Dynamic shear jamming in dense granular suspensions under extension

Unlike dry granular materials, a dense granular suspension like cornstarch in water can strongly resist extensional flows. At low extension rates, such a suspension behaves like a viscous fluid, but rapid extension results in a response where stresses far exceed the predictions of lubrication hydrodynamics and capillarity. To understand this remarkable mechanical response, we experimentally measure the normal force imparted by a large bulk of the suspension on a plate moving vertically upward at a controlled velocity. We observe that, above a velocity threshold, the peak force increases by orders of magnitude. Using fast ultrasound imaging we map out the local velocity profiles inside the suspension, which reveal the formation of a growing jammed region under rapid extension. This region interacts with the rigid boundaries of the container through strong velocity gradients, suggesting a direct connection to the recently proposed shear-jamming mechanism.

Figure 1

(a) Schematic of the experimental setup. (b) Formation of a growing jamming front (indicated by the darker region) under pulling. (c) Normal force and stress as a function of time for two pulling velocities. For $v=8$ mm/s, cases of weaker (filled squares) and stronger (filled circles) adhesion between plate and suspension are shown; $\varphi =0.52, {\eta }_s=63\mathrm{mPa}\mathrm{s}$.

Figure 2

(a) Geometrical parameters of the frustum shape formed by the suspension. (b) Schematic of the volume pulled out of the bulk and free surface depression under extensile deformation. The frustum shape is approximated by drawing a tangent in the vertical plane on the side surface of the jammed region. The instantaneous point of intersection of the tangent at the plate gives $r$ and the intersection point on the free surface of the suspension gives $R\left(t\right)$. The lower dashed horizontal red line gives the initial position of the free surface of the suspension. (c) Force response as a function of time for different pulling velocities. Arrows indicate the onset of a rapid increase in force. Solid lines represent the model predictions. Here, $\varphi =0.51$ and ${\eta }_s=50\mathrm{mPa}\mathrm{s}$. (d) Force response as a function of displacement for different initial depths ($h_0$) of the suspension. For each depth, two consecutive measurements are shown. Here, $\varphi =0.52$ and $v=8$ mm/s. (e) Critical plate velocity ($v_c$) for the onset of a sharply increasing normal force for different volume fractions $\varphi$; $h_0=27.5$ mm. Error bars correspond to the minimum velocity step size in going from a viscouslike to a sharply increasing force response (each measured three times). The solvent viscosity ${\eta }_s=20\mathrm{mPa}\mathrm{s}$ in both (d) and (e).

Figure 3

Schematics of $\mathit{in}$ situ ultrasound imaging. The plane of propagation of ultrasound is the same as the vertical plane passing through the center of the top plate. A typical ultrasound image with PIV vectors estimated over the region of interest (ROI) is superposed at a particular instant. The angle of the frustum at the free surface is found to be $\sim {16}^{\circ }$. The tangent of this angle (or the slope of the sidewall of the frustum, $\tan {16}^{\circ }\sim 0.3$) gives a value very close to the strain required to jam the system. Here, $\varphi =0.50$ and ${\eta }_s=10\mathrm{mPa}\mathrm{s}$. The slope of the sidewall of the frustum is found to decrease with increasing packing fraction and pulling velocity.

Figure 4

(a)–(d) Time evolution of the flow field under extensional flow. The top plate starts to move at $t=0$ s at velocity $v=50$ mm/s. The dark-red region in the PIV window, close to the top plate, indicates the jammed region (velocity vectors pointing upward). The scale bar indicates the vertical components ($-v_z$) of the velocity. (e)–(h) The same vector field as before, but with the scale bar indicating the radial ($-v_r$) velocity components. The yellow region indicates higher magnitudes of the radial velocity components. Time evolution of (i) $-v_z$ and (j) $-v_r$ as a function of $z$ (computed along the dashed vertical line in each panel); dark- to light-colored lines in (i) and (j) indicate increasing times from $t=0.009$ s to $t=0.104$ s in steps of 0.005 s. Here, $\varphi =0.50$ and ${\eta }_s=10\mathrm{mPa}\mathrm{s}$.

Figure 5

(a) Vertical velocity components ($-v_z$) estimated along the vertical line passing through the center of the plate at different instants in time. (b) Radial components of the velocity ($-v_r$) along the vertical line that contains the maximum magnitude $|v_r|$ at different instants. In both (a) and (b), the pulling velocity is 2 mm/s and the dark to light colors indicate increasing times from $t=0$ s to $t=0.75$ s, with a time step of 0.125 s. These plots indicate that the high-velocity region never reaches the boundary for slow pulling velocities where no rapid force increase is observed. Here, $\varphi =0.50$ and ${\eta }_s=10\mathrm{mPa}\mathrm{s}$. Also, both $|\frac{\delta v_z}{\delta z}|$ and $|\frac{\delta v_r}{\delta z}|$ remain $\ll 5{\mathrm{s}}^{-1}$, indicating that the stress scales are well below the jamming transition.

Figure 6

(a) Local strain rate (averaged over three consecutive PIV vectors in the vertical direction) as a function of time during jamming front propagation. Parameter values are the same as in Fig. 4. Error bars are given by the standard deviation of the strain rate at each time estimated at these three windows. The integrated area under the curve gives the total local strain deformation to be $\sim 0.25$. (b) Strain evolution as a function of time for applied stresses (indicated in the legend) high enough to jam the system. These experiments are done on the same sample as in (a) but in a narrow-gap, parallel-plate rheometer. The onset strain for jamming (indicated by a sudden decrease in slope) is found to be very close to that obtained in (a).

