Cavity method for force transmission in jammed disordered packings of hard particles

Lin, Bo; Mari, Romain; Ci, Song; Makse, Hernán A.

doi:10.1039/c4sm00667d

Cited by 17 publications

(39 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For an isostatic system at jamming the force network ensemble is not needed, since the contact geometry uniquely defines the contact forces (Charbonneau et al, 2015b;Gendelman et al, 2016;Lerner et al, 2013). In this case, an approximation of P (f ) can be calculated with the cavity method assuming a locally tree-like contact geometry corresponding to an assumption of replica symmetry (RS) (Bo et al, 2014). We note that a correct determination of P (f ) requires one to take into account subtle correlations between particle positions that exist at jamming beyond RS, and that are neglected in (Bo et al, 2014), which in the end fails to account for the non-trivial power-laws of P (f ) at jamming.…”

Section: Force Network Ensemblementioning

confidence: 99%

“…(47), the inequality γ ≥ 1/(2 + θ) is still observed, except for one packing A which is presumably hyperstatic. On the other limit of sparse graphs, replica symmetry calculations gives θ = 0 in the thermodynamic limit using population dynamics implying that RS calculations do not capture the full physics of the jamming point (Bo et al, 2014) Assuming ergodicity for a jammed system of grains as proposed by Edwards (see Sec. II.C) seems contradictory at first, but has become meaningful in the first place in light of certain seminal compaction experiments developed over the years starting from the work of Nowak et at.…”

Section: Force Statisticsmentioning

confidence: 99%

See 1 more Smart Citation

Edwards statistical mechanics for jammed granular matter

et al. 2018

Self Cite

View full text Add to dashboard Cite

In 1989, Sir Sam Edwards made the visionary proposition to treat jammed granular materials using a volume ensemble of equiprobable jammed states in analogy to thermal equilibrium statistical mechanics, despite their inherent athermal features. Since then, the statistical mechanics approach for jammed matter -one of the very few generalizations of Gibbs-Boltzmann statistical mechanics to out of equilibrium matter -has garnered an extraordinary amount of attention by both theorists and experimentalists. Its importance stems from the fact that jammed states of matter are ubiquitous in nature appearing in a broad range of granular and soft materials such as colloids, emulsions, glasses, and biomatter. Indeed, despite being one of the simplest states of matter -primarily governed by the steric interactions between the constitutive particles -a theoretical understanding based on first principles has proved exceedingly challenging. Here, we review a systematic approach to jammed matter based on the Edwards statistical mechanical ensemble. We discuss the construction of microcanonical and canonical ensembles based on the volume function, which replaces the Hamiltonian in jammed systems. The importance of approximation schemes at various levels is emphasized leading to quantitative predictions for ensemble averaged quantities such as packing fractions and contact force distributions. An overview of the phenomenology of jammed states and experiments, simulations, and theoretical models scrutinizing the strong assumptions underlying Edwards' approach is given including recent results suggesting the validity of Edwards ergodic hypothesis for jammed states. A theoretical framework for packings whose constitutive particles range from spherical to non-spherical shapes like dimers, polymers, ellipsoids, spherocylinders or tetrahedra, hard and soft, frictional, frictionless and adhesive, monodisperse and polydisperse particles in any dimensions is discussed providing insight into an unifying phase diagram for all jammed matter. Furthermore, the connection between the Edwards' ensemble of metastable jammed states and metastability in spin-glasses is established. This highlights that the packing problem can be understood as a constraint satisfaction problem for excluded volume and force and torque balance leading to a unifying framework between the Edwards ensemble of equiprobable jammed states and out-of-equilibrium spin-glasses.

show abstract

Section: Force Network Ensemblementioning

confidence: 99%

Section: Force Statisticsmentioning

confidence: 99%

Edwards statistical mechanics for jammed granular matter

et al. 2018

Self Cite

View full text Add to dashboard Cite

show abstract

“…Also, the low-force end of the force distribution has a power law, like in the CLM simulations, but the exponent is much larger in the bi-q-model, we observe θ ≈ 4. The underestimation of the number of low forces is a known deficiency of the q-model [50]. At q 1 = 0.92, that is, when the contrast between force transmission in the unthickened and thickened states is large, the large-force tail in the thickened state is signif- icantly altered and decays much slower than at lower q 1 .…”

Section: < L a T E X I T S H A 1 _ B A S E 6 4 = " V / H / H Z 4 8 + mentioning

confidence: 99%

Force transmission and the order parameter of shear thickening

Mari

Seto

2019

Soft Matter

Self Cite

View full text Add to dashboard Cite

The origin of the abrupt shear thickening observed in some dense suspensions has been recently argued to be a transition from frictionless (lubricated) to frictional interactions between immersed particles. The Wyart-Cates rheological model, built on this scenario, introduced the concept of fraction of frictional contacts f as the relevant order parameter for the shear thickening transition. Central to the model is the "equation-of-state" relating f to the applied stress σ, which is directly linked to the distribution of the normal components of non-hydrodynamics interparticle forces. Here, we develop a model for this force distribution, based on the so-called q-model that we borrow from granular physics. This model explains the known f (σ) in the simple case of sphere contacts displaying only sliding friction, but also predicts strong deviation from this "usual" form when stronger kinds of constraints are applied on relative motion. We verify these predictions in the case of contacts with rolling friction, in particular a broadening of the stress range over which shear thickening occurs. We finally discuss how a similar approach can be followed to predict f (σ) in systems with other variations from the canonical system of monodisperse spheres with sliding friction, in particular the case of large bidispersity.ing the jamming volume fraction φ J , as is often proposed close to the jamming transition [17][18][19][20]. The value of the exponent α is debated in the literature [14,[20][21][22], but this is not the focus of the present discussion; the most typical value in the literature is α = 2 [22][23][24]. The specificity of the WC theory is that φ J depends on a microscopic order parameter f , the so-called fraction of frictional contacts, that is, the proportion of nearest neighbor interactions which are frictional, as opposed to lubricated. The jamming volume fraction is linearly moving with f as φ J (f ) = (1 − f )φ 0 J + f φ 1 J , in between two limiting values, φ 0 J for only lubricated interactions and φ 1 J < φ 0 J when only frictional contacts are present. Finally, the competition between confining forces due to arXiv:1906.02103v1 [cond-mat.soft]

show abstract

“…Although missing from many theoretical approaches (5,34), the distribution of contact forces at ϕ c is known empirically to follow Pðf Þ ∼ f θ f at small f, with θ f ≈ 0:2 (20,35). This directly yields a diverging distribution of stiffnesses: PðkÞ = Pðf Þdf =dk ∼ k α , with α = −ð1 − θ f Þ=2 < 0.…”

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.

122

166

View full text Add to dashboard Cite

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...

show abstract

Cavity method for force transmission in jammed disordered packings of hard particles

Cited by 17 publications

References 60 publications

Edwards statistical mechanics for jammed granular matter

Edwards statistical mechanics for jammed granular matter

Force transmission and the order parameter of shear thickening

Force distribution affects vibrational properties in hard-sphere glasses

Contact Info

Product

Resources

About