Incompressibility of polydisperse random-close-packed colloidal particles

We use confocal microscopy to study the compressibility of a random-close-packed sample of colloidal particles. To do this, we introduce an algorithm to estimate the size of each particle. Taking into account their sizes, we compute the compressibility of the sample as a function of wave vector $q$, and find that this compressibility vanishes linearly as $q\to 0$, showing that the packing structure is incompressible. The particle sizes must be considered to calculate the compressibility properly. These results also suggest that the experimental packing is hyperuniform.

Figure 1

(a) Scatter plot of the calculated radius $a_c^i$ from our method [Eq. (1)] as a function of the given radius $a_g^i$ using data from a simulated packing with polydispersity 7$\%$. The solid line corresponds to $a_c^i=a_g^i$. (b) The particle size uncertainty $\Delta a$ found by analyzing simulation data from packings with a given polydispersity, both without noise (circles) and with noise added to the particle positions (triangles). The dashed line corresponds $\Delta a=p$.

Figure 2

The uncertainty of the particle radius $\Delta a$ is plotted as a function of the number of contacting neighbors $Z$ used in the algorithm. The data are from the simulated packings with a polydispersity $p=0.07$ with positional noise. Circles correspond to Eq. (1) using ${\langle {\langle a_k\rangle }_k\rangle }_j=\overline{a}$ with no iteration. Squares correspond to five iterations of Eq. (1), and triangles correspond to ten iterations.

Figure 3

Probability of particle sizes in our experimental sample. The average size is 1.265 $\mu$m and polydispersity is 6.7$\%$.

Figure 4

(a) The static structure factor $S\left(q\right)$. (b) $\rho k_{B}T{\chi }_0\left(q\right)$ (no approximation) and $\rho k_{B}T{\chi }_1\left(q\right)$ (first-order approximation of Ref. [11]), from the experimental data. Square symbols correspond to $\rho k_{B}T{\chi }_0\left(q\right)$, which is proportional to $S\left(q\right)$ at small $q$. Circle symbols correspond to $\rho k_{B}T{\chi }_1\left(q\right)$. The error bars (drawn for every fifth point) are from using different windowing functions in the Fourier transform. The lines are linear fits to the data for $0.2<q\overline{a}/\pi <0.5$.