Jamming Transition in Granular Systems

Abstract

Recent simulations have predicted that near jamming for collections of spherical particles, there will be a discontinuous increase in the mean contact number Z at a critical volume fraction c . Above c , Z and the pressure P are predicted to increase as power laws in ÿ c . In experiments using photoelastic disks we corroborate a rapid increase in Z at c and power-law behavior above c for Z and P. Specifically we find a power-law increase as a function of ÿ c for Z ÿ Z c with an exponent around 0.5, and for P with an exponent around 1.1. These exponents are in good agreement with simulations. We also find reasonable agreement with a recent mean-field theory for frictionless particles. DOI: 10.1103/PhysRevLett.98.058001 PACS numbers: 45.70.ÿn, 64.60.ÿi, 83.80.Fg A solid, in contrast to a fluid, is characterized by mechanical stability that implies a finite resistance to shear and isotropic deformation. While such stability can originate from long-range crystalline order, there is no general agreement on how mechanical stability arises for disordered systems, such as molecular and colloidal glasses, gels, foams, and granular packings [1]. For a granular system, in particular, a key question concerns how stability occurs when the packing fraction increases from below to above a critical value c for which there are just enough contacts per particle Z to satisfy the conditions of mechanical stability. In recent simulations on frictionless systems it was found that Z exhibits a discontinuity at c followed by a power-law increase for > c [2 -5]. The pressure is also predicted to increase as a power law above c .A number of recent theoretical studies address jamming, and we note work that may be relevant to granular systems. Silbert, O'Hern et al. have shown in computer simulations of frictionless particles [2 -4] that (a) for increasing , Z increases discontinuously at the transition point from zero to a finite number, Z c , corresponding to the isostatic value (needed for mechanical stability), (b) for both two-and three-dimensional systems, Z is expected to continue increasing as ÿ c above c , where 0:5, (c) the pressure P is expected to grow above c as ÿ c , where f ÿ 1 in the simulations, and f is the exponent for the interparticle potential. More recent simulations by Donev et al. for hard spheres in three dimensions found a slightly higher value for , 0:6, in maximally random jammed packings [5]. It is interesting to note that a model for foam exhibits quite similar behavior for Z [6]. Henkes and Chakraborty [7] constructed a mean-field theory of the jamming transition in 2D based on entropy arguments. These authors predict power-law scaling for P and Z in terms of a variable , which is the pressure derivative of the entropy. By eliminating , one obtains an algebraic relation between P and Z ÿ Z c from these predictions, which we present below in the context of our data.While the simulations agree among themselves at least qualitatively, so far, these novel features have not been identified in experiments. Hen...

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