Session K48: Athermal Systems and Statistical Mechanics

Percolation on correlated landscapes

Random landscapes with power-law height-height correlations are described by a Hurst exponent H and can be generated by the Fourier filtering method. The isoheight lines at the percolation threshold and their accessible perimeter as well as the shortest path on the percolation cluster are fractals for -1 < H < 0 with dimensions that depends on H. Exact results are recovered for H = -1 and H = 0. Analyzing numerically the winding angle, the left passage probability, and the driving function of these curves, it is found that for negative H there statistical properties are compatible with a Schramm-Loewner evolution (SLE) and therefore can be mapped to random walks, their fractal dimension determining the diffusion constant. By contrast, for positive H, we find that the random walk is not Markovian but strongly correlated in time. The watershed dividing drainage basins of the landscape can also be obtained from percolation by suppressing systematically the occupation of bridges, i.e. those sites or bonds which if occupied would create the spanning cluster. This delays the percolation threshold and produces at the end a connected line of bridges which is identical to the watershed, exhibiting trivially a first order percolation transition. At p_c, bridge percolation exhibits theta point scaling with a new tricritical exponent. Watersheds also are found to be compatible with SLE. For uncorrelated surfaces, i.e. H = -1, the corresponding diffusivities are κ = 1.734 +/- 0.005 for the watershed and κ = 1.04 +/- 0.02 for the shortest path, being the only examples known up to now with κ < 2. Both watershed and shortest path result from a global optimization process and identifying them, requires exploring an entire area. The perimeters of multi-layered and directed percolation clusters at criticality are the scaling limits of the Loewner evolution of an anomalous Brownian motion, being superdiffusive and subdiffusive, respectively.
  

Equation of state in a model granular suspension

We experimentally investigate the stationary sedimentation of an assembly of buoyant cylinders in a rotating Hele-Shaw cell. The resulting tunable centripetal force allows exploring various states of the system. We find that local density fluctuations are in one-to-one correspondence with its mean, irrespective of the potential. From this relation, we derive a statistical pressure that appears to be proportional to a hydrodynamic pressure built from a mixture model over the whole range of explored packing fractions, allowing us to define an effective temperature in the medium. This quantity, which only depends on the density contrast between the grains and the liquid, entertains a simple relation to the dissipated energy.   

Thermal Ensemble for Stresses in Dense Suspensions

We develop a predictive framework for the macroscopic stress properties of dense suspensions
in two dimensions, based on correlations in a dual space representing forces. Working with the
ensemble of steady state configurations obtained from simulations of suspensions, we find an
emerging anisotropy in the pair correlation function in force space as the confining shear stress
(σ_xy) and the packing fraction (φ) are varied. Using these microscopic correlations, we build
a statistical theory of the forces and relate this to the macroscopic stress tensor of the system.
We find that (i) the stress anisotropy τ /P decreases as φ is increased and (ii) the normal stress
N₁ = σ_xx− σ_yy changes sign near the discontinuous shear thickening transition (DST). We provide
evidence for a first order transition occurring near the DST point, with a coexistence of ‘fluid’
regions with low stresses and ‘solid’ regions with high stresses.   

Memory in Solid-Solid Interfaces

The interface between two solid, static bodies - your chair and the floor, your cup and the table, two books in a stack - is in fact continuously evolving. Due to small-scale roughness, ostensibly flat surfaces typically have a real area of contact several orders of magnitude smaller than apparent area. As a result, these contact points experience enormous pressures, and slowly deform in time.

By measuring the evolution of the real area of contact, we demonstrate that these multi-contact interfaces (MCIs) store a memory of the pressures they experienced. Unlike simple relaxation, e.g. a spring and dashpot system, which depends only on its current state, MCIs evolve according to their entire loading history.

Understanding of MCIs is not only useful for knowing how much coffee has been in your cup; the real area of contact also determines the friction between two bodies. As a result, the coefficient of static friction also evolves according to this interfacial memory. These effects suggests that MCIs and frictional systems belong in a universality class of glassy systems characterized by disorder and slow relaxation.   

