Creep in Colloidal Glasses

We investigate the nonlinear response to shear stress of a colloidal hard-sphere glass, identifying several regimes depending on time, sample age, and the magnitude of applied stress. This emphasizes a connection between stress-imposed deformation of soft and hard matter, in particular, colloidal and metallic systems. A generalized Maxwell model rationalizes logarithmic creep for long times and low stresses. We identify diverging time scales approaching a critical yield stress. At intermediate times, strong aging effects are seen, which we link to a stress overshoot seen in stress-strain curves.

Figure 1

(a) Deformation $\gamma (t)$ after application of a fixed stress $\widehat{\sigma }=\sigma /(k_{\mathrm{B}}T/R^3)=0.9$, 0.5, 0.4693, 0.4224, 0.3943, 0.3755, 0.2628, and 0.1 (top to bottom: all batch A except $\widehat{\sigma }=0.9$, 0.5, 0.1); waiting time $t_w=600\mathrm{s}$. Dotted (dashed) lines: short-time (glass-modulus) elastic response. Labels (i) to (v) mark regions discussed in the text. (b) Shear rate $\dot{\gamma }(t)$ obtained from differentiating the measured $\gamma (t)$ (smoothened for $\sigma =0.1$). Dotted lines: $\delta \dot{\gamma }(0)$, see text. Dashed line: $\dot{\gamma }\propto t^{-1}$.

Figure 2

Steady-state flow curve $\dot{\gamma }(\sigma )$ (right axis) from constant-rate experiments in increasing and decreasing sequence (crosses; $t_w=10\mathrm{s}$), and from the stress-controlled mode of the rheometer (small diamonds). Values read off from Fig. 1 are shown as large diamonds and circles (batches $A$ and $B$). Solid line: Herschel-Bulkley law, $\sigma -{\sigma }_c\propto {\dot{\gamma }}^n$, with $1/n\approx 1.85$, ${\sigma }_c\approx 0.38$. Left axis: time scale ${\tau }_{\mathrm{cr}}$ for logarithmic creep from the data (symbols) and a generalized Maxwell model (line), see text.

Figure 3

(a) Stress-strain curves: Startup stress $\sigma (t)$ at constant shear rate ${\dot{\gamma }}_{\mathrm{so}}$ (symbols with lines, batch $B$), as a function of strain $\gamma ={\dot{\gamma }}_{\mathrm{so}}t$ after switch-on, waiting times as in Fig. 4. (b) Ratio $r={\sigma }_{\max }/\sigma (\dot{\gamma })-1$ (open symbols) and $r^{\prime }=1/(G_{\infty }{\dot{\gamma }}_{\min }{\tau }_f)$ from the creep experiment (filled) as a function of $t_w$; see text. Dashed line: $0.06\ln t_w/{\tau }_0$. (c) Shear modulus $G(t)$ for ${\dot{\gamma }}_{\mathrm{so}}{\tau }_0=2.85\times {10}^{-3}$ and $t_w=6000\mathrm{s}$ (solid line), and from $\gamma (t)$ (dash-dotted in Fig. 4) and the convolution approximation.

Figure 4

(a) Deformation $\gamma (t)$ for $\sigma /(k_{\mathrm{B}}T/R^3)=0.9$, 0.5, and 0.1 (top to bottom, batch $B$), for waiting times $t_w=60$, 600, 3600, and 6000 s (left to right). Dash-dotted: fit used to evaluate $G(t)$, see text. (b) Associated shear rate $\dot{\gamma }(t)$. Dash-dotted line: Andrade creep law, $\dot{\gamma }(t)\sim t^{-2/3}$; dashed: logarithmic creep, $\dot{\gamma }(t)\sim 1/t$.