content_image/380945.jpg
# Spatial distribution functions of random packed granular spheres obtained by direct particle imaging 

Department of Physics, Clark University, Worcester, MA 01610 (Dated: (August 30, 2018)) 

Andreea Panaitescu and Arshad Kudrolli 

We measure the two-point density correlations and Voronoi cell distributions of cyclically sheared granular spheres obtained with a fluorescence technique and compare them with random packing of frictionless spheres. We find that the radial distribution function g ( r ) is captured by the Percus- Yevick equation for initial volume fraction φ = 0 . 59. However, small but systematic deviations are observed because of the splitting of the second peak as φ is increased towards random close packing. The distribution of the Voronoi free volumes deviates from postulated Γ distributions, and the orientational order metric Q 6 shows disorder compared to numerical results reported for frictionless spheres. Overall, these measures show significant similarity of random packing of granular and frictionless spheres, but some systematic differences as well. 

The packing of spheres is one of the enduring prob- lems in physics, and a basis to understand the structure and strength of granular matter. Assuming dominance of steric interactions, dense packing of steel spheres was first used to understand structure of simple liquids with the radial distribution function g ( r ) and the orientation order metric Q$_{6}$ [1]. However, experimental measurements at boundaries [2] and computer simulations in the bulk [3] have since shown that inter-particle friction can affect granular packing. Friction between particles changes the fundamental stability condition at contact from the fric- tionless case, causing a packing to be protocol dependent and the system to be out-of-equilibrium. 

The difficulty of accurately measuring significant num- ber of particle positions in the bulk away from the influ- ence of boundaries has also stymied progress. Recent ex- perimental studies [4, 5] have examined packing of gran- ular spheres and find that the associated free volume dis- tributions are described by a Γ distribution with two fit- ting parameters which were then given a thermodynamic interpretation [5]. These results are puzzling in light of earlier analytical work in one-dimension and simulations in two and three dimensions that show a Γ distribution with 3 fitting parameters is needed to describe a broad range of volume fraction for elastic particles [6]. 

Here, we discuss new experiments with spherical gran- ular particles which enable us to directly determine sta- tistical measures to understand the effect of friction, test the effect of shear, and perform a rigorous comparison with frictionless hard sphere packing. Using a fluores- cence technique [4, 7, 8], we obtain the packing of glass spheres before and after application of cyclic shear, and compare with random packing of frictionless spheres. We find that the overall shape of g ( r ) for volume fraction φ ∼ 0 . 6 is captured by the Percus-Yevick equation [9] which assumes random packing of spheres without angu- lar correlations. But, systematic deviations are observed because of the splitting of the second peak as φ is in- creased toward random close packing, Q$_{6}$ shows partial hexagonal order, and the distribution of the Voronoi free 

PACS numbers: 45.70.Qj, 05.65.+b 

<img src='content_image/380945.jpg'>

volumes shows enhanced probabilities at higher values compared with Γ distributions postulated [6] for random packing of spheres. 

The experiments to measure structure of granular packing are performed using a shear cell shown schemat- ically in Fig. 1(a). The parallelepiped shaped cell con- sists of a rigid front, back, and bottom transparent glass boundary, and side boundaries that can be tilted through a prescribed angle θ to cyclically shear and perturb the packing. Glass spheres with density ρ$_{g}$ = 2 . 5 × 10 3 

