Geometrical properties of rigid frictionless granular packings as a function of particle size and shape

Three-dimensional discrete numerical simulation is used to investigate the properties of close-packed frictionless granular assemblies as a function of particle polydispersity and shape. Unlike some experimental results, simulations show that disordered packings of pinacoids (eight-face convex polyhedra) achieve higher solid fraction values than amorphous packings of spherical or rounded particles, thus fulfilling the analog of Ulam's conjecture stated by Jiao and co-workers for random packings [Y. Jiao and S. Torquato, Phys. Rev. E 84, 041309 (2011)]. This seeming discrepancy between experimental and numerical results is believed to result from difficulties in overcoming inter particle friction through experimental densification processes. Moreover, solid fraction is shown to increase further with bidispersity and peak when the volume proportion of small particles reaches $30\%$. Contrarily, substituting up to $50\%$ of flat pinacoids for isometric ones yields solid fraction decrease, especially when flat particles are also elongated. Nevertheless, particle shape seems to play a minor role in packing solid fraction compared to polydispersity. Additional investigations focused on the packing microstructure confirm that pinacoid packings fulfill the isostatic conjecture and that they are free of order except beyond 30% to $50\%$ of flat or flat-elongated polyhedra in the packing. This order increase progressively takes the form of a nematic phase caused by the reorientation of flat or flat-elongated particles to minimize the packing potential energy. Simultaneously, this reorientation seems to increase the solid fraction value slightly above the maximum achieved by monodisperse isometric pinacoids, as well as the coordination number. Finally, partial substitution of elongated pinacoids for isometric ones has limited effect on packing solid fraction or order.

Figure 1

Pinacoid, a model polyhedron characterized by its length $L$, width $G$, height $E$ and angle $\alpha$. This polyhedron has three symmetry planes, each perpendicular to an inertia axis $\vec{u}, \vec{v}$, or $\vec{w}$.

Figure 2

Snapshot of the pinacoids used in this study: (a) isometric, (b) elongated, (c) flat, and (d) flat-elongated.

Figure 3

3D snapshot of a packing incorporating $29\%$ by volume of flat-elongated pinacoids. Biperiodic boundary conditions apply in $x$ and $y$ horizontal directions

Figure 4

Proportions of small (solid lines) and large (dashed lines) particles in $d$-thick layers along the $z$ axis for bidisperse packings incorporating $X_{d/3}=13\%$ by mass of spheres ($•$) or pinacoids ($\blacksquare$). Proportions of large isometric pinacoids (solid line) and flat-elongated pinacoids (dashed line) in $d$-thick layers for packings incorporating $X_{PO}=29\%$ by mass of flat-elongated ($▴$) particles. Error bars denote the standard deviation.

Figure 5

Maximum solid fraction $\varphi$ as a function of the volume proportion $X_S$ of (a) small spheres or rounded particles and (b) small pinacoids or crushed particles in bidisperse packings. Error bars denote the standard deviation.

Figure 6

Maximum solid fraction $\varphi$ vs the proportion by volume of (a) elongated ($X_P$), (b) flat ($X_O$), and (c) flat-elongated ($X_{PO}$) particles in mixtures with large isometric pinacoids. Error bars denote the standard deviation.

Figure 7

[Color online) Pair correlation function $g\left(r\right)$ for several proportions by volume of (a) small pinacoids $X_S$ and (b) elongated $X_P$, flat $X_O$ or flat-elongated particles $X_{PO}$. For each mixture, $R_{min}$ stands for the radius of the sphere circumscribed to the smallest pinacoid (e.g., $R_{min}=d/6$ for mixtures of large and small isometric pinacoids).

Figure 8

Cross-sectional view of the arrangement due to contact between trapezoidal and triangular faces with a distance between pinacoid centers of $r\simeq (2/\sqrt{6}+1/2)R_{min}\simeq 1.3R_{min}$.

Figure 9

Nematic order parameter $\left(Q_{00}^2\right)$ of pinacoid packings incorporating various proportions by volume of small ($X_S$), elongated ($X_P$), flat ($X_O$) or flat-elongated ($X_{PO}$) pinacoids. Error bars denote the standard deviation. Initial state refers to $Q_{00}^2$ values calculated during sample preparation just before applying gravity.

Figure 10

3D snapshot of a packing incorporating $100\%$ by volume of flat-elongated pinacoids. Biperiodic boundary conditions apply in $x$ and $y$ directions

Figure 11

Coordination number $N$ of (a) all, (b) simple, (c) double, and (d) triple contacts as a function of the proportion by volume of small ($X_S$), elongated ($X_P$), flat ($X_O$), or flat-elongated ($X_{PO}$) pinacoids in the packing. Error bars denote the standard deviation

Figure 12

Coordination number $N$ vs the nematic order parameter $\left(Q_{00}^2\right)$ of pinacoid packings incorporating various proportions by volume of small ($X_S$), elongated ($X_P$), flat ($X_O$), or flat-elongated ($X_{PO}$) pinacoids. The fit equation is $N\left(Q_{00}^2\right)=7.4+(9.2-7.4)[1-exp(-Q_{00}^2/0.2)]$ with $R^2=0.92$. Error bars denote the standard deviation.