressure of steady-state shear flow in a soft-disk model in two dimensions at zero temperature in the vicinity of the jamming transition $\phi_J$. We use critical point scaling analyses to determine the critical behavior at jamming, and we find that it is crucial to include {\it corrections to scaling} for a reliable analysis. We find that the relative size of these corrections are much smaller for pressure than for shear stress. We furthermore find a superlinear behavior for pressure and shear stress above $\phi_J$, both from the scaling analysis and from a direct analysis of pressure data extrapolated to the limit of vanishing shear rate.   

Glassiness, Rigidity and Jamming of Frictionless Soft Core Disks

The jamming of frictionless bi-disperse soft core disks is considered, using a variety of different protocols to produce the jammed state. We find, consistent with earlier works, that cooling and compression can lead to a broad range of jamming packing fractions $\phi_J$, depending on cooling or compression rate and on initial configuration. Such $\phi_J$ show no clear upper bound as the cooling or compression rate decreases. In contrast, we show that shearing leads to a jamming transition to a disordered solid, with a well-defined, non-trivial, value of $\phi_J$ as the shearing rate vanishes. We show that shearing breaks up the particle clustering (the precursor to phase separation) that can lead to increasing values of $\phi_J$ under slow cooling or compression, and argue that the process of shearing creates a well-defined ensemble that is independent of the starting configuration.   

Exact tools for 2D granular packings

Building on the loop force formulation of Ball and Blumenfeld\footnote{PRL 88 115505, 2002}, a new, exact potential formulation is given for two dimensional, static packings of frictional, monodisperse disks. Using degree-of-freedom counting and explicit constructions, it is shown that the natural graph for analysis of stress distribution in such packings is the Delaunay triangulation. Edges of this graph which do not correspond to contacts yield ``virtual contact'' vectors, which are shown to be of great physical importance. In particular, the new potential satisfies force and torque balance identically and is subject only to the Coulomb constraint and a new set of physically transparent constraints on the ``virtual contacts.'' Using the new coordinates, previous results on the contact force distribution are rationalized, and a unified framework is presented for understanding the sources of correlation between contact forces. A new maximum-entropy argument is presented to derive the contact force distribution, and the dependence on shear, friction, and coordination number is discussed.   

Particle-to-Particle Dynamics in a Granular Pile Subject to Cyclic Shear

We report a study of the particulate motions of a granular pile under cyclic shear and how they relate to the bulk rheological properties of the pile. Using a laser sheet scanning technique, we track the trajectories of all of the particles within a section of a split-bottom shear cell. We shear the pile quasistatically to ensure rate independence of shear stress. Immediately after reversal of the shear direction, we observe a transient drop in shear stress of the pile over a characteristic strain. We construct a network of nearest neighbors that roll or slide past one another between frames. We find that, for strain amplitudes less than the aforementioned characteristic strain, rolling/sliding links are extinguished with higher frequency than for larger amplitudes. We also report other particle level measures, such as mean squared displacements, for various amplitudes of oscillatory shear.   

Ratio of effective temperature to pressure controls the dynamics of sheared hard spheres

Using molecular dynamics simulations, we calculate the effective temperature, $T_{\rm eff}$, and the pressure, $p$, of steadily sheared mixtures of hard spheres of mass $m$ and diameters $\sigma$ and $1.4 \sigma$ in contact with a thermal reservoir at temperature $T$. We vary the packing fraction, $\phi$, and the shear stress, $\Sigma$. We define $T_{\rm eff}$ from the ratio of correlations to response and show that different correlation-response relations yield a consistent numerical value $T_{\rm eff}\ge T$ that reduces to $T_{\rm eff}=T$ when $\Sigma=0$. We show that the effective temperature represents the limiting value of the effective temperature for soft spheres in the limit $p\sigma^3/\epsilon\rightarrow 0$, where $\epsilon$ is the repulsive energy scale. We find that the dimensionless ratio $T_{\rm eff}/p \sigma^3$ controls the dynamic jamming transition that occurs with decreasing shear stress and increasing packing fraction. In particular, we find that the dependence of the dimensionless relaxation time, $\tau \sqrt{p \sigma/m}$, on $T_{\rm eff}/p \sigma^3$ as shear stress is varied is quantitatively similar to the dependence of $\tau \sqrt{p \sigma/m}$ on $T/p \sigma^3$ in equilibrium.   

