Microstructure of polydisperse colloidal gels

TL;DR

Polydisperse colloidal gels pose a challenge to understanding structure–property relations because particle size variability can both randomize neighbor connections and enable rigid local motifs. The authors image gels with a continuous size distribution by confocal microscopy, quantify the fractal dimension $D_f$, analyze the contact network, and identify tetrahedral motifs to connect microstructure to mechanical rigidity. They report $D_f = 2.5 \pm 0.1$, find that neighbor pairings are approximately random with probability ∝ $P(R)$ but larger particles have more contacts and participate in tetrahedra, and show that tetrahedra can rearrange before arrest, providing a path to rigidity. A Soddy-radius analysis indicates the geometric constraint is negligible for their distribution but would become important for distributions with more large particles, highlighting how polydispersity and rearrangements tune gel mechanics.

Abstract

We use confocal microscopy to image colloidal gels formed from highly polydisperse particles. We suspend our polydisperse particles in a density matched solvent, and let the particles spontaneously aggregate through the van der Waals force. The particle size distribution $P(R)$ is roughly log-normal, with the largest particles more than 15 times the size of the smallest particles. The pairing of nearest neighbor particles is consistent with a null hypothesis that pairings are made randomly, that is, any two particle sizes have a probability of being neighbors consistent with their proportionality in $P(R)$. That being said, as expected, larger particles have more nearest neighbors than small ones. This leads to an over-representation of large particles in tetrahedral structures where four particles are mutually nearest neighbors, showing that large particles help provide rigidity to the gel structure. The tetrahedral structures also suggest that particles are able to rearrange during the gelation process, until their motion is stabilized by the multiple contacts with their neighbors. We discuss the implications of how other size distributions $P(R)$ would affect the gel structure.

Figures (12)

• Figure 1: (a) Scanning Electron Microscope image of colloidal PMMA particles (image courtesy of the Robert P. Apkarian Integrated Electron Microscopy Core at Emory University). The scale bar is 10 $\mu$m. (b) 2-dimensional confocal slice of fluorescently labelled colloidal particles. The scale bar is 20 $\mu$m and the volume fraction is $\phi=0.26$.
• Figure 2: Measured probability distribution of the colloidal particle radii. The dashed black line corresponds to a log-normal fit (Eq. \ref{['lognorm']} with $\sigma=0.378$). The mean size of the particles is $\mu=2.69$$\mu$m with a polydispersity of $\delta=37\%$. The total range of particle sizes goes from $1.15$$\mu$m$\leq r \leq 17.6$$\mu$m.
• Figure 3: Graph of box counting method to estimate $D_f$ for one image stack of a gel at $\phi \sim 0.15$. To simplify the calculation of $D_f$, the aspect ratio of the voxel is made uniform. The slope of inverse box size ($s$) against the number of boxes with a bright voxel ($N(s)$) gives the fractal dimension of the aggregate. The dotted black line fits to the equation $N(s)=s^{-D_f}$ where $D_f=2.43$. We repeat this measurement for three other image stacks and report a mean value of $D_f=2.5 \pm 0.1$.
• Figure 4: (a) The average contact number $\langle Z \rangle$ as a function of the radius of the particle $R$. The volume fraction $\phi$ increases monotonically from bottom to top as labeled. The dashed black curve represents data gathered from RCP simulation which pack at $\phi_{RCP}=0.67$. (b) The data from (a) is replotted as $[ \langle Z \rangle \bar{R}^2] / [ \langle Z(\bar{R} \rangle R^2]$, using the mean particle size $\bar{R} = 2.69$$\mu$m. This normalizes the data by the contact number corresponding to $\bar{R}$, as well as dividing by $R^2$ to remove the influence of the surface area of each particle. For both (a) and (b), the dotted lines follow a power law with slopes $\alpha$ as indicated.
• Figure 5: Histograms of the particle size distribution given that it is neighbors with a particle of size $3.0<R<5.0$$\mu$m (green) and $10.0<R<18.0$$\mu$m (pink) for the experimental data. The blue distribution is the "contact former" distribution, i.e., the list of particle sizes that form contacts.