Experiments on the Random Packing of Tetrahedral Dice

Tetrahedra may be the ultimate frustrating, disordered glass forming units. Our experiments on tetrahedral dice indicate the densest (volume fraction $\varphi =0.76\pm .02$, compared with ${\varphi }_{\mathrm{sphere}}=0.64$), most disordered, experimental, random packing of any set of congruent convex objects to date. Analysis of MRI scans yield translational and orientational correlation functions which decay as soon as particles do not touch, much more rapidly than the $\sim 6$ diameters for sphere correlations to decay. Although there are only $6.3\pm .5$ touching neighbors on average, face-face and edge-face contacts provide enough additional constraints, $12\pm 1.6$ total, to roughly bring the structure to the isostatic limit for frictionless particles. Randomly jammed tetrahedra form a dense rigid highly uncorrelated material.

Figure 1

(a) Tetrahedral dice used for this study; top free surface of cylindrical packing. (b) Packing fraction of tetrahedral dice in cylinders of different radii. Extrapolated value for an infinite radius cylinder is $0.76\pm 0.02$. (c) Volume of water needed to fill a spherical container to a fraction of its height when packed with spheres or tetrahedral dice relative to the volume required when the container is empty of particles.

Figure 2

Left: A slice from an MRI scan of tetrahderal dice in a cylindrical container. Water is white, dice are black. Middle: The identified inscribed spheres. Right: The dice are replaced by ideal tetrahedra with identified position and orientation.

Figure 3

(a) Radial distribution function of dice centroids compared with published results for randomly packed spheres. $R_{\min }$ is a diameter for spheres, $R_{\min }=a/\sqrt{6}$ for tetrahedra where $a$ is an edge length. (b) Orientational correlation function for the tetrahedral dice, $S(r)$ is one for complete overlap after translation and is minimum at $1/3$. (c) Face-face correlation function, $F(r)$ is $-1$ for face-face contacts. All correlations decay in a distance of $\sim 3R_{\min }$ which is the furthest distance between the centers of touching tetrahedra.

Figure 4

Fraction of tetrahedra with specified particle-particle contacts. (a) is the total number of contacting neighbors per particle which averages to $6.3\pm .5$. (f) is the coordination number or number of constraints per particle obtained by weighting face-face contacts by three, edge-face contacts by two, and others by one. The average coordination $⟨z⟩$ is $12\pm 1.6$. The error bars shown in the figures result from the spread of criteria used in evaluating the types of contacts.