Microscopic ergodicity breaking governs the emergence and evolution of elasticity in glass-forming nanoclay suspensions

We report a study combining x-ray photon correlation spectroscopy (XPCS) with in situ rheology to investigate the microscopic dynamics and mechanical properties of aqueous suspensions of the synthetic hectorite clay Laponite, which is composed of charged, nanometer-scale, disk-shaped particles. The suspensions, with particle concentrations ranging from 3.25 to 3.75 wt %, evolve over time from a fluid to a soft glass that displays aging behavior. The XPCS measurements characterize the localization of the particles during the formation and aging of the soft-glass state. The fraction of localized particles, $f_0$, increases rapidly during the early formation stage and grows more slowly during subsequent aging, while the characteristic localization length $r_{\mathrm{loc}}$ steadily decreases. Despite the strongly varying rates of aging at different concentrations, both $f_0$ and $r_{\mathrm{loc}}$ scale with the elastic shear modulus $G^{\prime }$ in a manner independent of concentration. During the later aging stage, the scaling between $r_{\mathrm{loc}}$ and $G^{\prime }$ agrees quantitatively with a prediction of naive mode coupling theory. Breakdown of agreement with the theory during the early formation stage indicates the prevalence of dynamic heterogeneity, suggesting the soft solid forms through precursors of dynamically localized clusters.

Figure 1

Storage modulus $G^{\prime }$ as a function of age of Laponite suspensions with concentrations $C_w=$ 3.25 wt % (red squares), 3.5 wt % (black triangles), and 3.75 wt % (blue circles). Measurements were performed at strain amplitude 0.1% and frequency 0.1 Hz. The suspension with $C_w=$ 3.5 wt % was also subjected intermittently to larger-amplitude strains between ages of 18 940 and 29 860 s to test for rejuvenation, as described in the text. The data in this range are shown in grey and bracketed by the arrows.

Figure 2

Normalized XPCS correlation functions measured using the Rigaku (solid symbols) and LAMBDA (open symbols) detectors at wave vector $q=0.077{\mathrm{nm}}^{-1}$ at different ages of the suspension with $C_w=$ 3.25 wt %. The solid lines depict the results of fits using Eq. (3).

Figure 3

Logarithm of the intermediate plateau value $A$ of the dynamic structure as a function of the wave vector squared divided by the measurable structure factor of suspensions with (a) $C_w=$ 3.25 wt % and (b) $C_w=$ 3.5 wt % at various ages indicated in the legends. The lines depict the results of fits using a Debye-Waller relation [Eq. (4)].

Figure 4

(a) Localization length $r_{\mathrm{loc}}$ and (b) localized fraction $f_0$ as a function of age for suspensions with concentrations $C_w=$ 3.25 wt % (squares), 3.5 wt % (triangles), and 3.75 wt % (circles). The solid lines in (b) show the results of fits using piecewise logarithmic functions.

Figure 5

(a) $G^{\prime }{v}_0/k_{B}T$ (open symbols) and $6{\pi }^2\varphi /5r_{\mathrm{loc}}^2{q}_p^2$ (solid symbols), the two sides of Eq. (6), as functions of age, and (b) the ratio of the two sides of Eq. (6) as a function of age for suspensions with concentrations $C_w=$ 3.25 wt % (squares), 3.5 wt % (triangles), and 3.75 wt % (circles). The arrows indicate the ages of the crossing points between faster and slower logarithmic growth of $f_0$ depicted by the solid lines in Fig. 4.

Figure 6

Inverse of the localization length squared, $1/r_{\mathrm{loc}}^2$, versus the storage modulus of the three suspensions plotted on (a) linear and (b) logarithmic scales. The line in (a), which depicts the result of a linear fit to all points, has an intercept of $0.002{\mathrm{nm}}^{-2}$ and a slope of $1.0\times {10}^{-5}{\mathrm{nm}}^{-2}{\mathrm{Pa}}^{-1}$. The red, black, and blue lines in (b) show the relation between $r_{\mathrm{loc}}^{-2}$ and $G^{\prime }$ obtained from NMCT [Eq. (6)] for the suspensions with $C_w=$ 3.25, 3.5, and 3.75 wt %, respectively.

Figure 7

$f_0$, the nonergodicity parameter in the limit $q\to 0$, as a function of shear modulus. The solid line is a logarithmic fit to all points, $f_0=cln{G}^{\prime }+b$, with $c$ = 0.089 and $b$ = 0.20 with $G^{\prime }$ measured in units of Pa.

Figure 8

(a) Scattering intensity $I\left(q\right)$ of the Laponite suspensions at the three concentrations at late age and of the empty Couette cell obtained with the Rigaku detector. The intensity is averaged over wave-vector direction and summed over $1\times {10}^5$ frames. (b) Measurable structure factor $S_M\left(q\right)$ of the suspension with $C_w=$ 3.25 wt % at different ages. The arrow at $q=0.123{\mathrm{nm}}^{-1}$ indicates the position of the interparticle structure factor peak. The inset shows the theoretical form factor $\langle F\left(q\right)\rangle$ of a disk with 30 nm diameter and 1 nm thickness, averaged uniformly over orientations.

Figure 9

(a) Stretching exponent $\beta$ characterizing the shape of the terminal relaxation in $f(q,\tau )$ at $q=0.07{\mathrm{nm}}^{-1}$ as a function of age obtained from fits using Eq. (3) for the three Laponite concentrations. The solid horizontal lines show the mean values $\overline{\beta }$ at late ages used in fits to obtain other parameters in Eq. (3). Specifically, $\overline{\beta }(C_w=3.25\text{wt}\%)=0.35, \overline{\beta }(C_w=3.5\text{wt}\%)=0.41$, and $\overline{\beta }(C_w=3.75\text{wt}\%)=0.44$. (b) Relaxation time ${\tau }_2$ characterizing the terminal decay in $f(q,\tau )$ at $q=0.077{\mathrm{nm}}^{-1}$ as a function of age. The solid lines are exponential fits to ${\tau }_2$ at early age, ${\tau }_2={\tau }_0{e}^{a{t}_w}$, where ${\tau }_0=1.5$ s and $a=2.9\times {10}^{-4}{\mathrm{s}}^{-1}$ for $C_w=3.25$ wt %, and ${\tau }_0=0.79$ s and $a=3.7\times {10}^{-4}{\mathrm{s}}^{-1}$ for $C_w=3.5$ wt %. The dashed line has slope of 1.

Figure 10

The terminal relaxation time ${\tau }_2$ of the dynamic structure factor as a function of wave vector of suspensions with (a) $C_w=3.25$ wt % and (b) $C_w=3.5$ wt % at various ages indicated in the legends. The solid lines in both panels have slope of $-1$, and the dashed lines have slope of $-2$.