Shear thickening in dense bidisperse suspensions

Discrete-particle simulations of bidisperse shear thickening suspensions are reported. The work considers two packing parameters, the large-to-small particle radius ratio ranging from $\delta =1.4$ (nearly monodisperse) to $\delta =4$, and the large particle fraction of the total solid loading with values $\zeta =0.15$, 0.5, and 0.85. Particle-scale simulations are performed over a broad range of shear stresses using a simulation model for spherical particles accounting for short-range lubrication forces, frictional interaction, and repulsion between particles. The variation of rheological properties and the maximum packing fraction ${\varphi }_J$ with shear stress $\sigma$ are reported. At a fixed volume fraction $\varphi$, bidispersity decreases the suspension relative viscosity ${\eta }_r={\eta }_s/{\eta }_0$, where ${\eta }_s$ is the suspension viscosity and ${\eta }_0$ is the suspending fluid viscosity, over the entire range of shear stresses studied. However, under low shear stress conditions, the suspension exhibits an unusual rheological behavior: the minimum viscosity does not occur as expected at $\zeta \approx 0.5$, but instead decreases with further increase of $\zeta$ to $0.85$. The second normal stress difference $N_2$ acts similarly. This behavior is caused by particles ordering into a layered structure, as is also reflected by the zero slope with respect to time of the mean-square displacement in the velocity gradient direction. The relative viscosity ${\eta }_r$ of bidisperse rate-dependent suspensions can be predicted by a power law linking it to ${\varphi }_J$, ${\eta }_r={(1-\varphi /{\varphi }_J)}^{-2}$ in both low and high shear stress regimes. The agreement between the power law and experimental data from literature demonstrates that the model captures well the effect of particle size distribution, showing that viscosity roughly collapses onto a single master curve when plotted against the reduced volume fraction $\varphi /{\varphi }_J$.