Fragile Matter: Stress Networks and Stability of Athermal Solids

In this talk, I will discuss a theoretical framework for for understanding materials that are fragile. These are marginal solids that emerge out of thermal equilibrium in response to external stresses. Granular materials and non-Brownian colloidal suspensions are well-known examples, however, reconfigurable pathways of force transmission also play an important role in biological systems. In granular materials, external forces such as gravity create rigid and flowing states. The mechanical integrity of these marginal solids is reliant on a filamentary network of stress-bearing structures. An outstanding question in the field has been how the constraints of vectorial force balance influence the response of granular assemblies to stress, and create localized stress pathways. I will present results of recent work showing that the localized response is a consequence of the disorder in the underlying contact network, and can be mapped on to a ``localization'' problem. I will then discuss criterion for stability and yielding of these networks, based on a gauge-potential formalism that imposes all of the constraints of mechanical equilibrium.   

Field theory for athermal solids

Amorphous solids composed of macroscopic constituents, like granular media, foams, and emulsions, can be considered as zero temperature systems. Long ago, Edwards proposed that such systems could be treated by statistical mechanical methods by taking a flat average over all metastable states. Despite its attractive parsimony, this hypothesis has remained controversial and has led to conflicting results. I will present some new ideas that remove the ambiguity in Edwards' approach and make it a constructive, feasible route to derive macroscopic equations of state and field equations for athermal systems. In particular, I will show how shear strain can be included as a control parameter, explain the effect of preparation protocol, and show the utility of the renormalization group in this context.   

Emergent Strain-Stiffening in Interlocked Granular Chains

In nature and architecture, a wide variety of stable structures are formed from dense assemblies of randomly-distributed objects, as illustrated by the various shapes of bird nests. Most physical studies focus on highly anisotropic objects that ensure a solid-like collective behavior and thus the great stability of their assembly. In this presentation, we investigate the effective mechanical rigidity of granular-chain assemblies from indentation experiments. Our observations show an exponential growth of the resistance force with the product of two factors: the indentation depth and the square root of the number of beads per chain. The first factor is reminiscent of the self-amplification of friction in a capstan, as well as in interleaved assemblies of sheets, where the increase in resistance is created by, and proportional to, the force exerted by the operator. The second factor points towards the central role played by topological constraints, in a way that is similar to polymer physics. We propose a novel interlocking model based on those two ingredients, and confront it to the experimental data.   

Stress Tensor Measurements of Sheared Packings of Non-Spherical Particles

In processing apparatus for dense gas-solid flows, the motion of the constituent particles is dictated by interactions with the ambient gas and inter-collisions with neighbouring particles. Typically, these particles are non-spherical in nature, such as the elongated biomass particles used in the production of biofuels in fluidised bed reactors. While experiments at both the laboratory and industrial scale can be used to study particle dynamics, information pertaining to particle contact neighbourhood and the inter-particle forces in the bulk are difficult to ascertain, thus necessitating a numerical approach.

We present a CFD-DEM approach to calculate particle forces and hence stress tensors in packings of elongated particles with varying packing fraction, mutual alignment and shear rates. The packings are subject to shear rates comparable to the deformations experienced by particles in real fluidised bed reactors. We also compare the stress tensors calculated with those for packings of spherical particles. Such calculations will prove valuable in the derivation of expressions relevant for industrial-scale simulations.   

Stick-Slip Dynamics of an Intruder in a 2D Annular Bed of Photoelastic Disks

When a granular material is strained elastically, the system may resist granular rearrangement depending on interparticle friction, packing fraction, and surrounding boundary conditions. In these long time-scale "stick" phases, stresses develop in the granular bed until the system yields in a fast time-scale "slip" phase; an example is a slow build-up of stress between sliding tectonic plates and subsequent rapid unloading of stress in an earthquake. We investigate stick-slip dynamics of an elastically driven intruder moving through a 2D annular bed of bidisperse photoelastic disks using high speed imaging of force chains on the grain scale. We analyze both the topology of force networks and the global pressure in the bed to understand force chain self organization and contact rearrangement in slip events once a steady state has been reached. In particular, ongoing work focuses on spatial extent of force chains in front of the intruder and local particle density in its wake at varying packing fraction, interparticle friction, and driving spring elastic constant. Parallel simulations are being studied by E. Goldberg, L. Pugnaloni, M. Carlevaro, and L. Kondic.