Pore configuration landscape of granular crystallization

Abstract

Uncovering grain-scale mechanisms that underlie the disorder-order transition in assemblies of dissipative, athermal particles is a fundamental problem with technological relevance. To date, the study of granular crystallization has mainly focussed on the symmetry of crystalline patterns while their emergence and growth from irregular clusters of grains remains largely unexplored. Here crystallization of three-dimensional packings of frictional spheres is studied at the grain-scale using X-ray tomography and persistent homology. The latter produces a map of the topological configurations of grains within static partially crystallized packings. Using numerical simulations, we show that similar maps are measured dynamically during the melting of a perfect crystal. This map encodes new information on the formation process of tetrahedral and octahedral pores, the building blocks of perfect crystals. Four key formation mechanisms of these pores reproduce the main changes of the map during crystallization and provide continuous deformation pathways representative of the crystallization dynamics.

Figure 1. 3D rendering of spherical and cylindrical experimental packings.

Internal structure of partially crystallized packing with a global packing density of φ=0.685 in (a) a spherical container and (b) a cylindrical container. The scale bar length is ≈25 mm. In (a), the spherical packing contains 80,000 beads. In panel (b), two different 3D sections of a cylindrical packing made of 200,000 beads. Panel (b) highlights the heteregeneous structure of our packings composed of both random and highly crystalline clusters. By comparing the two 3D sections, it also shows that the rich polycrystalline structures of our packings are characterized by defects, such as grain vacancies and dislocations.

Figure 2. Persistence diagrams.

(a) Tetrahedral configuration of mathematical balls with growing radius α. Colour changes from blue to red with increasing z value. (b) Persistence diagram PD₂ as a probability density function of pores indicating via the colour map the occurence rate P of a given grain configuration at a specific (b,d) pair. PD₂ is plotted for a packing with a density of φ=0.685. (c) PD₂ for the same packing, the rate of occurence is now expressed in terms of the frequency index I_f=log((log(P+1))+1)/I_fM to highlight the fine details of the diagram.

Figure 3. Persistence diagram of experimental packings.

Persistence diagrams PD₂ of partially crystallized packings produced in: (a) a cylindrical container with a density φ=0.66 (N=61,000 and D=1.62 mm, note that the (b,d) coordinates have been rescaled to be directly comparable with data shown in panels (b,c) for which D=1 mm), (b) a spherical container with a density φ=0.685 (N=86,000 and D=1 mm), (c) a spherical container with a density φ=0.71 (N=86,000 and D=1 mm).

Figure 4. PD₂ of disordered, partially ordered and highly ordered sphere packings.

Representative PD₂ of sphere packings with density ranging from φ=0.60 to φ=0.73. Crystallization onset occurs at φ≈0.64. The diagrams in panels (a,b) have been computed over more than 150,000 beads and 500,000 cavities. The diagram in (c) has been computed over more than 200,000 beads and 800,000 cavities. The diagram in (d) has been computed over more than 4,000 beads and 20,000 cavities.

Figure 5. Quantification of regular cavities versus φ based on PD₂.

(a) PD₂ for a sphere packing with a density of φ=0.73. The blue squares indicate the regions of interest used to count the number of quasi regular tetrahedral and octahedral cavities. The size of these rectangular regions are defined by birth and death associated with grain polydispersity (D±0.025 mm) around (birth, death) tetrahedral and octahedral cavities. (b) Numbers N_tetra and N_octa of quasi regular tetrahedral and octahedral cavities (normalized by the total number of cavities) versus packing density φ. (c) Proportion P_tetra=N_tetra/(N_tetra+N_octa) and P_octa=N_octa/(N_tetra+N_octa) of quasi regular tetrahedral and octahedral cavities versus packing density φ.

Figure 6. Grain-scale tetrahedral and octahedral formation/deformation scenarios.

(a) PD₂ of a partially crystallized sphere packing with density φ=0.685. The superimposed curves correspond to analytically computed birth–death curves of the deformation scenarios shown in panels (b,c). (b) Top and side views of D1 and D2 deformations scenarios of a tetrahedral cavity. (c) Top and side views of D3 and D4 deformation scenarios of an octahedral cavity. The colour code indicates the relative height of sections of the grain with respect to the horizontal median plane.

Figure 7. Numerical simulation of the dynamics of the order–disorder transition of a bead packing under shear.

(a) Packing density versus time (expressed in inverse shear rate units). (b) Snapshot of the numerically generated packings as it gets disordered. (c–f) Temporal evolution of PD₂ at different packing density ranging from φ=0.72 to φ=0.63. These diagrams have been computed over >6,000 beads.

Figure 8. Evolution of persistence diagram as a function of packing density φ.

Persistence diagrams PD₂ of sphere packings (subsets of 4,000 beads) over the density range φ=0.60−0.73.