Common packing patterns for jammed particles of different power size distributions

We introduce a model for particles that are extremely polydisperse in size compared with monodisperse and bidisperse systems. In two dimensions (2D), size polydispersity inhibits crystallization and increases the packing fraction at jamming points. However, no packing pattern common to diverse polydisperse particles has been reported. We focused on polydisperse particles with a power size distribution $r^{-a}$ as a ubiquitous system that can be expected to be scale invariant. We experimentally and numerically constructed 2D random packing for various polydisperse particles with different size exponents $a$. Analysis of the packing pattern revealed a common contact number distribution for $a<3$ and a higher jamming point for $2<a<3$ than in monodisperse systems. These findings demonstrate that the ambiguity of the characteristic length provides common properties that lead to a classification scheme for polydisperse particles.

Figure 1

(a) Schematic of the experimental setup of polydisperse droplets confined in 2D space. (b) Microscopic image of the polydisperse droplets from the top (top panel) and cross-sectional image along the dashed line in the top panel (bottom panel). The scale bar is $100\mu \mathrm{m}$. (c) Micrograph of the packed droplets with the automatically detected contact lines. The scale bar is $200\mu \mathrm{m}$. (d) and (e) Microscopic images for bidisperse and polydisperse systems, respectively. The scale bars are 1 mm. (f) and (g) Droplet radius distribution $N\left(r\right)$ and contact number distribution $\nu \left(z\right)$, respectively, for a bidisperse system. The solid curve in the $N\left(r\right)$ plot is the fitting line for multiple Gaussians, indicating that there are two peaks. (h) and (i) $N\left(r\right)$ and $\nu \left(z\right)$ for a polydisperse system. The inset log-log graphs show their normalized cumulative distributions, $N_n\left(r\right)$ and ${\nu }_n\left(z\right)$. (i) The radius distribution $N\left(r\right)$ follows a power size distribution of $a\simeq 3$ in the region of one order of magnitude.

Figure 2

Cumulative radius distribution $N\left(r\right)$ for numerically produced polydisperse systems, $r^{-a}$. (a), (b), (c), (d), and (e) $a=1.5$, 2, 2.5, 3, and 3.5, respectively. The error bars represent the standard deviation of five multiple runs.

Figure 3

(a) Examples of numerically produced packing patterns for various polydisperse systems $r^{-a}$ and (b)–(f) corresponding normalized cumulative contact number distribution ${\nu }_n\left(z\right)$. For (b), (c), (d), (e), and (f), $a=1.5$, 2, 2.5, 3, and 3.5, respectively. The solid line indicates an exponent of 3.8. The error bars represent the standard deviation of five multiple runs.

Figure 4

(a) Dependence of pressure $P$ on the packing fraction $\varphi$ for various $a$. (b) Dependence of ${\varphi }_{\mathrm{c}}$ on $a$.