Ellák Somfai

Critical scaling in linear response of frictionless granular packings near jamming.

We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

Elastic wave propagation in confined granular systems

We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wave front followed by random oscillations made of multiply scattered waves. We find that the coherent wave front is insensitive to details of the packing: force chains do not play an important role in determining this wave front. The coherent wave propagates linearly in time, and its amplitude and width depend as a power law on distance, while its velocity is roughly compatible with the predictions of macroscopic elasticity. As there is at present no theory for the broadening and decay of the coherent wave, we numerically and analytically study pulse propagation in a one-dimensional chain of identical elastic balls. The results for the broadening and decay exponents of this system differ significantly from those of the random packings. In all our simulations, the speed of the coherent wave front scales with pressure as p1/6; we compare this result with experimental data on various granular systems where deviations from the p1/6 behavior are seen. We briefly discuss the eigenmodes of the system and effects of damping are investigated as well.

Critical and noncritical jamming of frictional grains

We probe the nature of the jamming transition of frictional granular media by studying their vibrational properties as a function of the applied pressure p and friction coefficient mu. The density of vibrational states exhibits a crossover from a plateau at frequencies omega > or similar to omega*(p,mu) to a linear growth for omega < or similar to omega*(p,mu). We show that omega* is proportional to Deltaz, the excess number of contacts per grain relative to the minimally allowed, isostatic value. For zero and infinitely large friction, typical packings at the jamming threshold have Deltaz-->0, and then exhibit critical scaling. We study the nature of the soft modes in these two limits, and find that the ratio of elastic moduli is governed by the distance from isostaticity.

A giant atomic slide-puzzle

Atoms in close-packed surfaces of metal crystals move around at surprisingly high rates, even though each atom is locked in tightly by its neighbours. Here we use a low density of indium atoms, embedded in the outermost atomic layer of a copper surface, as tracer particles for scanning tunnelling microscopy to reveal the high vacancy-assisted mobility of atoms in this surface. We believe that most close-packed surfaces of metals and other materials will exhibit a similar vacancy-assisted motion at room temperature, with such surfaces behaving like a gigantic atomic slide-puzzle.

Vacancy diffusion in the Cu(001) surface I: an STM study

We have used the indium/copper surface alloy to study the dynamics of surface vacancies on the Cu(0 0 1) surface. Individual indium atoms that are embedded within the first layer of the crystal, are used as probes to detect the rapid diffusion of surface vacancies. STM measurements show that these indium atoms make multi-lattice-spacing jumps separated by long time intervals. Temperature dependent waiting time distributions show that the creation and diffusion of thermal vacancies form an Arrhenius type process with individual long jumps being caused by one vacancy only. The length of the long jumps is shown to depend on the specific location of the indium atom and is directly related to the lifetime of vacancies at these sites on the surface. This observation is used to expose the role of step edges as emitting and absorbing boundaries for vacancies.

Vacancy diffusion in the Cu(0 0 1) surface II: Random walk theory

We develop a version of the vacancy mediated tracer diffusion model, which follows the properties of the physical system of In atoms diffusing within the top layer of Cu(001) terraces. This model differs from the classical tracer diffusion problem in that (i) the lattice is finite, (ii) the boundary is a trap for the vacancy, and (iii) the diffusion rate of the vacancy is different, in our case strongly enhanced, in the neighborhood of the tracer atom. A simple continuum solution is formulated for this problem, which together with the numerical solution of the discrete model compares well with our experimental results.

Effects of grain shape on packing and dilatancy of sheared granular materials

A granular material exposed to shear shows a variety of unique phenomena: Reynolds dilatancy, positional order and orientational order effects may compete in the shear zone. We study granular packing consisting of macroscopic prolate, oblate and spherical grains and compare their behaviour. X-ray tomography is used to determine the particle positions and orientations in a cylindrical split bottom shear cell. Packing densities and the arrangements of individual particles in the shear zone are evaluated. For anisometric particles, we observe the competition of two opposite effects. On the one hand, the sheared granules are dilated, on the other hand the particles reorient and align with respect to the streamlines. Even though aligned cylinders in principle may achieve higher packing densities, this alignment compensates for the effect of dilatancy only partially. The complex rearrangements lead to a depression of the surface above the well oriented region while neighbouring parts still show the effect of dilation in the form of heaps. For grains with isotropic shapes, the surface remains rather flat. Perfect monodisperse spheres crystallize in the shear zone, whereby positional order partially overcompensates dilatancy effects. However, even slight deviations from the ideal monodisperse sphere shape inhibit crystallization.

Force and weight distributions in granular media: Effects of contact geometry

The effect of boundaries on the force distributions in granular media is illustrated by simulations of 2D packings of frictionless, Hertzian spheres. To elucidate discrepancies between experimental observations and theoretical predictions, we distinguish between the weight distribution {\cal P} (w) measured in experiments and analyzed in the q-model, and the distribution of interparticle forces P(f). The latter one is robust, while {\cal P}(w) can be obtained once the local packing geometry and P(f) are known. By manipulating the (boundary) geometry, we show that {\cal P}(w) can be varied drastically.

Packing geometry and statistics of force networks in granular media

The relation between packing geometry and force network statistics is studied for granular media. Based on simulations of two-dimensional packings of Hertzian spheres, we develop a geometrical framework relating the distribution of interparticle forces P(f) to the weight distribution P(w), which is measured in experiments. We apply this framework to reinterpret recent experimental data on strongly deformed packings and suggest that the observed changes of P(w) are dominated by changes in contact network while P(f) remains relatively unaltered. We furthermore investigate the role of packing disorder in the context of the q model and address the question of how force fluctuations build up as a function of the distance beneath the top surface.

Scaling and crossovers in diffusion limited aggregation

We discuss the scaling of characteristic lengths in diffusion limited aggregation clusters in light of recent developments using conformal maps. We are led-to the:conjecture that the apparently anomalous scaling of lengths is due to one slow crossover. This is supported by an analytical argument for the scaling of the penetration depth of newly arrived random walkers, and by numerical evidence on the Laurent coefficients which uniquely determine each cluster. We find common crossover behavior for the squares of the characteristic lengths and the penetration depth of the form N-2/D(alpha + beta N-phi) With phi in the range -0.3 +/- 0.1 suggesting that there is a single: dominant correction to scaling.