The jamming scenario—an introduction and outlook

Liu, A.J.; Nagel, Sidney R.; Saarloos, Wim van; Wyart, Matthieu

doi:10.1093/acprof:oso/9780199691470.003.0009

Cited by 78 publications

(126 citation statements)

References 134 publications

(372 reference statements)

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“…7 and 12 this bound was observed to be saturated, and here we assume such marginal stability, ω p =ω c ∼ ðϕ c − ϕÞ 1=2 . From [3][4][5][6] and [8] we then deduce…”

Section: Hard Spheresmentioning

confidence: 99%

Force distribution affects vibrational properties in hard-sphere glasses

DeGiuli

Lerner

Brito

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

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We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force f soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting P(f ) ∼ f θe , the force distribution of such pairs and ϕ c the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale ω*, rising up to it as D(ω) ∼ ω 2+a , and decaying above ω* as D(ω) ∼ ω −a where a = (1 − θ e )=(3 + θ e ) and ω is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with 〈δR 2 〉 ∼ 1=μ ∼ (ϕ c − ϕ) κ , where κ = 2 − 2=(3 + θ e ), and (iii) continuum elasticity breaks down on a scale ℓ c ∼ 1= ffiffiffiffiffi δz p ∼ (ϕ c − ϕ) −b , where b = (1 + θ e )=(6 + 2θ e ) and δz = z − 2d, where z is the coordination and d the spatial dimension. We numerically test (i) and provide data supporting that θ e ≈ 0:41 in our bidisperse system, independently of system preparation in two and three dimensions, leading to κ ≈ 1:41, a ≈ 0:17, and b ≈ 0:21. Our results for the mean-square displacement are consistent with a recent exact replica computation for d = ∞, whereas some observations differ, as rationalized by the present approach.colloids | glass transition | marginal stability | boson peak | jamming T he emergence of rigidity near the glass transition is a fundamental and highly debated topic in condensed matter and is perhaps most surprising in hard-sphere glasses where rigidity is purely entropic in nature. The rapid growth of relaxation time around a packing fraction ϕ g ≈ 0:58 suggests that metastable states have appeared in the free-energy landscape, and that activation above barriers is required for the system to flow (1). This scenario is presumably what mode-coupling theory captures (2, 3) and can be rationalized via density functional theory (4) and via the replica method (5). Recently a real-space description of mechanical stability and elasticity in hard-sphere glasses has been proposed (6, 7), which is most easily tested at large pressure, deep in the glass phase. It is based on two results. First, in elastic networks and athermal packings of soft spheres (8-10), mechanical stability is controlled by the mean number of contacts per particle, or coordination z (as already discussed by Maxwell in ref. 11), and the applied compressive strain e (10). As one may intuitively expect, increasing coordination is stabilizing, whereas increasing pressure at fixed coordination is destabilizing. Second, within a long-lived metastable state the vibrational free energy of a hard-sphere system can be approximated as a sum of local interaction terms between pairs of colliding particles, which are said to be in contact. On a time scale that contains many...

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“…7 and 12 this bound was observed to be saturated, and here we assume such marginal stability, ω p =ω c ∼ ðϕ c − ϕÞ 1=2 . From [3][4][5][6] and [8] we then deduce…”

Section: Hard Spheresmentioning

confidence: 99%

Force distribution affects vibrational properties in hard-sphere glasses

DeGiuli

Lerner

Brito

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

122

166

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“…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].…”

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confidence: 95%

“…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].…”

mentioning

confidence: 95%

Marginal Stability Constrains Force and Pair Distributions at Random Close Packing

2012

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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...

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“…As the average connectivity increases beyond 2d, the network undergoes a phase transition marked by a continuous increase in the elasticity. Other examples of such transitions are the jamming transition [3][4][5][6] in granular materials and rigidity percolation [7][8][9][10] in disordered spring networks. Jamming exhibits signatures characteristic of both first-and second-order transitions, with discontinuous behavior of the bulk modulus and continuous variation of the shear modulus [5,11,12].…”

Section: Introductionmentioning

confidence: 99%

Strain-driven criticality underlies nonlinear mechanics of fibrous networks

et al. 2016

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Networks with only central force interactions are floppy when their average connectivity is below an isostatic threshold. Although such networks are mechanically unstable, they can become rigid when strained. It was recently shown that the transition from floppy to rigid states as a function of simple shear strain is continuous, with hallmark signatures of criticality [Sharma et al., Nature Phys. 12, 584 (2016)]. The nonlinear mechanical response of collagen networks was shown to be quantitatively described within the framework of such mechanical critical phenomenon. Here, we provide a more quantitative characterization of critical behavior in subisostatic networks. Using finite-size scaling we demonstrate the divergence of strain fluctuations in the network at well-defined critical strain. We show that the characteristic strain corresponding to the onset of strain stiffening is distinct from but related to this critical strain in a way that depends on critical exponents. We confirm this prediction experimentally for collagen networks. Moreover, we find that the apparent critical exponents are largely independent of the spatial dimensionality. With subisostaticity as the only required condition, strain-driven criticality is expected to be a general feature of biologically relevant fibrous networks.

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The jamming scenario—an introduction and outlook

Cited by 78 publications

References 134 publications

Force distribution affects vibrational properties in hard-sphere glasses

Force distribution affects vibrational properties in hard-sphere glasses

Marginal Stability Constrains Force and Pair Distributions at Random Close Packing

Strain-driven criticality underlies nonlinear mechanics of fibrous networks

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