Characterization of void space, large-scale structure, and transport properties of maximally random jammed packings of superballs

Abstract

Dense, disordered packings of particles are useful models of low-temperature amorphous phases of matter, biological systems, granular media, and colloidal systems. The study of dense packings of nonspherical particles enables one to ascertain how rotational degrees of freedom affect packing behavior. Here, we study superballs, a large family of deformations of the sphere, defined in three dimensions by $|{x}_{1}{|}^{2p}+|{x}_{2}{|}^{2p}+{|{x}_{3}|}^{2p}\ensuremath{\le}1$, where $p\ensuremath{\in}(0,\ensuremath{\infty})$ is a deformation parameter indicating to what extent the shape deviates from a sphere. As $p$ increases from the sphere point ($p=1$), the superball tends to a cuboidal shape and approaches a cube in the $p\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. As $p\ensuremath{\rightarrow}0.5$, it approaches an octahedron, becomes a concave body with octahedral symmetry for $p<0.5$, and approaches a three-dimensional cross in the limit $p\ensuremath{\rightarrow}0$. Previous characterization of superball packings has shown that they have a maximally random jammed (MRJ) state, whose properties (e.g., packing fraction $\ensuremath{\phi}$, average contact number $\overline{Z}$) vary nonanalytically as $p$ diverges from unity. Here, we use an event-driven molecular dynamics algorithm to produce MRJ superball packings with $0.85\ensuremath{\le}p\ensuremath{\le}1.50$. To supplement the previous work on such packings, we characterize their large-scale structure by examining the behaviors of their structure factors $S(Q)$ and spectral densities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{{}_{V}}(Q)$, as the wave number $Q$ tends to zero, and find that these packings are effectively hyperuniform for all values of $p$ examined. We show that the mean width $\overline{w}$ is a useful length scale to make distances dimensionless in order to compare systematically superballs of different shape. Moreover, we compute the complementary cumulative pore-size distribution $F(\ensuremath{\delta})$ and find that the pore sizes tend to decrease as $|1\ensuremath{-}p|$ increases. From $F(\ensuremath{\delta})$, we estimate how the fluid permeability, mean survival time, and principal diffusion relaxation time vary as a function of $p$. Additionally, we compute the diffusion ``spreadability'' $\mathcal{S}(t)$ [Torquato, Phys. Rev. E 104, 054102 (2021)] of these packings and find that the long-time power-law scaling indicates these packings are hyperuniform with a small-$Q$ power law scaling of the spectral density ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{{}_{V}}(Q)\ensuremath{\sim}{Q}^{\ensuremath{\alpha}}$ with an exponent $\ensuremath{\alpha}$ that ranges from 0.64 at the sphere point to 0.32 at $p=1.50$, and decreases as $|1\ensuremath{-}p|$ increases. Each of the structural characteristics computed here exhibits an extremum at the sphere point and varies nonanalytically as $p$ departs the sphere point. We find the nonanalytic behavior in $\ensuremath{\phi}$ on either side of the sphere point is nearly linear, and determine that the rattler fraction ${\ensuremath{\phi}}_{R}$ decreases rapidly as $|1\ensuremath{-}p|$ increases. Our results can be used to help inform the design of colloidal or granular materials with targeted densities and transport properties.