Dynamic self-assembly of non-Brownian spheres studied by molecular dynamics simulations

Granular self-assembly of confined non-Brownian spheres under gravity is studied by molecular dynamics simulations. Starting from a disordered phase, dry or cohesive spheres organize, by vibrational annealing, into body-centered-tetragonal or face-centered-cubic structures, respectively. During the self-assembling process, isothermal and isodense points are observed. The existence of such points indicates that both granular temperature and packing fraction undergo an inversion process that may be in the core of crystal nucleation. Around the isothermal point, a sudden growth of granular clusters having the maximum coordination number takes place, indicating the outcome of a first-order phase transition. We propose a heuristic equation that successfully describes the dynamic evolution of the local packing fraction in terms of the local granular temperature, along the entire crystallization process.

Figure 1

(a) Snapshots of the dry system at 22, 32, 42, and 62 Hz. The system crystallizes into a bct structure. (b) Granular temperature versus frequency for each ${\text{bin}}_k$ depicted in the inset. We clearly observe an inversion beyond the isothermal point. (c) and (d) Cohesive system crystallizing into a fcc structure.

Figure 2

Mean number of particles having the maximum coordination number. The maximum coordination number sharply grows around 34–36 Hz, when the temperature vs frequency profiles cross. The inset shows the same plot on a semilogarithmic scale, where the sharp transition can be better appreciated.

Figure 3

Local packing fraction for various frequencies as a function of height (${\text{bin}}_k$) using Voronoi tessellation for (a) a bct crystal and (b) a fcc crystal. Isodense points appear near the final heights for each lattice.

Figure 4

Packing fraction of the individual spheres in their final positions inside their corresponding crystal structure. Low values correspond to beads in corners and walls; 0.74 is for beads in the bulk of the fcc crystal (crosses).

Figure 5

Logarithms of the mean local packing fraction inside their corresponding ${\text{bin}}_k$, as a function of the equivalent mean local temperature (dots, also calculated in the same ${\text{bin}}_k$), and the model proposed in Eq. (6) (solid lines). Here $\alpha \left({\text{bin}}_k\right)$ is (a) 5.603 for the bct structure and (b) 5.541 for the fcc structure.