Nonmonotonic flow curves of shear thickening suspensions

The discontinuous shear thickening (DST) of dense suspensions is a remarkable phenomenon in which the viscosity can increase by several orders of magnitude at a critical shear rate. It has the appearance of a first-order phase transition between two hypothetical “states” that we have recently identified as Stokes flows with lubricated or frictional contacts, respectively. Here we extend the analogy further by means of stress-controlled simulations and show the existence of a nonmonotonic steady-state flow curve analogous to a nonmonotonic equation of state. While we associate DST with an $\mathsf{S}$-shaped flow curve, at volume fractions above the shear jamming transition the frictional state loses flowability and the flow curve reduces to an arch, permitting the system to flow only at small stresses. Whereas a thermodynamic transition leads to phase separation in the coexistence region, we observe a uniform shear flow all along the thickening transition. A stability analysis suggests that uniform shear may be mechanically stable for the small Reynolds numbers and system sizes in a rheometer.

Figure 1

Sketch of the relation between the shear rate $\dot{\gamma }$ and the shear stress $\sigma$ in a shear thickening suspension, with increasing volume fraction $\varphi$ from top to bottom, in solid lines. At low $\varphi ,\dot{\gamma }\left(\sigma \right)$ is a strictly monotonic function and the shear thickening is continuous, appearing as an inflection in the curve. As $\varphi$ increases, this inflection deepens, and at $\varphi ={\varphi }_{\mathrm{c}}$ a “critical” point $({\sigma }_{\mathrm{c}},{\dot{\gamma }}_{\mathrm{c}})$ appears in which $d\dot{\gamma }/d\sigma =0$. For $\varphi >{\varphi }_{\mathrm{c}}$ the shear thickening is discontinuous. The discontinuity region (with constant $\dot{\gamma })$ is analogous to a coexistence region (light gray region). (Hysteresis is sometimes observed, in which case the discontinuity region would be analogous to a spinodal region.) The discontinuity region is accessible only under stress-controlled conditions. The putative nonmonotonic flow curves are plotted as dashed lines in the coexistence region, in analogy to the nonmonotonic equations of state underlying a first-order thermodynamic transition.

Figure 2

Left: The shear rate $\dot{\gamma }$ as a function of the shear stress $\sigma$ for several values of the volume fraction $\varphi$ in our simulations at constant shear stress. Continuous shear thickening at the lowest $\varphi$ is associated with monotonic flow curves, discontinuous shear thickening appears here as an $\mathsf{S}$-shaped flow curve for $\varphi =0.56$–$0.58$, and shear jamming is indicated by arch-shaped flow curves with a vanishing shear rate at high stresses for $\varphi \ge 0.59$. Simulations are run with $n=500$ particles except for $\varphi =0.58$, for which we use $n=2000$. Two data points with $n=4500$, corresponding to the systems shown in Fig. 3, are indicated by red crosses. Right: The same data plotted as viscosity $\eta$ versus shear rate $\dot{\gamma }$ compared with simulations at constant shear rate (blue crosses).

Figure 3

Top: Instantaneous velocity profiles in the shear thickening regime $\left(\sigma /{\sigma }_0=2\right)$ for $\varphi =0.54$ (left), where $d\dot{\gamma }/d\sigma >0$, and for $\varphi =0.58$ (right), where $d\dot{\gamma }/d\sigma <0$. The different curves are taken at different strains separated by one strain unit. In both cases the shear flow is uniform. Bottom: Configurations of the system at $\varphi =0.54$ (left) and $\varphi =0.58$ (right) and stress $\sigma /{\sigma }_0=2$ as above, color coded with the norm of the nonaffine velocity, form dark gray (small value) to yellow (light gray, large value). Fluctuations are larger at higher volume fraction, but they do not show any sign of nonuniform flow or phase separation.

Figure 4

Strain series of the shear rate for $\varphi =0.58$ and $\sigma /{\sigma }_0=2$, for which the steady-state flow curve shows $d\dot{\gamma }/d\sigma <0$. The system shows no intermittency or exotic fluctuating or chaotic behavior and is in a simple steady state. Three sizes are shown: $n=500,2000,4500$, from which we can see that the variance of the signal decreases with increasing $n$, as is expected for a steady uniform shear flow.

Figure 5

Typical strain response to a step increment of applied stress in the region $d{\dot{\gamma }}^{*}/d\sigma <0$. These curves are obtained by averaging 100 simulations at $\varphi =0.58$, for which we impose a stress $\sigma /{\sigma }_0=2$ for strains $\gamma <2$ and increment to $\sigma /{\sigma }_0=3$ at $\gamma =2$ (top panel). The average contact number per particle $n_{\mathrm{c}}$ and the viscosity $\eta$ increase (middle panels), with an exponential-like relaxation, but the shear rate $\dot{\gamma }$ has a nonmonotonic behavior, first increasing and then relaxing to a lower value (bottom panel). The dashed line is a fit to an exponential relaxation, giving a relaxation strain of $c^{-1}\approx 0.023$ for these conditions. Hence at short time scales, and in particular on the inertial time scale, the shear rate increases with the shear stress and the system is mechanically stable.