Epitaxial growth of ordered and disordered granular sphere packings

We demonstrate that epitaxy can be used to obtain a wide range of ordered to disordered granular packings by simply changing the deposition flux. We show that a defect-free face-centered-cubic (fcc) monocrystal can be obtained by depositing athermal granular spheres randomly into a container with a templated surface in a gravitational field without direct manipulation. This packing corresponds to the maximum sphere packing fraction and is obtained when the substrate is templated corresponding to the (100) plane of a fcc crystal and the container side is an integer multiple of the sphere diameter. We find that the maximum sphere packing is obtained when the deposited grains come to rest, one at a time, without damaging the substrate. A transition to a disordered packing is observed when the flux is increased. Using micro x-ray computed tomography, we find that defects nucleate at the boundaries of the container in which the packing is grown as grains cooperatively come to rest above their local potential minimum. This leads to a transition from ordered to disordered loose packings that grow in the form of an inverted cone, with the apex located at the defect nucleation site. We capture the observed decrease in order using a minimal model in which a defect leads to growth of further defects in the neighboring sites in the layer above with a probability that increases with the deposition flux.

Figure 1

(a) A fully ordered fcc packing observed by dropping spheres for sufficiently low deposition flux $\Gamma =11$, which is normalized by the template potential well energy. A transition to disordered packings is observed as the deposition flux is increased to $\Gamma =16$ (b), $\Gamma =22$ (c), $\Gamma =109$ (d), and $\Gamma =1090$ (e). (f) Example of a disordered packing observed when all the spheres are dropped from a height of the container where $\Gamma =3.6\times {10}^5$. The spheres are rendered using their measured positions with ordered [orange (light gray)] and disordered (gray) phase denoted by using $Q_{6,\text{local}}$ as a criterion.

Figure 2

(a)–(f) The density profile $\rho \left(z\right)$ as a function of height $z$ corresponding to packings shown in Figs. 1– 1.

Figure 3

(a)–(f) The scatter plot of $Q_{4,\text{local}}$ vs $Q_{6,\text{local}}$ corresponding to packings shown in Figs. 1– 1. Each point corresponds to a particular sphere in the packings. The expected values for a fcc monocrystal of the size of the container is indicated by a red . Insets: the corresponding probability distribution function (PDF) of $Q_{6,\text{local}}$. (a) A sharp peak is observed corresponding to the theoretical $Q_{6,\text{local}}$ in the bulk of a fcc monocrystal, and a much smaller peak due the ones next to the boundaries. As the deposition flux $\Gamma$ is increased, the fcc peak is observed to decrease in value, and a broader distribution is observed that is fitted by a Gaussian [dashed red (gray) line] with mean 0.45 (b), 0.43 (c), 0.42 (d), and 0.43 (e). (f) The fcc peak is observed to be completely absent and the distribution is fitted by a Gaussian with mean 0.37.

Figure 4

(a) Top view of the (100) plane of a fcc crystal with open blue (dark gray) circles and orange (light gray) cross symbols denoting the lattice sites in consecutive planes. The sites linked by dashed blue (dark gray) lines correspond to layer A and solid orange (light gray) lines to layer B, respectively. A defect leading to one of the spheres being slightly off its potential minima and away from the side wall is shown as well. In this case, the two points indicated by the solid orange (light gray) circles form small minima in which spheres can lodge, leading to defects, which in turn can exclude spheres from lodging into sites indicated by arrows. (b) Examples of defects that can arise because of imperfections in the template are indicated by circles with dashed-line style. A wide range of incipient arches can be supported with granular spheres because of the presence of friction.

Figure 5

The number of ordered spheres as a function of height $z$ is observed to decrease, increasing faster as the deposition flux is increased. The small layer-to-layer oscillation in each curve arises because the total number of fcc ordered spheres is different because of the finite size of the container. The overall trends are captured by a minimal model in which defects are nucleated at the boundaries, or by the presence of a defect in the layer below with a probability that increases with $\Gamma$ (see text).

Figure 6

(a)–(f) The density-density radial correlation function for the six packings shown in Fig. 1, which span the fully ordered to random packings. The experimental data are compared and contrasted with the $g\left(r\right)$ obtained for a fcc packed crystal that fits in the container used in the experiments [the red (gray) dotted line]. The location of the peaks and the relative amplitude is observed to compare well in the case of the ordered packing. The peaks in the experiments are systematically lower and broader to the polydispersity and error in locating the center of the grains.

Figure 7

A computer-generated face-centered-cubic monocrystal: The particles are divided into eight categories based on their coordination number, $Z$, and the local packing volume fraction, ${\varphi }_{\text{local}}$, and bond orientational parameter, $Q_{6,\text{local}}$, are reported for each category. $N$ is the number of particles with the same $Z$. (a) The bulk particles have $Z=12$ and ${\varphi }_{\text{local}}$ = ${\varphi }^{\text{fcc}}$, $Q_{6,\text{local}}$ = $Q_6^{\text{fcc}}$. (b)–(h) The spheres on the crystal surface have fewer than 12 nearest neighbors, and consequently deviation from the expected value of $Q_6^{\text{fcc}}$ and ${\varphi }^{\text{fcc}}$ can be observed in their case.