The jamming transition

Substantial evidence indicates that disordered granular materials feature a critical tran- sition as a function of packing fraction, applied stress, or temperature known as the jamming transition [31, 32]. At this transition, characterized by a volumetric compression, the pack- ing changes from a loose and free-flowing state to a rigid and fully confined state. Notably, unlike the conventional notion of a phase transition [33], no change in overall structure occurs, aside from the aforementioned compression.

Despite the lack of an instantaneous change in structure, the jamming transition is

nevertheless an instantaneous transition under certain assumptions1. The critical condition

governing this transition is that all degrees of freedom of the system must be constrained [31]. For frictionless spheres in three-dimensional space, an average of six contacting neighbors per sphere [30, 31] are needed to fully constrain the three translational degrees of freedom for each particle in the system, whereas for frictionless disks in two-dimensional space, an average of four contacting neighbors per disk are needed.

1

These assumptions include that the constituents are rigid, smooth, and frictionless, that the applied load is constant, the loading rate is quasistatic, and the boundary conditions fully confine the granular packing.

The critical number of contacts per constituent can be determined for spheres or disks with friction as well, but is more dependent on the assumptions made regarding the nature of contact at the interface. For example, accounting for sliding, rolling, or torsional friction as well as adhesion can eliminate additional degrees of freedom. In these cases, every contact eliminates more than one degree of freedom. Importantly, by incorporating friction, one must also account for rotational degrees of freedom of the constituents, thus increasing the total degrees of freedom of the system. For disks in two-dimensional space with friction, the critical number of contacts is approximately three [32, 1], and for spheres with sliding and torsional friction but no adhesion, the critical number of contacts is four [30], but these figures can vary depending on the friction and adhesion interactions [32].

For a sufficiently large system of frictionless, monodisperse spheres, the jamming transi-

tion occurs at a packing fraction of approximately ΦJ = 64% [34, 35, 30, 36]. This value may

be compared with the crystalline close packing limit, which is 74.1% [34]. The crystalline close packing limit benefits from using long-range order to find a particular configuration where constituents are packed tightly; therefore, crystalline close packing also corresponds with a greater number of contacts per constituent—twelve—compared with six for random close packing. Still, elastic disks or spheres may be compressed to higher packing fractions beyond the jamming transition, corresponding to the jammed region in Figure 1.1a. An

increase in shear stress τ can cause the packing to unjam and flow.

Simulations of elastic spheres have demonstrated that polydispersity in sphere sizes can result in a packing fraction greater than the random close-packing limit [36]. The presence of smaller spheres among a matrix of larger spheres allows for the smaller spheres to fill in voids that are too small for the larger spheres. Conversely, incorporating friction or adhesion allows for a disordered packing to transition to a rigid state below the random close packing limit [36], because fewer than six contacts per constituent are needed to eliminate all degrees of freedom, as described previously. Armed with this knowledge, it is possible to surmise some information about a rigid packing by determining its porosity or packing fraction alone. Specifically, a rigid disordered packing with a packing fraction

(a) (b)

Figure 1.1: Schematic phase diagrams of the jamming transition for (a) frictionless granular materials and (a) granular materials with friction, showing the presence of the fragile and shear-jammed states which develop from particles with friction. The vertical axis indicates the shear stress, while the horizontal axis indicates the packing fraction. Figure reproduced with permission from Ref. [1].

greater than the random close packing limit must necessarily have either polydispersity, irregularity in constituent shapes, or close-packed crystalline regions. A rigid packing with a packing fraction less than the random close packing limit must include friction, and in absence of external confinement, must contain adhesion as well.

In addition to allowing for jamming below the close packing limit, friction also allows for the introduction of an additional state known as shear jamming [1]. This state was identified by performing shearing and isotropic compression experiments on two-dimensional packings of photoelastic disks. Crossed polarizers on each side of the packing allowed the authors to identify the disks which transmitted the greatest forces. In the shear-jammed state, depicted in the phase diagram in Figure 1.1b, the packing remains rigid only in the presence of shear

stress. This state may be sustained down to a packing fraction ΦS <ΦJ under some shear

stress. Unlike the jammed state, the shear-jammed state relies on inter-constituent friction to maintain structural stability. The distinction between the shear-jammed state and the fragile state also shown in Figure 1.1b is related to the shape and distribution of the force network that transmits the applied stress state through the material [1]. Specifically, the force network in the shear jammed state fully spans the system and the system remains

stable if the applied shear stress is partially relaxed. In the fragile state, the force network does not fully span the system, and the system may become unstable if the stress state is partially relaxed.

At the jamming transition, an abrupt change in mechanical properties occurs; specifi- cally, the material behavior transitions from viscous-like flow to an elastic response [31]. The shear modulus and bulk modulus both become defined at this transition, and any further compression will result in an increase in the moduli as well as an increase in the number of contacts per constituent above and beyond the critical number for stability.