Three-Dimensional Imaging of Colloidal Glasses under Steady Shear

Using fast confocal microscopy we image the three-dimensional dynamics of particles in a yielded hard-sphere colloidal glass under steady shear. The structural relaxation, observed in regions with uniform shear, is nearly isotropic but is distinctly different from that of quiescent metastable colloidal fluids. The inverse relaxation time ${\tau }_{\alpha }^{-1}$ and diffusion constant $D$, as functions of the local shear rate $\dot{\gamma }$, show marked shear thinning with ${\tau }_{\alpha }^{-1}\propto D\propto {\dot{\gamma }}^{0.8}$ over more than two decades in $\dot{\gamma }$. In contrast, the global rheology of the system displays Herschel-Bulkley behavior. We discuss the possible role of large scale shear localization and other mechanisms in generating this difference.

Figure 1

Colloid trajectories for $\dot{\gamma }=9.3\times {10}^{-4}{\mathrm{s}}^{-1}$. (a) $1.5\mu \mathrm{m}$ thick slice in the $x$, $z$ plane for 160 s. The start of each trajectory is shown by ○, the end by □. The big arrow marks the shear direction. (b) As in (a) but in the desheared, $\tilde{x}$, $z$, reference frame, with ${\tilde{x}}_i=x_i-\dot{\gamma }{\int }_0^t{z}_i(t^{\prime })d{t}^{\prime }$. (c) $y$, $z$ plane over 160 s. (d) $\tilde{x}$, $y$ plane over 160 s. (e) Single trajectory in the $\tilde{x}$, $y$ plane over 800 s. Dotted circles mark rattling in several cages (not the particle size), gray dots show the locations at $t=0$, 200, 400, 600, 800 s.

Figure 2

Selected incoherent scattering functions $F_s(Q_m,t)$, with $\dot{\gamma }$ increasing from right to left. Lines for $\dot{\gamma }=0.93\times {10}^{-3}{\mathrm{s}}^{-1}$ show two curves used in the average with start times $t_0$ spaced by 180 s. The dashed line schematizes initial relaxation. Inset: data collapse using $f_s(Q_m,t/{\tau }_{\alpha })$. Line: $f_s\propto \exp (-t/{\tau }_{\alpha })$.

Figure 3

(a) Relaxation time (●) and the time ${\tau }_2$ characterizing the crossover from caged to diffusive behavior (■) vs $\dot{\gamma }$; dashed line: ${\tau }_{\alpha }\propto {\dot{\gamma }}^{-0.8}$; full line: ${\tau }_2\propto {\dot{\gamma }}^{-0.65}$. Inset: $y$-mean-squared displacement for shear rates as in Fig. 2. Line: $⟨d{y}^2(t)⟩=2D_{y}t$ for $\dot{\gamma }=1.48\times {10}^{-3}{\mathrm{s}}^{-1}$. (b) (●) Diffusion constant $D_y$ vs $\dot{\gamma }$. Line: $D_y\propto {\dot{\gamma }}^{0.8}$. (c) The scaled diffusion constant $2D_j{\tau }_{\alpha }\simeq ⟨d{r}_j^2({\tau }_{\alpha })⟩$ vs $\dot{\gamma }$ for $j=x$, $y$, $z$. Line: the value $⟨d{y}^2({\tau }_{\alpha })⟩=2/Q_m^2$ expected for Gaussian behavior.

Figure 4

(a) Non-Gaussian parameter ${\alpha }_2(t)$ of the probability distribution $P[dy(t)]$, for several shear rates. (b) $P[dy(t={\tau }_2(\dot{\gamma }))]$ for the corresponding $\dot{\gamma }$, showing a near collapse of the data (each involving $>{10}^5$ displacements). Line: best Gaussian fit.

Figure 5

(□) Local “flow curve” $\overline{\sigma }=G_0\dot{\gamma }{\tau }_{\alpha }$, with $G_0=8.5\mathrm{Pa}$, vs $\dot{\gamma }$ (dashed line: $\overline{\sigma }\propto {\dot{\gamma }}^{0.2}$) compared with the macroscopic flow curve $\sigma ({\dot{\gamma }}_a)$ measured in cone-plate geometry (●). Full line: fit to the Herschel-Bulkley model $\sigma =1.36\mathrm{Pa}+A{\dot{\gamma }}_a^{0.56}$.