Jamming Transition as Probed by Quasistatic Shear Flow

Abstract

We study the rheology of amorphous packings of soft, frictionless particles close to jamming. Implementing a quasistatic simulation method we generate a well defined ensemble of states that directly samples the system at its yield-stress. A continuous jamming transition from a freely-flowing state to a yield stress situation takes place at a well defined packing fraction, where the scaling laws characteristic of isostatic solids are observed. We propose that long-range correlations observed below the transition are dominated by this isostatic point, while those that are observed above the transition are characteristic of dense, disordered elastic media.PACS numbers: 83.80. Fg,83.50.Ax,62.20.de A collection of spherical particles, interacting via a finite-range repulsive (contact) potential, unjams from a solid to a non-rigid state when being decompressed below a critical volume-fraction, φ c [1,2]. This transition, which has been given the name "point J", is accompanied by several interesting and nontrivial scaling relations in the solid phase [1,3]. There, pressure and linear elastic shear modulus vanish as does the ratio of shear to bulk modulus. The average number of particle contacts jumps from a finite value z 0 at point J to zero just below the transition. The value of z 0 is compatible with Maxwell's estimate for the rigidity transition and signals the fact that at point J each particle has just enough contacts for a rigid/solid state to exist (which is called "isostatic" state) [4,5]. Above the transition, additional contacts are generated according to the surprisingly universal law, δz ∼ δφ 1/2 . While the system moves away from its isostatic state the effective size of the remaining isostatic regions, l ⋆ ∼ δz −1 [6,7] has been argued to provide the diverging length-scale that endows point J with a certain "criticality". Based on these findings, jamming has been regarded as a "mixed" transition that shares properties of both, discontinuous (jump in contact number) and continuous phase-transitions (diverging length-scale) [8,9].A different route to approach point J from the fluid phase has been used in the flow simulations of Refs. [10,11,12]. Several scaling relations have been reported, some of which depend on model details. Some others seem to share the universality encountered in the solid phase. In contrast to the linear elastic properties in the solid phase, only little understanding about the flow properties and their relation to nearby point J has been achieved up to now.In this Letter, we employ a quasistatic simulation technique that studies the borderline between fluid and solid state. As we will see, it combines both aspects, the elasticity of a solid and the flow of a fluid, in one simulation. This fact allows us to get insight into how point J, and its isostatic state, affects the flow properties close by. In particular we show that jamming (i.e. the development of a finite shear stress) as probed by shear flow should be viewed as a continuous transition, with isostatic effects showing ...

Finite-size scaling at the jamming transition: Corrections to scaling and the correlation-length critical exponent

“…Hatano [10] obtained ν = 0.73 ± 0.05 from simulations of shear relaxation in 3D, while relaxation of random initial states to mechanical equilibrium in 2D led Head [11] to ν = 0.57 ± 0.05. Heussinger and Barat [12] estimate ν = 0.8 − 1.0 from displacement correlations in a 2D system under quasistatic shearing, while Heussinger et al [13] find a dynamic correlation length in 2D with exponent ν = 0.9. Establishing the precise value of ν and determining whether all these correlations lengths are the same thus remains a crucial theoretical objective.…”

“…We consider two different jamming ensembles: (i) quench from random initial positions (RAND) and (ii) quasistatic shearing (QS) [12]. By considering the fraction of jammed states f as a function of packing fraction φ for systems with different numbers of particles N , we demonstrate that the correlation length critical exponent in both ensembles is ν ≈ 1.…”

Shear Flow of Non-Brownian Suspensions Close to Jamming

“…Above a yield stress σ Y , vanishing at φ c , they present a non-Newtonian rheology, for which several different interpretations have been proposed, based on (i) an analogy with the glassy dynamics of a system presenting scale-free energy distributions [7], (ii) interacting plastic events [8], (iii) the critical scaling laws of the shear modulus and of the coordination number [9]. Together with conventional molecular dynamics simulations (MD), quasistatic methods (QS) have been applied to study the plastic flow of athermal amorphous solids at the yield-stress σ Y [10][11][12][13][14]. It is generally assumed that QS accurately describe the dynamics of the true system in the limit of asymptotically small shear rateγ.…”

“…All observables below are given in units of d, m and ǫ. We compare the divergence of the viscosity for φ < φ c = 0.843 [13,19] using two different dynamics: non-equilibrium dissipative molecular dynamics (MD) and quasistatic simulations (QS).…”

Rheology near jamming: The influence of lubrication forces

“…Progress in this direction has been made, for example, by defining an underlying contact network [7,25] in analogy to what has been achieved for the jammed packings; or by establishing macro-micro correspondences [26] or scaling relations [27][28][29] between exponents characterizing different aspects of the jamming singularity.…”