Jamming of soft particles: geometry, mechanics, scaling and isostaticity

Abstract

Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. Here we review the current understanding of the transition to jamming and the nature of the jammed state for disordered packings of particles that act through repulsive contact interactions and are at zero temperature and zero shear stress. We first discuss the breakdown of affine assumptions that underlies the rich mechanics near jamming. We then extensively discuss jamming of frictionless soft spheres. At the jamming point, these systems are marginally stable (isostatic) in the sense of constraint counting, and many geometric and mechanical properties scale with distance to this jamming point. Finally, we discuss current explorations of jamming of frictional and non-spherical (ellipsoidal) particles. Both friction and asphericity tune the contact number at jamming away from the isostatic limit, but in opposite directions. This allows one to disentangle the distance to jamming and the distance to isostaticity. The picture that emerges is that most quantities are governed by the contact number and scale with the distance to isostaticity, while the contact number itself scales with the distance to jamming.

Z-cone model for the energy of an ordered foam

“…13 Theories of jamming oen invoke the kind of quadratic or Hertzian forces 14,15 that we now know to be inappropriate for foams, in the wet limit. 5,8 In this way, the cone model ties together a number of previous results with a single coherent picture, with the correct asymptotic forms, based entirely on analytic expressions.…”

Soliton Attenuation and Emergent Hydrodynamics in Fragile Matter

“…For comparison, we plot the corresponding longitudinal (black squares) and transverse (black circles) dispersion curves for a highly compressed jammed packing (far from the critical density), prepared at a pressure of P ∼ 10 −1 . For precompressed jammed packings with an average initial overlap between disks δ and an interaction potential of the form δ α , the pressure is related to the overlap via the relation P ∼ δ α−1 , while the shear modulus is expected to scale as G ∼ δ α−ð3=2Þ and the bulk modulus as B ∼ δ α−2 [1]. For Hertzian interaction with α ¼ 5 2 , the shear and bulk moduli therefore scale as G ∼ δ 1 and B ∼ δ 1=2 respectively.…”

“…For Hertzian interaction with α ¼ 5 2 , the shear and bulk moduli therefore scale as G ∼ δ 1 and B ∼ δ 1=2 respectively. In addition, the energy of the packing due to precompression is E ∼ δ α , and therefore, G ∼ E 2=5 and B ∼ E 1=5 [1].…”

“…However, it is the geometry and topology of a material's architecture that ultimately controls the strength of the elastic response. For example, if we remodel a solid block of steel into a granular aggregate of steel balls just in contact with their nearest neighbors, both the linear elastic moduli and sound speed of the aggregate drop to zero [1,2]. This extreme softness, which we term fragility, can have a local or global origin.…”

“…The critical point at which a packing of soft repulsive grains become fragile (unjams) can be accessed experimentally by progressively reducing the pressure, or alternatively, the average overlap, to zero [1]. Right at the critical point, even the tiniest mechanical strains must propagate in the form of supersonic solitons and shocks since the linear sound speed is zero [7,8].…”