Structure of sticky-hard-sphere random aggregates: The viewpoint of contact coordination and tetrahedra

We study more than ${10}^4$ random aggregates of ${10}^6$ monodisperse sticky hard spheres each, generated by various static algorithms. Their packing fraction varies from 0.370 up to 0.593. These aggregates are shown to be based on two types of disordered structures: random regular polytetrahedra and random aggregates, the former giving rise to $\delta$ peaks on pair distribution functions. Distortion of structural (Delaunay) tetrahedra is studied by two parameters, which show some similarities and some differences in terms of overall tendencies. Isotropy of aggregates is characterized by the nematic order parameter. The overall structure is then studied by distinguishing spheres in function of their contact coordination number (CCN). Distributions of average CCN around spheres of a given CCN value show trends that depend on packing fraction and building algorithms. The radial dependence of the average CCN turns out to be dependent upon the CCN of the central sphere and shows discontinuities that resemble those of the pair distribution function. Moreover, it is shown that structural details appear when the CCN is used as pseudochemical parameter, such as various angular distribution of bond angles, partial pair distribution functions, Ashcroft-Langreth and Bhatia-Thornton partial structure factors. These allow distinguishing aggregates with the same values of packing fraction or average tetrahedral distortion or even similar global pair distribution function, indicative of the great interest of paying attention to contact coordination numbers to study more precisely the structure of random aggregates.

Figure 1

The $\gamma (R_{\text{SC}}/R_{\text{ag}})$ for several (a) MAX-1 aggregates and (b) RMIN-MAX-1 aggregates built with various values of $a$ and a single $N$-RMIN-MAX-1 aggregate (seed composed of 600 spheres), for the sake of comparison, with $\mathcal{S}$ centered in (0,0,0).

Figure 2

Packing fraction of shells with thickness $w=0.1r_s$ on the densest aggregates built with three strategies: MAX-1 (three-sphere seed), RMIN-MAX-1 (three-sphere seed, RMIN), and $N$-RMIN-MAX-1 with a spherical seed of radius 10 extracted from a previously generated aggregate with the same value for $a$ and $N_{\text{ins}}$ for (a) relatively small $r$ ($r/R_{\text{ag}}\in [0,0.5]$) and (b) large $r$ ($r/R_{\text{ag}}\in [0.5,1]$).

Figure 3

Slices of aggregates generated by two algorithms: (a) RMIN-MAX-1 with $a=4$ and (b) $N$-RMIN-MAX-1 with $a=4$ and a seed consisting of $N=400$ spheres. Colors (grayscale) correspond to the contact coordination number: Brighter spheres have a higher CCN. (These figures were generated with ovito software [13].)

Figure 4

Slices of aggregates generated by two algorithms: (a) RMIN-MAX-1 with $a=4$ and (b) RRPA. Colors (grayscale) correspond to the contact coordination number: Brighter spheres have a higher CCN. (These figures were generated with ovito software [13].)

Figure 5

Normalized distributions of (a) ${\kappa }_D$ and (b) $L_{\text{max}}$ for various packing fractions. Aggregates were generated with the MAX-1 algorithm, with the exception of the RRPA. Error bars are smaller than the point size.

Figure 6

Dependence of (a) ${\overline{\kappa }}_D$ and (b) ${\overline{L}}_{\text{max}}$ with packing fraction. Error bars are smaller than the point size.

Figure 7

(a) Dependence of the volume fraction ${\mathrm{\Phi }}_V$ of perfect tetrahedra ($L_{\text{max}}=2$) with packing fraction. (b) Dependence of the volume fraction ${\mathrm{\Phi }}_V$ of tetrahedra with $L_{\text{max}}<2.3$ for the aggregates of the present study and results obtained by Anikeenko and Medvedev in [7].

Figure 8

Normalized distributions of ${\eta }_{ij}$ (lines are mere guides for the eye) for aggregates produced by the RMIN-MAX-1 algorithm with (a) $\gamma =0.593$, (b) $\gamma =0.538$, and (c) $\gamma =0.378$ and for (d) the RRPA with $\gamma =0.456$.

Figure 9

The $\langle {\eta }_{ij}\rangle$ for various values of $i$ in the case of the (a) MAX-1 and (b) RMIN-MAX-1 algorithms.

Figure 10

Radial value of the average CCN around $i$ coordinated spheres in RMIN-MAX-1 aggregates with (a) $\gamma =0.593$ and (b) $\gamma =0.378$.

