Direct observation of crystal nucleation and growth in a quasi-two-dimensional nonvibrating granular system

We study a quasi-two-dimensional macroscopic system of magnetic spherical particles settled on a shallow concave dish under a temporally oscillating magnetic field. The system reaches a stationary state where the energy losses from collisions and friction with the concave dish surface are compensated by the continuous energy input coming from the oscillating magnetic field. Random particle motions show some similarities with the motions of atoms and molecules in a glass or a crystal-forming fluid. Because of the curvature of the surface, particles experience an additional force toward the center of the concave dish. When decreasing the magnetic field, the effective temperature is decreased and diffusive particle motion slows. For slow cooling rates we observe crystallization, where the particles organize into a hexagonal lattice. We study the birth of the crystalline nucleus and the subsequent growth of the crystal. Our observations support nonclassical theories of crystal formation. Initially a dense amorphous aggregate of particles forms, and then in a second stage this aggregate rearranges internally to form the crystalline nucleus. As the aggregate grows, the crystal grows in its interior. After a certain size, all the aggregated particles are part of the crystal and after that crystal growth follows the classical theory for crystal growth.

Figure 1

Experimental setup. The Helmholtz coils have an inner diameter of 15 cm and an outer diameter of 20 cm.

Figure 2

Sequence of photos from experiment E1, showing evolution from a fluid-like configuration to crystal configuration. The amplitude of the forcing $B_o$ is noted in the corner of each image.

Figure 3

(a) A typical photo at a certain stage of the experiment, (b) trajectories of particles over 1.66 s (0.83 s before and 0.83 s after the position shown), (c) Delaunay triangulation, (d) ${\psi }_6$ order parameter, (e) number of bonded neighbors $N_B$, and (f) the bond order parameter ${\psi }_6^{\prime }$, which is calculated based only on touching neighbors (those counted by $N_B$). This configuration is the same as shown in Fig. 2.

Figure 4

Final configurations of experiments E1, E2, and E3. The three aggregates have an ordered configuration corresponding to the highest level in the color code of the sixth bond configurational order parameter ${\psi }_6^{\prime }$, depicted in Fig. 3.

Figure 5

(a) Average of the configurational order parameter ${\psi }_6$ for each frame as a function of magnetic field. (b) Average of the number of bonded neighbors of particles for each frame as a function of magnetic field. (c) Effective system radius in each frame as a function of magnetic field, in units of the particle diameter $\sigma$.

Figure 6

Mean order parameter $\overline{{\psi }_6}$ as function of the mean bond number. Each point represents a different time in the experiment, and the averages are over all particles at that time.

Figure 7

Effective diffusion coefficient measured in time windows of 4.16 s every 166.66 s. The open symbols correspond to gaseous particles and the filled symbols are averaged over all particles (both gaseous and aggregated). The units are ${\sigma }^2/s$ in terms of the particle diameter $\sigma$.

Figure 8

(a) Aggregate size (number of particles $N$) as a function of the magnetic field. (b) Mean ${\psi }_6^{\prime }$ order parameter for the aggregate particles as a function of the magnetic field. (c) Mean number of bonded neighbors $N_B$ for the aggregate particles as a function of the magnetic field.

Figure 9

Average number of bonds $N_B$ as function of the number of particles $N$ in the aggregate. The color indicates the mean value of ${\psi }_6^{\prime }$, matching the color key of Fig. 3.

Figure 10

Sequence of images showing a notable change of structural characteristics. The data are from experiment E3 and the color indicates ${\psi }_6^{\prime }$ of each particle, matching the color key of Fig. 3.

Figure 11

Number of particles in the aggregate with five and six bonds as a function of the magnetic field, for the three experiments studied here.

Figure 12

Number of particles in the aggregate as a function of the magnetic field amplitude with a time resolution of 1/60 s. The horizontal lines separate out the early stage, middle stage, and late stage of growth. The color indicates the value of ${\psi }_6^{\prime }$ as indicated in the legend.

Figure 13

Sequence of particle configurations in the early formation of a nucleus, corresponding to the data in Fig. 12. The earliest stage has $N<16$ particles in the aggregate and corresponds to $B_o\ge 5.716$ mT. The second stage has $16\le N\le 24$ and corresponds to 5.710 mT $\le B_o<5.716$ mT. The third stage has $N>24$ and $B_o<5.710$ mT. The color of each particle indicates ${\psi }_6^{\prime }$ matching the color key of Fig. 3.

Figure 14

Comparison of the behavior of the aggregate size as a function of the time elapsed from its formation. Recall the magnetic field amplitude decreases by 0.1 mT in 50 s.

Figure 15

Comparison between the stable aggregate at two different times from experiment E1. The growth suggests particles are added at concave regions where the new particles can contact more neighbors that help stabilize their presence in the aggregate, and particles with only tenuous connections to the aggregate are easier to “evaporate.”