Elasticity of dilatant particle suspensions during flow

Dense suspensions under sufficiently high shear stress can exhibit a dramatic transition to a solidlike state. This is known as extreme shear thickening and is sometimes accompanied by dilatancy. This behavior is contradictory; the material is solidlike but only when flowing. To probe the elasticity of the dilatant state, we measure the transient inertio-elastic oscillations that occur in response to step changes in applied stress. As the volume fraction of solids increases the apparent flow elasticity also increases, and the suspension flows more slowly, approaching an elastic solid, which will support stress statically.

Figure 1

(Color online) Both $N_1^{\left(ap\right)}$ and $\eta$ show dramatic increases when $\sigma >{\sigma }_c$. For the $D1$ samples shown here, ${\sigma }_c\approx 200–300\text{Pa}$. Both spheres and dumbbells showed qualitatively similar flow curves.

Figure 2

At the end of the steady stress tests on sample of $D1$, $\varphi =0.611$, the rheometer reverses direction due to elasticity in the sample.

Figure 3

(Color online) Step stress pattern on sample $S1$, $\varphi =0.583$. Values of $N_1^{\left(ap\right)}$ are proportional to $\sigma$, even as $\overline{\dot{\gamma }}$ and $\Delta \dot{\gamma }$ remain constant, with the exception of occasional high shear rates at 200 Pa.

Figure 4

(Color online) Shear rate transients from sample $D1$ during alternating stress steps between 400 and 600 Pa. The heavy black lines show solutions of the Maxwell model, using representative values of $G$, ${\eta }_2$, and ${\gamma }_{10}$ as described in the text. Samples are the same as those shown in Fig. 1. Ascending from the low shear rates, they are $\varphi =0.611$, $G=9300$ (gray or darkest thin lines); $\varphi =0.606$, $G=5000$ (blue or lightest thin lines); $\varphi =0.606$, $G=1500$ (green or second lightest thin lines); and $\varphi =0.581$, $G=400\text{Pa}$ (red or third lightest thin lines).

Figure 5

(Color online) As the volume fraction, $\varphi$, increases the flow elasticity, $G$, increases for $S1$ (red triangles), $S2$ (blue circles), $D1$ (black squares), and $D2$ (green diamonds). Each symbol is calculated from a single stress step at either 400 or 600 Pa.

Figure 6

(Color online) As the volume fraction, $\varphi$, increases the mean shear rate, $\overline{\dot{\gamma }}$, and shear rate fluctuation amplitude, $\Delta \dot{\gamma }$, decrease. Symbols correspond to $\overline{\dot{\gamma }}$, and the bars show one standard deviation, $\Delta \dot{\gamma }$, calculated from the combined shear rate data from all stress steps, with separate symbols used for 400 and 600 Pa. Symbols represent $S1$ (red triangles), $S2$ (blue circles), $D1$ (black squares), and $D2$ (green diamonds).

Figure 7

(Color online) As the mean shear rate, $\overline{\dot{\gamma }}$, decreases, the flow elasticity, $G$, increases for $S1$ (red triangles), $S2$ (blue circles), $D1$ (black squares), and $D2$ (green diamonds). Each symbol is calculated from a single stress step at either 400 or 600 Pa.

Figure 8

(Color online) As the mean shear rate, $\overline{\dot{\gamma }}$, decreases, the shear rate fluctuation amplitude, $\Delta \dot{\gamma }$, also decreases for $S1$ (red triangles), $S2$ (blue circles), $D1$ (black squares), and $D2$ (green diamonds). Each symbol is calculated from a single stress step at either 400 or 600 Pa.

Figure 9

(Color online) The standard deviation of ${\sigma }_s$ is an increasing function of the mean shear rate, $\overline{\dot{\gamma }}$. Each symbol corresponds to statistics calculated from all stress steps at a given stress, ($\sigma =400\text{Pa}$ or $\sigma =600\text{Pa}$) performed on a single sample. Symbols represent $S1$ (red triangles), $S2$ (blue circles), $D1$ (black squares), and $D2$ (green diamonds).

Figure 10

(Color online) Statistics of flow parameters measured during step stress tests at $\sigma =400\text{Pa}$ (black circles, dark error bars) and $\sigma =600\text{Pa}$ (purple triangles, light error bars), performed at various gap heights, $d$, on the $S1$ sample at $\varphi =0.583$. In (a) and (b), the symbols represent average values and the bars represent standard deviations from the four steps applied at each stress level. In (c), flow statistics were obtained from the combined data from all stress steps at a give stress. The symbols represent $\overline{\dot{\gamma }}$ and the bars indicate $\Delta \dot{\gamma }$.

Figure 11

(Color online) Statistics of flow parameters measured during step stress tests at $\sigma =400\text{Pa}$ (black circles, dark error bars) and $\sigma =600\text{Pa}$ (purple triangles, light error bars), on the $D1$ sample at $\varphi =0.606$. Experiments were performed with standard rheometer inertia ($S$, at left), then inertia enhanced by a factor of 5.5 (E), then returned to the original rheometer inertia ($S$, at right). In (a) and (b), the symbols and bars correspond to average values and standard deviations, respectively, calculated from all stress steps at a given stress. In (c) and (d), flow statistics were obtained from the combined data from all stress steps at a given stress. In (d) the symbols represent $\overline{\dot{\gamma }}$ and the bars indicate $\Delta \dot{\gamma }$.