Size-dependent diffusion in an aging colloidal glass

Colloidal dispersions of Laponite platelets are known to age slowly from viscous sols to colloidal glasses. We follow this aging process by monitoring the diffusion of probe particles embedded in the sample via dynamic light scattering. Our results show that the time-dependent diffusion of the probe particles scales with their size at early times and for the larger particles. This implies that within this regime the measurements can be used to investigate the generalized fluctuation-dissipation theorem for this out-of-equilibrium system, where the bath temperature is replaced with an effective temperature. Simultaneous dynamic rheological measurements reveal that this effective temperature increases as a function of aging time and frequency which suggests the existence of two measured time-scale regimes. In accord with recent work by Abou et al. [Phys. Rev. Lett. 93, 160603 (2004)], our results suggest that at probed time scales longer than the characteristic relaxation time of the Laponite dispersion the system thermalizes with the bath, whereas at shorter time scales the system is out-of-equilibrium with an effective temperature greater than the bath temperature.

Figure 1

(Color online) Aging behavior of $G^{\prime }$ (thin lines) and $G^{″}$ (thick lines) of a $3.0\mathrm{wt.}\%$ Laponite suspension. Data are determined at set aging times by making measurements of the viscoelastic modulus at 25 frequencies between $0.1\mathrm{rad}∕\mathrm{s}$ and $100\mathrm{rad}∕\mathrm{s}$ every $10\min$. These data are then interpolated to specific aging times and frequencies.

Figure 2

(Color online) The two sets of correlation functions, $g_2(\mathbf{q},t)-1$, are measured at two different aging times for four different concentrations of $50\mathrm{nm}$ polystyrene spheres in a $3\mathrm{wt.}\%$ dispersion of Laponite.

Figure 3

(Color online) Intensity as a function of scattering vector for $7.4\times {10}^{-3}\mathrm{wt.}\%$ of $200\mathrm{nm}$ polystyrene spheres in water (squares) and in a $3\mathrm{wt.}\%$ dispersion of Laponite after aging for $193\min$ (triangles). The solid curve is the scattering intensity expected for $200\mathrm{nm}$ spheres and the dashed curve is for $400\mathrm{nm}$ spheres [18]. The curves are normalized to 1 at the lowest value of $q$.

Figure 4

(Color online) Correlation functions taken for a $3\mathrm{wt.}\%$ dispersion of Laponite with $7.4\times {10}^{-3}\mathrm{wt.}\%$ of $100\mathrm{nm}$ polystyrene spheres. The curve at left is the initial correlation function measured just after filtering while the 9 curves on the right are all started after the system has aged for $171\min$. These latter 9 correlation functions are determined by averaging over accumulation times ranging from $13\text{to}300\mathrm{s}$.

Figure 5

(Color online) Time dependent diffusion, $D\left(t\right)$, normalized with the radius, $a$, of the probe particles. The measurements were performed in portions taken from the the same $3\mathrm{wt.}\%$ Laponite dispersion as used in the rheological measurements of Fig. 1. The aging times are noted next to the curves with the wt. % noted in the key. A comparison to probe particles immersed at the same concentrations in pH 10 water is shown at the top of the plot. These correspond to $3.7\times {10}^{-2}$ and $7.4\times {10}^{-2}\mathrm{wt.}\%$ for the $50\mathrm{nm}$ spheres and $7.4\times {10}^{-3}\mathrm{wt.}\%$ for the 100 and $200\mathrm{nm}$ spheres. The gray line is the value expected for spheres immersed in water at $298\mathrm{K}$. The curves for the $50\mathrm{nm}$ particles at $t_w=146\min$ are omitted for clarity of the plot.

Figure 6

(Color online) $aD\left(t\right)$ at $t=1\mathrm{ms}$ for the Laponite dispersion in Fig. 5.

Figure 7

(Color online) $T_{\mathit{eff}}$ determined from the rheological measurements in Fig. 1 and the time-dependent diffusion of $200\mathrm{nm}$ diameter probe particles shown in Fig. 5.