Particle-scale structure in frozen colloidal suspensions from small-angle x-ray scattering

During directional solidification of the solvent in a colloidal suspension, the colloidal particles segregate from the growing solid, forming high-particle-density regions with structure on a hierarchy of length scales ranging from that of the particle-scale packing to the large-scale spacing between these regions. Previous work has concentrated mostly on the medium- to large-length scale structure, as it is the most accessible and thought to be more technologically relevant. However, the packing of the colloids at the particle scale is an important component not only in theoretical descriptions of the segregation process, but also to the utility of freeze-cast materials for new applications. Here we present the results of experiments in which we investigated this structure across a wide range of length scales using a combination of small-angle x-ray scattering and direct optical imaging. As expected, during freezing the particles were concentrated into regions between ice dendrites forming a microscopic pattern of high- and low-particle-density regions. X-ray scattering indicates that the particles in the high-density regions were so closely packed as to be touching. However, the arrangement of the particles does not conform to that predicted by standard interparticle pair potentials, suggesting that the particle packing induced by freezing differs from that formed during equilibrium densification processes.

Figure 1

The image on the left shows the entire sample cell with the sample chamber and cold finger tip enlarged in the top right-hand corner. In the lower right-hand corner, the schematic diagram shows a plan-view of the cold finger tip with the approximate locations of x-ray data collection (LE $=$ left edge, LC $=$ left center, C $=$ center, RC $=$ right center, and RE $=$ right edge).

Figure 2

These images show two sets of before (time $t=0$ s) and after ($t>0$ s) snapshots from movies of stage I ice growth. The images in (a) show pure water, whereas those in (b) show a colloidal solution of silica spheres as described above, but with particle radius $142$ nm. We note that we did not observe any significant differences in the direct imaging experiments between the behavior of solutions of these larger particles and solutions of the smaller particles (as used in the x-ray scattering). Ice dendrites are visible in both sets: dark in (a) and lighter areas between dark regions of concentrated particles in (b).

Figure 3

These images show a sequence of snapshots of stage II solidification for a sample of $142$-nm particles. In areas where stage II ice has formed, the sample appears darker. The stage II ice edge is marked by a dark rim of particles being pushed ahead of the ice, which form a dark spot at the center upon complete solidification.

Figure 4

These images show a sequence of snapshots of a frozen solution of $32$-nm particles over time with the sample ages and temperatures indicated. The small dark spots near the cold finger are air bubbles. Because the water was not degassed before freezing, air gradually exolves from the ice.

Figure 5

SAXS intensity vs scattering vector taken at the center position from sample $1$ before being frozen (circles) and at $T=-2.00\hspace{0.5em}{}^{°}$C when frozen (squares). The solid curve represents the fit of the unfrozen data to a polydisperse sphere form factor and monodisperse hard-sphere structure factor with $R=32.4$ nm, $z=31$, $A=294$, $R_{\text{HS}}=53.0$ nm, and ${\varphi }_{\text{HS}}=0.073$ as described in the text. For comparison, the dotted line shows only the form factor with the same parameters as above, but an arbitrary amplitude. For clarity, the unfrozen data have been offset from the frozen data by multiplication with a constant coefficient.

Figure 6

Measured structure factors vs scattering vector taken at the left edge position from sample $4$ at $T=-1.20\hspace{0.5em}{}^{°}$C (circles) and $T=-0.60\hspace{0.5em}{}^{°}$C (squares). Solid curves represent the Gaussian fits of the main peak as described in the text. Along the top of the plot, the horizontal axis is labeled in units of $qR$.

Figure 7

Peak fit parameters vs temperature from the center position of two different samples (circles, sample $1$; squares, sample $2$). The peak position is in (a), the peak width is in (b), the peak amplitude is in (c), and the peak offset is in (d).