Structural Transformation between a Nematic Loose Packing and a Randomly Stacked Close Packing of Granular Disks

Packing structures of granular disks are reconstructed using magnetic resonance imaging techniques. As packing fraction increases, the packing structure transforms from a nematic loose packing to a dense packing with randomly oriented stacks. According to our model based on Edwards’ volume ensemble, stack structures are statistically favored when the effective temperature decreases, which has a lower structural anisotropy than single disks, and brings down the global orientational order consequently. This mechanism identified in athermal granular materials can help us understand the nonergodic characteristics of disklike particle assemblies such as discotic mesogens and clays.

Figure 1

(a) The nematic order parameters for packings with different $\mathrm{\Phi }$. (b),(c) Reconstructed packing structures with (b) $\mathrm{\Phi }=0.53$ and (c) 0.66. The disk color represents the angle between a disk’s orientation and the vertical direction.

Figure 2

(a) Probability distributions of stack size $N_s$ for packings with different $\mathrm{\Phi }$: 0.65 (hexagons), 0.63 (pentagons), 0.60 (diamonds), and 0.56 (triangles). The solid lines are exponential fits. (b) Average stack size for packings with different $\mathrm{\Phi }$. The dotted line is the theoretical ensemble-averaging result.

Figure 3

(a) Average numbers of different forms of contacts $⟨z⟩$ for packings with different $\mathrm{\Phi }$. (b) Averaged constraint number of type-$\alpha$ disks in each packing: ${⟨c⟩}_{\alpha }$, where the average is taken over disks of type 0 (diamonds), 1 (triangles), and 2 (squares), respectively. (c) Averaged constraint number $⟨c⟩$ of all disks in each packing. In panels (b) and (c), green symbols represent the average constraint numbers of all contacts, and red symbols represents those only counting contacts on disk faces. The solid lines in (b) represent the typical constraint numbers $C_0$ and $C_2$, and the one in (c) marks a generalized isostatic condition $⟨c⟩=6$.

Figure 4

(a) PDFs of Voronoi cell volume $V_{\mathrm{cell}}$ for packings with different $\mathrm{\Phi }$. The solid lines present PDFs of all disks’ $V_{\mathrm{cell}}$, and the symbols represent separated PDFs of disks of type 0 (diamonds), 1 (triangles), and 2 (squares). The vertical dashed lines represent mean values of the separated PDFs. The gray regions’ dotted borders mark the smallest and largest values of $V_{\mathrm{cell}}$: $V_d/0.92$ and $V_d/0.37$. (b) The PDFs of $V_{\mathrm{cell}}$ normalized by their averaged values. (c) The equation of state of compactivity versus packing fraction obtained by a thermodynamic integration of packing volume fluctuation (solid line) and ensemble averaging (dotted line).

Figure 5

(a) Correlation coefficients $\gamma$ between $s_{\mathrm{disk}}$ and $\varphi$ for disks of type 0 (diamonds), 1 (triangles), and 2 (squares), and (b) $\gamma$ between $s_{\text{stack}}$ and ${\varphi }_{\text{stack}}$, for packings with different $\mathrm{\Phi }$.