Formation and liquid permeability of dense colloidal cube packings

The liquid permeability of dense random packings of cubic colloids with rounded corners is studied for solid hematite cubes and hollow microporous silica cubes. The permeabilities of these two types of packings are similar, confirming that the micropores in the silica shell of the hollow cubes do not contribute to the permeability. From the Brinkman screening length $\sqrt{k}$ of $\sim 16$ nm, we infer that the relevant pores are indeed intercube pores. Furthermore, we relate the permeability to the volume fraction and specific solid volume of the cubes using the Kozeny-Carman relation. The Kozeny-Carman relation contains a constant that accounts for the topology and size distribution of the pores in the medium. The constant obtained from our study with aspherical particles is of the same order of magnitude as those from studies with spherical and ellipsoidal particles, which supports the notion that the Kozeny-Carman relation is applicable for any dense particle packing with (statistically) isotropic microstructures, irrespective of the particle shape.

Figure 1

Representative TEM images of the cubic colloids used for the packing formation and permeation experiments: (a) solid hematite cubes without silica coating (1477 nm) and (b) hollow microporous silica cubes (940 nm). The scale bars represent $2 \mu \mathrm{m}$.

Figure 2

The colloidal cubes in this study have a superball shape of which the precise shape depends on the deformation parameter $m$. (a) The shape changes from a sphere $\left(m=2\right)$ via cubes with rounded corners to a perfect cube $\left(m\to \infty \right)$. (b) Cross-sectional views of superballs at given $m$ values indicating the particle radius $r$; $d=2r$ with $d$ the edge length of a cube.

Figure 3

Schematic image of the setup used for packing formation and permeation experiments. A nitrogen gas source (I) is connected to a glass solvent reservoir (II). The solvent enters the glass dispersion cell (III) via valve vI. The cubic colloids inside the dispersion cell are forced down by the excess pressure exerted by the solvent and packed onto the Millipore filter at the bottom of the dispersion cell. The solvent passes through the packing and Millipore filter via valve vII and is collected in a beaker (IV). The permeated solvent mass is accurately recorded by a balance (V).

Figure 4

Typical plots of the permeated solvent mass $W$ as a function of time, measured during packing formation and permeation experiments. (a) During packing formation, the permeated mass deviates from the $\sqrt{t}$-scaling relation of Eq. (6) (dashed line) due to settling of the cubes. (b) During permeation experiments, the permeated mass grows linearly in time with a mass flow velocity of $U$, which is equal to the slope of the linear fit shown with the dashed line. From the constant slope it can be inferred that the microstructure of the packing does not change in time.

Figure 5

Representative images at increasing magnification of packings after drying. Photos of dry packings of (a) solid hematite cubes and (b) hollow silica cubes. The white disk underneath each packing is the Millipore filter substrate with a diameter of 2.5 cm. (c) The packings were broken into pieces to analyze them with SEM. The scale bar represents $500 \mu \mathrm{m}$. (d) The solid hematite cubes are disordered within a layer. Scale bar is $10 \mu \mathrm{m}$. (e) High-magnification SEM image of the upper layers of a dry packing. The tilted square (left) indicates a shifted squarelike arrangement of cubes. The square (right) indicates a squarelike arrangement of cubes. The resulting pores are marked by the red circles. The scale bar is $5 \mu \mathrm{m}$. (f) Magnifications of the arrangements and resulting pores indicated in (e). Pore diameters are $\sim 180$ nm (left) and $\sim 300$ nm (right). Scale bars represent $1 \mu \mathrm{m}$.

Figure 6

(a) Typical SEM image of the visible top layer of the bottom of a packing formed by hollow silica cubes. The cracks were created during drying and were not present during the permeation measurements. The scale bar represents $20 \mu \mathrm{m}$. (b) ${\Psi }_4$ analysis of the packing in (a). The colors of the dots indicate the extent of cubic ordering, where yellow to green signifies high cubic ordering and red to purple low cubic ordering. (c) ${\Psi }_6$ analysis of the packing in (a). The colors of the dots indicate the extent of hexagonal ordering, where red to purple signifies high hexagonal ordering and yellow to green low hexagonal ordering. Clearly there is significant short-range cubic as well as hexagonal order within a layer.

Figure 7

Mass flow velocities $U$ as a function of the applied pressure $\Delta P$ for packings of (a) solid hematite cubes and (b) hollow silica cubes. The mass flow velocity increases with increasing $\Delta P$ and decreasing amount of cubes. The drawn lines are linear fits through the data points and their slope is $U_{\mathrm{m}} (U_{\mathrm{m}}=U/\Delta P)$ as listed in Table 2.

Figure 8

The specific volume squared ${\nu }^2$ is equal to ${\left(\frac{1}{3}r\right)}^2$ in the case of a sphere $\left(m=2\right)$ and a perfect cube $\left(m\to \infty \right)$, as indicated by the drawn solid line. For $m$ values between $m=2$ and 5, ${\nu }^2$ increases. In these calculations, $r=500$ nm, corresponding to an edge length of 1000 nm. The dashed lines mark the range of $m$ values of the cubes used in the experiments.

Figure 9

The $C^{+}$ values calculated by the Kozeny-Carman relation (2) for the solid and hollow cubes. The values are scattered around $C^{+}=8.1$ and 9.1 for solid and hollow cubes, respectively.

Figure 10

Schematic illustration of the dispersion cell during packing formation. Initially, the cell is filled to height $H$ with dispersion with particle volume fraction ${\varphi }_{\mathrm{d}}$. During packing formation, the liquid level decreases by $h\left(t\right)$ and a packing of thickness $L\left(t\right)$ and particle volume fraction ${\varphi }_{\mathrm{p}}$ is formed.

Figure 11

Numerically calculated volume $V$ and surface area $S$ of a superball with $r=1$ and increasing deformation parameter $m$. The analytically known cases are for $m=2$ (sphere) and $m\to \infty$ (perfect cube).