Figure 7

Shear stress vs shear rate measured under simple shear in a narrow-gap rheometer. The shaded region indicates the shear-jammed state. Error bars indicate the standard deviations of three consecutive measurements. Here, $\varphi =0.50$ and ${\eta }_s=10\mathrm{mPa}\mathrm{s}$.

Figure 8

Jamming-front propagation under extension in a container having a diameter of 10 cm, which is $\sim 17$ times larger than the diameter of the pulling rod (6 mm). (a) PIV vector field at time $t=10$ ms after the top plate starts to move at a velocity $v=0.2$ m/s. The vertical components of the velocity are indicated in the scale bar. The dark-red region at the center of the PIV window shows that the velocity inside the jammed region (positive values in the scale bar corresponding to the velocity vectors pointing upward) is close to the pulling velocity. (b) Change in speed of sound through the suspension during jamming-front propagation. The plate starts to move at time $t=0$ s at a velocity $v=0.2$ m/s and detaches at $t\sim 0.02$ s. Data for three trials are shown. (c)–(e) Components of the strain rate tensor (${\dot{\epsilon }}_{r\theta z}$) in a cylindrical coordinate system ($\theta$ is the azimuthal angle) for the velocity field shown in (a): (c) magnitude of the simple shear component $|{\dot{\epsilon }}_{rz}|$, (d) magnitude of the pure shear component $|{\dot{\epsilon }}_{zz}|$, and (e) magnitude of the rate of expansion component $|{\dot{\epsilon }}_{rr}+{\dot{\epsilon }}_{\theta \theta }+{\dot{\epsilon }}_{zz}|$. Thin dashed lines in (c)–(e) are the contours of points where the vertical velocity drops from $0.9v$ (uppermost contour) to $0.1v$ (lowermost contour), in steps of $0.1v$. The thick dashed line indicates the contour where the velocity drops to $0.5v$, giving the position of the jamming front. All plots correspond to $t=10$ ms after the start of extensional flow. Here, $\varphi =0.49$ and ${\eta }_s=5\mathrm{mPa}\mathrm{s}$.

Figure 9

Immediate reversal of the pulling direction causes the jammed region to disappear. The flow field obtained by ultrasound imaging when the direction of the pulling rod (indicated by the dashed box) is reversed from upward (a) to downward (b). The dark-red region at the center in (a) shows that the velocity inside the jammed region (positive values in the scale bar) is similar to the pulling velocity. In (b), the light-blue region at the center indicates the downward velocity (negative values in the scale bar), indicating a much weaker velocity field compared to (a). Here, $\varphi =0.49$ and ${\eta }_s=5\mathrm{mPa}\mathrm{s}$.

Figure 10

Stress response with (filled diamonds) and without (filled circles) a solvent layer on top. Here, $\varphi =0.51, {\eta }_s=50\mathrm{mPa}\mathrm{s}$, and $v=8\mathrm{mm}/\mathrm{s}$.

Figure 11

(a) Peak force as a function of the volume fraction in the weak adhesion limit (the plate just touches the surface) for different pulling velocities as indicated. The dashed line indicates a function $\propto \frac{{\varphi }^2}{{(1-\varphi )}^3}$; the trend is consistent with the K-C relation. Although an overall increasing trend of the peak force is observed as a function of $\varphi$, due to the large error bars and small range of $\varphi (0.50\le \varphi \le 0.54)$ probed experimentally, the exact functional form is difficult to estimate accurately. Nevertheless, for higher values of pulling velocities and $\varphi$, the K-C prediction seems to match the trend in the experimental data. (b) Peak stress normalized by the solvent viscosity as a function of the pulling velocity. The data collapse indicates that the peak stress increases linearly with the solvent viscosity and pulling velocity.

Figure 12

Stress response under extension for two slightly different initial positions of the plate: (a) the plate just touches the suspension surface, and (b) the plate is slightly ($\approx 2$ mm) pushed inside the suspension. (c) Peak stress under extension ($v=8\mathrm{mm}/\mathrm{s}$) for the initial conditions given in (a) (filled diamonds) and (b) (filled circles). The measured peak stress is more than three times larger when the plate is pushed inside, indicating that the peak stress is governed by the coupling between the plate and the suspension. Here, $\varphi =0.54$ and ${\eta }_s=20\mathrm{mPa}\mathrm{s}$. Error bars indicate the average standard deviation of three consecutive measurements.

Figure 13

(a) Schematic showing the flow of solvent through the jammed granular pack of particles; solid red lines indicate typical capillary paths consistent with the Kozeny-Carman scheme. The surface texture: before force shoot-up (b), at the onset of force shoot-up (c), and after detachment (d).

Figure 14

Schematics showing drainage through the sidewall of the frustum, close to the plate.

Figure 15

Peak stress as a function of time for cornstarch and rice-starch particles. Volume fractions are indicated for a pulling velocity (a) $v=8$ mm/s and (b) $v=3$ mm/s. The solvent viscosity ${\eta }_s=8\mathrm{mPa}\mathrm{s}$ in both cases. Error bars indicate the standard deviation of three independent measurements on the same sample.