Coupling effects of particle size and shape on improving the density of disordered polydisperse packings

Abstract

It is well established that the packing density (volume fraction) of the random close packed (RCP) state of congruent three-dimensional spheres, i.e., ${\ensuremath{\varphi}}_{c}\ensuremath{\sim}0.64$, can be improved by introducing particle size polydispersity. In addition, the RCP density ${\ensuremath{\varphi}}_{c}$ can also be increased by perturbing the particle shape from a perfect sphere to nonspherical shapes (e.g., superballs or ellipsoids). In this paper, we numerically investigate the coupling effects of particle size and shape on improving the density of disordered polydisperse particle packings in a quantitative manner. A previously introduced concept of ``equivalent diameter'' (${D}_{e}$), which encodes information of both the particle volume and shape, is reexamined and utilized to quantify the effective size of a nonspherical particle in the disordered packing. In a highly disordered packing of mixed shapes (i.e., polydispersity in particle shapes) with particles of identical ${D}_{e}$, i.e., no size dispersity effects, we find that the overall specific volume $e$ (reciprocal of ${\ensuremath{\varphi}}_{c}$) can be expressed as a linear combination of the specific volume ${e}_{k}$ for each component $k$ (particles with identical shape), weighted by its corresponding volume fraction ${X}_{k}$ in the mixture, i.e., $e={\ensuremath{\sum}}_{k}{X}_{k}{e}_{k}$. In this case, the mixed-shape packing can be considered as a superposition of RCP packings of each component (shape) as implied by a set Voronoi tessellation and contact number analysis. When size polydispersity is added, i.e., ${D}_{e}$ of particles varies, the overall packing density can be decomposed as ${\ensuremath{\varphi}}_{c}={\ensuremath{\varphi}}_{L}+{f}_{\mathrm{inc}}$, where ${\ensuremath{\varphi}}_{L}$ is the linear part determined by the superposition law, i.e., ${\ensuremath{\varphi}}_{L}=1/{\ensuremath{\sum}}_{k}{X}_{k}{e}_{k}$, and ${f}_{\mathrm{inc}}$ is the incremental part owing to the size polydispersity. We empirically estimate ${f}_{\mathrm{inc}}$ using two distribution parameters, and apply a shape-dependent modification to improve the accuracy from $\ensuremath{\sim}0.01$ to $\ensuremath{\sim}0.005$. Especially for nearly spherical particles, ${f}_{\mathrm{inc}}$ is only weakly coupled with the particle shape. Generalized polydisperse packings even with a moderate size ratio ($\ensuremath{\sim}4$) can achieve a relatively high density ${\ensuremath{\varphi}}_{c}\ensuremath{\sim}0.8$ compared with polydisperse sphere packings. Our results also have implications for the rational design of granular materials and model glass formers.