Edan Lerner

A unified framework for non-Brownian suspension flows and soft amorphous solids

While the rheology of non-Brownian suspensions in the dilute regime is well understood, their behavior in the dense limit remains mystifying. As the packing fraction of particles increases, particle motion becomes more collective, leading to a growing length scale and scaling properties in the rheology as the material approaches the jamming transition. There is no accepted microscopic description of this phenomenon. However, in recent years it has been understood that the elasticity of simple amorphous solids is governed by a critical point, the unjamming transition where the pressure vanishes, and where elastic properties display scaling and a diverging length scale. The correspondence between these two transitions is at present unclear. Here we show that for a simple model of dense flow, which we argue captures the essential physics near the jamming threshold, a formal analogy can be made between the rheology of the flow and the elasticity of simple networks. This analogy leads to a new conceptual framework to relate microscopic structure to rheology. It enables us to define and compute numerically normal modes and a density of states. We find striking similarities between the density of states in flow, and that of amorphous solids near unjamming: both display a plateau above some frequency scale ω∗ ∼ |zc - z|, where z is the coordination of the network of particle in contact, zc = 2D where D is the spatial dimension. However, a spectacular difference appears: the density of states in flow displays a single mode at another frequency scale ωmin ≪ ω∗ governing the divergence of the viscosity.

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Significance Yield stress solids flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield. Here we propose a scaling description of the yielding transition that relates the flow curve, the statistics of the avalanches of plasticity observed at threshold, and the density of local zones that are about to yield. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. Numerical simulations on a simple elasto-plastic model find good agreement with our predictions. Yield stress materials flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield, even in the simplest situations in which temperature is negligible and in which flow inhomogeneities such as shear bands or fractures are absent. Here we propose a scaling description of the yielding transition in amorphous solids made of soft particles at zero temperature. Our description makes a connection between the Herschel–Bulkley exponent characterizing the singularity of the flow curve near the yield stress Σc, the extension and duration of the avalanches of plasticity observed at threshold, and the density P(x) of soft spots, or shear transformation zones, as a function of the stress increment x beyond which they yield. We argue that the critical exponents of the yielding transition may be expressed in terms of three independent exponents, θ, df, and z, characterizing, respectively, the density of soft spots, the fractal dimension of the avalanches, and their duration. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. We test our arguments in an elasto-plastic model, an automaton model similar to those used in depinning, but with a different interaction kernel, and find satisfying agreement with our predictions in both two and three dimensions.

Locality and nonlocality in elastoplastic responses of amorphous solids.

A number of current theories of plasticity in amorphous solids assume at their basis that plastic deformations are spatially localized. We present in this paper a series of numerical experiments to test the degree of locality of plastic deformation. These experiments increase in terms of the stringency of the removal of elastic contributions to the observed elastoplastic deformations. It is concluded that for all our simulational protocols the plastic deformations are not localized, and their scaling is subextensive. We offer a number of measures of the magnitude of the plastic deformation, all of which display subextensive scaling characterized by nontrivial exponents. We provide some evidence that the scaling exponents governing the subextensive scaling laws are nonuniversal, depending on the degree of disorder and on the parameters of the systems. Nevertheless, understanding what determines these exponents should shed considerable light on the physics of amorphous solids.

Breakdown of continuum elasticity in amorphous solids

We show numerically that the response of simple amorphous solids (elastic networks and particle packings) to a local force dipole is characterized by a lengthscale lc that diverges as unjamming is approached as lc ∼ (z - 2d)(-1/2), where z ≥ 2d is the mean coordination, and d is the spatial dimension, at odds with previous numerical claims. We also show how the magnitude of the lengthscale lc is amplified by the presence of internal stresses in the disordered solid. Our data suggests a divergence of lc ∼ (pc - p)(-1/4) with proximity to a critical internal stress pc at which soft elastic modes become unstable.

Statistical Physics of the Yielding Transition in Amorphous Solids

The art of making structural, polymeric, and metallic glasses is rapidly developing with many applications. A limitation is that under increasing external strain all amorphous solids (like their crystalline counterparts) have a finite yield stress which cannot be exceeded without effecting a plastic response which typically leads to mechanical failure. Understanding this is crucial for assessing the risk of failure of glassy materials under mechanical loads. Here we show that the statistics of the energy barriers ΔE that need to be surmounted changes from a probability distribution function that goes smoothly to zero as ΔE=0 to a pdf which is finite at ΔE=0 . This fundamental change implies a dramatic transition in the mechanical stability properties with respect to external strain. We derive exact results for the scaling exponents that characterize the magnitudes of average energy and stress drops in plastic events as a function of system size.

