Hydrodynamic and Contact Contributions to Continuous Shear Thickening in Colloidal Suspensions

Shear thickening is a widespread phenomenon in suspension flow that, despite sustained study, is still the subject of much debate. The longstanding view that shear thickening is due to hydrodynamic clusters has been challenged by recent theory and simulations suggesting that contact forces dominate, not only in discontinuous, but also in continuous shear thickening. Here, we settle this dispute using shear reversal experiments on micron-sized silica and latex particles to measure directly the hydrodynamic and contact force contributions to shear thickening. We find that contact forces dominate even continuous shear thickening. Computer simulations show that these forces most likely arise from frictional interactions.

Figure 1

(a) Schematic of our shear reversal protocol. The applied strain $\gamma$ is a triangle wave with peak-to-peak amplitude of 10. The applied strain rate $\dot{\gamma }(t)$ is a square wave whose amplitude we vary in our experiments. The instantaneous stress response $\sigma$ of a viscous fluid and a contact-dominated fluid are shown in the green and red, respectively. (The same color scheme applies throughout all parts of this figure.) The instantaneous relative viscosity $\eta (t)=\sigma (t)/[\dot{\gamma }(t){\eta }_0]$. (b) Schematic illustrating the difference between hydrodynamic forces, which stay the same in magnitude immediately upon reversal, and contact forces, which drop to zero immediately upon reversal.

Figure 2

Relative viscosity $\eta$ vs strain rate $\dot{\gamma }$ for the (a) silica and (b) PMMA suspensions. Shown are rate-sweep measurements (thick gray line), steady-state relative viscosities from flow reversal (red) decomposed into contributions from hydrodynamic (blue) and contact (green) interactions. (b),(d) Instantaneous relative viscosity after shear reversal, ${\eta }_{\mathrm{rev}}(\gamma )=\sigma (\gamma )/(\dot{\gamma }{\eta }_0)$, vs strain $\gamma$ at different applied $\dot{\gamma }$. The hydrodynamic component of the relative viscosity ${\eta }_{\mathrm{rev}}^0$ is taken to be the average of the relative viscosity over the range $0.01<\gamma <0.2$. The contact component is taken to be the difference between the steady-state value at $\gamma =10$, ${\eta }_{\mathrm{rev}}^{\infty }$, and ${\eta }_{\mathrm{rev}}^0$.

Figure 3

Critical load model simulations of particles with hydrodynamic and frictional interactions. (a) Relative viscosity $\eta (\gamma )$ vs strain $\gamma$ at four different shear rates (given in the legend, see text for the definition of ${\dot{\gamma }}_0$); compare experimental results in Figs. 2 and 2. (b) Directly calculated hydrodynamic and contact contributions to the total viscosity as functions of the accumulated strain after reversal for the case of $\dot{\gamma }=0.74{\dot{\gamma }}_0$. (c) The total viscosity as a function of shear rate, and its decomposition into hydrodynamic and contact contributions done in two ways: directly calculated from simulations, and “read off” the data shown in part (a) in the same way as the experimental data are analyzed into these contributions, for which see Fig. 2.

Figure 4

Scaling of the relative viscosity with shear stress. (a),(c) Viscosity vs shear rate for silica ($\varphi =0.43$) and PMMA ($\varphi =0.46$) suspensions at different temperatures. Solvent viscosities and temperatures are as labeled in (b) and (d) (for silica, we used a different solvent composition from Fig. 1; see Supplemental Material [28]). Because of the temperature dependence of the solvent viscosity the initial suspension viscosity and onset strain rate (arrows) of the thickening are substantially altered. (b),(d) Relative viscosity vs stress showing collapse of the thickening data onto a master curve.