Figure 11

(a) Bond angle distributions for high- and low-density aggregates (values for $\alpha =\pi /3$ fall out of the range of the figure). (b) Dependence of the 1.9106-rad $\delta$ peak in the bond angle distribution with the irregularity index of building tetrahedra ${\overline{\kappa }}_{\text{BT}}$.

Figure 12

Partial bond angle distributions for various MAX-1 aggregates with (a) $\gamma =0.586$, (b) $\gamma =0.500$, and (c) $\gamma =0.370$, and (d) one RRPA with $\gamma =0.456$. Values for $\alpha =\pi /3$ fall out of the range of the figure.

Figure 13

Global and partial pair distribution functions $\left[P\right(r)$ and $P_{ii}\left(r\right)$ for $i=4$, 6, and 8] for a random regular polytetrahedral aggregate.

Figure 14

Samples $P\left(r\right)$ and $P_{ii}\left(r\right)$ obtained for aggregates generated by the algorithms (a) MAX-1 ($\gamma =0.586$) and (b) MAX-1 ($\gamma =0.370$).

Figure 15

The $P_i\left(r\right)$ for two packing fractions generated by the algorithm MAX-1.

Figure 16

(a) Local packing fraction and (b) local packing fraction divided by the average packing fraction for various MAX-1 aggregates. The radial evolution of the packing fraction around $i$ coordinated spheres is shown for MAX-1 aggregates with (c) $\gamma =0.586$ and (d) $\gamma =0.370$. Also shown is the radial evolution of the packing fraction around $i$ coordinated spheres without contact neighbors [the relation (14)] for MAX-1 aggregates with (e) $\gamma =0.586$ [the bottom lines represent the difference between the highest and lowest bound, i.e., ${\gamma }_{\text{WCN},3}\left(R_{\text{SC}}\right)-{\gamma }_{\text{WCN},10}\left(R_{\text{SC}}\right)$, and intermediate, i.e., ${\gamma }_{\text{WCN},6}\left(R_{\text{SC}}\right)-{\gamma }_{\text{WCN},10}\left(R_{\text{SC}}\right)]$ and (f) $\gamma =0.370$.

Figure 17

(a) Structure factor of two RRPAs and experimental results obtained on amorphous Ge (a-Ge) (data obtained in [23]) at small $Q$ and (b) large-$Q$ behavior of the structure factor of the RRPAs and one (high-packing-fraction) RA.

Figure 18

Ashcroft-Langreth partial structure factor for (a) highest-packing-fraction RIPAs and (b) RRPAs.

Figure 19

(a) Amplitude of the $S_{4–8}^{CC}$ minimum ($Q\in [1.7–1.9]$). (b) Amplitude of the small-$Q$ maxima in Bhatia-Thornton $S_{4–8}^{CC}$ structure factors ($Q<1$).

Figure 20

Pair angle distributions with respect to $x$, $y$, and $z$ for the (a) most and (b) least dense aggregates produced by the MAX-1 algorithm. (c) Polar angle distribution relative to the favorable direction in the uniaxial anisotropy case for different values of the nematic order parameter $\lambda$ defined as the largest eigenvalue of the tensor $\overline{Q}$ [Eq. (A1)].

Figure 21

Samples $P\left(r\right)$ and $P_{ii}\left(r\right)$ obtained for aggregates generated by the algorithms (a) MAX-1 ($\gamma =0.483$), (b) RAN-1 ($\gamma =0.482$), (c) MAX-9 (${\overline{\kappa }}_D=1.771$), (d) RAN-1 (${\overline{\kappa }}_D=1.771$), (e) RAN-4, and (f) RMIN-MAX-1.

Figure 22

Sample of $P_{ij}\left(r\right)$ for various packing fractions generated by the algorithm MAX-1: (a) $P_{4,5}\left(r\right)$, (b) $P_{4,8}\left(r\right)$, (c) $P_{6,8}\left(r\right)$, and (d) $P_{8,9}\left(r\right)$.

Figure 23

Sample Ashcroft-Langreth structure factors for MAX-1 aggregates: (a) $S_{44}$, (b) $S_{66}$, and (c) $S_{88}$.

Figure 24

Bhatia-Thornton partial structure factors for (a) 4-5, (b) 7-8, and (c) 4-8 for MAX-1 aggregates and (d) the RRPA. The $S_{CC}$ oscillates around 1 but has been shifted to 1.5 for clarity.