Experimental study of random-close-packed colloidal particles

A collection of spherical particles can be packed tightly together into an amorphous packing known as “random close packing” (RCP). This structure is of interest as a model for the arrangement of molecules in simple liquids and glasses, as well as the arrangement of particles in sand piles. We use confocal microscopy to study the arrangement of colloidal particles in an experimentally realized RCP state. We image a large volume containing more than $450000$ particles with a resolution of each particle position to better than 0.02 particle diameters. While the arrangement of the particles satisfies multiple criteria for being random, we also observe a small fraction (less than 3%) of tiny crystallites (4 particles or fewer). These regions pack slightly better and are thus associated with locally higher densities. The structure factor of our sample at long length scales is nonzero, $S\left(0\right)=0.049\pm 0.008$, suggesting that there are long wavelength density fluctuations in our sample. These may be due to polydispersity or tiny crystallites. Our results suggest that experimentally realizable RCP systems may be different from simulated RCP systems, in particular, with the presence of these long wavelength density fluctuations.

Figure 1

Confocal micrograph of the colloidal sediment in $\left(x,y\right)$ plane. The image was taken $30\mu \text{m}$ inside the sample. The scale bar represents $10\mu \text{m}$. The arrow indicates the direction of gravity during sedimentation.

Figure 2

(Color online) (a) The image plot of the fraction of successfully superimposed particles $f$ in a plane of $\left(\Delta x,\Delta y\right)$. The dark central region corresponds to $f=1$, meaning that all possible particle overlaps are successful. (b) The circles, triangles, squares and crosses correspond to particle positions obtained from 4 separate 3D images. (c) The local number of particles observed $N$ normalized by the average, as a function of each axis after connecting the images. We add an offset to the data of $y$ and $z$ so they can be seen clearly.

Figure 3

(Color online) (a) The probability of the number of ordered neighbors $N_o$. When a particle has $N_o^i\ge 8$, it is classed as crystalline. The circles correspond to local volume fraction calculated from Voronoi cell volume, averaged over all particles with $N_o$ having the given value. (b) The spatial distribution of crystalline particles in 3D image. This image is $115\times 115\times 23.5\mu {\text{m}}^3$. The other particles are not drawn, to better show the crystalline particles.

Figure 4

(Color online) (a) Distribution of the Voronoi cell volumes plotted as a function of $\left(V-V_{\text{min}}\right)/\left(⟨V⟩-V_{\text{min}}\right)$. The solid line is a fitting line with Eq. (7) using $k=13.1$. (b) Distribution of the position differences between the particle position and the center of mass of the Voronoi cell. Almost all particles are located at the center of mass of the Voronoi cell. The values on the horizontal axis only go from $-0.04$ to $+0.04\mu \text{m}$, much less than the particle diameter $d=2.53\mu \text{m}$.

Figure 5

(Color online) (a) The structure factor $S\left(q\right)$ as a function of wavenumber $q$. The inset is the quarter image of the structure factor in a plane of $\left(q_x,q_y\right)$. (b) The expanded view of $S\left(q\right)$ (circles) and $S_c\left(q\right)$ (squares) near $q=0$. $S\left(q\right)$ increases near $q=0$ because of computational artifacts due to the finite size of our data set; the data are reliable for $qd/2\pi >0.2$, indicated by the vertical dashed line. The solid line is a fitting line with a linear function over the data with $0.2<qd/2\pi <0.5$. We obtain $S\left(0\right)=0.049$ by interpolating the fitting line to $q=0$. On the other hand, the structure factor for the crystalline particles $S_c\left(q\right)$ decreases with larger $q$ and it is well fitted by Ornstein-Zernike function (dashed line) over $0.2<qd/2\pi <0.6$.

Figure 6

The pair correlation function $g\left(r\right)$ of the sediment. The inset is enlarged view at $2<r/d<6$. The solid line is a fitting line with Eq. (8) over $r/d>2$.