Contribution of Slow Clusters to the Bulk Elasticity Near the Colloidal Glass Transition

We use confocal microscopy to visualize individual particles near the colloidal glass transition. We identify the most slowly-relaxing particles and show that they form spatially correlated clusters that percolate across the sample. In supercooled fluids, the largest cluster spans the system on short time scales but breaks up on longer time scales. In contrast, in glasses, a percolating cluster exists on all accessible time scales. Using molecular dynamics simulation, we show that these clusters make the dominant contribution to the bulk elasticity of the sample.

Figure 1

Comparison of slow clusters between experiment [(a)–(f)] and simulation [(g)–(l)]. Only slow particles are shown and are shaded by cluster. Simulation times are in rescaled units. $\varphi =0.52$, $\tau$: (a) 600 s, (b) 1500 s. $\varphi =0.56$, $\tau$: (c) 1000 s, (d) 4000 s. $\varphi =0.60$, $\tau$: (e) 3000 s, (f) 40000 s. $\varphi =0.55$, $\tau$ (g) 12.5, (h) 25. $\varphi =0.585$, $\tau$: (i) 150, (j) 600. $\varphi =0.60$, $\tau$: (k) 2000, (l) 73000.

Figure 2

Time-dependent properties of slow clusters for experiment [(a)–(c)] and simulation [(d)–(f)]. (a), (d): Fraction of particles in percolating slow cluster, $f$, versus time scale $\tau$. (b), (e): Sample-averaged MSD (lines) and MSD of slow particles (symbols) versus $\tau$. (c), (f): Sample-averaged ${\alpha }_2(\tau )$ (lines) and ${\alpha }_2(\tau )$ of slow particles (symbols) versus $\tau$. Experiment: $\mathrm{\text{solid lines}}+\mathrm{\text{squares}}$: $\varphi =0.52$; $\mathrm{\text{dash}}+\mathrm{\text{circles}}$: $\varphi =0.53$; $\mathrm{\text{dot}}+\mathrm{\text{triangles}}$: $\varphi =0.56$; $\mathrm{\text{dash}}\mathrm{\text{−}}\mathrm{\text{dot}}+\mathrm{\text{diamonds}}$: $\varphi =0.60$; $\mathrm{\text{dash}}\mathrm{\text{−}}\mathrm{\text{dot}}\mathrm{\text{−}}\mathrm{\text{dot}}+\mathrm{\text{stars}}$: $\varphi =0.61$. Simulation: $\mathrm{\text{solid}}+\mathrm{\text{squares}}$: $\varphi =0.57$; $\mathrm{\text{dash}}+\mathrm{\text{circles}}$: $\varphi =0.585$; $\mathrm{\text{dot}}+\mathrm{\text{triangles}}$: $\varphi =0.59$; $\mathrm{\text{dash}}\mathrm{\text{−}}\mathrm{\text{dot}}+\mathrm{\text{diamonds}}$: $\varphi =0.595$; $\mathrm{\text{dash}}\mathrm{\text{−}}\mathrm{\text{dot}}\mathrm{\text{−}}\mathrm{\text{dot}}+\mathrm{\text{stars}}$: $\varphi =0.60$.

Figure 3

Fractal dimension, $d_f$, of slow clusters at the cluster break-up time scale ($\varphi <{\varphi }_g$) or at the longest accessible time scale ($\varphi >{\varphi }_g$) as a function of $\varphi$ for (a) experiment and (b) simulation. Error bars indicate standard error of the slope from linear regression.

Figure 4

Comparison of $G^{\prime }(\omega )$ and $G^{\prime \prime }(\omega )$ of (a)–(b) bulk samples and (c)–(d) slow clusters, calculated from ${\eta }_{mct}^{\mathrm{slow}}$ and ${\eta }_{mct}^{\mathrm{bulk}}$. Stars indicate break-up time scale of percolating cluster of slow particles. Note that this breakup coincides with the $\omega$ where $G^{\prime }(\omega )>G^{\prime \prime }(\omega )$. Solid line: $\varphi =0.57$; dashed line: $\varphi =0.585$; dotted line: $\varphi =0.59$; dashed-dotted line: $\varphi =0.595$.