Shear-jammed states in granular materials

For frictionless particles with purely repulsive interactions, there is a critical packing fraction $\phi_J$ below which no jammed states exist. Experiments by Zhang \& Behringer on physical granular systems show jammed states in the regime of $\phi < \phi_J$ can be created by the application of shear stress. Compared to the states above $\phi_J$, the shear-jammed states are mechanically more fragile, but they resist shear. These shear-jammed states cannot exist under isotropic stress. Rather, their formation require the anisotropic contact network as a backbone which is created by an applied shear stress. The anisotropic components of the stress tensor and contact network fabric tensor form a classic hysteresis loop suggesting an analogy to ferromagnetic behavior and critical phenomena. These new states must be incorporated into a more general jamming picture. We also carry out extensive analysis on shear-jammed states and find local stress fluctuations are controlled by their respective global pressures. To explain the scaling of local stress fluctuations, we construct a mean-field model based on the entropy of stress configurations.   

Quasistatic flows near jamming: The role of inertia and dissipation

We perform massively parallel computer simulations of granular particles at fixed shearing rate and density near the onset of jamming. The microscopic dynamical model contains two types of damping; one which damps the \emph{absolute} motion of a particle with respect to a homogeneously shearing background (as in SLLOD type approaches) and another which damps the \emph{relative} motion of a particle with respect to its near-neighbors (as in discrete element approaches). We study how the damping mechanism and its strength affects the collective particle dynamics through the statistics of local particle displacements and local strains. In particular, we show that for strong, \emph{absolute} damping, the single particle displacement statistics can be similar for systems at different distances from jamming while the short-time plastic activity can vary dramatically.   

Viscoelastic response near the jamming transition

We use numerical and theoretical methods to investigate oscillatory rheology in soft sphere packings, which serve as a minimal model for foams, emulsions, and other complex fluids that undergo a jamming transition. Although the zero frequency (elastic) properties of jammed media are well documented, far less is known about their viscoelastic response. We demonstrate that the frequency-dependent storage and loss moduli display critical scaling with distance to the jamming point. This behavior is governed by a diverging time scale that separates quasistatic response from a critical regime in which viscous and elastic forces contribute equally to the stress. We provide scaling arguments for all of the relevant critical exponents.   

Phase transition kinetics in the site dilute Ising model

We consider the phase transition kinetics of a quenched site dilute Ising model. To date, most studies of this model have focused on dilution-averaged quantities, such as the critical temperature and the associated critical exponents. In this talk we study how the spatial distribution of the dilution affects the local growth of the stable phase after an instantaneous quench. For an off critical quench, we find growth occurs most rapidly in areas of increased dilution for both unstable state decay and nucleation. Conversely, growth after a critical quench is accelerated in environments with relatively few vacant sites. Additionally, we consider the role of the range of interaction in these processes.   

A first-order phase transition defines the random close packing of hard spheres

Randomly packing spheres of equal size into a container consistently results in a static configuration with a density of $\sim$64\%. The ubiquity of random close packing (RCP) rather than the optimal crystalline array at 74\% begs the question of the physical law behind this empirically deduced state. Indeed, there is no signature of any macroscopic quantity with a discontinuity associated with the observed packing limit. Here we show that RCP can be interpreted as a manifestation of a thermodynamic singularity, which defines it as the ``freezing point'' in a first-order phase transition between ordered and disordered packing phases. Despite the athermal nature of granular matter, we show the thermodynamic character of the transition in that it is accompanied by sharp discontinuities in volume and entropy. This occurs at a critical compactivity, which is the intensive variable that plays the role of temperature in granular matter. This approach is useful since it maps out-of-equilibrium problems in complex systems onto simpler established frameworks in statistical mechanics.