Statistical characterization of microstructure of packings of polydisperse hard cubes

Polydisperse packings of cubic particles arise in several important problems. Examples include zeolite microcubes that represent catalytic materials, fluidization of such microcubes in catalytic reactors, fabrication of new classes of porous materials with precise control of their morphology, and several others. We present the results of detailed and extensive simulation and microstructural characterization of packings of nonoverlapping polydisperse cubic particles. The packings are generated via a modified random sequential-addition algorithm. Two probability density functions (PDFs) for the particle-size distribution, the Schulz and log-normal PDFs, are used. The packings are analyzed, and their random close-packing density is computed as a function of the parameters of the two PDFs. The maximum packing fraction for the highest degree of polydispersivity is estimated to be about 0.81, much higher than 0.57 for the monodisperse packings. In addition, a variety of microstructural descriptors have been calculated and analyzed. In particular, we show that (i) an approximate analytical expression for the structure factor of Percus-Yevick fluids of polydisperse hard spheres with the Schulz PDF also predicts all the qualitative features of the structure factor of the packings that we study; (ii) as the packings become more polydisperse, their behavior resembles increasingly that of an ideal system—“ideal gas”—with little or no correlations; and (iii) the mean survival time and mean relaxation time of a diffusing species in the packings increase with increasing degrees of polydispersivity.

Figure 1

The SEM images of (a) cubic aluminosilicate zeolite particles (Advera® 401) suspended in a Newtonian fluid of polyethylene glycol (after Ref. [33]), and (b) self-synthesized LTA zeolite crystals (after Ref. [34]).

Figure 2

The size distributions of the particles. (a) The Schulz distribution, and (b) the log-normal distribution.

Figure 3

Examples of the packings. Parts (a), (b), and (c) show packings with the Schulz distribution with, respectively, $m=0$, 2, and 10, while parts (d), (e), and (f) present those generated with the log-normal PDF with standard deviations that are, respectively, $\sigma =1$, 0.6, and 0.2.

Figure 4

Random close-packing density vs (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal PDF.

Figure 5

Dependence of the radial distribution function $g\left(r\right)$ on (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal distribution. $m=\infty$ and $\sigma =0$ represent the monodisperse packings.

Figure 6

Dependence of the structure factor $S\left(k\right)$ on (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal distribution. $m=\infty$ and $\sigma =0$ represent the monodisperse packings.

Figure 7

Comparison of the structure factor $S\left(k\right)$, computed by the numerical simulation, and the predictions of the analytical approximation, Eq. (12). The sizes of the particles follow the Schulz distribution with the polydispersivity parameter $m$. The monodisperse packings correspond to $m=\infty$.

Figure 8

Loss of order in the polydisperse packings, as characterized by the order metric $\tau$ defined by Eq. (12).

Figure 9

Dependence of the two-point probability function $S_2^{\left(1\right)}\left(r\right)$ for the pore space of the packings on (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal PDF. $m=\infty$ and $\sigma =0$ represent the monodisperse packings.

Figure 10

The surface area ratio $A$ vs (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal PDF.

Figure 11

Dependence of the lineal-path function $L^{\left(1\right)}\left(z\right)$ of the pore space of the packings on (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal distribution. $m=\infty$ and $\sigma =0$ represent the monodisperse packings.

Figure 12

Dependence of the cumulative pore-size distribution function $F\left(\delta \right)$ on (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal distribution. $m=\infty$ and $\sigma =0$ represent the monodisperse packings.

Figure 13

Lower bounds for the mean survival time $T_m$ and the principal relaxation time $T_1$ of a diffusing species in the two types of packings.

Figure 14

The scaled mean nearest-neighbor distance $L_P$ vs (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal PDF.

Figure 15

Dependence of the specific surface area $s$ on the porosity for various values of (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviation $\sigma$ of the log-normal PDF. $m=\infty$ and $\sigma =0$ represent the monodisperse packings.

Figure 16

Dependence of the mean nearest-neighbor distance $l_p$ on the porosity of the packings for several values of (a) the polydispersivity parameter $m$ of the Schulz distribution, and (b) the standard deviations $\sigma$ of the log-normal PDF. $m=\infty$ and $\sigma =0$ represent the monodisperse packings.