Low-energy non-linear excitations in sphere packings

We study theoretically and numerically how hard frictionless particles in random packings can rearrange. We demonstrate the existence of two distinct unstable non-linear modes of rearrangement, both associated with the opening and the closing of contacts. The first mode, whose density is characterized by some exponent θ′, corresponds to motions of particles extending throughout the entire system. The second mode, whose density is characterized by an exponent θ ≠ θ′, corresponds to the local buckling of a few particles. Extended modes are shown to yield at a much higher rate than local modes when a stress is applied. We show that the distribution of contact forces follows P(f) ∼ fmin(θ′,θ), and that imposing the restriction that the packing cannot be densified further leads to the bounds and , where γ characterizes the singularity of the pair distribution function g(r) at contact. These results extend the theoretical analysis of [Wyart, Phys. Rev. Lett., 2012, 109, 125502] where the existence of local modes was not considered. We perform numerics that support that these bounds are saturated with γ ≈ 0.38, θ ≈ 0.17 and θ′ ≈ 0.44. We measure systematically the stability of all such modes in packings, and confirm their marginal stability. The principle of marginal stability thus allows us to make clearcut predictions on the ensemble of configurations visited in these out-of-equilibrium systems, and on the contact forces and pair distribution functions. It also reveals the excitations that need to be included in a description of plasticity or flow near jamming, and suggests a new path to study two-level systems and soft spots in simple amorphous solids of repulsive particles.

Direct estimate of the static length-scale accompanying the glass transition

Glasses are liquids whose viscosity has increased so much that they cannot flow. Accordingly, there have been many attempts to define a static length-scale associated with the dramatic slowing down of supercooled liquid with decreasing temperature. Here we present a simple method to extract the desired length-scale which is highly accessible both for experiments and for numerical simulations. The fundamental new idea is that low lying vibrational frequencies come in two types, those related to elastic response and those determined by plastic instabilities. The minimal observed frequency is determined by one or the other, crossing at a typical length-scale which is growing with the approach of the glass transition. This length-scale characterizes the correlated disorder in the system: on longer length-scales the details of the disorder become irrelevant, dominated by the Debye model of elastic modes. To connect the newly defined length-scale to relaxation dynamics near the glass transition, we show that supercooled liquids in which there exist random pinning sites of density ρim∼1/ξsd exhibit complete jamming of all dynamics. This is a direct demonstration that the proposed length scale is indeed the static length that was long sought-after.

Force distribution affects vibrational properties in hard-sphere glasses

Significance How a liquid becomes rigid at the glass transition is a central problem in condensed matter physics. In many scenarios of the glass transition, liquids go through a critical temperature below which minima of free energy appear. However, even in the simplest glass, hard spheres, what confers mechanical stability at large density is highly debated. In this work we show that to quantitatively understand stability at a microscopic level, the presence of weakly interacting pairs of particles must be included. This approach allows us to predict various nontrivial scaling behavior of the elasticity and vibrational properties of colloidal glasses that can be tested experimentally. It also gives a spatial interpretation to recent, exact calculations in infinite dimensions. 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)∼fθe, the force distribution of such pairs and ϕc 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)∼ω2+a, and decaying above ω* as D(ω)∼ω−a where a=(1−θe)/(3+θe) and ω is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with 〈δR2〉∼1/μ∼(ϕc−ϕ)κ, where κ=2−2/(3+θe), and (iii) continuum elasticity breaks down on a scale ℓc∼1/δz∼(ϕc−ϕ)−b, where b=(1+θe)/(6+2θe) and δz=z−2d, where z is the coordination and d the spatial dimension. We numerically test (i) and provide data supporting that θe≈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.

On the density of shear transformations in amorphous solids

We study the stability of amorphous solids, focussing on the distribution P(x) of the local stress increase x that would lead to an instability. We argue that this distribution behaves as , where the exponent θ is larger than zero if the elastic interaction between rearranging regions is non-monotonic, and increases with the interaction range. For a class of finite-dimensional models we show that stability implies a lower bound on θ, which is found to lie near saturation. For quadrupolar interactions these models yield for d = 2 and in d = 3 where d is the spatial dimension, accurately capturing previously unresolved observations in atomistic models, both in quasi-static flow and after a fast quench. In addition, we compute the Herschel-Buckley exponent in these models and show that it depends on a subtle choice of dynamical rules, whereas the exponent θ does not.

Do athermal amorphous solids exist

We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist, one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of nonaffine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient B(2) has anomalous fluctuations and the second nonlinear coefficient B(3) and all the higher order coefficients (which are nonzero by symmetry) diverge in the thermodynamic limit. These results call into question the existence of elasticity (or solidity) of amorphous solids at finite strains, even at zero temperature. We discuss the physical meaning of these results and propose that in these systems elasticity can never be decoupled from plasticity: the nonlinear response must be very substantially plastic.