Marginal Stability Constrains Force and Pair Distributions at Random Close Packing

Abstract

The requirement that packings of hard particles, arguably the simplest structural glass, cannot be compressed by rearranging their network of contacts is shown to yield a new constraint on their microscopic structure. This constraint takes the form a bound between the distribution of contact forces P (f ) and the pair distribution function g(r): if P (f ) ∼ f θ and g(r) ∼ (r − σ0) −γ , where σ0 is the particle diameter, one finds that γ ≥ 1/(2 + θ). This bound plays a role similar to those found in some glassy materials with long-range interactions, such as the Coulomb gap in Anderson insulators or the distribution of local fields in mean-field spin glasses. There is ground to believe that this bound is saturated, offering an explanation for the presence of avalanches of rearrangements with power-law statistics observed in packings.PACS numbers: 63.50.Lm,Amorphous materials are perhaps the simplest example of glasses, in which the dynamics is so slow that thermal equilibrium cannot be reached. In these systems properties are history-dependent, and configurations of equal energy are not equiprobable. What principles then govern which part of the configuration space is explored, for example when a pile of sand is prepared? One approach was proposed by Edwards in the context of granular matter [1], and is based on the hypothesis that all mechanically stable states are equiprobable. Another line of thought assumes that the configurations generated by the dynamics are linearly stable, but only marginally [2,3]: the microscopic structure is such that soft elastic modes are present at vanishingly small frequencies. This view can explain [2][3][4] in particular the singularities occurring in the coordination number and in the elasticity of amorphous solids made of repulsive particles near the unjamming threshold [5][6][7] where rigidity disappears. Despite these successes, the hypothesis of linear marginal stability yields an incomplete insight on the non-linear processes occurring in amorphous materials, which are critical to understand plasticity, thermal activation or granular flows [7]. When interactions are short-range one key source of non-linearity is the creation or destruction of contacts between particles [8,9]. Combe and Roux have observed numerically [8] that such rearrangements occur intermittently, in bursts or avalanches whose size is power-law distributed, a kind of dynamics referred to as crackling noise [10].Interestingly some glassy systems with long-range interactions display such dynamics, in particular Coulomb glasses [11] and mean-field spin glasses [12]. In both cases the requirement of stability toward discrete excitations (flipping two spins or moving one electron) leads to bounds on important physical quantities: Efros and Shklovskii showed that the density of states in a Coulomb glass must vanish at the Fermi energy [13], implying the presence of the so-called Coulomb gap. Thouless, Anderson and Palmer [14] demonstrated for mean-field spin glasses that the distribution of local fields must v...

Effect of particle collisions in dense suspension flows

“…It was recently shown that mechanical stability requires the distribution of contact forces in packings of frictionless spheres to vanish at small forces [23], as observed in Ref. [25].…”

“…Finally, in Sec. VIII the marginal stability criterion observed for jammed packing [23,24] is rederived in flows.…”

An experimental investigation of the force network ensemble

“…For small forces we find, P(F) to rise exponentially, as approximately F 3/2 . Wyart [15], showed that the exponent is determined by the pair distribution function g(r). However, in our small system with a fixed particle configuration, g(r) is undersampled, so the question arises what the exponent is set by here.…”

Unifying Suspension and Granular flows near Jamming

“…Moreover, its value does not depend on the preparation protocol of the isostatic state: up to error bars, equal values are found from compression of hard spheres [32], shear-jammed hard disks [35], and decompression of soft spheres [33,36] 1 . The exponent θ e can be shown to control the stability of the solid phase [32,37,38]. Recently replica calculations in infinite dimension on the force distribution [36,39,40] led to the prediction [33]:…”