Rigid clusters in shear-thickening suspensions: a nonequilibrium critical transition

The onset and growth of rigid clusters in a two-dimensional (2D) suspension in shear flow are studied by numerical simulations. The suspension exhibits the lubricated-to-frictional rheology transition, but the key results here are for stresses above the levels that cause extreme shear-thickening. At large solid fraction, $\phi$, but below the stress-dependent jamming fraction, we find a critical ${\phi }_c(\sigma ,\mu )$ where $\sigma$ is a dimensionless shear stress and $\mu$ is the interparticle friction coefficient. For $\phi >{\phi }_c$, the proportion of particles in rigid clusters grows sharply, as $f_{\mathrm{r}\mathrm{i}\mathrm{g}}\sim \mathrm{\mid }\phi -{\phi }_c{\mathrm{\mid }}^{\beta }$ with $\beta =1\mathrm{/}8$. The fluctuations in the fraction of particles in rigid clusters yield a susceptibility measure ${\chi }_{\mathrm{r}\mathrm{i}\mathrm{g}}\sim \mathrm{\mid }\phi -{\phi }_c{\mathrm{\mid }}^{-\gamma }$ with $\gamma =7\mathrm{/}4$. The system is thus found to exhibit criticality. The results are shown to depend on an effective field $h(\mu )$, which provides data collapse near ${\phi }_c$ for both $f_{\mathrm{r}\mathrm{i}\mathrm{g}}$ and ${\chi }_{\mathrm{r}\mathrm{i}\mathrm{g}}$. This behavior occurs over a range of stresses, with ${\phi }_c(\sigma ,\mu )$ increasing as the stress decreases.