Discontinuous Shear Thickening of Frictional Hard-Sphere Suspensions

Discontinuous shear thickening (DST) observed in many dense athermal suspensions has proven difficult to understand and to reproduce by numerical simulation. By introducing a numerical scheme including both relevant hydrodynamic interactions and granularlike contacts, we show that contact friction is essential for having DST. Above a critical volume fraction, we observe the existence of two states: a low viscosity, contactless (hence, frictionless) state, and a high viscosity frictional shear jammed state. These two states are separated by a critical shear stress, associated with a critical shear rate where DST occurs. The shear jammed state is reminiscent of the jamming phase of granular matter. Continuous shear thickening is seen as a lower volume fraction vestige of the jamming transition.

Figure 1

(a), (b) Shear rate and stress dependence of the relative viscosity ${\eta }_r$, respectively. $\dot{\Gamma }$ is the dimensionless shear rate. The open and filled symbols indicate the results of $n=512$ and 2048, and the volume fractions are shown in the graphs. The friction coefficient is $\mu =1$ except for the dashed and dotted blue lines, for which $\mu =0.1$ and 0, respectively. Red symbols show the results with 1.5 times stiffer particles. (c) DST (red line) and CST (dashed line) are shown in the phase diagram. The former is expected to reach the jamming point ${\varphi }_J$ for $\dot{\Gamma }\to 0$, which is not seen because of log scale. The contour lines for $\varphi <0.56$ are labeled by the relative viscosity, ${\eta }_r(\varphi ,\dot{\Gamma })$. Before jamming (black domain), the shear jammed states (gray domain) exist. Observed flowing and jammed states at $\varphi =0.58$ are shown by circles and crosses, respectively.

Figure 2

(a) Particle contacts are visualized as bonds at the two shear rates $\dot{\Gamma }=0.05$ and $\dot{\Gamma }=0.1$ exhibiting low and high viscosity states, respectively ($\mu =1$, $\varphi =0.56$, $n=2048$). (b) The stress eigenvalues, ${\Lambda }_1$ (circles), ${\Lambda }_2$ (triangles), and ${\Lambda }_3$ (squares), normalized by the trace, are shown ($\varphi =0.56$, $n=512$). The solid lines (red) indicate the contact stress ${\mathit{\sigma }}^c$, and dashed lines (black) the total stress $\mathit{\sigma }$ ($\mu =1$). The dotted lines (blue) show the contact stress of the frictionless system ($\mu =0$). (c) The largest contact clusters for $\mu =1$ (solid red) and $\mu =0$ (dotted blue) are compared, where $n_{\mathrm{cc}}$ is number of particles in a cluster.