Microscopic Structure and Elasticity of Weakly Aggregated Colloidal Gels

We directly probe the microscopic structure, connectivity, and elasticity of colloidal gels using confocal microscopy. We show that the gel is a random network of one-dimensional chains of particles. By measuring thermal fluctuations, we determine the effective spring constant between pairs of particles as a function of separation; this is in agreement with the theory for fractal chains. Long-range attractions between particles lead to freely rotating bonds, and the gel is stabilized by multiple connections among the chains. By contrast, short-range attractions lead to bonds that resist bending, with dramatically suppressed formation of loops of particles.

Figure 1

Three-dimensional image of a colloidal gel reconstructed from confocal microscopy. The dark gray (blue) particles form the shortest chain connecting the two particles $\alpha$ and $\beta$ (enlarged, in stripes). The second-shortest path is shown in medium gray (red). Inset: Raw image of a slice through the same gel. The scale bar is $20\mu \mathrm{m}$.

Figure 2

(a) Pair distribution function $g$ vs separation $r$. The dotted line corresponds to a fractal dimension, $d_f=2.1$. (b) Probability of finding a second path between two particles, $f_2$, as a function of their shortest-path length, $L$.

Figure 3

(a) Plot of $U(r)$ measured for pairs of particles with $N=18$ in a sample with long-ranged interaction ($R_p=35\mathrm{nm}$); the dotted curve is a fit with elastic constant $\kappa =146\pm 6k_{B}T/\mu {\mathrm{m}}^2$. Inset: Illustration of a chain with $L=9$ and $N=13$. The numbers show the lengths $L^{\alpha \gamma }/L^{\beta \gamma }$ of the shortest path between each particle ($\gamma$) and the particles $\alpha$ and $\beta$. (b) Scaling of spring constant, $\kappa$, with the number of particles in a chain, $N$, for a sample with medium-range depletion attraction ($R_p=25\mathrm{nm}$, $\varphi =0.04$). The straight line represents $\kappa \propto 1/N$. Inset: Log-log plot of $\kappa N$ vs $r_{\perp }^2$. The dashed line shows a slope of $-1$, which is expected to apply for rigid chains.

Figure 4

Log-scale plot of the mean $\kappa$ vs Euclidean separation, $r$ (squares), for a sample with short-range depletion attraction ($R_{\mathrm{p}}=6\mathrm{nm}$). Also plotted are $1585/⟨N⟩$ and $3162/⟨L{r}_{\perp }^2⟩$; the numerical coefficients were chosen to overlay the data with $\kappa (r)$.